Reducing the two nested GMRES solvers to just one. Also implemented a complex version of AMS.

PA for GradientInterpolator and IdentityInterpolator

.gitignore

@@ -215,6 +215,7 @@ miniapps/meshing/polar-nc.mesh
 
 miniapps/navier/navier_mms
 miniapps/navier/navier_kovasznay
+miniapps/navier/navier_kovasznay_vs
 miniapps/navier/navier_tgv
 miniapps/navier/navier_shear
 miniapps/navier/navier_3dfoc

CHANGELOG

@@ -11,6 +11,12 @@
 Version 4.2.1 (development)
 ===========================
 
+- Added high-order matrix-free auxiliary Maxwell solver for H(curl) problems,
+  as described in Barker and Kolev 2020 (https://doi.org/10.1002/nla.2348).
+
+- Added matrix-free GPU-enabled implementations of GradientInterpolator and
+  IdentityInterpolator.
+
 - Added interface to MUMPS direct solver. Its usage is demonstrated in ex25p.
   See http://mumps.enseeiht.fr/ for more details. Supported versions >= 5.1.1.
 
@@ -20,11 +26,41 @@ Version 4.2.1 (development)
 - Added a "scaled Jacobian" visualization option in the Mesh Explorer miniapp to
   help identify elements with poor mesh quality.
 
+- Added support for the "BR2" discontinuous Galerkin discretization for
+  diffusion via DGDiffusionBR2Integrator (see Example 14/14p).
+
+- Generalized the Multigrid class to support non-geometric multigrid. The
+  previous functionality, based on FiniteElementSpaceHierarchy, is now available
+  in the derived class GeometricMultigrid.
+
 - Upgraded the Catch unit test framework from version 2.13.0 to version 2.13.2.
 
 - Implemented a filter method for the Navier miniapp to stabilize highly
   turbulent flows in direct numerical simulation.
 
+- Added partial assembly and device support to Example 25/25p, with diagonal
+  preconditioning.
+
+- Implemented a variable step-size IMEX (VSSIMEX) method for the Navier miniapp.
+
+- Added new mesh quality metrics and improved the untangling capabilities of the
+  TMOP-based mesh optimization algorithms.
+
+- Changed the interface for the error estimator.
+
+- Implemented the parallel Kelly error indicator for scalar-valued problems.
+
+- Added new classes DenseSymmetricMatrix and SymmetricMatrixCoefficient for
+  efficient evaluation of symmetric matrix coefficients. This replaces the now
+  deprecated EvalSymmetric in MatrixCoefficient. Added DiagonalMatrixCoefficient
+  for clarity, which is a typedef of VectorCoefficient.
+
+- Implemented an adaptive linear solver tolerance option for NewtonSolver based
+  on the algorithm of Eisenstat and Walker.
+
+- Extending support for L2 basis functions using MapTypes VALUE and INTEGRAL in
+  linear interpolators and GridFunction "GetValue" methods.
+
 
 Version 4.2, released on October 30, 2020
 =========================================

CMakeLists.txt

@@ -9,7 +9,8 @@
 # terms of the BSD-3 license. We welcome feedback and contributions, see file
 # CONTRIBUTING.md for details.
 
-cmake_minimum_required(VERSION 2.8.11)
+# The variable CMAKE_CXX_STANDARD and related were introduced in CMake v3.1
+cmake_minimum_required(VERSION 3.1)
 set(USER_CONFIG "${CMAKE_CURRENT_SOURCE_DIR}/config/user.cmake" CACHE PATH
   "Path to optional user configuration file.")
 

INSTALL

@@ -710,10 +710,10 @@ The specific libraries and their options are:
   Versions: libCEED >= 0.7.
 
 - RAJA (optional), used when MFEM_USE_RAJA = YES.
-  Beginning with MFEM v4.1, only RAJA v0.10.0+ is supported.
+  Beginning with MFEM v4.3, only RAJA v0.13.0+ is supported.
   URL: https://github.com/LLNL/RAJA
   Options: RAJA_DIR, RAJA_OPT, RAJA_LIB.
-  Versions: RAJA >= 0.10.0.
+  Versions: RAJA >= 0.13.0.
 
 - Umpire, used when MFEM_USE_UMPIRE = YES.
   URL: https://github.com/LLNL/Umpire

config/cmake/modules/FindMETIS.cmake

@@ -25,16 +25,16 @@ mfem_find_package(METIS METIS METIS_DIR "include;Lib" "metis.h"
 
 int main()
 {
-    int n = 10;
-    int nparts = 5;
-    int edgecut;
-    int* partitioning = new int[10];
-    int* I = partitioning,
+    idx_t n = 10;
+    idx_t nparts = 5;
+    idx_t edgecut;
+    idx_t* partitioning = new idx_t[10];
+    idx_t* I = partitioning,
        * J = partitioning;
 
-    int ncon = 1;
+    idx_t ncon = 1;
     int err;
-    int options[40];
+    idx_t options[40];
 
     METIS_SetDefaultOptions(options);
     options[10] = 1; // set METIS_OPTION_CONTIG

config/cmake/modules/MfemCmakeUtilities.cmake

@@ -754,7 +754,13 @@ function(mfem_export_mk_files)
   set(MFEM_CXX ${CMAKE_CXX_COMPILER})
   set(MFEM_HOST_CXX ${MFEM_CXX})
   set(MFEM_CPPFLAGS "")
-  string(STRIP "${CMAKE_CXX_FLAGS_${BUILD_TYPE}} ${CMAKE_CXX_FLAGS}"
+  get_target_property(cxx_std mfem CXX_STANDARD)
+  # For now, we ignore the setting of the CXX_EXTENSIONS property. If this
+  # property is set, then we need to use a variable like:
+  #    CMAKE_CXX11_EXTENSION_COMPILE_OPTION
+  set(cxx_std_flag ${CMAKE_CXX${cxx_std}_STANDARD_COMPILE_OPTION})
+  string(STRIP
+         "${cxx_std_flag} ${CMAKE_CXX_FLAGS_${BUILD_TYPE}} ${CMAKE_CXX_FLAGS}"
          MFEM_CXXFLAGS)
   set(MFEM_TPLFLAGS "")
   foreach(dir ${MFEM_TPL_INCLUDE_DIRS})

config/config.hpp

@@ -34,6 +34,12 @@
 // Macro needed to get defines like M_PI from <cmath>. (Visual Studio C++ only?)
 #define _USE_MATH_DEFINES
 #endif
+// On Cygwin the option -std=c++11 prevents the definition of M_PI. Defining
+// the following macro allows us to get M_PI and some needed functions, e.g.
+// posix_memalign(), strdup(), strerror_r().
+#ifdef __CYGWIN__
+#define _XOPEN_SOURCE 600
+#endif
 
 // Check dependencies:
 

config/defaults.mk

@@ -161,7 +161,7 @@ endif
 ZLIB_DIR =
 ZLIB_OPT = $(if $(ZLIB_DIR),-I$(ZLIB_DIR)/include)
 ZLIB_LIB = $(if $(ZLIB_DIR),$(ZLIB_RPATH) -L$(ZLIB_DIR)/lib ,)-lz
-ZLIB_RPATH = -Wl,-rpath,$(ZLIB_DIR)/lib
+ZLIB_RPATH = $(XLINKER)-rpath,$(ZLIB_DIR)/lib
 
 LIBUNWIND_OPT = -g
 LIBUNWIND_LIB = $(if $(NOTMAC),-lunwind -ldl,)
@@ -231,19 +231,21 @@ MESQUITE_LIB = -L$(MESQUITE_DIR)/lib -lmesquite
 LIB_RT = $(if $(NOTMAC),-lrt,)
 SUITESPARSE_DIR = @MFEM_DIR@/../SuiteSparse
 SUITESPARSE_OPT = -I$(SUITESPARSE_DIR)/include
-SUITESPARSE_LIB = -Wl,-rpath,$(SUITESPARSE_DIR)/lib -L$(SUITESPARSE_DIR)/lib\
- -lklu -lbtf -lumfpack -lcholmod -lcolamd -lamd -lcamd -lccolamd\
- -lsuitesparseconfig $(LIB_RT) $(METIS_LIB) $(LAPACK_LIB)
+SUITESPARSE_LIB = $(XLINKER)-rpath,$(SUITESPARSE_DIR)/lib\
+ -L$(SUITESPARSE_DIR)/lib -lklu -lbtf -lumfpack -lcholmod -lcolamd -lamd -lcamd\
+ -lccolamd -lsuitesparseconfig $(LIB_RT) $(METIS_LIB) $(LAPACK_LIB)
 
 # SuperLU library configuration
 ifeq ($(MFEM_USE_SUPERLU5),YES)
    SUPERLU_DIR = @MFEM_DIR@/../SuperLU_DIST_5.1.0
    SUPERLU_OPT = -I$(SUPERLU_DIR)/include
-   SUPERLU_LIB = -Wl,-rpath,$(SUPERLU_DIR)/lib -L$(SUPERLU_DIR)/lib -lsuperlu_dist_5.1.0
+   SUPERLU_LIB = $(XLINKER)-rpath,$(SUPERLU_DIR)/lib -L$(SUPERLU_DIR)/lib\
+      -lsuperlu_dist_5.1.0
 else
    SUPERLU_DIR = @MFEM_DIR@/../SuperLU_DIST_6.3.1
    SUPERLU_OPT = -I$(SUPERLU_DIR)/include
-   SUPERLU_LIB = -Wl,-rpath,$(SUPERLU_DIR)/lib64 -L$(SUPERLU_DIR)/lib64 -lsuperlu_dist -lblas
+   SUPERLU_LIB = $(XLINKER)-rpath,$(SUPERLU_DIR)/lib64 -L$(SUPERLU_DIR)/lib64\
+      -lsuperlu_dist -lblas
 endif
 
 # SCOTCH library configuration (required by STRUMPACK <= v2.1.0, optional in
@@ -269,7 +271,7 @@ MPI_FORTRAN_LIB = -lmpifort
 # MUMPS library configuration
 MUMPS_DIR = @MFEM_DIR@/../MUMPS_5.2.0
 MUMPS_OPT = -I$(MUMPS_DIR)/include
-MUMPS_LIB = -Wl,-rpath,$(MUMPS_DIR)/lib -L$(MUMPS_DIR)/lib -ldmumps\
+MUMPS_LIB = $(XLINKER)-rpath,$(MUMPS_DIR)/lib -L$(MUMPS_DIR)/lib -ldmumps\
  -lmumps_common -lpord $(SCALAPACK_LIB) $(LAPACK_LIB) $(MPI_FORTRAN_LIB)
 
 # STRUMPACK library configuration
@@ -299,8 +301,8 @@ GNUTLS_LIB = -lgnutls
 NETCDF_DIR = $(HOME)/local
 HDF5_DIR   = $(HOME)/local
 NETCDF_OPT = -I$(NETCDF_DIR)/include -I$(HDF5_DIR)/include $(ZLIB_OPT)
-NETCDF_LIB = -Wl,-rpath,$(NETCDF_DIR)/lib -L$(NETCDF_DIR)/lib\
- -Wl,-rpath,$(HDF5_DIR)/lib -L$(HDF5_DIR)/lib\
+NETCDF_LIB = $(XLINKER)-rpath,$(NETCDF_DIR)/lib -L$(NETCDF_DIR)/lib\
+ $(XLINKER)-rpath,$(HDF5_DIR)/lib -L$(HDF5_DIR)/lib\
  -lnetcdf -lhdf5_hl -lhdf5 $(ZLIB_LIB)
 
 # PETSc library configuration (version greater or equal to 3.8 or the dev branch)
@@ -312,9 +314,10 @@ PETSC_INC_VAR = PETSC_CC_INCLUDES
 PETSC_LIB_VAR = PETSC_EXTERNAL_LIB_BASIC
 ifeq ($(PETSC_FOUND),YES)
    PETSC_OPT := $(shell sed -n "s/$(PETSC_INC_VAR) = *//p" $(PETSC_VARS))
-   PETSC_LIB := $(shell sed -n "s/$(PETSC_LIB_VAR) = *//p" $(PETSC_VARS))
-   PETSC_LIB := -Wl,-rpath,$(abspath $(PETSC_DIR))/lib\
-      -L$(abspath $(PETSC_DIR))/lib -lpetsc $(PETSC_LIB)
+   PETSC_DEP := $(shell sed -n "s/$(PETSC_LIB_VAR) = *//p" $(PETSC_VARS))
+   PETSC_LIB = $(XLINKER)-rpath,$(abspath $(PETSC_DIR))/lib\
+      -L$(abspath $(PETSC_DIR))/lib -lpetsc\
+      $(subst $(CXX_XLINKER),$(XLINKER),$(PETSC_DEP))
 endif
 
 SLEPC_DIR := $(MFEM_DIR)/../slepc
@@ -326,9 +329,10 @@ ifeq ($(SLEPC_FOUND),YES)
    SLEPC_OPT := $(shell sed -n "s/$(SLEPC_INC_VAR) *= *//p" $(SLEPC_VARS))
    # Some additional external libraries might be defined in this file
    -include ${SLEPC_DIR}/${PETSC_ARCH}/lib/slepc/conf/slepcvariables
-   SLEPC_LIB := $(shell sed -n "s/$(SLEPC_LIB_VAR) *= *//p" $(SLEPC_VARS))
-   SLEPC_LIB := -Wl,-rpath,$(abspath $(SLEPC_DIR))/$(PETSC_ARCH)/lib\
-      -L$(abspath $(SLEPC_DIR))/$(PETSC_ARCH)/lib -lslepc $(SLEPC_LIB)
+   SLEPC_DEP := $(shell sed -n "s/$(SLEPC_LIB_VAR) *= *//p" $(SLEPC_VARS))
+   SLEPC_LIB = $(XLINKER)-rpath,$(abspath $(SLEPC_DIR))/$(PETSC_ARCH)/lib\
+      -L$(abspath $(SLEPC_DIR))/$(PETSC_ARCH)/lib -lslepc\
+      $(subst $(CXX_XLINKER),$(XLINKER),$(SLEPC_DEP))
 endif
 
 # MPFR library configuration
@@ -339,7 +343,7 @@ MPFR_LIB = -lmpfr
 CONDUIT_DIR = @MFEM_DIR@/../conduit
 CONDUIT_OPT = -I$(CONDUIT_DIR)/include/conduit
 CONDUIT_LIB = \
-   -Wl,-rpath,$(CONDUIT_DIR)/lib -L$(CONDUIT_DIR)/lib \
+   $(XLINKER)-rpath,$(CONDUIT_DIR)/lib -L$(CONDUIT_DIR)/lib \
    -lconduit -lconduit_relay -lconduit_blueprint  -ldl
 
 # Check if Conduit was built with hdf5 support, by looking
@@ -347,7 +351,7 @@ CONDUIT_LIB = \
 CONDUIT_HDF5_HEADER=$(CONDUIT_DIR)/include/conduit/conduit_relay_hdf5.hpp
 ifneq (,$(wildcard $(CONDUIT_HDF5_HEADER)))
    CONDUIT_OPT += -I$(HDF5_DIR)/include
-   CONDUIT_LIB += -Wl,-rpath,$(HDF5_DIR)/lib -L$(HDF5_DIR)/lib \
+   CONDUIT_LIB += $(XLINKER)-rpath,$(HDF5_DIR)/lib -L$(HDF5_DIR)/lib \
                   -lhdf5 $(ZLIB_LIB)
 endif
 
@@ -357,9 +361,9 @@ SIDRE_DIR = @MFEM_DIR@/../axom
 SIDRE_OPT = -I$(SIDRE_DIR)/include -I$(CONDUIT_DIR)/include/conduit\
  -I$(HDF5_DIR)/include
 SIDRE_LIB = \
-   -Wl,-rpath,$(SIDRE_DIR)/lib -L$(SIDRE_DIR)/lib \
-   -Wl,-rpath,$(CONDUIT_DIR)/lib -L$(CONDUIT_DIR)/lib \
-   -Wl,-rpath,$(HDF5_DIR)/lib -L$(HDF5_DIR)/lib \
+   $(XLINKER)-rpath,$(SIDRE_DIR)/lib -L$(SIDRE_DIR)/lib \
+   $(XLINKER)-rpath,$(CONDUIT_DIR)/lib -L$(CONDUIT_DIR)/lib \
+   $(XLINKER)-rpath,$(HDF5_DIR)/lib -L$(HDF5_DIR)/lib \
    -laxom -lconduit -lconduit_relay -lconduit_blueprint -lhdf5 $(ZLIB_LIB) -ldl
 
 # PUMI
@@ -415,9 +419,9 @@ MKL_CPARDISO_DIR ?=
 MKL_MPI_WRAPPER ?= mkl_blacs_mpich_lp64
 MKL_LIBRARY_SUBDIR ?= lib
 MKL_CPARDISO_OPT = -I$(MKL_CPARDISO_DIR)/include
-MKL_CPARDISO_LIB = -Wl,-rpath,$(MKL_CPARDISO_DIR)/$(MKL_LIBRARY_SUBDIR)\
-                   -L$(MKL_CPARDISO_DIR)/$(MKL_LIBRARY_SUBDIR) -l$(MKL_MPI_WRAPPER)\
-                   -lmkl_intel_lp64 -lmkl_sequential -lmkl_core
+MKL_CPARDISO_LIB = $(XLINKER)-rpath,$(MKL_CPARDISO_DIR)/$(MKL_LIBRARY_SUBDIR)\
+   -L$(MKL_CPARDISO_DIR)/$(MKL_LIBRARY_SUBDIR) -l$(MKL_MPI_WRAPPER)\
+   -lmkl_intel_lp64 -lmkl_sequential -lmkl_core
 
 # If YES, enable some informational messages
 VERBOSE = NO

data/beam-wedge.vtk

@@ -31,14 +31,14 @@ POINTS 27 double
 7 0.5 1
 8 0.5 1
 CELLS 8 56
-6 0 9 18 1 10 19
-6 1 10 19 2 11 20
-6 2 11 20 3 12 21
-6 3 12 21 4 13 22
-6 4 13 22 5 14 23
-6 5 14 23 6 15 24
-6 6 15 24 7 16 25
-6 7 16 25 8 17 26
+6 0 18 9 1 19 10
+6 1 19 10 2 20 11
+6 2 20 11 3 21 12
+6 3 21 12 4 22 13
+6 4 22 13 5 23 14
+6 5 23 14 6 24 15
+6 6 24 15 7 25 16
+6 7 25 16 8 26 17
 CELL_TYPES 8
 13
 13

examples/amgx/ex1.cpp

@@ -205,10 +205,14 @@ int main(int argc, char *argv[])
 
       if (amgx_solver)
       {
+         amgx.SetConvergenceCheck(true);
          amgx.Mult(B,X);
       }
       else
       {
+         // Omit convergence check at the AmgX level when using as a
+         // preconditioner.
+         amgx.SetConvergenceCheck(false);
          PCG(*A.As<SparseMatrix>(), amgx, B, X, 3, 40, 1e-12, 0.0);
       }
    }

examples/amgx/ex1p.cpp

@@ -264,6 +264,7 @@ int main(int argc, char *argv[])
       }
 
       amgx.SetOperator(*A.As<HypreParMatrix>());
+      amgx.SetConvergenceCheck(true);
       amgx.Mult(B, X);
 
       // Release MPI communicators and resources created by AmgX

examples/ex1.cpp

@@ -30,6 +30,7 @@
 // Device sample runs:
 //               ex1 -pa -d cuda
 //               ex1 -pa -d raja-cuda
+//             * ex1 -pa -d raja-hip
 //               ex1 -pa -d occa-cuda
 //               ex1 -pa -d raja-omp
 //               ex1 -pa -d occa-omp

examples/ex10p.cpp

@@ -178,6 +178,7 @@ int main(int argc, char *argv[])
    double visc = 1e-2;
    double mu = 0.25;
    double K = 5.0;
+   bool adaptive_lin_rtol = true;
    bool visualization = true;
    int vis_steps = 1;
 
@@ -206,6 +207,9 @@ int main(int argc, char *argv[])
                   "Shear modulus in the Neo-Hookean hyperelastic model.");
    args.AddOption(&K, "-K", "--bulk-modulus",
                   "Bulk modulus in the Neo-Hookean hyperelastic model.");
+   args.AddOption(&adaptive_lin_rtol, "-alrtol", "--adaptive-lin-rtol",
+                  "-no-alrtol", "--no-adaptive-lin-rtol",
+                  "Enable or disable adaptive linear solver rtol.");
    args.AddOption(&visualization, "-vis", "--visualization", "-no-vis",
                   "--no-visualization",
                   "Enable or disable GLVis visualization.");
@@ -574,6 +578,7 @@ HyperelasticOperator::HyperelasticOperator(ParFiniteElementSpace &f,
    newton_solver.SetPrintLevel(1); // print Newton iterations
    newton_solver.SetRelTol(rel_tol);
    newton_solver.SetAbsTol(0.0);
+   newton_solver.SetAdaptiveLinRtol(2, 0.5, 0.9);
    newton_solver.SetMaxIter(10);
 }
 

examples/ex14.cpp

@@ -5,6 +5,7 @@
 // Sample runs:  ex14 -m ../data/inline-quad.mesh -o 0
 //               ex14 -m ../data/star.mesh -r 4 -o 2
 //               ex14 -m ../data/star-mixed.mesh -r 4 -o 2
+//               ex14 -m ../data/star-mixed.mesh -r 2 -o 2 -k 0 -e 1
 //               ex14 -m ../data/escher.mesh -s 1
 //               ex14 -m ../data/fichera.mesh -s 1 -k 1
 //               ex14 -m ../data/fichera-mixed.mesh -s 1 -k 1
@@ -44,6 +45,7 @@ int main(int argc, char *argv[])
    int order = 1;
    double sigma = -1.0;
    double kappa = -1.0;
+   double eta = 0.0;
    bool visualization = 1;
 
    OptionsParser args(argc, argv);
@@ -54,11 +56,12 @@ int main(int argc, char *argv[])
    args.AddOption(&order, "-o", "--order",
                   "Finite element order (polynomial degree) >= 0.");
    args.AddOption(&sigma, "-s", "--sigma",
-                  "One of the two DG penalty parameters, typically +1/-1."
+                  "One of the three DG penalty parameters, typically +1/-1."
                   " See the documentation of class DGDiffusionIntegrator.");
    args.AddOption(&kappa, "-k", "--kappa",
-                  "One of the two DG penalty parameters, should be positive."
+                  "One of the three DG penalty parameters, should be positive."
                   " Negative values are replaced with (order+1)^2.");
+   args.AddOption(&eta, "-e", "--eta", "BR2 penalty parameter.");
    args.AddOption(&visualization, "-vis", "--visualization", "-no-vis",
                   "--no-visualization",
                   "Enable or disable GLVis visualization.");
@@ -130,6 +133,11 @@ int main(int argc, char *argv[])
    a->AddDomainIntegrator(new DiffusionIntegrator(one));
    a->AddInteriorFaceIntegrator(new DGDiffusionIntegrator(one, sigma, kappa));
    a->AddBdrFaceIntegrator(new DGDiffusionIntegrator(one, sigma, kappa));
+   if (eta > 0)
+   {
+      a->AddInteriorFaceIntegrator(new DGDiffusionBR2Integrator(fespace, eta));
+      a->AddBdrFaceIntegrator(new DGDiffusionBR2Integrator(fespace, eta));
+   }
    a->Assemble();
    a->Finalize();
    const SparseMatrix &A = a->SpMat();

examples/ex14p.cpp

@@ -5,6 +5,7 @@
 // Sample runs:  mpirun -np 4 ex14p -m ../data/inline-quad.mesh -o 0
 //               mpirun -np 4 ex14p -m ../data/star.mesh -o 2
 //               mpirun -np 4 ex14p -m ../data/star-mixed.mesh -o 2
+//               mpirun -np 4 ex14p -m ../data/star-mixed.mesh -o 2 -k 0 -e 1
 //               mpirun -np 4 ex14p -m ../data/escher.mesh -s 1
 //               mpirun -np 4 ex14p -m ../data/fichera.mesh -s 1 -k 1
 //               mpirun -np 4 ex14p -m ../data/fichera-mixed.mesh -s 1 -k 1
@@ -82,6 +83,7 @@ int main(int argc, char *argv[])
    int order = 1;
    double sigma = -1.0;
    double kappa = -1.0;
+   double eta = 0.0;
    bool visualization = 1;
 
    OptionsParser args(argc, argv);
@@ -95,11 +97,12 @@ int main(int argc, char *argv[])
    args.AddOption(&order, "-o", "--order",
                   "Finite element order (polynomial degree) >= 0.");
    args.AddOption(&sigma, "-s", "--sigma",
-                  "One of the two DG penalty parameters, typically +1/-1."
+                  "One of the three DG penalty parameters, typically +1/-1."
                   " See the documentation of class DGDiffusionIntegrator.");
    args.AddOption(&kappa, "-k", "--kappa",
-                  "One of the two DG penalty parameters, should be positive."
+                  "One of the three DG penalty parameters, should be positive."
                   " Negative values are replaced with (order+1)^2.");
+   args.AddOption(&eta, "-e", "--eta", "BR2 penalty parameter.");
    args.AddOption(&visualization, "-vis", "--visualization", "-no-vis",
                   "--no-visualization",
                   "Enable or disable GLVis visualization.");
@@ -194,6 +197,11 @@ int main(int argc, char *argv[])
    a->AddDomainIntegrator(new DiffusionIntegrator(one));
    a->AddInteriorFaceIntegrator(new DGDiffusionIntegrator(one, sigma, kappa));
    a->AddBdrFaceIntegrator(new DGDiffusionIntegrator(one, sigma, kappa));
+   if (eta > 0)
+   {
+      a->AddInteriorFaceIntegrator(new DGDiffusionBR2Integrator(fespace, eta));
+      a->AddBdrFaceIntegrator(new DGDiffusionBR2Integrator(fespace, eta));
+   }
    a->Assemble();
    a->Finalize();
 

examples/ex15p.cpp

@@ -19,6 +19,13 @@
 //               mpirun -np 4 ex15p -m ../data/square-disc.mesh
 //               mpirun -np 4 ex15p -m ../data/escher.mesh -r 2 -tf 0.3
 //
+//               Different estimators:
+//
+//               mpirun -np 4 ex15p -est 0 -e 1e-4
+//               mpirun -np 4 ex15p -est 1 -e 1e-6
+//               mpirun -np 4 ex15p -est 1 -o 3 -tf 0.3
+//               mpirun -np 4 ex15p -est 2 -o 2
+//
 // Description:  Building on Example 6, this example demonstrates dynamic AMR.
 //               The mesh is adapted to a time-dependent solution by refinement
 //               as well as by derefinement. For simplicity, the solution is
@@ -28,8 +35,11 @@
 //               At each outer iteration the right hand side function is changed
 //               to mimic a time dependent problem.  Within each inner iteration
 //               the problem is solved on a sequence of meshes which are locally
-//               refined according to a simple ZZ error estimator.  At the end
-//               of the inner iteration the error estimates are also used to
+//               refined according to a chosen error estimator. Currently there
+//               are three error estimators supported: A L2 formulation of the
+//               Zienkiewicz-Zhu error estimator (0), a Kelly error indicator (1)
+//               and a traditional Zienkiewicz-Zhu error estimator (2). At the
+//               end of the inner iteration the error estimates are also used to
 //               identify any elements which may be over-refined and a single
 //               derefinement step is performed.  After each refinement or
 //               derefinement step a rebalance operation is performed to keep
@@ -87,6 +97,7 @@ int main(int argc, char *argv[])
    int nc_limit = 3;         // maximum level of hanging nodes
    bool visualization = true;
    bool visit = false;
+   int which_estimator = 0;
 
    OptionsParser args(argc, argv);
    args.AddOption(&mesh_file, "-m", "--mesh",
@@ -107,6 +118,9 @@ int main(int argc, char *argv[])
                   "Maximum level of hanging nodes.");
    args.AddOption(&t_final, "-tf", "--t-final",
                   "Final time; start time is 0.");
+   args.AddOption(&which_estimator, "-est", "--estimator",
+                  "Which estimator to use: "
+                  "0 = L2ZZ, 1 = Kelly, 2 = ZZ. Defaults to L2ZZ.");
    args.AddOption(&visualization, "-vis", "--visualization", "-no-vis",
                   "--no-visualization",
                   "Enable or disable GLVis visualization.");
@@ -214,17 +228,43 @@ int main(int argc, char *argv[])
    //     provide the method ComputeElementFlux. We supply an L2 space for the
    //     discontinuous flux and an H(div) space for the smoothed flux.
    L2_FECollection flux_fec(order, dim);
-   ParFiniteElementSpace flux_fes(&pmesh, &flux_fec, sdim);
    RT_FECollection smooth_flux_fec(order-1, dim);
-   ParFiniteElementSpace smooth_flux_fes(&pmesh, &smooth_flux_fec);
-   L2ZienkiewiczZhuEstimator estimator(*integ, x, flux_fes, smooth_flux_fes);
+   ErrorEstimator* estimator;
+   switch (which_estimator)
+   {
+      case 1:
+      {
+         auto flux_fes = new ParFiniteElementSpace(&pmesh, &flux_fec, sdim);
+         estimator = new KellyErrorEstimator(*integ, x, flux_fes);
+         break;
+      }
+      case 2:
+      {
+         auto flux_fes = new ParFiniteElementSpace(&pmesh, &fec, sdim);
+         estimator = new ZienkiewiczZhuEstimator(*integ, x, flux_fes);
+         break;
+      }
+
+      default:
+         if (myid == 0)
+         {
+            std::cout << "Unkown estimator. Falling back to L2ZZ." << std::endl;
+         }
+      case 0:
+      {
+         auto flux_fes = new ParFiniteElementSpace(&pmesh, &flux_fec, sdim);
+         auto smooth_flux_fes = new ParFiniteElementSpace(&pmesh, &smooth_flux_fec);
+         estimator = new L2ZienkiewiczZhuEstimator(*integ, x, flux_fes, smooth_flux_fes);
+         break;
+      }
+   }
 
    // 11. As in Example 6p, we also need a refiner. This time the refinement
    //     strategy is based on a fixed threshold that is applied locally to each
    //     element. The global threshold is turned off by setting the total error
    //     fraction to zero. We also enforce a maximum refinement ratio between
    //     adjacent elements.
-   ThresholdRefiner refiner(estimator);
+   ThresholdRefiner refiner(*estimator);
    refiner.SetTotalErrorFraction(0.0); // use purely local threshold
    refiner.SetLocalErrorGoal(max_elem_error);
    refiner.PreferConformingRefinement();
@@ -233,7 +273,7 @@ int main(int argc, char *argv[])
    // 12. A derefiner selects groups of elements that can be coarsened to form
    //     a larger element. A conservative enough threshold needs to be set to
    //     prevent derefining elements that would immediately be refined again.
-   ThresholdDerefiner derefiner(estimator);
+   ThresholdDerefiner derefiner(*estimator);
    derefiner.SetThreshold(hysteresis * max_elem_error);
    derefiner.SetNCLimit(nc_limit);
 
@@ -316,7 +356,7 @@ int main(int argc, char *argv[])
          refiner.Apply(pmesh);
          if (myid == 0)
          {
-            cout << ", total error: " << estimator.GetTotalError() << endl;
+            cout << ", total error: " << estimator->GetTotalError() << endl;
          }
 
          // 21. Quit the AMR loop if the termination criterion has been met
@@ -346,6 +386,8 @@ int main(int argc, char *argv[])
       }
    }
 
+   delete estimator;
+
    // 25. Exit
    MPI_Finalize();
    return 0;

examples/ex1p_complex.cpp

@@ -0,0 +1,435 @@
+//                       MFEM Example 3 - Parallel Version
+//
+// Compile with: make ex3p_complex
+//
+
+#include "mfem.hpp"
+#include <fstream>
+#include <iostream>
+
+using namespace std;
+using namespace mfem;
+
+// Exact solution, E, and r.h.s., f. See below for implementation.
+double E_exact(const Vector &);
+void gradE_exact(const Vector &, Vector &);
+double f_exact(const Vector &);
+double freq = 1.0, kappa;
+int dim;
+
+
+#define COMPLEX_VERSION
+#define NEUMANN
+
+const double omega = 1.4;
+const double eps = 1.0e-8;
+
+
+int main(int argc, char *argv[])
+{
+   // 1. Initialize MPI.
+   int num_procs, myid;
+   MPI_Init(&argc, &argv);
+   MPI_Comm_size(MPI_COMM_WORLD, &num_procs);
+   MPI_Comm_rank(MPI_COMM_WORLD, &myid);
+
+   // 2. Parse command-line options.
+   const char *mesh_file = "../data/beam-tet.mesh";
+   int order = 1;
+   bool static_cond = false;
+   bool pa = false;
+   const char *device_config = "cpu";
+   bool visualization = 1;
+
+   OptionsParser args(argc, argv);
+   args.AddOption(&mesh_file, "-m", "--mesh",
+                  "Mesh file to use.");
+   args.AddOption(&order, "-o", "--order",
+                  "Finite element order (polynomial degree).");
+   args.AddOption(&freq, "-f", "--frequency", "Set the frequency for the exact"
+                  " solution.");
+   args.AddOption(&static_cond, "-sc", "--static-condensation", "-no-sc",
+                  "--no-static-condensation", "Enable static condensation.");
+   args.AddOption(&pa, "-pa", "--partial-assembly", "-no-pa",
+                  "--no-partial-assembly", "Enable Partial Assembly.");
+   args.AddOption(&device_config, "-d", "--device",
+                  "Device configuration string, see Device::Configure().");
+   args.AddOption(&visualization, "-vis", "--visualization", "-no-vis",
+                  "--no-visualization",
+                  "Enable or disable GLVis visualization.");
+
+   args.Parse();
+   if (!args.Good())
+   {
+      if (myid == 0)
+      {
+         args.PrintUsage(cout);
+      }
+      MPI_Finalize();
+      return 1;
+   }
+   if (myid == 0)
+   {
+      args.PrintOptions(cout);
+   }
+   kappa = freq * M_PI;
+
+   // 3. Enable hardware devices such as GPUs, and programming models such as
+   //    CUDA, OCCA, RAJA and OpenMP based on command line options.
+   Device device(device_config);
+   if (myid == 0) { device.Print(); }
+
+   // 4. Read the (serial) mesh from the given mesh file on all processors.  We
+   //    can handle triangular, quadrilateral, tetrahedral, hexahedral, surface
+   //    and volume meshes with the same code.
+   Mesh *mesh = new Mesh(mesh_file, 1, 1);
+   dim = mesh->Dimension();
+   int sdim = mesh->SpaceDimension();
+
+   // 5. Refine the serial mesh on all processors to increase the resolution. In
+   //    this example we do 'ref_levels' of uniform refinement. We choose
+   //    'ref_levels' to be the largest number that gives a final mesh with no
+   //    more than 1,000 elements.
+   {
+      int ref_levels = (int)floor(log(1000./mesh->GetNE())/log(2.)/dim);
+      for (int l = 0; l < ref_levels; l++)
+      {
+         mesh->UniformRefinement();
+      }
+   }
+
+   // 6. Define a parallel mesh by a partitioning of the serial mesh. Refine
+   //    this mesh further in parallel to increase the resolution. Once the
+   //    parallel mesh is defined, the serial mesh can be deleted. Tetrahedral
+   //    meshes need to be reoriented before we can define high-order Nedelec
+   //    spaces on them.
+   ParMesh *pmesh = new ParMesh(MPI_COMM_WORLD, *mesh);
+   delete mesh;
+   {
+      int par_ref_levels = 0;
+      for (int l = 0; l < par_ref_levels; l++)
+      {
+         pmesh->UniformRefinement();
+      }
+   }
+   pmesh->ReorientTetMesh();
+
+   // 7. Define a parallel finite element space on the parallel mesh. Here we
+   //    use the Nedelec finite elements of the specified order.
+   FiniteElementCollection *fec = new H1_FECollection(order, dim);
+   ParFiniteElementSpace *fespace = new ParFiniteElementSpace(pmesh, fec);
+   HYPRE_Int size = fespace->GlobalTrueVSize();
+   if (myid == 0)
+   {
+      cout << "Number of finite element unknowns: " << size << endl;
+   }
+
+   // 8. Determine the list of true (i.e. parallel conforming) essential
+   //    boundary dofs. In this example, the boundary conditions are defined
+   //    by marking all the boundary attributes from the mesh as essential
+   //    (Dirichlet) and converting them to a list of true dofs.
+   Array<int> ess_tdof_list;
+
+#ifndef NEUMANN
+   if (pmesh->bdr_attributes.Size())
+   {
+      Array<int> ess_bdr(pmesh->bdr_attributes.Max());
+      ess_bdr = 1;
+      fespace->GetEssentialTrueDofs(ess_bdr, ess_tdof_list);
+   }
+#endif
+
+   //const double imscale = 0.0;
+   const double imscale = -omega;
+
+   Coefficient *im = new ConstantCoefficient(imscale);  // im part
+   //Coefficient *im = new ConstantCoefficient(0.0);  // im part
+
+   FunctionCoefficient E_coef(E_exact);
+   VectorFunctionCoefficient grad_E(sdim, gradE_exact);
+   ProductCoefficient omegaE(imscale, E_coef);
+
+   // 9. Set up the parallel linear form b(.) which corresponds to the
+   //    right-hand side of the FEM linear system, which in this case is
+   //    (f,phi_i) where f is given by the function f_exact and phi_i are the
+   //    basis functions in the finite element fespace.
+   FunctionCoefficient f(f_exact);
+#ifdef COMPLEX_VERSION
+   ParComplexLinearForm *b = new ParComplexLinearForm(fespace);
+   b->AddDomainIntegrator(new DomainLFIntegrator(f), NULL);
+#ifdef NEUMANN
+   b->AddBoundaryIntegrator(NULL, new BoundaryNormalLFIntegrator(grad_E));
+#endif
+   b->AddBoundaryIntegrator(NULL, new BoundaryLFIntegrator(omegaE)); // im part
+#endif
+
+   b->Assemble();
+
+   // 10. Define the solution vector x as a parallel finite element grid function
+   //     corresponding to fespace. Initialize x by projecting the exact
+   //     solution. Note that only values from the boundary edges will be used
+   //     when eliminating the non-homogeneous boundary condition to modify the
+   //     r.h.s. vector b.
+   /*
+   ParGridFunction x(fespace);
+   VectorFunctionCoefficient E(sdim, E_exact);
+   x.ProjectCoefficient(E);
+   */
+
+#ifdef COMPLEX_VERSION
+   // Complex version
+   ParComplexGridFunction x(fespace);
+   x = 0.0;
+   ConstantCoefficient E_im(0.0);
+   //x.ProjectBdrCoefficientTangent(E_Re, E_Im, ess_bdr);
+   x.ProjectCoefficient(E_coef, E_im);
+#endif
+
+   // 11. Set up the parallel bilinear form corresponding to the EM diffusion
+   //     operator curl muinv curl + sigma I, by adding the curl-curl and the
+   //     mass domain integrators.
+   Coefficient *muinv = new ConstantCoefficient(1.0);
+   Coefficient *epscoef = new ConstantCoefficient(eps);
+   Coefficient *imabs = new ConstantCoefficient(fabs(imscale));  // im part
+
+#ifdef COMPLEX_VERSION
+   // Complex version
+   ParSesquilinearForm *a = new ParSesquilinearForm(fespace);
+   if (pa) { a->SetAssemblyLevel(AssemblyLevel::PARTIAL); }
+   a->AddDomainIntegrator(new DiffusionIntegrator(*muinv), NULL);
+   a->AddDomainIntegrator(new MassIntegrator(*epscoef), NULL);
+   a->AddBoundaryIntegrator(NULL, new MassIntegrator(*im));  // im part
+#endif
+
+   // 12. Assemble the parallel bilinear form and the corresponding linear
+   //     system, applying any necessary transformations such as: parallel
+   //     assembly, eliminating boundary conditions, applying conforming
+   //     constraints for non-conforming AMR, static condensation, etc.
+   //if (static_cond) { a->EnableStaticCondensation(); }
+   a->Assemble();
+
+   OperatorPtr A;
+   Vector B, X;
+   a->FormLinearSystem(ess_tdof_list, x, *b, A, X, B);
+
+   ParBilinearForm a_Re(fespace);
+   a_Re.AddDomainIntegrator(new DiffusionIntegrator(*muinv));
+   a_Re.AddDomainIntegrator(new MassIntegrator(*epscoef));
+
+   if (pa) { a_Re.SetAssemblyLevel(AssemblyLevel::PARTIAL); }
+   a_Re.Assemble();
+
+   OperatorPtr A_Re;
+   a_Re.FormSystemMatrix(ess_tdof_list, A_Re);
+
+   ParBilinearForm a_Im(fespace);
+   a_Im.AddBoundaryIntegrator(new MassIntegrator(*imabs));
+   a_Im.Assemble();
+
+   OperatorPtr A_Im;
+   a_Im.FormSystemMatrix(ess_tdof_list, A_Im);
+
+   // 13. Solve the system AX=B using PCG with the AMS preconditioner from hypre
+   //     (in the full assembly case) or CG with Jacobi preconditioner (in the
+   //     partial assembly case).
+
+   Array<int> offsets(3);
+   offsets[0] = 0;
+   offsets[1] = fespace->GetTrueVSize();
+   offsets[2] = fespace->GetTrueVSize();
+   offsets.PartialSum();
+
+   //OperatorJacobiSmoother massJacobi(a_Im, ess_tdof_list);
+
+   StopWatch sw;
+   sw.Clear();
+   sw.Start();
+
+   if (pa) // Jacobi preconditioning in partial assembly mode
+   {
+      MFEM_VERIFY(false, "TODO");
+      //OperatorJacobiSmoother Jacobi(*a, ess_tdof_list);
+
+      CGSolver cg(MPI_COMM_WORLD);
+      cg.SetRelTol(1e-12);
+      cg.SetMaxIter(1000);
+      cg.SetPrintLevel(1);
+      cg.SetOperator(*A);
+      //cg.SetPreconditioner(Jacobi);
+      cg.Mult(B, X);
+   }
+   else
+   {
+      if (myid == 0)
+      {
+         cout << "Size of linear system: "
+              << A.As<HypreParMatrix>()->GetGlobalNumRows() << endl;
+      }
+
+      HypreBoomerAMG amg(*A_Re.As<HypreParMatrix>());
+
+#ifdef COMPLEX_VERSION
+      BlockDiagonalPreconditioner BlockDP(offsets);
+      BlockDP.SetDiagonalBlock(0, &amg);
+      BlockDP.SetDiagonalBlock(1, &amg);
+
+      Complex_PMHSS PMHSS(A_Re.Ptr(), A_Im.Ptr(), &BlockDP, NULL, 1.0);
+
+      ComplexOperator AspdComplex(A_Re.Ptr(), A_Im.Ptr(), false, false);
+
+      GMRESSolver PMHSSgmres(MPI_COMM_WORLD);
+      PMHSSgmres.SetPrintLevel(1);
+      PMHSSgmres.SetKDim(100);
+      PMHSSgmres.SetMaxIter(100);
+      PMHSSgmres.SetRelTol(1e-6);
+      PMHSSgmres.SetAbsTol(0.0);
+      PMHSSgmres.SetOperator(AspdComplex);
+      PMHSSgmres.SetPreconditioner(PMHSS);
+
+      GMRESSolver gmres(MPI_COMM_WORLD);
+      gmres.SetPrintLevel(1);
+      gmres.SetKDim(1000);
+      gmres.SetMaxIter(100);
+      gmres.SetRelTol(1e-8);
+      gmres.SetAbsTol(0.0);
+      gmres.SetOperator(*A);
+      //gmres.SetPreconditioner(BlockDP);
+      gmres.SetPreconditioner(PMHSS);
+      //gmres.SetPreconditioner(PMHSSgmres);
+#else
+      GMRESSolver gmres(MPI_COMM_WORLD);
+      gmres.SetPrintLevel(1);
+      gmres.SetKDim(1000);
+      gmres.SetMaxIter(100);
+      gmres.SetRelTol(1e-8);
+      gmres.SetAbsTol(0.0);
+      gmres.SetOperator(*A);
+      gmres.SetPreconditioner(ams);
+#endif
+
+      gmres.Mult(B, X);
+   }
+
+   sw.Stop();
+   mfem::out << "Total solve time " <<sw.RealTime() << endl;
+
+   // 14. Recover the parallel grid function corresponding to X. This is the
+   //     local finite element solution on each processor.
+   a->RecoverFEMSolution(X, *b, x);
+
+   // 15. Compute and print the L^2 norm of the error.
+   {
+#ifdef COMPLEX_VERSION
+      double err = x.real().ComputeL2Error(E_coef);
+#else
+      double err = x.ComputeL2Error(E_coef);
+#endif
+      if (myid == 0)
+      {
+         cout << "\n|| E_h - E ||_{L^2} = " << err << '\n' << endl;
+      }
+   }
+
+   // 16. Save the refined mesh and the solution in parallel. This output can
+   //     be viewed later using GLVis: "glvis -np <np> -m mesh -g sol".
+   {
+      ostringstream mesh_name, sol_name;
+      mesh_name << "mesh." << setfill('0') << setw(6) << myid;
+      sol_name << "sol." << setfill('0') << setw(6) << myid;
+
+      ofstream mesh_ofs(mesh_name.str().c_str());
+      mesh_ofs.precision(8);
+      pmesh->Print(mesh_ofs);
+
+      ofstream sol_ofs(sol_name.str().c_str());
+      sol_ofs.precision(8);
+#ifdef COMPLEX_VERSION
+      x.real().Save(sol_ofs);
+#else
+      x.Save(sol_ofs);
+#endif
+   }
+
+   // 17. Send the solution by socket to a GLVis server.
+   if (visualization)
+   {
+      char vishost[] = "localhost";
+      int  visport   = 19916;
+      socketstream sol_sock(vishost, visport);
+      sol_sock << "parallel " << num_procs << " " << myid << "\n";
+      sol_sock.precision(8);
+#ifdef COMPLEX_VERSION
+      sol_sock << "solution\n" << *pmesh << x.real() << flush;
+#else
+      sol_sock << "solution\n" << *pmesh << x << flush;
+#endif
+   }
+
+   // 18. Free the used memory.
+   delete a;
+   delete muinv;
+   delete b;
+   delete fespace;
+   delete fec;
+   delete pmesh;
+
+   MPI_Finalize();
+
+   return 0;
+}
+
+#define VERSION_COS
+
+double E_exact(const Vector &x)
+{
+   if (dim == 3)
+   {
+#ifdef VERSION_COS
+      return cos(kappa * x(0)) * cos(kappa * x(1)) * cos(kappa * x(2));
+#else
+      return sin(kappa * x(0)) * sin(kappa * x(1)) * sin(kappa * x(2));
+#endif
+   }
+   else
+   {
+      return 0.0;
+   }
+}
+
+void gradE_exact(const Vector &x, Vector &grad)
+{
+   if (dim == 3)
+   {
+#ifdef VERSION_COS
+      grad(0) = -kappa * sin(kappa * x(0)) * cos(kappa * x(1)) * cos(kappa * x(2));
+      grad(1) = -kappa * sin(kappa * x(1)) * cos(kappa * x(0)) * cos(kappa * x(2));
+      grad(2) = -kappa * sin(kappa * x(2)) * cos(kappa * x(0)) * cos(kappa * x(1));
+#else
+      grad(0) = kappa * cos(kappa * x(0)) * sin(kappa * x(1)) * sin(kappa * x(2));
+      grad(1) = kappa * cos(kappa * x(1)) * sin(kappa * x(0)) * sin(kappa * x(2));
+      grad(2) = kappa * cos(kappa * x(2)) * sin(kappa * x(0)) * sin(kappa * x(1));
+#endif
+   }
+   else
+   {
+      MFEM_VERIFY(false, "");
+   }
+}
+
+// (grad u, grad v) + eps (u, v) = <grad u . n, v> - (div grad u, v) + eps (u, v)
+double f_exact(const Vector &x)
+{
+   if (dim == 3)
+   {
+      const double c = 3.0 * kappa * kappa;
+#ifdef VERSION_COS
+      return (eps + c) * cos(kappa * x(0)) * cos(kappa * x(1)) * cos(kappa * x(2));
+#else
+      return (eps + c) * sin(kappa * x(0)) * sin(kappa * x(1)) * sin(kappa * x(2));
+#endif
+   }
+   else
+   {
+      return 0.0;
+   }
+}

examples/ex25.cpp

@@ -10,6 +10,10 @@
 //               ex25 -o 2 -f 8.0 -ref 3 -prob 4 -m ../data/inline-quad.mesh
 //               ex25 -o 2 -f 2.0 -ref 1 -prob 4 -m ../data/inline-hex.mesh
 //
+// Device sample runs:
+//               ex25 -o 2 -f 8.0 -ref 3 -prob 4 -m ../data/inline-quad.mesh -pa -d cuda
+//               ex25 -o 2 -f 2.0 -ref 1 -prob 4 -m ../data/inline-hex.mesh -pa -d cuda
+//
 // Description:  This example code solves a simple electromagnetic wave
 //               propagation problem corresponding to the second order
 //               indefinite Maxwell equation
@@ -157,7 +161,10 @@ int main(int argc, char *argv[])
    int iprob = 4;
    double freq = 5.0;
    bool herm_conv = true;
+   bool umf_solver = false;
    bool visualization = 1;
+   bool pa = false;
+   const char *device_config = "cpu";
 
    OptionsParser args(argc, argv);
    args.AddOption(&mesh_file, "-m", "--mesh",
@@ -176,15 +183,28 @@ int main(int argc, char *argv[])
                   "Frequency (in Hz).");
    args.AddOption(&herm_conv, "-herm", "--hermitian", "-no-herm",
                   "--no-hermitian", "Use convention for Hermitian operators.");
+#ifdef MFEM_USE_SUITESPARSE
+   args.AddOption(&umf_solver, "-umf", "--umfpack", "-no-umf",
+                  "--no-umfpack", "Use the UMFPack Solver.");
+#endif
    args.AddOption(&visualization, "-vis", "--visualization", "-no-vis",
                   "--no-visualization",
                   "Enable or disable GLVis visualization.");
+   args.AddOption(&pa, "-pa", "--partial-assembly", "-no-pa",
+                  "--no-partial-assembly", "Enable Partial Assembly.");
+   args.AddOption(&device_config, "-d", "--device",
+                  "Device configuration string, see Device::Configure().");
    args.Parse();
 
    if (iprob > 4) { iprob = 4; }
    prob = (prob_type)iprob;
 
-   // 2. Setup the mesh
+   // 2. Enable hardware devices such as GPUs, and programming models such as
+   //    CUDA, OCCA, RAJA and OpenMP based on command line options.
+   Device device(device_config);
+   device.Print();
+
+   // 3. Setup the mesh
    if (!mesh_file)
    {
       exact_known = true;
@@ -225,7 +245,7 @@ int main(int argc, char *argv[])
    // Setup PML length
    Array2D<double> length(dim, 2); length = 0.0;
 
-   // 3. Setup the Cartesian PML region.
+   // 4. Setup the Cartesian PML region.
    switch (prob)
    {
       case disc:
@@ -251,19 +271,19 @@ int main(int argc, char *argv[])
    comp_domain_bdr = pml->GetCompDomainBdr();
    domain_bdr = pml->GetDomainBdr();
 
-   // 4. Refine the mesh to increase the resolution.
+   // 5. Refine the mesh to increase the resolution.
    for (int l = 0; l < ref_levels; l++)
    {
       mesh->UniformRefinement();
    }
 
-   // 5. Reorient mesh in case of a tet mesh
+   // 6. Reorient mesh in case of a tet mesh
    mesh->ReorientTetMesh();
 
    // Set element attributes in order to distinguish elements in the PML region
    pml->SetAttributes(mesh);
 
-   // 6. Define a finite element space on the mesh. Here we use the Nedelec
+   // 7. Define a finite element space on the mesh. Here we use the Nedelec
    //    finite elements of the specified order.
    FiniteElementCollection *fec = new ND_FECollection(order, dim);
    FiniteElementSpace *fespace = new FiniteElementSpace(mesh, fec);
@@ -271,7 +291,7 @@ int main(int argc, char *argv[])
 
    cout << "Number of finite element unknowns: " << size << endl;
 
-   // 7. Determine the list of true essential boundary dofs. In this example,
+   // 8. Determine the list of true essential boundary dofs. In this example,
    //    the boundary conditions are defined based on the specific mesh and the
    //    problem type.
    Array<int> ess_tdof_list;
@@ -313,12 +333,12 @@ int main(int argc, char *argv[])
    }
    fespace->GetEssentialTrueDofs(ess_bdr, ess_tdof_list);
 
-   // 8. Setup Complex Operator convention
+   // 9. Setup Complex Operator convention
    ComplexOperator::Convention conv =
       herm_conv ? ComplexOperator::HERMITIAN : ComplexOperator::BLOCK_SYMMETRIC;
 
-   // 9. Set up the linear form b(.) which corresponds to the right-hand side of
-   //    the FEM linear system.
+   // 10. Set up the linear form b(.) which corresponds to the right-hand side of
+   //     the FEM linear system.
    VectorFunctionCoefficient f(dim, source);
    ComplexLinearForm b(fespace, conv);
    if (prob == load_src)
@@ -328,7 +348,7 @@ int main(int argc, char *argv[])
    b.Vector::operator=(0.0);
    b.Assemble();
 
-   // 10. Define the solution vector x as a complex finite element grid function
+   // 11. Define the solution vector x as a complex finite element grid function
    //     corresponding to fespace.
    ComplexGridFunction x(fespace);
    x = 0.0;
@@ -336,7 +356,7 @@ int main(int argc, char *argv[])
    VectorFunctionCoefficient E_Im(dim, E_bdr_data_Im);
    x.ProjectBdrCoefficientTangent(E_Re, E_Im, ess_bdr);
 
-   // 11. Set up the sesquilinear form a(.,.)
+   // 12. Set up the sesquilinear form a(.,.)
    //
    //     In Comp
    //     Domain:   1/mu (Curl E, Curl F) - omega^2 * epsilon (E,F)
@@ -390,32 +410,35 @@ int main(int argc, char *argv[])
    a.AddDomainIntegrator(new VectorFEMassIntegrator(restr_c2_Re),
                          new VectorFEMassIntegrator(restr_c2_Im));
 
-   // 12. Assemble the bilinear form and the corresponding linear system,
+   // 13. Assemble the bilinear form and the corresponding linear system,
    //     applying any necessary transformations such as: assembly, eliminating
    //     boundary conditions, applying conforming constraints for
    //     non-conforming AMR, etc.
+   if (pa) { a.SetAssemblyLevel(AssemblyLevel::PARTIAL); }
    a.Assemble(0);
 
    OperatorPtr A;
    Vector B, X;
    a.FormLinearSystem(ess_tdof_list, x, b, A, X, B);
 
-   // 13. Solve using a direct or an iterative solver
+   // 14. Solve using a direct or an iterative solver
 #ifdef MFEM_USE_SUITESPARSE
+   if (!pa && umf_solver)
    {
       ComplexUMFPackSolver csolver(*A.As<ComplexSparseMatrix>());
       csolver.Control[UMFPACK_ORDERING] = UMFPACK_ORDERING_METIS;
       csolver.SetPrintLevel(1);
       csolver.Mult(B, X);
    }
-#else
-   // 13a. Set up the Bilinear form a(.,.) for the preconditioner
+#endif
+   // 14a. Set up the Bilinear form a(.,.) for the preconditioner
    //
    //    In Comp
    //    Domain:   1/mu (Curl E, Curl F) + omega^2 * epsilon (E,F)
    //
    //    In PML:   1/mu (abs(1/det(J) J^T J) Curl E, Curl F)
    //              + omega^2 * epsilon (abs(det(J) * (J^T J)^-1) * E, F)
+   if (pa || !umf_solver)
    {
       ConstantCoefficient absomeg(pow(omega, 2) * epsilon);
       RestrictedCoefficient restr_absomeg(absomeg,attr);
@@ -435,39 +458,57 @@ int main(int argc, char *argv[])
       prec.AddDomainIntegrator(new CurlCurlIntegrator(restr_c1_abs));
       prec.AddDomainIntegrator(new VectorFEMassIntegrator(restr_c2_abs));
 
+      if (pa) { prec.SetAssemblyLevel(AssemblyLevel::PARTIAL); }
       prec.Assemble();
 
-      OperatorPtr PCOpAh;
-      prec.FormSystemMatrix(ess_tdof_list, PCOpAh);
-
-      // 13b. Define and apply a GMRES solver for AU=B with a block diagonal
-      //      preconditioner based on the Gauss-Seidel sparse smoother.
+      // 14b. Define and apply a GMRES solver for AU=B with a block diagonal
+      //      preconditioner based on the Gauss-Seidel or Jacobi sparse smoother.
       Array<int> offsets(3);
       offsets[0] = 0;
       offsets[1] = fespace->GetTrueVSize();
       offsets[2] = fespace->GetTrueVSize();
       offsets.PartialSum();
 
-      GSSmoother gs00(*PCOpAh.As<SparseMatrix>());
-      BlockDiagonalPreconditioner BlockGS(offsets);
-      ScaledOperator gs11(&gs00,
-                          (conv == ComplexOperator::HERMITIAN) ? -1.0 : 1.0);
-      BlockGS.SetDiagonalBlock(0,&gs00);
-      BlockGS.SetDiagonalBlock(1,&gs11);
+      Operator *pc_r = nullptr;
+      Operator *pc_i = nullptr;
+      int s = (conv == ComplexOperator::HERMITIAN) ? -1.0 : 1.0;
+      if (pa)
+      {
+         // Jacobi Smoother
+         OperatorJacobiSmoother *d00 = new OperatorJacobiSmoother(prec, ess_tdof_list);
+         ScaledOperator *d11 = new ScaledOperator(d00, s);
+         pc_r = d00;
+         pc_i = d11;
+      }
+      else
+      {
+         OperatorPtr PCOpAh;
+         prec.SetDiagonalPolicy(mfem::Operator::DIAG_ONE);
+         prec.FormSystemMatrix(ess_tdof_list, PCOpAh);
+
+         // Gauss-Seidel Smoother
+         GSSmoother *gs00 = new GSSmoother(*PCOpAh.As<SparseMatrix>());
+         ScaledOperator *gs11 = new ScaledOperator(gs00, s);
+         pc_r = gs00;
+         pc_i = gs11;
+      }
+
+      BlockDiagonalPreconditioner BlockDP(offsets);
+      BlockDP.SetDiagonalBlock(0, pc_r);
+      BlockDP.SetDiagonalBlock(1, pc_i);
 
       GMRESSolver gmres;
       gmres.SetPrintLevel(1);
       gmres.SetKDim(200);
-      gmres.SetMaxIter(2000);
+      gmres.SetMaxIter(pa ? 5000 : 2000);
       gmres.SetRelTol(1e-5);
       gmres.SetAbsTol(0.0);
       gmres.SetOperator(*A);
-      gmres.SetPreconditioner(BlockGS);
+      gmres.SetPreconditioner(BlockDP);
       gmres.Mult(B, X);
    }
-#endif
 
-   // 14. Recover the solution as a finite element grid function and compute the
+   // 15. Recover the solution as a finite element grid function and compute the
    //     errors if the exact solution is known.
    a.RecoverFEMSolution(X, b, x);
 
@@ -504,7 +545,7 @@ int main(int argc, char *argv[])
            << sqrt(L2Error_Re*L2Error_Re + L2Error_Im*L2Error_Im) << "\n\n";
    }
 
-   // 15. Save the refined mesh and the solution. This output can be viewed
+   // 16. Save the refined mesh and the solution. This output can be viewed
    //     later using GLVis: "glvis -m mesh -g sol".
    {
       ofstream mesh_ofs("ex25.mesh");
@@ -519,7 +560,7 @@ int main(int argc, char *argv[])
       x.imag().Save(sol_i_ofs);
    }
 
-   // 16. Send the solution by socket to a GLVis server.
+   // 17. Send the solution by socket to a GLVis server.
    if (visualization)
    {
       // Define visualization keys for GLVis (see GLVis documentation)
@@ -570,7 +611,7 @@ int main(int argc, char *argv[])
       }
    }
 
-   // 17. Free the used memory.
+   // 18. Free the used memory.
    delete pml;
    delete fespace;
    delete fec;
@@ -916,7 +957,14 @@ void CartesianPML::SetBoundaries()
 
 void CartesianPML::SetAttributes(Mesh *mesh_)
 {
+   // Initialize bdr attributes
+   for (int i = 0; i < mesh_->GetNBE(); ++i)
+   {
+      mesh_->GetBdrElement(i)->SetAttribute(i+1);
+   }
+
    int nrelem = mesh_->GetNE();
+
    elems.SetSize(nrelem);
 
    // Loop through the elements and identify which of them are in the PML

examples/ex25p.cpp

@@ -10,6 +10,10 @@
 //               mpirun -np 4 ex25p -o 2 -f 8.0 -rs 2 -rp 2 -prob 4 -m ../data/inline-quad.mesh
 //               mpirun -np 4 ex25p -o 2 -f 2.0 -rs 1 -rp 1 -prob 4 -m ../data/inline-hex.mesh
 //
+// Device sample runs:
+//               mpirun -np 4 ex25p -o 1 -f 3.0 -rs 3 -rp 1 -prob 2 -pa -d cuda
+//               mpirun -np 4 ex25p -o 2 -f 1.0 -rs 1 -rp 1 -prob 3 -pa -d cuda
+//
 // Description:  This example code solves a simple electromagnetic wave
 //               propagation problem corresponding to the second order
 //               indefinite Maxwell equation
@@ -167,6 +171,8 @@ int main(int argc, char *argv[])
    bool slu_solver  = false;
    bool mumps_solver = false;
    bool visualization = 1;
+   bool pa = false;
+   const char *device_config = "cpu";
 
    OptionsParser args(argc, argv);
    args.AddOption(&mesh_file, "-m", "--mesh",
@@ -198,6 +204,10 @@ int main(int argc, char *argv[])
    args.AddOption(&visualization, "-vis", "--visualization", "-no-vis",
                   "--no-visualization",
                   "Enable or disable GLVis visualization.");
+   args.AddOption(&pa, "-pa", "--partial-assembly", "-no-pa",
+                  "--no-partial-assembly", "Enable Partial Assembly.");
+   args.AddOption(&device_config, "-d", "--device",
+                  "Device configuration string, see Device::Configure().");
    args.Parse();
    if (slu_solver && mumps_solver)
    {
@@ -211,7 +221,12 @@ int main(int argc, char *argv[])
    if (iprob > 4) { iprob = 4; }
    prob = (prob_type)iprob;
 
-   // 3. Setup the (serial) mesh on all processors.
+   // 3. Enable hardware devices such as GPUs, and programming models such as
+   //    CUDA, OCCA, RAJA and OpenMP based on command line options.
+   Device device(device_config);
+   if (myid == 0) { device.Print(); }
+
+   // 4. Setup the (serial) mesh on all processors.
    if (!mesh_file)
    {
       exact_known = true;
@@ -259,7 +274,7 @@ int main(int argc, char *argv[])
    // Setup PML length
    Array2D<double> length(dim, 2); length = 0.0;
 
-   // 4. Setup the Cartesian PML region.
+   // 5. Setup the Cartesian PML region.
    switch (prob)
    {
       case disc:
@@ -285,13 +300,13 @@ int main(int argc, char *argv[])
    comp_domain_bdr = pml->GetCompDomainBdr();
    domain_bdr = pml->GetDomainBdr();
 
-   // 5. Refine the serial mesh on all processors to increase the resolution.
+   // 6. Refine the serial mesh on all processors to increase the resolution.
    for (int l = 0; l < ref_levels; l++)
    {
       mesh->UniformRefinement();
    }
 
-   // 6. Define a parallel mesh by a partitioning of the serial mesh.
+   // 7. Define a parallel mesh by a partitioning of the serial mesh.
    ParMesh *pmesh = new ParMesh(MPI_COMM_WORLD, *mesh);
    delete mesh;
    {
@@ -301,13 +316,13 @@ int main(int argc, char *argv[])
       }
    }
 
-   // 6a. Reorient mesh in case of a tet mesh
+   // 7a. Reorient mesh in case of a tet mesh
    pmesh->ReorientTetMesh();
 
-   // 7. Set element attributes in order to distinguish elements in the PML
+   // 8. Set element attributes in order to distinguish elements in the PML
    pml->SetAttributes(pmesh);
 
-   // 8. Define a parallel finite element space on the parallel mesh. Here we
+   // 9. Define a parallel finite element space on the parallel mesh. Here we
    //    use the Nedelec finite elements of the specified order.
    FiniteElementCollection *fec = new ND_FECollection(order, dim);
    ParFiniteElementSpace *fespace = new ParFiniteElementSpace(pmesh, fec);
@@ -317,9 +332,9 @@ int main(int argc, char *argv[])
       cout << "Number of finite element unknowns: " << size << endl;
    }
 
-   // 9. Determine the list of true (i.e. parallel conforming) essential
-   //    boundary dofs. In this example, the boundary conditions are defined
-   //    based on the specific mesh and the problem type.
+   // 10. Determine the list of true (i.e. parallel conforming) essential
+   //     boundary dofs. In this example, the boundary conditions are defined
+   //     based on the specific mesh and the problem type.
    Array<int> ess_tdof_list;
    Array<int> ess_bdr;
    if (pmesh->bdr_attributes.Size())
@@ -359,11 +374,11 @@ int main(int argc, char *argv[])
    }
    fespace->GetEssentialTrueDofs(ess_bdr, ess_tdof_list);
 
-   // 10. Setup Complex Operator convention
+   // 11. Setup Complex Operator convention
    ComplexOperator::Convention conv =
       herm_conv ? ComplexOperator::HERMITIAN : ComplexOperator::BLOCK_SYMMETRIC;
 
-   // 11. Set up the parallel linear form b(.) which corresponds to the
+   // 12. Set up the parallel linear form b(.) which corresponds to the
    //     right-hand side of the FEM linear system.
    VectorFunctionCoefficient f(dim, source);
    ParComplexLinearForm b(fespace, conv);
@@ -374,7 +389,7 @@ int main(int argc, char *argv[])
    b.Vector::operator=(0.0);
    b.Assemble();
 
-   // 12. Define the solution vector x as a parallel complex finite element grid
+   // 13. Define the solution vector x as a parallel complex finite element grid
    //     function corresponding to fespace.
    ParComplexGridFunction x(fespace);
    x = 0.0;
@@ -382,7 +397,7 @@ int main(int argc, char *argv[])
    VectorFunctionCoefficient E_Im(dim, E_bdr_data_Im);
    x.ProjectBdrCoefficientTangent(E_Re, E_Im, ess_bdr);
 
-   // 13. Set up the parallel sesquilinear form a(.,.)
+   // 14. Set up the parallel sesquilinear form a(.,.)
    //
    //     In Comp
    //     Domain:   1/mu (Curl E, Curl F) - omega^2 * epsilon (E,F)
@@ -436,19 +451,20 @@ int main(int argc, char *argv[])
    a.AddDomainIntegrator(new VectorFEMassIntegrator(restr_c2_Re),
                          new VectorFEMassIntegrator(restr_c2_Im));
 
-   // 14. Assemble the parallel bilinear form and the corresponding linear
+   // 15. Assemble the parallel bilinear form and the corresponding linear
    //     system, applying any necessary transformations such as: parallel
    //     assembly, eliminating boundary conditions, applying conforming
    //     constraints for non-conforming AMR, etc.
+   if (pa) { a.SetAssemblyLevel(AssemblyLevel::PARTIAL); }
    a.Assemble();
 
    OperatorPtr Ah;
    Vector B, X;
    a.FormLinearSystem(ess_tdof_list, x, b, Ah, X, B);
 
-   // 15. Solve using a direct or an iterative solver
+   // 16. Solve using a direct or an iterative solver
 #ifdef MFEM_USE_SUPERLU
-   if (slu_solver)
+   if (!pa && slu_solver)
    {
       // Transform to monolithic HypreParMatrix
       HypreParMatrix *A = Ah.As<ComplexHypreParMatrix>()->GetSystemMatrix();
@@ -463,7 +479,7 @@ int main(int argc, char *argv[])
    }
 #endif
 #ifdef MFEM_USE_MUMPS
-   if (mumps_solver)
+   if (!pa && mumps_solver)
    {
       HypreParMatrix *A = Ah.As<ComplexHypreParMatrix>()->GetSystemMatrix();
       MUMPSSolver mumps;
@@ -481,7 +497,7 @@ int main(int argc, char *argv[])
    //
    //    In PML:   1/mu (abs(1/det(J) J^T J) Curl E, Curl F)
    //              + omega^2 * epsilon (abs(det(J) * (J^T J)^-1) * E, F)
-   if (!slu_solver && !mumps_solver)
+   if (pa || (!slu_solver && !mumps_solver))
    {
       ConstantCoefficient absomeg(pow(omega, 2) * epsilon);
       RestrictedCoefficient restr_absomeg(absomeg,attr);
@@ -501,11 +517,9 @@ int main(int argc, char *argv[])
       prec.AddDomainIntegrator(new CurlCurlIntegrator(restr_c1_abs));
       prec.AddDomainIntegrator(new VectorFEMassIntegrator(restr_c2_abs));
 
+      if (pa) { prec.SetAssemblyLevel(AssemblyLevel::PARTIAL); }
       prec.Assemble();
 
-      OperatorPtr PCOpAh;
-      prec.FormSystemMatrix(ess_tdof_list, PCOpAh);
-
       // 16b. Define and apply a parallel GMRES solver for AU=B with a block
       //      diagonal preconditioner based on hypre's AMS preconditioner.
       Array<int> offsets(3);
@@ -514,21 +528,41 @@ int main(int argc, char *argv[])
       offsets[2] = fespace->GetTrueVSize();
       offsets.PartialSum();
 
-      HypreAMS ams00(*PCOpAh.As<HypreParMatrix>(),fespace);
-      BlockDiagonalPreconditioner BlockAMS(offsets);
-      ScaledOperator ams11(&ams00,
-                           (conv == ComplexOperator::HERMITIAN) ? -1.0 : 1.0);
-      BlockAMS.SetDiagonalBlock(0,&ams00);
-      BlockAMS.SetDiagonalBlock(1,&ams11);
+      Operator *pc_r = nullptr;
+      Operator *pc_i = nullptr;
+      int s = (conv == ComplexOperator::HERMITIAN) ? -1.0 : 1.0;
+      if (pa)
+      {
+         // Jacobi Smoother
+         OperatorJacobiSmoother *d00 = new OperatorJacobiSmoother(prec, ess_tdof_list);
+         ScaledOperator *d11 = new ScaledOperator(d00, s);
+         pc_r = d00;
+         pc_i = d11;
+      }
+      else
+      {
+         OperatorPtr PCOpAh;
+         prec.FormSystemMatrix(ess_tdof_list, PCOpAh);
+
+         // Hypre AMS
+         HypreAMS *ams00 = new HypreAMS(*PCOpAh.As<HypreParMatrix>(), fespace);
+         ScaledOperator *ams11 = new ScaledOperator(ams00, s);
+         pc_r = ams00;
+         pc_i = ams11;
+      }
+
+      BlockDiagonalPreconditioner BlockDP(offsets);
+      BlockDP.SetDiagonalBlock(0, pc_r);
+      BlockDP.SetDiagonalBlock(1, pc_i);
 
       GMRESSolver gmres(MPI_COMM_WORLD);
       gmres.SetPrintLevel(1);
       gmres.SetKDim(200);
-      gmres.SetMaxIter(2000);
+      gmres.SetMaxIter(pa ? 5000 : 2000);
       gmres.SetRelTol(1e-5);
       gmres.SetAbsTol(0.0);
       gmres.SetOperator(*Ah);
-      gmres.SetPreconditioner(BlockAMS);
+      gmres.SetPreconditioner(BlockDP);
       gmres.Mult(B, X);
    }
 
@@ -1003,8 +1037,12 @@ void CartesianPML::SetBoundaries()
 
 void CartesianPML::SetAttributes(ParMesh *pmesh)
 {
-   int myid;
-   MPI_Comm_rank(MPI_COMM_WORLD,&myid);
+   // Initialize bdr attributes
+   for (int i = 0; i < pmesh->GetNBE(); ++i)
+   {
+      pmesh->GetBdrElement(i)->SetAttribute(i+1);
+   }
+
    int nrelem = pmesh->GetNE();
 
    // Initialize list with 1

examples/ex26.cpp

@@ -40,7 +40,7 @@ using namespace mfem;
 // in the FiniteElementSpaceHierarchy. The preconditioner uses a CG solver on
 // the coarsest level and second order Chebyshev accelerated smoothers on the
 // other levels.
-class DiffusionMultigrid : public Multigrid
+class DiffusionMultigrid : public GeometricMultigrid
 {
 private:
    ConstantCoefficient one;
@@ -49,7 +49,7 @@ public:
    // Constructs a diffusion multigrid for the given FiniteElementSpaceHierarchy
    // and the array of essential boundaries
    DiffusionMultigrid(FiniteElementSpaceHierarchy& fespaces, Array<int>& ess_bdr)
-      : Multigrid(fespaces), one(1.0)
+      : GeometricMultigrid(fespaces), one(1.0)
    {
       ConstructCoarseOperatorAndSolver(fespaces.GetFESpaceAtLevel(0), ess_bdr);
 

examples/ex26p.cpp

@@ -37,7 +37,7 @@ using namespace mfem;
 // all spaces except the coarsest one in the ParFiniteElementSpaceHierarchy.
 // The multigrid uses a PCG solver preconditioned with AMG on the coarsest level
 // and second order Chebyshev accelerated smoothers on the other levels.
-class DiffusionMultigrid : public Multigrid
+class DiffusionMultigrid : public GeometricMultigrid
 {
 private:
    ConstantCoefficient one;
@@ -48,7 +48,7 @@ public:
    // and the array of essential boundaries
    DiffusionMultigrid(ParFiniteElementSpaceHierarchy& fespaces,
                       Array<int>& ess_bdr)
-      : Multigrid(fespaces), one(1.0)
+      : GeometricMultigrid(fespaces), one(1.0)
    {
       ConstructCoarseOperatorAndSolver(fespaces.GetFESpaceAtLevel(0), ess_bdr);
 

examples/ex3p.cpp

@@ -22,6 +22,7 @@
 //
 // Device sample runs:
 //               mpirun -np 4 ex3p -m ../data/star.mesh -pa -d cuda
+//               mpirun -np 4 ex3p -m ../data/star.mesh -no-pa -d cuda
 //               mpirun -np 4 ex3p -m ../data/star.mesh -pa -d raja-cuda
 //               mpirun -np 4 ex3p -m ../data/star.mesh -pa -d raja-omp
 //               mpirun -np 4 ex3p -m ../data/beam-hex.mesh -pa -d cuda
@@ -68,7 +69,10 @@ int main(int argc, char *argv[])
    bool static_cond = false;
    bool pa = false;
    const char *device_config = "cpu";
-   bool visualization = 1;
+   bool visualization = true;
+#ifdef MFEM_USE_AMGX
+   bool useAmgX = false;
+#endif
 
    OptionsParser args(argc, argv);
    args.AddOption(&mesh_file, "-m", "--mesh",
@@ -86,6 +90,11 @@ int main(int argc, char *argv[])
    args.AddOption(&visualization, "-vis", "--visualization", "-no-vis",
                   "--no-visualization",
                   "Enable or disable GLVis visualization.");
+#ifdef MFEM_USE_AMGX
+   args.AddOption(&useAmgX, "-amgx", "--useAmgX", "-no-amgx",
+                  "--no-useAmgX",
+                  "Enable or disable AmgX in MatrixFreeAMS.");
+#endif
 
    args.Parse();
    if (!args.Good())
@@ -158,9 +167,10 @@ int main(int argc, char *argv[])
    //    by marking all the boundary attributes from the mesh as essential
    //    (Dirichlet) and converting them to a list of true dofs.
    Array<int> ess_tdof_list;
+   Array<int> ess_bdr;
    if (pmesh->bdr_attributes.Size())
    {
-      Array<int> ess_bdr(pmesh->bdr_attributes.Max());
+      ess_bdr.SetSize(pmesh->bdr_attributes.Max());
       ess_bdr = 1;
       fespace->GetEssentialTrueDofs(ess_bdr, ess_tdof_list);
    }
@@ -204,20 +214,20 @@ int main(int argc, char *argv[])
    Vector B, X;
    a->FormLinearSystem(ess_tdof_list, x, *b, A, X, B);
 
-   // 13. Solve the system AX=B using PCG with the AMS preconditioner from hypre
-   //     (in the full assembly case) or CG with Jacobi preconditioner (in the
-   //     partial assembly case).
-
-   if (pa) // Jacobi preconditioning in partial assembly mode
+   // 13. Solve the system AX=B using PCG with an AMS preconditioner.
+   if (pa)
    {
-      OperatorJacobiSmoother Jacobi(*a, ess_tdof_list);
-
+#ifdef MFEM_USE_AMGX
+      MatrixFreeAMS ams(*a, *A, *fespace, muinv, sigma, NULL, ess_bdr, useAmgX);
+#else
+      MatrixFreeAMS ams(*a, *A, *fespace, muinv, sigma, NULL, ess_bdr);
+#endif
       CGSolver cg(MPI_COMM_WORLD);
       cg.SetRelTol(1e-12);
       cg.SetMaxIter(1000);
       cg.SetPrintLevel(1);
       cg.SetOperator(*A);
-      cg.SetPreconditioner(Jacobi);
+      cg.SetPreconditioner(ams);
       cg.Mult(B, X);
    }
    else

examples/ex3p_complex.cpp

@@ -0,0 +1,517 @@
+//                       MFEM Example 3 - Parallel Version
+//
+// Compile with: make ex3p_complex
+//
+
+#include "mfem.hpp"
+#include <fstream>
+#include <iostream>
+
+using namespace std;
+using namespace mfem;
+
+// Exact solution, E, and r.h.s., f. See below for implementation.
+void E_exact(const Vector &, Vector &);
+void curlE_exact(const Vector &, Vector &);
+void f_exact(const Vector &, Vector &);
+double freq = 1.0, kappa;
+int dim;
+
+
+#define COMPLEX_VERSION
+#define NEUMANN
+#define INDEFINITE
+
+const double omega = 1.4;
+
+
+int main(int argc, char *argv[])
+{
+   // 1. Initialize MPI.
+   int num_procs, myid;
+   MPI_Init(&argc, &argv);
+   MPI_Comm_size(MPI_COMM_WORLD, &num_procs);
+   MPI_Comm_rank(MPI_COMM_WORLD, &myid);
+
+   // 2. Parse command-line options.
+   const char *mesh_file = "../data/beam-tet.mesh";
+   int order = 1;
+   bool static_cond = false;
+   bool pa = false;
+   const char *device_config = "cpu";
+   bool visualization = 1;
+
+   OptionsParser args(argc, argv);
+   args.AddOption(&mesh_file, "-m", "--mesh",
+                  "Mesh file to use.");
+   args.AddOption(&order, "-o", "--order",
+                  "Finite element order (polynomial degree).");
+   args.AddOption(&freq, "-f", "--frequency", "Set the frequency for the exact"
+                  " solution.");
+   args.AddOption(&static_cond, "-sc", "--static-condensation", "-no-sc",
+                  "--no-static-condensation", "Enable static condensation.");
+   args.AddOption(&pa, "-pa", "--partial-assembly", "-no-pa",
+                  "--no-partial-assembly", "Enable Partial Assembly.");
+   args.AddOption(&device_config, "-d", "--device",
+                  "Device configuration string, see Device::Configure().");
+   args.AddOption(&visualization, "-vis", "--visualization", "-no-vis",
+                  "--no-visualization",
+                  "Enable or disable GLVis visualization.");
+
+   args.Parse();
+   if (!args.Good())
+   {
+      if (myid == 0)
+      {
+         args.PrintUsage(cout);
+      }
+      MPI_Finalize();
+      return 1;
+   }
+   if (myid == 0)
+   {
+      args.PrintOptions(cout);
+   }
+   kappa = freq * M_PI;
+
+   // 3. Enable hardware devices such as GPUs, and programming models such as
+   //    CUDA, OCCA, RAJA and OpenMP based on command line options.
+   Device device(device_config);
+   if (myid == 0) { device.Print(); }
+
+   // 4. Read the (serial) mesh from the given mesh file on all processors.  We
+   //    can handle triangular, quadrilateral, tetrahedral, hexahedral, surface
+   //    and volume meshes with the same code.
+   Mesh *mesh = new Mesh(mesh_file, 1, 1);
+   dim = mesh->Dimension();
+   int sdim = mesh->SpaceDimension();
+
+   // 5. Refine the serial mesh on all processors to increase the resolution. In
+   //    this example we do 'ref_levels' of uniform refinement. We choose
+   //    'ref_levels' to be the largest number that gives a final mesh with no
+   //    more than 1,000 elements.
+   {
+      int ref_levels = (int)floor(log(1000./mesh->GetNE())/log(2.)/dim);
+      for (int l = 0; l < ref_levels; l++)
+      {
+         mesh->UniformRefinement();
+      }
+   }
+
+   // 6. Define a parallel mesh by a partitioning of the serial mesh. Refine
+   //    this mesh further in parallel to increase the resolution. Once the
+   //    parallel mesh is defined, the serial mesh can be deleted. Tetrahedral
+   //    meshes need to be reoriented before we can define high-order Nedelec
+   //    spaces on them.
+   ParMesh *pmesh = new ParMesh(MPI_COMM_WORLD, *mesh);
+   delete mesh;
+   {
+      int par_ref_levels = 0;
+      for (int l = 0; l < par_ref_levels; l++)
+      {
+         pmesh->UniformRefinement();
+      }
+   }
+   pmesh->ReorientTetMesh();
+
+   // 7. Define a parallel finite element space on the parallel mesh. Here we
+   //    use the Nedelec finite elements of the specified order.
+   FiniteElementCollection *fec = new ND_FECollection(order, dim);
+   ParFiniteElementSpace *fespace = new ParFiniteElementSpace(pmesh, fec);
+   HYPRE_Int size = fespace->GlobalTrueVSize();
+   if (myid == 0)
+   {
+      cout << "Number of finite element unknowns: " << size << endl;
+   }
+
+   // 8. Determine the list of true (i.e. parallel conforming) essential
+   //    boundary dofs. In this example, the boundary conditions are defined
+   //    by marking all the boundary attributes from the mesh as essential
+   //    (Dirichlet) and converting them to a list of true dofs.
+   Array<int> ess_tdof_list;
+   Array<int> ess_bdr;
+   ess_bdr.SetSize(pmesh->bdr_attributes.Max());
+   ess_bdr = 0;
+
+#ifndef NEUMANN
+   if (pmesh->bdr_attributes.Size())
+   {
+      ess_bdr = 1;
+      fespace->GetEssentialTrueDofs(ess_bdr, ess_tdof_list);
+   }
+#endif
+
+   //const double imscale = 0.0;
+   const double imscale = omega;
+
+   Coefficient *im = new ConstantCoefficient(imscale);  // im part
+   //Coefficient *im = new ConstantCoefficient(0.0);  // im part
+
+   VectorFunctionCoefficient E_Re(sdim, E_exact);
+   VectorFunctionCoefficient curlE_Re(sdim, curlE_exact);
+
+   ScalarVectorProductCoefficient omegaE(imscale, E_Re);  // im part
+   //ScalarVectorProductCoefficient omegaE(0.0, E_Re);  // im part
+
+   // 9. Set up the parallel linear form b(.) which corresponds to the
+   //    right-hand side of the FEM linear system, which in this case is
+   //    (f,phi_i) where f is given by the function f_exact and phi_i are the
+   //    basis functions in the finite element fespace.
+   VectorFunctionCoefficient f(sdim, f_exact);
+#ifdef COMPLEX_VERSION
+   ParComplexLinearForm *b = new ParComplexLinearForm(fespace);
+   b->AddDomainIntegrator(new VectorFEDomainLFIntegrator(f), NULL);
+   b->AddBoundaryIntegrator(NULL,
+                            new VectorFEDomainLFIntegrator(omegaE));  // im part
+#else
+   // Real version
+   ParLinearForm *b = new ParLinearForm(fespace);
+   b->AddDomainIntegrator(new VectorFEDomainLFIntegrator(f));
+#endif
+
+#ifdef NEUMANN
+   b->AddBoundaryIntegrator(new VectorFEBoundaryTangentLFIntegrator(curlE_Re),
+                            NULL);
+#endif
+
+   b->Assemble();
+
+   // 10. Define the solution vector x as a parallel finite element grid function
+   //     corresponding to fespace. Initialize x by projecting the exact
+   //     solution. Note that only values from the boundary edges will be used
+   //     when eliminating the non-homogeneous boundary condition to modify the
+   //     r.h.s. vector b.
+   /*
+   ParGridFunction x(fespace);
+   VectorFunctionCoefficient E(sdim, E_exact);
+   x.ProjectCoefficient(E);
+   */
+
+#ifdef COMPLEX_VERSION
+   // Complex version
+   ParComplexGridFunction x(fespace);
+   x = 0.0;
+   Vector zero(sdim);
+   zero = 0.0;
+   VectorConstantCoefficient E_Im(zero);
+   //x.ProjectBdrCoefficientTangent(E_Re, E_Im, ess_bdr);
+   x.ProjectCoefficient(E_Re, E_Im);
+#else
+   ParGridFunction x(fespace);
+   x = 0.0;
+   x.ProjectCoefficient(E_Re);
+#endif
+
+   // 11. Set up the parallel bilinear form corresponding to the EM diffusion
+   //     operator curl muinv curl + sigma I, by adding the curl-curl and the
+   //     mass domain integrators.
+   Coefficient *muinv = new ConstantCoefficient(1.0);
+#ifdef INDEFINITE
+   Coefficient *sigma = new ConstantCoefficient(
+      -omega*omega); // indefinite -, definite +
+#else
+   Coefficient *sigma = new ConstantCoefficient(
+      omega*omega); // indefinite -, definite +
+#endif
+   Coefficient *abssigma = new ConstantCoefficient(omega*omega);
+   Coefficient *imabs = new ConstantCoefficient(imscale);  // im part
+   //Coefficient *imabs = new ConstantCoefficient(0.0);  // im part
+   //Coefficient *im = new ConstantCoefficient(0.0);
+
+#ifdef COMPLEX_VERSION
+   // Complex version
+   ParSesquilinearForm *a = new ParSesquilinearForm(fespace);
+   if (pa) { a->SetAssemblyLevel(AssemblyLevel::PARTIAL); }
+   a->AddDomainIntegrator(new CurlCurlIntegrator(*muinv), NULL);
+   //a->AddDomainIntegrator(new VectorFEMassIntegrator(*sigma), new VectorFEMassIntegrator(*im));
+   a->AddDomainIntegrator(new VectorFEMassIntegrator(*sigma), NULL);
+   a->AddBoundaryIntegrator(NULL, new VectorFEMassIntegrator(*im));  // im part
+#else
+   // Real version
+   ParBilinearForm *a = new ParBilinearForm(fespace);
+   if (pa) { a->SetAssemblyLevel(AssemblyLevel::PARTIAL); }
+   a->AddDomainIntegrator(new CurlCurlIntegrator(*muinv));
+   //a->AddDomainIntegrator(new VectorFEMassIntegrator(*sigma), new VectorFEMassIntegrator(*im));
+   a->AddDomainIntegrator(new VectorFEMassIntegrator(*sigma));
+#endif
+
+   // 12. Assemble the parallel bilinear form and the corresponding linear
+   //     system, applying any necessary transformations such as: parallel
+   //     assembly, eliminating boundary conditions, applying conforming
+   //     constraints for non-conforming AMR, static condensation, etc.
+   //if (static_cond) { a->EnableStaticCondensation(); }
+   a->Assemble();
+
+   OperatorPtr A;
+   Vector B, X;
+   a->FormLinearSystem(ess_tdof_list, x, *b, A, X, B);
+
+   ParBilinearForm a_Re(fespace);
+   a_Re.AddDomainIntegrator(new CurlCurlIntegrator(*muinv));
+   a_Re.AddDomainIntegrator(new VectorFEMassIntegrator(*abssigma));
+
+   //if (pa) { a_Re.SetAssemblyLevel(AssemblyLevel::PARTIAL); }
+   a_Re.SetAssemblyLevel(AssemblyLevel::PARTIAL);
+   a_Re.Assemble();
+
+   OperatorPtr A_Re;
+   a_Re.FormSystemMatrix(ess_tdof_list, A_Re);
+
+   ParBilinearForm a_Im(fespace);
+   a_Im.AddBoundaryIntegrator(new VectorFEMassIntegrator(*imabs));
+   a_Im.Assemble();
+
+   OperatorPtr A_Im;
+   a_Im.FormSystemMatrix(ess_tdof_list, A_Im);
+
+   // 13. Solve the system AX=B using PCG with the AMS preconditioner from hypre
+   //     (in the full assembly case) or CG with Jacobi preconditioner (in the
+   //     partial assembly case).
+
+   Array<int> offsets(3);
+   offsets[0] = 0;
+   offsets[1] = fespace->GetTrueVSize();
+   offsets[2] = fespace->GetTrueVSize();
+   offsets.PartialSum();
+
+   //OperatorJacobiSmoother massJacobi(a_Im, ess_tdof_list);
+
+   StopWatch sw;
+   sw.Clear();
+   sw.Start();
+
+   if (pa) // Jacobi preconditioning in partial assembly mode
+   {
+      MFEM_VERIFY(false, "TODO");
+      //OperatorJacobiSmoother Jacobi(*a, ess_tdof_list);
+
+      CGSolver cg(MPI_COMM_WORLD);
+      cg.SetRelTol(1e-12);
+      cg.SetMaxIter(1000);
+      cg.SetPrintLevel(1);
+      cg.SetOperator(*A);
+      //cg.SetPreconditioner(Jacobi);
+      cg.Mult(B, X);
+   }
+   else
+   {
+      if (myid == 0)
+      {
+         cout << "Size of linear system: "
+              << A.As<HypreParMatrix>()->GetGlobalNumRows() << endl;
+      }
+
+      //HypreAMS ams(*A_Re.As<HypreParMatrix>(), fespace);
+
+      // One option is to use the standard real-valued MatrixFreeAMS to precondition
+      // the real part of the complex system in the PMHSS preconditioner (BlockDiagonalPreconditioner).
+      // Another option is to use complex MatrixFreeAMS to precondition the
+      // complex system without PMHSS and without a BlockDiagonalPreconditioner.
+      //#define COMPLEX_AMS
+
+#ifdef MFEM_USE_AMGX
+      bool useAmgX = false;
+      cout << "Built with AMGX, using AMGX " << useAmgX << endl;
+      MatrixFreeAMS ams(a_Re, *A_Re, *fespace, muinv, abssigma, im, imabs, NULL,
+                        ess_bdr, useAmgX);
+      MatrixFreeAMS ams(a_Re, *A_Re, *fespace, muinv, abssigma, NULL, NULL, ess_bdr,
+                        useAmgX);
+#ifdef COMPLEX_AMS
+      MFEM_VERIFY(false, "TODO");
+#endif
+
+#else
+      cout << "Not built with AMGX" << endl;
+#ifdef COMPLEX_AMS
+      MatrixFreeAMS ams(a_Re, *A_Re, A.Ptr(), *fespace, muinv, abssigma, im, imabs,
+                        NULL, ess_bdr);
+#else
+      MatrixFreeAMS ams(a_Re, *A_Re, NULL, *fespace, muinv, abssigma, NULL, NULL,
+                        NULL, ess_bdr);
+#endif
+#endif
+
+#ifdef COMPLEX_VERSION
+
+#ifdef COMPLEX_AMS
+      //MFEM_VERIFY(false, "TODO");
+#else
+      BlockDiagonalPreconditioner BlockDP(offsets);
+      BlockDP.SetDiagonalBlock(0, &ams);
+      BlockDP.SetDiagonalBlock(1, &ams);
+
+      /*
+      BlockDiagonalPreconditioner BlockDP_Im(offsets);
+      BlockDP_Im.SetDiagonalBlock(0, &massJacobi); // TODO: this won't work if it has zeros on diagonal
+      BlockDP_Im.SetDiagonalBlock(1, &massJacobi);
+      */
+
+      //Complex_PMHSS PMHSS(A_Re, A_Im, &BlockDP, &BlockDP_Im);
+      //Complex_PMHSS PMHSS(A_Re, A_Im, &BlockDP, NULL, 2.0 * omega);
+      //Complex_PMHSS PMHSS(A_Re, A_Im, &BlockDP, NULL, omega);
+      Complex_PMHSS PMHSS(A_Re.Ptr(), A_Im.Ptr(), &BlockDP, NULL, 1.0);
+
+      ComplexOperator AspdComplex(A_Re.Ptr(), A_Im.Ptr(), false, false);
+
+      GMRESSolver PMHSSgmres(MPI_COMM_WORLD);
+      PMHSSgmres.SetPrintLevel(1);
+      PMHSSgmres.SetKDim(100);
+      PMHSSgmres.SetMaxIter(100);
+      PMHSSgmres.SetRelTol(1e-6);
+      PMHSSgmres.SetAbsTol(0.0);
+      PMHSSgmres.SetOperator(AspdComplex);
+      PMHSSgmres.SetPreconditioner(PMHSS);
+#endif
+
+      GMRESSolver gmres(MPI_COMM_WORLD);
+      gmres.SetPrintLevel(1);
+      gmres.SetKDim(1000);
+      gmres.SetMaxIter(100);
+      gmres.SetRelTol(1e-8);
+      gmres.SetAbsTol(0.0);
+      gmres.SetOperator(*A);
+      //gmres.SetPreconditioner(BlockDP);
+#ifdef COMPLEX_AMS
+      //MFEM_VERIFY(false, "TODO");
+      gmres.SetPreconditioner(ams);
+#else
+      gmres.SetPreconditioner(PMHSS);
+      //gmres.SetPreconditioner(PMHSSgmres);
+#endif
+
+#else
+      GMRESSolver gmres(MPI_COMM_WORLD);
+      gmres.SetPrintLevel(1);
+      gmres.SetKDim(1000);
+      gmres.SetMaxIter(100);
+      gmres.SetRelTol(1e-8);
+      gmres.SetAbsTol(0.0);
+      gmres.SetOperator(*A);
+      gmres.SetPreconditioner(ams);
+#endif
+
+      gmres.Mult(B, X);
+   }
+
+   sw.Stop();
+   mfem::out << "Total solve time " <<sw.RealTime() << endl;
+
+   // 14. Recover the parallel grid function corresponding to X. This is the
+   //     local finite element solution on each processor.
+   a->RecoverFEMSolution(X, *b, x);
+
+   // 15. Compute and print the L^2 norm of the error.
+   {
+#ifdef COMPLEX_VERSION
+      double err = x.real().ComputeL2Error(E_Re);
+#else
+      double err = x.ComputeL2Error(E_Re);
+#endif
+      if (myid == 0)
+      {
+         cout << "\n|| E_h - E ||_{L^2} = " << err << '\n' << endl;
+      }
+   }
+
+   // 16. Save the refined mesh and the solution in parallel. This output can
+   //     be viewed later using GLVis: "glvis -np <np> -m mesh -g sol".
+   {
+      ostringstream mesh_name, sol_name;
+      mesh_name << "mesh." << setfill('0') << setw(6) << myid;
+      sol_name << "sol." << setfill('0') << setw(6) << myid;
+
+      ofstream mesh_ofs(mesh_name.str().c_str());
+      mesh_ofs.precision(8);
+      pmesh->Print(mesh_ofs);
+
+      ofstream sol_ofs(sol_name.str().c_str());
+      sol_ofs.precision(8);
+#ifdef COMPLEX_VERSION
+      x.real().Save(sol_ofs);
+#else
+      x.Save(sol_ofs);
+#endif
+   }
+
+   // 17. Send the solution by socket to a GLVis server.
+   if (visualization)
+   {
+      char vishost[] = "localhost";
+      int  visport   = 19916;
+      socketstream sol_sock(vishost, visport);
+      sol_sock << "parallel " << num_procs << " " << myid << "\n";
+      sol_sock.precision(8);
+#ifdef COMPLEX_VERSION
+      sol_sock << "solution\n" << *pmesh << x.real() << flush;
+#else
+      sol_sock << "solution\n" << *pmesh << x << flush;
+#endif
+   }
+
+   // 18. Free the used memory.
+   delete a;
+   delete sigma;
+   delete muinv;
+   delete b;
+   delete fespace;
+   delete fec;
+   delete pmesh;
+
+   MPI_Finalize();
+
+   return 0;
+}
+
+
+void E_exact(const Vector &x, Vector &E)
+{
+   if (dim == 3)
+   {
+      E(0) = sin(kappa * x(1));
+      E(1) = sin(kappa * x(2));
+      E(2) = sin(kappa * x(0));
+   }
+   else
+   {
+      E(0) = sin(kappa * x(1));
+      E(1) = sin(kappa * x(0));
+      if (x.Size() == 3) { E(2) = 0.0; }
+   }
+}
+
+void curlE_exact(const Vector &x, Vector &curl)
+{
+   if (dim == 3)
+   {
+      curl(0) = kappa * cos(kappa * x(2));
+      curl(1) = kappa * cos(kappa * x(0));
+      curl(2) = kappa * cos(kappa * x(1));
+   }
+   else
+   {
+      MFEM_VERIFY(false, "");
+   }
+}
+
+void f_exact(const Vector &x, Vector &f)
+{
+   if (dim == 3)
+   {
+      // indefinite -m, definite +m
+      const double c = kappa * kappa;
+#ifdef INDEFINITE
+      const double m = -omega * omega;
+#else
+      const double m = omega * omega;
+#endif
+      f(0) = (c + m) * sin(kappa * x(1));
+      f(1) = (c + m) * sin(kappa * x(2));
+      f(2) = (c + m) * sin(kappa * x(0));
+   }
+   else
+   {
+      f(0) = (1. + kappa * kappa) * sin(kappa * x(1));
+      f(1) = (1. + kappa * kappa) * sin(kappa * x(0));
+      if (x.Size() == 3) { f(2) = 0.0; }
+   }
+}

fem/CMakeLists.txt

@@ -13,6 +13,7 @@ set(SRCS
   bilinearform.cpp
   bilinearform_ext.cpp
   bilininteg.cpp
+  bilininteg_br2.cpp
   bilininteg_convection_pa.cpp
   bilininteg_convection_ea.cpp
   bilininteg_dgtrace_pa.cpp

fem/bilinearform.cpp

@@ -1437,9 +1437,9 @@ void MixedBilinearForm::Assemble (int skip_zeros)
          ftr = mesh->GetBdrFaceTransformations(i);
          if (ftr)
          {
-            trial_fes->GetFaceVDofs(i, tr_vdofs);
+            trial_fes->GetFaceVDofs(ftr->ElementNo, tr_vdofs);
             test_fes->GetElementVDofs(ftr->Elem1No, te_vdofs);
-            trial_face_fe = trial_fes->GetFaceElement(i);
+            trial_face_fe = trial_fes->GetFaceElement(ftr->ElementNo);
             test_fe1 = test_fes->GetFE(ftr->Elem1No);
             // The test_fe2 object is really a dummy and not used on the
             // boundaries, but we can't dereference a NULL pointer, and we don't
@@ -1770,9 +1770,41 @@ MixedBilinearForm::~MixedBilinearForm()
    delete ext;
 }
 
+void DiscreteLinearOperator::SetAssemblyLevel(AssemblyLevel assembly_level)
+{
+   if (ext)
+   {
+      MFEM_ABORT("the assembly level has already been set!");
+   }
+   assembly = assembly_level;
+   switch (assembly)
+   {
+      case AssemblyLevel::LEGACYFULL:
+      case AssemblyLevel::FULL:
+         // Use the original implementation for now
+         break;
+      case AssemblyLevel::ELEMENT:
+         mfem_error("Element assembly not supported yet... stay tuned!");
+         break;
+      case AssemblyLevel::PARTIAL:
+         ext = new PADiscreteLinearOperatorExtension(this);
+         break;
+      case AssemblyLevel::NONE:
+         mfem_error("Matrix-free action not supported yet... stay tuned!");
+         break;
+      default:
+         mfem_error("Unknown assembly level");
+   }
+}
 
 void DiscreteLinearOperator::Assemble(int skip_zeros)
 {
+   if (ext)
+   {
+      ext->Assemble();
+      return;
+   }
+
    Array<int> dom_vdofs, ran_vdofs;
    ElementTransformation *T;
    const FiniteElement *dom_fe, *ran_fe;

fem/bilinearform.hpp

@@ -376,6 +376,13 @@ public:
    /// Get the output finite element space prolongation matrix
    virtual const Operator *GetOutputProlongation() const
    { return GetProlongation(); }
+   /** @brief Returns the output fe space restriction matrix, transposed
+
+       Logically, this is the transpose of GetOutputRestriction, but in
+       practice it is convenient to have it in transposed form for
+       construction of RAP operators in matrix-free methods. */
+   virtual const Operator *GetOutputRestrictionTranspose() const
+   { return GetOutputProlongation(); }
    /// Get the output finite element space restriction matrix
    virtual const Operator *GetOutputRestriction() const
    { return GetRestriction(); }
@@ -977,9 +984,18 @@ public:
    /// Access all interpolators added with AddDomainInterpolator().
    Array<BilinearFormIntegrator*> *GetDI() { return &dbfi; }
 
+   /// Set the desired assembly level. The default is AssemblyLevel::FULL.
+   /** This method must be called before assembly. */
+   void SetAssemblyLevel(AssemblyLevel assembly_level);
+
    /** @brief Construct the internal matrix representation of the discrete
        linear operator. */
    virtual void Assemble(int skip_zeros = 1);
+
+   /** @brief Get the output finite element space restriction matrix in
+       transposed form. */
+   virtual const Operator *GetOutputRestrictionTranspose() const
+   { return test_fes->GetRestrictionTransposeOperator(); }
 };
 
 }

fem/bilinearform_ext.cpp

@@ -1021,7 +1021,6 @@ void PAMixedBilinearFormExtension::Update()
       localTrial.UseDevice(true);
       localTrial.SetSize(elem_restrict_trial->Height(),
                          Device::GetMemoryType());
-
    }
    if (elem_restrict_test)
    {
@@ -1221,4 +1220,134 @@ void PAMixedBilinearFormExtension::AssembleDiagonal_ADAt(const Vector &D,
    }
 }
 
+PADiscreteLinearOperatorExtension::PADiscreteLinearOperatorExtension(
+   DiscreteLinearOperator *linop) :
+   PAMixedBilinearFormExtension(linop)
+{
+}
+
+const
+Operator *PADiscreteLinearOperatorExtension::GetOutputRestrictionTranspose()
+const
+{
+   return a->GetOutputRestrictionTranspose();
+}
+
+void PADiscreteLinearOperatorExtension::Assemble()
+{
+   Array<BilinearFormIntegrator*> &integrators = *a->GetDBFI();
+   const int integratorCount = integrators.Size();
+   for (int i = 0; i < integratorCount; ++i)
+   {
+      integrators[i]->AssemblePA(*trialFes, *testFes);
+   }
+
+   test_multiplicity.UseDevice(true);
+   test_multiplicity.SetSize(elem_restrict_test->Width()); // l-vector
+   Vector ones(elem_restrict_test->Height()); // e-vector
+   ones = 1.0;
+
+   const ElementRestriction* elem_restrict =
+      dynamic_cast<const ElementRestriction*>(elem_restrict_test);
+   if (elem_restrict)
+   {
+      elem_restrict->MultTransposeUnsigned(ones, test_multiplicity);
+   }
+   else
+   {
+      mfem_error("A real ElementRestriction is required in this setting!");
+   }
+
+   auto tm = test_multiplicity.ReadWrite();
+   MFEM_FORALL(i, test_multiplicity.Size(),
+   {
+      tm[i] = 1.0 / tm[i];
+   });
+}
+
+void PADiscreteLinearOperatorExtension::AddMult(
+   const Vector &x, Vector &y, const double c) const
+{
+   Array<BilinearFormIntegrator*> &integrators = *a->GetDBFI();
+   const int iSz = integrators.Size();
+
+   // * G operation
+   SetupMultInputs(elem_restrict_trial, x, localTrial,
+                   elem_restrict_test, y, localTest, c);
+
+   // * B^TDB operation
+   for (int i = 0; i < iSz; ++i)
+   {
+      integrators[i]->AddMultPA(localTrial, localTest);
+   }
+
+   // do a kind of "set" rather than "add" in the below
+   // operation as compared to the BilinearForm case
+   // * G^T operation (kind of...)
+   const ElementRestriction* elem_restrict =
+      dynamic_cast<const ElementRestriction*>(elem_restrict_test);
+   if (elem_restrict)
+   {
+      tempY.SetSize(y.Size());
+      elem_restrict->MultLeftInverse(localTest, tempY);
+      y += tempY;
+   }
+   else
+   {
+      mfem_error("In this setting you need a real ElementRestriction!");
+   }
+}
+
+void PADiscreteLinearOperatorExtension::AddMultTranspose(
+   const Vector &x, Vector &y, const double c) const
+{
+   Array<BilinearFormIntegrator*> &integrators = *a->GetDBFI();
+   const int iSz = integrators.Size();
+
+   // do a kind of "set" rather than "add" in the below
+   // operation as compared to the BilinearForm case
+   // * G operation (kinda)
+   Vector xscaled(x);
+   MFEM_VERIFY(x.Size() == test_multiplicity.Size(), "Input vector of wrong size");
+   auto xs = xscaled.ReadWrite();
+   auto tm = test_multiplicity.Read();
+   MFEM_FORALL(i, x.Size(),
+   {
+      xs[i] *= tm[i];
+   });
+   SetupMultInputs(elem_restrict_test, xscaled, localTest,
+                   elem_restrict_trial, y, localTrial, c);
+
+   // * B^TD^TB operation
+   for (int i = 0; i < iSz; ++i)
+   {
+      integrators[i]->AddMultTransposePA(localTest, localTrial);
+   }
+
+   // * G^T operation
+   if (elem_restrict_trial)
+   {
+      tempY.SetSize(y.Size());
+      elem_restrict_trial->MultTranspose(localTrial, tempY);
+      y += tempY;
+   }
+   else
+   {
+      mfem_error("Trial ElementRestriction not defined");
+   }
+}
+
+void PADiscreteLinearOperatorExtension::FormRectangularSystemOperator(
+   const Array<int>& ess1, const Array<int>& ess2, OperatorHandle &A)
+{
+   const Operator *Pi = this->GetProlongation();
+   const Operator *RoT = this->GetOutputRestrictionTranspose();
+   Operator *rap = SetupRAP(Pi, RoT);
+
+   RectangularConstrainedOperator *Arco
+      = new RectangularConstrainedOperator(rap, ess1, ess2, rap != this);
+
+   A.Reset(Arco);
+}
+
 } // namespace mfem

fem/bilinearform_ext.hpp

@@ -21,6 +21,7 @@ namespace mfem
 
 class BilinearForm;
 class MixedBilinearForm;
+class DiscreteLinearOperator;
 
 /// Class extending the BilinearForm class to support different AssemblyLevels.
 /**  FA - Full Assembly
@@ -212,7 +213,7 @@ protected:
    mutable Vector localTrial, localTest, tempY;
    const Operator *elem_restrict_trial; // Not owned
    const Operator *elem_restrict_test;  // Not owned
-private:
+
    /// Helper function to set up inputs/outputs for Mult or MultTranspose
    void SetupMultInputs(const Operator *elem_restrict_x,
                         const Vector &x, Vector &localX,
@@ -258,6 +259,35 @@ public:
    void Update();
 };
 
+
+/**
+   @brief Partial assembly extension for DiscreteLinearOperator
+
+   This acts very much like PAMixedBilinearFormExtension, but its
+   FormRectangularSystemOperator implementation emulates 'Set' rather than
+   'Add' in the assembly case.
+*/
+class PADiscreteLinearOperatorExtension : public PAMixedBilinearFormExtension
+{
+public:
+   PADiscreteLinearOperatorExtension(DiscreteLinearOperator *linop);
+
+   /// Partial assembly of all internal integrators
+   void Assemble();
+
+   void AddMult(const Vector &x, Vector &y, const double c) const;
+
+   void AddMultTranspose(const Vector &x, Vector &y, const double c=1.0) const;
+
+   void FormRectangularSystemOperator(const Array<int>&, const Array<int>&,
+                                      OperatorHandle& A);
+
+   const Operator * GetOutputRestrictionTranspose() const;
+
+private:
+   Vector test_multiplicity;
+};
+
 }
 
 #endif

fem/bilininteg.cpp

@@ -22,14 +22,14 @@ namespace mfem
 
 void BilinearFormIntegrator::AssemblePA(const FiniteElementSpace&)
 {
-   mfem_error ("BilinearFormIntegrator::AssemblePA(...)\n"
+   mfem_error ("BilinearFormIntegrator::AssemblePA(fes)\n"
                "   is not implemented for this class.");
 }
 
 void BilinearFormIntegrator::AssemblePA(const FiniteElementSpace&,
                                         const FiniteElementSpace&)
 {
-   mfem_error ("BilinearFormIntegrator::AssemblePA(...)\n"
+   mfem_error ("BilinearFormIntegrator::AssemblePA(fes, fes)\n"
                "   is not implemented for this class.");
 }
 
@@ -92,7 +92,7 @@ void BilinearFormIntegrator::AddMultPA(const Vector &, Vector &) const
 
 void BilinearFormIntegrator::AddMultTransposePA(const Vector &, Vector &) const
 {
-   mfem_error ("BilinearFormIntegrator::MultAssembledTranspose(...)\n"
+   mfem_error ("BilinearFormIntegrator::AddMultTransposePA(...)\n"
                "   is not implemented for this class.");
 }
 
@@ -1913,12 +1913,12 @@ void VectorFEMassIntegrator::AssembleElementMatrix(
    double w;
 
 #ifdef MFEM_THREAD_SAFE
-   Vector D(VQ ? VQ->GetVDim() : 0);
+   Vector D(DQ ? DQ->GetVDim() : 0);
    DenseMatrix trial_vshape(dof, spaceDim);
    DenseMatrix K(MQ ? MQ->GetVDim() : 0, MQ ? MQ->GetVDim() : 0);
 #else
    trial_vshape.SetSize(dof, spaceDim);
-   D.SetSize(VQ ? VQ->GetVDim() : 0);
+   D.SetSize(DQ ? DQ->GetVDim() : 0);
    K.SetSize(MQ ? MQ->GetVDim() : 0, MQ ? MQ->GetVDim() : 0);
 #endif
    DenseMatrix tmp(trial_vshape.Height(), K.Width());
@@ -1950,9 +1950,9 @@ void VectorFEMassIntegrator::AssembleElementMatrix(
          Mult(trial_vshape,K,tmp);
          AddMultABt(tmp,trial_vshape,elmat);
       }
-      else if (VQ)
+      else if (DQ)
       {
-         VQ->Eval(D, Trans, ip);
+         DQ->Eval(D, Trans, ip);
          D *= w;
          AddMultADAt(trial_vshape, D, elmat);
       }
@@ -1984,12 +1984,12 @@ void VectorFEMassIntegrator::AssembleElementMatrix2(
 #ifdef MFEM_THREAD_SAFE
       DenseMatrix trial_vshape(trial_dof, spaceDim);
       Vector shape(test_dof);
-      Vector D(VQ ? VQ->GetVDim() : 0);
+      Vector D(DQ ? DQ->GetVDim() : 0);
       DenseMatrix K(MQ ? MQ->GetVDim() : 0, MQ ? MQ->GetVDim() : 0);
 #else
       trial_vshape.SetSize(trial_dof, spaceDim);
       shape.SetSize(test_dof);
-      D.SetSize(VQ ? VQ->GetVDim() : 0);
+      D.SetSize(DQ ? DQ->GetVDim() : 0);
       K.SetSize(MQ ? MQ->GetVDim() : 0, MQ ? MQ->GetVDim() : 0);
 #endif
 
@@ -2013,9 +2013,9 @@ void VectorFEMassIntegrator::AssembleElementMatrix2(
          test_fe.CalcShape(ip, shape);
 
          w = ip.weight * Trans.Weight();
-         if (VQ)
+         if (DQ)
          {
-            VQ->Eval(D, Trans, ip);
+            DQ->Eval(D, Trans, ip);
             D *= w;
             for (int d = 0; d < vdim; d++)
             {
@@ -2081,12 +2081,12 @@ void VectorFEMassIntegrator::AssembleElementMatrix2(
 #ifdef MFEM_THREAD_SAFE
       DenseMatrix trial_vshape(trial_dof,spaceDim);
       DenseMatrix test_vshape(test_dof,spaceDim);
-      Vector D(VQ ? VQ->GetVDim() : 0);
+      Vector D(DQ ? DQ->GetVDim() : 0);
       DenseMatrix K(MQ ? MQ->GetVDim() : 0, MQ ? MQ->GetVDim() : 0);
 #else
       trial_vshape.SetSize(trial_dof,spaceDim);
       test_vshape.SetSize(test_dof,spaceDim);
-      D.SetSize(VQ ? VQ->GetVDim() : 0);
+      D.SetSize(DQ ? DQ->GetVDim() : 0);
       K.SetSize(MQ ? MQ->GetVDim() : 0, MQ ? MQ->GetVDim() : 0);
 #endif
       DenseMatrix tmp(test_vshape.Height(), K.Width());
@@ -2118,9 +2118,9 @@ void VectorFEMassIntegrator::AssembleElementMatrix2(
             Mult(test_vshape,K,tmp);
             AddMultABt(tmp,trial_vshape,elmat);
          }
-         else if (VQ)
+         else if (DQ)
          {
-            VQ->Eval(D, Trans, ip);
+            DQ->Eval(D, Trans, ip);
             D *= w;
             AddMultADBt(test_vshape,D,trial_vshape,elmat);
          }
@@ -3515,6 +3515,47 @@ VectorScalarProductInterpolator::AssembleElementMatrix2(
 }
 
 
+void
+ScalarCrossProductInterpolator::AssembleElementMatrix2(
+   const FiniteElement &dom_fe,
+   const FiniteElement &ran_fe,
+   ElementTransformation &Trans,
+   DenseMatrix &elmat)
+{
+   // Vector coefficient product with vector shape functions
+   struct VCrossVShapeCoefficient : public VectorCoefficient
+   {
+      VectorCoefficient &VQ;
+      const FiniteElement &fe;
+      DenseMatrix vshape;
+      Vector vc;
+
+      VCrossVShapeCoefficient(VectorCoefficient &vq, const FiniteElement &fe_)
+         : VectorCoefficient(fe_.GetDof()), VQ(vq), fe(fe_),
+           vshape(vdim, vq.GetVDim()), vc(vq.GetVDim()) { }
+
+      virtual void Eval(Vector &V, ElementTransformation &T,
+                        const IntegrationPoint &ip)
+      {
+         V.SetSize(vdim);
+         VQ.Eval(vc, T, ip);
+         fe.CalcPhysVShape(T, vshape);
+         for (int k = 0; k < vdim; k++)
+         {
+            V(k) = vc(0) * vshape(k,1) - vc(1) * vshape(k,0);
+         }
+      }
+   };
+
+   VCrossVShapeCoefficient dom_shape_coeff(*VQ, dom_fe);
+
+   elmat.SetSize(ran_fe.GetDof(),dom_fe.GetDof());
+
+   Vector elmat_as_vec(elmat.Data(), elmat.Height()*elmat.Width());
+
+   ran_fe.Project(dom_shape_coeff, Trans, elmat_as_vec);
+}
+
 void
 VectorCrossProductInterpolator::AssembleElementMatrix2(
    const FiniteElement &dom_fe,

fem/bilininteg.hpp

@@ -465,8 +465,8 @@ protected:
       : same_calc_shape(false), Q(NULL), VQ(NULL), DQ(NULL), MQ(NULL) {}
    MixedVectorIntegrator(Coefficient &q)
       : same_calc_shape(false), Q(&q), VQ(NULL), DQ(NULL), MQ(NULL) {}
-   MixedVectorIntegrator(VectorCoefficient &dq, bool diag = true)
-      : same_calc_shape(false), Q(NULL), VQ(diag?NULL:&dq), DQ(diag?&dq:NULL),
+   MixedVectorIntegrator(VectorCoefficient &vq, bool diag = true)
+      : same_calc_shape(false), Q(NULL), VQ(diag?NULL:&vq), DQ(diag?&vq:NULL),
         MQ(NULL) {}
    MixedVectorIntegrator(MatrixCoefficient &mq)
       : same_calc_shape(false), Q(NULL), VQ(NULL), DQ(NULL), MQ(&mq) {}
@@ -503,7 +503,7 @@ protected:
 
    Coefficient *Q;
    VectorCoefficient *VQ;
-   VectorCoefficient *DQ;
+   DiagonalMatrixCoefficient *DQ;
    MatrixCoefficient *MQ;
 
 private:
@@ -901,7 +901,7 @@ public:
    MixedVectorMassIntegrator() { same_calc_shape = true; }
    MixedVectorMassIntegrator(Coefficient &q)
       : MixedVectorIntegrator(q) { same_calc_shape = true; }
-   MixedVectorMassIntegrator(VectorCoefficient &dq)
+   MixedVectorMassIntegrator(DiagonalMatrixCoefficient &dq)
       : MixedVectorIntegrator(dq, true) { same_calc_shape = true; }
    MixedVectorMassIntegrator(MatrixCoefficient &mq)
       : MixedVectorIntegrator(mq) { same_calc_shape = true; }
@@ -1019,7 +1019,7 @@ public:
    MixedGradGradIntegrator() { same_calc_shape = true; }
    MixedGradGradIntegrator(Coefficient &q)
       : MixedVectorIntegrator(q) { same_calc_shape = true; }
-   MixedGradGradIntegrator(VectorCoefficient &dq)
+   MixedGradGradIntegrator(DiagonalMatrixCoefficient &dq)
       : MixedVectorIntegrator(dq, true) { same_calc_shape = true; }
    MixedGradGradIntegrator(MatrixCoefficient &mq)
       : MixedVectorIntegrator(mq) { same_calc_shape = true; }
@@ -1107,7 +1107,7 @@ public:
    MixedCurlCurlIntegrator() { same_calc_shape = true; }
    MixedCurlCurlIntegrator(Coefficient &q)
       : MixedVectorIntegrator(q) { same_calc_shape = true; }
-   MixedCurlCurlIntegrator(VectorCoefficient &dq)
+   MixedCurlCurlIntegrator(DiagonalMatrixCoefficient &dq)
       : MixedVectorIntegrator(dq, true) { same_calc_shape = true; }
    MixedCurlCurlIntegrator(MatrixCoefficient &mq)
       : MixedVectorIntegrator(mq) { same_calc_shape = true; }
@@ -1651,7 +1651,7 @@ public:
    MixedVectorGradientIntegrator() {}
    MixedVectorGradientIntegrator(Coefficient &q)
       : MixedVectorIntegrator(q) {}
-   MixedVectorGradientIntegrator(VectorCoefficient &dq)
+   MixedVectorGradientIntegrator(DiagonalMatrixCoefficient &dq)
       : MixedVectorIntegrator(dq, true) {}
    MixedVectorGradientIntegrator(MatrixCoefficient &mq)
       : MixedVectorIntegrator(mq) {}
@@ -1705,7 +1705,7 @@ public:
    MixedVectorCurlIntegrator() {}
    MixedVectorCurlIntegrator(Coefficient &q)
       : MixedVectorIntegrator(q) {}
-   MixedVectorCurlIntegrator(VectorCoefficient &dq)
+   MixedVectorCurlIntegrator(DiagonalMatrixCoefficient &dq)
       : MixedVectorIntegrator(dq, true) {}
    MixedVectorCurlIntegrator(MatrixCoefficient &mq)
       : MixedVectorIntegrator(mq) {}
@@ -1760,7 +1760,7 @@ public:
    MixedVectorWeakCurlIntegrator() {}
    MixedVectorWeakCurlIntegrator(Coefficient &q)
       : MixedVectorIntegrator(q) {}
-   MixedVectorWeakCurlIntegrator(VectorCoefficient &dq)
+   MixedVectorWeakCurlIntegrator(DiagonalMatrixCoefficient &dq)
       : MixedVectorIntegrator(dq, true) {}
    MixedVectorWeakCurlIntegrator(MatrixCoefficient &mq)
       : MixedVectorIntegrator(mq) {}
@@ -1813,7 +1813,7 @@ public:
    MixedVectorWeakDivergenceIntegrator() {}
    MixedVectorWeakDivergenceIntegrator(Coefficient &q)
       : MixedVectorIntegrator(q) {}
-   MixedVectorWeakDivergenceIntegrator(VectorCoefficient &dq)
+   MixedVectorWeakDivergenceIntegrator(DiagonalMatrixCoefficient &dq)
       : MixedVectorIntegrator(dq, true) {}
    MixedVectorWeakDivergenceIntegrator(MatrixCoefficient &mq)
       : MixedVectorIntegrator(mq) {}
@@ -1844,8 +1844,10 @@ protected:
 };
 
 /** Class for integrating the bilinear form a(u,v) := (Q grad u, v) where Q is a
-    scalar coefficient, and v is a vector with components v_i in the same space
-    as u. */
+    scalar coefficient, and v is a vector with components v_i in the same (H1) space
+    as u.
+
+    See also MixedVectorGradientIntegrator when v is in H(curl). */
 class GradientIntegrator : public BilinearFormIntegrator
 {
 protected:
@@ -1900,6 +1902,7 @@ protected:
    Coefficient *Q;
    VectorCoefficient *VQ;
    MatrixCoefficient *MQ;
+   SymmetricMatrixCoefficient *SMQ;
 
 private:
    Vector vec, pointflux, shape;
@@ -1922,19 +1925,28 @@ private:
 public:
    /// Construct a diffusion integrator with coefficient Q = 1
    DiffusionIntegrator()
-      : Q(NULL), VQ(NULL), MQ(NULL), maps(NULL), geom(NULL), ceedDataPtr(NULL) { }
+      : Q(NULL), VQ(NULL), MQ(NULL), SMQ(NULL), maps(NULL), geom(NULL),
+        ceedDataPtr(NULL) { }
 
    /// Construct a diffusion integrator with a scalar coefficient q
    DiffusionIntegrator(Coefficient &q)
-      : Q(&q), VQ(NULL), MQ(NULL), maps(NULL), geom(NULL), ceedDataPtr(NULL) { }
+      : Q(&q), VQ(NULL), MQ(NULL), SMQ(NULL), maps(NULL), geom(NULL),
+        ceedDataPtr(NULL) { }
 
    /// Construct a diffusion integrator with a vector coefficient q
    DiffusionIntegrator(VectorCoefficient &q)
-      : Q(NULL), VQ(&q), MQ(NULL), maps(NULL), geom(NULL), ceedDataPtr(NULL) { }
+      : Q(NULL), VQ(&q), MQ(NULL), SMQ(NULL), maps(NULL), geom(NULL),
+        ceedDataPtr(NULL) { }
 
    /// Construct a diffusion integrator with a matrix coefficient q
    DiffusionIntegrator(MatrixCoefficient &q)
-      : Q(NULL), VQ(NULL), MQ(&q), maps(NULL), geom(NULL), ceedDataPtr(NULL) { }
+      : Q(NULL), VQ(NULL), MQ(&q), SMQ(NULL), maps(NULL), geom(NULL),
+        ceedDataPtr(NULL) { }
+
+   /// Construct a diffusion integrator with a symmetric matrix coefficient q
+   DiffusionIntegrator(SymmetricMatrixCoefficient &q)
+      : Q(NULL), VQ(NULL), MQ(NULL), SMQ(&q), maps(NULL), geom(NULL),
+        ceedDataPtr(NULL) { }
 
    virtual ~DiffusionIntegrator()
    {
@@ -2050,8 +2062,6 @@ public:
    static const IntegrationRule &GetRule(const FiniteElement &trial_fe,
                                          const FiniteElement &test_fe,
                                          ElementTransformation &Trans);
-
-   void SetupPA(const FiniteElementSpace &fes);
 };
 
 /** Mass integrator (u, v) restricted to the boundary of a domain */
@@ -2335,8 +2345,9 @@ private:
 
 protected:
    Coefficient *Q;
-   VectorCoefficient *DQ;
+   DiagonalMatrixCoefficient *DQ;
    MatrixCoefficient *MQ;
+   SymmetricMatrixCoefficient *SMQ;
 
    // PA extension
    Vector pa_data;
@@ -2347,14 +2358,18 @@ protected:
    bool symmetric = true; ///< False if using a nonsymmetric matrix coefficient
 
 public:
-   CurlCurlIntegrator() { Q = NULL; DQ = NULL; MQ = NULL; }
+   CurlCurlIntegrator() { Q = NULL; DQ = NULL; MQ = NULL; SMQ = NULL; }
    /// Construct a bilinear form integrator for Nedelec elements
    CurlCurlIntegrator(Coefficient &q, const IntegrationRule *ir = NULL) :
-      BilinearFormIntegrator(ir), Q(&q) { DQ = NULL; MQ = NULL; }
-   CurlCurlIntegrator(VectorCoefficient &dq, const IntegrationRule *ir = NULL) :
-      BilinearFormIntegrator(ir), DQ(&dq) { Q = NULL; MQ = NULL; }
+      BilinearFormIntegrator(ir), Q(&q), DQ(NULL), MQ(NULL), SMQ(NULL) { }
+   CurlCurlIntegrator(DiagonalMatrixCoefficient &dq,
+                      const IntegrationRule *ir = NULL) :
+      BilinearFormIntegrator(ir), Q(NULL), DQ(&dq), MQ(NULL), SMQ(NULL) { }
    CurlCurlIntegrator(MatrixCoefficient &mq, const IntegrationRule *ir = NULL) :
-      BilinearFormIntegrator(ir), MQ(&mq) { Q = NULL; DQ = NULL; }
+      BilinearFormIntegrator(ir), Q(NULL), DQ(NULL), MQ(&mq), SMQ(NULL) { }
+   CurlCurlIntegrator(SymmetricMatrixCoefficient &smq,
+                      const IntegrationRule *ir = NULL) :
+      BilinearFormIntegrator(ir), Q(NULL), DQ(NULL), MQ(NULL), SMQ(&smq) { }
 
    /* Given a particular Finite Element, compute the
       element curl-curl matrix elmat */
@@ -2411,8 +2426,9 @@ public:
 class VectorFEMassIntegrator: public BilinearFormIntegrator
 {
 private:
-   void Init(Coefficient *q, VectorCoefficient *vq, MatrixCoefficient *mq)
-   { Q = q; VQ = vq; MQ = mq; }
+   void Init(Coefficient *q, DiagonalMatrixCoefficient *dq, MatrixCoefficient *mq,
+             SymmetricMatrixCoefficient *smq)
+   { Q = q; DQ = dq; MQ = mq; SMQ = smq; }
 
 #ifndef MFEM_THREAD_SAFE
    Vector shape;
@@ -2425,8 +2441,9 @@ private:
 
 protected:
    Coefficient *Q;
-   VectorCoefficient *VQ;
+   DiagonalMatrixCoefficient *DQ;
    MatrixCoefficient *MQ;
+   SymmetricMatrixCoefficient *SMQ;
 
    // PA extension
    Vector pa_data;
@@ -2439,13 +2456,15 @@ protected:
    bool symmetric = true; ///< False if using a nonsymmetric matrix coefficient
 
 public:
-   VectorFEMassIntegrator() { Init(NULL, NULL, NULL); }
-   VectorFEMassIntegrator(Coefficient *_q) { Init(_q, NULL, NULL); }
-   VectorFEMassIntegrator(Coefficient &q) { Init(&q, NULL, NULL); }
-   VectorFEMassIntegrator(VectorCoefficient *_vq) { Init(NULL, _vq, NULL); }
-   VectorFEMassIntegrator(VectorCoefficient &vq) { Init(NULL, &vq, NULL); }
-   VectorFEMassIntegrator(MatrixCoefficient *_mq) { Init(NULL, NULL, _mq); }
-   VectorFEMassIntegrator(MatrixCoefficient &mq) { Init(NULL, NULL, &mq); }
+   VectorFEMassIntegrator() { Init(NULL, NULL, NULL, NULL); }
+   VectorFEMassIntegrator(Coefficient *_q) { Init(_q, NULL, NULL, NULL); }
+   VectorFEMassIntegrator(Coefficient &q) { Init(&q, NULL, NULL, NULL); }
+   VectorFEMassIntegrator(DiagonalMatrixCoefficient *_dq) { Init(NULL, _dq, NULL, NULL); }
+   VectorFEMassIntegrator(DiagonalMatrixCoefficient &dq) { Init(NULL, &dq, NULL, NULL); }
+   VectorFEMassIntegrator(MatrixCoefficient *_mq) { Init(NULL, NULL, _mq, NULL); }
+   VectorFEMassIntegrator(MatrixCoefficient &mq) { Init(NULL, NULL, &mq, NULL); }
+   VectorFEMassIntegrator(SymmetricMatrixCoefficient &smq) { Init(NULL, NULL, NULL, &smq); }
+   VectorFEMassIntegrator(SymmetricMatrixCoefficient *smq) { Init(NULL, NULL, NULL, smq); }
 
    virtual void AssembleElementMatrix(const FiniteElement &el,
                                       ElementTransformation &Trans,
@@ -2654,11 +2673,28 @@ public:
 /** Integrator for the DG form:
     alpha < rho_u (u.n) {v},[w] > + beta < rho_u |u.n| [v],[w] >,
     where v and w are the trial and test variables, respectively, and rho/u are
-    given scalar/vector coefficients. The vector coefficient, u, is assumed to
-    be continuous across the faces and when given the scalar coefficient, rho,
-    is assumed to be discontinuous. The integrator uses the upwind value of rho,
-    rho_u, which is value from the side into which the vector coefficient, u,
-    points. */
+    given scalar/vector coefficients. {v} represents the average value of v on
+    the face and [v] is the jump such that {v}=(v1+v2)/2 and [v]=(v1-v2) for the
+    face between elements 1 and 2. For boundary elements, v2=0. The vector
+    coefficient, u, is assumed to be continuous across the faces and when given
+    the scalar coefficient, rho, is assumed to be discontinuous. The integrator
+    uses the upwind value of rho, rho_u, which is value from the side into which
+    the vector coefficient, u, points.
+
+    One use case for this integrator is to discretize the operator -u.grad(v)
+    with a DG formulation. The resulting formulation uses the
+    ConvectionIntegrator (with coefficient u, and parameter alpha = -1) and the
+    transpose of the DGTraceIntegrator (with coefficient u, and parameters
+    alpha = 1, beta = -1/2 to use the upwind face flux). This discretization and
+    the handling of the inflow and outflow boundaries is illustrated in Example
+    9/9p.
+
+    Another use case for this integrator is to discretize the operator -div(u v)
+    with a DG formulation. The resulting formulation is conservative and
+    consists of the transpose of the ConvectionIntegrator (with coefficient u,
+    and parameter alpha = 1) plus the DGTraceIntegrator (with coefficient u, and
+    parameters alpha = -1, beta = -1/2 to use the upwind face flux).
+    */
 class DGTraceIntegrator : public BilinearFormIntegrator
 {
 protected:
@@ -2752,6 +2788,51 @@ public:
                                    DenseMatrix &elmat);
 };
 
+/** Integrator for the "BR2" diffusion stabilization term
+
+    sum_e eta (r_e([u]), r_e([v]))
+
+    where r_e is the lifting operator defined on each edge e. The parameter eta
+    can be chosen to be one to obtain a stable discretization. The constructor
+    for this integrator requires the finite element space because the lifting
+    operator depends on the element-wise inverse mass matrix.
+
+    BR2 stands for the second method of Bassi and Rebay:
+
+    - F. Bassi and S. Rebay. A high order discontinuous Galerkin method for
+      compressible turbulent flows. In B. Cockburn, G. E. Karniadakis, and
+      C.-W. Shu, editors, Discontinuous Galerkin Methods, pages 77–88. Springer
+      Berlin Heidelberg, 2000.
+    - D. N. Arnold, F. Brezzi, B. Cockburn, and L. D. Marini. Unified analysis
+      of discontinuous Galerkin methods for elliptic problems. SIAM Journal on
+      Numerical Analysis, 39(5):1749–1779, 2002.
+*/
+class DGDiffusionBR2Integrator : public BilinearFormIntegrator
+{
+protected:
+   double eta;
+
+   // Block factorizations of local mass matrices, with offsets for the case of
+   // not equally sized blocks (mixed meshes, p-refinement)
+   Array<double> Minv;
+   Array<int> ipiv;
+   Array<int> ipiv_offsets, Minv_offsets;
+
+   Vector shape1, shape2;
+
+   DenseMatrix R11, R12, R21, R22;
+   DenseMatrix MinvR11, MinvR12, MinvR21, MinvR22;
+   DenseMatrix Re, MinvRe;
+
+public:
+   DGDiffusionBR2Integrator(class FiniteElementSpace *fes, double e = 1.0);
+   using BilinearFormIntegrator::AssembleFaceMatrix;
+   virtual void AssembleFaceMatrix(const FiniteElement &el1,
+                                   const FiniteElement &el2,
+                                   FaceElementTransformations &Trans,
+                                   DenseMatrix &elmat);
+};
+
 /** Integrator for the DG elasticity form, for the formulations see:
     - PhD Thesis of Jonas De Basabe, High-Order Finite %Element Methods for
       Seismic Wave Propagation, UT Austin, 2009, p. 23, and references therein
@@ -2910,11 +2991,36 @@ class DiscreteInterpolator : public BilinearFormIntegrator { };
 class GradientInterpolator : public DiscreteInterpolator
 {
 public:
+   GradientInterpolator() : dofquad_fe(NULL) { }
+   virtual ~GradientInterpolator() { delete dofquad_fe; }
+
    virtual void AssembleElementMatrix2(const FiniteElement &h1_fe,
                                        const FiniteElement &nd_fe,
                                        ElementTransformation &Trans,
                                        DenseMatrix &elmat)
    { nd_fe.ProjectGrad(h1_fe, Trans, elmat); }
+
+   using BilinearFormIntegrator::AssemblePA;
+
+   /** @brief Setup method for PA data.
+
+       @param[in] trial_fes   H1 Lagrange space
+       @param[in] test_fes    H(curl) Nedelec space
+    */
+   virtual void AssemblePA(const FiniteElementSpace &trial_fes,
+                           const FiniteElementSpace &test_fes);
+
+   virtual void AddMultPA(const Vector &x, Vector &y) const;
+   virtual void AddMultTransposePA(const Vector &x, Vector &y) const;
+
+private:
+   /// 1D finite element that generates and owns the 1D DofToQuad maps below
+   FiniteElement * dofquad_fe;
+
+   bool B_id; // is the B basis operator (maps_C_C) the identity?
+   const DofToQuad *maps_C_C; // one-d map with Lobatto rows, Lobatto columns
+   const DofToQuad *maps_O_C; // one-d map with Legendre rows, Lobatto columns
+   int dim, ne, o_dofs1D, c_dofs1D;
 };
 
 
@@ -2929,6 +3035,24 @@ public:
                                        ElementTransformation &Trans,
                                        DenseMatrix &elmat)
    { ran_fe.Project(dom_fe, Trans, elmat); }
+
+   using BilinearFormIntegrator::AssemblePA;
+
+   virtual void AssemblePA(const FiniteElementSpace &trial_fes,
+                           const FiniteElementSpace &test_fes);
+
+   virtual void AddMultPA(const Vector &x, Vector &y) const;
+   virtual void AddMultTransposePA(const Vector &x, Vector &y) const;
+
+private:
+   /// 1D finite element that generates and owns the 1D DofToQuad maps below
+   FiniteElement * dofquad_fe;
+
+   const DofToQuad *maps_C_C; // one-d map with Lobatto rows, Lobatto columns
+   const DofToQuad *maps_O_C; // one-d map with Legendre rows, Lobatto columns
+   int dim, ne, o_dofs1D, c_dofs1D;
+
+   Vector pa_data;
 };
 
 
@@ -3028,6 +3152,22 @@ protected:
    VectorCoefficient *VQ;
 };
 
+/** Interpolator of the 2D cross product between a vector coefficient and an
+    H(curl)-conforming field onto an L2-conforming field. */
+class ScalarCrossProductInterpolator : public DiscreteInterpolator
+{
+public:
+   ScalarCrossProductInterpolator(VectorCoefficient & vc)
+      : VQ(&vc) { }
+
+   virtual void AssembleElementMatrix2(const FiniteElement &nd_fe,
+                                       const FiniteElement &l2_fe,
+                                       ElementTransformation &Trans,
+                                       DenseMatrix &elmat);
+protected:
+   VectorCoefficient *VQ;
+};
+
 /** Interpolator of the cross product between a vector coefficient and an
     H(curl)-conforming field onto an H(div)-conforming field. The range space
     can also be vector L2. */

fem/bilininteg_br2.cpp

@@ -0,0 +1,242 @@
+// Copyright (c) 2010-2020, Lawrence Livermore National Security, LLC. Produced
+// at the Lawrence Livermore National Laboratory. All Rights reserved. See files
+// LICENSE and NOTICE for details. LLNL-CODE-806117.
+//
+// This file is part of the MFEM library. For more information and source code
+// availability visit https://mfem.org.
+//
+// MFEM is free software; you can redistribute it and/or modify it under the
+// terms of the BSD-3 license. We welcome feedback and contributions, see file
+// CONTRIBUTING.md for details.
+
+#include "bilininteg.hpp"
+#include "pfespace.hpp"
+#include <algorithm>
+
+namespace mfem
+{
+
+DGDiffusionBR2Integrator::DGDiffusionBR2Integrator(FiniteElementSpace *fes,
+                                                   double e) : eta(e)
+{
+   // Precompute local mass matrix inverses needed for the lifting operators
+   // First compute offsets and total size needed (e.g. for mixed meshes or
+   // p-refinement)
+   int nel = fes->GetNE();
+   Minv_offsets.SetSize(nel+1);
+   ipiv_offsets.SetSize(nel+1);
+   ipiv_offsets[0] = 0;
+   Minv_offsets[0] = 0;
+   for (int i=0; i<nel; ++i)
+   {
+      int dof = fes->GetFE(i)->GetDof();
+      ipiv_offsets[i+1] = ipiv_offsets[i] + dof;
+      Minv_offsets[i+1] = Minv_offsets[i] + dof*dof;
+   }
+
+#ifdef MFEM_USE_MPI
+   // When running in parallel, we also need to compute the local mass matrices
+   // of face neighbor elements
+   ParFiniteElementSpace *pfes = dynamic_cast<ParFiniteElementSpace *>(fes);
+   if (pfes != NULL)
+   {
+      ParMesh *pmesh = pfes->GetParMesh();
+      pfes->ExchangeFaceNbrData();
+      int nel_nbr = pmesh->GetNFaceNeighborElements();
+      Minv_offsets.SetSize(nel+nel_nbr+1);
+      ipiv_offsets.SetSize(nel+nel_nbr+1);
+      for (int i=0; i<nel_nbr; ++i)
+      {
+         int dof = pfes->GetFaceNbrFE(i)->GetDof();
+         ipiv_offsets[nel+i+1] = ipiv_offsets[nel+i] + dof;
+         Minv_offsets[nel+i+1] = Minv_offsets[nel+i] + dof*dof;
+      }
+      nel += nel_nbr;
+   }
+#endif
+   // The final "offset" is the total size of all the blocks
+   Minv.SetSize(Minv_offsets[nel]);
+   ipiv.SetSize(ipiv_offsets[nel]);
+
+   // Assemble the local mass matrices and compute LU factorization
+   MassIntegrator mi;
+   for (int i=0; i<nel; ++i)
+   {
+      const FiniteElement *fe = NULL;
+      ElementTransformation *tr = NULL;
+      if (i < fes->GetNE())
+      {
+         fe = fes->GetFE(i);
+         tr = fes->GetElementTransformation(i);
+      }
+      else
+      {
+#ifdef MFEM_USE_MPI
+         int inbr = i - fes->GetNE();
+         fe = pfes->GetFaceNbrFE(inbr);
+         tr = pfes->GetParMesh()->GetFaceNbrElementTransformation(inbr);
+#endif
+      }
+      int dof = fe->GetDof();
+      double *Minv_el = &Minv[Minv_offsets[i]];
+      int *ipiv_el = &ipiv[ipiv_offsets[i]];
+      DenseMatrix Me(Minv_el, dof, dof);
+      mi.AssembleElementMatrix(*fe, *tr, Me);
+      LUFactors lu(Minv_el, ipiv_el);
+      lu.Factor(dof);
+   }
+}
+
+void DGDiffusionBR2Integrator::AssembleFaceMatrix(
+   const FiniteElement &el1, const FiniteElement &el2,
+   FaceElementTransformations &Trans, DenseMatrix &elmat)
+{
+   int ndof1 = el1.GetDof();
+   shape1.SetSize(ndof1);
+
+   R11.SetSize(ndof1, ndof1);
+   R11 = 0.0;
+   LUFactors M1inv(&Minv[Minv_offsets[Trans.Elem1No]],
+                   &ipiv[ipiv_offsets[Trans.Elem1No]]);
+   LUFactors M2inv;
+
+   double factor = Geometries.NumBdr(Trans.Elem1->GetGeometryType());
+
+   int ndof2;
+   if (Trans.Elem2No >= 0)
+   {
+      ndof2 = el2.GetDof();
+      shape2.SetSize(ndof2);
+      R12.SetSize(ndof1, ndof2);
+      R21.SetSize(ndof2, ndof1);
+      R22.SetSize(ndof2, ndof2);
+      M2inv.data = &Minv[Minv_offsets[Trans.Elem2No]];
+      M2inv.ipiv = &ipiv[ipiv_offsets[Trans.Elem2No]];
+
+      R12 = 0.0;
+      R21 = 0.0;
+      R22 = 0.0;
+
+      Geometry::Type geom2 = Trans.Elem2->GetGeometryType();
+      factor = std::max(factor, double(Geometries.NumBdr(geom2)));
+   }
+   else
+   {
+      ndof2 = 0;
+   }
+
+   int ndofs = ndof1 + ndof2;
+
+   Re.SetSize(ndofs, ndofs);
+   MinvRe.SetSize(ndofs, ndofs);
+
+   elmat.SetSize(ndofs);
+   elmat = 0.0;
+
+   const IntegrationRule *ir = IntRule;
+   if (ir == NULL)
+   {
+      int order;
+      if (ndof2)
+      {
+         order = 2*std::max(el1.GetOrder(), el2.GetOrder());
+      }
+      else
+      {
+         order = 2*el1.GetOrder();
+      }
+      ir = &IntRules.Get(Trans.FaceGeom, order);
+   }
+
+   for (int p = 0; p < ir->GetNPoints(); p++)
+   {
+      const IntegrationPoint &ip = ir->IntPoint(p);
+      IntegrationPoint eip1, eip2;
+
+      Trans.Loc1.Transform(ip, eip1);
+      el1.CalcShape(eip1, shape1);
+      if (ndof2)
+      {
+         Trans.Loc2.Transform(ip, eip2);
+         el2.CalcShape(eip2, shape2);
+      }
+
+      double w = factor*sqrt(eta)*ip.weight*Trans.Face->Weight();
+      if (ndof2)
+      {
+         w /= 2;
+      }
+
+      for (int i = 0; i < ndof1; i++)
+      {
+         const double wsi = w*shape1(i);
+         for (int j = 0; j < ndof1; j++)
+         {
+            R11(i, j) += wsi*shape1(j);
+         }
+      }
+
+      if (ndof2)
+      {
+         for (int i = 0; i < ndof2; i++)
+         {
+            const double wsi = w*shape2(i);
+            for (int j = 0; j < ndof1; j++)
+            {
+               R21(i, j) += wsi*shape1(j);
+               R12(j, i) -= wsi*shape1(j);
+            }
+            for (int j = 0; j < ndof2; j++)
+            {
+               R22(i, j) -= wsi*shape2(j);
+            }
+         }
+      }
+   }
+
+   MinvR11 = R11;
+   M1inv.Solve(ndof1, ndof1, MinvR11.Data());
+   for (int i = 0; i < ndof1; i++)
+   {
+      for (int j = 0; j < ndof1; j++)
+      {
+         Re(i, j) = R11(i, j);
+         MinvRe(i, j) = MinvR11(i, j);
+      }
+   }
+
+   if (ndof2)
+   {
+      MinvR12 = R12;
+      MinvR21 = R21;
+      MinvR22 = R22;
+      M1inv.Solve(ndof1, ndof2, MinvR12.Data());
+      M2inv.Solve(ndof2, ndof1, MinvR21.Data());
+      M2inv.Solve(ndof2, ndof2, MinvR22.Data());
+
+      for (int i = 0; i < ndof2; i++)
+      {
+         for (int j = 0; j < ndof1; j++)
+         {
+            Re(ndof1 + i, j) = R21(i, j);
+            MinvRe(ndof1 + i, j) = MinvR21(i, j);
+
+            Re(j, ndof1 + i) = R12(j, i);
+            MinvRe(j, ndof1 + i) = MinvR12(j, i);
+         }
+         for (int j = 0; j < ndof2; j++)
+         {
+            Re(ndof1 + i, ndof1 + j) = R22(i, j);
+            MinvRe(ndof1 + i, ndof1 + j) = MinvR22(i, j);
+         }
+      }
+   }
+
+   // Compute the matrix associated with (r_e([u]), r_e([u])).
+   // The matrix for r_e([u]) is `MinvRe`, and so we need to form the product
+   // `(MinvRe)^T M MinvRe`. Using `Minv^T M = Minv M = I`, we obtain
+   // `Re^T MinvRe`.
+   MultAtB(Re, MinvRe, elmat);
+}
+
+}

fem/bilininteg_diffusion_pa.cpp

@@ -379,51 +379,54 @@ void DiffusionIntegrator::AssemblePA(const FiniteElementSpace &fes)
    const int MQfullDim = MQ ? MQ->GetHeight() * MQ->GetWidth() : 0;
    if (MQ)
    {
+      symmetric = false;
       MFEM_VERIFY(MQ->GetHeight() == dim && MQ->GetWidth() == dim, "");
-      const int MQsymmDim = MQ->GetWidth() * (MQ->GetWidth() + 1) / 2;
 
-      const int MQdim = MQ->IsSymmetric() ? MQsymmDim : MQfullDim;
-      coeffDim = MQdim;
+      coeffDim = MQfullDim;
 
-      coeff.SetSize(MQdim * nq * ne);
-      symmetric = MQ ? MQ->IsSymmetric() : true;
+      coeff.SetSize(MQfullDim * nq * ne);
 
       DenseMatrix M;
-      Vector Msymm;
-      if (symmetric)
-      {
-         Msymm.SetSize(MQsymmDim);
-      }
-      else
-      {
-         M.SetSize(dim);
-      }
+      M.SetSize(dim);
 
-      auto C = Reshape(coeff.HostWrite(), MQdim, nq, ne);
+      auto C = Reshape(coeff.HostWrite(), MQfullDim, nq, ne);
       for (int e=0; e<ne; ++e)
       {
          ElementTransformation *tr = mesh->GetElementTransformation(e);
          for (int p=0; p<nq; ++p)
          {
-            if (MQ->IsSymmetric())
-            {
-               MQ->EvalSymmetric(Msymm, *tr, ir->IntPoint(p));
-
-               for (int i=0; i<MQsymmDim; ++i)
+            MQ->Eval(M, *tr, ir->IntPoint(p));
+            for (int i=0; i<dim; ++i)
+               for (int j=0; j<dim; ++j)
                {
-                  C(i, p, e) = Msymm[i];
+                  C(j+(i*dim), p, e) = M(i,j);
                }
-            }
-            else
-            {
-               MQ->Eval(M, *tr, ir->IntPoint(p));
+         }
+      }
+   }
+   else if (SMQ)
+   {
+      MFEM_VERIFY(SMQ->GetSize() == dim, "");
+      coeffDim = symmDims;
+      coeff.SetSize(symmDims * nq * ne);
 
-               for (int i=0; i<dim; ++i)
-                  for (int j=0; j<dim; ++j)
-                  {
-                     C(j+(i*dim), p, e) = M(i,j);
-                  }
-            }
+      DenseSymmetricMatrix M;
+      M.SetSize(dim);
+
+      auto C = Reshape(coeff.HostWrite(), symmDims, nq, ne);
+
+      for (int e=0; e<ne; ++e)
+      {
+         ElementTransformation *tr = mesh->GetElementTransformation(e);
+         for (int p=0; p<nq; ++p)
+         {
+            SMQ->Eval(M, *tr, ir->IntPoint(p));
+            int cnt = 0;
+            for (int i=0; i<dim; ++i)
+               for (int j=i; j<dim; ++j, ++cnt)
+               {
+                  C(cnt, p, e) = M(i,j);
+               }
          }
       }
    }

fem/bilininteg_gradient.cpp

@@ -334,7 +334,7 @@ static void PAGradientApplyTranspose2D(const int NE,
                                        const int q1d = 0)
 {
    // TODO
-   MFEM_ASSERT(false, "GradientPAApplyTranspose 3D not implemented.");
+   MFEM_ASSERT(false, "PAGradientApplyTranspose2D not implemented.");
 }
 
 // PA Gradient Apply 3D kernel

fem/bilininteg_vecmass.cpp

@@ -106,7 +106,7 @@ void VectorMassIntegrator::AssemblePA(const FiniteElementSpace &fes)
 template<const int T_D1D = 0,
          const int T_Q1D = 0>
 static void PAVectorMassApply2D(const int NE,
-                                const Array<double> &_B,
+                                const Array<double> &B_,
                                 const Array<double> &_Bt,
                                 const Vector &_op,
                                 const Vector &_x,
@@ -119,7 +119,7 @@ static void PAVectorMassApply2D(const int NE,
    constexpr int VDIM = 2;
    MFEM_VERIFY(D1D <= MAX_D1D, "");
    MFEM_VERIFY(Q1D <= MAX_Q1D, "");
-   auto B = Reshape(_B.Read(), Q1D, D1D);
+   auto B = Reshape(B_.Read(), Q1D, D1D);
    auto Bt = Reshape(_Bt.Read(), D1D, Q1D);
    auto op = Reshape(_op.Read(), Q1D, Q1D, NE);
    auto x = Reshape(_x.Read(), D1D, D1D, VDIM, NE);
@@ -203,7 +203,7 @@ static void PAVectorMassApply2D(const int NE,
 template<const int T_D1D = 0,
          const int T_Q1D = 0>
 static void PAVectorMassApply3D(const int NE,
-                                const Array<double> &_B,
+                                const Array<double> &B_,
                                 const Array<double> &_Bt,
                                 const Vector &_op,
                                 const Vector &_x,
@@ -216,7 +216,7 @@ static void PAVectorMassApply3D(const int NE,
    constexpr int VDIM = 3;
    MFEM_VERIFY(D1D <= MAX_D1D, "");
    MFEM_VERIFY(Q1D <= MAX_Q1D, "");
-   auto B = Reshape(_B.Read(), Q1D, D1D);
+   auto B = Reshape(B_.Read(), Q1D, D1D);
    auto Bt = Reshape(_Bt.Read(), D1D, Q1D);
    auto op = Reshape(_op.Read(), Q1D, Q1D, Q1D, NE);
    auto x = Reshape(_x.Read(), D1D, D1D, D1D, VDIM, NE);
@@ -381,7 +381,7 @@ void VectorMassIntegrator::AddMultPA(const Vector &x, Vector &y) const
 
 template<const int T_D1D = 0, const int T_Q1D = 0>
 static void PAVectorMassAssembleDiagonal2D(const int NE,
-                                           const Array<double> &_B,
+                                           const Array<double> &B_,
                                            const Array<double> &_Bt,
                                            const Vector &_op,
                                            Vector &_diag,
@@ -393,7 +393,7 @@ static void PAVectorMassAssembleDiagonal2D(const int NE,
    constexpr int VDIM = 2;
    MFEM_VERIFY(D1D <= MAX_D1D, "");
    MFEM_VERIFY(Q1D <= MAX_Q1D, "");
-   auto B = Reshape(_B.Read(), Q1D, D1D);
+   auto B = Reshape(B_.Read(), Q1D, D1D);
    auto op = Reshape(_op.Read(), Q1D, Q1D, NE);
    auto y = Reshape(_diag.ReadWrite(), D1D, D1D, VDIM, NE);
    MFEM_FORALL(e, NE,
@@ -433,7 +433,7 @@ static void PAVectorMassAssembleDiagonal2D(const int NE,
 
 template<const int T_D1D = 0, const int T_Q1D = 0>
 static void PAVectorMassAssembleDiagonal3D(const int NE,
-                                           const Array<double> &_B,
+                                           const Array<double> &B_,
                                            const Array<double> &_Bt,
                                            const Vector &_op,
                                            Vector &_diag,
@@ -445,7 +445,7 @@ static void PAVectorMassAssembleDiagonal3D(const int NE,
    constexpr int VDIM = 3;
    MFEM_VERIFY(D1D <= MAX_D1D, "");
    MFEM_VERIFY(Q1D <= MAX_Q1D, "");
-   auto B = Reshape(_B.Read(), Q1D, D1D);
+   auto B = Reshape(B_.Read(), Q1D, D1D);
    auto op = Reshape(_op.Read(), Q1D, Q1D, Q1D, NE);
    auto y = Reshape(_diag.ReadWrite(), D1D, D1D, D1D, VDIM, NE);
    MFEM_FORALL(e, NE,

fem/bilininteg_vectorfe.cpp

@@ -761,12 +761,12 @@ void VectorFEMassIntegrator::AssemblePA(const FiniteElementSpace &trial_fes,
    trial_fetype = trial_el->GetDerivType();
    test_fetype = test_el->GetDerivType();
 
-   const int MQsymmDim = MQ ? (MQ->GetWidth() * (MQ->GetWidth() + 1)) / 2 : 0;
+   const int MQsymmDim = SMQ ? (SMQ->GetSize() * (SMQ->GetSize() + 1)) / 2 : 0;
    const int MQfullDim = MQ ? (MQ->GetHeight() * MQ->GetWidth()) : 0;
-   const int MQdim = MQ ? (MQ->IsSymmetric() ? MQsymmDim : MQfullDim) : 0;
-   const int coeffDim = MQ ? MQdim : (VQ ? VQ->GetVDim() : 1);
+   const int MQdim = MQ ? MQfullDim : MQsymmDim;
+   const int coeffDim = (MQ || SMQ) ? MQdim : (DQ ? DQ->GetVDim() : 1);
 
-   symmetric = MQ ? MQ->IsSymmetric() : true;
+   symmetric = (MQ == NULL);
 
    const bool trial_curl = (trial_fetype == mfem::FiniteElement::CURL);
    const bool trial_div = (trial_fetype == mfem::FiniteElement::DIV);
@@ -783,24 +783,13 @@ void VectorFEMassIntegrator::AssemblePA(const FiniteElementSpace &trial_fes,
    Vector coeff(coeffDim * ne * nq);
    coeff = 1.0;
    auto coeffh = Reshape(coeff.HostWrite(), coeffDim, nq, ne);
-   if (Q || VQ || MQ)
+   if (Q || DQ || MQ || SMQ)
    {
-      Vector D(VQ ? coeffDim : 0);
+      Vector D(DQ ? coeffDim : 0);
       DenseMatrix M;
-      Vector Msymm;
-      if (MQ)
-      {
-         if (symmetric)
-         {
-            Msymm.SetSize(MQsymmDim);
-         }
-         else
-         {
-            M.SetSize(dim);
-         }
-      }
+      DenseSymmetricMatrix SM;
 
-      if (VQ)
+      if (DQ)
       {
          MFEM_VERIFY(coeffDim == dim, "");
       }
@@ -808,6 +797,12 @@ void VectorFEMassIntegrator::AssemblePA(const FiniteElementSpace &trial_fes,
       {
          MFEM_VERIFY(coeffDim == MQdim, "");
          MFEM_VERIFY(MQ->GetHeight() == dim && MQ->GetWidth() == dim, "");
+         M.SetSize(dim);
+      }
+      if (SMQ)
+      {
+         MFEM_VERIFY(SMQ->GetSize() == dim, "");
+         SM.SetSize(dim);
       }
 
       for (int e=0; e<ne; ++e)
@@ -817,29 +812,27 @@ void VectorFEMassIntegrator::AssemblePA(const FiniteElementSpace &trial_fes,
          {
             if (MQ)
             {
-               if (MQ->IsSymmetric())
-               {
-                  MQ->EvalSymmetric(Msymm, *tr, ir->IntPoint(p));
+               MQ->Eval(M, *tr, ir->IntPoint(p));
 
-                  for (int i=0; i<MQsymmDim; ++i)
+               for (int i=0; i<dim; ++i)
+                  for (int j=0; j<dim; ++j)
                   {
-                     coeffh(i, p, e) = Msymm[i];
+                     coeffh(j+(i*dim), p, e) = M(i,j);
                   }
-               }
-               else
-               {
-                  MQ->Eval(M, *tr, ir->IntPoint(p));
-
-                  for (int i=0; i<dim; ++i)
-                     for (int j=0; j<dim; ++j)
-                     {
-                        coeffh(j+(i*dim), p, e) = M(i,j);
-                     }
-               }
             }
-            else if (VQ)
+            else if (SMQ)
             {
-               VQ->Eval(D, *tr, ir->IntPoint(p));
+               SMQ->Eval(SM, *tr, ir->IntPoint(p));
+               int cnt = 0;
+               for (int i=0; i<dim; ++i)
+                  for (int j=i; j<dim; ++j, ++cnt)
+                  {
+                     coeffh(cnt, p, e) = SM(i,j);
+                  }
+            }
+            else if (DQ)
+            {
+               DQ->Eval(D, *tr, ir->IntPoint(p));
                for (int i=0; i<coeffDim; ++i)
                {
                   coeffh(i, p, e) = D[i];
@@ -1007,14 +1000,14 @@ void VectorFEMassIntegrator::AddMultPA(const Vector &x, Vector &y) const
       }
       else if (trial_curl && test_div)
       {
-         const bool scalarCoeff = !(VQ || MQ);
+         const bool scalarCoeff = !(DQ || MQ || SMQ);
          PAHcurlHdivMassApply3D(dofs1D, dofs1Dtest, quad1D, ne, scalarCoeff,
                                 true, mapsO->B, mapsC->B, mapsOtest->Bt,
                                 mapsCtest->Bt, pa_data, x, y);
       }
       else if (trial_div && test_curl)
       {
-         const bool scalarCoeff = !(VQ || MQ);
+         const bool scalarCoeff = !(DQ || MQ || SMQ);
          PAHcurlHdivMassApply3D(dofs1D, dofs1Dtest, quad1D, ne, scalarCoeff,
                                 false, mapsO->B, mapsC->B, mapsOtest->Bt,
                                 mapsCtest->Bt, pa_data, x, y);
@@ -1038,7 +1031,7 @@ void VectorFEMassIntegrator::AddMultPA(const Vector &x, Vector &y) const
       }
       else if ((trial_curl && test_div) || (trial_div && test_curl))
       {
-         const bool scalarCoeff = !(VQ || MQ);
+         const bool scalarCoeff = !(DQ || MQ || SMQ);
          PAHcurlHdivMassApply2D(dofs1D, dofs1Dtest, quad1D, ne, scalarCoeff,
                                 trial_curl, mapsO->B, mapsC->B, mapsOtest->Bt,
                                 mapsCtest->Bt, pa_data, x, y);

fem/coefficient.cpp

@@ -301,7 +301,7 @@ void MatrixFunctionCoefficient::Eval(DenseMatrix &K, ElementTransformation &T,
 
    K.SetSize(height, width);
 
-   if (symmetric) // Use SymmFunction
+   if (symmetric) // Use SymmFunction (deprecated version)
    {
       MFEM_VERIFY(height == width && SymmFunction,
                   "MatrixFunctionCoefficient is not symmetric");
@@ -371,6 +371,36 @@ void MatrixFunctionCoefficient::EvalSymmetric(Vector &K,
    }
 }
 
+void SymmetricMatrixFunctionCoefficient::Eval(DenseSymmetricMatrix &K,
+                                              ElementTransformation &T,
+                                              const IntegrationPoint &ip)
+{
+   double x[3];
+   Vector transip(x, 3);
+
+   T.Transform(ip, transip);
+
+   K.SetSize(dim);
+
+   if (Function)
+   {
+      Function(transip, K);
+   }
+   else if (TDFunction)
+   {
+      TDFunction(transip, GetTime(), K);
+   }
+   else
+   {
+      K = mat;
+   }
+
+   if (Q)
+   {
+      K *= Q->Eval(T, ip, GetTime());
+   }
+}
+
 MatrixArrayCoefficient::MatrixArrayCoefficient (int dim)
    : MatrixCoefficient (dim)
 {
@@ -485,32 +515,32 @@ VectorSumCoefficient::VectorSumCoefficient(int dim)
 }
 
 VectorSumCoefficient::VectorSumCoefficient(VectorCoefficient &_A,
-                                           VectorCoefficient &_B,
+                                           VectorCoefficient &B_,
                                            double _alpha, double _beta)
    : VectorCoefficient(_A.GetVDim()),
-     ACoef(&_A), BCoef(&_B),
+     ACoef(&_A), BCoef(&B_),
      A(_A.GetVDim()), B(_A.GetVDim()),
      alphaCoef(NULL), betaCoef(NULL),
      alpha(_alpha), beta(_beta)
 {
-   MFEM_ASSERT(_A.GetVDim() == _B.GetVDim(),
+   MFEM_ASSERT(_A.GetVDim() == B_.GetVDim(),
                "VectorSumCoefficient:  "
                "Arguments must have the same dimension.");
 }
 
 VectorSumCoefficient::VectorSumCoefficient(VectorCoefficient &_A,
-                                           VectorCoefficient &_B,
+                                           VectorCoefficient &B_,
                                            Coefficient &_alpha,
                                            Coefficient &_beta)
    : VectorCoefficient(_A.GetVDim()),
-     ACoef(&_A), BCoef(&_B),
+     ACoef(&_A), BCoef(&B_),
      A(_A.GetVDim()),
      B(_A.GetVDim()),
      alphaCoef(&_alpha),
      betaCoef(&_beta),
      alpha(0.0), beta(0.0)
 {
-   MFEM_ASSERT(_A.GetVDim() == _B.GetVDim(),
+   MFEM_ASSERT(_A.GetVDim() == B_.GetVDim(),
                "VectorSumCoefficient:  "
                "Arguments must have the same dimension.");
 }
@@ -595,6 +625,7 @@ void MatrixVectorProductCoefficient::Eval(Vector &V, ElementTransformation &T,
 {
    a->Eval(ma, T, ip);
    b->Eval(vb, T, ip);
+   V.SetSize(vdim);
    ma.Mult(vb, V);
 }
 
@@ -697,6 +728,11 @@ void OuterProductCoefficient::Eval(DenseMatrix &M, ElementTransformation &T,
    }
 }
 
+CrossCrossCoefficient::CrossCrossCoefficient(double A, VectorCoefficient &K)
+   : MatrixCoefficient(K.GetVDim(), K.GetVDim()), aConst(A), a(NULL), k(&K),
+     vk(K.GetVDim())
+{}
+
 CrossCrossCoefficient::CrossCrossCoefficient(Coefficient &A,
                                              VectorCoefficient &K)
    : MatrixCoefficient(K.GetVDim(), K.GetVDim()), aConst(0.0), a(&A), k(&K),

fem/coefficient.hpp

@@ -688,6 +688,7 @@ public:
                      const IntegrationRule &ir);
 };
 
+typedef VectorCoefficient DiagonalMatrixCoefficient;
 
 /// Base class for Matrix Coefficients that optionally depend on time and space.
 class MatrixCoefficient
@@ -695,7 +696,7 @@ class MatrixCoefficient
 protected:
    int height, width;
    double time;
-   bool symmetric;
+   bool symmetric;  // deprecated
 
 public:
    /// Construct a dim x dim matrix coefficient.
@@ -721,6 +722,7 @@ public:
    /// For backward compatibility get the width of the matrix.
    int GetVDim() const { return width; }
 
+   /** @deprecated Use SymmetricMatrixCoefficient instead */
    bool IsSymmetric() const { return symmetric; }
 
    /** @brief Evaluate the matrix coefficient in the element described by @a T
@@ -731,11 +733,13 @@ public:
    virtual void Eval(DenseMatrix &K, ElementTransformation &T,
                      const IntegrationPoint &ip) = 0;
 
+   /// (DEPRECATED) Evaluate a symmetric matrix coefficient.
    /** @brief Evaluate the upper triangular entries of the matrix coefficient
        in the symmetric case, similarly to Eval. Matrix entry (i,j) is stored
        in K[j - i + os_i] for 0 <= i <= j < width, os_0 = 0,
        os_{i+1} = os_i + width - i. That is, K = {M(0,0), ..., M(0,w-1),
-       M(1,1), ..., M(1,w-1), ..., M(w-1,w-1) with w = width. */
+       M(1,1), ..., M(1,w-1), ..., M(w-1,w-1) with w = width.
+       @deprecated Use Eval() instead. */
    virtual void EvalSymmetric(Vector &K, ElementTransformation &T,
                               const IntegrationPoint &ip)
    { mfem_error("MatrixCoefficient::EvalSymmetric"); }
@@ -767,7 +771,7 @@ class MatrixFunctionCoefficient : public MatrixCoefficient
 {
 private:
    std::function<void(const Vector &, DenseMatrix &)> Function;
-   std::function<void(const Vector &, Vector &)> SymmFunction;
+   std::function<void(const Vector &, Vector &)> SymmFunction;  // deprecated
    std::function<void(const Vector &, double, DenseMatrix &)> TDFunction;
 
    Coefficient *Q;
@@ -791,6 +795,18 @@ public:
       : MatrixCoefficient(m.Height(), m.Width()), Q(&q), mat(m)
    { }
 
+   /** @brief Define a time-independent symmetric square matrix coefficient from
+       a std function */
+   /** \param dim - the size of the matrix
+       \param SymmF - function used in EvalSymmetric
+       \param q - optional scalar Coefficient to scale the matrix coefficient
+       @deprecated Use another constructor without setting SymmFunction. */
+   MatrixFunctionCoefficient(int dim,
+                             std::function<void(const Vector &, Vector &)> SymmF,
+                             Coefficient *q = NULL)
+      : MatrixCoefficient(dim, true), SymmFunction(std::move(SymmF)), Q(q), mat(0)
+   { }
+
    /// Define a time-dependent square matrix coefficient from a std function
    /** \param dim - the size of the matrix
        \param TDF - time-dependent function
@@ -801,22 +817,12 @@ public:
       : MatrixCoefficient(dim), TDFunction(std::move(TDF)), Q(q)
    { }
 
-   /** @brief Define a time-independent symmetric square matrix coefficient from
-       a std function */
-   /** \param dim - the size of the matrix
-       \param SymmF - function used in EvalSymmetric
-       \param q - optional scalar Coefficient to scale the matrix coefficient */
-   MatrixFunctionCoefficient(int dim,
-                             std::function<void(const Vector &, Vector &)> SymmF,
-                             Coefficient *q = NULL)
-      : MatrixCoefficient(dim, true), SymmFunction(std::move(SymmF)), Q(q), mat(0)
-   { }
-
    /// Evaluate the matrix coefficient at @a ip.
    virtual void Eval(DenseMatrix &K, ElementTransformation &T,
                      const IntegrationPoint &ip);
 
-   /// Evaluate the symmetric matrix coefficient at @a ip.
+   /// (DEPRECATED) Evaluate the symmetric matrix coefficient at @a ip.
+   /** @deprecated Use Eval() instead. */
    virtual void EvalSymmetric(Vector &K, ElementTransformation &T,
                               const IntegrationPoint &ip);
 
@@ -824,7 +830,6 @@ public:
 };
 
 
-
 /** @brief Matrix coefficient defined by a matrix of scalar coefficients.
     Coefficients that are not set will evaluate to zero in the vector. The
     coefficient is stored as a flat Array with indexing (i,j) -> i*width+j. */
@@ -940,6 +945,106 @@ public:
    }
 };
 
+
+/// Base class for symmetric matrix coefficients that optionally depend on time and space.
+class SymmetricMatrixCoefficient
+{
+protected:
+   int dim;
+   double time;
+
+public:
+   /// Construct a dim x dim matrix coefficient.
+   explicit SymmetricMatrixCoefficient(int dimension)
+   { dim = dimension; time = 0.; }
+
+   /// Set the time for time dependent coefficients
+   void SetTime(double t) { time = t; }
+
+   /// Get the time for time dependent coefficients
+   double GetTime() { return time; }
+
+   /// Get the size of the matrix.
+   int GetSize() const { return dim; }
+
+   /** @brief Evaluate the matrix coefficient in the element described by @a T
+       at the point @a ip, storing the result in @a K. */
+   /** @note When this method is called, the caller must make sure that the
+       IntegrationPoint associated with @a T is the same as @a ip. This can be
+       achieved by calling T.SetIntPoint(&ip). */
+   virtual void Eval(DenseSymmetricMatrix &K, ElementTransformation &T,
+                     const IntegrationPoint &ip) = 0;
+
+   virtual ~SymmetricMatrixCoefficient() { }
+};
+
+
+/// A matrix coefficient that is constant in space and time.
+class SymmetricMatrixConstantCoefficient : public SymmetricMatrixCoefficient
+{
+private:
+   DenseSymmetricMatrix mat;
+
+public:
+   ///Construct using matrix @a m for the constant.
+   SymmetricMatrixConstantCoefficient(const DenseSymmetricMatrix &m)
+      : SymmetricMatrixCoefficient(m.Height()), mat(m) { }
+   using SymmetricMatrixCoefficient::Eval;
+   /// Evaluate the matrix coefficient at @a ip.
+   virtual void Eval(DenseSymmetricMatrix &M, ElementTransformation &T,
+                     const IntegrationPoint &ip) { M = mat; }
+};
+
+
+/** @brief A matrix coefficient with an optional scalar coefficient multiplier
+    \a q.  The matrix function can either be represented by a std function or
+    a constant matrix provided when constructing this object.  */
+class SymmetricMatrixFunctionCoefficient : public SymmetricMatrixCoefficient
+{
+private:
+   std::function<void(const Vector &, DenseSymmetricMatrix &)> Function;
+   std::function<void(const Vector &, double, DenseSymmetricMatrix &)> TDFunction;
+
+   Coefficient *Q;
+   DenseSymmetricMatrix mat;
+
+public:
+   /// Define a time-independent symmetric matrix coefficient from a std function
+   /** \param dim - the size of the matrix
+       \param F - time-independent function
+       \param q - optional scalar Coefficient to scale the matrix coefficient */
+   SymmetricMatrixFunctionCoefficient(int dim,
+                                      std::function<void(const Vector &, DenseSymmetricMatrix &)> F,
+                                      Coefficient *q = nullptr)
+      : SymmetricMatrixCoefficient(dim), Function(std::move(F)), Q(q), mat(0)
+   { }
+
+   /// Define a constant matrix coefficient times a scalar Coefficient
+   /** \param m - constant matrix
+       \param q - optional scalar Coefficient to scale the matrix coefficient */
+   SymmetricMatrixFunctionCoefficient(const DenseSymmetricMatrix &m,
+                                      Coefficient &q)
+      : SymmetricMatrixCoefficient(m.Height()), Q(&q), mat(m)
+   { }
+
+   /// Define a time-dependent square matrix coefficient from a std function
+   /** \param dim - the size of the matrix
+       \param TDF - time-dependent function
+       \param q - optional scalar Coefficient to scale the matrix coefficient */
+   SymmetricMatrixFunctionCoefficient(int dim,
+                                      std::function<void(const Vector &, double, DenseSymmetricMatrix &)> TDF,
+                                      Coefficient *q = nullptr)
+      : SymmetricMatrixCoefficient(dim), TDFunction(std::move(TDF)), Q(q)
+   { }
+
+   /// Evaluate the matrix coefficient at @a ip.
+   virtual void Eval(DenseSymmetricMatrix &K, ElementTransformation &T,
+                     const IntegrationPoint &ip);
+
+   virtual ~SymmetricMatrixFunctionCoefficient() { }
+};
+
+
 /** @brief Scalar coefficient defined as the product of two scalar coefficients
     or a scalar and a scalar coefficient. */
 class ProductCoefficient : public Coefficient
@@ -1171,8 +1276,8 @@ public:
                         double _alpha = 1.0, double _beta = 1.0);
 
    /** Constructor with scalar coefficients.
-       Result is _alpha * _A + _beta * _B */
-   VectorSumCoefficient(VectorCoefficient &_A, VectorCoefficient &_B,
+       Result is _alpha * _A + _beta * B_ */
+   VectorSumCoefficient(VectorCoefficient &_A, VectorCoefficient &B_,
                         Coefficient &_alpha, Coefficient &_beta);
 
    /// Reset the first vector coefficient
@@ -1201,7 +1306,7 @@ public:
    const Vector & GetA() const { return A; }
 
    /// Reset the second vector as a constant
-   void SetB(const Vector &_B) { B = _B; BCoef = NULL; }
+   void SetB(const Vector &B_) { B = B_; BCoef = NULL; }
    /// Return the second vector constant
    const Vector & GetB() const { return B; }
 

fem/complex_fem.cpp

@@ -1204,6 +1204,7 @@ ParSesquilinearForm::FormLinearSystem(const Array<int> &ess_tdof_list,
       });
       // Modify offdiagonal blocks (imaginary parts of the matrix) to conform
       // with standard essential BC treatment
+      ess_tdof_list.HostRead();
       if (A_i.Type() == Operator::Hypre_ParCSR)
       {
          HypreParMatrix * Ah;

fem/conduitdatacollection.cpp

@@ -981,6 +981,11 @@ ConduitDataCollection::SaveRootFile(int num_domains,
    n_root["file_pattern"]     = MeshFilePattern(relay_protocol);
    n_root["tree_pattern"]     = "";
 
+   // Add the time, time step, and cycle
+   n_root["blueprint_index/mesh/state/time"] = time;
+   n_root["blueprint_index/mesh/state/time_step"] = time_step;
+   n_root["blueprint_index/mesh/state/cycle"] = cycle;
+
    relay::io::save(n_root, RootFileName(), root_proto);
 }
 

fem/convergence.cpp

@@ -110,7 +110,8 @@ void ConvergenceStudy::AddL2Error(GridFunction *gf,
 
 void ConvergenceStudy::AddGf(GridFunction *gf, Coefficient *scalar_u,
                              VectorCoefficient *grad,
-                             Coefficient *ell_coeff, double Nu)
+                             Coefficient *ell_coeff,
+                             JumpScaling jump_scaling)
 {
    cont_type = gf->FESpace()->FEColl()->GetContType();
 
@@ -140,7 +141,7 @@ void ConvergenceStudy::AddGf(GridFunction *gf, Coefficient *scalar_u,
 
    if (cont_type == mfem::FiniteElementCollection::DISCONTINUOUS && ell_coeff)
    {
-      double DGErr = gf->ComputeDGFaceJumpError(scalar_u,ell_coeff,Nu);
+      double DGErr = gf->ComputeDGFaceJumpError(scalar_u,ell_coeff,jump_scaling);
       DGFaceErrors.Append(DGErr);
       // Compute the rate of convergence by:
       // rate = log (||u - u_h|| / ||u - u_{h/2}||)/log(2)
@@ -270,26 +271,26 @@ void ConvergenceStudy::Print(bool relative, std::ostream &out)
             }
             out << "\n";
          }
-         if (cont_type == 3 && fcounter)
+      }
+      if (cont_type == 3 && fcounter)
+      {
+         out << " -------------------------------------------" << "\n";
+         out << "            DG Face Jump Error          " << "\n";
+         out << " -------------------------------------------"
+             << "\n";
+         out << std::right<< std::setw(11)<< "DOFs "<< std::setw(13);
+         out << "Error ";
+         out << std::setw(15) << "Rate " << "\n";
+         out << " -------------------------------------------"
+             << "\n";
+         out << std::setprecision(4);
+         for (int i =0; i<fcounter; i++)
          {
-            out << " -------------------------------------------" << "\n";
-            out << "            DG Face Jump Error          " << "\n";
-            out << " -------------------------------------------"
-                << "\n";
-            out << std::right<< std::setw(11)<< "DOFs "<< std::setw(13);
-            out << "Error ";
-            out << std::setw(15) << "Rate " << "\n";
-            out << " -------------------------------------------"
-                << "\n";
-            out << std::setprecision(4);
-            for (int i =0; i<fcounter; i++)
-            {
-               out << std::right << std::setw(10)<< ndofs[i] << std::setw(16)
-                   << std::scientific << DGFaceErrors[i] << std::setw(13)
-                   << std::fixed << DGFaceRates[i] << "\n";
-            }
-            out << "\n";
+            out << std::right << std::setw(10)<< ndofs[i] << std::setw(16)
+                << std::scientific << DGFaceErrors[i] << std::setw(13)
+                << std::fixed << DGFaceRates[i] << "\n";
          }
+         out << "\n";
       }
    }
 }

fem/convergence.hpp

@@ -59,7 +59,8 @@ private:
                    VectorCoefficient *vector_u);
    void AddGf(GridFunction *gf, Coefficient *scalar_u,
               VectorCoefficient *grad=nullptr,
-              Coefficient *ell_coeff=nullptr, double Nu=1.0);
+              Coefficient *ell_coeff=nullptr,
+              JumpScaling jump_scaling = {1.0, JumpScaling::ONE_OVER_H});
    void AddGf(GridFunction *gf, VectorCoefficient *vector_u,
               VectorCoefficient *curl, Coefficient *div);
    // returns the L2-norm of scalar_u or vector_u
@@ -75,9 +76,10 @@ public:
    /// DG face jumps parameters
    void AddL2GridFunction(GridFunction *gf, Coefficient *scalar_u,
                           VectorCoefficient *grad=nullptr,
-                          Coefficient *ell_coeff=nullptr, double Nu=1.0)
+                          Coefficient *ell_coeff=nullptr,
+                          JumpScaling jump_scaling = {1.0, JumpScaling::ONE_OVER_H})
    {
-      AddGf(gf, scalar_u, grad, ell_coeff, Nu);
+      AddGf(gf, scalar_u, grad, ell_coeff, jump_scaling);
    }
 
    /// Add H1 GridFunction, the exact solution and possibly its gradient

fem/estimators.cpp

@@ -48,8 +48,373 @@ void L2ZienkiewiczZhuEstimator::ComputeEstimates()
    current_sequence = solution->FESpace()->GetMesh()->GetSequence();
 }
 
+
+KellyErrorEstimator::KellyErrorEstimator(BilinearFormIntegrator& di_,
+                                         ParGridFunction& sol_,
+                                         ParFiniteElementSpace& flux_fespace_,
+                                         const Array<int> &attributes_)
+   : attributes(attributes_)
+   , flux_integrator(&di_)
+   , solution(&sol_)
+   , flux_space(&flux_fespace_)
+   , own_flux_fespace(false)
+{
+   ResetCoefficientFunctions();
+}
+
+KellyErrorEstimator::KellyErrorEstimator(BilinearFormIntegrator& di_,
+                                         ParGridFunction& sol_,
+                                         ParFiniteElementSpace* flux_fespace_,
+                                         const Array<int> &attributes_)
+   : attributes(attributes_)
+   , flux_integrator(&di_)
+   , solution(&sol_)
+   , flux_space(flux_fespace_)
+   , own_flux_fespace(true)
+{
+   ResetCoefficientFunctions();
+}
+
+KellyErrorEstimator::~KellyErrorEstimator()
+{
+   if (own_flux_fespace)
+   {
+      delete flux_space;
+   }
+}
+
+void KellyErrorEstimator::ResetCoefficientFunctions()
+{
+   compute_element_coefficient = [](ParMesh* pmesh, const int e)
+   {
+      return 1.0;
+   };
+
+   compute_face_coefficient = [](ParMesh* pmesh, const int f,
+                                 const bool shared_face)
+   {
+      auto FT = [&]()
+      {
+         if (shared_face)
+         {
+            return pmesh->GetSharedFaceTransformations(f);
+         }
+         return pmesh->GetFaceElementTransformations(f);
+      }();
+      const auto order = FT->GetFE()->GetOrder();
+
+      // Poor man's face diameter.
+      double diameter = 0.0;
+
+      Vector p1(pmesh->SpaceDimension());
+      Vector p2(pmesh->SpaceDimension());
+      // NOTE: We have no direct access to vertices for shared faces,
+      // so we fall back to compute the positions from the element.
+      // This can also be modified to compute the diameter for non-linear
+      // geometries by sampling along geometry-specific lines.
+      auto vtx_intrule = Geometries.GetVertices(FT->GetGeometryType());
+      const auto nip = vtx_intrule->GetNPoints();
+      for (int i = 0; i < nip; i++)
+      {
+         // Evaluate flux vector at integration point
+         auto fip1 = vtx_intrule->IntPoint(i);
+         FT->Transform(fip1, p1);
+
+         for (int j = 0; j < nip; j++)
+         {
+            auto fip2 = vtx_intrule->IntPoint(j);
+            FT->Transform(fip2, p2);
+
+            diameter = std::max<double>(diameter, p2.DistanceTo(p1));
+         }
+      }
+      return diameter/(2.0*order);
+   };
+}
+
+void KellyErrorEstimator::ComputeEstimates()
+{
+   // Remarks:
+   // For some context you may have to consult the documentation of
+   // the FaceInfo class [1]. Also, the FaceElementTransformations
+   // documentation [2] may be helpful to grasp what is going on. Note
+   // that the FaceElementTransformations also works in the non-
+   // conforming case to transfer the gauss points from the slave to
+   // the master element.
+   // [1]
+   // https://github.com/mfem/mfem/blob/02d0bfe9c18ce049c3c93a6a4208080fcfc96991/mesh/mesh.hpp#L94
+   // [2]
+   // https://github.com/mfem/mfem/blob/02d0bfe9c18ce049c3c93a6a4208080fcfc96991/fem/eltrans.hpp#L435
+
+   flux_space->Update(false);
+
+   auto xfes      = solution->ParFESpace();
+   MFEM_ASSERT(xfes->GetVDim() == 1,
+               "Estimation for vector-valued problems not implemented yet.");
+   auto pmesh     = xfes->GetParMesh();
+
+   this->error_estimates.SetSize(xfes->GetNE());
+   this->error_estimates = 0.0;
+
+   // 1. Compute fluxes in discontinuous space
+   ParGridFunction flux(flux_space);
+   flux = 0.0;
+
+   // We pre-sort the array to speed up the search in the following loops.
+   if (attributes.Size())
+   {
+      attributes.Sort();
+   }
+
+   Array<int> xdofs, fdofs;
+   Vector el_x, el_f;
+   for (int e = 0; e < xfes->GetNE(); e++)
+   {
+      auto attr = xfes->GetAttribute(e);
+      if (attributes.Size() && attributes.FindSorted(attr) == -1)
+      {
+         continue;
+      }
+
+      xfes->GetElementVDofs(e, xdofs);
+      solution->GetSubVector(xdofs, el_x);
+
+      ElementTransformation* Transf = xfes->GetElementTransformation(e);
+      flux_integrator->ComputeElementFlux(*xfes->GetFE(e), *Transf, el_x,
+                                          *flux_space->GetFE(e), el_f, true);
+
+      flux_space->GetElementVDofs(e, fdofs);
+      flux.AddElementVector(fdofs, el_f);
+   }
+
+   // 2. Add error contribution from local interior faces
+   for (int f = 0; f < pmesh->GetNumFaces(); f++)
+   {
+      auto FT = pmesh->GetFaceElementTransformations(f);
+
+      auto &int_rule = IntRules.Get(FT->FaceGeom, 2 * xfes->GetFaceOrder(f));
+      const auto nip = int_rule.GetNPoints();
+
+      if (pmesh->FaceIsInterior(f))
+      {
+         int Inf1, Inf2, NCFace;
+         pmesh->GetFaceInfos(f, &Inf1, &Inf2, &NCFace);
+
+         // Convention
+         // * Conforming face: Face side with smaller element id handles
+         // the integration
+         // * Non-conforming face: The slave handles the integration.
+         // See FaceInfo documentation for details.
+         bool isNCSlave    = FT->Elem2No >= 0 && NCFace >= 0;
+         bool isConforming = FT->Elem2No >= 0 && NCFace == -1;
+         if ((FT->Elem1No < FT->Elem2No && isConforming) || isNCSlave)
+         {
+            if (attributes.Size() &&
+                (attributes.FindSorted(FT->Elem1->Attribute) == -1
+                 || attributes.FindSorted(FT->Elem2->Attribute) == -1))
+            {
+               continue;
+            }
+
+            IntegrationRule eir;
+            Vector jumps(nip);
+
+            // Integral over local half face on the side of e₁
+            // i.e. the numerical integration of ∫ flux ⋅ n dS₁
+            for (int i = 0; i < nip; i++)
+            {
+               // Evaluate flux at IP
+               auto &fip = int_rule.IntPoint(i);
+               IntegrationPoint ip;
+               FT->Loc1.Transform(fip, ip);
+
+               Vector val(flux_space->GetVDim());
+               flux.GetVectorValue(FT->Elem1No, ip, val);
+
+               // And build scalar product with normal
+               Vector normal(pmesh->SpaceDimension());
+               FT->Face->SetIntPoint(&fip);
+               if (pmesh->Dimension() == pmesh->SpaceDimension())
+               {
+                  CalcOrtho(FT->Face->Jacobian(), normal);
+               }
+               else
+               {
+                  Vector ref_normal(pmesh->Dimension());
+                  FT->Loc1.Transf.SetIntPoint(&fip);
+                  CalcOrtho(FT->Loc1.Transf.Jacobian(), ref_normal);
+                  auto &e1 = FT->GetElement1Transformation();
+                  e1.AdjugateJacobian().MultTranspose(ref_normal, normal);
+                  normal /= e1.Weight();
+               }
+               jumps(i) = val * normal * fip.weight * FT->Face->Weight();
+            }
+
+            // Subtract integral over half face of e₂
+            // i.e. the numerical integration of ∫ flux ⋅ n dS₂
+            for (int i = 0; i < nip; i++)
+            {
+               // Evaluate flux vector at IP
+               auto &fip = int_rule.IntPoint(i);
+               IntegrationPoint ip;
+               FT->Loc2.Transform(fip, ip);
+
+               Vector val(flux_space->GetVDim());
+               flux.GetVectorValue(FT->Elem2No, ip, val);
+
+               // And build scalar product with normal
+               Vector normal(pmesh->SpaceDimension());
+               FT->Face->SetIntPoint(&fip);
+               if (pmesh->Dimension() == pmesh->SpaceDimension())
+               {
+                  CalcOrtho(FT->Face->Jacobian(), normal);
+               }
+               else
+               {
+                  Vector ref_normal(pmesh->Dimension());
+                  FT->Loc1.Transf.SetIntPoint(&fip);
+                  CalcOrtho(FT->Loc1.Transf.Jacobian(), ref_normal);
+                  auto &e1 = FT->GetElement1Transformation();
+                  e1.AdjugateJacobian().MultTranspose(ref_normal, normal);
+                  normal /= e1.Weight();
+               }
+
+               jumps(i) -= val * normal * fip.weight * FT->Face->Weight();
+            }
+
+            // Finalize "local" L₂ contribution
+            for (int i = 0; i < nip; i++)
+            {
+               jumps(i) *= jumps(i);
+            }
+            auto h_k_face = compute_face_coefficient(pmesh, f, false);
+            double jump_integral = h_k_face*jumps.Sum();
+
+            // A local face is shared between two local elements, so we
+            // can get away with integrating the jump only once and add
+            // it to both elements. To minimize communication, the jump
+            // of shared faces is computed locally by each process.
+            error_estimates(FT->Elem1No) += jump_integral;
+            error_estimates(FT->Elem2No) += jump_integral;
+         }
+      }
+   }
+
+   // 3. Add error contribution from shared interior faces
+   // Synchronize face data.
+   flux.ExchangeFaceNbrData();
+
+   for (int sf = 0; sf < pmesh->GetNSharedFaces(); sf++)
+   {
+      auto FT = pmesh->GetSharedFaceTransformations(sf, true);
+      if (attributes.Size() &&
+          (attributes.FindSorted(FT->Elem1->Attribute) == -1
+           || attributes.FindSorted(FT->Elem2->Attribute) == -1))
+      {
+         continue;
+      }
+
+      auto &int_rule = IntRules.Get(FT->FaceGeom, 2 * xfes->GetFaceOrder(0));
+      const auto nip = int_rule.GetNPoints();
+
+      IntegrationRule eir;
+      Vector jumps(nip);
+
+      // Integral over local half face on the side of e₁
+      // i.e. the numerical integration of ∫ flux ⋅ n dS₁
+      for (int i = 0; i < nip; i++)
+      {
+         // Evaluate flux vector at integration point
+         auto &fip = int_rule.IntPoint(i);
+         IntegrationPoint ip;
+         FT->Loc1.Transform(fip, ip);
+
+         Vector val(flux_space->GetVDim());
+         flux.GetVectorValue(FT->Elem1No, ip, val);
+
+         Vector normal(pmesh->SpaceDimension());
+         FT->Face->SetIntPoint(&fip);
+         if (pmesh->Dimension() == pmesh->SpaceDimension())
+         {
+            CalcOrtho(FT->Face->Jacobian(), normal);
+         }
+         else
+         {
+            Vector ref_normal(pmesh->Dimension());
+            FT->Loc1.Transf.SetIntPoint(&fip);
+            CalcOrtho(FT->Loc1.Transf.Jacobian(), ref_normal);
+            auto &e1 = FT->GetElement1Transformation();
+            e1.AdjugateJacobian().MultTranspose(ref_normal, normal);
+            normal /= e1.Weight();
+         }
+
+         jumps(i) = val * normal * fip.weight * FT->Face->Weight();
+      }
+
+      // Subtract integral over non-local half face of e₂
+      // i.e. the numerical integration of ∫ flux ⋅ n dS₂
+      for (int i = 0; i < nip; i++)
+      {
+         // Evaluate flux vector at integration point
+         auto &fip = int_rule.IntPoint(i);
+         IntegrationPoint ip;
+         FT->Loc2.Transform(fip, ip);
+
+         Vector val(flux_space->GetVDim());
+         flux.GetVectorValue(FT->Elem2No, ip, val);
+
+         // Evaluate gauss point
+         Vector normal(pmesh->SpaceDimension());
+         FT->Face->SetIntPoint(&fip);
+         if (pmesh->Dimension() == pmesh->SpaceDimension())
+         {
+            CalcOrtho(FT->Face->Jacobian(), normal);
+         }
+         else
+         {
+            Vector ref_normal(pmesh->Dimension());
+            CalcOrtho(FT->Loc1.Transf.Jacobian(), ref_normal);
+            auto &e1 = FT->GetElement1Transformation();
+            e1.AdjugateJacobian().MultTranspose(ref_normal, normal);
+            normal /= e1.Weight();
+         }
+
+         jumps(i) -= val * normal * fip.weight * FT->Face->Weight();
+      }
+
+      // Finalize "local" L₂ contribution
+      for (int i = 0; i < nip; i++)
+      {
+         jumps(i) *= jumps(i);
+      }
+      auto h_k_face = compute_face_coefficient(pmesh, sf, true);
+      double jump_integral = h_k_face*jumps.Sum();
+
+      error_estimates(FT->Elem1No) += jump_integral;
+      // We skip "error_estimates(FT->Elem2No) += jump_integral"
+      // because the error is stored on the remote process and
+      // recomputed there.
+   }
+
+   // Finalize element errors
+   for (int e = 0; e < xfes->GetNE(); e++)
+   {
+      auto factor = compute_element_coefficient(pmesh, e);
+      // The sqrt belongs to the norm and hₑ to the indicator.
+      error_estimates(e) = sqrt(factor * error_estimates(e));
+   }
+
+   current_sequence = solution->FESpace()->GetMesh()->GetSequence();
+
+   // Finish by computing the global error.
+   double process_local_error = error_estimates.Sum();
+   MPI_Allreduce(&process_local_error, &total_error, 1, MPI_DOUBLE,
+                 MPI_SUM, xfes->GetComm());
+}
+
 #endif // MFEM_USE_MPI
 
+
 void LpErrorEstimator::ComputeEstimates()
 {
    MFEM_VERIFY(coef != NULL || vcoef != NULL,
@@ -64,6 +429,17 @@ void LpErrorEstimator::ComputeEstimates()
    {
       sol->ComputeElementLpErrors(local_norm_p, *vcoef, error_estimates);
    }
+#ifdef MFEM_USE_MPI
+   total_error = error_estimates.Sum();
+   auto pfes = dynamic_cast<ParFiniteElementSpace*>(sol->FESpace());
+   if (pfes)
+   {
+      auto process_local_error = total_error;
+      MPI_Allreduce(&process_local_error, &total_error, 1, MPI_DOUBLE,
+                    MPI_SUM, pfes->GetComm());
+   }
+#endif // MFEM_USE_MPI
+   total_error = pow(total_error, 1.0/local_norm_p);
    current_sequence = sol->FESpace()->GetMesh()->GetSequence();
 }
 

fem/estimators.hpp

@@ -12,6 +12,8 @@
 #ifndef MFEM_ERROR_ESTIMATORS
 #define MFEM_ERROR_ESTIMATORS
 
+#include <functional>
+
 #include "../config/config.hpp"
 #include "../linalg/vector.hpp"
 #include "bilinearform.hpp"
@@ -39,6 +41,11 @@ public:
 class ErrorEstimator : public AbstractErrorEstimator
 {
 public:
+   /// Return the total error from the last error estimate.
+   /** @note This method is optional for derived classes to override and the
+       base class implementation simply returns 0. */
+   virtual double GetTotalError() const { return 0.0; }
+
    /// Get a Vector with all element errors.
    virtual const Vector &GetLocalErrors() = 0;
 
@@ -148,8 +155,8 @@ public:
         own_flux_fes(false)
    { }
 
-   /** @brief Consider the coefficient in BilinearFormIntegrator to calculate the
-       fluxes for the error estimator.*/
+   /** @brief Consider the coefficient in BilinearFormIntegrator to calculate
+       the fluxes for the error estimator.*/
    void SetWithCoeff(bool w_coeff = true) { with_coeff = w_coeff; }
 
    /** @brief Enable/disable anisotropic estimates. To enable this option, the
@@ -166,10 +173,10 @@ public:
    void SetFluxAveraging(int fa) { flux_averaging = fa; }
 
    /// Return the total error from the last error estimate.
-   double GetTotalError() const { return total_error; }
+   virtual double GetTotalError() const override { return total_error; }
 
    /// Get a Vector with all element errors.
-   virtual const Vector &GetLocalErrors()
+   virtual const Vector &GetLocalErrors() override
    {
       if (MeshIsModified()) { ComputeEstimates(); }
       return error_estimates;
@@ -178,14 +185,14 @@ public:
    /** @brief Get an Array<int> with anisotropic flags for all mesh elements.
        Return an empty array when anisotropic estimates are not available or
        enabled. */
-   virtual const Array<int> &GetAnisotropicFlags()
+   virtual const Array<int> &GetAnisotropicFlags() override
    {
       if (MeshIsModified()) { ComputeEstimates(); }
       return aniso_flags;
    }
 
    /// Reset the error estimator.
-   virtual void Reset() { current_sequence = -1; }
+   virtual void Reset() override { current_sequence = -1; }
 
    /** @brief Destroy a ZienkiewiczZhuEstimator object. Destroys, if owned, the
        FiniteElementSpace, flux_space. */
@@ -292,17 +299,17 @@ public:
    void SetLocalErrorNormP(int p) { local_norm_p = p; }
 
    /// Return the total error from the last error estimate.
-   double GetTotalError() const { return total_error; }
+   virtual double GetTotalError() const override { return total_error; }
 
    /// Get a Vector with all element errors.
-   virtual const Vector &GetLocalErrors()
+   virtual const Vector &GetLocalErrors() override
    {
       if (MeshIsModified()) { ComputeEstimates(); }
       return error_estimates;
    }
 
    /// Reset the error estimator.
-   virtual void Reset() { current_sequence = -1; }
+   virtual void Reset() override { current_sequence = -1; }
 
    /** @brief Destroy a L2ZienkiewiczZhuEstimator object. Destroys, if owned,
        the FiniteElementSpace, flux_space. */
@@ -314,6 +321,7 @@ public:
 
 #endif // MFEM_USE_MPI
 
+
 /** @brief The LpErrorEstimator class compares the solution to a known
     coefficient.
 
@@ -332,6 +340,8 @@ protected:
    int local_norm_p;
    Vector error_estimates;
 
+   double total_error = 0.0;
+
    Coefficient * coef;
    VectorCoefficient * vcoef;
    GridFunction * sol;
@@ -383,10 +393,10 @@ public:
    void SetCoef(VectorCoefficient &A) { vcoef = &A; }
 
    /// Reset the error estimator.
-   virtual void Reset() { current_sequence = -1; }
+   virtual void Reset() override { current_sequence = -1; }
 
    /// Get a Vector with all element errors.
-   virtual const Vector &GetLocalErrors()
+   virtual const Vector &GetLocalErrors() override
    {
       if (MeshIsModified()) { ComputeEstimates(); }
       return error_estimates;
@@ -396,6 +406,174 @@ public:
    virtual ~LpErrorEstimator() {}
 };
 
+
+#ifdef MFEM_USE_MPI
+/** @brief The KellyErrorEstimator class provides a fast error indication
+    strategy for smooth scalar parallel problems.
+
+    The Kelly error indicator is based on the following papers:
+
+    Kelly, D. W., et al. "A posteriori error analysis and adaptive processes in
+    the finite element method: Part I—Error analysis." International journal for
+    numerical methods in engineering 19.11 (1983): 1593-1619.
+
+    De SR Gago, J. P., et al. "A posteriori error analysis and adaptive
+    processes in the finite element method: Part II—Adaptive mesh refinement."
+    International journal for numerical methods in engineering 19.11 (1983):
+    1621-1656.
+
+    It can be roughly described by:
+        ||∇(u-uₕ)||ₑ ≦ √( C hₑ ∑ₖ (hₖ ∫ |J[∇uₕ]|²) dS )
+    where "e" denotes an element, ||⋅||ₑ the corresponding local norm and k the
+    corresponding faces. u is the analytic solution and uₕ the discretized
+    solution. hₖ and hₑ are factors dependend on the face and element geometry.
+    J is the jump function, i.e. the difference between the limits at each point
+    for each side of the face. A custom method to compute hₖ can be provided. It
+    is also possible to estimate the error only on a subspace by feeding this
+    class an attribute array describing the subspace.
+
+    @note This algorithm is only for Poisson problems a proper error esimator.
+    The current implementation does not reflect this, because the "C" factor is
+    not included.
+    It further assumes that the approximation error at the boundary is small
+    enough, as the implementation ignores boundary faces.
+*/
+class KellyErrorEstimator final : public ErrorEstimator
+{
+public:
+   /// Function type to compute the local coefficient hₑ of an element.
+   using ElementCoefficientFunction =
+      std::function<double(ParMesh*, const int)>;
+   /** @brief Function type to compute the local coefficient hₖ of a face. The
+       third argument is true for shared faces and false for local faces. */
+   using FaceCoefficientFunction =
+      std::function<double(ParMesh*, const int, const bool)>;
+
+private:
+   int current_sequence = -1;
+
+   Vector error_estimates;
+
+   double total_error = 0.0;
+
+   Array<int> attributes;
+
+   /** @brief A method to compute hₑ on per-element basis.
+
+       This method weights the error approximation on the element level.
+
+       Defaults to hₑ=1.0.
+   */
+   ElementCoefficientFunction compute_element_coefficient;
+
+   /** @brief A method to compute hₖ on per-face basis.
+
+       This method weights the error approximation on the face level. The
+       background here is that classical Kelly error estimator implementations
+       approximate the geometrical characteristic hₖ with the face diameter,
+       which should be also be a possibility in this implementation.
+
+       Defaults to hₖ=diameter/2p.
+   */
+   FaceCoefficientFunction compute_face_coefficient;
+
+   BilinearFormIntegrator* flux_integrator; ///< Not owned.
+   ParGridFunction* solution;               ///< Not owned.
+
+   ParFiniteElementSpace*
+   flux_space; /**< @brief Ownership based on own_flux_fes. */
+   bool own_flux_fespace; ///< Ownership flag for flux_space.
+
+   /// Check if the mesh of the solution was modified.
+   bool MeshIsModified()
+   {
+      long mesh_sequence = solution->FESpace()->GetMesh()->GetSequence();
+      MFEM_ASSERT(mesh_sequence >= current_sequence,
+                  "improper mesh update sequence");
+      return (mesh_sequence > current_sequence);
+   }
+
+   /** @brief Compute the element error estimates.
+
+       Algorithm outline:
+       1. Compute flux field for each element
+       2. Add error contribution from local interior faces
+       3. Add error contribution from shared interior faces
+       4. Finalize by computing hₖ and scale errors.
+   */
+   void ComputeEstimates();
+
+public:
+   /** @brief Construct a new KellyErrorEstimator object for a scalar field.
+       @param di_         The bilinearform to compute the interface flux.
+       @param sol_        The solution field whose error is to be estimated.
+       @param flux_fes_   The finite element space for the interface flux.
+       @param attributes_ The attributes of the subdomain(s) for which the
+                          error should be estimated. An empty array results in
+                          estimating the error over the complete domain.
+   */
+   KellyErrorEstimator(BilinearFormIntegrator& di_, ParGridFunction& sol_,
+                       ParFiniteElementSpace& flux_fes_,
+                       const Array<int> &attributes_ = Array<int>());
+
+   /** @brief Construct a new KellyErrorEstimator object for a scalar field.
+       @param di_         The bilinearform to compute the interface flux.
+       @param sol_        The solution field whose error is to be estimated.
+       @param flux_fes_   The finite element space for the interface flux.
+       @param attributes_ The attributes of the subdomain(s) for which the
+                          error should be estimated. An empty array results in
+                          estimating the error over the complete domain.
+   */
+   KellyErrorEstimator(BilinearFormIntegrator& di_, ParGridFunction& sol_,
+                       ParFiniteElementSpace* flux_fes_,
+                       const Array<int> &attributes_ = Array<int>());
+
+   ~KellyErrorEstimator();
+
+   /// Get a Vector with all element errors.
+   const Vector& GetLocalErrors() override
+   {
+      if (MeshIsModified())
+      {
+         ComputeEstimates();
+      }
+      return error_estimates;
+   }
+
+   /// Reset the error estimator.
+   void Reset() override { current_sequence = -1; };
+
+   virtual double GetTotalError() const override { return total_error; }
+
+   /** @brief Change the method to compute hₑ on a per-element basis.
+       @param compute_element_coefficient_
+                        A function taking a mesh and an element index to
+                        compute the local hₑ for the element.
+   */
+   void SetElementCoefficientFunction(ElementCoefficientFunction
+                                      compute_element_coefficient_)
+   {
+      compute_element_coefficient = compute_element_coefficient_;
+   }
+
+   /** @brief Change the method to compute hₖ on a per-element basis.
+       @param compute_face_coefficient_
+                        A function taking a mesh and a face index to
+                        compute the local hₖ for the face.
+   */
+   void SetFaceCoefficientFunction(
+      FaceCoefficientFunction
+      compute_face_coefficient_)
+   {
+      compute_face_coefficient = compute_face_coefficient_;
+   }
+
+   /// Change the coefficients back to default as described above.
+   void ResetCoefficientFunctions();
+};
+
+#endif // MFEM_USE_MPI
+
 } // namespace mfem
 
 #endif // MFEM_ERROR_ESTIMATORS

fem/fe.cpp

@@ -435,7 +435,7 @@ void ScalarFiniteElement::ScalarLocalInterpolation(
    IntegrationPoint f_ip;
 
    const int fs = fine_fe.GetDof(), cs = this->GetDof();
-   I.SetSize(fs, cs );
+   I.SetSize(fs, cs);
    Vector fine_shape(fs), coarse_shape(cs);
    DenseMatrix fine_mass(fs), fine_coarse_mass(fs, cs); // initialized with 0
    const int ir_order = GetOrder() + fine_fe.GetOrder();
@@ -464,6 +464,44 @@ void ScalarFiniteElement::ScalarLocalInterpolation(
    }
 }
 
+void ScalarFiniteElement::ScalarLocalRestriction(
+   ElementTransformation &Trans, DenseMatrix &R,
+   const ScalarFiniteElement &coarse_fe) const
+{
+   // General "restriction", defined by L2 projection
+   double v[Geometry::MaxDim];
+   Vector vv (v, dim);
+   IntegrationPoint f_ip;
+
+   const int cs = coarse_fe.GetDof(), fs = this->GetDof();
+   R.SetSize(cs, fs);
+   Vector fine_shape(fs), coarse_shape(cs);
+   DenseMatrix coarse_mass(cs), coarse_fine_mass(cs, fs); // initialized with 0
+   const int ir_order = GetOrder() + coarse_fe.GetOrder();
+   const IntegrationRule &ir = IntRules.Get(coarse_fe.GetGeomType(), ir_order);
+
+   for (int i = 0; i < ir.GetNPoints(); i++)
+   {
+      const IntegrationPoint &ip = ir.IntPoint(i);
+      this->CalcShape(ip, fine_shape);
+      Trans.Transform(ip, vv);
+      f_ip.Set(v, dim);
+      coarse_fe.CalcShape(f_ip, coarse_shape);
+
+      AddMult_a_VVt(ip.weight, coarse_shape, coarse_mass);
+      AddMult_a_VWt(ip.weight, coarse_shape, fine_shape, coarse_fine_mass);
+   }
+
+   DenseMatrixInverse coarse_mass_inv(coarse_mass);
+   coarse_mass_inv.Mult(coarse_fine_mass, R);
+
+   if (map_type == INTEGRAL)
+   {
+      // assuming Trans is linear; this should be ok for all refinement types
+      Trans.SetIntPoint(&Geometries.GetCenter(geom_type));
+      R *= 1.0 / Trans.Weight();
+   }
+}
 const DofToQuad &ScalarFiniteElement::GetDofToQuad(const IntegrationRule &ir,
                                                    DofToQuad::Mode mode) const
 {
@@ -558,17 +596,22 @@ void NodalFiniteElement::ProjectCurl_2D(
    const FiniteElement &fe, ElementTransformation &Trans,
    DenseMatrix &curl) const
 {
-   MFEM_ASSERT(GetMapType() == FiniteElement::INTEGRAL, "");
-
    DenseMatrix curl_shape(fe.GetDof(), 1);
 
    curl.SetSize(dof, fe.GetDof());
    for (int i = 0; i < dof; i++)
    {
       fe.CalcCurlShape(Nodes.IntPoint(i), curl_shape);
+
+      double w = 1.0;
+      if (GetMapType() == FiniteElement::VALUE)
+      {
+         Trans.SetIntPoint(&Nodes.IntPoint(i));
+         w /= Trans.Weight();
+      }
       for (int j = 0; j < fe.GetDof(); j++)
       {
-         curl(i,j) = curl_shape(j,0);
+         curl(i,j) = w * curl_shape(j,0);
       }
    }
 }
@@ -684,17 +727,34 @@ void NodalFiniteElement::Project(
 {
    if (fe.GetRangeType() == SCALAR)
    {
-      MFEM_ASSERT(map_type == fe.GetMapType(), "");
-
       Vector shape(fe.GetDof());
 
       I.SetSize(dof, fe.GetDof());
-      for (int k = 0; k < dof; k++)
+      if (map_type == fe.GetMapType())
       {
-         fe.CalcShape(Nodes.IntPoint(k), shape);
-         for (int j = 0; j < shape.Size(); j++)
+         for (int k = 0; k < dof; k++)
          {
-            I(k,j) = (fabs(shape(j)) < 1e-12) ? 0.0 : shape(j);
+            fe.CalcShape(Nodes.IntPoint(k), shape);
+            for (int j = 0; j < shape.Size(); j++)
+            {
+               I(k,j) = (fabs(shape(j)) < 1e-12) ? 0.0 : shape(j);
+            }
+         }
+      }
+      else
+      {
+         for (int k = 0; k < dof; k++)
+         {
+            Trans.SetIntPoint(&Nodes.IntPoint(k));
+            fe.CalcPhysShape(Trans, shape);
+            if (map_type == INTEGRAL)
+            {
+               shape *= Trans.Weight();
+            }
+            for (int j = 0; j < shape.Size(); j++)
+            {
+               I(k,j) = (fabs(shape(j)) < 1e-12) ? 0.0 : shape(j);
+            }
          }
       }
    }
@@ -993,6 +1053,8 @@ void VectorFiniteElement::Project_RT(
 
          fe.CalcShape(ip, shape);
          Trans.SetIntPoint(&ip);
+         // Transform RT face normals from reference to physical space
+         // vk = adj(J)^T nk
          Trans.AdjugateJacobian().MultTranspose(nk + d2n[k]*dim, vk);
          if (fe.GetMapType() == INTEGRAL)
          {
@@ -1010,6 +1072,8 @@ void VectorFiniteElement::Project_RT(
             {
                s = 0.0;
             }
+            // Project scalar basis function multiplied by each coordinate
+            // direction onto the transformed face normals
             for (int d = 0; d < sdim; d++)
             {
                I(k,j+d*shape.Size()) = s*vk[d];
@@ -1019,7 +1083,31 @@ void VectorFiniteElement::Project_RT(
    }
    else
    {
-      mfem_error("VectorFiniteElement::Project_RT (fe version)");
+      int sdim = Trans.GetSpaceDim();
+      double vk[Geometry::MaxDim];
+      DenseMatrix vshape(fe.GetDof(), sdim);
+      Vector vshapenk(fe.GetDof());
+      const bool square_J = (dim == sdim);
+
+      I.SetSize(dof, fe.GetDof());
+      for (int k = 0; k < dof; k++)
+      {
+         const IntegrationPoint &ip = Nodes.IntPoint(k);
+
+         Trans.SetIntPoint(&ip);
+         // Transform RT face normals from reference to physical space
+         // vk = adj(J)^T nk
+         Trans.AdjugateJacobian().MultTranspose(nk + d2n[k]*dim, vk);
+         // Compute fe basis functions in physical space
+         fe.CalcVShape(Trans, vshape);
+         // Project fe basis functions onto transformed face normals
+         vshape.Mult(vk, vshapenk);
+         if (!square_J) { vshapenk /= Trans.Weight(); }
+         for (int j=0; j<vshapenk.Size(); j++)
+         {
+            I(k,j) = vshapenk(j);
+         }
+      }
    }
 }
 
@@ -1180,6 +1268,8 @@ void VectorFiniteElement::Project_ND(
 
          fe.CalcShape(ip, shape);
          Trans.SetIntPoint(&ip);
+         // Transform ND edge tengents from reference to physical space
+         // vk = J tk
          Trans.Jacobian().Mult(tk + d2t[k]*dim, vk);
          if (fe.GetMapType() == INTEGRAL)
          {
@@ -1197,6 +1287,8 @@ void VectorFiniteElement::Project_ND(
             {
                s = 0.0;
             }
+            // Project scalar basis function multiplied by each coordinate
+            // direction onto the transformed edge tangents
             for (int d = 0; d < sdim; d++)
             {
                I(k, j + d*shape.Size()) = s*vk[d];
@@ -1206,7 +1298,29 @@ void VectorFiniteElement::Project_ND(
    }
    else
    {
-      mfem_error("VectorFiniteElement::Project_ND (fe version)");
+      int sdim = Trans.GetSpaceDim();
+      double vk[Geometry::MaxDim];
+      DenseMatrix vshape(fe.GetDof(), sdim);
+      Vector vshapetk(fe.GetDof());
+
+      I.SetSize(dof, fe.GetDof());
+      for (int k = 0; k < dof; k++)
+      {
+         const IntegrationPoint &ip = Nodes.IntPoint(k);
+
+         Trans.SetIntPoint(&ip);
+         // Transform ND edge tangents from reference to physical space
+         // vk = J tk
+         Trans.Jacobian().Mult(tk + d2t[k]*dim, vk);
+         // Compute fe basis functions in physical space
+         fe.CalcVShape(Trans, vshape);
+         // Project fe basis functions onto transformed edge tangents
+         vshape.Mult(vk, vshapetk);
+         for (int j=0; j<vshapetk.Size(); j++)
+         {
+            I(k, j) = vshapetk(j);
+         }
+      }
    }
 }
 

fem/fe.hpp

@@ -504,12 +504,18 @@ public:
    /** @brief Given a coefficient and a transformation, compute its projection
        (approximation) in the local finite dimensional space in terms
        of the degrees of freedom. */
+   /** The approximation used to project is usually local interpolation of
+       degrees of freedom. The derived class could use other methods not
+       implemented yet, e.g. local L2 projection. */
    virtual void Project(Coefficient &coeff,
                         ElementTransformation &Trans, Vector &dofs) const;
 
    /** @brief Given a vector coefficient and a transformation, compute its
        projection (approximation) in the local finite dimensional space
        in terms of the degrees of freedom. (VectorFiniteElements) */
+   /** The approximation used to project is usually local interpolation of
+       degrees of freedom. The derived class could use other methods not
+       implemented yet, e.g. local L2 projection. */
    virtual void Project(VectorCoefficient &vc,
                         ElementTransformation &Trans, Vector &dofs) const;
 
@@ -666,7 +672,7 @@ public:
                                 const ScalarFiniteElement &fine_fe) const;
 
    /** @brief Get matrix @a I "Interpolation" defined through local
-       L2-projection in the space defined by the @a fine_fe.  */
+       L2-projection in the space defined by the @a fine_fe. */
    /** If the "fine" elements cannot represent all basis functions of the
        "coarse" element, then boundary values from different sub-elements are
        generally different. */
@@ -674,6 +680,15 @@ public:
                                  DenseMatrix &I,
                                  const ScalarFiniteElement &fine_fe) const;
 
+   /** @brief Get restriction matrix @a R defined through local L2-projection
+        in the space defined by the @a coarse_fe. */
+   /** If the "fine" elements cannot represent all basis functions of the
+       "coarse" element, then boundary values from different sub-elements are
+       generally different. */
+   void ScalarLocalRestriction(ElementTransformation &Trans,
+                               DenseMatrix &R,
+                               const ScalarFiniteElement &coarse_fe) const;
+
    virtual const DofToQuad &GetDofToQuad(const IntegrationRule &ir,
                                          DofToQuad::Mode mode) const;
 };
@@ -754,6 +769,10 @@ public:
                                       DenseMatrix &I) const
    { ScalarLocalInterpolation(Trans, I, *this); }
 
+   virtual void GetLocalRestriction(ElementTransformation &Trans,
+                                    DenseMatrix &R) const
+   { ScalarLocalRestriction(Trans, R, *this); }
+
    virtual void GetTransferMatrix(const FiniteElement &fe,
                                   ElementTransformation &Trans,
                                   DenseMatrix &I) const
@@ -800,11 +819,25 @@ protected:
    void CalcVShape_ND(ElementTransformation &Trans,
                       DenseMatrix &shape) const;
 
+   /** @brief Project a vector coefficient onto the RT basis functions
+       @param nk    Face normal vectors for this element type
+       @param d2n   Offset into nk for each degree of freedom
+       @param vc    Vector coefficient to be projected
+       @param Trans Transformation from reference to physical coordinates
+       @param dofs  Expansion coefficients for the approximation of vc
+   */
    void Project_RT(const double *nk, const Array<int> &d2n,
                    VectorCoefficient &vc, ElementTransformation &Trans,
                    Vector &dofs) const;
 
    /// Projects the vector of values given at FE nodes to RT space
+   /** Project vector values onto the RT basis functions
+       @param nk    Face normal vectors for this element type
+       @param d2n   Offset into nk for each degree of freedom
+       @param vc    Vector values at each interpolation point
+       @param Trans Transformation from reference to physical coordinates
+       @param dofs  Expansion coefficients for the approximation of vc
+   */
    void Project_RT(const double *nk, const Array<int> &d2n,
                    Vector &vc, ElementTransformation &Trans,
                    Vector &dofs) const;
@@ -814,6 +847,19 @@ protected:
       const double *nk, const Array<int> &d2n,
       MatrixCoefficient &mc, ElementTransformation &T, Vector &dofs) const;
 
+   /** @brief Project vector-valued basis functions onto the RT basis functions
+       @param nk    Face normal vectors for this element type
+       @param d2n   Offset into nk for each degree of freedom
+       @param fe    Vector-valued finite element basis
+       @param Trans Transformation from reference to physical coordinates
+       @param I     Expansion coefficients for the approximation of each basis
+                    function
+
+       Note: If the FiniteElement, fe, is scalar-valued the projection will
+             assume that a FiniteElementSpace is being used to define a vector
+             field using the scalar basis functions for each component of the
+             vector field.
+   */
    void Project_RT(const double *nk, const Array<int> &d2n,
                    const FiniteElement &fe, ElementTransformation &Trans,
                    DenseMatrix &I) const;
@@ -833,11 +879,25 @@ protected:
                        const FiniteElement &fe, ElementTransformation &Trans,
                        DenseMatrix &curl) const;
 
+   /** @brief Project a vector coefficient onto the ND basis functions
+       @param tk    Edge tangent vectors for this element type
+       @param d2t   Offset into tk for each degree of freedom
+       @param vc    Vector coefficient to be projected
+       @param Trans Transformation from reference to physical coordinates
+       @param dofs  Expansion coefficients for the approximation of vc
+   */
    void Project_ND(const double *tk, const Array<int> &d2t,
                    VectorCoefficient &vc, ElementTransformation &Trans,
                    Vector &dofs) const;
 
    /// Projects the vector of values given at FE nodes to ND space
+   /** Project vector values onto the ND basis functions
+       @param tk    Edge tangent vectors for this element type
+       @param d2t   Offset into tk for each degree of freedom
+       @param vc    Vector values at each interpolation point
+       @param Trans Transformation from reference to physical coordinates
+       @param dofs  Expansion coefficients for the approximation of vc
+   */
    void Project_ND(const double *tk, const Array<int> &d2t,
                    Vector &vc, ElementTransformation &Trans,
                    Vector &dofs) const;
@@ -847,6 +907,19 @@ protected:
       const double *tk, const Array<int> &d2t,
       MatrixCoefficient &mc, ElementTransformation &T, Vector &dofs) const;
 
+   /** @brief Project vector-valued basis functions onto the ND basis functions
+       @param tk    Edge tangent vectors for this element type
+       @param d2t   Offset into tk for each degree of freedom
+       @param fe    Vector-valued finite element basis
+       @param Trans Transformation from reference to physical coordinates
+       @param I     Expansion coefficients for the approximation of each basis
+                    function
+
+       Note: If the FiniteElement, fe, is scalar-valued the projection will
+             assume that a FiniteElementSpace is being used to define a vector
+             field using the scalar basis functions for each component of the
+             vector field.
+   */
    void Project_ND(const double *tk, const Array<int> &d2t,
                    const FiniteElement &fe, ElementTransformation &Trans,
                    DenseMatrix &I) const;

fem/fespace.cpp

@@ -2608,9 +2608,9 @@ const Operator &GridTransfer::MakeTrueOperator(
    else // Parallel() == true
    {
 #ifdef MFEM_USE_MPI
-      const SparseMatrix *out_R = fes_out.GetRestrictionMatrix();
       if (oper_type == Operator::Hypre_ParCSR)
       {
+         const SparseMatrix *out_R = fes_out.GetRestrictionMatrix();
          const ParFiniteElementSpace *pfes_in =
             dynamic_cast<const ParFiniteElementSpace *>(&fes_in);
          const ParFiniteElementSpace *pfes_out =
@@ -2638,6 +2638,7 @@ const Operator &GridTransfer::MakeTrueOperator(
       }
       else if (oper_type == Operator::ANY_TYPE)
       {
+         const Operator *out_R = fes_out.GetRestrictionOperator();
          t_oper.Reset(new TripleProductOperator(
                          out_R, &oper, fes_in.GetProlongationMatrix(),
                          false, false, false));

fem/fespace.hpp

@@ -330,6 +330,18 @@ public:
    virtual const Operator *GetProlongationMatrix() const
    { return GetConformingProlongation(); }
 
+   /// Return an operator that performs the transpose of GetRestrictionOperator
+   /** The returned operator is owned by the FiniteElementSpace. In serial this
+       is the same as GetProlongationMatrix() */
+   virtual const Operator *GetRestrictionTransposeOperator() const
+   { return GetConformingProlongation(); }
+
+   /// An abstract operator that performs the same action as GetRestrictionMatrix
+   /** In some cases this is an optimized matrix-free implementation. The
+       returned operator is owned by the FiniteElementSpace. */
+   virtual const Operator *GetRestrictionOperator() const
+   { return GetConformingRestriction(); }
+
    /// The returned SparseMatrix is owned by the FiniteElementSpace.
    virtual const SparseMatrix *GetRestrictionMatrix() const
    { return GetConformingRestriction(); }
@@ -571,7 +583,7 @@ public:
 
    /** @brief Returns pointer to the FiniteElement in the FiniteElementCollection
         associated with i'th element in the mesh object. */
-   const FiniteElement *GetFE(int i) const;
+   virtual const FiniteElement *GetFE(int i) const;
 
    /** @brief Returns pointer to the FiniteElement in the FiniteElementCollection
         associated with i'th boundary face in the mesh object. */
@@ -756,6 +768,9 @@ public:
    /// Return the total number of quadrature points.
    int GetSize() const { return size; }
 
+   /// Return the order of the quadrature rule(s) used by all elements.
+   int GetOrder() const { return order; }
+
    /// Returns the mesh
    inline Mesh *GetMesh() const { return mesh; }
 

fem/gridfunc.cpp

@@ -15,6 +15,10 @@
 #include "../mesh/nurbs.hpp"
 #include "../general/text.hpp"
 
+#ifdef MFEM_USE_MPI
+#include "pfespace.hpp"
+#endif
+
 #include <limits>
 #include <cstring>
 #include <string>
@@ -22,6 +26,7 @@
 #include <iostream>
 #include <algorithm>
 
+
 namespace mfem
 {
 
@@ -462,15 +467,26 @@ const
    fes->GetElementDofs(i, dofs);
    fes->DofsToVDofs(vdim-1, dofs);
    const FiniteElement *FElem = fes->GetFE(i);
-   MFEM_ASSERT(FElem->GetMapType() == FiniteElement::VALUE,
-               "invalid FE map type");
    int dof = FElem->GetDof();
    Vector DofVal(dof), loc_data(dof);
    GetSubVector(dofs, loc_data);
-   for (int k = 0; k < n; k++)
+   if (FElem->GetMapType() == FiniteElement::VALUE)
    {
-      FElem->CalcShape(ir.IntPoint(k), DofVal);
-      vals(k) = DofVal * loc_data;
+      for (int k = 0; k < n; k++)
+      {
+         FElem->CalcShape(ir.IntPoint(k), DofVal);
+         vals(k) = DofVal * loc_data;
+      }
+   }
+   else
+   {
+      ElementTransformation *Tr = fes->GetElementTransformation(i);
+      for (int k = 0; k < n; k++)
+      {
+         Tr->SetIntPoint(&ir.IntPoint(k));
+         FElem->CalcPhysShape(*Tr, DofVal);
+         vals(k) = DofVal * loc_data;
+      }
    }
 }
 
@@ -984,15 +1000,14 @@ void GridFunction::GetVectorValues(ElementTransformation &T,
 
    if (FElem->GetRangeType() == FiniteElement::SCALAR)
    {
-      MFEM_ASSERT(FElem->GetMapType() == FiniteElement::VALUE,
-                  "invalid FE map type");
       Vector shape(dof);
       int vdim = fes->GetVDim();
       vals.SetSize(vdim, nip);
       for (int j = 0; j < nip; j++)
       {
          const IntegrationPoint &ip = ir.IntPoint(j);
-         FElem->CalcShape(ip, shape);
+         T.SetIntPoint(&ip);
+         FElem->CalcPhysShape(T, shape);
 
          for (int k = 0; k < vdim; k++)
          {
@@ -1550,18 +1565,16 @@ void GridFunction::GetGradient(ElementTransformation &T, Vector &grad) const
    {
       case ElementTransformation::ELEMENT:
       {
-         const FiniteElement * fe = fes->GetFE(T.ElementNo);
+         const FiniteElement *fe = fes->GetFE(T.ElementNo);
          MFEM_ASSERT(fe->GetMapType() == FiniteElement::VALUE,
                      "invalid FE map type");
          int spaceDim = fes->GetMesh()->SpaceDimension();
          int dim = fe->GetDim(), dof = fe->GetDof();
          DenseMatrix dshape(dof, dim);
          Vector lval, gh(dim);
-         Array<int> dofs;
 
          grad.SetSize(spaceDim);
-         fes->GetElementDofs(T.ElementNo, dofs);
-         GetSubVector(dofs, lval);
+         GetElementDofValues(T.ElementNo, lval);
          fe->CalcDShape(T.GetIntPoint(), dshape);
          dshape.MultTranspose(lval, gh);
          T.InverseJacobian().MultTranspose(gh, grad);
@@ -1731,6 +1744,13 @@ void GridFunction::GetElementAverages(GridFunction &avgs) const
    }
 }
 
+void GridFunction::GetElementDofValues(int el, Vector &dof_vals) const
+{
+   Array<int> dof_idx;
+   fes->GetElementVDofs(el, dof_idx);
+   GetSubVector(dof_idx, dof_vals);
+}
+
 void GridFunction::ProjectGridFunction(const GridFunction &src)
 {
    Mesh *mesh = fes->GetMesh();
@@ -2777,10 +2797,11 @@ double GridFunction::ComputeDivError(
 }
 
 double GridFunction::ComputeDGFaceJumpError(Coefficient *exsol,
-                                            Coefficient *ell_coeff, double Nu,
+                                            Coefficient *ell_coeff,
+                                            class JumpScaling jump_scaling,
                                             const IntegrationRule *irs[])  const
 {
-   int fdof, dim, intorder, k;
+   int fdof, intorder, k;
    Mesh *mesh;
    const FiniteElement *fe;
    ElementTransformation *transf;
@@ -2791,20 +2812,24 @@ double GridFunction::ComputeDGFaceJumpError(Coefficient *exsol,
    double error = 0.0;
 
    mesh = fes->GetMesh();
-   dim = mesh->Dimension();
 
    for (int i = 0; i < mesh->GetNumFaces(); i++)
    {
-      face_elem_transf = mesh->GetFaceElementTransformations(i, 5);
-      int i1 = face_elem_transf->Elem1No;
-      int i2 = face_elem_transf->Elem2No;
+      int i1, i2;
+      mesh->GetFaceElements(i, &i1, &i2);
+      double h = mesh->GetElementSize(i1);
       intorder = fes->GetFE(i1)->GetOrder();
       if (i2 >= 0)
+      {
          if ( (k = fes->GetFE(i2)->GetOrder()) > intorder )
          {
             intorder = k;
          }
+         h = std::min(h, mesh->GetElementSize(i2));
+      }
+      int p = intorder;
       intorder = 2 * intorder;  // <-------------
+      face_elem_transf = mesh->GetFaceElementTransformations(i, 5);
       const IntegrationRule *ir;
       if (irs)
       {
@@ -2875,8 +2900,9 @@ double GridFunction::ComputeDGFaceJumpError(Coefficient *exsol,
       {
          const IntegrationPoint &ip = ir->IntPoint(j);
          transf->SetIntPoint(&ip);
-         error += (ip.weight * Nu * ell_coeff_val(j) *
-                   pow(transf->Weight(), 1.0-1.0/(dim-1)) *
+         double nu = jump_scaling.Eval(h, p);
+         error += (ip.weight * nu * ell_coeff_val(j) *
+                   transf->Weight() *
                    err_val(j) * err_val(j));
       }
    }
@@ -2884,6 +2910,15 @@ double GridFunction::ComputeDGFaceJumpError(Coefficient *exsol,
    return (error < 0.0) ? -sqrt(-error) : sqrt(error);
 }
 
+double GridFunction::ComputeDGFaceJumpError(Coefficient *exsol,
+                                            Coefficient *ell_coeff,
+                                            double Nu,
+                                            const IntegrationRule *irs[])  const
+{
+   return ComputeDGFaceJumpError(
+             exsol, ell_coeff, {Nu, JumpScaling::ONE_OVER_H}, irs);
+}
+
 double GridFunction::ComputeH1Error(Coefficient *exsol,
                                     VectorCoefficient *exgrad,
                                     Coefficient *ell_coef, double Nu,
@@ -2892,7 +2927,11 @@ double GridFunction::ComputeH1Error(Coefficient *exsol,
    double error1 = 0.0;
    double error2 = 0.0;
    if (norm_type & 1) { error1 = GridFunction::ComputeGradError(exgrad); }
-   if (norm_type & 2) { error2 = GridFunction::ComputeDGFaceJumpError(exsol,ell_coef,Nu); }
+   if (norm_type & 2)
+   {
+      error2 = GridFunction::ComputeDGFaceJumpError(
+                  exsol, ell_coef, {Nu, JumpScaling::ONE_OVER_H});
+   }
 
    return sqrt(error1 * error1 + error2 * error2);
 }
@@ -3670,7 +3709,7 @@ QuadratureFunction & QuadratureFunction::operator=(double value)
 
 QuadratureFunction & QuadratureFunction::operator=(const Vector &v)
 {
-   MFEM_ASSERT(qspace && v.Size() == qspace->GetSize(), "");
+   MFEM_ASSERT(qspace && v.Size() == this->Size(), "");
    Vector::operator=(v);
    return *this;
 }
@@ -3774,7 +3813,15 @@ double ZZErrorEstimator(BilinearFormIntegrator &blfi,
          }
       }
    }
-
+#ifdef MFEM_USE_MPI
+   auto pfes = dynamic_cast<ParFiniteElementSpace*>(ufes);
+   if (pfes)
+   {
+      auto process_local_error = total_error;
+      MPI_Allreduce(&process_local_error, &total_error, 1, MPI_DOUBLE,
+                    MPI_SUM, pfes->GetComm());
+   }
+#endif // MFEM_USE_MPI
    return std::sqrt(total_error);
 }
 

fem/gridfunc.hpp

@@ -325,6 +325,10 @@ public:
        Both FE spaces should be scalar and on the same mesh. */
    void GetElementAverages(GridFunction &avgs) const;
 
+   /** Sets the output vector @a dof_vals to the values of the degrees of
+       freedom of element @a el. */
+   virtual void GetElementDofValues(int el, Vector &dof_vals) const;
+
    /** Impose the given bounds on the function's DOFs while preserving its local
     *  integral (described in terms of the given weights) on the i'th element
     *  through SLBPQ optimization.
@@ -345,14 +349,30 @@ public:
        projection matrix. */
    void ProjectGridFunction(const GridFunction &src);
 
+   /** @brief Project @a coeff Coefficient to @a this GridFunction. The
+       projection computation depends on the choice of the FiniteElementSpace
+       #fes. Note that this is usually interpolation at the degrees of freedom
+       in each element (not L2 projection). */
    virtual void ProjectCoefficient(Coefficient &coeff);
 
+   /** @brief Project @a coeff Coefficient to @a this GridFunction, using one
+       element for each degree of freedom in @a dofs and nodal interpolation on
+       that element. */
    void ProjectCoefficient(Coefficient &coeff, Array<int> &dofs, int vd = 0);
 
+   /** @brief Project @a vcoeff VectorCoefficient to @a this GridFunction. The
+       projection computation depends on the choice of the FiniteElementSpace
+       #fes. Note that this is usually interpolation at the degrees of freedom
+       in each element (not L2 projection).*/
    void ProjectCoefficient(VectorCoefficient &vcoeff);
 
+   /** @brief Project @a vcoeff VectorCoefficient to @a this GridFunction, using
+       one element for each degree of freedom in @a dofs and nodal interpolation
+       on that element. */
    void ProjectCoefficient(VectorCoefficient &vcoeff, Array<int> &dofs);
 
+   /** @brief Analogous to the version with argument @a vcoeff VectorCoefficient
+       but using an array of scalar coefficients for each component. */
    void ProjectCoefficient(Coefficient *coeff[]);
 
    /** @brief Project a discontinuous vector coefficient as a grid function on
@@ -451,13 +471,22 @@ public:
    virtual double ComputeDivError(Coefficient *exdiv,
                                   const IntegrationRule *irs[] = NULL) const;
 
-   /// Returns the Face Jumps error for L2 elements
+   /// Returns the Face Jumps error for L2 elements. The error can be weighted
+   /// by a constant nu, by nu/h, or nu*p^2/h, depending on the value of
+   /// @a jump_scaling.
    virtual double ComputeDGFaceJumpError(Coefficient *exsol,
                                          Coefficient *ell_coeff,
-                                         double Nu,
+                                         class JumpScaling jump_scaling,
                                          const IntegrationRule *irs[] = NULL)
    const;
 
+   /// Returns the Face Jumps error for L2 elements, with 1/h scaling.
+   MFEM_DEPRECATED
+   double ComputeDGFaceJumpError(Coefficient *exsol,
+                                 Coefficient *ell_coeff,
+                                 double Nu,
+                                 const IntegrationRule *irs[] = NULL) const;
+
    /** This method is kept for backward compatibility.
 
        Returns either the H1-seminorm, or the DG face jumps error, or both
@@ -664,6 +693,32 @@ public:
     derived class ParGridFunction */
 std::ostream &operator<<(std::ostream &out, const GridFunction &sol);
 
+/// Class used to specify how the jump terms in
+/// GridFunction::ComputeDGFaceJumpError are scaled.
+class JumpScaling
+{
+public:
+   enum JumpScalingType
+   {
+      CONSTANT,
+      ONE_OVER_H,
+      P_SQUARED_OVER_H
+   };
+private:
+   double nu;
+   JumpScalingType type;
+public:
+   JumpScaling(double nu_=1.0, JumpScalingType type_=CONSTANT)
+      : nu(nu_), type(type_) { }
+   double Eval(double h, int p) const
+   {
+      double val = nu;
+      if (type != CONSTANT) { val /= h; }
+      if (type == P_SQUARED_OVER_H) { val *= p*p; }
+      return val;
+   }
+};
+
 
 /** @brief Class representing a function through its values (scalar or vector)
     at quadrature points. */
@@ -752,7 +807,8 @@ public:
 
    /// Copy the data from @a v.
    /** The size of @a v must be equal to the size of the associated
-       QuadratureSpace #qspace. */
+       QuadratureSpace #qspace times the QuadratureFunction dimension
+       i.e. QuadratureFunction::Size(). */
    QuadratureFunction &operator=(const Vector &v);
 
    /// Copy assignment. Only the data of the base class Vector is copied.

fem/lininteg.cpp

@@ -10,8 +10,8 @@
 // CONTRIBUTING.md for details.
 
 
-#include <cmath>
 #include "fem.hpp"
+#include <cmath>
 
 namespace mfem
 {

fem/multigrid.cpp

@@ -14,11 +14,40 @@
 namespace mfem
 {
 
-Multigrid::Multigrid(const FiniteElementSpaceHierarchy& fespaces_)
-   : fespaces(fespaces_), cycleType(CycleType::VCYCLE), preSmoothingSteps(1),
-     postSmoothingSteps(1)
+Multigrid::Multigrid()
+   : cycleType(CycleType::VCYCLE), preSmoothingSteps(1), postSmoothingSteps(1)
 {}
 
+Multigrid::Multigrid(const Array<Operator*>& operators_,
+                     const Array<Solver*>& smoothers_,
+                     const Array<Operator*>& prolongations_,
+                     const Array<bool>& ownedOperators_,
+                     const Array<bool>& ownedSmoothers_,
+                     const Array<bool>& ownedProlongations_)
+   : Solver(operators_.Last()->NumRows()), cycleType(CycleType::VCYCLE),
+     preSmoothingSteps(1), postSmoothingSteps(1),
+     X(operators_.Size()), Y(X.Size()), R(X.Size()), Z(X.Size())
+{
+   operators_.Copy(operators);
+   smoothers_.Copy(smoothers);
+   prolongations_.Copy(prolongations);
+   ownedOperators_.Copy(ownedOperators);
+   ownedSmoothers_.Copy(ownedSmoothers);
+   ownedProlongations_.Copy(ownedProlongations);
+
+   for (int level = 0; level < operators.Size(); ++level)
+   {
+      X[level] = new Vector(operators[level]->NumRows());
+      *X[level] = 0.0;
+      Y[level] = new Vector(operators[level]->NumRows());
+      *Y[level] = 0.0;
+      R[level] = new Vector(operators[level]->NumRows());
+      *R[level] = 0.0;
+      Z[level] = new Vector(operators[level]->NumRows());
+      *Z[level] = 0.0;
+   }
+}
+
 Multigrid::~Multigrid()
 {
    for (int i = 0; i < operators.Size(); ++i)
@@ -37,26 +66,21 @@ Multigrid::~Multigrid()
       delete Z[i];
    }
 
+   for (int i = 0; i < prolongations.Size(); ++i)
+   {
+      if (ownedProlongations[i])
+      {
+         delete prolongations[i];
+      }
+   }
+
    operators.DeleteAll();
    smoothers.DeleteAll();
+   prolongations.DeleteAll();
    X.DeleteAll();
    Y.DeleteAll();
    R.DeleteAll();
    Z.DeleteAll();
-
-   for (int i = 0; i < bfs.Size(); ++i)
-   {
-      delete bfs[i];
-   }
-
-   bfs.DeleteAll();
-
-   for (int i = 0; i < essentialTrueDofs.Size(); ++i)
-   {
-      delete essentialTrueDofs[i];
-   }
-
-   essentialTrueDofs.DeleteAll();
 }
 
 void Multigrid::AddLevel(Operator* opr, Solver* smoother, bool ownOperator,
@@ -168,8 +192,7 @@ void Multigrid::Cycle(int level) const
    subtract(*X[level], *R[level], *R[level]);
 
    // Restrict residual
-   fespaces.GetProlongationAtLevel(level - 1)->MultTranspose(*R[level],
-                                                             *X[level - 1]);
+   GetProlongationAtLevel(level - 1)->MultTranspose(*R[level], *X[level - 1]);
 
    // Init zeros
    *Y[level - 1] = 0.0;
@@ -186,7 +209,7 @@ void Multigrid::Cycle(int level) const
    }
 
    // Prolongate
-   fespaces.GetProlongationAtLevel(level - 1)->Mult(*Y[level - 1], *R[level]);
+   GetProlongationAtLevel(level - 1)->Mult(*Y[level - 1], *R[level]);
 
    // Add update
    *Y[level] += *R[level];
@@ -198,16 +221,44 @@ void Multigrid::Cycle(int level) const
    }
 }
 
-void Multigrid::FormFineLinearSystem(Vector& x, Vector& b, OperatorHandle& A,
-                                     Vector& X, Vector& B)
+const Operator* Multigrid::GetProlongationAtLevel(int level) const
+{
+   return prolongations[level];
+}
+
+GeometricMultigrid::~GeometricMultigrid()
+{
+   for (int i = 0; i < bfs.Size(); ++i)
+   {
+      delete bfs[i];
+   }
+
+   bfs.DeleteAll();
+
+   for (int i = 0; i < essentialTrueDofs.Size(); ++i)
+   {
+      delete essentialTrueDofs[i];
+   }
+
+   essentialTrueDofs.DeleteAll();
+}
+
+void GeometricMultigrid::FormFineLinearSystem(Vector& x, Vector& b,
+                                              OperatorHandle& A,
+                                              Vector& X, Vector& B)
 {
    bfs.Last()->FormLinearSystem(*essentialTrueDofs.Last(), x, b, A, X, B);
 }
 
-void Multigrid::RecoverFineFEMSolution(const Vector& X, const Vector& b,
-                                       Vector& x)
+void GeometricMultigrid::RecoverFineFEMSolution(const Vector& X,
+                                                const Vector& b, Vector& x)
 {
    bfs.Last()->RecoverFEMSolution(X, b, x);
 }
 
+const Operator* GeometricMultigrid::GetProlongationAtLevel(int level) const
+{
+   return fespaces.GetProlongationAtLevel(level);
+}
+
 } // namespace mfem

fem/multigrid.hpp

@@ -32,16 +32,13 @@ public:
    };
 
 protected:
-   const FiniteElementSpaceHierarchy& fespaces;
-   Array<Array<int>*> essentialTrueDofs;
-   Array<BilinearForm*> bfs;
-
-private:
    Array<Operator*> operators;
    Array<Solver*> smoothers;
+   Array<Operator*> prolongations;
 
    Array<bool> ownedOperators;
    Array<bool> ownedSmoothers;
+   Array<bool> ownedProlongations;
 
    CycleType cycleType;
    int preSmoothingSteps;
@@ -53,8 +50,16 @@ private:
    mutable Array<Vector*> Z;
 
 public:
-   /// Constructs an empty multigrid for the given FiniteElementSpaceHierarchy
-   Multigrid(const FiniteElementSpaceHierarchy& fespaces_);
+   /// Constructs an empty multigrid hierarchy.
+   Multigrid();
+
+   /// Constructs a multigrid hierarchy from the given inputs.
+   /** Inputs include operators and smoothers on all levels, prolongation
+       operators that go from coarser to finer levels, and ownership of the
+       given operators, smoothers, and prolongations. */
+   Multigrid(const Array<Operator*>& operators_, const Array<Solver*>& smoothers_,
+             const Array<Operator*>& prolongations_, const Array<bool>& ownedOperators_,
+             const Array<bool>& ownedSmoothers_, const Array<bool>& ownedProlongations_);
 
    /// Destructor
    virtual ~Multigrid();
@@ -89,7 +94,7 @@ public:
    /// Returns smoother at given level
    Solver* GetSmootherAtLevel(int level);
 
-   /// Set the cycle type and number of pre- and post-smoothing steps used by Mult
+   /// Set cycle type and number of pre- and post-smoothing steps used by Mult
    void SetCycleType(CycleType cycleType_, int preSmoothingSteps_,
                      int postSmoothingSteps_);
 
@@ -99,7 +104,36 @@ public:
    /// Not supported for multigrid
    virtual void SetOperator(const Operator& op) override;
 
-   /// Form the linear system A X = B, corresponding to the operator on the finest level
+private:
+   /// Application of a smoothing step at particular level
+   void SmoothingStep(int level, bool transpose) const;
+
+   /// Application of a multigrid cycle at particular level
+   void Cycle(int level) const;
+
+   /// Returns prolongation operator at given level
+   virtual const Operator* GetProlongationAtLevel(int level) const;
+};
+
+/// Geometric multigrid associated with a hierarchy of finite element spaces
+class GeometricMultigrid : public Multigrid
+{
+protected:
+   const FiniteElementSpaceHierarchy& fespaces;
+   Array<Array<int>*> essentialTrueDofs;
+   Array<BilinearForm*> bfs;
+
+public:
+   /** Construct an empty multigrid object for the given finite element space
+       hierarchy @a fespaces_ */
+   GeometricMultigrid(const FiniteElementSpaceHierarchy& fespaces_)
+      : Multigrid(), fespaces(fespaces_) { }
+
+   /// Destructor
+   virtual ~GeometricMultigrid();
+
+   /** Form the linear system A X = B, corresponding to the operator on the
+       finest level of the geometric multigrid hierarchy */
    void FormFineLinearSystem(Vector& x, Vector& b, OperatorHandle& A, Vector& X,
                              Vector& B);
 
@@ -107,11 +141,8 @@ public:
    void RecoverFineFEMSolution(const Vector& X, const Vector& b, Vector& x);
 
 private:
-   /// Application of a smoothing step at particular level
-   void SmoothingStep(int level, bool transpose) const;
-
-   /// Application of a cycle at particular level
-   void Cycle(int level) const;
+   /// Returns prolongation operator at given level
+   virtual const Operator* GetProlongationAtLevel(int level) const override;
 };
 
 } // namespace mfem

fem/pbilinearform.cpp

@@ -565,6 +565,38 @@ HypreParMatrix* ParDiscreteLinearOperator::ParallelAssemble() const
    return RAP;
 }
 
+void ParDiscreteLinearOperator::ParallelAssemble(OperatorHandle &A)
+{
+   // construct the rectangular block-diagonal matrix dA
+   OperatorHandle dA(A.Type());
+   dA.MakeRectangularBlockDiag(domain_fes->GetComm(),
+                               range_fes->GlobalVSize(),
+                               domain_fes->GlobalVSize(),
+                               range_fes->GetDofOffsets(),
+                               domain_fes->GetDofOffsets(),
+                               mat);
+
+   OperatorHandle R_test_transpose(A.Type()), P_trial(A.Type());
+
+   // TODO - construct the Dof_TrueDof_Matrix directly in the required format.
+   R_test_transpose.ConvertFrom(range_fes->Dof_TrueDof_Matrix());
+   P_trial.ConvertFrom(domain_fes->Dof_TrueDof_Matrix());
+
+   A.MakeRAP(R_test_transpose, dA, P_trial);
+}
+
+void ParDiscreteLinearOperator::FormRectangularSystemMatrix(OperatorHandle &A)
+{
+   if (ext)
+   {
+      Array<int> empty;
+      ext->FormRectangularSystemOperator(empty, empty, A);
+      return;
+   }
+
+   mfem_error("not implemented!");
+}
+
 void ParDiscreteLinearOperator::GetParBlocks(Array2D<HypreParMatrix *> &blocks)
 const
 {

fem/pbilinearform.hpp

@@ -160,6 +160,9 @@ public:
    /// Get the parallel finite element space prolongation matrix
    virtual const Operator *GetProlongation() const
    { return pfes->GetProlongationMatrix(); }
+   /// Get the transpose of GetRestriction, useful for matrix-free RAP
+   virtual const Operator *GetRestrictionTranspose() const
+   { return pfes->GetRestrictionTransposeOperator(); }
    /// Get the parallel finite element space restriction matrix
    virtual const Operator *GetRestriction() const
    { return pfes->GetRestrictionMatrix(); }
@@ -301,10 +304,18 @@ public:
    /// Returns the matrix "assembled" on the true dofs
    HypreParMatrix *ParallelAssemble() const;
 
+   /** @brief Returns the matrix assembled on the true dofs, i.e.
+       @a A = R_test A_local P_trial, in the format (type id) specified by
+       @a A. */
+   void ParallelAssemble(OperatorHandle &A);
+
    /** Extract the parallel blocks corresponding to the vector dimensions of the
        domain and range parallel finite element spaces */
    void GetParBlocks(Array2D<HypreParMatrix *> &blocks) const;
 
+   /** @brief Return in @a A a parallel (on truedofs) version of this operator. */
+   virtual void FormRectangularSystemMatrix(OperatorHandle &A);
+
    virtual ~ParDiscreteLinearOperator() { }
 };
 

fem/pfespace.cpp

@@ -101,6 +101,8 @@ void ParFiniteElementSpace::ParInit(ParMesh *pm)
 
    P = NULL;
    Pconf = NULL;
+   Rconf = NULL;
+   R_transpose = NULL;
    R = NULL;
 
    num_face_nbr_dofs = -1;
@@ -499,6 +501,12 @@ void ParFiniteElementSpace::GetFaceDofs(int i, Array<int> &dofs) const
    }
 }
 
+const FiniteElement *ParFiniteElementSpace::GetFE(int i) const
+{
+   int ne = mesh->GetNE();
+   if (i >= ne) { return GetFaceNbrFE(i - ne); }
+   else { return FiniteElementSpace::GetFE(i); }
+}
 
 const Operator *ParFiniteElementSpace::GetFaceRestriction(
    ElementDofOrdering e_ordering, FaceType type, L2FaceValues mul) const
@@ -921,6 +929,45 @@ const Operator *ParFiniteElementSpace::GetProlongationMatrix() const
    }
 }
 
+const Operator *ParFiniteElementSpace::GetRestrictionOperator() const
+{
+   if (Conforming())
+   {
+      if (Rconf) { return Rconf; }
+
+      if (NRanks == 1)
+      {
+         R_transpose = new IdentityOperator(GetTrueVSize());
+      }
+      else
+      {
+         if (!Device::Allows(Backend::DEVICE_MASK))
+         {
+            R_transpose = new ConformingProlongationOperator(*this, true);
+         }
+         else
+         {
+            R_transpose =
+               new DeviceConformingProlongationOperator(*this, true);
+         }
+      }
+      Rconf = new TransposeOperator(R_transpose);
+      return Rconf;
+   }
+   else
+   {
+      Dof_TrueDof_Matrix();
+      R_transpose = new TransposeOperator(R);
+      return R;
+   }
+}
+
+const Operator *ParFiniteElementSpace::GetRestrictionTransposeOperator() const
+{
+   GetRestrictionOperator();
+   return R_transpose;
+}
+
 void ParFiniteElementSpace::ExchangeFaceNbrData()
 {
    if (num_face_nbr_dofs >= 0) { return; }
@@ -2834,6 +2881,8 @@ void ParFiniteElementSpace::Destroy()
 
    delete P; P = NULL;
    delete Pconf; Pconf = NULL;
+   delete Rconf; Rconf = NULL;
+   delete R_transpose; R_transpose = NULL;
    delete R; R = NULL;
 
    delete gcomm; gcomm = NULL;
@@ -2959,12 +3008,12 @@ void ParFiniteElementSpace::Update(bool want_transform)
    }
 }
 
-
 ConformingProlongationOperator::ConformingProlongationOperator(
-   const ParFiniteElementSpace &pfes)
+   const ParFiniteElementSpace &pfes, bool local_)
    : Operator(pfes.GetVSize(), pfes.GetTrueVSize()),
      external_ldofs(),
-     gc(pfes.GroupComm())
+     gc(pfes.GroupComm()),
+     local(local_)
 {
    MFEM_VERIFY(pfes.Conforming(), "");
    const Table &group_ldof = gc.GroupLDofTable();
@@ -3013,7 +3062,14 @@ void ConformingProlongationOperator::Mult(const Vector &x, Vector &y) const
    const int m = external_ldofs.Size();
 
    const int in_layout = 2; // 2 - input is ltdofs array
-   gc.BcastBegin(const_cast<double*>(xdata), in_layout);
+   if (local)
+   {
+      y = 0.0;
+   }
+   else
+   {
+      gc.BcastBegin(const_cast<double*>(xdata), in_layout);
+   }
 
    int j = 0;
    for (int i = 0; i < m; i++)
@@ -3025,7 +3081,10 @@ void ConformingProlongationOperator::Mult(const Vector &x, Vector &y) const
    std::copy(xdata+j-m, xdata+Width(), ydata+j);
 
    const int out_layout = 0; // 0 - output is ldofs array
-   gc.BcastEnd(ydata, out_layout);
+   if (!local)
+   {
+      gc.BcastEnd(ydata, out_layout);
+   }
 }
 
 void ConformingProlongationOperator::MultTranspose(
@@ -3038,7 +3097,10 @@ void ConformingProlongationOperator::MultTranspose(
    double *ydata = y.HostWrite();
    const int m = external_ldofs.Size();
 
-   gc.ReduceBegin(xdata);
+   if (!local)
+   {
+      gc.ReduceBegin(xdata);
+   }
 
    int j = 0;
    for (int i = 0; i < m; i++)
@@ -3050,13 +3112,18 @@ void ConformingProlongationOperator::MultTranspose(
    std::copy(xdata+j, xdata+Height(), ydata+j-m);
 
    const int out_layout = 2; // 2 - output is an array on all ltdofs
-   gc.ReduceEnd<double>(ydata, out_layout, GroupCommunicator::Sum);
+   if (!local)
+   {
+      gc.ReduceEnd<double>(ydata, out_layout, GroupCommunicator::Sum);
+   }
 }
 
 DeviceConformingProlongationOperator::DeviceConformingProlongationOperator(
-   const ParFiniteElementSpace &pfes) :
+   const ParFiniteElementSpace &pfes,
+   bool local_) :
    ConformingProlongationOperator(pfes),
-   mpi_gpu_aware(Device::GetGPUAwareMPI())
+   mpi_gpu_aware(Device::GetGPUAwareMPI()),
+   local(local_)
 {
    MFEM_ASSERT(pfes.Conforming(), "internal error");
    const SparseMatrix *R = pfes.GetRestrictionMatrix();
@@ -3173,32 +3240,42 @@ void DeviceConformingProlongationOperator::Mult(const Vector &x,
                                                 Vector &y) const
 {
    const GroupTopology &gtopo = gc.GetGroupTopology();
-   BcastBeginCopy(x); // copy to 'shr_buf'
    int req_counter = 0;
-   for (int nbr = 1; nbr < gtopo.GetNumNeighbors(); nbr++)
+   if (local)
    {
-      const int send_offset = shr_buf_offsets[nbr];
-      const int send_size = shr_buf_offsets[nbr+1] - send_offset;
-      if (send_size > 0)
+      y = 0.0;
+   }
+   else
+   {
+      BcastBeginCopy(x); // copy to 'shr_buf'
+      for (int nbr = 1; nbr < gtopo.GetNumNeighbors(); nbr++)
       {
-         auto send_buf = mpi_gpu_aware ? shr_buf.Read() : shr_buf.HostRead();
-         MPI_Isend(send_buf + send_offset, send_size, MPI_DOUBLE,
-                   gtopo.GetNeighborRank(nbr), 41822,
-                   gtopo.GetComm(), &requests[req_counter++]);
-      }
-      const int recv_offset = ext_buf_offsets[nbr];
-      const int recv_size = ext_buf_offsets[nbr+1] - recv_offset;
-      if (recv_size > 0)
-      {
-         auto recv_buf = mpi_gpu_aware ? ext_buf.Write() : ext_buf.HostWrite();
-         MPI_Irecv(recv_buf + recv_offset, recv_size, MPI_DOUBLE,
-                   gtopo.GetNeighborRank(nbr), 41822,
-                   gtopo.GetComm(), &requests[req_counter++]);
+         const int send_offset = shr_buf_offsets[nbr];
+         const int send_size = shr_buf_offsets[nbr+1] - send_offset;
+         if (send_size > 0)
+         {
+            auto send_buf = mpi_gpu_aware ? shr_buf.Read() : shr_buf.HostRead();
+            MPI_Isend(send_buf + send_offset, send_size, MPI_DOUBLE,
+                      gtopo.GetNeighborRank(nbr), 41822,
+                      gtopo.GetComm(), &requests[req_counter++]);
+         }
+         const int recv_offset = ext_buf_offsets[nbr];
+         const int recv_size = ext_buf_offsets[nbr+1] - recv_offset;
+         if (recv_size > 0)
+         {
+            auto recv_buf = mpi_gpu_aware ? ext_buf.Write() : ext_buf.HostWrite();
+            MPI_Irecv(recv_buf + recv_offset, recv_size, MPI_DOUBLE,
+                      gtopo.GetNeighborRank(nbr), 41822,
+                      gtopo.GetComm(), &requests[req_counter++]);
+         }
       }
    }
    BcastLocalCopy(x, y);
-   MPI_Waitall(req_counter, requests, MPI_STATUSES_IGNORE);
-   BcastEndCopy(y); // copy from 'ext_buf'
+   if (!local)
+   {
+      MPI_Waitall(req_counter, requests, MPI_STATUSES_IGNORE);
+      BcastEndCopy(y); // copy from 'ext_buf'
+   }
 }
 
 DeviceConformingProlongationOperator::~DeviceConformingProlongationOperator()
@@ -3261,32 +3338,38 @@ void DeviceConformingProlongationOperator::MultTranspose(const Vector &x,
                                                          Vector &y) const
 {
    const GroupTopology &gtopo = gc.GetGroupTopology();
-   ReduceBeginCopy(x); // copy to 'ext_buf'
    int req_counter = 0;
-   for (int nbr = 1; nbr < gtopo.GetNumNeighbors(); nbr++)
+   if (!local)
    {
-      const int send_offset = ext_buf_offsets[nbr];
-      const int send_size = ext_buf_offsets[nbr+1] - send_offset;
-      if (send_size > 0)
+      ReduceBeginCopy(x); // copy to 'ext_buf'
+      for (int nbr = 1; nbr < gtopo.GetNumNeighbors(); nbr++)
       {
-         auto send_buf = mpi_gpu_aware ? ext_buf.Read() : ext_buf.HostRead();
-         MPI_Isend(send_buf + send_offset, send_size, MPI_DOUBLE,
-                   gtopo.GetNeighborRank(nbr), 41823,
-                   gtopo.GetComm(), &requests[req_counter++]);
-      }
-      const int recv_offset = shr_buf_offsets[nbr];
-      const int recv_size = shr_buf_offsets[nbr+1] - recv_offset;
-      if (recv_size > 0)
-      {
-         auto recv_buf = mpi_gpu_aware ? shr_buf.Write() : shr_buf.HostWrite();
-         MPI_Irecv(recv_buf + recv_offset, recv_size, MPI_DOUBLE,
-                   gtopo.GetNeighborRank(nbr), 41823,
-                   gtopo.GetComm(), &requests[req_counter++]);
+         const int send_offset = ext_buf_offsets[nbr];
+         const int send_size = ext_buf_offsets[nbr+1] - send_offset;
+         if (send_size > 0)
+         {
+            auto send_buf = mpi_gpu_aware ? ext_buf.Read() : ext_buf.HostRead();
+            MPI_Isend(send_buf + send_offset, send_size, MPI_DOUBLE,
+                      gtopo.GetNeighborRank(nbr), 41823,
+                      gtopo.GetComm(), &requests[req_counter++]);
+         }
+         const int recv_offset = shr_buf_offsets[nbr];
+         const int recv_size = shr_buf_offsets[nbr+1] - recv_offset;
+         if (recv_size > 0)
+         {
+            auto recv_buf = mpi_gpu_aware ? shr_buf.Write() : shr_buf.HostWrite();
+            MPI_Irecv(recv_buf + recv_offset, recv_size, MPI_DOUBLE,
+                      gtopo.GetNeighborRank(nbr), 41823,
+                      gtopo.GetComm(), &requests[req_counter++]);
+         }
       }
    }
    ReduceLocalCopy(x, y);
-   MPI_Waitall(req_counter, requests, MPI_STATUSES_IGNORE);
-   ReduceEndAssemble(y); // assemble from 'shr_buf'
+   if (!local)
+   {
+      MPI_Waitall(req_counter, requests, MPI_STATUSES_IGNORE);
+      ReduceEndAssemble(y); // assemble from 'shr_buf'
+   }
 }
 
 } // namespace mfem

fem/pfespace.hpp

@@ -75,6 +75,12 @@ private:
 
    /// The (block-diagonal) matrix R (restriction of dof to true dof). Owned.
    mutable SparseMatrix *R;
+   /// Optimized action-only restriction operator for conforming meshes. Owned.
+   mutable Operator *Rconf;
+   /** Transpose of R or Rconf. For conforming mesh, this is a matrix-free
+       (Device)ConformingProlongationOperator, for a non-conforming mesh
+       this is a TransposeOperator wrapping R. */
+   mutable Operator *R_transpose;
 
    ParNURBSExtension *pNURBSext() const
    { return dynamic_cast<ParNURBSExtension *>(NURBSext); }
@@ -264,6 +270,12 @@ public:
        including the dofs for the edges and the vertices of the face. */
    virtual void GetFaceDofs(int i, Array<int> &dofs) const;
 
+   /** Returns pointer to the FiniteElement in the FiniteElementCollection
+       associated with i'th element in the mesh object. If @a i is greater than
+       or equal to the number of local mesh elements, @a i will be interpreted
+       as a shifted index of a face neigbor element. */
+   virtual const FiniteElement *GetFE(int i) const;
+
    /** Returns an Operator that converts L-vectors to E-vectors on each face.
        The parallel version is different from the serial one because of the
        presence of shared faces. Shared faces are treated as interior faces,
@@ -335,6 +347,16 @@ public:
    HYPRE_Int GetMyTDofOffset() const;
 
    virtual const Operator *GetProlongationMatrix() const;
+   /** @brief Return logical transpose of restriction matrix, but in
+       non-assembled optimized matrix-free form.
+
+       The implementation is like GetProlongationMatrix, but it sets local
+       DOFs to the true DOF values if owned locally, otherwise zero. */
+   virtual const Operator *GetRestrictionTransposeOperator() const;
+   /** Get an Operator that performs the action of GetRestrictionMatrix(),
+       but potentially with a non-assembled optimized matrix-free
+       implementation. */
+   virtual const Operator *GetRestrictionOperator() const;
    /// Get the R matrix which restricts a local dof vector to true dof vector.
    virtual const SparseMatrix *GetRestrictionMatrix() const
    { Dof_TrueDof_Matrix(); return R; }
@@ -389,9 +411,11 @@ class ConformingProlongationOperator : public Operator
 protected:
    Array<int> external_ldofs;
    const GroupCommunicator &gc;
+   bool local;
 
 public:
-   ConformingProlongationOperator(const ParFiniteElementSpace &pfes);
+   ConformingProlongationOperator(const ParFiniteElementSpace &pfes,
+                                  bool local_=false);
 
    virtual void Mult(const Vector &x, Vector &y) const;
 
@@ -410,6 +434,8 @@ protected:
    Array<int> ltdof_ldof, unq_ltdof;
    Array<int> unq_shr_i, unq_shr_j;
    MPI_Request *requests;
+   bool local;
+
    // Kernel: copy ltdofs from 'src' to 'shr_buf' - prepare for send.
    //         shr_buf[i] = src[shr_ltdof[i]]
    void BcastBeginCopy(const Vector &src) const;
@@ -435,7 +461,8 @@ protected:
    void ReduceEndAssemble(Vector &dst) const;
 
 public:
-   DeviceConformingProlongationOperator(const ParFiniteElementSpace &pfes);
+   DeviceConformingProlongationOperator(const ParFiniteElementSpace &pfes,
+                                        bool local_=false);
 
    virtual ~DeviceConformingProlongationOperator();
 

fem/pgridfunc.cpp

@@ -471,6 +471,25 @@ void ParGridFunction::GetVectorValue(ElementTransformation &T,
    }
 }
 
+void ParGridFunction::GetElementDofValues(int el, Vector &dof_vals) const
+{
+   int ne = fes->GetNE();
+   if (el >= ne)
+   {
+      MFEM_ASSERT(face_nbr_data.Size() > 0,
+                  "ParGridFunction::GetElementDofValues: ExchangeFaceNbrData "
+                  "must be called before accessing face neighbor elements.");
+      // Face neighbor element
+      Array<int> dof_idx;
+      pfes->GetFaceNbrElementVDofs(el - ne, dof_idx);
+      face_nbr_data.GetSubVector(dof_idx, dof_vals);
+   }
+   else
+   {
+      GridFunction::GetElementDofValues(el, dof_vals);
+   }
+}
+
 void ParGridFunction::ProjectCoefficient(Coefficient &coeff)
 {
    DeltaCoefficient *delta_c = dynamic_cast<DeltaCoefficient *>(&coeff);
@@ -657,12 +676,12 @@ void ParGridFunction::ProjectBdrCoefficientTangent(VectorCoefficient &vcoeff,
 
 double ParGridFunction::ComputeDGFaceJumpError(Coefficient *exsol,
                                                Coefficient *ell_coeff,
-                                               double Nu,
+                                               JumpScaling jump_scaling,
                                                const IntegrationRule *irs[]) const
 {
    const_cast<ParGridFunction *>(this)->ExchangeFaceNbrData();
 
-   int fdof, dim, intorder, k;
+   int fdof, intorder, k;
    ElementTransformation *transf;
    Vector shape, el_dofs, err_val, ell_coeff_val;
    Array<int> vdofs;
@@ -670,7 +689,6 @@ double ParGridFunction::ComputeDGFaceJumpError(Coefficient *exsol,
    double error = 0.0;
 
    ParMesh *mesh = pfes->GetParMesh();
-   dim = mesh->Dimension();
 
    std::map<int,int> local_to_shared;
    for (int i = 0; i < mesh->GetNSharedFaces(); ++i)
@@ -687,6 +705,7 @@ double ParGridFunction::ComputeDGFaceJumpError(Coefficient *exsol,
       mesh->GetFaceElements(i, &iel1, &iel2);
       mesh->GetFaceInfos(i, &info1, &info2);
 
+      double h = mesh->GetElementSize(iel1);
       intorder = fes->GetFE(iel1)->GetOrder();
 
       FaceElementTransformations *face_elem_transf;
@@ -703,11 +722,10 @@ double ParGridFunction::ComputeDGFaceJumpError(Coefficient *exsol,
          }
          shared_face = true;
          shared_face_factor = 0.5;
+         h = std::min(h, mesh->GetFaceNbrElementSize(iel2));
       }
       else
       {
-         face_elem_transf = mesh->GetFaceElementTransformations(i);
-
          if (iel2 >= 0)
          {
             fe2 = pfes->GetFE(iel2);
@@ -715,12 +733,15 @@ double ParGridFunction::ComputeDGFaceJumpError(Coefficient *exsol,
             {
                intorder = k;
             }
+            h = std::min(h, mesh->GetElementSize(iel2));
          }
          else
          {
             fe2 = NULL;
          }
+         face_elem_transf = mesh->GetFaceElementTransformations(i);
       }
+      int p = intorder;
 
       intorder = 2 * intorder;  // <-------------
       const IntegrationRule *ir;
@@ -806,8 +827,9 @@ double ParGridFunction::ComputeDGFaceJumpError(Coefficient *exsol,
       {
          const IntegrationPoint &ip = ir->IntPoint(j);
          transf->SetIntPoint(&ip);
-         error += shared_face_factor*(ip.weight * Nu * ell_coeff_val(j) *
-                                      pow(transf->Weight(), 1.0-1.0/(dim-1)) *
+         double nu = jump_scaling.Eval(h, p);
+         error += shared_face_factor*(ip.weight * nu * ell_coeff_val(j) *
+                                      transf->Weight() *
                                       err_val(j) * err_val(j));
       }
    }

fem/pgridfunc.hpp

@@ -221,6 +221,12 @@ public:
                                const IntegrationPoint &ip,
                                Vector &val, Vector *tr = NULL) const;
 
+   /** Sets the output vector @a dof_vals to the values of the degrees of
+       freedom of element @a el. If @a el is greater than or equal to the number
+       of local elements, it will be interpreted as a shifted index of a face
+       neighbor element. */
+   virtual void GetElementDofValues(int el, Vector &dof_vals) const;
+
    using GridFunction::ProjectCoefficient;
    virtual void ProjectCoefficient(Coefficient &coeff);
 
@@ -310,7 +316,7 @@ public:
    /// Returns the Face Jumps error for L2 elements
    virtual double ComputeDGFaceJumpError(Coefficient *exsol,
                                          Coefficient *ell_coeff,
-                                         double Nu,
+                                         JumpScaling jump_scaling,
                                          const IntegrationRule *irs[]=NULL)
    const;
 

fem/quadinterpolator.cpp

@@ -420,6 +420,35 @@ void QuadratureInterpolator::MultTranspose(
    MFEM_ABORT("this method is not implemented yet");
 }
 
+static void D2QValues1D(const int NE,
+                        const Array<double> &b_,
+                        const Vector &x_,
+                        Vector &y_,
+                        const int vdim = 1,
+                        const int d1d = 0,
+                        const int q1d = 0)
+{
+   auto b = Reshape(b_.Read(), q1d, d1d);
+   auto x = Reshape(x_.Read(), d1d, vdim, NE);
+   auto y = Reshape(y_.Write(), vdim, q1d, NE);
+
+   MFEM_FORALL(e, NE,
+   {
+      for (int c = 0; c < vdim; c++)
+      {
+         for (int q = 0; q < q1d; ++q)
+         {
+            double val = 0.0;
+            for (int d = 0; d < d1d; ++d)
+            {
+               val += b(q, d) * x(d, c, e);
+            }
+            y(c, q, e) = val;
+         }
+      }
+   });
+}
+
 
 template<int T_VDIM = 0, int T_D1D = 0, int T_Q1D = 0, int T_NBZ = 0>
 static void D2QValues2D(const int NE,
@@ -631,6 +660,15 @@ static void D2QValues(const FiniteElementSpace &fes,
    const int Q1D = maps->nqpt;
    const int id = (vdim<<8) | (D1D<<4) | Q1D;
 
+   if (dim == 1)
+   {
+      MFEM_VERIFY(D1D <= MAX_D1D, "Orders higher than " << MAX_D1D-1
+                  << " are not supported!");
+      MFEM_VERIFY(Q1D <= MAX_Q1D, "Quadrature rules with more than "
+                  << MAX_Q1D << " 1D points are not supported!");
+      D2QValues1D(NE, maps->B, e_vec, q_val, vdim, D1D, Q1D);
+      return;
+   }
    if (dim == 2)
    {
       switch (id)

fem/restriction.cpp

@@ -195,6 +195,31 @@ void ElementRestriction::MultTransposeUnsigned(const Vector& x, Vector& y) const
    });
 }
 
+void ElementRestriction::MultLeftInverse(const Vector& x, Vector& y) const
+{
+   // Assumes all elements have the same number of dofs
+   const int nd = dof;
+   const int vd = vdim;
+   const bool t = byvdim;
+   auto d_offsets = offsets.Read();
+   auto d_indices = indices.Read();
+   auto d_x = Reshape(x.Read(), nd, vd, ne);
+   auto d_y = Reshape(y.Write(), t?vd:ndofs, t?ndofs:vd);
+   MFEM_FORALL(i, ndofs,
+   {
+      const int nextOffset = d_offsets[i + 1];
+      for (int c = 0; c < vd; ++c)
+      {
+         double dofValue = 0;
+         const int j = nextOffset - 1;
+         const int idx_j = (d_indices[j] >= 0) ? d_indices[j] : -1 - d_indices[j];
+         dofValue = (d_indices[j] >= 0) ? d_x(idx_j % nd, c, idx_j / nd) :
+         -d_x(idx_j % nd, c, idx_j / nd);
+         d_y(t?c:i,t?i:c) = dofValue;
+      }
+   });
+}
+
 void ElementRestriction::BooleanMask(Vector& y) const
 {
    // Assumes all elements have the same number of dofs

fem/restriction.hpp

@@ -57,6 +57,10 @@ public:
    /// Compute MultTranspose without applying signs based on DOF orientations.
    void MultTransposeUnsigned(const Vector &x, Vector &y) const;
 
+   /// Compute MultTranspose by setting (rather than adding) element
+   /// contributions; this is a left inverse of the Mult() operation
+   void MultLeftInverse(const Vector &x, Vector &y) const;
+
    /// @brief Fills the E-vector y with `boolean` values 0.0 and 1.0 such that each
    /// each entry of the L-vector is uniquely represented in `y`.
    /** This means, the sum of the E-vector `y` is equal to the sum of the

fem/tmop.cpp

@@ -19,6 +19,42 @@ namespace mfem
 
 // Target-matrix optimization paradigm (TMOP) mesh quality metrics.
 
+double TMOP_Combo_QualityMetric::EvalW(const DenseMatrix &Jpt) const
+{
+   double metric = 0.;
+   for (int i = 0; i < tmop_q_arr.Size(); i++)
+   {
+      metric += wt_arr[i]*tmop_q_arr[i]->EvalW(Jpt);
+   }
+   return metric;
+}
+
+void TMOP_Combo_QualityMetric::EvalP(const DenseMatrix &Jpt,
+                                     DenseMatrix &P) const
+{
+   DenseMatrix Pt(P.Size());
+   for (int i = 0; i < tmop_q_arr.Size(); i++)
+   {
+      tmop_q_arr[i]->EvalP(Jpt, Pt);
+      Pt *= wt_arr[i];
+      P += Pt;
+   }
+}
+
+void TMOP_Combo_QualityMetric::AssembleH(const DenseMatrix &Jpt,
+                                         const DenseMatrix &DS,
+                                         const double weight,
+                                         DenseMatrix &A) const
+{
+   DenseMatrix At(A.Size());
+   for (int i = 0; i < tmop_q_arr.Size(); i++)
+   {
+      tmop_q_arr[i]->AssembleH(Jpt, DS, weight, At);
+      At *= wt_arr[i];
+      A += At;
+   }
+}
+
 double TMOP_Metric_001::EvalW(const DenseMatrix &Jpt) const
 {
    ie.SetJacobian(Jpt.GetData());
@@ -160,23 +196,6 @@ double TMOP_Metric_aspratio3D::EvalW(const DenseMatrix &Jpt) const
           ) / 3.0;
 }
 
-// mu_14 = |T-I|^2
-double TMOP_Metric_SSA2D::EvalW(const DenseMatrix &Jpt) const
-{
-   MFEM_VERIFY(Jtr != NULL,
-               "Requires a target Jacobian, use SetTargetJacobian().");
-
-   DenseMatrix Id(2,2);
-
-   Id(0,0) = 1; Id(0,1) = 0;
-   Id(1,0) = 0; Id(1,1) = 1;
-
-   DenseMatrix Mat(2,2);
-   Mat = Jpt;
-   Mat.Add(-1,Id);
-   return Mat.FNorm2();
-}
-
 double TMOP_Metric_002::EvalW(const DenseMatrix &Jpt) const
 {
    ie.SetJacobian(Jpt.GetData());
@@ -272,13 +291,41 @@ void TMOP_Metric_009::AssembleH(const DenseMatrix &Jpt,
    ie.Assemble_ddI1b(weight, A.GetData());
 }
 
+// mu_14 = |T-I|^2
+double TMOP_Metric_014::EvalW(const DenseMatrix &Jpt) const
+{
+   MFEM_VERIFY(Jtr != NULL,
+               "Requires a target Jacobian, use SetTargetJacobian().");
+
+   DenseMatrix Id(2,2);
+
+   Id(0,0) = 1; Id(0,1) = 0;
+   Id(1,0) = 0; Id(1,1) = 1;
+
+   DenseMatrix Mat(2,2);
+   Mat = Jpt;
+   Mat.Add(-1,Id);
+   return Mat.FNorm2();
+}
+
 double TMOP_Metric_022::EvalW(const DenseMatrix &Jpt) const
 {
    // mu_22 = (0.5*|J|^2 - det(J)) / (det(J) - tau0)
    //       = (0.5*I1 - I2b) / (I2b - tau0)
    ie.SetJacobian(Jpt.GetData());
    const double I2b = ie.Get_I2b();
-   return (0.5*ie.Get_I1() - I2b) / (I2b - tau0);
+
+   double d = I2b - min_detT;
+   if (d < 0.0 && min_detT == 0.0)
+   {
+      // The mesh has been untangled, but it's still possible to get negative
+      // detJ in FD calculations, as they move the nodes around with some small
+      // increments and can produce negative determinants. Thus we put a small
+      // value in the denominator. Note that here I2b < 0.
+      d = - I2b * 0.1;
+   }
+
+   return (0.5*ie.Get_I1() - I2b) / d;
 }
 
 void TMOP_Metric_022::EvalP(const DenseMatrix &Jpt, DenseMatrix &P) const
@@ -287,8 +334,8 @@ void TMOP_Metric_022::EvalP(const DenseMatrix &Jpt, DenseMatrix &P) const
    // P = 1/(I2b - tau0)*(0.5*dI1 - dI2b) - (0.5*I1 - I2b)/(I2b - tau0)^2*dI2b
    //   = 0.5/(I2b - tau0)*dI1 + (tau0 - 0.5*I1)/(I2b - tau0)^2*dI2b
    ie.SetJacobian(Jpt.GetData());
-   const double c1 = 1.0/(ie.Get_I2b() - tau0);
-   Add(c1/2, ie.Get_dI1(), (tau0 - ie.Get_I1()/2)*c1*c1, ie.Get_dI2b(), P);
+   const double c1 = 1.0/(ie.Get_I2b() - min_detT);
+   Add(c1/2, ie.Get_dI1(), (min_detT - ie.Get_I1()/2)*c1*c1, ie.Get_dI2b(), P);
 }
 
 void TMOP_Metric_022::AssembleH(const DenseMatrix &Jpt,
@@ -308,10 +355,10 @@ void TMOP_Metric_022::AssembleH(const DenseMatrix &Jpt,
    //      +0.5/(I2b - tau0)*ddI1 + z*ddI2b
    ie.SetJacobian(Jpt.GetData());
    ie.SetDerivativeMatrix(DS.Height(), DS.GetData());
-   const double c1 = 1.0/(ie.Get_I2b() - tau0);
+   const double c1 = 1.0/(ie.Get_I2b() - min_detT);
    const double c2 = weight*c1/2;
    const double c3 = c1*c2;
-   const double c4 = (2*tau0 - ie.Get_I1())*c3; // weight*z
+   const double c4 = (2*min_detT - ie.Get_I1())*c3; // weight*z
    ie.Assemble_TProd(-c3, ie.Get_dI1(), ie.Get_dI2b(), A.GetData());
    ie.Assemble_TProd(-2*c1*c4, ie.Get_dI2b(), A.GetData());
    ie.Assemble_ddI1(c2, A.GetData());
@@ -484,6 +531,23 @@ double TMOP_Metric_085::EvalW(const DenseMatrix &Jpt) const
    return Mat.FNorm2();
 }
 
+// mu_98 = 1/(tau)|T-I|^2
+double TMOP_Metric_098::EvalW(const DenseMatrix &Jpt) const
+{
+   MFEM_VERIFY(Jtr != NULL,
+               "Requires a target Jacobian, use SetTargetJacobian().");
+
+   DenseMatrix Id(2,2);
+
+   Id(0,0) = 1; Id(0,1) = 0;
+   Id(1,0) = 0; Id(1,1) = 1;
+
+   DenseMatrix Mat(2,2);
+   Mat = Jpt;
+   Mat.Add(-1,Id);
+   return Mat.FNorm2()/Jtr->Det();
+}
+
 double TMOP_Metric_211::EvalW(const DenseMatrix &Jpt) const
 {
    // mu_211 = (det(J) - 1)^2 - det(J) + (det(J)^2 + eps)^{1/2}
@@ -650,6 +714,71 @@ void TMOP_Metric_303::AssembleH(const DenseMatrix &Jpt,
    ie.Assemble_ddI1b(weight/3., A.GetData());
 }
 
+double TMOP_Metric_311::EvalW(const DenseMatrix &Jpt) const
+{
+   // mu_311 = (det(J) - 1)^2 - det(J) + (det(J)^2 + eps)^{1/2}
+   //        = (I3b - 1)^2 - I3b + sqrt(I3b^2 + eps)
+   ie.SetJacobian(Jpt.GetData());
+   const double I3b = ie.Get_I3b();
+   return (I3b - 1.0)*(I3b - 1.0) - I3b + std::sqrt(I3b*I3b + eps);
+}
+
+void TMOP_Metric_311::EvalP(const DenseMatrix &Jpt, DenseMatrix &P) const
+{
+   ie.SetJacobian(Jpt.GetData());
+   const double I3b = ie.Get_I3b();
+   const double c = 2*I3b-3+(I3b)/(std::pow((I3b*I3b+eps),0.5));
+   P.Set(c, ie.Get_dI3b());
+}
+
+void TMOP_Metric_311::AssembleH(const DenseMatrix &Jpt,
+                                const DenseMatrix &DS,
+                                const double weight,
+                                DenseMatrix &A) const
+{
+   ie.SetJacobian(Jpt.GetData());
+   ie.SetDerivativeMatrix(DS.Height(), DS.GetData());
+   const double I3b = ie.Get_I3b();
+   const double c0 = I3b*I3b+eps;
+   const double c1 = 2 + 1/(pow(c0,0.5)) - I3b*I3b/(pow(c0,1.5));
+   const double c2 = 2*I3b - 3 + I3b/(pow(c0,0.5));
+   ie.Assemble_TProd(weight*c1, ie.Get_dI3b(), A.GetData());
+   ie.Assemble_ddI3b(c2*weight, A.GetData());
+}
+
+double TMOP_Metric_313::EvalW(const DenseMatrix &Jpt) const
+{
+   ie.SetJacobian(Jpt.GetData());
+
+   const double I3b = ie.Get_I3b();
+   double d = I3b - min_detT;
+   if (d < 0.0 && min_detT == 0.0)
+   {
+      // The mesh has been untangled, but it's still possible to get negative
+      // detJ in FD calculations, as they move the nodes around with some small
+      // increments and can produce negative determinants. Thus we put a small
+      // value in the denominator. Note that here I3b < 0.
+      d = - I3b * 0.1;
+   }
+
+   const double c = std::pow(d, -2.0/3.0);
+
+   return ie.Get_I1() * c / 3.0;
+}
+
+void TMOP_Metric_313::EvalP(const DenseMatrix &Jpt, DenseMatrix &P) const
+{
+   MFEM_ABORT("Metric not implemented yet.");
+}
+
+void TMOP_Metric_313::AssembleH(const DenseMatrix &Jpt,
+                                const DenseMatrix &DS,
+                                const double weight,
+                                DenseMatrix &A) const
+{
+   MFEM_ABORT("Metric not implemented yet.");
+}
+
 double TMOP_Metric_315::EvalW(const DenseMatrix &Jpt) const
 {
    // mu_315 = mu_15_3D = (det(J) - 1)^2
@@ -800,6 +929,84 @@ void TMOP_Metric_352::AssembleH(const DenseMatrix &Jpt,
    ie.Assemble_ddI3b(weight*(c - 0.5*c*c), A.GetData());
 }
 
+double TMOP_AMetric_011::EvalW(const DenseMatrix &Jpt) const
+{
+   MFEM_VERIFY(Jtr != NULL,
+               "Requires a target Jacobian, use SetTargetJacobian().");
+
+   int dim = Jpt.Size();
+
+   DenseMatrix Jpr(dim, dim);
+   Mult(Jpt, *Jtr, Jpr);
+
+   double alpha = Jpr.Det(),
+          omega = Jtr->Det();
+
+   DenseMatrix AdjAt(dim), WtW(dim), WRK(dim), Jtrt(dim);
+   CalcAdjugateTranspose(Jpr, AdjAt);
+   Jtrt.Transpose(*Jtr);
+   MultAAt(Jtrt, WtW);
+   WtW *= 1./omega;
+   Mult(AdjAt, WtW, WRK);
+
+   WRK -= Jpr;
+   WRK *= -1.;
+
+   return (0.25/alpha)*WRK.FNorm2();
+}
+
+double TMOP_AMetric_014a::EvalW(const DenseMatrix &Jpt) const
+{
+   MFEM_VERIFY(Jtr != NULL,
+               "Requires a target Jacobian, use SetTargetJacobian().");
+
+   int dim = Jpt.Size();
+
+   DenseMatrix Jpr(dim, dim);
+   Mult(Jpt, *Jtr, Jpr);
+
+   double sqalpha = pow(Jpr.Det(), 0.5),
+          sqomega = pow(Jtr->Det(), 0.5);
+
+   return 0.5*pow(sqalpha/sqomega - sqomega/sqalpha, 2.);
+}
+
+double TMOP_AMetric_036::EvalW(const DenseMatrix &Jpt) const
+{
+   MFEM_VERIFY(Jtr != NULL,
+               "Requires a target Jacobian, use SetTargetJacobian().");
+
+   int dim = Jpt.Size();
+
+   DenseMatrix Jpr(dim, dim);
+   Mult(Jpt, *Jtr, Jpr); // T*W = A
+
+   double alpha = Jpr.Det(); // det(A)
+   Jpr -= *Jtr; // A-W
+
+   return (1./alpha)*(Jpr.FNorm2()); //(1/alpha)*(|A-W|^2)
+}
+
+double TMOP_AMetric_107a::EvalW(const DenseMatrix &Jpt) const
+{
+   MFEM_VERIFY(Jtr != NULL,
+               "Requires a target Jacobian, use SetTargetJacobian().");
+
+   int dim = Jpt.Size();
+
+   DenseMatrix Jpr(dim, dim);
+   Mult(Jpt, *Jtr, Jpr);
+
+   double alpha = Jpr.Det(),
+          aw = Jpr.FNorm()/Jtr->FNorm();
+
+   DenseMatrix W = *Jtr;
+   W *= aw;
+   Jpr -= W;
+
+   return (0.5/alpha)*Jpr.FNorm2();
+}
+
 
 void TargetConstructor::ComputeAvgVolume() const
 {

fem/tmop.hpp

@@ -40,7 +40,7 @@ public:
        The specified Jacobian matrix, #Jtr, can be used by metrics that cannot
        be written just as a function of the target->physical Jacobian matrix,
        Jpt. */
-   void SetTargetJacobian(const DenseMatrix &_Jtr) { Jtr = &_Jtr; }
+   virtual void SetTargetJacobian(const DenseMatrix &_Jtr) { Jtr = &_Jtr; }
 
    /** @brief Evaluate the strain energy density function, W = W(Jpt).
        @param[in] Jpt  Represents the target->physical transformation
@@ -70,8 +70,37 @@ public:
                           const double weight, DenseMatrix &A) const = 0;
 };
 
+/// Abstract class used to define combination of metrics with constant coefficients.
+class TMOP_Combo_QualityMetric : public TMOP_QualityMetric
+{
+protected:
+   Array<TMOP_QualityMetric *> tmop_q_arr; //not owned
+   Array<double> wt_arr;
 
-/// Metric without a type, 2D
+public:
+   virtual void AddQualityMetric(TMOP_QualityMetric *tq, double wt = 1.0)
+   {
+      tmop_q_arr.Append(tq);
+      wt_arr.Append(wt);
+   }
+
+   virtual void SetTargetJacobian(const DenseMatrix &_Jtr)
+   {
+      for (int i = 0; i < tmop_q_arr.Size(); i++)
+      {
+         tmop_q_arr[i]->SetTargetJacobian(_Jtr);
+      }
+   }
+
+   virtual double EvalW(const DenseMatrix &Jpt) const;
+
+   virtual void EvalP(const DenseMatrix &Jpt, DenseMatrix &P) const;
+
+   virtual void AssembleH(const DenseMatrix &Jpt, const DenseMatrix &DS,
+                          const double weight, DenseMatrix &A) const;
+};
+
+/// 2D non-barrier metric without a type.
 class TMOP_Metric_001 : public TMOP_QualityMetric
 {
 protected:
@@ -87,7 +116,7 @@ public:
                           const double weight, DenseMatrix &A) const;
 };
 
-/// Skew metric, 2D.
+/// 2D non-barrier Skew metric.
 class TMOP_Metric_skew2D : public TMOP_QualityMetric
 {
 public:
@@ -102,7 +131,7 @@ public:
    { MFEM_ABORT("Not implemented"); }
 };
 
-/// Skew metric, 3D.
+/// 3D non-barrier Skew metric.
 class TMOP_Metric_skew3D : public TMOP_QualityMetric
 {
 public:
@@ -117,7 +146,7 @@ public:
    { MFEM_ABORT("Not implemented"); }
 };
 
-/// Aspect ratio metric, 2D.
+/// 2D non-barrier Aspect ratio metric.
 class TMOP_Metric_aspratio2D : public TMOP_QualityMetric
 {
 public:
@@ -132,7 +161,7 @@ public:
    { MFEM_ABORT("Not implemented"); }
 };
 
-/// Aspect ratio metric, 3D.
+/// 3D non-barrier Aspect ratio metric.
 class TMOP_Metric_aspratio3D : public TMOP_QualityMetric
 {
 public:
@@ -147,22 +176,7 @@ public:
    { MFEM_ABORT("Not implemented"); }
 };
 
-/// Shape+Size+Orientation metric, 2D.
-class TMOP_Metric_SSA2D : public TMOP_QualityMetric
-{
-public:
-   // W = 0.5 (1 - cos(theta_Jpr - theta_Jtr)).
-   virtual double EvalW(const DenseMatrix &Jpt) const;
-
-   virtual void EvalP(const DenseMatrix &Jpt, DenseMatrix &P) const
-   { MFEM_ABORT("Not implemented"); }
-
-   virtual void AssembleH(const DenseMatrix &Jpt, const DenseMatrix &DS,
-                          const double weight, DenseMatrix &A) const
-   { MFEM_ABORT("Not implemented"); }
-};
-
-/// Shape, ideal barrier metric, 2D
+/// 2D barrier shape (S) metric (polyconvex).
 class TMOP_Metric_002 : public TMOP_QualityMetric
 {
 protected:
@@ -178,7 +192,7 @@ public:
                           const double weight, DenseMatrix &A) const;
 };
 
-/// Shape & area, ideal barrier metric, 2D
+/// 2D barrier Shape+Size (VS) metric (not polyconvex).
 class TMOP_Metric_007 : public TMOP_QualityMetric
 {
 protected:
@@ -194,7 +208,7 @@ public:
                           const double weight, DenseMatrix &A) const;
 };
 
-/// Shape & area metric, 2D
+/// 2D barrier Shape+Size (VS) metric (not polyconvex).
 class TMOP_Metric_009 : public TMOP_QualityMetric
 {
 protected:
@@ -210,15 +224,30 @@ public:
                           const double weight, DenseMatrix &A) const;
 };
 
-/// Shifted barrier form of metric 2 (shape, ideal barrier metric), 2D
+/// 2D non-barrier Shape+Size+Orientation (VOS) metric (polyconvex).
+class TMOP_Metric_014 : public TMOP_QualityMetric
+{
+public:
+   // W = |T-I|^2.
+   virtual double EvalW(const DenseMatrix &Jpt) const;
+
+   virtual void EvalP(const DenseMatrix &Jpt, DenseMatrix &P) const
+   { MFEM_ABORT("Not implemented"); }
+
+   virtual void AssembleH(const DenseMatrix &Jpt, const DenseMatrix &DS,
+                          const double weight, DenseMatrix &A) const
+   { MFEM_ABORT("Not implemented"); }
+};
+
+/// 2D Shifted barrier form of shape metric (mu_2).
 class TMOP_Metric_022 : public TMOP_QualityMetric
 {
 protected:
-   double &tau0;
+   double &min_detT;
    mutable InvariantsEvaluator2D<double> ie;
 
 public:
-   TMOP_Metric_022(double &t0): tau0(t0) {}
+   TMOP_Metric_022(double &t0): min_detT(t0) {}
 
    // W = 0.5(|J|^2 - 2det(J)) / (det(J) - tau0).
    virtual double EvalW(const DenseMatrix &Jpt) const;
@@ -229,7 +258,7 @@ public:
                           const double weight, DenseMatrix &A) const;
 };
 
-/// Shape, ideal barrier metric, 2D
+/// 2D barrier (not a shape) metric (polyconvex).
 class TMOP_Metric_050 : public TMOP_QualityMetric
 {
 protected:
@@ -245,7 +274,7 @@ public:
                           const double weight, DenseMatrix &A) const;
 };
 
-/// Area metric, 2D
+/// 2D non-barrier size (V) metric (not polyconvex).
 class TMOP_Metric_055 : public TMOP_QualityMetric
 {
 protected:
@@ -262,7 +291,7 @@ public:
 
 };
 
-/// Area, ideal barrier metric, 2D
+/// 2D barrier size (V) metric (polyconvex).
 class TMOP_Metric_056 : public TMOP_QualityMetric
 {
 protected:
@@ -281,7 +310,7 @@ public:
 
 };
 
-/// Shape, ideal barrier metric, 2D
+/// 2D barrier shape (S) metric (not polyconvex).
 class TMOP_Metric_058 : public TMOP_QualityMetric
 {
 protected:
@@ -299,7 +328,7 @@ public:
 
 };
 
-/// Area, ideal barrier metric, 2D
+/// 2D barrier size (V) metric (polyconvex).
 class TMOP_Metric_077 : public TMOP_QualityMetric
 {
 protected:
@@ -316,7 +345,28 @@ public:
 
 };
 
-/// Shape & orientation metric, 2D.
+/// 2D barrier Shape+Size (VS) metric (polyconvex).
+class TMOP_Metric_080 : public TMOP_Combo_QualityMetric
+{
+protected:
+   mutable InvariantsEvaluator2D<double> ie;
+   double gamma;
+   TMOP_QualityMetric *sh_metric, *sz_metric;
+
+public:
+   TMOP_Metric_080(double gamma_) : gamma(gamma_),
+      sh_metric(new TMOP_Metric_002),
+      sz_metric(new TMOP_Metric_077)
+   {
+      // (1-gamma) mu_2 + gamma mu_77
+      AddQualityMetric(sh_metric, 1.-gamma_);
+      AddQualityMetric(sz_metric, gamma_);
+   }
+
+   virtual ~TMOP_Metric_080() { delete sh_metric; delete sz_metric; }
+};
+
+/// 2D barrier Shape+Orientation (OS) metric (polyconvex).
 class TMOP_Metric_085 : public TMOP_QualityMetric
 {
 public:
@@ -331,7 +381,22 @@ public:
    { MFEM_ABORT("Not implemented"); }
 };
 
-/// Untangling metric, 2D
+/// 2D barrier Shape+Size+Orientation (VOS) metric (polyconvex).
+class TMOP_Metric_098 : public TMOP_QualityMetric
+{
+public:
+   // W = 1/tau |T-I|^2.
+   virtual double EvalW(const DenseMatrix &Jpt) const;
+
+   virtual void EvalP(const DenseMatrix &Jpt, DenseMatrix &P) const
+   { MFEM_ABORT("Not implemented"); }
+
+   virtual void AssembleH(const DenseMatrix &Jpt, const DenseMatrix &DS,
+                          const double weight, DenseMatrix &A) const
+   { MFEM_ABORT("Not implemented"); }
+};
+
+/// 2D untangling metric.
 class TMOP_Metric_211 : public TMOP_QualityMetric
 {
 protected:
@@ -370,7 +435,7 @@ public:
                           const double weight, DenseMatrix &A) const;
 };
 
-/// Shape, ideal barrier metric, 3D
+/// 3D barrier Shape (S) metric.
 class TMOP_Metric_301 : public TMOP_QualityMetric
 {
 protected:
@@ -386,7 +451,7 @@ public:
                           const double weight, DenseMatrix &A) const;
 };
 
-/// Shape, ideal barrier metric, 3D
+/// 3D barrier Shape (S) metric.
 class TMOP_Metric_302 : public TMOP_QualityMetric
 {
 protected:
@@ -402,14 +467,14 @@ public:
                           const double weight, DenseMatrix &A) const;
 };
 
-/// Shape, ideal barrier metric, 3D
+/// 3D barrier Shape (S) metric.
 class TMOP_Metric_303 : public TMOP_QualityMetric
 {
 protected:
    mutable InvariantsEvaluator3D<double> ie;
 
 public:
-   // W = |J|^2 / 3 * det(J)^(2/3) - 1.
+   // W = |J|^2 / 3 * det(J)^(-2/3) - 1.
    virtual double EvalW(const DenseMatrix &Jpt) const;
 
    virtual void EvalP(const DenseMatrix &Jpt, DenseMatrix &P) const;
@@ -418,7 +483,45 @@ public:
                           const double weight, DenseMatrix &A) const;
 };
 
-/// Volume metric, 3D
+/// 3D Size (V) untangling metric.
+class TMOP_Metric_311 : public TMOP_QualityMetric
+{
+protected:
+   const double eps;
+   mutable InvariantsEvaluator3D<double> ie;
+
+public:
+   TMOP_Metric_311(double epsilon = 1e-4) : eps(epsilon) { }
+
+   // W = (det(J) - 1)^2 - det(J)  + (det(J)^2 + eps)^(1/2).
+   virtual double EvalW(const DenseMatrix &Jpt) const;
+
+   virtual void EvalP(const DenseMatrix &Jpt, DenseMatrix &P) const;
+
+   virtual void AssembleH(const DenseMatrix &Jpt, const DenseMatrix &DS,
+                          const double weight, DenseMatrix &A) const;
+};
+
+/// 3D Shape (S) metric, untangling version of 303.
+class TMOP_Metric_313 : public TMOP_QualityMetric
+{
+protected:
+   double &min_detT;
+   mutable InvariantsEvaluator3D<double> ie;
+
+public:
+   TMOP_Metric_313(double &mindet) : min_detT(mindet) { }
+
+   // W = 1/3 |J|^2 / [det(J)-tau0]^(-2/3).
+   virtual double EvalW(const DenseMatrix &Jpt) const;
+
+   virtual void EvalP(const DenseMatrix &Jpt, DenseMatrix &P) const;
+
+   virtual void AssembleH(const DenseMatrix &Jpt, const DenseMatrix &DS,
+                          const double weight, DenseMatrix &A) const;
+};
+
+/// 3D non-barrier Size (V) metric.
 class TMOP_Metric_315 : public TMOP_QualityMetric
 {
 protected:
@@ -434,7 +537,7 @@ public:
                           const double weight, DenseMatrix &A) const;
 };
 
-/// Volume, ideal barrier metric, 3D
+/// 3D barrier Size (V) metric.
 class TMOP_Metric_316 : public TMOP_QualityMetric
 {
 protected:
@@ -452,7 +555,7 @@ public:
                           const double weight, DenseMatrix &A) const;
 };
 
-/// Shape & volume, ideal barrier metric, 3D
+/// 3D barrier Shape+Size (VS) metric.
 class TMOP_Metric_321 : public TMOP_QualityMetric
 {
 protected:
@@ -487,6 +590,99 @@ public:
                           const double weight, DenseMatrix &A) const;
 };
 
+/// A-metrics
+/// 2D barrier Shape (S) metric (polyconvex).
+class TMOP_AMetric_011 : public TMOP_QualityMetric
+{
+protected:
+   mutable InvariantsEvaluator3D<double> ie;
+
+public:
+   // (1/4 alpha) | A - (adj A)^t W^t W / omega |^2
+   virtual double EvalW(const DenseMatrix &Jpt) const;
+
+   virtual void EvalP(const DenseMatrix &Jpt, DenseMatrix &P) const
+   { MFEM_ABORT("Not implemented"); }
+
+   virtual void AssembleH(const DenseMatrix &Jpt, const DenseMatrix &DS,
+                          const double weight, DenseMatrix &A) const
+   { MFEM_ABORT("Not implemented"); }
+};
+
+/// 2D barrier Size (V) metric (polyconvex).
+class TMOP_AMetric_014a : public TMOP_QualityMetric
+{
+protected:
+   mutable InvariantsEvaluator3D<double> ie;
+
+public:
+   // 0.5 * ( sqrt(alpha/omega) - sqrt(omega/alpha) )^2
+   virtual double EvalW(const DenseMatrix &Jpt) const;
+
+   virtual void EvalP(const DenseMatrix &Jpt, DenseMatrix &P) const
+   { MFEM_ABORT("Not implemented"); }
+
+   virtual void AssembleH(const DenseMatrix &Jpt, const DenseMatrix &DS,
+                          const double weight, DenseMatrix &A) const
+   { MFEM_ABORT("Not implemented"); }
+};
+
+/// 2D barrier Shape+Size+Orientation (VOS) metric (polyconvex).
+class TMOP_AMetric_036 : public TMOP_QualityMetric
+{
+protected:
+   mutable InvariantsEvaluator3D<double> ie;
+
+public:
+   // (1/alpha) | A - W |^2
+   virtual double EvalW(const DenseMatrix &Jpt) const;
+
+   virtual void EvalP(const DenseMatrix &Jpt, DenseMatrix &P) const
+   { MFEM_ABORT("Not implemented"); }
+
+   virtual void AssembleH(const DenseMatrix &Jpt, const DenseMatrix &DS,
+                          const double weight, DenseMatrix &A) const
+   { MFEM_ABORT("Not implemented"); }
+};
+
+/// 2D barrier Shape+Orientation (OS) metric (polyconvex).
+class TMOP_AMetric_107a : public TMOP_QualityMetric
+{
+protected:
+   mutable InvariantsEvaluator3D<double> ie;
+
+public:
+   // (1/2 alpha) | A - (|A|/|W|) W |^2
+   virtual double EvalW(const DenseMatrix &Jpt) const;
+
+   virtual void EvalP(const DenseMatrix &Jpt, DenseMatrix &P) const
+   { MFEM_ABORT("Not implemented"); }
+
+   virtual void AssembleH(const DenseMatrix &Jpt, const DenseMatrix &DS,
+                          const double weight, DenseMatrix &A) const
+   { MFEM_ABORT("Not implemented"); }
+};
+
+/// 2D barrier Shape+Size (VS) metric (polyconvex).
+class TMOP_AMetric_126 : public TMOP_Combo_QualityMetric
+{
+protected:
+   mutable InvariantsEvaluator2D<double> ie;
+   double gamma;
+   TMOP_QualityMetric *sh_metric, *sz_metric;
+
+public:
+   TMOP_AMetric_126(double gamma_) : gamma(gamma_),
+      sh_metric(new TMOP_AMetric_011),
+      sz_metric(new TMOP_AMetric_014a)
+   {
+      // (1-gamma) nu_11 + gamma nu_14
+      AddQualityMetric(sh_metric, 1.-gamma_);
+      AddQualityMetric(sz_metric, gamma_);
+   }
+
+   virtual ~TMOP_AMetric_126() { delete sh_metric; delete sz_metric; }
+};
 
 /// Base class for limiting functions to be used in class TMOP_Integrator.
 /** This class represents a scalar function f(x, x0, d), where x and x0 are

fem/tmop_tools.cpp

@@ -331,10 +331,6 @@ double TMOPNewtonSolver::ComputeScalingFactor(const Vector &x,
       energy_in = nlf->GetEnergy(x);
    }
 
-   const int NE = fes->GetMesh()->GetNE(), dim = fes->GetMesh()->Dimension();
-   Array<int> xdofs;
-   DenseMatrix Jpr(dim);
-
    // Get the local prolongation of the solution vector.
    Vector x_out_loc(fes->GetVSize());
    if (serial)
@@ -350,51 +346,32 @@ double TMOPNewtonSolver::ComputeScalingFactor(const Vector &x,
    }
 #endif
 
-   // Check if the starting mesh (given by x) is inverted.
-   // Note that x hasn't been modified by the Newton update yet.
-   double min_detJ = infinity();
-   for (int i = 0; i < NE; i++)
+   // Check if the starting mesh (given by x) is inverted. Note that x hasn't
+   // been modified by the Newton update yet.
+   const double min_detT_in = ComputeMinDet(x_out_loc, *fes);
+   const bool untangling = (min_detT_in <= 0.0) ? true : false;
+   const double untangle_factor = 1.5;
+   if (untangling)
    {
-      const int dof = fes->GetFE(i)->GetDof();
-      DenseMatrix dshape(dof, dim), pos(dof, dim);
-      Vector posV(pos.Data(), dof * dim);
-
-      fes->GetElementVDofs(i, xdofs);
-      x_out_loc.GetSubVector(xdofs, posV);
-
-      const IntegrationRule &irule = GetIntegrationRule(*fes->GetFE(i));
-      const int nsp = irule.GetNPoints();
-      for (int j = 0; j < nsp; j++)
-      {
-         fes->GetFE(i)->CalcDShape(irule.IntPoint(j), dshape);
-         MultAtB(pos, dshape, Jpr);
-         min_detJ = std::min(min_detJ, Jpr.Det());
-      }
+      // Needed for the line search below. The untangling metrics see this
+      // reference to detect deteriorations.
+      *min_det_ptr = untangle_factor * min_detT_in;
    }
-   double min_detJ_all = min_detJ;
-#ifdef MFEM_USE_MPI
-   if (parallel)
-   {
-      MPI_Allreduce(&min_detJ, &min_detJ_all, 1, MPI_DOUBLE, MPI_MIN,
-                    p_nlf->ParFESpace()->GetComm());
-   }
-#endif
-   const bool untangling = (min_detJ_all <= 0) ? true : false;
 
    const bool have_b = (b.Size() == Height());
 
    Vector x_out(x.Size());
    bool x_out_ok = false;
-   double scale = 1.0, energy_out = 0.0;
-   const double norm0 = Norm(r);
+   double scale = 1.0, energy_out = 0.0, min_detT_out;
+   const double norm_in = Norm(r);
 
    const double detJ_factor = (solver_type == 1) ? 0.25 : 0.5;
 
    // Perform the line search.
    for (int i = 0; i < 12; i++)
    {
+      // Update the mesh and get the L-vector in x_out_loc.
       add(x, -scale, c, x_out);
-
       if (serial)
       {
          const SparseMatrix *cP = fes->GetConformingProlongation();
@@ -408,47 +385,29 @@ double TMOPNewtonSolver::ComputeScalingFactor(const Vector &x,
       }
 #endif
 
-      // Check det(Jpr) > 0.
-      if (!untangling)
+      // Check the changes in detJ.
+      min_detT_out = ComputeMinDet(x_out_loc, *fes);
+      if (untangling == false && min_detT_out < 0.0)
       {
-         int jac_ok = 1;
-         for (int i = 0; i < NE; i++)
-         {
-            const int dof = fes->GetFE(i)->GetDof();
-            DenseMatrix dshape(dof, dim), pos(dof, dim);
-            Vector posV(pos.Data(), dof * dim);
+         // No untangling, and detJ got negative -- no good.
+         if (print_level >= 0)
+         { mfem::out << "Scale = " << scale << " Neg det(J) found.\n"; }
+         scale *= detJ_factor; continue;
+      }
+      if (untangling == true && min_detT_out < *min_det_ptr)
+      {
+         // Untangling, and detJ got even more negative -- no good.
+         if (print_level >= 0)
+         { mfem::out << "Scale = " << scale << " Neg det(J) decreased.\n"; }
+         scale *= detJ_factor; continue;
+      }
 
-            fes->GetElementVDofs(i, xdofs);
-            x_out_loc.GetSubVector(xdofs, posV);
-
-            const IntegrationRule &irule = GetIntegrationRule(*fes->GetFE(i));
-            const int nsp = irule.GetNPoints();
-            for (int j = 0; j < nsp; j++)
-            {
-               fes->GetFE(i)->CalcDShape(irule.IntPoint(j), dshape);
-               MultAtB(pos, dshape, Jpr);
-               if (Jpr.Det() <= 0.0) { jac_ok = 0; goto break2; }
-            }
-         }
-
-      break2:
-         int jac_ok_all = jac_ok;
-#ifdef MFEM_USE_MPI
-         if (parallel)
-         {
-            MPI_Allreduce(&jac_ok, &jac_ok_all, 1, MPI_INT, MPI_LAND,
-                          p_nlf->ParFESpace()->GetComm());
-         }
-#endif
-
-         if (jac_ok_all == 0)
-         {
-            if (print_level >= 0)
-            { mfem::out << "Scale = " << scale << " Neg det(J) found.\n"; }
-            scale *= detJ_factor; continue;
-         }
-      } // endif(!untangling)
+      // Skip the energy and residual checks when we're untangling. The
+      // untangling metrics change their denominators, which can affect the
+      // energy and residual, so their increase/decrease is not relevant.
+      if (untangling) { x_out_ok = true; break; }
 
+      // Check the changes in total energy.
       ProcessNewState(x_out);
       if (serial)
       {
@@ -460,43 +419,55 @@ double TMOPNewtonSolver::ComputeScalingFactor(const Vector &x,
          energy_out = p_nlf->GetParGridFunctionEnergy(x_out_loc);
       }
 #endif
-
-      if (untangling)
+      if (energy_out > 1.2*energy_in || std::isnan(energy_out) != 0)
       {
-         if (energy_out > energy_in || std::isnan(energy_out) != 0)
+         if (print_level >= 0)
          {
-            scale *= 0.5;
+            mfem::out << "Scale = " << scale << " Increasing energy.\n";
          }
-         else { x_out_ok = true; break; }
+         scale *= 0.5; continue;
       }
-      else
+
+      // Check the changes in the Newton residual.
+      oper->Mult(x_out, r);
+      if (have_b) { r -= b; }
+      double norm_out = Norm(r);
+
+      if (norm_out > 1.2*norm_in)
       {
-         if (energy_out > 1.2*energy_in || std::isnan(energy_out) != 0)
-         {
-            if (print_level >= 0)
-            { mfem::out << "Scale = " << scale << " Increasing energy.\n"; }
-            scale *= 0.5; continue;
-         }
+         if (print_level >= 0)
+         { mfem::out << "Scale = " << scale << " Norm increased.\n"; }
+         scale *= 0.5; continue;
+      }
+      else { x_out_ok = true; break; }
+   } // end line search
 
-         oper->Mult(x_out, r);
-         if (have_b) { r -= b; }
-         double norm = Norm(r);
-
-         if (norm > 1.2*norm0)
-         {
-            if (print_level >= 0)
-            { mfem::out << "Scale = " << scale << " Norm increased.\n"; }
-            scale *= 0.5; continue;
-         }
-         else { x_out_ok = true; break; }
-      } // endif (untangling)
-   } // enddo (i)
+   if (untangling)
+   {
+      // Update the global min detJ. Untangling metrics see this min_det_ptr.
+      if (min_detT_out > 0.0)
+      {
+         *min_det_ptr = 0.0;
+         if (print_level >= 0)
+         { mfem::out << "The mesh has been untangled at the used points!\n"; }
+      }
+      else { *min_det_ptr = untangle_factor * min_detT_out; }
+   }
 
    if (print_level >= 0)
    {
-      mfem::out << "Energy decrease: "
-                << (energy_in - energy_out) / energy_in * 100.0
-                << "% with " << scale << " scaling.\n";
+      if (untangling)
+      {
+         mfem::out << "Min det(T) change: "
+                   << min_detT_in << " -> " << min_detT_out
+                   << " with " << scale << " scaling.\n";
+      }
+      else
+      {
+         mfem::out << "Energy decrease: "
+                   << (energy_in - energy_out) / energy_in * 100.0
+                   << "% with " << scale << " scaling.\n";
+      }
    }
 
    if (x_out_ok == false) { scale = 0.0; }
@@ -508,8 +479,8 @@ void TMOPNewtonSolver::ProcessNewState(const Vector &x) const
    const NonlinearForm *nlf = dynamic_cast<const NonlinearForm *>(oper);
    const Array<NonlinearFormIntegrator*> &integs = *nlf->GetDNFI();
 
-   // Reset the update flags of all TargetConstructors.
-   // This is done to avoid repeated updates of shared TargetConstructors.
+   // Reset the update flags of all TargetConstructors. This is done to avoid
+   // repeated updates of shared TargetConstructors.
    TMOP_Integrator *ti  = NULL;
    TMOPComboIntegrator *co = NULL;
    DiscreteAdaptTC *dtc = NULL;
@@ -617,6 +588,47 @@ void TMOPNewtonSolver::UpdateDiscreteTC(const TMOP_Integrator &ti,
    }
 }
 
+double TMOPNewtonSolver::ComputeMinDet(const Vector &x_loc,
+                                       const FiniteElementSpace &fes) const
+{
+   double min_detJ = infinity();
+   const int NE = fes.GetNE(), dim = fes.GetMesh()->Dimension();
+   Array<int> xdofs;
+   DenseMatrix Jpr(dim);
+   for (int i = 0; i < NE; i++)
+   {
+      const int dof = fes.GetFE(i)->GetDof();
+      DenseMatrix dshape(dof, dim), pos(dof, dim);
+      Vector posV(pos.Data(), dof * dim);
+
+      fes.GetElementVDofs(i, xdofs);
+      x_loc.GetSubVector(xdofs, posV);
+
+      const IntegrationRule &irule = GetIntegrationRule(*fes.GetFE(i));
+      const int nsp = irule.GetNPoints();
+      for (int j = 0; j < nsp; j++)
+      {
+         fes.GetFE(i)->CalcDShape(irule.IntPoint(j), dshape);
+         MultAtB(pos, dshape, Jpr);
+         min_detJ = std::min(min_detJ, Jpr.Det());
+      }
+   }
+   double min_detT_all = min_detJ;
+#ifdef MFEM_USE_MPI
+   if (parallel)
+   {
+      auto p_nlf = dynamic_cast<const ParNonlinearForm *>(oper);
+      MPI_Allreduce(&min_detJ, &min_detT_all, 1, MPI_DOUBLE, MPI_MIN,
+                    p_nlf->ParFESpace()->GetComm());
+   }
+#endif
+   const DenseMatrix &Wideal =
+      Geometries.GetGeomToPerfGeomJac(fes.GetFE(0)->GetGeomType());
+   min_detT_all /= Wideal.Det();
+
+   return min_detT_all;
+}
+
 #ifdef MFEM_USE_MPI
 // Metric values are visualized by creating an L2 finite element functions and
 // computing the metric values at the nodes.

fem/tmop_tools.hpp

@@ -114,6 +114,9 @@ protected:
    int solver_type;
    bool parallel;
 
+   // Minimum determinant over the whole mesh. Used for mesh untangling.
+   double *min_det_ptr = nullptr;
+
    // Quadrature points that are checked for negative Jacobians etc.
    const IntegrationRule &ir;
    // These fields are relevant for mixed meshes.
@@ -131,6 +134,9 @@ protected:
 
    void UpdateDiscreteTC(const TMOP_Integrator &ti, const Vector &x_new) const;
 
+   double ComputeMinDet(const Vector &x_loc,
+                        const FiniteElementSpace &fes) const;
+
 public:
 #ifdef MFEM_USE_MPI
    TMOPNewtonSolver(MPI_Comm comm, const IntegrationRule &irule, int type = 0)
@@ -150,6 +156,8 @@ public:
       integ_order = order;
    }
 
+   void SetMinDetPtr(double *md_ptr) { min_det_ptr = md_ptr; }
+
    virtual double ComputeScalingFactor(const Vector &x, const Vector &b) const;
 
    virtual void ProcessNewState(const Vector &x) const;

general/device.cpp

@@ -43,7 +43,7 @@ CeedRestrMap ceed_restr_map;
 static const Backend::Id backend_list[Backend::NUM_BACKENDS] =
 {
    Backend::CEED_CUDA, Backend::OCCA_CUDA, Backend::RAJA_CUDA, Backend::CUDA,
-   Backend::CEED_HIP, Backend::HIP, Backend::DEBUG_DEVICE,
+   Backend::CEED_HIP, Backend::RAJA_HIP, Backend::HIP, Backend::DEBUG_DEVICE,
    Backend::OCCA_OMP, Backend::RAJA_OMP, Backend::OMP,
    Backend::CEED_CPU, Backend::OCCA_CPU, Backend::RAJA_CPU, Backend::CPU
 };
@@ -52,7 +52,7 @@ static const Backend::Id backend_list[Backend::NUM_BACKENDS] =
 static const char *backend_name[Backend::NUM_BACKENDS] =
 {
    "ceed-cuda", "occa-cuda", "raja-cuda", "cuda",
-   "ceed-hip", "hip", "debug",
+   "ceed-hip", "raja-hip", "hip", "debug",
    "occa-omp", "raja-omp", "omp",
    "ceed-cpu", "occa-cpu", "raja-cpu", "cpu"
 };
@@ -394,6 +394,8 @@ static void RajaDeviceSetup(const int dev, int &ngpu)
 {
 #ifdef MFEM_USE_CUDA
    if (ngpu <= 0) { DeviceSetup(dev, ngpu); }
+#elif defined(MFEM_USE_HIP)
+   HipDeviceSetup(dev, ngpu);
 #else
    MFEM_CONTRACT_VAR(dev);
    MFEM_CONTRACT_VAR(ngpu);
@@ -507,7 +509,8 @@ void Device::Setup(const int device)
 #endif
    if (Allows(Backend::CUDA)) { CudaDeviceSetup(dev, ngpu); }
    if (Allows(Backend::HIP)) { HipDeviceSetup(dev, ngpu); }
-   if (Allows(Backend::RAJA_CUDA)) { RajaDeviceSetup(dev, ngpu); }
+   if (Allows(Backend::RAJA_CUDA) || Allows(Backend::RAJA_HIP))
+   { RajaDeviceSetup(dev, ngpu); }
    // The check for MFEM_USE_OCCA is in the function OccaDeviceSetup().
    if (Allows(Backend::OCCA_MASK)) { OccaDeviceSetup(dev); }
    if (Allows(Backend::CEED_CPU))

general/device.hpp

@@ -46,30 +46,33 @@ struct Backend
       /** @brief [device] RAJA CUDA backend. Enabled when MFEM_USE_RAJA = YES
           and MFEM_USE_CUDA = YES. */
       RAJA_CUDA = 1 << 6,
+      /** @brief [device] RAJA HIP backend. Enabled when MFEM_USE_RAJA = YES
+          and MFEM_USE_HIP = YES. */
+      RAJA_HIP = 1 << 7,
       /** @brief [host] OCCA CPU backend: sequential execution on each MPI rank.
           Enabled when MFEM_USE_OCCA = YES. */
-      OCCA_CPU = 1 << 7,
+      OCCA_CPU = 1 << 8,
       /// [host] OCCA OpenMP backend. Enabled when MFEM_USE_OCCA = YES.
-      OCCA_OMP = 1 << 8,
+      OCCA_OMP = 1 << 9,
       /** @brief [device] OCCA CUDA backend. Enabled when MFEM_USE_OCCA = YES
           and MFEM_USE_CUDA = YES. */
-      OCCA_CUDA = 1 << 9,
+      OCCA_CUDA = 1 << 10,
       /** @brief [host] CEED CPU backend. GPU backends can still be used, but
           with expensive memory transfers. Enabled when MFEM_USE_CEED = YES. */
-      CEED_CPU  = 1 << 10,
+      CEED_CPU  = 1 << 11,
       /** @brief [device] CEED CUDA backend working together with the CUDA
           backend. Enabled when MFEM_USE_CEED = YES and MFEM_USE_CUDA = YES.
           NOTE: The current default libCEED CUDA backend is non-deterministic! */
-      CEED_CUDA = 1 << 11,
+      CEED_CUDA = 1 << 12,
       /** @brief [device] CEED HIP backend working together with the HIP
           backend. Enabled when MFEM_USE_CEED = YES and MFEM_USE_HIP = YES. */
-      CEED_HIP = 1 << 12,
+      CEED_HIP = 1 << 13,
       /** @brief [device] Debug backend: host memory is READ/WRITE protected
           while a device is in use. It allows to test the "device" code-path
           (using separate host/device memory pools and host <-> device
           transfers) without any GPU hardware. As 'DEBUG' is sometimes used
           as a macro, `_DEVICE` has been added to avoid conflicts. */
-      DEBUG_DEVICE = 1 << 13
+      DEBUG_DEVICE = 1 << 14
    };
 
    /** @brief Additional useful constants. For example, the *_MASK constants can
@@ -77,14 +80,14 @@ struct Backend
    enum
    {
       /// Number of backends: from (1 << 0) to (1 << (NUM_BACKENDS-1)).
-      NUM_BACKENDS = 14,
+      NUM_BACKENDS = 15,
 
       /// Biwise-OR of all CPU backends
       CPU_MASK = CPU | RAJA_CPU | OCCA_CPU | CEED_CPU,
       /// Biwise-OR of all CUDA backends
       CUDA_MASK = CUDA | RAJA_CUDA | OCCA_CUDA | CEED_CUDA,
       /// Biwise-OR of all HIP backends
-      HIP_MASK = HIP | CEED_HIP,
+      HIP_MASK = HIP | RAJA_HIP | CEED_HIP,
       /// Biwise-OR of all OpenMP backends
       OMP_MASK = OMP | RAJA_OMP | OCCA_OMP,
       /// Bitwise-OR of all CEED backends
@@ -93,7 +96,7 @@ struct Backend
       DEVICE_MASK = CUDA_MASK | HIP_MASK | DEBUG_DEVICE,
 
       /// Biwise-OR of all RAJA backends
-      RAJA_MASK = RAJA_CPU | RAJA_OMP | RAJA_CUDA,
+      RAJA_MASK = RAJA_CPU | RAJA_OMP | RAJA_CUDA | RAJA_HIP,
       /// Biwise-OR of all OCCA backends
       OCCA_MASK = OCCA_CPU | OCCA_OMP | OCCA_CUDA
    };

general/forall.hpp

@@ -86,77 +86,157 @@ void OmpWrap(const int N, HBODY &&h_body)
 }
 
 
-/// RAJA Cuda backend
+/// RAJA Cuda and Hip backends
 #if defined(MFEM_USE_RAJA) && defined(RAJA_ENABLE_CUDA)
-
-#if RAJA_VERSION_MAJOR == 0 && RAJA_VERSION_MINOR < 12
-using RAJA::statement::Segs;
-#else
-using RAJA::Segs;
+using cuda_launch_policy =
+   RAJA::expt::LaunchPolicy<RAJA::expt::null_launch_t, RAJA::expt::cuda_launch_t<false>>;
+using cuda_teams_x =
+   RAJA::expt::LoopPolicy<RAJA::loop_exec,RAJA::cuda_block_x_direct>;
+using cuda_threads_z =
+   RAJA::expt::LoopPolicy<RAJA::loop_exec,RAJA::cuda_thread_z_direct>;
 #endif
 
+#if defined(MFEM_USE_RAJA) && defined(RAJA_ENABLE_HIP)
+using hip_launch_policy =
+   RAJA::expt::LaunchPolicy<RAJA::expt::null_launch_t, RAJA::expt::hip_launch_t<false>>;
+using hip_teams_x =
+   RAJA::expt::LoopPolicy<RAJA::loop_exec,RAJA::hip_block_x_direct>;
+using hip_threads_z =
+   RAJA::expt::LoopPolicy<RAJA::loop_exec,RAJA::hip_thread_z_direct>;
+#endif
+
+#if defined(MFEM_USE_RAJA) && defined(RAJA_ENABLE_CUDA)
 template <const int BLOCKS = MFEM_CUDA_BLOCKS, typename DBODY>
-void RajaCudaWrap1D(const int N, DBODY &&d_body)
+void RajaCuWrap1D(const int N, DBODY &&d_body)
 {
-   // true denotes asynchronous kernel
+   //true denotes asynchronous kernel
    RAJA::forall<RAJA::cuda_exec<BLOCKS,true>>(RAJA::RangeSegment(0,N),d_body);
 }
 
 template <typename DBODY>
-void RajaCudaWrap2D(const int N, DBODY &&d_body,
-                    const int X, const int Y, const int BZ)
+void RajaCuWrap2D(const int N, DBODY &&d_body,
+                  const int X, const int Y, const int BZ)
 {
    MFEM_VERIFY(N>0, "");
    MFEM_VERIFY(BZ>0, "");
    const int G = (N+BZ-1)/BZ;
-   RAJA::kernel<RAJA::KernelPolicy<
-   RAJA::statement::CudaKernelAsync<
-   RAJA::statement::For<0, RAJA::cuda_block_x_direct,
-        RAJA::statement::For<1, RAJA::cuda_thread_x_direct,
-        RAJA::statement::For<2, RAJA::cuda_thread_y_direct,
-        RAJA::statement::For<3, RAJA::cuda_thread_z_direct,
-        RAJA::statement::Lambda<0, Segs<0>>>>>>>>>
-        (RAJA::make_tuple(RAJA::RangeSegment(0,G), RAJA::RangeSegment(0,X),
-                          RAJA::RangeSegment(0,Y), RAJA::RangeSegment(0,BZ)),
-         [=] RAJA_DEVICE (const int n)
+
+   using namespace RAJA::expt;
+   using RAJA::RangeSegment;
+
+   launch<cuda_launch_policy>
+   (DEVICE, Resources(Teams(G), Threads(X, Y, BZ)),
+    [=] RAJA_DEVICE (LaunchContext ctx)
    {
-      const int k = n*BZ + threadIdx.z;
-      if (k >= N) { return; }
-      d_body(k);
+
+      loop<cuda_teams_x>(ctx, RangeSegment(0, G), [&] (const int n)
+      {
+
+         loop<cuda_threads_z>(ctx, RangeSegment(0, BZ), [&] (const int tz)
+         {
+
+            const int k = n*BZ + tz;
+            if (k >= N) { return; }
+            d_body(k);
+
+         });
+
+      });
+
    });
+
    MFEM_GPU_CHECK(cudaGetLastError());
 }
 
 template <typename DBODY>
-void RajaCudaWrap3D(const int N, DBODY &&d_body,
-                    const int X, const int Y, const int Z)
+void RajaCuWrap3D(const int N, DBODY &&d_body,
+                  const int X, const int Y, const int Z)
 {
    MFEM_VERIFY(N>0, "");
-   RAJA::kernel<RAJA::KernelPolicy<
-   RAJA::statement::CudaKernelAsync<
-   RAJA::statement::For<0, RAJA::cuda_block_x_direct,
-        RAJA::statement::For<1, RAJA::cuda_thread_x_direct,
-        RAJA::statement::For<2, RAJA::cuda_thread_y_direct,
-        RAJA::statement::For<3, RAJA::cuda_thread_z_direct,
-        RAJA::statement::Lambda<0, Segs<0>>>>>>>>>
-        (RAJA::make_tuple(RAJA::RangeSegment(0,N), RAJA::RangeSegment(0,X),
-                          RAJA::RangeSegment(0,Y), RAJA::RangeSegment(0,Z)),
-   [=] RAJA_DEVICE (const int k) { d_body(k); });
+   using namespace RAJA::expt;
+   using RAJA::RangeSegment;
+
+   launch<cuda_launch_policy>
+   (DEVICE, Resources(Teams(N), Threads(X, Y, Z)),
+    [=] RAJA_DEVICE (LaunchContext ctx)
+   {
+
+      loop<cuda_teams_x>(ctx, RangeSegment(0, N), d_body);
+
+   });
+
    MFEM_GPU_CHECK(cudaGetLastError());
 }
 
 #endif
 
+#if defined(MFEM_USE_RAJA) && defined(RAJA_ENABLE_HIP)
+template <const int BLOCKS = MFEM_HIP_BLOCKS, typename DBODY>
+void RajaHipWrap1D(const int N, DBODY &&d_body)
+{
+   //true denotes asynchronous kernel
+   RAJA::forall<RAJA::hip_exec<BLOCKS,true>>(RAJA::RangeSegment(0,N),d_body);
+}
+
+template <typename DBODY>
+void RajaHipWrap2D(const int N, DBODY &&d_body,
+                   const int X, const int Y, const int BZ)
+{
+   MFEM_VERIFY(N>0, "");
+   MFEM_VERIFY(BZ>0, "");
+   const int G = (N+BZ-1)/BZ;
+
+   using namespace RAJA::expt;
+   using RAJA::RangeSegment;
+
+   launch<hip_launch_policy>
+   (DEVICE, Resources(Teams(G), Threads(X, Y, BZ)),
+    [=] RAJA_DEVICE (LaunchContext ctx)
+   {
+
+      loop<hip_teams_x>(ctx, RangeSegment(0, G), [&] (const int n)
+      {
+
+         loop<hip_threads_z>(ctx, RangeSegment(0, BZ), [&] (const int tz)
+         {
+
+            const int k = n*BZ + tz;
+            if (k >= N) { return; }
+            d_body(k);
+
+         });
+
+      });
+
+   });
+
+   MFEM_GPU_CHECK(hipGetLastError());
+}
+
+template <typename DBODY>
+void RajaHipWrap3D(const int N, DBODY &&d_body,
+                   const int X, const int Y, const int Z)
+{
+   MFEM_VERIFY(N>0, "");
+   using namespace RAJA::expt;
+   using RAJA::RangeSegment;
+
+   launch<hip_launch_policy>
+   (DEVICE, Resources(Teams(N), Threads(X, Y, Z)),
+    [=] RAJA_DEVICE (LaunchContext ctx)
+   {
+
+      loop<hip_teams_x>(ctx, RangeSegment(0, N), d_body);
+
+   });
+
+   MFEM_GPU_CHECK(hipGetLastError());
+}
+#endif
 
 /// RAJA OpenMP backend
 #if defined(MFEM_USE_RAJA) && defined(RAJA_ENABLE_OPENMP)
 
-#if RAJA_VERSION_MAJOR == 0 && RAJA_VERSION_MINOR < 12
-using RAJA::statement::Segs;
-#else
-using RAJA::Segs;
-#endif
-
 template <typename HBODY>
 void RajaOmpWrap(const int N, HBODY &&h_body)
 {
@@ -319,9 +399,19 @@ inline void ForallWrap(const bool use_dev, const int N,
    // If Backend::RAJA_CUDA is allowed, use it
    if (Device::Allows(Backend::RAJA_CUDA))
    {
-      if (DIM == 1) { return RajaCudaWrap1D(N, d_body); }
-      if (DIM == 2) { return RajaCudaWrap2D(N, d_body, X, Y, Z); }
-      if (DIM == 3) { return RajaCudaWrap3D(N, d_body, X, Y, Z); }
+      if (DIM == 1) { return RajaCuWrap1D(N, d_body); }
+      if (DIM == 2) { return RajaCuWrap2D(N, d_body, X, Y, Z); }
+      if (DIM == 3) { return RajaCuWrap3D(N, d_body, X, Y, Z); }
+   }
+#endif
+
+#if defined(MFEM_USE_RAJA) && defined(RAJA_ENABLE_HIP)
+   // If Backend::RAJA_HIP is allowed, use it
+   if (Device::Allows(Backend::RAJA_HIP))
+   {
+      if (DIM == 1) { return RajaHipWrap1D(N, d_body); }
+      if (DIM == 2) { return RajaHipWrap2D(N, d_body, X, Y, Z); }
+      if (DIM == 3) { return RajaHipWrap3D(N, d_body, X, Y, Z); }
    }
 #endif
 

general/hip.cpp

@@ -125,7 +125,7 @@ void* HipMemcpyDtoDAsync(void* dst, const void *src, size_t bytes)
 void* HipMemcpyDtoH(void *dst, const void *src, size_t bytes)
 {
 #ifdef MFEM_USE_HIP
-#ifdef MFEM_TRACK_HPI_MEM
+#ifdef MFEM_TRACK_HIP_MEM
    mfem::out << "HipMemcpyDtoH(): copying " << bytes << " bytes from "
              << src << " to " << dst << " ... " << std::flush;
 #endif

general/mem_manager.cpp

@@ -482,7 +482,8 @@ public:
       HostMemorySpace(),
       name(mm.GetUmpireAllocatorHostName()),
       rm(umpire::ResourceManager::getInstance()),
-      h_allocator(rm.isAllocator(name)? rm.getAllocator(name):
+      h_allocator((!std::strcmp(name, "HOST") || rm.isAllocator(name)) ?
+                  rm.getAllocator(name) :
                   rm.makeAllocator<umpire::strategy::DynamicPool>
                   (name, rm.getAllocator("HOST"))),
       strat(h_allocator.getAllocationStrategy()) { }
@@ -506,7 +507,8 @@ public:
       DeviceMemorySpace(),
       name(mm.GetUmpireAllocatorDeviceName()),
       rm(umpire::ResourceManager::getInstance()),
-      d_allocator(rm.isAllocator(name)? rm.getAllocator(name):
+      d_allocator((!std::strcmp(name, "DEVICE") || rm.isAllocator(name)) ?
+                  rm.getAllocator(name) :
                   rm.makeAllocator<umpire::strategy::DynamicPool>
                   (name, rm.getAllocator("DEVICE"))) { }
    void Alloc(Memory &base) { base.d_ptr = d_allocator.allocate(base.bytes); }

general/text.hpp

@@ -12,6 +12,7 @@
 #ifndef MFEM_TEXT
 #define MFEM_TEXT
 
+#include "../config/config.hpp"
 #include <istream>
 #include <iomanip>
 #include <sstream>
@@ -24,6 +25,8 @@ namespace mfem
 
 // Utilities for text parsing
 
+using std::to_string;
+
 /// Check if the stream starts with @a comment_char. If so skip it.
 inline void skip_comment_lines(std::istream &is, const char comment_char)
 {
@@ -47,18 +50,6 @@ inline void filter_dos(std::string &line)
    }
 }
 
-/// Convert an integer to an std::string.
-inline std::string to_string(int i)
-{
-   std::stringstream ss;
-   ss << i;
-
-   // trim leading spaces
-   std::string out_str = ss.str();
-   out_str = out_str.substr(out_str.find_first_not_of(" \t"));
-   return out_str;
-}
-
 /// Convert an integer to a 0-padded string with the given number of @a digits
 inline std::string to_padded_string(int i, int digits)
 {

general/zstr.hpp

@@ -71,7 +71,7 @@ static std::string strerror()
    }
 #elif (_POSIX_C_SOURCE >= 200112L || _XOPEN_SOURCE >= 600) && ! _GNU_SOURCE || \
       defined(__APPLE__) || defined(__FreeBSD__) || defined(__OpenBSD__) || \
-      defined(__NetBSD__) || defined(__DragonFly__)
+      defined(__NetBSD__) || defined(__DragonFly__) || defined(__EMSCRIPTEN__)
    // XSI-compliant strerror_r()
    if (strerror_r(errno, &buff[0], buff.size()) != 0)
    {

linalg/CMakeLists.txt

@@ -10,11 +10,13 @@
 # CONTRIBUTING.md for details.
 
 list(APPEND SRCS
+  auxiliary.cpp
   blockmatrix.cpp
   blockoperator.cpp
   blockvector.cpp
   complex_operator.cpp
   densemat.cpp
+  symmat.cpp
   handle.cpp
   matrix.cpp
   ode.cpp
@@ -26,11 +28,13 @@ list(APPEND SRCS
   )
 
 list(APPEND HDRS
+  auxiliary.hpp
   blockmatrix.hpp
   blockoperator.hpp
   blockvector.hpp
   complex_operator.hpp
   densemat.hpp
+  symmat.hpp
   dtensor.hpp
   handle.hpp
   invariants.hpp

linalg/amgxsolver.cpp

@@ -30,10 +30,16 @@ int AmgXSolver::count = 0;
 
 AMGX_resources_handle AmgXSolver::rsrc = nullptr;
 
+AmgXSolver::AmgXSolver()
+   : ConvergenceCheck(false) {};
+
 AmgXSolver::AmgXSolver(const AMGX_MODE amgxMode_, const bool verbose)
 {
    amgxMode = amgxMode_;
 
+   if (amgxMode == AmgXSolver::SOLVER) { ConvergenceCheck = true;}
+   else { ConvergenceCheck = false;}
+
    DefaultParameters(amgxMode, verbose);
 
    InitSerial();
@@ -47,6 +53,9 @@ AmgXSolver::AmgXSolver(const MPI_Comm &comm,
    std::string config;
    amgxMode = amgxMode_;
 
+   if (amgxMode == AmgXSolver::SOLVER) { ConvergenceCheck = true;}
+   else { ConvergenceCheck = false;}
+
    DefaultParameters(amgxMode, verbose);
 
    InitExclusiveGPU(comm);
@@ -58,6 +67,9 @@ AmgXSolver::AmgXSolver(const MPI_Comm &comm, const int nDevs,
    std::string config;
    amgxMode = amgxMode_;
 
+   if (amgxMode == AmgXSolver::SOLVER) { ConvergenceCheck = true;}
+   else { ConvergenceCheck = false;}
+
    DefaultParameters(amgxMode_, verbose);
 
    InitMPITeams(comm, nDevs);
@@ -178,6 +190,11 @@ void AmgXSolver::ReadParameters(const std::string config,
    configSrc = source;
 }
 
+void AmgXSolver::SetConvergenceCheck(bool setConvergenceCheck_)
+{
+   ConvergenceCheck = setConvergenceCheck_;
+}
+
 void AmgXSolver::DefaultParameters(const AMGX_MODE amgxMode_,
                                    const bool verbose)
 {
@@ -201,8 +218,8 @@ void AmgXSolver::DefaultParameters(const AMGX_MODE amgxMode_,
       {
          amgx_config = amgx_config + ",\n"
                        "   \"obtain_timings\": 1, \n"
-                       "   \"monitor_residual\": 1, \n"
                        "   \"print_grid_stats\": 1, \n"
+                       "   \"monitor_residual\": 1, \n"
                        "   \"print_solve_stats\": 1 \n";
       }
       else
@@ -238,12 +255,12 @@ void AmgXSolver::DefaultParameters(const AMGX_MODE amgxMode_,
                     "  \"convergence\": \"RELATIVE_MAX\", \n"
                     "  \"scope\": \"main\", \n"
                     "  \"tolerance\": 1e-12, \n"
+                    "  \"monitor_residual\": 1, \n"
                     "  \"norm\": \"L2\" ";
       if (verbose)
       {
          amgx_config = amgx_config + ", \n"
                        "        \"obtain_timings\": 1, \n"
-                       "        \"monitor_residual\": 1, \n"
                        "        \"print_grid_stats\": 1, \n"
                        "        \"print_solve_stats\": 1 \n";
       }
@@ -884,7 +901,7 @@ void AmgXSolver::Mult(const Vector& B, Vector& X) const
 
       AMGX_SOLVE_STATUS   status;
       AMGX_solver_get_status(solver, &status);
-      if (status != AMGX_SOLVE_SUCCESS && amgxMode == SOLVER)
+      if (status != AMGX_SOLVE_SUCCESS && ConvergenceCheck)
       {
          if (status == AMGX_SOLVE_DIVERGED)
          {

linalg/amgxsolver.hpp

@@ -73,13 +73,16 @@ public:
    /// Flags to configure AmgXSolver as a solver or preconditioner
    enum AMGX_MODE {SOLVER, PRECONDITIONER};
 
+   /// Flag to check for convergence
+   bool ConvergenceCheck;
+
    /**
       Flags to determine whether user solver settings are defined internally in
       the source code or will be read through an external JSON file.
    */
    enum CONFIG_SRC {INTERNAL, EXTERNAL, UNDEFINED};
 
-   AmgXSolver() = default;
+   AmgXSolver();
 
    /**
       Configures AmgX with a default configuration based on the AmgX mode, and
@@ -162,6 +165,9 @@ public:
    */
    void DefaultParameters(const AMGX_MODE amgxMode_, const bool verbose);
 
+   /// Add a check for convergence after applying Mult.
+   void SetConvergenceCheck(bool setConvergenceCheck_=true);
+
    ~AmgXSolver();
 
    void Finalize();

linalg/auxiliary.hpp

@@ -0,0 +1,285 @@
+// Copyright (c) 2010-2020, Lawrence Livermore National Security, LLC. Produced
+// at the Lawrence Livermore National Laboratory. All Rights reserved. See files
+// LICENSE and NOTICE for details. LLNL-CODE-806117.
+//
+// This file is part of the MFEM library. For more information and source code
+// availability visit https://mfem.org.
+//
+// MFEM is free software; you can redistribute it and/or modify it under the
+// terms of the BSD-3 license. We welcome feedback and contributions, see file
+// CONTRIBUTING.md for details.
+
+#ifndef MFEM_AUXILIARY
+#define MFEM_AUXILIARY
+
+#include "../config/config.hpp"
+
+#ifdef MFEM_USE_MPI
+
+#include "../general/tic_toc.hpp"
+#include "solvers.hpp"
+
+namespace mfem
+{
+
+// forward declarations
+class Coefficient;
+class MatrixCoefficient;
+class ParMesh;
+class ParBilinearForm;
+class ParDiscreteLinearOperator;
+
+/** @brief Auxiliary space solvers for MatrixFreeAMS preconditioner
+
+    Given an operator A and a transfer G, this will create a solver
+    that approximates (G^T A G)^{-1}. Used for two different
+    auxiliary spaces in the AMS cycle.
+
+    The produced solver is based on a low-order refined discretization
+    for the high-order H1 problem. */
+class MatrixFreeAuxiliarySpace : public Solver
+{
+public:
+   /** @brief Pi space constructor
+
+       In the AMS framework this auxiliary space has two coefficients.
+
+       @param mesh_lor       Low-order refined auxiliary mesh
+       @param alpha_coeff    coefficient on curl-curl term (1 if null)
+       @param beta_coeff     coefficient on mass term (1 if null)
+       @param beta_mcoeff    matrix coefficient on mass term
+       @param ess_bdr        attributes for essential boundaries
+       @param curlcurl_oper  High-order operator for the system
+       @param pi             Intentity interpolation operator
+       @param useAmgX_       Use AmgX instead of hypre for auxiliary solves
+       @param cg_iterations  number of CG iterations used to invert
+                             auxiliary system, choosing 0 means to use a
+                             single V-cycle
+   */
+   MatrixFreeAuxiliarySpace(
+      ParMesh& mesh_lor, Coefficient* alpha_coeff, Coefficient* beta_coeff,
+      MatrixCoefficient* beta_mcoeff,
+      Array<int>& ess_bdr, Operator& curlcurl_oper, Operator& pi,
+#ifdef MFEM_USE_AMGX
+      bool useAmgX_,
+#endif
+      int cg_iterations = 0);
+
+   // Complex Pi space constructor
+   MatrixFreeAuxiliarySpace(
+      ParMesh& mesh_lor, Coefficient* alpha_coeff, Coefficient* beta_coeff,
+      Coefficient* beta_imag, Coefficient* abs_beta_imag,
+      MatrixCoefficient* beta_mcoeff,
+      Array<int>& ess_bdr, Operator& curlcurl_oper, Operator *oper_complex,
+      Operator& pi,
+#ifdef MFEM_USE_AMGX
+      bool useAmgX_,
+#endif
+      int cg_iterations = 0);
+
+   /** @brief G space constructor
+
+       This has one coefficient in the AMS framework.
+
+       @param mesh_lor       Low-order refined auxiliary mesh
+       @param beta_coeff     coefficient on mass term (1 if null)
+       @param beta_mcoeff    matrix coefficient on mass term
+       @param ess_bdr        attributes for essential boundaries
+       @param curlcurl_oper  High-order operator for the system
+       @param g              Gradient interpolation operator
+       @param useAmgX_       Use AmgX instead of hypre for auxiliary solves
+       @param cg_iterations  number of CG iterations used to invert
+                             auxiliary system, choosing 0 means to
+                             use a single V-cycle
+   */
+   MatrixFreeAuxiliarySpace(
+      ParMesh& mesh_lor, Coefficient* beta_coeff,
+      MatrixCoefficient* beta_mcoeff, Array<int>& ess_bdr,
+      Operator& curlcurl_oper, Operator& g,
+#ifdef MFEM_USE_AMGX
+      bool useAmgX_,
+#endif
+      int cg_iterations = 1);
+
+   // Complex G space constructor
+   MatrixFreeAuxiliarySpace(
+      ParMesh& mesh_lor, Coefficient* beta_coeff, Coefficient* beta_imag,
+      Coefficient* abs_beta_imag,
+      MatrixCoefficient* beta_mcoeff, Array<int>& ess_bdr,
+      Operator& curlcurl_oper, Operator *oper_complex, Operator& g,
+#ifdef MFEM_USE_AMGX
+      bool useAmgX_,
+#endif
+      int cg_iterations = 1);
+
+   ~MatrixFreeAuxiliarySpace();
+
+   void Mult(const Vector& x, Vector& y) const;
+
+   void SetOperator(const Operator& op) {}
+
+private:
+   /** @brief Helper routine for constructors.
+
+       @param system_dimension is passed to HypreBoomerAMG::SetSystemsOptions
+   */
+   void SetupAMG(int system_dimension);
+   void SetupVCycle();
+
+   /// inner_cg_iterations > 99 applies an exact solve here
+   void SetupCG(Operator& curlcurl_oper, Operator& conn,
+                int inner_cg_iterations);
+
+   void SetupGMRES(Operator& curlcurl_oper, Operator& conn);
+
+   void SetupPMHSS();
+
+   MPI_Comm comm;
+   Array<int> ess_tdof_list;
+   HypreParMatrix * aspacematrix;
+   HypreParMatrix * aspacematrix_complex;
+   HypreParMatrix * aspacematrix_imag;
+   Solver * aspacepc;
+   Operator* matfree;
+   CGSolver* cg;
+   GMRESSolver* gmres;
+   GMRESSolver* gmres_PMHSS;
+   Operator* aspacewrapper;
+#ifdef MFEM_USE_AMGX
+   const bool useAmgX;
+#endif
+   mutable int inner_aux_iterations;
+
+   const bool imagBdry;
+
+   Complex_PMHSS *PMHSS = NULL;
+
+   Array<int> offsets;
+   Array<int> offsets_nd;
+   BlockDiagonalPreconditioner *BlockDP;
+
+   BlockOperator *conn_block;
+};
+
+
+/** @brief Perform AMS cycle with generic Operator objects.
+
+    Most users should use MatrixFreeAMS, which wraps this. */
+class GeneralAMS : public Solver
+{
+public:
+   /** @brief Constructor.
+
+       Most of these arguments just need a Mult() operation,
+       but pi and g also require MultTranspose() */
+   GeneralAMS(const Operator& curlcurl_op_,
+              Operator *oper_complex,
+              const Operator& pi_,
+              const Operator& gradient_,
+              const Operator& pispacesolver_,
+              const Operator& gspacesolver_,
+              const Operator& smoother_,
+              const Array<int>& ess_tdof_list_);
+   virtual ~GeneralAMS();
+
+   /// in principle this should set A_ = op;
+   void SetOperator(const Operator &op) {}
+
+   virtual void Mult(const Vector& x, Vector& y) const;
+
+private:
+   const Operator& curlcurl_op;
+   Operator *oper_complex;
+   const Operator& pi;
+   const Operator& gradient;
+   const Operator& pispacesolver;
+   const Operator& gspacesolver;
+   const Operator& smoother;
+   const Array<int> ess_tdof_list;
+
+   void FormResidual(const Vector& rhs, const Vector& x,
+                     Vector& residual) const;
+};
+
+
+/** @brief An auxiliary Maxwell solver for a high-order curl-curl
+    system without high-order assembly.
+
+    The auxiliary space solves are done using a low-order refined approach,
+    but all the interpolation operators, residuals, etc. are done in a
+    matrix-free manner.
+
+    See Barker and Kolev, Matrix-free preconditioning for high-order H(curl)
+    discretizations (https://doi.org/10.1002/nla.2348) */
+class MatrixFreeAMS : public Solver
+{
+public:
+   /** @brief Construct matrix-free AMS preconditioner
+
+       @param aform        BilinearForm for curl-curl problem, generally will
+                           have a CurlCurlIntegrator and possibly a
+                           VectorFEMassIntegrator.
+       @param oper         Operator to precondition.
+       @param nd_fespace   Underlying Nedelec finite element space.
+       @param alpha_coeff  coefficient on curl-curl term in Maxwell problem
+                           (can be null, in which case constant 1 is assumed)
+       @param beta_coeff   (scalar) coefficient on mass term in Maxwell problem
+       @param beta_mcoeff  (matrix) coefficient on mass term
+       @param ess_bdr      boundary *attributes* that are marked essential. In
+                           contrast to other MFEM cases, these are *attributes*
+                           not dofs, because we need to apply these boundary
+                           conditions to different bilinear forms.
+       @param useAmgX      use AmgX (instead of hypre) for LOR problems
+       @param inner_pi_its number of CG iterations on auxiliary pi space,
+                           may need more for difficult coefficients
+       @param inner_g_its  number of CG iterations on auxiliary g space,
+                           may need more for difficult coefficients
+       @param nd_smoother  optional user-provided smoother for Nedelec space,
+                           this object takes ownership and will delete.
+   */
+   MatrixFreeAMS(ParBilinearForm& aform, Operator& oper, Operator *oper_complex,
+                 ParFiniteElementSpace& nd_fespace, Coefficient* alpha_coeff,
+                 Coefficient* beta_coeff, Coefficient* beta_imag,
+                 Coefficient* abs_beta_imag, MatrixCoefficient* beta_mcoeff,
+                 Array<int>& ess_bdr,
+#ifdef MFEM_USE_AMGX
+                 bool useAmgX = false,
+#endif
+                 int inner_pi_its = 0, int inner_g_its = 1,
+                 Solver* nd_smoother = NULL);
+
+   ~MatrixFreeAMS();
+
+   void SetOperator(const Operator &op) {}
+
+   void Mult(const Vector& x, Vector& y) const { general_ams->Mult(x, y); }
+
+private:
+   GeneralAMS * general_ams;
+
+   Solver * smoother;
+   ParDiscreteLinearOperator * pa_grad;
+   OperatorPtr Gradient;
+   ParDiscreteLinearOperator * pa_interp;
+   OperatorPtr Pi;
+
+   Solver * Gspacesolver;
+   Solver * Pispacesolver;
+
+   ParFiniteElementSpace * h1_fespace;
+   ParFiniteElementSpace * h1_fespace_d;
+
+   Array<int> offsets_nd;
+   Array<int> offsets_vector;
+   Array<int> offsets_scalar;
+
+   BlockOperator *Pi_block;
+   BlockOperator *Gradient_block;
+   BlockOperator *smoother_block;
+};
+
+} // namespace mfem
+
+#endif // MFEM_USE_MPI
+
+#endif

linalg/blockvector.cpp

@@ -101,6 +101,7 @@ void BlockVector::Update(Vector & data, const Array<int> & bOffsets)
    {
       blocks[i].MakeRef(data, blockOffsets[i], BlockSize(i));
    }
+   MakeRef(data, 0, blockOffsets[numBlocks]);
 }
 
 void BlockVector::Update(const Array<int> &bOffsets)

linalg/complex_operator.cpp

@@ -287,10 +287,7 @@ void ComplexUMFPackSolver::Init()
 
 void ComplexUMFPackSolver::SetOperator(const Operator &op)
 {
-   int *Ap, *Ai;
    void *Symbolic;
-   double *Ax;
-   double *Az;
 
    if (Numeric)
    {
@@ -322,10 +319,11 @@ void ComplexUMFPackSolver::SetOperator(const Operator &op)
    width = mat->real().Width();
    MFEM_VERIFY(width == height, "not a square matrix");
 
-   Ap = mat->real().GetI(); // assuming real and imag have the same sparsity
-   Ai = mat->real().GetJ();
-   Ax = mat->real().GetData();
-   Az = mat->imag().GetData();
+   const int * Ap =
+      mat->real().HostReadI(); // assuming real and imag have the same sparsity
+   const int * Ai = mat->real().HostReadJ();
+   const double * Ax = mat->real().HostReadData();
+   const double * Az = mat->imag().HostReadData();
 
    if (!use_long_ints)
    {
@@ -395,6 +393,10 @@ void ComplexUMFPackSolver::Mult(const Vector &b, Vector &x) const
    if (mat == NULL)
       mfem_error("ComplexUMFPackSolver::Mult : matrix is not set!"
                  " Call SetOperator first!");
+
+   b.HostRead();
+   x.HostReadWrite();
+
    int n = b.Size()/2;
    double * datax = x.GetData();
    double * datab = b.GetData();
@@ -413,8 +415,8 @@ void ComplexUMFPackSolver::Mult(const Vector &b, Vector &x) const
    if (!use_long_ints)
    {
       int status =
-         umfpack_zi_solve(UMFPACK_Aat, mat->real().GetI(), mat->real().GetJ(),
-                          mat->real().GetData(), mat->imag().GetData(),
+         umfpack_zi_solve(UMFPACK_Aat, mat->real().HostReadI(), mat->real().HostReadJ(),
+                          mat->real().HostReadData(), mat->imag().HostReadData(),
                           datax, &datax[n], datab, &datab[n], Numeric, Control, Info);
       umfpack_zi_report_info(Control, Info);
       if (status < 0)
@@ -426,8 +428,8 @@ void ComplexUMFPackSolver::Mult(const Vector &b, Vector &x) const
    else
    {
       SuiteSparse_long status =
-         umfpack_zl_solve(UMFPACK_Aat,AI,AJ,mat->real().GetData(),
-                          mat->imag().GetData(),
+         umfpack_zl_solve(UMFPACK_Aat,AI,AJ,mat->real().HostReadData(),
+                          mat->imag().HostReadData(),
                           datax,&datax[n],datab,&datab[n],Numeric,Control,Info);
 
       umfpack_zl_report_info(Control, Info);
@@ -448,6 +450,8 @@ void ComplexUMFPackSolver::MultTranspose(const Vector &b, Vector &x) const
    if (mat == NULL)
       mfem_error("ComplexUMFPackSolver::Mult : matrix is not set!"
                  " Call SetOperator first!");
+   b.HostRead();
+   x.HostReadWrite();
    int n = b.Size()/2;
    double * datax = x.GetData();
    double * datab = b.GetData();
@@ -467,8 +471,8 @@ void ComplexUMFPackSolver::MultTranspose(const Vector &b, Vector &x) const
    if (!use_long_ints)
    {
       int status =
-         umfpack_zi_solve(UMFPACK_A, mat->real().GetI(), mat->real().GetJ(),
-                          mat->real().GetData(), mat->imag().GetData(),
+         umfpack_zi_solve(UMFPACK_A, mat->real().HostReadI(), mat->real().HostReadJ(),
+                          mat->real().HostReadData(), mat->imag().HostReadData(),
                           datax, &datax[n], datab, &datab[n], Numeric, Control, Info);
       umfpack_zi_report_info(Control, Info);
       if (status < 0)
@@ -480,8 +484,8 @@ void ComplexUMFPackSolver::MultTranspose(const Vector &b, Vector &x) const
    else
    {
       SuiteSparse_long status =
-         umfpack_zl_solve(UMFPACK_A,AI,AJ,mat->real().GetData(),
-                          mat->imag().GetData(),
+         umfpack_zl_solve(UMFPACK_A,AI,AJ,mat->real().HostReadData(),
+                          mat->imag().HostReadData(),
                           datax,&datax[n],datab,&datab[n],Numeric,Control,Info);
 
       umfpack_zl_report_info(Control, Info);

linalg/hypre.cpp

@@ -1279,6 +1279,7 @@ void HypreParMatrix::operator*=(double s)
 static void get_sorted_rows_cols(const Array<int> &rows_cols,
                                  Array<HYPRE_Int> &hypre_sorted)
 {
+   rows_cols.HostRead();
    hypre_sorted.SetSize(rows_cols.Size());
    bool sorted = true;
    for (int i = 0; i < rows_cols.Size(); i++)
@@ -1995,6 +1996,8 @@ void EliminateBC(HypreParMatrix &A, HypreParMatrix &Ae,
    double *data_offd = hypre_CSRMatrixData(A_offd);
 #endif
 
+   ess_dof_list.HostRead();
+
    for (int i = 0; i < ess_dof_list.Size(); i++)
    {
       int r = ess_dof_list[i];
@@ -2161,6 +2164,7 @@ HypreSmoother::HypreSmoother() : Solver()
    B = X = V = Z = NULL;
    X0 = X1 = NULL;
    fir_coeffs = NULL;
+   A_is_symmetric = false;
 }
 
 HypreSmoother::HypreSmoother(HypreParMatrix &_A, int _type,
@@ -2180,6 +2184,7 @@ HypreSmoother::HypreSmoother(HypreParMatrix &_A, int _type,
    B = X = V = Z = NULL;
    X0 = X1 = NULL;
    fir_coeffs = NULL;
+   A_is_symmetric = false;
 
    SetOperator(_A);
 }
@@ -2467,6 +2472,16 @@ void HypreSmoother::Mult(const Vector &b, Vector &x) const
    Mult(*B, *X);
 }
 
+void HypreSmoother::MultTranspose(const Vector &b, Vector &x) const
+{
+   if (A_is_symmetric || type == 0 || type == 1 || type == 5)
+   {
+      Mult(b, x);
+      return;
+   }
+   mfem_error("HypreSmoother::MultTranspose (...) : undefined!\n");
+}
+
 HypreSmoother::~HypreSmoother()
 {
    if (B) { delete B; }
@@ -2511,6 +2526,14 @@ void HypreSolver::Mult(const HypreParVector &b, HypreParVector &x) const
       mfem_error("HypreSolver::Mult (...) : HypreParMatrix A is missing");
       return;
    }
+
+   if (!iterative_mode)
+   {
+      x = 0.0;
+   }
+
+   b.HostRead();
+   x.HostReadWrite();
    if (!setup_called)
    {
       err = SetupFcn()(*this, *A, b, x);
@@ -2526,10 +2549,6 @@ void HypreSolver::Mult(const HypreParVector &b, HypreParVector &x) const
       setup_called = 1;
    }
 
-   if (!iterative_mode)
-   {
-      x = 0.0;
-   }
    err = SolveFcn()(*this, *A, b, x);
    if (error_mode == WARN_HYPRE_ERRORS)
    {
@@ -2550,7 +2569,7 @@ void HypreSolver::Mult(const Vector &b, Vector &x) const
       return;
    }
    auto b_data = b.HostRead();
-   auto x_data = x.HostWrite();
+   auto x_data = iterative_mode ? x.HostReadWrite() : x.HostWrite();
    if (B == NULL)
    {
       B = new HypreParVector(A->GetComm(),
@@ -2673,6 +2692,11 @@ void HyprePCG::Mult(const HypreParVector &b, HypreParVector &x) const
 
    HYPRE_ParCSRMatrixGetComm(*A, &comm);
 
+   if (!iterative_mode)
+   {
+      x = 0.0;
+   }
+
    if (!setup_called)
    {
       if (print_level > 0 && print_level < 3)
@@ -2681,6 +2705,8 @@ void HyprePCG::Mult(const HypreParVector &b, HypreParVector &x) const
          hypre_BeginTiming(time_index);
       }
 
+      b.HostRead();
+      x.HostReadWrite();
       HYPRE_ParCSRPCGSetup(pcg_solver, *A, b, x);
       setup_called = 1;
 
@@ -2699,14 +2725,8 @@ void HyprePCG::Mult(const HypreParVector &b, HypreParVector &x) const
       hypre_BeginTiming(time_index);
    }
 
-   if (!iterative_mode)
-   {
-      x = 0.0;
-   }
-
    b.HostRead();
    x.HostReadWrite();
-
    HYPRE_ParCSRPCGSolve(pcg_solver, *A, b, x);
 
    if (print_level > 0)
@@ -3703,6 +3723,9 @@ void HypreAMS::Init(ParFiniteElementSpace *edge_fespace)
       }
       x = x_coord.ParallelProject();
       y = y_coord.ParallelProject();
+
+      x->HostReadWrite();
+      y->HostReadWrite();
       if (sdim == 2)
       {
          z = NULL;
@@ -3711,6 +3734,7 @@ void HypreAMS::Init(ParFiniteElementSpace *edge_fespace)
       else
       {
          z = z_coord.ParallelProject();
+         z->HostReadWrite();
          HYPRE_AMSSetCoordinateVectors(ams, *x, *y, *z);
       }
    }

linalg/hypre.hpp

@@ -638,6 +638,9 @@ protected:
    /// Combined coefficients for windowing and Chebyshev polynomials.
    double* fir_coeffs;
 
+   /// A flag that indicates whether the linear system matrix A is symmetric
+   bool A_is_symmetric;
+
 public:
    /** Hypre smoother types:
        0    = Jacobi
@@ -684,6 +687,12 @@ public:
        entries in the associated matrix. */
    void SetPositiveDiagonal(bool pos = true) { pos_l1_norms = pos; }
 
+   /** Explicitly indicate whether the linear system matrix A is symmetric. If A
+       is symmetric, the smoother will also be symmetric. In this case, calling
+       MultTranspose will be redirected to Mult. (This is also done if the
+       smoother is diagonal.) By default, A is assumed to be nonsymmetric. */
+   void SetOperatorSymmetry(bool is_sym) { A_is_symmetric = is_sym; }
+
    /** Set/update the associated operator. Must be called after setting the
        HypreSmoother type and options. */
    virtual void SetOperator(const Operator &op);
@@ -692,6 +701,9 @@ public:
    virtual void Mult(const HypreParVector &b, HypreParVector &x) const;
    virtual void Mult(const Vector &b, Vector &x) const;
 
+   /// Apply transpose of the smoother to relax the linear system Ax=b
+   virtual void MultTranspose(const Vector &b, Vector &x) const;
+
    virtual ~HypreSmoother();
 };
 

linalg/invariants.hpp

@@ -62,8 +62,6 @@ protected:
    const scalar_t *D; // Always points to external data or is empty
    scalar_t *DaJ, *DJt, *DXt, *DYt;
 
-   scalar_t sign_detJ;
-
    enum EvalMasks
    {
       HAVE_I1   = 1,
@@ -96,8 +94,7 @@ protected:
    {
       eval_state |= HAVE_I2b;
       const scalar_t det = J[0]*J[3] - J[1]*J[2];
-      sign_detJ = scalar_ops::sign(det);
-      I2b = sign_detJ*det;
+      I2b = det;
    }
    void Eval_dI1()
    {
@@ -136,10 +133,10 @@ protected:
       // I2b = det(J)
       // dI2b = adj(J)^T
       Get_I2b();
-      dI2b[0] =  sign_detJ*J[3];
-      dI2b[1] = -sign_detJ*J[2];
-      dI2b[2] = -sign_detJ*J[1];
-      dI2b[3] =  sign_detJ*J[0];
+      dI2b[0] =  J[3];
+      dI2b[1] = -J[2];
+      dI2b[2] = -J[1];
+      dI2b[3] =  J[0];
    }
    void Eval_DaJ() // D adj(J) = D dI2b^t
    {
@@ -516,8 +513,6 @@ protected:
    const scalar_t *D; // Always points to external data or is empty
    scalar_t *DaJ, *DJt, *DdI2t, *DXt, *DYt;
 
-   scalar_t sign_detJ;
-
    enum EvalMasks
    {
       HAVE_I1     = 1,
@@ -585,8 +580,6 @@ protected:
       eval_state |= HAVE_I3b;
       I3b = J[0]*(J[4]*J[8] - J[7]*J[5]) - J[1]*(J[3]*J[8] - J[5]*J[6]) +
             J[2]*(J[3]*J[7] - J[4]*J[6]);
-      sign_detJ = scalar_ops::sign(I3b);
-      I3b = sign_detJ*I3b;
    }
    scalar_t Get_I3b_p()  // I3b^{-2/3}
    {
@@ -594,7 +587,7 @@ protected:
       {
          eval_state |= HAVE_I3b_p;
          const scalar_t i3b = Get_I3b();
-         I3b_p = sign_detJ*scalar_ops::pow(i3b, -2, 3);
+         I3b_p = scalar_ops::pow(i3b, -2, 3);
       }
       return I3b_p;
    }
@@ -680,15 +673,15 @@ protected:
       eval_state |= HAVE_dI3b;
       // I3b = det(J)
       // dI3b = adj(J)^T
-      dI3b[0] = sign_detJ*(J[4]*J[8] - J[5]*J[7]);  // 0  3  6
-      dI3b[1] = sign_detJ*(J[5]*J[6] - J[3]*J[8]);  // 1  4  7
-      dI3b[2] = sign_detJ*(J[3]*J[7] - J[4]*J[6]);  // 2  5  8
-      dI3b[3] = sign_detJ*(J[2]*J[7] - J[1]*J[8]);
-      dI3b[4] = sign_detJ*(J[0]*J[8] - J[2]*J[6]);
-      dI3b[5] = sign_detJ*(J[1]*J[6] - J[0]*J[7]);
-      dI3b[6] = sign_detJ*(J[1]*J[5] - J[2]*J[4]);
-      dI3b[7] = sign_detJ*(J[2]*J[3] - J[0]*J[5]);
-      dI3b[8] = sign_detJ*(J[0]*J[4] - J[1]*J[3]);
+      dI3b[0] = J[4]*J[8] - J[5]*J[7];  // 0  3  6
+      dI3b[1] = J[5]*J[6] - J[3]*J[8];  // 1  4  7
+      dI3b[2] = J[3]*J[7] - J[4]*J[6];  // 2  5  8
+      dI3b[3] = J[2]*J[7] - J[1]*J[8];
+      dI3b[4] = J[0]*J[8] - J[2]*J[6];
+      dI3b[5] = J[1]*J[6] - J[0]*J[7];
+      dI3b[6] = J[1]*J[5] - J[2]*J[4];
+      dI3b[7] = J[2]*J[3] - J[0]*J[5];
+      dI3b[8] = J[0]*J[4] - J[1]*J[3];
    }
    void Eval_DZt(const scalar_t *Z, scalar_t **DZt_ptr)
    {

linalg/kernels.hpp

@@ -12,11 +12,6 @@
 #ifndef MFEM_KERNELS_HPP
 #define MFEM_KERNELS_HPP
 
-#ifdef _WIN32
-#define _USE_MATH_DEFINES
-#include <cmath>
-#endif
-
 #include "../config/config.hpp"
 #include "../general/backends.hpp"
 #include "../general/globals.hpp"

linalg/linalg.hpp

@@ -24,10 +24,12 @@
 #include "blockoperator.hpp"
 #include "sparsesmoothers.hpp"
 #include "densemat.hpp"
+#include "symmat.hpp"
 #include "ode.hpp"
 #include "solvers.hpp"
 #include "handle.hpp"
 #include "invariants.hpp"
+#include "auxiliary.hpp"
 
 #ifdef MFEM_USE_AMGX
 #include "amgxsolver.hpp"

linalg/mumps.cpp

@@ -196,6 +196,8 @@ void MUMPSSolver::SetOperator(const Operator &op)
 
 void MUMPSSolver::Mult(const Vector &x, Vector &y) const
 {
+   x.HostRead();
+   y.HostReadWrite();
 #if MFEM_MUMPS_VERSION >= 530
 
    id->nloc_rhs = x.Size();

linalg/operator.hpp

@@ -37,7 +37,8 @@ protected:
       const Array<int> &test_tdof_list,
       RectangularConstrainedOperator* &Aout);
 
-   /// Returns RAP Operator of this, taking in input/output Prolongation matrices
+   /** @brief Returns RAP Operator of this, using input/output Prolongation matrices
+       @a Pi corresponds to "P", @a Po corresponds to "Rt" */
    Operator *SetupRAP(const Operator *Pi, const Operator *Po);
 
 public:
@@ -112,6 +113,11 @@ public:
    {
       return GetProlongation(); // Assume square unless specialized
    }
+   /** @brief Transpose of GetOutputRestriction, directly available in this
+       form to facilitate matrix-free RAP-type operators.
+
+       `NULL` means identity. */
+   virtual const Operator *GetOutputRestrictionTranspose() const { return NULL; }
    /** @brief Restriction operator from output vectors for the operator to linear
        algebra (linear system) vectors. `NULL` means identity. */
    virtual const Operator *GetOutputRestriction() const
@@ -606,23 +612,22 @@ public:
 
    using TimeDependentOperator::ImplicitSolve;
    /** @brief Solve the equation:
-       @a k = f(@a x + 1/2 @a dt0^2 @a k, @a dxdt + @a dt1 @a k, t), for the
+       @a k = f(@a x + @a fac0 @a k, @a dxdt + @a fac1 @a k, t), for the
        unknown @a k at the current time t.
 
        For general F and G, the equation for @a k becomes:
-       F(@a x + 1/2 @a dt0^2 @a k, @a dxdt + @a dt1 @a k, t)
-                        = G(@a x + 1/2 @a dt0^2 @a k, @a dxdt + @a dt1 @a k, t).
+       F(@a x +  @a fac0 @a k, @a dxdt + @a fac1 @a k, t)
+                        = G(@a x +  @a fac0 @a k, @a dxdt + @a fac1 @a k, t).
 
-       The input vector @a x corresponds to time index (or cycle) n, while the
+       The input vectors @a x and @a dxdt corresponds to time index (or cycle) n, while the
        currently set time, #t, and the result vector @a k correspond to time
-       index n+1. The time step @a dt corresponds to the time interval between
-       cycles n and n+1.
+       index n+1.
 
        This method allows for the abstract implementation of some time
        integration methods.
 
        If not re-implemented, this method simply generates an error. */
-   virtual void ImplicitSolve(const double dt0, const double dt1,
+   virtual void ImplicitSolve(const double fac0, const double fac1,
                               const Vector &x, const Vector &dxdt, Vector &k);
 
 
@@ -691,6 +696,42 @@ public:
    { A_.Mult(x, y); y *= a_; }
 };
 
+/// General sum operator: x -> A(x)+B(x)
+class SumOperator : public Operator
+{
+   const Operator *A, *B;
+   bool ownA, ownB;
+   mutable Vector z, w;
+   double cA, cB;
+
+public:
+   SumOperator(const Operator *A_, const Operator *B_,
+               bool ownA_, bool ownB_, double cA_, double cB_)
+      : Operator(A_->Height(), B_->Width()),
+        A(A_), B(B_), ownA(ownA_), ownB(ownB_), z(A_->Height()), w(A_->Width()),
+        cA(cA_), cB(cB_)
+   {
+      MFEM_VERIFY(A->Width() == B->Width() && A->Height() == B->Height(),
+                  "incompatible Operators: A->Width() = " << A->Width()
+                  << ", B->Height() = " << B->Height());
+
+      z.UseDevice(true);
+      w.UseDevice(true);
+   }
+
+   ~SumOperator()
+   {
+      if (ownA) { delete A; }
+      if (ownB) { delete B; }
+   }
+
+   virtual void Mult(const Vector &x, Vector &y) const
+   { B->Mult(x, z); A->Mult(x, y); y *= cA; z *= cB; y += z;}
+
+   virtual void MultTranspose(const Vector &x, Vector &y) const
+   { B->MultTranspose(x, w); A->MultTranspose(x, y); y *= cA; w *= cB; y += w;}
+
+};
 
 /** @brief The transpose of a given operator. Switches the roles of the methods
     Mult() and MultTranspose(). */

linalg/solvers.cpp

@@ -1556,6 +1556,7 @@ void NewtonSolver::SetOperator(const Operator &op)
    width = op.Width();
    MFEM_ASSERT(height == width, "square Operator is required.");
 
+   xcur.SetSize(width);
    r.SetSize(width);
    c.SetSize(width);
 }
@@ -1615,9 +1616,20 @@ void NewtonSolver::Mult(const Vector &b, Vector &x) const
          break;
       }
 
-      prec->SetOperator(oper->GetGradient(x));
+      grad = &oper->GetGradient(x);
+      prec->SetOperator(*grad);
 
-      prec->Mult(r, c);  // c = [DF(x_i)]^{-1} [F(x_i)-b]
+      if (lin_rtol_type)
+      {
+         AdaptiveLinRtolPreSolve(x, it, norm);
+      }
+
+      prec->Mult(r, c); // c = [DF(x_i)]^{-1} [F(x_i)-b]
+
+      if (lin_rtol_type)
+      {
+         AdaptiveLinRtolPostSolve(c, r, it, norm);
+      }
 
       const double c_scale = ComputeScalingFactor(x, b);
       if (c_scale == 0.0)
@@ -1641,6 +1653,86 @@ void NewtonSolver::Mult(const Vector &b, Vector &x) const
    final_norm = norm;
 }
 
+void NewtonSolver::SetAdaptiveLinRtol(const int type,
+                                      const double rtol0,
+                                      const double rtol_max,
+                                      const double alpha,
+                                      const double gamma)
+{
+   lin_rtol_type = type;
+   lin_rtol0 = rtol0;
+   lin_rtol_max = rtol_max;
+   this->alpha = alpha;
+   this->gamma = gamma;
+}
+
+void NewtonSolver::AdaptiveLinRtolPreSolve(const Vector &x,
+                                           const int it,
+                                           const double fnorm) const
+{
+   // Assume that when adaptive linear solver relative tolerance is activated,
+   // we are working with an iterative solver.
+   auto iterative_solver = static_cast<IterativeSolver *>(prec);
+   // Adaptive linear solver relative tolerance
+   double eta;
+   // Safeguard threshold
+   double sg_threshold = 0.1;
+
+   if (it == 0)
+   {
+      eta = lin_rtol0;
+   }
+   else
+   {
+      if (lin_rtol_type == 1)
+      {
+         // eta = gamma * abs(||F(x1)|| - ||F(x0) + DF(x0) s0||) / ||F(x0)||
+         eta = gamma * abs(fnorm - lnorm_last) / fnorm_last;
+      }
+      else if (lin_rtol_type == 2)
+      {
+         // eta = gamma * (||F(x1)|| / ||F(x0)||)^alpha
+         eta = gamma * pow(fnorm / fnorm_last, alpha);
+      }
+      else
+      {
+         MFEM_ABORT("Unknown adaptive linear solver rtol version");
+      }
+
+      // Safeguard rtol from "oversolving" ?!
+      const double sg_eta = gamma * pow(eta_last, alpha);
+      if (sg_eta > sg_threshold) { eta = std::max(eta, sg_eta); }
+   }
+
+   eta = std::min(eta, lin_rtol_max);
+   iterative_solver->SetRelTol(eta);
+   eta_last = eta;
+   if (print_level >= 0)
+   {
+      mfem::out << "Eisenstat-Walker rtol = " << eta << "\n";
+   }
+}
+
+void NewtonSolver::AdaptiveLinRtolPostSolve(const Vector &x,
+                                            const Vector &b,
+                                            const int it,
+                                            const double fnorm) const
+{
+   fnorm_last = fnorm;
+
+   // If version 1 is chosen, the true linear residual norm has to be computed
+   // and in most cases we can only retrieve the preconditioned linear residual
+   // norm.
+   if (lin_rtol_type == 1)
+   {
+      // lnorm_last = ||F(x0) + DF(x0) s0||
+      Vector linres(x.Size());
+      grad->Mult(x, linres);
+      linres -= b;
+      lnorm_last = Norm(linres);
+   }
+}
+
 void LBFGSSolver::Mult(const Vector &b, Vector &x) const
 {
    MFEM_VERIFY(oper != NULL, "the Operator is not set (use SetOperator).");
@@ -2720,9 +2812,7 @@ void UMFPackSolver::Init()
 
 void UMFPackSolver::SetOperator(const Operator &op)
 {
-   int *Ap, *Ai;
    void *Symbolic;
-   double *Ax;
 
    if (Numeric)
    {
@@ -2748,9 +2838,9 @@ void UMFPackSolver::SetOperator(const Operator &op)
    width = mat->Width();
    MFEM_VERIFY(width == height, "not a square matrix");
 
-   Ap = mat->GetI();
-   Ai = mat->GetJ();
-   Ax = mat->GetData();
+   const int * Ap = mat->HostReadI();
+   const int * Ai = mat->HostReadJ();
+   const double * Ax = mat->HostReadData();
 
    if (!use_long_ints)
    {
@@ -2820,12 +2910,13 @@ void UMFPackSolver::Mult(const Vector &b, Vector &x) const
    if (mat == NULL)
       mfem_error("UMFPackSolver::Mult : matrix is not set!"
                  " Call SetOperator first!");
-
+   b.HostRead();
+   x.HostReadWrite();
    if (!use_long_ints)
    {
       int status =
-         umfpack_di_solve(UMFPACK_At, mat->GetI(), mat->GetJ(),
-                          mat->GetData(), x, b, Numeric, Control, Info);
+         umfpack_di_solve(UMFPACK_At, mat->HostReadI(), mat->HostReadJ(),
+                          mat->HostReadData(), x, b, Numeric, Control, Info);
       umfpack_di_report_info(Control, Info);
       if (status < 0)
       {
@@ -2836,7 +2927,7 @@ void UMFPackSolver::Mult(const Vector &b, Vector &x) const
    else
    {
       SuiteSparse_long status =
-         umfpack_dl_solve(UMFPACK_At, AI, AJ, mat->GetData(), x, b,
+         umfpack_dl_solve(UMFPACK_At, AI, AJ, mat->HostReadData(), x, b,
                           Numeric, Control, Info);
       umfpack_dl_report_info(Control, Info);
       if (status < 0)
@@ -2852,12 +2943,13 @@ void UMFPackSolver::MultTranspose(const Vector &b, Vector &x) const
    if (mat == NULL)
       mfem_error("UMFPackSolver::MultTranspose : matrix is not set!"
                  " Call SetOperator first!");
-
+   b.HostRead();
+   x.HostReadWrite();
    if (!use_long_ints)
    {
       int status =
-         umfpack_di_solve(UMFPACK_A, mat->GetI(), mat->GetJ(),
-                          mat->GetData(), x, b, Numeric, Control, Info);
+         umfpack_di_solve(UMFPACK_A, mat->HostReadI(), mat->HostReadJ(),
+                          mat->HostReadData(), x, b, Numeric, Control, Info);
       umfpack_di_report_info(Control, Info);
       if (status < 0)
       {
@@ -2869,7 +2961,7 @@ void UMFPackSolver::MultTranspose(const Vector &b, Vector &x) const
    else
    {
       SuiteSparse_long status =
-         umfpack_dl_solve(UMFPACK_A, AI, AJ, mat->GetData(), x, b,
+         umfpack_dl_solve(UMFPACK_A, AI, AJ, mat->HostReadData(), x, b,
                           Numeric, Control, Info);
       umfpack_dl_report_info(Control, Info);
       if (status < 0)

linalg/solvers.hpp

@@ -406,7 +406,40 @@ void MINRES(const Operator &A, Solver &B, const Vector &b, Vector &x,
 class NewtonSolver : public IterativeSolver
 {
 protected:
-   mutable Vector r, c;
+   mutable Vector xcur, r, c;
+   mutable Operator *grad;
+
+   // Adaptive linear solver rtol variables
+
+   // Method to determine rtol, 0 means the adaptive algorithm is deactivated.
+   int lin_rtol_type = 0;
+   // rtol to use in first iteration
+   double lin_rtol0;
+   // Maximum rtol
+   double lin_rtol_max;
+   // Function norm ||F(x)|| of the previous iterate
+   mutable double fnorm_last = 0.0;
+   // Linear residual norm of the previous iterate
+   mutable double lnorm_last = 0.0;
+   // Forcing term (linear residual rtol) from the previous iterate
+   mutable double eta_last = 0.0;
+   // Eisenstat-Walker factor gamma
+   double gamma;
+   // Eisenstat-Walker factor alpha
+   double alpha;
+
+   /** @brief Method for the adaptive linear solver rtol invoked before the
+       linear solve. */
+   void AdaptiveLinRtolPreSolve(const Vector &x,
+                                const int it,
+                                const double fnorm) const;
+
+   /** @brief Method for the adaptive linear solver rtol invoked after the
+       linear solve. */
+   void AdaptiveLinRtolPostSolve(const Vector &x,
+                                 const Vector &b,
+                                 const int it,
+                                 const double fnorm) const;
 
 public:
    NewtonSolver() { }
@@ -434,6 +467,26 @@ public:
    /** @brief This method can be overloaded in derived classes to perform
        computations that need knowledge of the newest Newton state. */
    virtual void ProcessNewState(const Vector &x) const { }
+
+   const Vector &GetCurrentResidual() const { return r; }
+   const Vector &GetCurrentIterate() const { return xcur; }
+
+   /// Enable adaptive linear solver relative tolerance algorithm.
+   /** Compute a relative tolerance for the Krylov method after each nonlinear
+    iteration, based on the algorithm presented in [1].
+
+    The maximum linear solver relative tolerance @a rtol_max should be < 1. For
+    @a type 1 the parameters @a alpha and @a gamma are ignored. For @a type 2
+    @a alpha has to be between 0 and 1 and @a gamma between 1 and 2.
+
+    [1] Eisenstat, Stanley C., and Homer F. Walker. "Choosing the forcing terms
+    in an inexact Newton method."
+    */
+   void SetAdaptiveLinRtol(const int type = 2,
+                           const double rtol0 = 0.5,
+                           const double rtol_max = 0.9,
+                           const double alpha = 0.5 * (1.0 + sqrt(5.0)),
+                           const double gamma = 1.0);
 };
 
 /** L-BFGS method for solving F(x)=b for a given operator F, by minimizing
@@ -799,6 +852,169 @@ public:
 
 #endif // MFEM_USE_SUITESPARSE
 
+class Complex_PMHSS : public Solver
+{
+public:
+   Complex_PMHSS(Operator *Re, Operator *Im, Solver *prec_Re, Solver *prec_Im,
+                 double a_)
+      : Solver(2*Re->Height()), a(a_), A(Re, Im, false, false),
+        A_Re(Re, NULL, false, false),
+        A_Im(Im, NULL, false, false), u(2*Re->Height()), rhs(2*Re->Height()),
+        n(Re->Height())
+   {
+      MFEM_VERIFY(Re->Height() == Im->Height() && Re->Height() == Re->Width() &&
+                  Im->Height() == Im->Width(), "");
+      MFEM_VERIFY(this->Height() == A.Height(), "");
+
+      // Create CG solver for real operator aV + A_Re in complex space.
+
+      V = useIdentityV ? (Operator*) new IdentityOperator(this->Height()) :
+          (Operator*) &A_Re;
+
+      // In the case V = A_Re, it is faster to use a scaled operator than a SumOperator
+      Operator *sumOpRe = useIdentityV ? (Operator*) new SumOperator(V, &A_Re, false,
+                                                                     false, a, 1.0)
+                          : (Operator*) new ScaledOperator(&A_Re, a + 1.0);
+
+      SumOperator *sumOpIm = new SumOperator(V, &A_Im, false, false, a, 1.0);
+
+      CGSolver *cg = new CGSolver(MPI_COMM_WORLD);
+      cg->SetRelTol(1e-6);
+      cg->SetMaxIter(1000);
+      cg->SetPrintLevel(0);
+      cg->SetOperator(*sumOpRe);
+      cg->SetPreconditioner(*prec_Re);
+      cg->iterative_mode = false;
+
+      SRe = cg;
+
+      CGSolver *cgi = new CGSolver(MPI_COMM_WORLD);
+      cgi->SetRelTol(1e-6);
+      cgi->SetMaxIter(1000);
+      cgi->SetPrintLevel(0);
+      cgi->SetOperator(*sumOpIm);
+      if (prec_Im && useIdentityV) { cgi->SetPreconditioner(*prec_Im); }
+      if (!useIdentityV) { cgi->SetPreconditioner(*prec_Re); }
+      cgi->iterative_mode = false;
+
+      /*
+      // For negative definite imaginary part, but then PMHSS does not work?
+      MINRESSolver *cgi = new MINRESSolver(MPI_COMM_WORLD);
+      cgi->SetRelTol(1e-12);
+      cgi->SetMaxIter(1000);
+      cgi->SetPrintLevel(0);
+      cgi->SetOperator(*sumOpIm);
+      if (prec_Im) cgi->SetPreconditioner(*prec_Im);
+      */
+
+      SIm = cgi;
+   }
+
+   void SetOperator(const Operator &op)
+   {
+      MFEM_VERIFY(false, "Don't call SetOperator");
+   }
+
+   void ComputeResidual(const Vector &b, const Vector &sol, Vector &res) const
+   {
+      A.Mult(sol, res);
+      res -= b;
+   }
+
+   void Mult(const Vector &x, Vector &y) const
+   {
+      if (!(x.Size() == Height() && y.Size() == Height()))
+      {
+         std::cout << "bug";
+      }
+
+      MFEM_VERIFY(x.Size() == Height() && y.Size() == Height(), "");
+
+      const double initNorm = x.Norml2();
+      mfem::out << "MHSS RHS norm " << initNorm << '\n';
+
+      // With V = I, use modified HSS (MHSS) from Bai, Benzi, Chen 2010.
+      y = 0.0;
+
+      for (int it=0; it<maxiter; ++it)
+      {
+         // Solve (aI + Re) u = (aI - i Im) y + x
+
+         if (it == 0)
+         {
+            // Optimize the first iteration, when the initial guess is y=0.
+            SRe->Mult(x, u);
+         }
+         else
+         {
+            A_Im.Mult(y, u);  // u = Im y
+            // Set rhs = -i Im y = -i u
+            for (int j=0; j<n; ++j)
+            {
+               rhs[j] = u[n+j];
+               rhs[n+j] = -u[j];
+            }
+
+            rhs += x;
+
+            V->Mult(y, u);
+            rhs.Add(a, u);
+
+            SRe->Mult(rhs, u);
+         }
+
+         // Solve (aI + Im) y = (aI + i Re) u - i x
+
+         A_Re.Mult(u, y);  // y = Re u
+         // Set rhs = i (Re u - x) = i (y - x)
+         for (int j=0; j<n; ++j)
+         {
+            rhs[j] = -(y[n+j] - x[n+j]);
+            rhs[n+j] = y[j] - x[j];
+         }
+
+         if (useIdentityV)
+         {
+            //V->Mult(u, y);
+            //rhs.Add(a, y);
+            rhs.Add(a, u);
+         }
+         else
+         {
+            // Using V = A_Re
+            rhs.Add(a, y);
+         }
+
+         SIm->Mult(rhs, y);
+
+         ComputeResidual(x, y, rhs);
+         const double resNorm = rhs.Norml2();
+         mfem::out << "MHSS iter " << it << " residual norm " << resNorm << '\n';
+
+         if (resNorm / initNorm < tol)
+         {
+            mfem::out << "MHSS converged\n";
+            break;
+         }
+      }
+   }
+
+private:
+   const double a;
+   const int maxiter = 1;
+   ComplexOperator A, A_Re, A_Im;
+   mutable Vector u, rhs;
+   const int n;
+
+   const double tol = 1.0e-8;
+
+   const bool useIdentityV = false;
+   Operator *V = NULL;
+
+   Solver *SRe = NULL;
+   Solver *SIm = NULL;
+};
+
 }
 
 #endif // MFEM_SOLVERS

linalg/sparsemat.cpp

@@ -415,10 +415,14 @@ void SparseMatrix::SortColumnIndices()
       return;
    }
 
+   const int * Ip=HostReadI();
+   HostReadWriteJ();
+   HostReadWriteData();
+
    Array<Pair<int,double> > row;
    for (int j = 0, i = 0; i < height; i++)
    {
-      int end = I[i+1];
+      int end = Ip[i+1];
       row.SetSize(end - j);
       for (int k = 0; k < row.Size(); k++)
       {
@@ -3358,12 +3362,12 @@ SparseMatrix *Mult (const SparseMatrix &A, const SparseMatrix &B,
                "number of columns of A (" << ncolsA
                << ") must equal number of rows of B (" << nrowsB << ")");
 
-   A_i    = A.GetI();
-   A_j    = A.GetJ();
-   A_data = A.GetData();
-   B_i    = B.GetI();
-   B_j    = B.GetJ();
-   B_data = B.GetData();
+   A_i    = A.HostReadI();
+   A_j    = A.HostReadJ();
+   A_data = A.HostReadData();
+   B_i    = B.HostReadI();
+   B_j    = B.HostReadJ();
+   B_data = B.HostReadData();
 
    B_marker = new int[ncolsB];
 
@@ -3409,16 +3413,16 @@ SparseMatrix *Mult (const SparseMatrix &A, const SparseMatrix &B,
    {
       C = OAB;
 
-      MFEM_VERIFY(nrowsA == C -> Height() && ncolsB == C -> Width(),
+      MFEM_VERIFY(nrowsA == C->Height() && ncolsB == C->Width(),
                   "Input matrix sizes do not match output sizes"
                   << " nrowsA = " << nrowsA
                   << ", C->Height() = " << C->Height()
                   << " ncolsB = " << ncolsB
                   << ", C->Width() = " << C->Width());
 
-      // C_i    = C -> GetI(); // not used
-      C_j    = C -> GetJ();
-      C_data = C -> GetData();
+      // C_i    = C->HostReadI(); // not used
+      C_j    = C->HostWriteJ();
+      C_data = C->HostWriteData();
    }
 
    counter = 0;

linalg/superlu.cpp

@@ -523,9 +523,10 @@ void SuperLUSolver::Mult( const Vector & x, Vector & y ) const
    // SuperLU overwrites x with y, so copy x to y and pass that to the solve
    // routine.
 
-   y = x;
+   const double *xPtr = x.HostRead();
+   y = xPtr;
+   double * yPtr = y.HostReadWrite();
 
-   double*  yPtr = (double*)y;
    int      info = -1, locSize = y.Size();
 
    // Solve the system

linalg/symmat.cpp

@@ -0,0 +1,109 @@
+// Copyright (c) 2010-2020, Lawrence Livermore National Security, LLC. Produced
+// at the Lawrence Livermore National Laboratory. All Rights reserved. See files
+// LICENSE and NOTICE for details. LLNL-CODE-806117.
+//
+// This file is part of the MFEM library. For more information and source code
+// availability visit https://mfem.org.
+//
+// MFEM is free software; you can redistribute it and/or modify it under the
+// terms of the BSD-3 license. We welcome feedback and contributions, see file
+// CONTRIBUTING.md for details.
+
+
+// Implementation of data type DenseSymmetricMatrix
+
+#include "symmat.hpp"
+
+namespace mfem
+{
+
+DenseSymmetricMatrix::DenseSymmetricMatrix() : Matrix(0)
+{
+   data.Reset();
+}
+
+DenseSymmetricMatrix::DenseSymmetricMatrix(int s) : Matrix(s)
+{
+   MFEM_ASSERT(s >= 0, "invalid DenseSymmetricMatrix size: " << s);
+   if (s > 0)
+   {
+      data.New((s*(s+1))/2);
+      *this = 0.0; // init with zeroes
+   }
+   else
+   {
+      data.Reset();
+   }
+}
+
+void DenseSymmetricMatrix::SetSize(int s)
+{
+   MFEM_ASSERT(s >= 0,
+               "invalid DenseSymmetricMatrix size: " << s);
+   if (Height() == s)
+   {
+      return;
+   }
+   height = s;
+   width = s;
+   const int s2 = (s*(s+1))/2;
+   if (s2 > data.Capacity())
+   {
+      data.Delete();
+      data.New(s2);
+      *this = 0.0; // init with zeroes
+   }
+}
+
+DenseSymmetricMatrix &DenseSymmetricMatrix::operator=(double c)
+{
+   const int s = (Height()*(Height()+1))/2;
+   for (int i = 0; i < s; i++)
+   {
+      data[i] = c;
+   }
+   return *this;
+}
+
+double &DenseSymmetricMatrix::Elem(int i, int j)
+{
+   return (*this)(i,j);
+}
+
+const double &DenseSymmetricMatrix::Elem(int i, int j) const
+{
+   return (*this)(i,j);
+}
+
+DenseSymmetricMatrix &DenseSymmetricMatrix::operator*=(double c)
+{
+   int s = Height()*(Height()+1)/2;
+   for (int i = 0; i < s; i++)
+   {
+      data[i] *= c;
+   }
+   return *this;
+}
+
+void DenseSymmetricMatrix::Mult(const Vector &x, Vector &y) const
+{
+   mfem_error("DenseSymmetricMatrix::Mult() not implemented!");
+}
+
+MatrixInverse *DenseSymmetricMatrix::Inverse() const
+{
+   mfem_error("DenseSymmetricMatrix::Inverse() not implemented!");
+   return nullptr;
+}
+
+void DenseSymmetricMatrix::Print (std::ostream & out, int width_) const
+{
+   mfem_error("DenseSymmetricMatrix::Print() not implemented!");
+}
+
+DenseSymmetricMatrix::~DenseSymmetricMatrix()
+{
+   data.Delete();
+}
+
+}

linalg/symmat.hpp

@@ -0,0 +1,175 @@
+// Copyright (c) 2010-2020, Lawrence Livermore National Security, LLC. Produced
+// at the Lawrence Livermore National Laboratory. All Rights reserved. See files
+// LICENSE and NOTICE for details. LLNL-CODE-806117.
+//
+// This file is part of the MFEM library. For more information and source code
+// availability visit https://mfem.org.
+//
+// MFEM is free software; you can redistribute it and/or modify it under the
+// terms of the BSD-3 license. We welcome feedback and contributions, see file
+// CONTRIBUTING.md for details.
+
+#ifndef MFEM_SYMMETRICMAT
+#define MFEM_SYMMETRICMAT
+
+#include "../config/config.hpp"
+#include "../general/globals.hpp"
+#include "matrix.hpp"
+
+namespace mfem
+{
+
+/// Dense symmetric matrix storing the upper triangular part. This class so far
+/// has little functionality beyond storage.
+class DenseSymmetricMatrix : public Matrix
+{
+private:
+   Memory<double> data;
+
+public:
+
+   /** Default constructor for DenseSymmetricMatrix.
+       Sets data = NULL and height = width = 0. */
+   DenseSymmetricMatrix();
+
+   /// Creates square matrix of size s.
+   explicit DenseSymmetricMatrix(int s);
+
+   /// Construct a DenseSymmetricMatrix using an existing data array.
+   /** The DenseSymmetricMatrix does not assume ownership of the data array, i.e. it will
+       not delete the array. */
+   DenseSymmetricMatrix(double *d, int s)
+      : Matrix(s, s) { UseExternalData(d, s); }
+
+   /// Change the data array and the size of the DenseSymmetricMatrix.
+   /** The DenseSymmetricMatrix does not assume ownership of the data array, i.e. it will
+       not delete the data array @a d. This method should not be used with
+       DenseSymmetricMatrix that owns its current data array. */
+   void UseExternalData(double *d, int s)
+   {
+      data.Wrap(d, (s*(s+1))/2, false);
+      height = s; width = s;
+   }
+
+   /// Change the data array and the size of the DenseSymmetricMatrix.
+   /** The DenseSymmetricMatrix does not assume ownership of the data array, i.e. it will
+       not delete the new array @a d. This method will delete the current data
+       array, if owned. */
+   void Reset(double *d, int s)
+   { if (OwnsData()) { data.Delete(); } UseExternalData(d, s); }
+
+   /** Clear the data array and the dimensions of the DenseSymmetricMatrix. This method
+       should not be used with DenseSymmetricMatrix that owns its current data array. */
+   void ClearExternalData() { data.Reset(); height = width = 0; }
+
+   /// Delete the matrix data array (if owned) and reset the matrix state.
+   void Clear()
+   { if (OwnsData()) { data.Delete(); } ClearExternalData(); }
+
+   /// Change the size of the DenseSymmetricMatrix to s x s.
+   void SetSize(int s);
+
+   /// Returns the matrix data array.
+   inline double *Data() const
+   { return const_cast<double*>((const double*)data);}
+
+   /// Returns the matrix data array.
+   inline double *GetData() const { return Data(); }
+
+   Memory<double> &GetMemory() { return data; }
+   const Memory<double> &GetMemory() const { return data; }
+
+   /// Return the DenseSymmetricMatrix data (host pointer) ownership flag.
+   inline bool OwnsData() const { return data.OwnsHostPtr(); }
+
+   /// Returns reference to a_{ij}.
+   inline double &operator()(int i, int j);
+
+   /// Returns constant reference to a_{ij}.
+   inline const double &operator()(int i, int j) const;
+
+   /// Returns reference to a_{ij}.
+   virtual double &Elem(int i, int j);
+
+   /// Returns constant reference to a_{ij}.
+   virtual const double &Elem(int i, int j) const;
+
+   /// Sets the matrix elements equal to constant c
+   DenseSymmetricMatrix &operator=(double c);
+
+   DenseSymmetricMatrix &operator*=(double c);
+
+   long MemoryUsage() const { return data.Capacity() * sizeof(double); }
+
+   /// Shortcut for mfem::Read( GetMemory(), TotalSize(), on_dev).
+   const double *Read(bool on_dev = true) const
+   { return mfem::Read(data, Height()*Width(), on_dev); }
+
+   /// Shortcut for mfem::Read(GetMemory(), TotalSize(), false).
+   const double *HostRead() const
+   { return mfem::Read(data, Height()*Width(), false); }
+
+   /// Shortcut for mfem::Write(GetMemory(), TotalSize(), on_dev).
+   double *Write(bool on_dev = true)
+   { return mfem::Write(data, Height()*Width(), on_dev); }
+
+   /// Shortcut for mfem::Write(GetMemory(), TotalSize(), false).
+   double *HostWrite()
+   { return mfem::Write(data, Height()*Width(), false); }
+
+   /// Shortcut for mfem::ReadWrite(GetMemory(), TotalSize(), on_dev).
+   double *ReadWrite(bool on_dev = true)
+   { return mfem::ReadWrite(data, Height()*Width(), on_dev); }
+
+   /// Shortcut for mfem::ReadWrite(GetMemory(), TotalSize(), false).
+   double *HostReadWrite()
+   { return mfem::ReadWrite(data, Height()*Width(), false); }
+
+   /// Matrix vector multiplication.
+   virtual void Mult(const Vector &x, Vector &y) const;
+
+   /// Returns a pointer to (an approximation) of the matrix inverse.
+   virtual MatrixInverse *Inverse() const;
+
+   /// Prints matrix to stream out.
+   virtual void Print (std::ostream & out = mfem::out, int width_ = 4) const;
+
+   /// Destroys the symmetric matrix.
+   virtual ~DenseSymmetricMatrix();
+};
+
+// Inline methods
+
+// The number of entries stored in rows 1,...,k is
+// n + n-1 + n-2 + ... + n-k+1, where there are k terms. This equals
+// kn - sum_{i=1}^{k-1} i = kn - (k-1)k/2
+// This formula is used for the offset for each row.
+inline double &DenseSymmetricMatrix::operator()(int i, int j)
+{
+   MFEM_ASSERT(data && i >= 0 && i < height && j >= 0 && j < width, "");
+   if (i > j)  // reverse i and j
+   {
+      return data[(j*height) - (((j-1)*j)/2) + i - j];
+   }
+   else
+   {
+      return data[(i*height) - (((i-1)*i)/2) + j - i];
+   }
+}
+
+inline const double &DenseSymmetricMatrix::operator()(int i, int j) const
+{
+   MFEM_ASSERT(data && i >= 0 && i < height && j >= 0 && j < width, "");
+   if (i > j)  // reverse i and j
+   {
+      return data[(j*height) - (((j-1)*j)/2) + i - j];
+   }
+   else
+   {
+      return data[(i*height) - (((i-1)*i)/2) + j - i];
+   }
+}
+
+} // namespace mfem
+
+#endif

linalg/vector.cpp

@@ -22,6 +22,10 @@
 #endif
 #endif
 
+#ifdef MFEM_USE_OPENMP
+#include <omp.h>
+#endif
+
 #include <iostream>
 #include <iomanip>
 #include <cmath>
@@ -1076,6 +1080,30 @@ double Vector::operator*(const Vector &v) const
 #ifdef MFEM_USE_OPENMP
    if (Device::Allows(Backend::OMP_MASK))
    {
+#define MFEM_USE_OPENMP_DETERMINISTIC_DOT
+#ifdef MFEM_USE_OPENMP_DETERMINISTIC_DOT
+      // By default, use a deterministic way of computing the dot product
+      static Vector th_dot;
+      #pragma omp parallel
+      {
+         const int nt = omp_get_num_threads();
+         #pragma omp master
+         th_dot.SetSize(nt);
+         const int tid    = omp_get_thread_num();
+         const int stride = (size + nt - 1)/nt;
+         const int start  = tid*stride;
+         const int stop   = std::min(start + stride, size);
+         double my_dot = 0.0;
+         for (int i = start; i < stop; i++)
+         {
+            my_dot += m_data[i] * v_data[i];
+         }
+         #pragma omp barrier
+         th_dot(tid) = my_dot;
+      }
+      return th_dot.Sum();
+#else
+      // The standard way of computing the dot product is non-deterministic
       double prod = 0.0;
       #pragma omp parallel for reduction(+:prod)
       for (int i = 0; i < size; i++)
@@ -1083,8 +1111,9 @@ double Vector::operator*(const Vector &v) const
          prod += m_data[i] * v_data[i];
       }
       return prod;
+#endif // MFEM_USE_OPENMP_DETERMINISTIC_DOT
    }
-#endif
+#endif // MFEM_USE_OPENMP
    if (Device::Allows(Backend::DEBUG_DEVICE))
    {
       const int N = size;

linalg/vector.hpp

@@ -42,10 +42,19 @@ namespace mfem
 inline int CheckFinite(const double *v, const int n);
 
 /// Define a shortcut for std::numeric_limits<double>::infinity()
+#ifndef __CYGWIN__
 inline double infinity()
 {
    return std::numeric_limits<double>::infinity();
 }
+#else
+// On Cygwin math.h defines a function 'infinity()' which will conflict with the
+// above definition if we have 'using namespace mfem;' and try to use something
+// like 'double a = infinity();'. This 'infinity()' function is non-standard and
+// is defined by the Newlib C standard library implementation used by Cygwin,
+// see https://en.wikipedia.org/wiki/Newlib, http://www.sourceware.org/newlib.
+using ::infinity;
+#endif
 
 /// Vector data type.
 class Vector

makefile

@@ -536,7 +536,7 @@ clean: $(addsuffix /clean,$(EM_DIRS) $(TEST_DIRS))
 distclean: clean config/clean doc/clean
 	rm -rf mfem/
 
-INSTALL_SHARED_LIB = $(MFEM_CXX) $(MFEM_BUILD_FLAGS) $(INSTALL_SOFLAGS)\
+INSTALL_SHARED_LIB = $(MFEM_CXX) $(MFEM_LINK_FLAGS) $(INSTALL_SOFLAGS)\
    $(OBJECT_FILES) $(EXT_LIBS) -o $(PREFIX_LIB)/libmfem.$(SO_VER) && \
    cd $(PREFIX_LIB) && ln -sf libmfem.$(SO_VER) libmfem.$(SO_EXT)
 

mesh/mesh.cpp

@@ -71,10 +71,14 @@ void Mesh::GetElementCenter(int i, Vector &center)
    eltransf->Transform(Geometries.GetCenter(geom), center);
 }
 
-double Mesh::GetElementSize(int i, int type)
+double Mesh::GetElementSize(ElementTransformation *T, int type)
 {
    DenseMatrix J(Dim);
-   GetElementJacobian(i, J);
+
+   Geometry::Type geom = T->GetGeometryType();
+   T->SetIntPoint(&Geometries.GetCenter(geom));
+   Geometries.JacToPerfJac(geom, T->Jacobian(), J);
+
    if (type == 0)
    {
       return pow(fabs(J.Det()), 1./Dim);
@@ -89,6 +93,11 @@ double Mesh::GetElementSize(int i, int type)
    }
 }
 
+double Mesh::GetElementSize(int i, int type)
+{
+   return GetElementSize(GetElementTransformation(i), type);
+}
+
 double Mesh::GetElementSize(int i, const Vector &dir)
 {
    DenseMatrix J(Dim);
@@ -1040,6 +1049,13 @@ void Mesh::GetFaceInfos(int Face, int *Inf1, int *Inf2) const
    *Inf2 = faces_info[Face].Elem2Inf;
 }
 
+void Mesh::GetFaceInfos(int Face, int *Inf1, int *Inf2, int *NCFace) const
+{
+   *Inf1   = faces_info[Face].Elem1Inf;
+   *Inf2   = faces_info[Face].Elem2Inf;
+   *NCFace = faces_info[Face].NCFace;
+}
+
 Geometry::Type Mesh::GetFaceGeometryType(int Face) const
 {
    switch (Dim)
@@ -1463,6 +1479,13 @@ void Mesh::AddBdrQuadAsTriangles(const int *vi, int attr)
    }
 }
 
+int Mesh::AddBdrPoint(int v, int attr)
+{
+   CheckEnlarge(boundary, NumOfBdrElements);
+   boundary[NumOfBdrElements] = new Point(&v, attr);
+   return NumOfBdrElements++;
+}
+
 void Mesh::GenerateBoundaryElements()
 {
    int i, j;
@@ -5967,6 +5990,7 @@ int *Mesh::GeneratePartitioning(int nparts, int part_method)
    el_to_el = NULL;
 
    // Check for empty partitionings (a "feature" in METIS)
+   if (nparts > 1 && NumOfElements > nparts)
    {
       Array< Pair<int,int> > psize(nparts);
       int empty_parts;
@@ -8913,9 +8937,11 @@ void Mesh::PrintVTK(std::ostream &out)
          const int *v = elements[i]->GetVertices();
          const int nv = elements[i]->GetNVertices();
          out << nv;
+         Geometry::Type geom = elements[i]->GetGeometryType();
+         const int *perm = (geom == Geometry::PRISM) ? vtk_prism_perm : NULL;
          for (int j = 0; j < nv; j++)
          {
-            out << ' ' << v[j];
+            out << ' ' << v[perm ? perm[j] : j];
          }
          out << '\n';
       }
@@ -9243,9 +9269,10 @@ void Mesh::PrintVTU(std::ostream &out, int ref, VTKFormat format,
          {
             coff = coff+nv;
             offset.push_back(coff);
+            const int *p = (geom == Geometry::PRISM) ? vtk_prism_perm : NULL;
             for (int k = 0; k < nv; k++, j++)
             {
-               WriteBinaryOrASCII(out, buf, np + RG[j], " ", format);
+               WriteBinaryOrASCII(out, buf, np + RG[p ? p[j] : j], " ", format);
             }
             if (format == VTKFormat::ASCII) { out << '\n'; }
          }

mesh/mesh.hpp

@@ -471,6 +471,8 @@ protected:
    void GetElementData(const Array<Element*> &elem_array, int geom,
                        Array<int> &elem_vtx, Array<int> &attr) const;
 
+   double GetElementSize(ElementTransformation *T, int type = 0);
+
 public:
 
    Mesh() { SetEmpty(); }
@@ -555,6 +557,8 @@ public:
    int AddBdrQuad(const int *vi, int attr = 1);
    void AddBdrQuadAsTriangles(const int *vi, int attr = 1);
 
+   int AddBdrPoint(int v, int attr = 1);
+
    void GenerateBoundaryElements();
    /// Finalize the construction of a triangular Mesh.
    void FinalizeTriMesh(int generate_edges = 0, int refine = 0,
@@ -1032,6 +1036,7 @@ public:
    }
    void GetFaceElements (int Face, int *Elem1, int *Elem2) const;
    void GetFaceInfos (int Face, int *Inf1, int *Inf2) const;
+   void GetFaceInfos (int Face, int *Inf1, int *Inf2, int *NCFace) const;
 
    Geometry::Type GetFaceGeometryType(int Face) const;
    Element::Type  GetFaceElementType(int Face) const;

mesh/pmesh.cpp

@@ -1735,6 +1735,11 @@ void ParMesh::GetFaceNbrElementTransformation(
    }
 }
 
+double ParMesh::GetFaceNbrElementSize(int i, int type)
+{
+   return GetElementSize(GetFaceNbrElementTransformation(i), type);
+}
+
 void ParMesh::DeleteFaceNbrData()
 {
    if (!have_face_nbr_data)

mesh/pmesh.hpp

@@ -284,6 +284,7 @@ public:
                              int ordering = 1);
 
    int GetNFaceNeighbors() const { return face_nbr_group.Size(); }
+   int GetNFaceNeighborElements() const { return face_nbr_elements.Size(); }
    int GetFaceNbrGroup(int fn) const { return face_nbr_group[fn]; }
    int GetFaceNbrRank(int fn) const;
 
@@ -305,6 +306,10 @@ public:
       return &FaceNbrTransformation;
    }
 
+   /// Get the size of the i-th face neighbor element relative to the reference
+   /// element.
+   double GetFaceNbrElementSize(int i, int type=0);
+
    /// Return the number of shared faces (3D), edges (2D), vertices (1D)
    int GetNSharedFaces() const;