Merge pull request #4983 from dnpham23/master

Methods for extracting boundary edge DOFs from a 3D ParMesh and for synchronizing boundary conditions across processor boundaries

CHANGELOG

@@ -53,6 +53,16 @@ Discretization improvements
   ComplexHypreParMatrix::GetSystemMatrix, which previously assumed equal
   trial and test spaces.
 
+- Added FiniteElementSpace::GetBoundaryLoopEdgeDofs to extract the edge DOFs on
+  the perimeter loop of a set of boundary elements, with a ParFiniteElementSpace
+  overload that reconciles the selection across processor boundaries so the
+  result is partition invariant. This is useful for imposing boundary conditions
+  on boundary edge DOFs.
+
+- Added a MaxAbs reduction to GroupCommunicator that selects the signed value of
+  largest magnitude across a group, keeping its sign. Equal-magnitude ties
+  resolve deterministically to the positive value.
+
 Meshing improvements
 --------------------
 - Added support for nonuniform anisotropic mesh refinement on parallel quad/hex
@@ -79,6 +89,11 @@ Linear and nonlinear solvers
   PRefinement multigrid methods for problems posed on trace spaces (see e.g. the
   DPG miniapps).
 
+- Added new class MultiVector: an array of Vectors of different sizes where each
+  Vector can be allocated independently. Also, added associated methods in class
+  Operator: MultMV, MultTransposeMV, and GetGradientMV, that use MultiVector
+  objects for input and/or output parameters. [PR #5249]
+
 GPU computing
 -------------
 - Improved partial assembly for VectorDivergenceIntegrator with shared-memory
@@ -92,6 +107,22 @@ GPU computing
 
 - Added device assembly support for 3D H(curl) VectorFEDomainLFIntegrator.
 
+- Added partial assembly support for MixedScalarWeakGradientIntegrator.
+
+- Added partial assembly support for MixedDotProductIntegrator.
+
+- Added partial assembly support for MixedScalarCrossProductIntegrator.
+
+- Added partial assembly support for MixedScalarWeakCrossProductIntegrator.
+
+- Added partial assembly support for MixedVectorGradientIntegrator for H1->RT.
+
+- Added support for device partial assembly CurlInterpolator.
+  This supports 2D and 3D variants:
+    2D H1 (out-of-plane) to RT (in-plane)
+    2D ND (in-plane) to Integral L2 (out-of-plane)
+    3D ND to RT
+
 - Added NVIDIA cuDSS library interface. Implementation examples have been
   added to ex1 and ex1p. See https://developer.nvidia.com/cudss for more
   details. Supported versions >= 0.6.0.
@@ -104,6 +135,9 @@ GPU computing
 - Added support for FiniteElement::MapType::INTEGRAL spaces to
   QuadratureInterpolator.
 
+- Added support for FiniteElement::MapType::INTEGRAL spaces to
+  MixedScalarCurlIntegrator.
+
 New and updated examples and miniapps
 -------------------------------------
 - The Lorentz miniapp (in miniapps/electromagnetics) has been updated to
@@ -118,6 +152,20 @@ Miscellaneous
   using the new method ApplyDofSigns() in class ParFiniteElementSpace: the
   method will return immediately if no sign flips are needed.
 
+- Added support for coefficient-weighted LOR transfer in
+  L2ProjectionGridTransfer. The transfer conserves the weighted mass, for
+  example when transferring velocity while conserving density-weighted momentum.
+  This is illustrated in the lor-transfer and plor-transfer miniapps.
+
+- Added support for saving DataCollection output on the node-local storage,
+  instead of requiring that the filesystem is shared among all the ranks.
+
+API changes
+-----------
+- Removed ProjectGrad from 2D RT elements. Users should use ProjectCurl instead.
+  This also fixes a bug where ProjectCurl was returning the negative curl,
+  identical to ProjectGrad.
+
 
 Version 4.9, released on Dec 11, 2025
 =====================================

CMakeLists.txt

@@ -89,7 +89,7 @@ if (MFEM_USE_STRUMPACK OR MFEM_USE_MUMPS)
   set(XSDK_ENABLE_Fortran ON)
 endif()
 # RAJA requires C++20:
-if (MFEM_USE_RAJA AND ("${CMAKE_CXX_STANDARD}" LESS "20"))
+if ((MFEM_USE_UMPIRE OR MFEM_USE_RAJA) AND ("${CMAKE_CXX_STANDARD}" LESS "20"))
   set(CMAKE_CXX_STANDARD 20 CACHE STRING "C++ standard to use." FORCE)
 endif()
 

config/defaults.mk

@@ -28,11 +28,8 @@ MPICXX = mpicxx
 BASE_FLAGS  = -std=c++17
 OPTIM_FLAGS = -O3 $(BASE_FLAGS)
 
-# Shadow warnings for clang only; GCC's -Wshadow flags more.
-SHADOW_WARNING_FLAG = $(if $(findstring clang,\
- $(shell $(MFEM_HOST_CXX) --version 2>/dev/null)),-Wshadow,)
-WARNING_FLAGS = -pedantic -Wall $(SHADOW_WARNING_FLAG)
-
+# The variable WARNING_FLAGS depends on which compiler is used, and is defined
+# later in this file.
 DEBUG_FLAGS = $(strip -g $(addprefix $(XCOMPILER),$(WARNING_FLAGS)) $(BASE_FLAGS))
 
 # Prefixes for passing flags to the compiler and linker when using CXX or MPICXX
@@ -52,6 +49,10 @@ SHARED = NO
 #
 # If you set MFEM_USE_ENZYME=YES, must use CUDA_CXX=clang++
 CUDA_CXX = nvcc
+# CUDA compute capability used during compilation, e.g. sm_60. Multiple
+# architectures can be requested as a comma-separated list, e.g. sm_70,sm_80.
+# A single value may also be one of the nvcc special values "all",
+# "all-major", or "native".
 CUDA_ARCH = sm_60
 # Base CUDA install directory, only needed if building with clang+cuda:
 # The default setting is:
@@ -60,11 +61,23 @@ CUDA_ARCH = sm_60
 # 3. Use /usr/local/cuda
 CUDA_DIR = $(or $(CUDA_HOME),$(patsubst %/,%,$(dir \
  $(patsubst %/,%,$(dir $(shell command -v nvcc))))),/usr/local/cuda)
+# Derive nvcc/clang architecture flags from CUDA_ARCH. A comma-separated list
+# expands into one -gencode / --cuda-gpu-arch flag per architecture; otherwise
+# use the -arch / --cuda-gpu-arch shorthand.
+MFEM_COMMA := ,
+CUDA_ARCH_NUMS = $(patsubst sm_%,%,$(subst $(MFEM_COMMA), ,$(CUDA_ARCH)))
+NVCC_ARCH_FLAGS = $(strip $(if $(findstring $(MFEM_COMMA),$(CUDA_ARCH)),\
+ $(foreach arch,$(CUDA_ARCH_NUMS),\
+ -gencode arch=compute_$(arch)$(MFEM_COMMA)code=sm_$(arch)),\
+ -arch=$(CUDA_ARCH)))
+CLANG_ARCH_FLAGS = $(strip $(if $(findstring $(MFEM_COMMA),$(CUDA_ARCH)),\
+ $(foreach arch,$(CUDA_ARCH_NUMS),--cuda-gpu-arch=sm_$(arch)),\
+ --cuda-gpu-arch=$(CUDA_ARCH)))
 # flags for clang+cuda
-CLANG_CUDA_FLAGS = -xcuda --cuda-path=$(CUDA_DIR) --cuda-gpu-arch=$(CUDA_ARCH)
+CLANG_CUDA_FLAGS = -xcuda --cuda-path=$(CUDA_DIR) $(CLANG_ARCH_FLAGS)
 # flags for nvcc
 NVCC_FLAGS = -x=cu --expt-extended-lambda --expt-relaxed-constexpr \
- -arch=$(CUDA_ARCH) -isystem "$(CUDA_DIR)/include"
+ $(NVCC_ARCH_FLAGS) -isystem "$(CUDA_DIR)/include"
 # Prefixes for passing flags to the host compiler and linker when using
 # CUDA_CXX=nvcc
 CUDA_XCOMPILER = -Xcompiler=
@@ -382,7 +395,7 @@ CUDSS_LIBRARY_DIR = $(CUDSS_DIR)/lib
 CUDSS_OPT         = -I$(CUDSS_INCLUDE_DIR)
 CUDSS_LIB         = \
  $(XLINKER)-rpath,$(CUDSS_LIBRARY_DIR) -L$(CUDSS_LIBRARY_DIR) -lcudss
-# The cuDSS communication and threading libraries. 
+# The cuDSS communication and threading libraries.
 MFEM_CUDSS_COMM_LIB = $(abspath $(wildcard $(or $(CUDSS_COMM_LIB),\
    $(subst @MFEM_DIR@,$(MFEM_DIR), $(CUDSS_LIBRARY_DIR)/libcudss_commlayer_openmpi.so))))
 MFEM_CUDSS_THREADING_LIB = $(abspath $(wildcard $(or $(CUDSS_THREADING_LIB),\
@@ -665,3 +678,15 @@ VERBOSE = NO
 
 # Optional build tag
 MFEM_BUILD_TAG = $(shell uname -snm)
+
+# Enable -pedantic flag only for gcc or clang. nvcc complains with -pedantic
+# because of line directives.
+PEDANTIC_FLAG = $(if \
+   $(findstring NVIDIA,$(shell $(MFEM_CXX) --version 2>&1)),, \
+   $(if $(or \
+      $(findstring gcc version,$(shell $(MFEM_CXX) -v 2>&1)), \
+      $(findstring clang version,$(shell $(MFEM_CXX) -v 2>&1))),-pedantic,))
+# Enable shadow warnings for clang only; GCC's -Wshadow flags more.
+SHADOW_WARNING_FLAG = $(if $(findstring clang,\
+ $(shell $(MFEM_HOST_CXX) --version 2>/dev/null)),-Wshadow,)
+WARNING_FLAGS = $(PEDANTIC_FLAG) -Wall $(SHADOW_WARNING_FLAG)

data/nested_cubes.geo

@@ -0,0 +1,131 @@
+// Define the cube sizes
+L_outer = 1.0;
+L_inner = 0.5;
+
+// Set mesh size and algorithm
+mesh_size = 0.4;
+Mesh.Algorithm3D = 1; // Delaunay algorithm for 3D mesh
+Mesh.CharacteristicLengthFactor = 1.0;
+Mesh.MshFileVersion = 2.2;
+
+// Define center point for concentric cubes
+cx = 0.5;
+cy = 0.5;
+cz = 0.5;
+
+// Define the points (vertices of the outer cube)
+Point(1) = {cx-L_outer/2, cy-L_outer/2, cz-L_outer/2, mesh_size};
+Point(2) = {cx+L_outer/2, cy-L_outer/2, cz-L_outer/2, mesh_size};
+Point(3) = {cx+L_outer/2, cy+L_outer/2, cz-L_outer/2, mesh_size};
+Point(4) = {cx-L_outer/2, cy+L_outer/2, cz-L_outer/2, mesh_size};
+Point(5) = {cx-L_outer/2, cy-L_outer/2, cz+L_outer/2, mesh_size};
+Point(6) = {cx+L_outer/2, cy-L_outer/2, cz+L_outer/2, mesh_size};
+Point(7) = {cx+L_outer/2, cy+L_outer/2, cz+L_outer/2, mesh_size};
+Point(8) = {cx-L_outer/2, cy+L_outer/2, cz+L_outer/2, mesh_size};
+
+// Define the points (vertices of the inner cube)
+Point(9) = {cx-L_inner/2, cy-L_inner/2, cz-L_inner/2, mesh_size};
+Point(10) = {cx+L_inner/2, cy-L_inner/2, cz-L_inner/2, mesh_size};
+Point(11) = {cx+L_inner/2, cy+L_inner/2, cz-L_inner/2, mesh_size};
+Point(12) = {cx-L_inner/2, cy+L_inner/2, cz-L_inner/2, mesh_size};
+Point(13) = {cx-L_inner/2, cy-L_inner/2, cz+L_inner/2, mesh_size};
+Point(14) = {cx+L_inner/2, cy-L_inner/2, cz+L_inner/2, mesh_size};
+Point(15) = {cx+L_inner/2, cy+L_inner/2, cz+L_inner/2, mesh_size};
+Point(16) = {cx-L_inner/2, cy+L_inner/2, cz+L_inner/2, mesh_size};
+
+// Define the lines (edges of the outer cube)
+Line(1) = {1, 2};
+Line(2) = {2, 3};
+Line(3) = {3, 4};
+Line(4) = {4, 1};
+Line(5) = {5, 6};
+Line(6) = {6, 7};
+Line(7) = {7, 8};
+Line(8) = {8, 5};
+Line(9) = {1, 5};
+Line(10) = {2, 6};
+Line(11) = {3, 7};
+Line(12) = {4, 8};
+
+// Define the lines (edges of the inner cube)
+Line(13) = {9, 10};
+Line(14) = {10, 11};
+Line(15) = {11, 12};
+Line(16) = {12, 9};
+Line(17) = {13, 14};
+Line(18) = {14, 15};
+Line(19) = {15, 16};
+Line(20) = {16, 13};
+Line(21) = {9, 13};
+Line(22) = {10, 14};
+Line(23) = {11, 15};
+Line(24) = {12, 16};
+
+// Define the surfaces (faces of the outer cube)
+Line Loop(1) = {1, 2, 3, 4};
+Plane Surface(1) = {1};
+
+Line Loop(2) = {5, 6, 7, 8};
+Plane Surface(2) = {2};
+
+Line Loop(3) = {9, 5, -10, -1};
+Plane Surface(3) = {3};
+
+Line Loop(4) = {10, 6, -11, -2};
+Plane Surface(4) = {4};
+
+Line Loop(5) = {11, 7, -12, -3};
+Plane Surface(5) = {5};
+
+Line Loop(6) = {12, 8, -9, -4};
+Plane Surface(6) = {6};
+
+// Define the surfaces (faces of the inner cube)
+Line Loop(7) = {13, 14, 15, 16};
+Plane Surface(7) = {7};
+
+Line Loop(8) = {17, 18, 19, 20};
+Plane Surface(8) = {8};
+
+Line Loop(9) = {21, 17, -22, -13};
+Plane Surface(9) = {9};
+
+Line Loop(10) = {22, 18, -23, -14};
+Plane Surface(10) = {10};
+
+Line Loop(11) = {23, 19, -24, -15};
+Plane Surface(11) = {11};
+
+Line Loop(12) = {24, 20, -21, -16};
+Plane Surface(12) = {12};
+
+// Define the volumes
+Surface Loop(1) = {1, 2, 3, 4, 5, 6};
+Surface Loop(2) = {7, 8, 9, 10, 11, 12};
+Volume(1) = {1, 2}; // Outer volume with inner hole
+Volume(2) = {2};    // Inner volume
+
+// Assign physical groups
+Physical Volume(1) = {1}; // Outer volume
+Physical Volume(2) = {2}; // Inner volume
+
+// Outer cube surfaces
+Physical Surface(1) = {1}; // Outer bottom
+Physical Surface(2) = {2}; // Outer top
+Physical Surface(3) = {3}; // Outer front
+Physical Surface(4) = {4}; // Outer right
+Physical Surface(5) = {5}; // Outer back
+Physical Surface(6) = {6}; // Outer left
+
+// Inner cube surfaces
+Physical Surface(7) = {7};  // Inner bottom (-xy)
+Physical Surface(8) = {8};  // Inner top (+xy)
+Physical Surface(9) = {9};  // Inner front (-xz)
+Physical Surface(10) = {10}; // Inner right (+yz)
+Physical Surface(11) = {11}; // Inner back (+xz)
+Physical Surface(12) = {12}; // Inner left (-yz)
+
+// Mesh control
+Mesh.OptimizeNetgen = 1;
+Mesh.Optimize = 1;
+Mesh.ElementOrder = 1;

data/nested_cubes.msh

@@ -0,0 +1,907 @@
+$MeshFormat
+2.2 0 8
+$EndMeshFormat
+$Nodes
+138
+1 0 0 0
+2 1 0 0
+3 1 1 0
+4 0 1 0
+5 0 0 1
+6 1 0 1
+7 1 1 1
+8 0 1 1
+9 0.25 0.25 0.25
+10 0.75 0.25 0.25
+11 0.75 0.75 0.25
+12 0.25 0.75 0.25
+13 0.25 0.25 0.75
+14 0.75 0.25 0.75
+15 0.75 0.75 0.75
+16 0.25 0.75 0.75
+17 0.3333333333325025 0 0
+18 0.6666666666657889 0 0
+19 1 0.3333333333325025 0
+20 1 0.6666666666657889 0
+21 0.6666666666675911 1 0
+22 0.3333333333347203 1 0
+23 0 0.6666666666675911 0
+24 0 0.3333333333347203 0
+25 0.3333333333325025 0 1
+26 0.6666666666657889 0 1
+27 1 0.3333333333325025 1
+28 1 0.6666666666657889 1
+29 0.6666666666675911 1 1
+30 0.3333333333347203 1 1
+31 0 0.6666666666675911 1
+32 0 0.3333333333347203 1
+33 0 0 0.3333333333325025
+34 0 0 0.6666666666657889
+35 1 0 0.3333333333325025
+36 1 0 0.6666666666657889
+37 1 1 0.3333333333325025
+38 1 1 0.6666666666657889
+39 0 1 0.3333333333325025
+40 0 1 0.6666666666657889
+41 0.5000000000003468 0.25 0.25
+42 0.75 0.5000000000003468 0.25
+43 0.5000000000013763 0.75 0.25
+44 0.25 0.5000000000013763 0.25
+45 0.5000000000003468 0.25 0.75
+46 0.75 0.5000000000003468 0.75
+47 0.5000000000013763 0.75 0.75
+48 0.25 0.5000000000013763 0.75
+49 0.25 0.25 0.5000000000003468
+50 0.75 0.25 0.5000000000003468
+51 0.75 0.75 0.5000000000003468
+52 0.25 0.75 0.5000000000003468
+53 0.7113248654055673 0.4999999999991457 0
+54 0.2886751345942123 0.5000000000011557 0
+55 0.5000000000006117 0.7525600817161773 0
+56 0.4999999999993867 0.2474399182839603 0
+57 0.2423197548524782 0.7576802451481532 0
+58 0.757680245147464 0.2423197548520695 0
+59 0.2423197548507857 0.2423197548513912 0
+60 0.7576802451491019 0.7576802451486099 0
+61 0.7113248654055673 0.4999999999991457 1
+62 0.2886751345942123 0.5000000000011557 1
+63 0.5000000000006117 0.7525600817161773 1
+64 0.4999999999993867 0.2474399182839603 1
+65 0.2423197548524782 0.7576802451481532 1
+66 0.757680245147464 0.2423197548520695 1
+67 0.2423197548507857 0.2423197548513912 1
+68 0.7576802451491019 0.7576802451486099 1
+69 0.4999999999993203 0 0.301447615129799
+70 0.4999999999992795 0 0.7028666213189801
+71 0.7525600817158393 0 0.5007190394076877
+72 0.2474399182836191 0 0.5007190394076877
+73 0.7576802451479793 0 0.7576802451479793
+74 0.2423197548517962 0 0.7576802451477375
+75 0.7576802451484569 0 0.2423197548510767
+76 0.2423197548513188 0 0.2423197548513187
+77 1 0.4999999999993203 0.301447615129799
+78 1 0.4999999999992795 0.7028666213189801
+79 1 0.7525600817158394 0.5007190394076877
+80 1 0.2474399182836191 0.5007190394076877
+81 1 0.7576802451479794 0.7576802451479794
+82 1 0.2423197548517962 0.7576802451477376
+83 1 0.7576802451484569 0.2423197548510768
+84 1 0.2423197548513188 0.2423197548513188
+85 0.5000000000008327 1 0.3014476151298047
+86 0.500000000000961 1 0.7028666213191928
+87 0.2474399182842484 1 0.5007190394077241
+88 0.7525600817164384 1 0.5007190394078933
+89 0.2423197548520873 1 0.7576802451480517
+90 0.7576802451481496 1 0.2423197548518761
+91 0.2423197548516099 1 0.2423197548510044
+92 0.7576802451486874 1 0.7576802451481952
+93 0 0.5000000000008327 0.3014476151298047
+94 0 0.500000000000961 0.7028666213191928
+95 0 0.2474399182842484 0.5007190394077241
+96 0 0.7525600817164384 0.5007190394078933
+97 0 0.2423197548520873 0.7576802451480517
+98 0 0.7576802451481496 0.2423197548518761
+99 0 0.2423197548516099 0.2423197548510044
+100 0 0.7576802451486874 0.7576802451481952
+101 0.3968750000003409 0.603125000000244 0.25
+102 0.4374999999998713 0.4375000000001287 0.25
+103 0.5739583333335919 0.5718750000001767 0.25
+104 0.6093749999999631 0.3906250000003402 0.25
+105 0.3968750000003409 0.603125000000244 0.75
+106 0.4374999999998713 0.4375000000001287 0.75
+107 0.5739583333335919 0.5718750000001767 0.75
+108 0.6093749999999631 0.3906250000003402 0.75
+109 0.3806942419826734 0.25 0.3806942419826734
+110 0.5625000000001735 0.25 0.4375000000000001
+111 0.4254282069971791 0.25 0.5712615403304835
+112 0.6093749999998808 0.25 0.6093749999998808
+113 0.75 0.3806942419826734 0.3806942419826734
+114 0.75 0.5625000000001735 0.4375000000000001
+115 0.75 0.4254282069971791 0.5712615403304835
+116 0.75 0.6093749999998808 0.6093749999998808
+117 0.3968750000004991 0.75 0.3968750000002804
+118 0.4375000000000001 0.75 0.5625000000004308
+119 0.5739583333336153 0.75 0.4281250000000707
+120 0.6093749999998166 0.75 0.6093749999993661
+121 0.25 0.3968750000004991 0.3968750000002804
+122 0.25 0.4375000000000001 0.5625000000004308
+123 0.25 0.5739583333336153 0.4281250000000707
+124 0.25 0.6093749999998166 0.6093749999993661
+125 0.4962939304035875 0.5214350017087855 0.4925553109323813
+126 0.3432581985549767 0.6471275530923523 0.3554206606168006
+127 0.6442168181713744 0.5929232373214773 0.3593785632300704
+128 0.625174517421737 0.3491579444372839 0.3604636129462503
+129 0.6130544111091688 0.6576364353993367 0.5046720648819553
+130 0.4281518698369243 0.3632662294430484 0.3548726263205102
+131 0.3639383531355198 0.3520949250221278 0.4978389662613994
+132 0.629585530087249 0.3489878162230438 0.5124654846339122
+133 0.3710853378652663 0.6517586292698121 0.6382643241302075
+134 0.5917018263727056 0.6522525456211955 0.6390764961119443
+135 0.3530810314228338 0.4582477062424107 0.6430239639637545
+136 0.6571010904289998 0.5300774811423468 0.6603970500567977
+137 0.6484596018596915 0.361399676127967 0.6360267588157177
+138 0.4782020887035478 0.3534611476388013 0.6141275027013793
+$EndNodes
+$Elements
+760
+1 2 2 1 1 1 17 59
+2 2 2 1 1 24 1 59
+3 2 2 1 1 18 2 58
+4 2 2 1 1 2 19 58
+5 2 2 1 1 20 3 60
+6 2 2 1 1 3 21 60
+7 2 2 1 1 22 4 57
+8 2 2 1 1 4 23 57
+9 2 2 1 1 17 18 56
+10 2 2 1 1 17 56 59
+11 2 2 1 1 56 18 58
+12 2 2 1 1 19 20 53
+13 2 2 1 1 19 53 58
+14 2 2 1 1 53 20 60
+15 2 2 1 1 21 22 55
+16 2 2 1 1 21 55 60
+17 2 2 1 1 55 22 57
+18 2 2 1 1 23 24 54
+19 2 2 1 1 23 54 57
+20 2 2 1 1 54 24 59
+21 2 2 1 1 54 53 55
+22 2 2 1 1 53 54 56
+23 2 2 1 1 55 53 60
+24 2 2 1 1 53 56 58
+25 2 2 1 1 54 55 57
+26 2 2 1 1 56 54 59
+27 2 2 2 2 5 25 67
+28 2 2 2 2 32 5 67
+29 2 2 2 2 26 6 66
+30 2 2 2 2 6 27 66
+31 2 2 2 2 28 7 68
+32 2 2 2 2 7 29 68
+33 2 2 2 2 30 8 65
+34 2 2 2 2 8 31 65
+35 2 2 2 2 25 26 64
+36 2 2 2 2 25 64 67
+37 2 2 2 2 64 26 66
+38 2 2 2 2 27 28 61
+39 2 2 2 2 27 61 66
+40 2 2 2 2 61 28 68
+41 2 2 2 2 29 30 63
+42 2 2 2 2 29 63 68
+43 2 2 2 2 63 30 65
+44 2 2 2 2 31 32 62
+45 2 2 2 2 31 62 65
+46 2 2 2 2 62 32 67
+47 2 2 2 2 62 61 63
+48 2 2 2 2 61 62 64
+49 2 2 2 2 63 61 68
+50 2 2 2 2 61 64 66
+51 2 2 2 2 62 63 65
+52 2 2 2 2 64 62 67
+53 2 2 3 3 17 1 76
+54 2 2 3 3 1 33 76
+55 2 2 3 3 2 18 75
+56 2 2 3 3 35 2 75
+57 2 2 3 3 5 25 74
+58 2 2 3 3 34 5 74
+59 2 2 3 3 26 6 73
+60 2 2 3 3 6 36 73
+61 2 2 3 3 18 17 69
+62 2 2 3 3 69 17 76
+63 2 2 3 3 18 69 75
+64 2 2 3 3 25 26 70
+65 2 2 3 3 25 70 74
+66 2 2 3 3 70 26 73
+67 2 2 3 3 33 34 72
+68 2 2 3 3 33 72 76
+69 2 2 3 3 72 34 74
+70 2 2 3 3 36 35 71
+71 2 2 3 3 71 35 75
+72 2 2 3 3 36 71 73
+73 2 2 3 3 69 70 71
+74 2 2 3 3 70 69 72
+75 2 2 3 3 69 71 75
+76 2 2 3 3 72 69 76
+77 2 2 3 3 71 70 73
+78 2 2 3 3 70 72 74
+79 2 2 4 4 19 2 84
+80 2 2 4 4 2 35 84
+81 2 2 4 4 3 20 83
+82 2 2 4 4 37 3 83
+83 2 2 4 4 6 27 82
+84 2 2 4 4 36 6 82
+85 2 2 4 4 28 7 81
+86 2 2 4 4 7 38 81
+87 2 2 4 4 20 19 77
+88 2 2 4 4 77 19 84
+89 2 2 4 4 20 77 83
+90 2 2 4 4 27 28 78
+91 2 2 4 4 27 78 82
+92 2 2 4 4 78 28 81
+93 2 2 4 4 35 36 80
+94 2 2 4 4 35 80 84
+95 2 2 4 4 80 36 82
+96 2 2 4 4 38 37 79
+97 2 2 4 4 79 37 83
+98 2 2 4 4 38 79 81
+99 2 2 4 4 77 78 79
+100 2 2 4 4 78 77 80
+101 2 2 4 4 77 79 83
+102 2 2 4 4 80 77 84
+103 2 2 4 4 79 78 81
+104 2 2 4 4 78 80 82
+105 2 2 5 5 21 3 90
+106 2 2 5 5 3 37 90
+107 2 2 5 5 4 22 91
+108 2 2 5 5 39 4 91
+109 2 2 5 5 7 29 92
+110 2 2 5 5 38 7 92
+111 2 2 5 5 30 8 89
+112 2 2 5 5 8 40 89
+113 2 2 5 5 22 21 85
+114 2 2 5 5 85 21 90
+115 2 2 5 5 22 85 91
+116 2 2 5 5 29 30 86
+117 2 2 5 5 29 86 92
+118 2 2 5 5 86 30 89
+119 2 2 5 5 37 38 88
+120 2 2 5 5 37 88 90
+121 2 2 5 5 88 38 92
+122 2 2 5 5 40 39 87
+123 2 2 5 5 87 39 91
+124 2 2 5 5 40 87 89
+125 2 2 5 5 85 86 87
+126 2 2 5 5 86 85 88
+127 2 2 5 5 85 87 91
+128 2 2 5 5 88 85 90
+129 2 2 5 5 87 86 89
+130 2 2 5 5 86 88 92
+131 2 2 6 6 1 24 99
+132 2 2 6 6 33 1 99
+133 2 2 6 6 23 4 98
+134 2 2 6 6 4 39 98
+135 2 2 6 6 32 5 97
+136 2 2 6 6 5 34 97
+137 2 2 6 6 8 31 100
+138 2 2 6 6 40 8 100
+139 2 2 6 6 24 23 93
+140 2 2 6 6 93 23 98
+141 2 2 6 6 24 93 99
+142 2 2 6 6 31 32 94
+143 2 2 6 6 31 94 100
+144 2 2 6 6 94 32 97
+145 2 2 6 6 34 33 95
+146 2 2 6 6 95 33 99
+147 2 2 6 6 34 95 97
+148 2 2 6 6 39 40 96
+149 2 2 6 6 39 96 98
+150 2 2 6 6 96 40 100
+151 2 2 6 6 93 94 95
+152 2 2 6 6 94 93 96
+153 2 2 6 6 93 95 99
+154 2 2 6 6 96 93 98
+155 2 2 6 6 95 94 97
+156 2 2 6 6 94 96 100
+157 2 2 7 7 9 41 102
+158 2 2 7 7 44 9 102
+159 2 2 7 7 41 10 104
+160 2 2 7 7 10 42 104
+161 2 2 7 7 42 11 103
+162 2 2 7 7 11 43 103
+163 2 2 7 7 43 12 101
+164 2 2 7 7 12 44 101
+165 2 2 7 7 102 41 104
+166 2 2 7 7 42 103 104
+167 2 2 7 7 43 101 103
+168 2 2 7 7 101 44 102
+169 2 2 7 7 101 102 103
+170 2 2 7 7 103 102 104
+171 2 2 8 8 13 45 106
+172 2 2 8 8 48 13 106
+173 2 2 8 8 45 14 108
+174 2 2 8 8 14 46 108
+175 2 2 8 8 46 15 107
+176 2 2 8 8 15 47 107
+177 2 2 8 8 47 16 105
+178 2 2 8 8 16 48 105
+179 2 2 8 8 106 45 108
+180 2 2 8 8 46 107 108
+181 2 2 8 8 47 105 107
+182 2 2 8 8 105 48 106
+183 2 2 8 8 105 106 107
+184 2 2 8 8 107 106 108
+185 2 2 9 9 41 9 109
+186 2 2 9 9 9 49 109
+187 2 2 9 9 10 41 110
+188 2 2 9 9 50 10 110
+189 2 2 9 9 13 45 111
+190 2 2 9 9 49 13 111
+191 2 2 9 9 45 14 112
+192 2 2 9 9 14 50 112
+193 2 2 9 9 41 109 110
+194 2 2 9 9 111 45 112
+195 2 2 9 9 109 49 111
+196 2 2 9 9 50 110 112
+197 2 2 9 9 110 109 111
+198 2 2 9 9 110 111 112
+199 2 2 10 10 42 10 113
+200 2 2 10 10 10 50 113
+201 2 2 10 10 11 42 114
+202 2 2 10 10 51 11 114
+203 2 2 10 10 14 46 115
+204 2 2 10 10 50 14 115
+205 2 2 10 10 46 15 116
+206 2 2 10 10 15 51 116
+207 2 2 10 10 42 113 114
+208 2 2 10 10 115 46 116
+209 2 2 10 10 113 50 115
+210 2 2 10 10 51 114 116
+211 2 2 10 10 114 113 115
+212 2 2 10 10 114 115 116
+213 2 2 11 11 43 11 119
+214 2 2 11 11 11 51 119
+215 2 2 11 11 12 43 117
+216 2 2 11 11 52 12 117
+217 2 2 11 11 15 47 120
+218 2 2 11 11 51 15 120
+219 2 2 11 11 47 16 118
+220 2 2 11 11 16 52 118
+221 2 2 11 11 117 43 119
+222 2 2 11 11 47 118 120
+223 2 2 11 11 119 51 120
+224 2 2 11 11 52 117 118
+225 2 2 11 11 118 117 119
+226 2 2 11 11 118 119 120
+227 2 2 12 12 9 44 121
+228 2 2 12 12 49 9 121
+229 2 2 12 12 44 12 123
+230 2 2 12 12 12 52 123
+231 2 2 12 12 48 13 122
+232 2 2 12 12 13 49 122
+233 2 2 12 12 16 48 124
+234 2 2 12 12 52 16 124
+235 2 2 12 12 121 44 123
+236 2 2 12 12 48 122 124
+237 2 2 12 12 49 121 122
+238 2 2 12 12 123 52 124
+239 2 2 12 12 122 121 123
+240 2 2 12 12 122 123 124
+241 4 2 1 1 105 62 106 107
+242 4 2 1 1 102 54 101 103
+243 4 2 1 1 118 86 120 119
+244 4 2 1 1 124 94 123 122
+245 4 2 1 1 52 39 12 96
+246 4 2 1 1 52 12 39 87
+247 4 2 1 1 88 38 15 51
+248 4 2 1 1 79 15 38 51
+249 4 2 1 1 80 14 50 36
+250 4 2 1 1 71 50 14 36
+251 4 2 1 1 120 88 15 51
+252 4 2 1 1 116 15 79 51
+253 4 2 1 1 50 14 112 71
+254 4 2 1 1 52 117 12 87
+255 4 2 1 1 111 69 109 110
+256 4 2 1 1 114 77 115 113
+257 4 2 1 1 124 123 94 96
+258 4 2 1 1 120 86 88 119
+259 4 2 1 1 14 45 64 108
+260 4 2 1 1 103 54 101 55
+261 4 2 1 1 62 105 63 107
+262 4 2 1 1 63 29 15 47
+263 4 2 1 1 14 45 26 64
+264 4 2 1 1 51 11 88 37
+265 4 2 1 1 80 50 10 35
+266 4 2 1 1 49 33 9 72
+267 4 2 1 1 71 10 50 35
+268 4 2 1 1 79 11 51 37
+269 4 2 1 1 49 9 33 95
+270 4 2 1 1 12 123 52 96
+271 4 2 1 1 96 16 52 40
+272 4 2 1 1 49 13 34 72
+273 4 2 1 1 87 52 16 40
+274 4 2 1 1 13 49 34 95
+275 4 2 1 1 43 101 12 55
+276 4 2 1 1 43 12 22 55
+277 4 2 1 1 106 61 108 107
+278 4 2 1 1 102 103 104 53
+279 4 2 1 1 50 115 14 80
+280 4 2 1 1 80 10 50 113
+281 4 2 1 1 9 109 49 72
+282 4 2 1 1 47 15 63 107
+283 4 2 1 1 124 16 52 96
+284 4 2 1 1 49 121 9 95
+285 4 2 1 1 15 28 68 46
+286 4 2 1 1 81 28 15 46
+287 4 2 1 1 92 15 29 47
+288 4 2 1 1 27 82 14 46
+289 4 2 1 1 14 66 27 46
+290 4 2 1 1 26 45 14 73
+291 4 2 1 1 78 115 116 114
+292 4 2 1 1 112 70 111 110
+293 4 2 1 1 79 11 114 51
+294 4 2 1 1 71 50 10 110
+295 4 2 1 1 10 41 104 56
+296 4 2 1 1 77 80 115 113
+297 4 2 1 1 69 109 72 111
+298 4 2 1 1 63 16 30 47
+299 4 2 1 1 25 45 13 64
+300 4 2 1 1 52 16 118 87
+301 4 2 1 1 95 13 49 122
+302 4 2 1 1 57 12 23 44
+303 4 2 1 1 119 11 88 51
+304 4 2 1 1 41 17 9 56
+305 4 2 1 1 18 41 10 56
+306 4 2 1 1 21 11 43 55
+307 4 2 1 1 63 105 16 47
+308 4 2 1 1 13 45 106 64
+309 4 2 1 1 9 102 41 56
+310 4 2 1 1 72 49 13 111
+311 4 2 1 1 11 103 43 55
+312 4 2 1 1 70 112 71 110
+313 4 2 1 1 116 79 78 114
+314 4 2 1 1 32 13 48 67
+315 4 2 1 1 16 89 30 47
+316 4 2 1 1 32 48 13 97
+317 4 2 1 1 48 16 100 31
+318 4 2 1 1 121 123 93 122
+319 4 2 1 1 119 117 118 85
+320 4 2 1 1 10 19 42 84
+321 4 2 1 1 17 9 76 41
+322 4 2 1 1 24 9 59 44
+323 4 2 1 1 42 19 10 58
+324 4 2 1 1 10 41 18 75
+325 4 2 1 1 20 83 11 42
+326 4 2 1 1 90 11 43 21
+327 4 2 1 1 53 104 102 56
+328 4 2 1 1 106 61 64 108
+329 4 2 1 1 87 118 117 85
+330 4 2 1 1 95 121 93 122
+331 4 2 1 1 80 50 115 113
+332 4 2 1 1 109 49 72 111
+333 4 2 1 1 78 27 28 46
+334 4 2 1 1 31 94 48 32
+335 4 2 1 1 19 77 20 42
+336 4 2 1 1 23 24 44 93
+337 4 2 1 1 123 124 52 96
+338 4 2 1 1 88 120 119 51
+339 4 2 1 1 91 43 85 117
+340 4 2 1 1 39 98 12 96
+341 4 2 1 1 12 91 39 87
+342 4 2 1 1 15 92 38 88
+343 4 2 1 1 38 81 15 79
+344 4 2 1 1 36 80 14 82
+345 4 2 1 1 14 71 36 73
+346 4 2 1 1 103 102 54 53
+347 4 2 1 1 61 106 62 107
+348 4 2 1 1 86 29 30 47
+349 4 2 1 1 26 45 70 25
+350 4 2 1 1 21 85 22 43
+351 4 2 1 1 94 93 123 122
+352 4 2 1 1 119 118 86 85
+353 4 2 1 1 12 96 93 123
+354 4 2 1 1 115 78 77 114
+355 4 2 1 1 69 111 70 110
+356 4 2 1 1 28 27 61 46
+357 4 2 1 1 48 62 31 32
+358 4 2 1 1 88 15 92 120
+359 4 2 1 1 79 81 15 116
+360 4 2 1 1 71 14 112 73
+361 4 2 1 1 114 116 79 51
+362 4 2 1 1 112 50 71 110
+363 4 2 1 1 91 12 117 87
+364 4 2 1 1 80 115 14 82
+365 4 2 1 1 63 15 29 68
+366 4 2 1 1 64 26 14 66
+367 4 2 1 1 47 63 105 107
+368 4 2 1 1 43 103 101 55
+369 4 2 1 1 35 10 80 84
+370 4 2 1 1 33 76 9 72
+371 4 2 1 1 37 11 88 90
+372 4 2 1 1 35 71 10 75
+373 4 2 1 1 37 79 11 83
+374 4 2 1 1 9 99 33 95
+375 4 2 1 1 68 63 15 107
+376 4 2 1 1 99 44 93 121
+377 4 2 1 1 18 69 41 17
+378 4 2 1 1 10 35 2 84
+379 4 2 1 1 15 7 28 81
+380 4 2 1 1 3 37 90 11
+381 4 2 1 1 108 64 14 66
+382 4 2 1 1 51 38 79 37
+383 4 2 1 1 12 4 23 98
+384 4 2 1 1 12 39 91 4
+385 4 2 1 1 49 33 72 34
+386 4 2 1 1 10 71 110 75
+387 4 2 1 1 114 11 79 83
+388 4 2 1 1 40 96 16 100
+389 4 2 1 1 34 13 74 72
+390 4 2 1 1 16 87 40 89
+391 4 2 1 1 34 97 13 95
+392 4 2 1 1 20 19 42 53
+393 4 2 1 1 54 23 24 44
+394 4 2 1 1 87 16 118 89
+395 4 2 1 1 95 97 13 122
+396 4 2 1 1 57 22 12 55
+397 4 2 1 1 41 102 104 56
+398 4 2 1 1 64 45 106 108
+399 4 2 1 1 101 57 12 55
+400 4 2 1 1 74 13 5 25
+401 4 2 1 1 13 34 74 5
+402 4 2 1 1 8 40 16 100
+403 4 2 1 1 16 31 65 8
+404 4 2 1 1 82 36 6 14
+405 4 2 1 1 102 9 59 56
+406 4 2 1 1 106 67 13 64
+407 4 2 1 1 119 88 11 90
+408 4 2 1 1 95 49 121 122
+409 4 2 1 1 52 118 117 87
+410 4 2 1 1 124 16 96 100
+411 4 2 1 1 80 10 113 84
+412 4 2 1 1 109 9 76 72
+413 4 2 1 1 72 13 70 111
+414 4 2 1 1 99 9 121 95
+415 4 2 1 1 30 65 16 63
+416 4 2 1 1 13 67 25 64
+417 4 2 1 1 11 55 53 103
+418 4 2 1 1 123 12 44 93
+419 4 2 1 1 56 104 10 58
+420 4 2 1 1 10 18 56 58
+421 4 2 1 1 21 60 11 55
+422 4 2 1 1 17 59 9 56
+423 4 2 1 1 16 63 62 105
+424 4 2 1 1 48 105 16 62
+425 4 2 1 1 111 45 13 70
+426 4 2 1 1 119 86 88 85
+427 4 2 1 1 123 93 94 96
+428 4 2 1 1 115 78 80 77
+429 4 2 1 1 111 72 69 70
+430 4 2 1 1 55 103 54 53
+431 4 2 1 1 63 61 62 107
+432 4 2 1 1 14 27 66 6
+433 4 2 1 1 99 9 1 24
+434 4 2 1 1 49 33 34 95
+435 4 2 1 1 96 52 39 40
+436 4 2 1 1 22 12 91 4
+437 4 2 1 1 20 11 83 3
+438 4 2 1 1 29 15 92 7
+439 4 2 1 1 18 10 75 2
+440 4 2 1 1 36 50 80 35
+441 4 2 1 1 18 17 41 56
+442 4 2 1 1 51 88 38 37
+443 4 2 1 1 62 31 16 48
+444 4 2 1 1 62 16 31 65
+445 4 2 1 1 26 45 25 64
+446 4 2 1 1 104 58 42 10
+447 4 2 1 1 29 63 30 47
+448 4 2 1 1 77 78 79 114
+449 4 2 1 1 71 69 70 110
+450 4 2 1 1 42 103 11 53
+451 4 2 1 1 93 94 95 122
+452 4 2 1 1 118 87 86 85
+453 4 2 1 1 61 62 106 64
+454 4 2 1 1 53 102 54 56
+455 4 2 1 1 78 82 27 46
+456 4 2 1 1 94 48 32 97
+457 4 2 1 1 86 30 89 47
+458 4 2 1 1 78 28 81 46
+459 4 2 1 1 26 70 45 73
+460 4 2 1 1 100 94 48 31
+461 4 2 1 1 86 92 29 47
+462 4 2 1 1 76 69 17 41
+463 4 2 1 1 42 19 77 84
+464 4 2 1 1 77 83 20 42
+465 4 2 1 1 18 41 69 75
+466 4 2 1 1 43 85 90 21
+467 4 2 1 1 91 43 117 12
+468 4 2 1 1 48 62 32 67
+469 4 2 1 1 27 66 61 46
+470 4 2 1 1 68 28 61 46
+471 4 2 1 1 54 24 59 44
+472 4 2 1 1 19 42 53 58
+473 4 2 1 1 23 54 57 44
+474 4 2 1 1 12 93 96 98
+475 4 2 1 1 99 44 121 9
+476 4 2 1 1 42 58 104 53
+477 4 2 1 1 21 43 22 55
+478 4 2 1 1 53 20 11 42
+479 4 2 1 1 53 11 20 60
+480 4 2 1 1 106 67 48 13
+481 4 2 1 1 36 71 50 35
+482 4 2 1 1 52 87 39 40
+483 4 2 1 1 48 67 106 62
+484 4 2 1 1 72 70 13 74
+485 4 2 1 1 9 33 99 1
+486 4 2 1 1 3 37 11 83
+487 4 2 1 1 13 34 5 97
+488 4 2 1 1 12 39 4 98
+489 4 2 1 1 14 36 6 73
+490 4 2 1 1 15 38 92 7
+491 4 2 1 1 8 40 89 16
+492 4 2 1 1 75 35 2 10
+493 4 2 1 1 30 16 65 8
+494 4 2 1 1 14 27 6 82
+495 4 2 1 1 26 14 66 6
+496 4 2 1 1 16 31 8 100
+497 4 2 1 1 32 13 67 5
+498 4 2 1 1 13 67 5 25
+499 4 2 1 1 68 7 28 15
+500 4 2 1 1 29 15 7 68
+501 4 2 1 1 57 4 23 12
+502 4 2 1 1 22 12 4 57
+503 4 2 1 1 10 19 84 2
+504 4 2 1 1 9 59 1 24
+505 4 2 1 1 17 9 59 1
+506 4 2 1 1 18 10 2 58
+507 4 2 1 1 11 21 90 3
+508 4 2 1 1 20 11 3 60
+509 4 2 1 1 16 62 63 65
+510 4 2 1 1 11 53 55 60
+511 4 2 1 1 78 81 79 116
+512 4 2 1 1 71 112 70 73
+513 4 2 1 1 86 88 92 120
+514 4 2 1 1 96 94 124 100
+515 4 2 1 1 94 97 95 122
+516 4 2 1 1 89 118 87 86
+517 4 2 1 1 15 38 7 81
+518 4 2 1 1 102 44 59 9
+519 4 2 1 1 59 44 102 54
+520 4 2 1 1 110 71 69 75
+521 4 2 1 1 79 77 114 83
+522 4 2 1 1 76 69 109 72
+523 4 2 1 1 113 77 80 84
+524 4 2 1 1 93 99 121 95
+525 4 2 1 1 87 117 91 85
+526 4 2 1 1 90 119 43 11
+527 4 2 1 1 80 78 115 82
+528 4 2 1 1 88 119 85 90
+529 4 2 1 1 90 43 119 85
+530 4 2 1 1 46 108 66 61
+531 4 2 1 1 46 66 108 14
+532 4 2 1 1 26 14 6 73
+533 4 2 1 1 44 101 57 12
+534 4 2 1 1 44 57 101 54
+535 4 2 1 1 46 107 68 15
+536 4 2 1 1 46 68 107 61
+537 4 2 1 1 9 33 1 76
+538 4 2 1 1 100 124 48 94
+539 4 2 1 1 100 48 124 16
+540 4 2 1 1 25 70 13 45
+541 4 2 1 1 56 53 104 58
+542 4 2 1 1 61 64 108 66
+543 4 2 1 1 106 62 67 64
+544 4 2 1 1 102 59 54 56
+545 4 2 1 1 32 13 5 97
+546 4 2 1 1 30 16 8 89
+547 4 2 1 1 13 70 25 74
+548 4 2 1 1 97 122 48 13
+549 4 2 1 1 75 110 41 69
+550 4 2 1 1 54 57 101 55
+551 4 2 1 1 97 48 122 94
+552 4 2 1 1 47 118 89 86
+553 4 2 1 1 47 89 118 16
+554 4 2 1 1 17 9 1 76
+555 4 2 1 1 11 21 3 60
+556 4 2 1 1 10 19 2 58
+557 4 2 1 1 75 41 110 10
+558 4 2 1 1 61 63 68 107
+559 4 2 1 1 70 111 45 112
+560 4 2 1 1 78 115 46 116
+561 4 2 1 1 77 114 42 113
+562 4 2 1 1 69 41 109 110
+563 4 2 1 1 107 61 108 46
+564 4 2 1 1 103 42 104 53
+565 4 2 1 1 118 86 47 120
+566 4 2 1 1 94 124 48 122
+567 4 2 1 1 117 119 43 85
+568 4 2 1 1 123 44 121 93
+569 4 2 1 1 101 54 102 44
+570 4 2 1 1 48 105 62 106
+571 4 2 1 1 91 43 12 22
+572 4 2 1 1 91 43 22 85
+573 4 2 1 1 93 12 23 98
+574 4 2 1 1 93 23 12 44
+575 4 2 1 1 46 81 116 15
+576 4 2 1 1 46 116 81 78
+577 4 2 1 1 92 47 120 15
+578 4 2 1 1 120 47 92 86
+579 4 2 1 1 46 82 115 78
+580 4 2 1 1 46 115 82 14
+581 4 2 1 1 73 45 112 70
+582 4 2 1 1 73 112 45 14
+583 4 2 1 1 99 44 9 24
+584 4 2 1 1 99 44 24 93
+585 4 2 1 1 84 42 113 77
+586 4 2 1 1 84 113 42 10
+587 4 2 1 1 76 41 109 69
+588 4 2 1 1 109 41 76 9
+589 4 2 1 1 42 83 114 77
+590 4 2 1 1 42 114 83 11
+591 4 2 2 2 135 13 122 131
+592 4 2 2 2 132 138 125 110
+593 4 2 2 2 138 108 125 106
+594 4 2 2 2 136 108 125 137
+595 4 2 2 2 110 138 125 131
+596 4 2 2 2 136 107 134 125
+597 4 2 2 2 122 135 131 125
+598 4 2 2 2 13 49 122 131
+599 4 2 2 2 136 108 137 46
+600 4 2 2 2 110 138 131 111
+601 4 2 2 2 132 138 110 112
+602 4 2 2 2 106 107 108 125
+603 4 2 2 2 138 112 137 45
+604 4 2 2 2 104 127 125 128
+605 4 2 2 2 113 115 114 125
+606 4 2 2 2 131 122 125 121
+607 4 2 2 2 112 45 14 137
+608 4 2 2 2 110 131 125 130
+609 4 2 2 2 115 116 114 125
+610 4 2 2 2 132 138 112 137
+611 4 2 2 2 50 137 115 132
+612 4 2 2 2 104 127 128 42
+613 4 2 2 2 105 134 125 133
+614 4 2 2 2 129 120 51 119
+615 4 2 2 2 104 130 128 125
+616 4 2 2 2 105 134 133 47
+617 4 2 2 2 135 106 138 125
+618 4 2 2 2 106 105 107 125
+619 4 2 2 2 138 108 106 45
+620 4 2 2 2 104 130 125 102
+621 4 2 2 2 104 127 42 103
+622 4 2 2 2 109 131 49 111
+623 4 2 2 2 106 135 48 105
+624 4 2 2 2 126 101 43 125
+625 4 2 2 2 138 137 125 108
+626 4 2 2 2 104 127 103 125
+627 4 2 2 2 116 114 129 51
+628 4 2 2 2 136 108 46 107
+629 4 2 2 2 136 108 107 125
+630 4 2 2 2 52 124 123 125
+631 4 2 2 2 132 112 110 50
+632 4 2 2 2 44 121 125 123
+633 4 2 2 2 131 122 121 49
+634 4 2 2 2 105 134 47 107
+635 4 2 2 2 106 13 135 138
+636 4 2 2 2 52 133 125 118
+637 4 2 2 2 103 43 101 125
+638 4 2 2 2 117 43 119 125
+639 4 2 2 2 105 134 107 125
+640 4 2 2 2 137 115 132 125
+641 4 2 2 2 50 137 132 112
+642 4 2 2 2 106 135 105 125
+643 4 2 2 2 102 101 44 125
+644 4 2 2 2 104 130 102 41
+645 4 2 2 2 138 45 13 111
+646 4 2 2 2 131 138 135 13
+647 4 2 2 2 138 112 45 111
+648 4 2 2 2 131 138 13 111
+649 4 2 2 2 107 134 15 136
+650 4 2 2 2 118 52 117 125
+651 4 2 2 2 101 126 12 44
+652 4 2 2 2 104 130 41 128
+653 4 2 2 2 138 137 108 45
+654 4 2 2 2 133 118 16 47
+655 4 2 2 2 44 101 126 125
+656 4 2 2 2 133 134 118 47
+657 4 2 2 2 133 134 125 118
+658 4 2 2 2 124 122 123 125
+659 4 2 2 2 129 118 119 125
+660 4 2 2 2 102 104 103 125
+661 4 2 2 2 135 48 13 106
+662 4 2 2 2 123 122 121 125
+663 4 2 2 2 103 127 43 125
+664 4 2 2 2 106 45 13 138
+665 4 2 2 2 103 11 127 42
+666 4 2 2 2 127 51 129 114
+667 4 2 2 2 13 49 131 111
+668 4 2 2 2 137 108 14 46
+669 4 2 2 2 15 47 134 107
+670 4 2 2 2 10 41 128 104
+671 4 2 2 2 16 118 133 52
+672 4 2 2 2 46 107 15 136
+673 4 2 2 2 41 102 9 130
+674 4 2 2 2 128 50 132 110
+675 4 2 2 2 10 41 110 128
+676 4 2 2 2 50 137 112 14
+677 4 2 2 2 130 102 9 44
+678 4 2 2 2 105 133 16 47
+679 4 2 2 2 127 11 103 43
+680 4 2 2 2 128 130 41 110
+681 4 2 2 2 116 134 15 51
+682 4 2 2 2 137 45 14 108
+683 4 2 2 2 12 126 101 43
+684 4 2 2 2 133 48 135 105
+685 4 2 2 2 128 42 10 104
+686 4 2 2 2 131 9 109 49
+687 4 2 2 2 118 117 119 125
+688 4 2 2 2 102 103 101 125
+689 4 2 2 2 129 118 125 134
+690 4 2 2 2 125 129 116 114
+691 4 2 2 2 117 126 43 125
+692 4 2 2 2 126 117 12 52
+693 4 2 2 2 52 126 117 125
+694 4 2 2 2 12 117 126 43
+695 4 2 2 2 127 51 114 11
+696 4 2 2 2 114 129 127 125
+697 4 2 2 2 127 119 43 125
+698 4 2 2 2 119 11 127 43
+699 4 2 2 2 127 113 114 125
+700 4 2 2 2 127 113 42 114
+701 4 2 2 2 128 130 110 125
+702 4 2 2 2 15 47 120 134
+703 4 2 2 2 127 11 114 42
+704 4 2 2 2 120 47 118 134
+705 4 2 2 2 44 130 102 125
+706 4 2 2 2 44 126 12 123
+707 4 2 2 2 123 44 126 125
+708 4 2 2 2 129 118 134 120
+709 4 2 2 2 110 132 128 125
+710 4 2 2 2 128 127 125 113
+711 4 2 2 2 128 127 113 42
+712 4 2 2 2 128 50 110 10
+713 4 2 2 2 113 42 10 128
+714 4 2 2 2 129 134 125 116
+715 4 2 2 2 127 51 11 119
+716 4 2 2 2 129 134 116 51
+717 4 2 2 2 127 51 119 129
+718 4 2 2 2 129 119 127 125
+719 4 2 2 2 110 131 130 109
+720 4 2 2 2 109 130 9 131
+721 4 2 2 2 121 130 9 44
+722 4 2 2 2 122 48 13 135
+723 4 2 2 2 44 121 130 125
+724 4 2 2 2 110 138 111 112
+725 4 2 2 2 131 138 125 135
+726 4 2 2 2 115 137 14 46
+727 4 2 2 2 126 52 12 123
+728 4 2 2 2 123 126 52 125
+729 4 2 2 2 50 137 14 115
+730 4 2 2 2 136 137 115 46
+731 4 2 2 2 121 9 131 49
+732 4 2 2 2 121 131 130 125
+733 4 2 2 2 131 130 9 121
+734 4 2 2 2 113 132 115 125
+735 4 2 2 2 113 50 115 132
+736 4 2 2 2 128 50 10 113
+737 4 2 2 2 132 113 128 125
+738 4 2 2 2 128 50 113 132
+739 4 2 2 2 133 105 135 125
+740 4 2 2 2 52 124 125 133
+741 4 2 2 2 133 48 105 16
+742 4 2 2 2 16 133 124 52
+743 4 2 2 2 136 137 125 115
+744 4 2 2 2 132 138 137 125
+745 4 2 2 2 41 130 9 109
+746 4 2 2 2 134 120 15 51
+747 4 2 2 2 129 118 120 119
+748 4 2 2 2 129 120 134 51
+749 4 2 2 2 135 122 124 125
+750 4 2 2 2 135 48 124 122
+751 4 2 2 2 133 48 16 124
+752 4 2 2 2 133 135 124 125
+753 4 2 2 2 133 48 124 135
+754 4 2 2 2 116 136 134 125
+755 4 2 2 2 115 136 116 125
+756 4 2 2 2 46 115 136 116
+757 4 2 2 2 136 134 15 116
+758 4 2 2 2 46 136 15 116
+759 4 2 2 2 109 41 130 110
+760 4 2 2 2 110 131 109 111
+$EndElements

data/square_in_square.geo

@@ -0,0 +1,77 @@
+// Square-in-square 2D geometry for MFEM
+// Creates concentric squares with different material attributes
+
+// Define the square sizes
+L_outer = 2.0;
+L_inner = 0.5;
+
+// Set mesh size and algorithm
+mesh_size = 1.0;
+Mesh.Algorithm = 6; // Frontal-Delaunay for 2D triangular mesh
+Mesh.CharacteristicLengthFactor = 1.0;
+Mesh.MshFileVersion = 2.2;
+
+// Define center point for concentric squares
+cx = 0.0;
+cy = 0.0;
+
+// Define the points (vertices of the outer square)
+Point(1) = {cx-L_outer/2, cy-L_outer/2, 0, mesh_size}; // bottom-left outer
+Point(2) = {cx+L_outer/2, cy-L_outer/2, 0, mesh_size}; // bottom-right outer
+Point(3) = {cx+L_outer/2, cy+L_outer/2, 0, mesh_size}; // top-right outer
+Point(4) = {cx-L_outer/2, cy+L_outer/2, 0, mesh_size}; // top-left outer
+
+// Define the points (vertices of the inner square)
+Point(5) = {cx-L_inner/2, cy-L_inner/2, 0, mesh_size}; // bottom-left inner
+Point(6) = {cx+L_inner/2, cy-L_inner/2, 0, mesh_size}; // bottom-right inner
+Point(7) = {cx+L_inner/2, cy+L_inner/2, 0, mesh_size}; // top-right inner
+Point(8) = {cx-L_inner/2, cy+L_inner/2, 0, mesh_size}; // top-left inner
+
+// Define the lines (edges of the outer square)
+Line(1) = {1, 2}; // bottom edge
+Line(2) = {2, 3}; // right edge
+Line(3) = {3, 4}; // top edge
+Line(4) = {4, 1}; // left edge
+
+// Define the lines (edges of the inner square)
+Line(5) = {5, 6}; // bottom edge
+Line(6) = {6, 7}; // right edge
+Line(7) = {7, 8}; // top edge
+Line(8) = {8, 5}; // left edge
+
+// Define the surfaces
+// Outer square boundary
+Line Loop(1) = {1, 2, 3, 4};
+
+// Inner square boundary (hole in the outer region)
+Line Loop(2) = {5, 6, 7, 8};
+
+// Define the surface areas
+// Outer region (annular region between squares)
+Plane Surface(1) = {1, 2}; // Outer loop minus inner loop (creates hole)
+
+// Inner region (solid inner square)
+Plane Surface(2) = {2}; // Inner loop only
+
+// Assign physical groups for materials
+Physical Surface(1) = {1}; // Outer material (annular region)
+Physical Surface(2) = {2}; // Inner material (solid square)
+
+// Physical lines for boundary conditions
+// Outer square boundary edges
+Physical Line(1) = {1}; // outer bottom
+Physical Line(2) = {2}; // outer right
+Physical Line(3) = {3}; // outer top
+Physical Line(4) = {4}; // outer left
+
+// Inner square boundary edges
+Physical Line(5) = {5}; // inner bottom
+Physical Line(6) = {6}; // inner right
+Physical Line(7) = {7}; // inner top
+Physical Line(8) = {8}; // inner left
+
+// Mesh control for quality
+Mesh.OptimizeNetgen = 1;
+Mesh.Optimize = 1;
+Mesh.ElementOrder = 1;
+Mesh.RecombineAll = 0; // Keep triangular elements (don't recombine to quads)

data/square_in_square.msh

@@ -0,0 +1,50 @@
+$MeshFormat
+2.2 0 8
+$EndMeshFormat
+$Nodes
+13
+1 -1 -1 0
+2 1 -1 0
+3 1 1 0
+4 -1 1 0
+5 -0.25 -0.25 0
+6 0.25 -0.25 0
+7 0.25 0.25 0
+8 -0.25 0.25 0
+9 -2.752797989558076e-12 -1 0
+10 1 -2.752797989558076e-12 0
+11 2.752797989558076e-12 1 0
+12 -1 2.752797989558076e-12 0
+13 0 0 0
+$EndNodes
+$Elements
+28
+1 1 2 1 1 1 9
+2 1 2 1 1 9 2
+3 1 2 2 2 2 10
+4 1 2 2 2 10 3
+5 1 2 3 3 3 11
+6 1 2 3 3 11 4
+7 1 2 4 4 4 12
+8 1 2 4 4 12 1
+9 1 2 5 5 5 6
+10 1 2 6 6 6 7
+11 1 2 7 7 7 8
+12 1 2 8 8 8 5
+13 2 2 1 1 6 5 9
+14 2 2 1 1 5 8 12
+15 2 2 1 1 7 6 10
+16 2 2 1 1 8 7 11
+17 2 2 1 1 9 5 1
+18 2 2 1 1 5 12 1
+19 2 2 1 1 6 9 2
+20 2 2 1 1 10 6 2
+21 2 2 1 1 7 10 3
+22 2 2 1 1 11 7 3
+23 2 2 1 1 8 11 4
+24 2 2 1 1 8 4 12
+25 2 2 2 2 5 6 13
+26 2 2 2 2 8 5 13
+27 2 2 2 2 6 7 13
+28 2 2 2 2 7 8 13
+$EndElements

doc/CodeDocumentation.conf.in

@@ -1083,7 +1083,8 @@ EXCLUDE_PATTERNS       =
 # ANamespace::AClass, ANamespace::*Test
 
 EXCLUDE_SYMBOLS        = mfem::internal \
-                         mfem::kernels::internal
+                         mfem::kernels::internal \
+                         mfem::future::detail
 
 # The EXAMPLE_PATH tag can be used to specify one or more files or directories
 # that contain example code fragments that are included (see the \include

fem/bilinearform.cpp

@@ -1255,6 +1255,31 @@ void BilinearForm::Mult(const Vector &x, Vector &y) const
    }
 }
 
+void BilinearForm::AddMult(const Vector &x, Vector &y, const real_t a) const
+{
+   if (ext)
+   {
+      ext->AddMult(x, y, a);
+   }
+   else
+   {
+      mat->AddMult(x, y, a);
+   }
+}
+
+void BilinearForm::AddMultTranspose(const Vector &x, Vector &y,
+                                    const real_t a) const
+{
+   if (ext)
+   {
+      ext->AddMultTranspose(x, y, a);
+   }
+   else
+   {
+      mat->AddMultTranspose(x, y, a);
+   }
+}
+
 void BilinearForm::MultTranspose(const Vector & x, Vector & y) const
 {
    if (ext)

fem/bilinearform.hpp

@@ -307,8 +307,8 @@ public:
    { mat->Mult(x, y); mat_e->AddMult(x, y); }
 
    /// Add the matrix vector multiple to a vector:  $ y += a M x $
-   void AddMult(const Vector &x, Vector &y, const real_t a = 1.0) const override
-   { mat -> AddMult (x, y, a); }
+   void AddMult(const Vector &x, Vector &y,
+                const real_t a = 1.0) const override;
 
    /** @brief Add the original uneliminated matrix vector multiple to a vector.
        The original matrix is $ M + Me $ so we have:
@@ -318,8 +318,7 @@ public:
 
    /// Add the matrix transpose vector multiplication:  $ y += a M^T x $
    void AddMultTranspose(const Vector & x, Vector & y,
-                         const real_t a = 1.0) const override
-   { mat->AddMultTranspose(x, y, a); }
+                         const real_t a = 1.0) const override;
 
    /** @brief Add the original uneliminated matrix transpose vector
        multiple to a vector. The original matrix is $ M + M_e $

fem/bilinearform_ext.cpp

@@ -1997,7 +1997,11 @@ void PADiscreteLinearOperatorExtension::Assemble()
    }
    else
    {
-      mfem_error("A real ElementRestriction is required in this setting!");
+      const L2ElementRestriction* l2_elem_restrict =
+         dynamic_cast<const L2ElementRestriction*>(elem_restrict_test);
+      MFEM_VERIFY(l2_elem_restrict,
+                  "A real ElementRestriction is required in this setting!");
+      test_multiplicity = 1.0;
    }
 
    auto tm = test_multiplicity.ReadWrite();
@@ -2036,7 +2040,13 @@ void PADiscreteLinearOperatorExtension::AddMult(
    }
    else
    {
-      mfem_error("In this setting you need a real ElementRestriction!");
+      const L2ElementRestriction* l2_elem_restrict =
+         dynamic_cast<const L2ElementRestriction*>(elem_restrict_test);
+      MFEM_VERIFY(l2_elem_restrict,
+                  "In this setting you need a real ElementRestriction!");
+      tempY.SetSize(y.Size());
+      l2_elem_restrict->MultTranspose(localTest, tempY);
+      y += tempY;
    }
 }
 

fem/coefficient.hpp

@@ -1055,7 +1055,8 @@ public:
 
 typedef VectorCoefficient DiagonalMatrixCoefficient;
 
-/// Base class for Matrix Coefficients that optionally depend on time and space.
+/** Base class for matrix-valued coefficients that optionally depend on time
+    and space. */
 class MatrixCoefficient
 {
 protected:
@@ -1102,6 +1103,9 @@ public:
    /// the quadrature points. The matrix will be transposed or not according to
    /// the boolean argument @a transpose.
    ///
+   /// The stored entries use the same row/column convention as `Eval()`,
+   /// unless `transpose == true`, in which case `K^T` is stored instead.
+   ///
    /// The @a vdim of the QuadratureFunction should be equal to the height times
    /// the width of the matrix.
    virtual void Project(QuadratureFunction &qf, bool transpose=false);

fem/complex_fem.cpp

@@ -588,6 +588,38 @@ SesquilinearForm::AssembleComplexSparseMatrix()
                                   false, false, conv);
 }
 
+void
+SesquilinearForm::BuildComplexOperator(OperatorHandle &A_r,
+                                       OperatorHandle &A_i,
+                                       OperatorHandle &A) const
+{
+   // A = A_r + i A_i
+   A.Clear();
+   if ((!A_r.Ptr() || A_r.Type() == Operator::MFEM_SPARSEMAT) &&
+       (!A_i.Ptr() || A_i.Type() == Operator::MFEM_SPARSEMAT))
+   {
+      ComplexSparseMatrix * A_sp =
+         new ComplexSparseMatrix(A_r.As<SparseMatrix>(),
+                                 A_i.As<SparseMatrix>(),
+                                 A_r.OwnsOperator(),
+                                 A_i.OwnsOperator(),
+                                 conv);
+      A.Reset<ComplexSparseMatrix>(A_sp, true);
+   }
+   else
+   {
+      ComplexOperator * A_op =
+         new ComplexOperator(A_r.Ptr(),
+                             A_i.Ptr(),
+                             A_r.OwnsOperator(),
+                             A_i.OwnsOperator(),
+                             conv);
+      A.Reset<ComplexOperator>(A_op, true);
+   }
+   A_r.SetOperatorOwner(false);
+   A_i.SetOperatorOwner(false);
+}
+
 void
 SesquilinearForm::FormLinearSystem(const Array<int> &ess_tdof_list,
                                    Vector &x, Vector &b,
@@ -716,31 +748,7 @@ SesquilinearForm::FormLinearSystem(const Array<int> &ess_tdof_list,
    B_r.SyncAliasMemory(B);
    B_i.SyncAliasMemory(B);
 
-   // A = A_r + i A_i
-   A.Clear();
-   if ((!A_r.Ptr() || A_r.Type() == Operator::MFEM_SPARSEMAT) &&
-       (!A_i.Ptr() || A_i.Type() == Operator::MFEM_SPARSEMAT))
-   {
-      ComplexSparseMatrix * A_sp =
-         new ComplexSparseMatrix(A_r.As<SparseMatrix>(),
-                                 A_i.As<SparseMatrix>(),
-                                 A_r.OwnsOperator(),
-                                 A_i.OwnsOperator(),
-                                 conv);
-      A.Reset<ComplexSparseMatrix>(A_sp, true);
-   }
-   else
-   {
-      ComplexOperator * A_op =
-         new ComplexOperator(A_r.Ptr(),
-                             A_i.Ptr(),
-                             A_r.OwnsOperator(),
-                             A_i.OwnsOperator(),
-                             conv);
-      A.Reset<ComplexOperator>(A_op, true);
-   }
-   A_r.SetOperatorOwner(false);
-   A_i.SetOperatorOwner(false);
+   BuildComplexOperator(A_r, A_i, A);
 }
 
 void
@@ -777,31 +785,7 @@ SesquilinearForm::FormSystemMatrix(const Array<int> &ess_tdof_list,
       }
    }
 
-   // A = A_r + i A_i
-   A.Clear();
-   if ((!A_r.Ptr() || A_r.Type() == Operator::MFEM_SPARSEMAT) &&
-       (!A_i.Ptr() || A_i.Type() == Operator::MFEM_SPARSEMAT))
-   {
-      ComplexSparseMatrix * A_sp =
-         new ComplexSparseMatrix(A_r.As<SparseMatrix>(),
-                                 A_i.As<SparseMatrix>(),
-                                 A_r.OwnsOperator(),
-                                 A_i.OwnsOperator(),
-                                 conv);
-      A.Reset<ComplexSparseMatrix>(A_sp, true);
-   }
-   else
-   {
-      ComplexOperator * A_op =
-         new ComplexOperator(A_r.Ptr(),
-                             A_i.Ptr(),
-                             A_r.OwnsOperator(),
-                             A_i.OwnsOperator(),
-                             conv);
-      A.Reset<ComplexOperator>(A_op, true);
-   }
-   A_r.SetOperatorOwner(false);
-   A_i.SetOperatorOwner(false);
+   BuildComplexOperator(A_r, A_i, A);
 }
 
 void
@@ -1893,6 +1877,81 @@ ParSesquilinearForm::ParallelAssemble()
                                     true, true, conv);
 }
 
+void
+ParSesquilinearForm::BuildComplexOperator(OperatorHandle &A_r,
+                                          OperatorHandle &A_i,
+                                          OperatorHandle &A) const
+{
+   // A = A_r + i A_i
+   A.Clear();
+   if ((!A_r.Ptr() || A_r.Type() == Operator::Hypre_ParCSR) &&
+       (!A_i.Ptr() || A_i.Type() == Operator::Hypre_ParCSR))
+   {
+      ComplexHypreParMatrix * A_hyp =
+         new ComplexHypreParMatrix(A_r.As<HypreParMatrix>(),
+                                   A_i.As<HypreParMatrix>(),
+                                   A_r.OwnsOperator(),
+                                   A_i.OwnsOperator(),
+                                   conv);
+      A.Reset<ComplexHypreParMatrix>(A_hyp, true);
+   }
+   else
+   {
+      ComplexOperator * A_op =
+         new ComplexOperator(A_r.As<Operator>(),
+                             A_i.As<Operator>(),
+                             A_r.OwnsOperator(),
+                             A_i.OwnsOperator(),
+                             conv);
+      A.Reset<ComplexOperator>(A_op, true);
+   }
+   A_r.SetOperatorOwner(false);
+   A_i.SetOperatorOwner(false);
+}
+
+namespace
+{
+struct ZeroDiagonalHypreKernel
+{
+   const int *ess_tdof_list;
+   const HYPRE_Int *diag_i;
+   real_t *diag_data;
+
+   void MFEM_HOST_DEVICE operator()(int k) const
+   {
+      const int j = ess_tdof_list[k];
+      diag_data[diag_i[j]] = 0.0;
+   }
+};
+}
+
+void
+ParSesquilinearForm::SetImaginaryEssentialDiagonalToZero(
+   const Array<int> &ess_tdof_list, OperatorHandle &A)
+{
+   if (A.Type() == Operator::Hypre_ParCSR)
+   {
+      const int n = ess_tdof_list.Size();
+      HypreParMatrix *Ah;
+      A.Get(Ah);
+      hypre_ParCSRMatrix *Aih = *Ah;
+      Ah->HypreReadWrite();
+      const int *d_ess_tdof_list =
+         ess_tdof_list.GetMemory().Read(GetHypreForallMemoryClass(), n);
+      HYPRE_Int *d_diag_i = Aih->diag->i;
+      real_t *d_diag_data = Aih->diag->data;
+      mfem::hypre_forall(n, ZeroDiagonalHypreKernel
+      {
+         d_ess_tdof_list, d_diag_i, d_diag_data
+      });
+   }
+   else
+   {
+      A.As<ConstrainedOperator>()->SetDiagonalPolicy
+      (mfem::Operator::DiagonalPolicy::DIAG_ZERO);
+   }
+}
+
 void
 ParSesquilinearForm::FormLinearSystem(const Array<int> &ess_tdof_list,
                                       Vector &x, Vector &b,
@@ -1993,27 +2052,7 @@ ParSesquilinearForm::FormLinearSystem(const Array<int> &ess_tdof_list,
       });
       // Modify off-diagonal blocks (imaginary parts of the matrix) to conform
       // with standard essential BC treatment
-      if (A_i.Type() == Operator::Hypre_ParCSR)
-      {
-         HypreParMatrix * Ah;
-         A_i.Get(Ah);
-         hypre_ParCSRMatrix *Aih = *Ah;
-         Ah->HypreReadWrite();
-         const int *d_ess_tdof_list =
-            ess_tdof_list.GetMemory().Read(GetHypreForallMemoryClass(), n);
-         HYPRE_Int *d_diag_i = Aih->diag->i;
-         real_t *d_diag_data = Aih->diag->data;
-         mfem::hypre_forall(n, [=] MFEM_HOST_DEVICE (int k)
-         {
-            const int j = d_ess_tdof_list[k];
-            d_diag_data[d_diag_i[j]] = 0.0;
-         });
-      }
-      else
-      {
-         A_i.As<ConstrainedOperator>()->SetDiagonalPolicy
-         (mfem::Operator::DiagonalPolicy::DIAG_ZERO);
-      }
+      SetImaginaryEssentialDiagonalToZero(ess_tdof_list, A_i);
    }
 
    if (conv == ComplexOperator::BLOCK_SYMMETRIC)
@@ -2032,31 +2071,7 @@ ParSesquilinearForm::FormLinearSystem(const Array<int> &ess_tdof_list,
    B_r.SyncAliasMemory(B);
    B_i.SyncAliasMemory(B);
 
-   // A = A_r + i A_i
-   A.Clear();
-   if ((!A_r.Ptr() || A_r.Type() == Operator::Hypre_ParCSR) &&
-       (!A_i.Ptr() || A_i.Type() == Operator::Hypre_ParCSR))
-   {
-      ComplexHypreParMatrix * A_hyp =
-         new ComplexHypreParMatrix(A_r.As<HypreParMatrix>(),
-                                   A_i.As<HypreParMatrix>(),
-                                   A_r.OwnsOperator(),
-                                   A_i.OwnsOperator(),
-                                   conv);
-      A.Reset<ComplexHypreParMatrix>(A_hyp, true);
-   }
-   else
-   {
-      ComplexOperator * A_op =
-         new ComplexOperator(A_r.As<Operator>(),
-                             A_i.As<Operator>(),
-                             A_r.OwnsOperator(),
-                             A_i.OwnsOperator(),
-                             conv);
-      A.Reset<ComplexOperator>(A_op, true);
-   }
-   A_r.SetOperatorOwner(false);
-   A_i.SetOperatorOwner(false);
+   BuildComplexOperator(A_r, A_i, A);
 }
 
 void
@@ -2081,50 +2096,10 @@ ParSesquilinearForm::FormSystemMatrix(const Array<int> &ess_tdof_list,
    {
       // Modify off-diagonal blocks (imaginary parts of the matrix) to conform
       // with standard essential BC treatment
-      if ( A_i.Type() == Operator::Hypre_ParCSR )
-      {
-         int n = ess_tdof_list.Size();
-         HypreParMatrix * Ah;
-         A_i.Get(Ah);
-         hypre_ParCSRMatrix * Aih = *Ah;
-         for (int k = 0; k < n; k++)
-         {
-            int j = ess_tdof_list[k];
-            Aih->diag->data[Aih->diag->i[j]] = 0.0;
-         }
-      }
-      else
-      {
-         A_i.As<ConstrainedOperator>()->SetDiagonalPolicy
-         (mfem::Operator::DiagonalPolicy::DIAG_ZERO);
-      }
+      SetImaginaryEssentialDiagonalToZero(ess_tdof_list, A_i);
    }
 
-   // A = A_r + i A_i
-   A.Clear();
-   if ((!A_r.Ptr() || A_r.Type() == Operator::Hypre_ParCSR) &&
-       (!A_i.Ptr() || A_i.Type() == Operator::Hypre_ParCSR))
-   {
-      ComplexHypreParMatrix * A_hyp =
-         new ComplexHypreParMatrix(A_r.As<HypreParMatrix>(),
-                                   A_i.As<HypreParMatrix>(),
-                                   A_r.OwnsOperator(),
-                                   A_i.OwnsOperator(),
-                                   conv);
-      A.Reset<ComplexHypreParMatrix>(A_hyp, true);
-   }
-   else
-   {
-      ComplexOperator * A_op =
-         new ComplexOperator(A_r.As<Operator>(),
-                             A_i.As<Operator>(),
-                             A_r.OwnsOperator(),
-                             A_i.OwnsOperator(),
-                             conv);
-      A.Reset<ComplexOperator>(A_op, true);
-   }
-   A_r.SetOperatorOwner(false);
-   A_i.SetOperatorOwner(false);
+   BuildComplexOperator(A_r, A_i, A);
 }
 
 void

fem/complex_fem.hpp

@@ -392,6 +392,9 @@ private:
    bool RealInteg();
    bool ImagInteg();
 
+   void BuildComplexOperator(OperatorHandle &A_r, OperatorHandle &A_i,
+                             OperatorHandle &A) const;
+
 public:
    SesquilinearForm(FiniteElementSpace *fes,
                     ComplexOperator::Convention
@@ -986,6 +989,12 @@ private:
    bool RealInteg();
    bool ImagInteg();
 
+   void SetImaginaryEssentialDiagonalToZero(
+      const Array<int> &ess_tdof_list, OperatorHandle &A);
+
+   void BuildComplexOperator(OperatorHandle &A_r, OperatorHandle &A_i,
+                             OperatorHandle &A) const;
+
 public:
    ParSesquilinearForm(ParFiniteElementSpace *pf,
                        ComplexOperator::Convention

fem/datacollection.cpp

@@ -38,9 +38,24 @@ int DataCollection::create_directory(const std::string &dir_name,
    // create directories recursively
    const char path_delim = '/';
    std::string::size_type pos = 0;
-   int err_flag;
+   int err_flag = 0;
 #ifdef MFEM_USE_MPI
    const ParMesh *pmesh = dynamic_cast<const ParMesh*>(mesh);
+   // In addition to the global root, let the lowest rank on each shared-memory
+   // node create the directory too, so that node-local (non-shared) filesystems
+   // get it on every node rather than only where the global root lives. On a
+   // shared filesystem the extra mkdir() hits EEXIST and is tolerated below.
+   bool node_root = true;
+   if (pmesh)
+   {
+      MPI_Comm node_comm;
+      MPI_Comm_split_type(pmesh->GetComm(), MPI_COMM_TYPE_SHARED, myid,
+                          MPI_INFO_NULL, &node_comm);
+      int node_rank;
+      MPI_Comm_rank(node_comm, &node_rank);
+      node_root = (node_rank == 0);
+      MPI_Comm_free(&node_comm);
+   }
 #endif
 
    do
@@ -52,7 +67,7 @@ int DataCollection::create_directory(const std::string &dir_name,
       err_flag = mkdir(subdir.c_str(), 0777);
       err_flag = (err_flag && (errno != EEXIST)) ? 1 : 0;
 #else
-      if (myid == 0 || pmesh == NULL)
+      if (node_root || pmesh == NULL)
       {
          err_flag = mkdir(subdir.c_str(), 0777);
          err_flag = (err_flag && (errno != EEXIST)) ? 1 : 0;
@@ -64,7 +79,8 @@ int DataCollection::create_directory(const std::string &dir_name,
 #ifdef MFEM_USE_MPI
    if (pmesh)
    {
-      MPI_Bcast(&err_flag, 1, MPI_INT, 0, pmesh->GetComm());
+      MPI_Allreduce(MPI_IN_PLACE, &err_flag, 1, MPI_INT, MPI_MAX,
+                    pmesh->GetComm());
    }
 #endif
 

fem/dfem/doperator.cpp

@@ -51,4 +51,52 @@ DifferentiableOperator::DifferentiableOperator(
    }
 }
 
+
+void FDJacobian::Mult(const Vector &v, Vector &y) const
+{
+   // See [1] for choice of eps.
+   //
+   // [1] Woodward, C.S., Gardner, D.J. and Evans, K.J., 2015. On the use of
+   // finite difference matrix-vector products in Newton-Krylov solvers for
+   // implicit climate dynamics with spectral elements. Procedia Computer
+   // Science, 51, pp.2036-2045.
+   real_t eps;
+   if (fixed_eps > 0.0)
+   {
+      eps = fixed_eps;
+   }
+   else
+   {
+      const real_t vnorm_local = v.Norml2();
+      real_t vnorm;
+      MPI_Allreduce(&vnorm_local, &vnorm, 1, MPITypeMap<real_t>::mpi_type, MPI_SUM,
+                    MPI_COMM_WORLD);
+      eps = lambda * (lambda + xnorm / vnorm);
+   }
+
+   // x + eps * v
+   {
+      const auto d_v = v.Read();
+      const auto d_x = x.Read();
+      auto d_xpev = xpev.Write();
+      mfem::forall(x.Size(), [=] MFEM_HOST_DEVICE (int i)
+      {
+         d_xpev[i] = d_x[i] + eps * d_v[i];
+      });
+   }
+
+   // y = f(x + eps * v)
+   op.Mult(xpev, y);
+
+   // y = (f(x + eps * v) - f(x)) / eps
+   {
+      const auto d_f = f.Read();
+      auto d_y = y.ReadWrite();
+      mfem::forall(f.Size(), [=] MFEM_HOST_DEVICE (int i)
+      {
+         d_y[i] = (d_y[i] - d_f[i]) / eps;
+      });
+   }
+}
+
 #endif // MFEM_USE_MPI

fem/dfem/doperator.hpp

@@ -697,17 +697,18 @@ void DifferentiableOperator::AddIntegrator(
 
    // The explicit captures are necessary to avoid dependency on
    // the specific instance of this class (this pointer).
-   restriction_callback =
-      [=, solutions = this->solutions, parameters = this->parameters]
-      (std::vector<Vector> &sol,
-       const std::vector<Vector> &par,
-       std::vector<Vector> &f)
+   restriction_callback = [element_dof_ordering,
+                           solutions_ = this->solutions,
+                           parameters_ = this->parameters]
+   (std::vector<Vector> &sol,
+    const std::vector<Vector> &par,
+    std::vector<Vector> &f)
    {
-      restriction<entity_t>(solutions, sol, f,
+      restriction<entity_t>(solutions_, sol, f,
                             element_dof_ordering);
-      restriction<entity_t>(parameters, par, f,
+      restriction<entity_t>(parameters_, par, f,
                             element_dof_ordering,
-                            solutions.size());
+                            solutions_.size());
    };
 
    prolongation_transpose = get_prolongation_transpose(
@@ -835,19 +836,19 @@ void DifferentiableOperator::AddIntegrator(
 
          // capture by ref:
          &restriction_cb = this->restriction_callback,
-         &fields_e = this->fields_e,
-         &residual_e = this->residual_e,
-         &output_restriction_transpose = this->output_restriction_transpose
+         &fields_e_ = this->fields_e,
+         &residual_e_ = this->residual_e,
+         &output_restriction_transpose_ = this->output_restriction_transpose
       ]
       (std::vector<Vector> &sol, const std::vector<Vector> &par, Vector &res)
       mutable // mutable: needed to modify 'shmem_cache'
    {
-      restriction_cb(sol, par, fields_e);
+      restriction_cb(sol, par, fields_e_);
 
-      residual_e = 0.0;
-      auto ye = Reshape(residual_e.ReadWrite(), test_vdim, num_test_dof, num_entities);
+      residual_e_ = 0.0;
+      auto ye = Reshape(residual_e_.ReadWrite(), test_vdim, num_test_dof, num_entities);
 
-      auto wrapped_fields_e = wrap_fields(fields_e,
+      auto wrapped_fields_e = wrap_fields(fields_e_,
                                           action_shmem_info.field_sizes,
                                           num_entities);
 
@@ -878,7 +879,7 @@ void DifferentiableOperator::AddIntegrator(
             y, fhat, output_fop, output_dtq_shmem[0],
             scratch_shmem, dimension, use_sum_factorization);
       }, num_entities, thread_blocks, action_shmem_info.total_size, shmem_cache.ReadWrite());
-      output_restriction_transpose(residual_e, res);
+      output_restriction_transpose_(residual_e_, res);
    });
 
    // Without this compile-time check, some valid instantiations of this method
@@ -1193,7 +1194,7 @@ void DifferentiableOperator::AddIntegrator(
 
                // capture by ref:
                &qpdc_mem = derivative_qp_caches_ref,
-               &fields = fields_ref
+               &fields_ = fields_ref
             ](std::vector<Vector> &f_e, SparseMatrix *&A) mutable
          {
             auto wrapped_fields_e = wrap_fields(f_e, shmem_info.field_sizes,
@@ -1241,14 +1242,14 @@ void DifferentiableOperator::AddIntegrator(
             {
                if (input_is_dependent[s])
                {
-                  trial_field = &fields[input_to_field[s]];
+                  trial_field = &fields_[input_to_field[s]];
                }
             }
 
             auto trial_fes = *std::get_if<const ParFiniteElementSpace *>
                              (&trial_field->data);
             auto test_fes = *std::get_if<const ParFiniteElementSpace *>
-                            (&fields[output_to_field[0]].data);
+                            (&fields_[output_to_field[0]].data);
 
             A = new SparseMatrix(test_fes->GetVSize(), trial_fes->GetVSize());
 
@@ -1334,7 +1335,7 @@ void DifferentiableOperator::AddIntegrator(
                input_to_field,
                output_to_field,
                &spmatcb = assemble_derivative_sparsematrix_callbacks_ref,
-               &fields = fields_ref
+               &fields_ = fields_ref
             ](std::vector<Vector> &f_e, HypreParMatrix *&A) mutable
          {
             SparseMatrix *spmat = nullptr;
@@ -1366,14 +1367,14 @@ void DifferentiableOperator::AddIntegrator(
             {
                if (input_is_dependent[s])
                {
-                  trial_field = &fields[input_to_field[s]];
+                  trial_field = &fields_[input_to_field[s]];
                }
             }
 
             auto trial_fes = *std::get_if<const ParFiniteElementSpace *>
                              (&trial_field->data);
             auto test_fes = *std::get_if<const ParFiniteElementSpace *>
-                            (&fields[output_to_field[0]].data);
+                            (&fields_[output_to_field[0]].data);
 
             if (same_test_and_trial)
             {

fem/dfem/util.hpp

@@ -597,7 +597,7 @@ struct ThreadBlocks
    int z = 1;
 };
 
-#if defined(MFEM_USE_CUDA_OR_HIP)
+#if defined(MFEM_USE_CUDA_OR_HIP_LANG)
 template <typename func_t>
 __global__ void forall_kernel_shmem(func_t f, int n)
 {
@@ -617,10 +617,11 @@ void forall(func_t f,
             int num_shmem = 0,
             real_t *shmem = nullptr)
 {
-   if (Device::Allows(Backend::CUDA_MASK) ||
-       Device::Allows(Backend::HIP_MASK))
+   internal::RequireKernelCompilation();
+
+#if defined(MFEM_USE_CUDA_OR_HIP_LANG)
+   if (Device::Allows(Backend::CUDA_MASK | Backend::HIP_MASK))
    {
-#if defined(MFEM_USE_CUDA_OR_HIP)
       // int gridsize = (N + Z - 1) / Z;
       int num_bytes = num_shmem * sizeof(decltype(shmem));
       dim3 block_size(blocks.x, blocks.y, blocks.z);
@@ -631,9 +632,10 @@ void forall(func_t f,
       MFEM_GPU_CHECK(hipGetLastError());
 #endif
       MFEM_DEVICE_SYNC;
-#endif
+      return;
    }
-   else if (Device::Allows(Backend::CPU_MASK))
+#endif
+   if (Device::Allows(Backend::CPU_MASK))
    {
       MFEM_ASSERT(!((bool)num_shmem != (bool)shmem),
                   "Backend::CPU needs a pre-allocated shared memory block");
@@ -671,52 +673,7 @@ public:
                     MPI_COMM_WORLD);
    }
 
-   void Mult(const Vector &v, Vector &y) const override
-   {
-      // See [1] for choice of eps.
-      //
-      // [1] Woodward, C.S., Gardner, D.J. and Evans, K.J., 2015. On the use of
-      // finite difference matrix-vector products in Newton-Krylov solvers for
-      // implicit climate dynamics with spectral elements. Procedia Computer
-      // Science, 51, pp.2036-2045.
-      real_t eps;
-      if (fixed_eps > 0.0)
-      {
-         eps = fixed_eps;
-      }
-      else
-      {
-         const real_t vnorm_local = v.Norml2();
-         real_t vnorm;
-         MPI_Allreduce(&vnorm_local, &vnorm, 1, MPITypeMap<real_t>::mpi_type, MPI_SUM,
-                       MPI_COMM_WORLD);
-         eps = lambda * (lambda + xnorm / vnorm);
-      }
-
-      // x + eps * v
-      {
-         const auto d_v = v.Read();
-         const auto d_x = x.Read();
-         auto d_xpev = xpev.Write();
-         mfem::forall(x.Size(), [=] MFEM_HOST_DEVICE (int i)
-         {
-            d_xpev[i] = d_x[i] + eps * d_v[i];
-         });
-      }
-
-      // y = f(x + eps * v)
-      op.Mult(xpev, y);
-
-      // y = (f(x + eps * v) - f(x)) / eps
-      {
-         const auto d_f = f.Read();
-         auto d_y = y.ReadWrite();
-         mfem::forall(f.Size(), [=] MFEM_HOST_DEVICE (int i)
-         {
-            d_y[i] = (d_y[i] - d_f[i]) / eps;
-         });
-      }
-   }
+   void Mult(const Vector &v, Vector &y) const override;
 
    virtual MemoryClass GetMemoryClass() const override
    {

fem/fe/fe_base.cpp

@@ -1316,13 +1316,14 @@ void VectorFiniteElement::Project_RT(
    }
 }
 
-void VectorFiniteElement::ProjectGrad_RT(
+void VectorFiniteElement::ProjectCurl2D_RT(
    const real_t *nk, const Array<int> &d2n, const FiniteElement &fe,
    ElementTransformation &Trans, DenseMatrix &grad) const
 {
+   // 2D "ProjectCurl_RT"
    if (dim != 2)
    {
-      mfem_error("VectorFiniteElement::ProjectGrad_RT works only in 2D!");
+      mfem_error("VectorFiniteElement::ProjectCurl2D_RT works only in 2D!");
    }
 
    DenseMatrix dshape(fe.GetDof(), fe.GetDim());
@@ -1333,8 +1334,8 @@ void VectorFiniteElement::ProjectGrad_RT(
    for (int k = 0; k < dof; k++)
    {
       fe.CalcDShape(Nodes.IntPoint(k), dshape);
-      tk[0] = nk[d2n[k]*dim+1];
-      tk[1] = -nk[d2n[k]*dim];
+      tk[0] = -nk[d2n[k]*dim+1];
+      tk[1] = nk[d2n[k]*dim];
       dshape.Mult(tk, grad_k);
       for (int j = 0; j < grad_k.Size(); j++)
       {
@@ -1381,7 +1382,7 @@ void VectorFiniteElement::ProjectCurl_ND(
    }
 }
 
-void VectorFiniteElement::ProjectCurl_RT(
+void VectorFiniteElement::ProjectCurl3D_RT(
    const real_t *nk, const Array<int> &d2n, const FiniteElement &fe,
    ElementTransformation &Trans, DenseMatrix &curl) const
 {

fem/fe/fe_base.hpp

@@ -957,10 +957,11 @@ protected:
                    const FiniteElement &fe, ElementTransformation &Trans,
                    DenseMatrix &I) const;
 
-   // rotated gradient in 2D
-   void ProjectGrad_RT(const real_t *nk, const Array<int> &d2n,
-                       const FiniteElement &fe, ElementTransformation &Trans,
-                       DenseMatrix &grad) const;
+   // Input is a scalar representing the Z (out of plane) component, Output is
+   // the X-Y (in-plane) RT curl
+   void ProjectCurl2D_RT(const real_t *nk, const Array<int> &d2n,
+                         const FiniteElement &fe, ElementTransformation &Trans,
+                         DenseMatrix &grad) const;
 
    // Compute the curl as a discrete operator from ND FE (fe) to ND FE (this).
    // The natural FE for the range is RT, so this is an approximation.
@@ -968,9 +969,9 @@ protected:
                        const FiniteElement &fe, ElementTransformation &Trans,
                        DenseMatrix &curl) const;
 
-   void ProjectCurl_RT(const real_t *nk, const Array<int> &d2n,
-                       const FiniteElement &fe, ElementTransformation &Trans,
-                       DenseMatrix &curl) const;
+   void ProjectCurl3D_RT(const real_t *nk, const Array<int> &d2n,
+                         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
@@ -1446,6 +1447,8 @@ public:
                                 dof2quad_array_open);
    }
 
+   const Poly_1D::Basis &GetOpenBasis1D() const { return obasis1d; }
+
    virtual ~VectorTensorFiniteElement();
 };
 

fem/fe/fe_rt.hpp

@@ -73,16 +73,11 @@ public:
    void Project(const FiniteElement &fe, ElementTransformation &Trans,
                 DenseMatrix &I) const override
    { Project_RT(nk, dof2nk, fe, Trans, I); }
-   // Gradient + rotation = Curl: H1 -> H(div)
-   void ProjectGrad(const FiniteElement &fe,
-                    ElementTransformation &Trans,
-                    DenseMatrix &grad) const override
-   { ProjectGrad_RT(nk, dof2nk, fe, Trans, grad); }
    // Curl = Gradient + rotation: H1 -> H(div)
    void ProjectCurl(const FiniteElement &fe,
                     ElementTransformation &Trans,
                     DenseMatrix &curl) const override
-   { ProjectGrad_RT(nk, dof2nk, fe, Trans, curl); }
+   { ProjectCurl2D_RT(nk, dof2nk, fe, Trans, curl); }
 
    void GetFaceMap(const int face_id, Array<int> &face_map) const override;
 
@@ -148,7 +143,7 @@ public:
    void ProjectCurl(const FiniteElement &fe,
                     ElementTransformation &Trans,
                     DenseMatrix &curl) const override
-   { ProjectCurl_RT(nk, dof2nk, fe, Trans, curl); }
+   { ProjectCurl3D_RT(nk, dof2nk, fe, Trans, curl); }
 
    /// @brief Return the mapping from lexicographically ordered face DOFs to
    /// lexicographically ordered element DOFs corresponding to local face
@@ -210,16 +205,11 @@ public:
    void Project(const FiniteElement &fe, ElementTransformation &Trans,
                 DenseMatrix &I) const override
    { Project_RT(nk, dof2nk, fe, Trans, I); }
-   // Gradient + rotation = Curl: H1 -> H(div)
-   void ProjectGrad(const FiniteElement &fe,
-                    ElementTransformation &Trans,
-                    DenseMatrix &grad) const override
-   { ProjectGrad_RT(nk, dof2nk, fe, Trans, grad); }
    // Curl = Gradient + rotation: H1 -> H(div)
    void ProjectCurl(const FiniteElement &fe,
                     ElementTransformation &Trans,
                     DenseMatrix &curl) const override
-   { ProjectGrad_RT(nk, dof2nk, fe, Trans, curl); }
+   { ProjectCurl2D_RT(nk, dof2nk, fe, Trans, curl); }
 };
 
 
@@ -274,7 +264,7 @@ public:
    void ProjectCurl(const FiniteElement &fe,
                     ElementTransformation &Trans,
                     DenseMatrix &curl) const override
-   { ProjectCurl_RT(nk, dof2nk, fe, Trans, curl); }
+   { ProjectCurl3D_RT(nk, dof2nk, fe, Trans, curl); }
 };
 
 class RT_WedgeElement : public VectorFiniteElement
@@ -332,7 +322,7 @@ public:
    void ProjectCurl(const FiniteElement &fe,
                     ElementTransformation &Trans,
                     DenseMatrix &curl) const override
-   { ProjectCurl_RT(nk, dof2nk, fe, Trans, curl); }
+   { ProjectCurl3D_RT(nk, dof2nk, fe, Trans, curl); }
 };
 
 /** Arbitrary order H(Div) basis functions defined on pyramid-shaped elements
@@ -428,7 +418,7 @@ public:
    virtual void ProjectCurl(const FiniteElement &fe,
                             ElementTransformation &Trans,
                             DenseMatrix &curl) const
-   { ProjectCurl_RT(nk, dof2nk, fe, Trans, curl); }
+   { ProjectCurl3D_RT(nk, dof2nk, fe, Trans, curl); }
 
    void CalcRawVShape(const IntegrationPoint &ip,
                       DenseMatrix &shape) const;

fem/fespace.cpp

@@ -22,6 +22,8 @@
 #include <algorithm>
 #include <cmath>
 #include <cstdarg>
+#include <unordered_map>
+#include <unordered_set>
 
 using namespace std;
 
@@ -4527,6 +4529,210 @@ void FiniteElementSpace
    }
 }
 
+void FiniteElementSpace::GetBoundaryLoopEdgeDofs(
+   const Array<int> &boundary_element_indices,
+   Array<int> &boundary_edge_dofs,
+   Array<int> *dof_edges,
+   Array<int> *dof_boundary_elements) const
+{
+   MFEM_VERIFY(mesh->Dimension() >= 2,
+               "GetBoundaryLoopEdgeDofs requires 2D or 3D meshes to find edge objects");
+
+   boundary_edge_dofs.SetSize(0);
+   if (dof_edges) { dof_edges->SetSize(0); }
+   if (dof_boundary_elements) { dof_boundary_elements->SetSize(0); }
+
+   // A DOF that appears in exactly one selected boundary element lies on the
+   // bounding loop; one appearing in two or more is interior to the boundary
+   // region and is dropped. Count occurrences of each DOF (using scratch maps,
+   // exposed only as parallel-indexed Array<int> below) and record, on first
+   // sight, the local edge and boundary element carrying it.
+   //
+   // The count is over GetEdgeDofs, which returns endpoint vertex DOFs as well
+   // as edge-interior DOFs (relevant for collections such as ND_R2D that carry
+   // vertex DOFs). Edge-interior DOFs occur once per edge, so the count mainly
+   // resolves vertex DOFs: a vertex shared by several elements is interior and
+   // dropped, while a genuine loop-corner (open-curve endpoint) vertex is kept.
+   // This is why we count GetEdgeDofs rather than collecting GetEdgeInteriorDofs,
+   // which would omit the endpoint vertex DOFs the method is documented to keep.
+   // The 3D removal criterion (any edge in two or more faces) matches the
+   // parallel version rather than a parity toggle.
+   std::unordered_map<int, int> dof_count, dof_edge, dof_belem;
+   Array<int> edge_dofs, edges, edge_orientations;
+
+   const int dim = mesh->Dimension();
+   for (int i = 0; i < boundary_element_indices.Size(); ++i)
+   {
+      const int boundary_element_idx = boundary_element_indices[i];
+      std::unordered_set<int> boundary_element_dofs;
+
+      if (dim == 3)
+      {
+         // Boundary elements are 2D faces; extract their 1D edges.
+         int face_index, face_orientation;
+         mesh->GetBdrElementFace(boundary_element_idx, &face_index,
+                                 &face_orientation);
+         mesh->GetFaceEdges(face_index, edges, edge_orientations);
+      }
+      else
+      {
+         // Boundary elements are 1D segments, each being a single edge.
+         mesh->GetBdrElementEdges(boundary_element_idx, edges, edge_orientations);
+         MFEM_VERIFY(edges.Size() == 1,
+                     "2D boundary element should have exactly one edge");
+      }
+
+      for (int j = 0; j < edges.Size(); ++j)
+      {
+         GetEdgeDofs(edges[j], edge_dofs);
+         for (int k = 0; k < edge_dofs.Size(); ++k)
+         {
+            const int dof = edge_dofs[k];
+            // Count each DOF once per boundary element and record metadata the
+            // first time it is seen, so H1 DOFs shared by multiple edges of the
+            // same element are not double counted.
+            if (boundary_element_dofs.insert(dof).second &&
+                dof_count[dof]++ == 0)
+            {
+               dof_edge[dof] = edges[j];
+               dof_belem[dof] = boundary_element_idx;
+            }
+         }
+      }
+   }
+
+   // Emit the DOFs seen in exactly one selected boundary element, in a
+   // deterministic (increasing DOF index) order shared by all output arrays.
+   std::vector<int> kept;
+   kept.reserve(dof_count.size());
+   for (const auto &[dof, count] : dof_count)
+   {
+      if (count == 1) { kept.push_back(dof); }
+   }
+   std::sort(kept.begin(), kept.end());
+
+   boundary_edge_dofs.Reserve(static_cast<int>(kept.size()));
+   if (dof_edges) { dof_edges->Reserve(static_cast<int>(kept.size())); }
+   if (dof_boundary_elements)
+   {
+      dof_boundary_elements->Reserve(static_cast<int>(kept.size()));
+   }
+   for (int dof : kept)
+   {
+      boundary_edge_dofs.Append(dof);
+      if (dof_edges) { dof_edges->Append(dof_edge[dof]); }
+      if (dof_boundary_elements) { dof_boundary_elements->Append(dof_belem[dof]); }
+   }
+}
+
+void FiniteElementSpace::GetBoundaryElementsByAttribute(
+   const Array<int> &bdr_attrs,
+   std::vector<Array<int>> &attr_to_elements)
+{
+   // One (initially empty) list of boundary elements per requested attribute,
+   // indexed to match bdr_attrs.
+   attr_to_elements.assign(bdr_attrs.Size(), Array<int>());
+
+   // Map attribute value -> position in bdr_attrs for quick lookup.
+   std::unordered_map<int, int> attr_to_index;
+   for (int i = 0; i < bdr_attrs.Size(); ++i)
+   {
+      attr_to_index[bdr_attrs[i]] = i;
+   }
+
+   // Bucket boundary elements by their attribute.
+   for (int i = 0; i < mesh->GetNBE(); ++i)
+   {
+      int attr = mesh->GetBdrElement(i)->GetAttribute();
+      auto it = attr_to_index.find(attr);
+      if (it != attr_to_index.end())
+      {
+         attr_to_elements[it->second].Append(i);
+      }
+   }
+}
+
+void FiniteElementSpace::GetBoundaryElementsByAttribute(int bdr_attr,
+                                                        Array<int> &boundary_elements)
+{
+   boundary_elements.SetSize(0);
+
+   for (int i = 0; i < mesh->GetNBE(); ++i)
+   {
+      if (mesh->GetBdrElement(i)->GetAttribute() == bdr_attr)
+      {
+         boundary_elements.Append(i);
+      }
+   }
+}
+
+void FiniteElementSpace::ComputeLoopEdgeOrientations(
+   const Array<int> &dof_edges,
+   const Array<int> &dof_boundary_elements,
+   const Vector &loop_normal,
+   Array<int> &dof_orientations) const
+{
+   MFEM_VERIFY(dof_edges.Size() == dof_boundary_elements.Size(),
+               "dof_edges and dof_boundary_elements must be parallel-indexed");
+
+   const int ndof = dof_edges.Size();
+   dof_orientations.SetSize(ndof);
+
+   Array<int> edge_verts, bdr_elem_verts;
+   Vector edge_vec(3), to_edge_vec(3), cross_product(3);
+   for (int i = 0; i < ndof; i++)
+   {
+      const int edge_id = dof_edges[i];
+      const int bdr_elem_idx = dof_boundary_elements[i];
+
+      // Get edge vertices
+      mesh->GetEdgeVertices(edge_id, edge_verts);
+
+      const real_t *v0 = mesh->GetVertex(edge_verts[0]);
+      const real_t *v1 = mesh->GetVertex(edge_verts[1]);
+
+      // Get boundary element vertices
+      mesh->GetBdrElement(bdr_elem_idx)->GetVertices(bdr_elem_verts);
+
+      // Find the third vertex (not part of the edge)
+      int third_vertex = -1;
+      for (int j = 0; j < bdr_elem_verts.Size(); j++)
+      {
+         int v = bdr_elem_verts[j];
+         if (v != edge_verts[0] && v != edge_verts[1])
+         {
+            third_vertex = v;
+            break;
+         }
+      }
+
+      if (third_vertex == -1)
+      {
+         MFEM_ABORT("Boundary element " << bdr_elem_idx << " has only 2 vertices, "
+                    "but 3D boundary elements must have at least 3 vertices");
+      }
+
+      const real_t *v2 = mesh->GetVertex(third_vertex);
+
+      // Edge vector
+      for (int j = 0; j < 3; j++) { edge_vec[j] = v1[j] - v0[j]; }
+
+      // Vector from third vertex to edge (use edge midpoint)
+      for (int j = 0; j < 3; j++)
+      {
+         real_t edge_midpoint = (v0[j] + v1[j]) * 0.5;
+         to_edge_vec[j] = edge_midpoint - v2[j];
+      }
+
+      // Cross product: to_edge × edge
+      to_edge_vec.cross3D(edge_vec, cross_product);
+
+      // Check alignment with loop normal
+      real_t dot_product = cross_product * loop_normal;
+      dof_orientations[i] = (dot_product > 0) ? 1 : -1;
+   }
+}
+
 FiniteElementCollection *FiniteElementSpace::Load(Mesh *m, std::istream &input)
 {
    string buff;

fem/fespace.hpp

@@ -22,6 +22,7 @@
 #include "restriction.hpp"
 #include <iostream>
 #include <unordered_map>
+#include <vector>
 
 namespace mfem
 {
@@ -1389,6 +1390,80 @@ public:
    virtual void GetExteriorTrueDofs(Array<int> &exterior_dofs,
                                     int component = -1) const;
 
+   /** @brief Extract the edge degrees of freedom of a boundary "loop".
+
+       Here a "loop" is the set of boundary edges bounding the region covered by
+       @a boundary_element_indices: in 3D the outer edges of a patch of boundary
+       faces, in 2D the boundary segments themselves. An edge that is shared by
+       two (or more) of the selected boundary elements is interior to that region
+       rather than on its bounding loop, so its DOFs are excluded from the result.
+       This exclusion of interior DOFs is the defining feature of the method.
+
+       The three output arrays share a single indexing: for each valid index @a i,
+       @a dof_edges[i] and @a dof_boundary_elements[i] describe the DOF
+       @a boundary_edge_dofs[i].
+
+       @param[in]  boundary_element_indices Boundary element indices spanning a
+                   boundary surface (3D) or curve (2D).
+       @param[out] boundary_edge_dofs Local DOF indices on the boundary loop.
+       @param[out] dof_edges Optional; local edge index carrying each DOF.
+       @param[out] dof_boundary_elements Optional; a boundary element containing
+                   each DOF.
+
+       @note In 3D the edge DOFs are extracted from the 1D edges of the 2D
+       boundary faces; in 2D they come directly from the 1D boundary segments, so
+       @a dof_edges then holds the boundary element (segment) edge indices.
+       @note This method uses GetEdgeDofs internally, which returns both vertex and
+       edge DOFs. Standard Nédélec elements (ND_FECollection) have no vertex DOFs,
+       so only genuine edge DOFs appear. Collections that carry vertex DOFs (e.g.
+       ND_R2D_FECollection) additionally contribute the vertex DOFs at loop
+       endpoints.
+       @note This is the serial version. For parallel meshes, use the parallel
+       version in ParFiniteElementSpace which handles processor boundaries
+       correctly.
+       @note Requires a 2D or 3D mesh to identify edge objects. The method will
+       assert if called on 1D meshes.
+       @note Only supports conforming meshes; non-conforming meshes are not
+       supported. */
+   void GetBoundaryLoopEdgeDofs(const Array<int> &boundary_element_indices,
+                                Array<int> &boundary_edge_dofs,
+                                Array<int> *dof_edges = nullptr,
+                                Array<int> *dof_boundary_elements = nullptr) const;
+
+   /** @brief Get boundary elements grouped by attribute.
+
+       For each attribute in @a bdr_attrs, collect the indices of all boundary
+       elements carrying that attribute. The result is indexed to match
+       @a bdr_attrs: @a attr_to_elements[i] holds the boundary elements with
+       attribute @a bdr_attrs[i]. */
+   void GetBoundaryElementsByAttribute(
+      const Array<int> &bdr_attrs,
+      std::vector<Array<int>> &attr_to_elements);
+
+   /** @brief Get all boundary elements with a specific attribute. */
+   void GetBoundaryElementsByAttribute(int bdr_attr,
+                                       Array<int> &boundary_elements);
+
+   /** @brief Compute edge orientations relative to a boundary loop direction.
+
+       For each boundary-loop DOF described by @a dof_edges and
+       @a dof_boundary_elements (see GetBoundaryLoopEdgeDofs), determine whether
+       the carrying edge is
+       traversed in the direction consistent with @a loop_normal, following the
+       right-hand rule. Intended for 3D meshes.
+
+       @param[in]  dof_edges Local edge index of each DOF (parallel-indexed with
+                   the boundary_edge_dofs output of GetBoundaryLoopEdgeDofs).
+       @param[in]  dof_boundary_elements A boundary element containing each DOF,
+                   using the same indexing as @a dof_edges.
+       @param[in]  loop_normal Normal vector defining the loop orientation.
+       @param[out] dof_orientations Orientation (+1 or -1) for each DOF, using the
+                   same indexing as @a dof_edges. */
+   void ComputeLoopEdgeOrientations(const Array<int> &dof_edges,
+                                    const Array<int> &dof_boundary_elements,
+                                    const Vector &loop_normal,
+                                    Array<int> &dof_orientations) const;
+
    /// Convert a Boolean marker array to a list containing all marked indices.
    static void MarkerToList(const Array<int> &marker, Array<int> &list);
 

fem/gslib.cpp

@@ -556,7 +556,7 @@ void obboxsurf_calc_3(Vector &bb,
    gslib::lagrange_fun *const lag = gslib::gll_lag_setup(work, n);
    lag(I0, work, n, 1, 0);
 
-   for (int ie = 0; ie < nel; ie++,x+=n2,y+=n2,z+=n2)
+   for (int ie = 0; (unsigned)ie < nel; ie++,x+=n2,y+=n2,z+=n2)
    {
       struct gslib::dbl_range ab[3];
       struct gslib::dbl_range tb[3];
@@ -780,7 +780,7 @@ void obboxedge_calc_2(Vector &bb,
    gslib::lagrange_fun *const lag = gslib::gll_lag_setup(work, nr);
    lag(I0r, work, nr,1, 0);
 
-   for (int ie = 0; ie < nel; ie++,x+=nr,y+=nr)
+   for (int ie = 0; (unsigned)ie < nel; ie++,x+=nr,y+=nr)
    {
       double x0[2], A[4];
       struct gslib::dbl_range ab[2], tb[2];
@@ -892,7 +892,7 @@ void obboxedge_calc_3(Vector &bb,
    gslib::lagrange_fun *const lag = gslib::gll_lag_setup(work, nr);
    lag(I0r, work, nr, 1, 0);
 
-   for (int ie = 0; ie < nel; ie++,x+=nr,y+=nr,z+=nr)
+   for (int ie = 0; (unsigned)ie < nel; ie++,x+=nr,y+=nr,z+=nr)
    {
       double x0[3], A[9], Ai[9];
       struct gslib::dbl_range ab[3], tb[3];
@@ -4518,7 +4518,7 @@ Mesh* FindPointsGSLIB::GetBoundingBoxMesh(int type)
    int eidx = 0;
    if (myid == save_rank)
    {
-      for (int p = 0; p < gsl_comm->np; p++)
+      for (int p = 0; (unsigned)p < gsl_comm->np; p++)
       {
          if (static_cast<unsigned int>(p) != save_rank)
          {

fem/integ/bilininteg_convection_pa.cpp

@@ -178,6 +178,8 @@ void ConvectionIntegrator::AssemblePA(const FiniteElementSpace &fes)
    // Assumes tensor-product elements
    Mesh *mesh = fes.GetMesh();
    const FiniteElement &el = *fes.GetTypicalFE();
+   MFEM_VERIFY(el.GetMapType() == FiniteElement::VALUE,
+               "Only value map type currently supported");
    ElementTransformation &Trans = *mesh->GetTypicalElementTransformation();
    const IntegrationRule *ir = IntRule ? IntRule : &GetRule(el, Trans);
    if (DeviceCanUseCeed())

fem/integ/bilininteg_hcurl_kernels.cpp

@@ -785,6 +785,23 @@ void PAHcurlL2Setup2D(const int Q1D,
    });
 }
 
+void PAHcurlL2IntSetup2D(const int Q1D, const int NE, const Array<real_t> &w,
+                         Vector &coeff, const Vector &detJ, Vector &op)
+{
+   const int NQ = Q1D*Q1D;
+   auto W = w.Read();
+   auto C = Reshape(coeff.Read(), NQ, NE);
+   auto J = Reshape(detJ.Read(), NQ, NE);
+   auto y = Reshape(op.Write(), NQ, NE);
+   mfem::forall(NE, [=] MFEM_HOST_DEVICE (int e)
+   {
+      for (int q = 0; q < NQ; ++q)
+      {
+         y(q,e) = W[q] * C(q,e) / J(q,e);
+      }
+   });
+}
+
 void PAHcurlL2Setup3D(const int NQ,
                       const int coeffDim,
                       const int NE,

fem/integ/bilininteg_hcurl_kernels.hpp

@@ -1889,13 +1889,17 @@ inline void SmemPACurlCurlApply3D(const int d1d,
    ForallWrap<3>(true, NE, device_kernel, host_kernel, Q1D, Q1D, Q1D);
 }
 
-// PA H(curl)-L2 Assemble 2D kernel
+// PA H(curl)-L2 value Assemble 2D kernel
 void PAHcurlL2Setup2D(const int Q1D,
                       const int NE,
                       const Array<real_t> &w,
                       Vector &coeff,
                       Vector &op);
 
+// PA H(curl)-L2 integral Assemble 2D kernel
+void PAHcurlL2IntSetup2D(const int Q1D, const int NE, const Array<real_t> &w,
+                         Vector &coeff, const Vector &detJ, Vector &op);
+
 // PA H(curl)-L2 Assemble 3D kernel
 void PAHcurlL2Setup3D(const int NQ,
                       const int coeffDim,

fem/integ/bilininteg_hcurlhdiv_kernels.hpp

@@ -864,8 +864,656 @@ inline void PAHcurlHdivApplyTranspose3D(const int d1d,
    }); // end of element loop
 }
 
+namespace curlinterp
+{
+constexpr int NBZ3D(int ndof_o, int nquad_o, int mdq)
+{
+   if (ndof_o <= 0 || nquad_o <= 0)
+   {
+      return 1;
+   }
+   int ndof_c = ndof_o + 1;
+   int nquad_c = nquad_o + 1;
+   // z dimension is capped at 64 on nvidia and amd gpus
+   int tmp =
+      std::min((128 + mdq * mdq * (mdq - 1) - 1) / (mdq * mdq * (mdq - 1)), 64);
+   int smem_req =
+      sizeof(mfem::real_t) *
+      ((3 * ndof_c * ndof_c * ndof_o + 2 * 2 * mdq * mdq * mdq) * tmp +
+       ndof_c * nquad_o + ndof_c * nquad_c + ndof_o * nquad_o);
+   // assume GPU has at least 48k shared memory
+   return std::max(std::min(tmp, (48 * 1024 + smem_req - 1) / smem_req), 1);
+}
+}
+
+template <int T_NDOF_O, int T_NQUAD_O>
+void CurlInterpolatorApply3DSmem(const int ne, const int ndof_o,
+                                 const int nquad_o, const Vector &pa,
+                                 const Vector &x_, Vector &y_)
+{
+   constexpr int mnd_o = T_NDOF_O ? T_NDOF_O : DofQuadLimits::HCURL_MAX_D1D - 1;
+   constexpr int mnq_o =
+      T_NQUAD_O ? T_NQUAD_O : DofQuadLimits::HDIV_MAX_D1D - 1;
+   constexpr int mndq = std::max(mnd_o + 1, mnq_o + 1);
+   constexpr int tbatch = curlinterp::NBZ3D(T_NDOF_O, T_NQUAD_O, mndq);
+   MFEM_VERIFY(ndof_o <= mnd_o, "Error: H(curl) order larger than supported");
+   MFEM_VERIFY(nquad_o <= mnq_o, "Error: H(div) order larger than supported");
+   int mnq = std::max(ndof_o + 1, nquad_o + 1);
+   auto pa_data = pa.Read();
+   auto x_d = x_.Read();
+   auto y_d = y_.ReadWrite();
+   mfem::forall_2D_batch<mndq * mndq * (mndq - 1) * tbatch>(
+      ne, mnq * mnq * (mnq - 1), 1, tbatch, [=] MFEM_HOST_DEVICE(int e)
+   {
+      constexpr int MND_O =
+         T_NDOF_O ? T_NDOF_O : DofQuadLimits::HCURL_MAX_D1D - 1;
+      constexpr int MNQ_O =
+         T_NQUAD_O ? T_NQUAD_O : DofQuadLimits::HDIV_MAX_D1D - 1;
+      constexpr int MNDQ = std::max(MND_O + 1, MNQ_O + 1);
+#if defined(__CUDA_ARCH__) || defined(__HIP_DEVICE_COMPILE__)
+      constexpr int nbz = curlinterp::NBZ3D(T_NDOF_O, T_NQUAD_O, MNDQ);
+      int tidz = MFEM_THREAD_ID(z);
+      // Make mnq a local variable since capturing would result in different
+      // captures between host/device versions, and spuriously fails
+      int mnq = std::max(ndof_o + 1, nquad_o + 1);
+#else
+      constexpr int nbz = 1;
+      constexpr int tidz = 0;
+#endif
+      const int NDOF_O = T_NDOF_O ? T_NDOF_O : ndof_o;
+      const int NQUAD_O = T_NQUAD_O ? T_NQUAD_O : nquad_o;
+      const int NDOF_C = NDOF_O + 1;
+      const int NQUAD_C = NQUAD_O + 1;
+      MFEM_SHARED real_t
+      sBG[(MND_O + 1) * MNQ_O + (MND_O + 1) * (MNQ_O + 1) + MND_O * MNQ_O];
+      auto X_ = Reshape(x_d, 3 * NDOF_C * NDOF_C * NDOF_O, ne);
+      auto Y = Reshape(y_d, 3 * NQUAD_C * NQUAD_O * NQUAD_O, ne);
+      auto Gco = Reshape(sBG, NQUAD_O, NDOF_C);
+      auto Bcc = Reshape(sBG + NDOF_C * NQUAD_O, NQUAD_C, NDOF_C);
+      auto Boo =
+         Reshape(sBG + NDOF_C * NQUAD_O + NDOF_C * NQUAD_C, NQUAD_O, NDOF_O);
+      MFEM_SHARED real_t X[3][nbz][MND_O * (MND_O + 1) * (MND_O + 1)];
+      MFEM_SHARED real_t sm0[nbz * 2 * MNDQ * MNDQ * MNDQ];
+      MFEM_SHARED real_t sm1[nbz * 2 * MNDQ * MNDQ * MNDQ];
+
+      // shapes of buffers always use MNDQ to mitigate shared memory bank
+      // conflicts
+      real_t(*DDQ)[nbz][MNDQ][MNDQ][MNDQ] =
+         (real_t(*)[nbz][MNDQ][MNDQ][MNDQ])(sm0);
+      real_t(*DQQ)[nbz][MNDQ][MNDQ][MNDQ] =
+         (real_t(*)[nbz][MNDQ][MNDQ][MNDQ])(sm1);
+      real_t(*QQQ)[nbz][MNDQ][MNDQ][MNDQ] =
+         (real_t(*)[nbz][MNDQ][MNDQ][MNDQ])(sm0);
+      const int offset = NDOF_O * NDOF_C * NDOF_C;
+      const int offsetq = NQUAD_C * NQUAD_O * NQUAD_O;
+      MFEM_FOREACH_THREAD_DIRECT(ix, x, offset)
+      {
+         for (int dim = 0; dim < 3; ++dim)
+         {
+            X[dim][tidz][ix] = X_(ix + dim * offset, e);
+         }
+      }
+      // load basis functions data
+      if (tidz == 0)
+      {
+         auto npts = NDOF_C * NQUAD_O + NDOF_C * NQUAD_C + NDOF_O * NQUAD_O;
+         MFEM_FOREACH_THREAD(ix, x, npts) { sBG[ix] = pa_data[ix]; }
+      }
+      MFEM_SYNC_THREAD;
+
+      // x: Vz Bcc Gco Boo - Vy Bcc Boo Gco
+      // threads assigned to mitigate bank conflicts
+      MFEM_FOREACH_THREAD_DIRECT_3D_OFFSET(qx, dy, dz, x, NQUAD_C, NDOF_C,
+                                           NDOF_O, mnq, mnq, mnq - 1)
+      {
+         real_t u = 0;
+         for (int dx = 0; dx < NDOF_C; ++dx)
+         {
+            u += X[2][tidz][dx + (dy + dz * NDOF_C) * NDOF_C] * Bcc(qx, dx);
+         }
+         DDQ[0][tidz][dz][dy][qx] = u;
+      }
+      // threads assigned to mitigate bank conflicts
+      MFEM_FOREACH_THREAD_DIRECT_3D_OFFSET(qx, dy, dz, x, NQUAD_C, NDOF_O,
+                                           NDOF_C, mnq, mnq - 1, mnq)
+      {
+         real_t u = 0;
+         for (int dx = 0; dx < NDOF_C; ++dx)
+         {
+            u += X[1][tidz][dx + (dy + dz * NDOF_O) * NDOF_C] * Bcc(qx, dx);
+         }
+         DDQ[1][tidz][dz][dy][qx] = u;
+      }
+      MFEM_SYNC_THREAD;
+
+      // threads assigned to mitigate bank conflicts
+      MFEM_FOREACH_THREAD_DIRECT_3D_OFFSET(qx, qy, dz, x, NQUAD_C, NQUAD_O,
+                                           NDOF_O, mnq, mnq, mnq - 1)
+      {
+         real_t u = 0;
+         for (int dy = 0; dy < NDOF_C; ++dy)
+         {
+            u += DDQ[0][tidz][dz][dy][qx] * Gco(qy, dy);
+         }
+         DQQ[0][tidz][dz][qy][qx] = u;
+      }
+      // threads assigned to mitigate bank conflicts
+      MFEM_FOREACH_THREAD_DIRECT_3D_OFFSET(qx, qy, dz, x, NQUAD_C, NQUAD_O,
+                                           NDOF_C, mnq, mnq - 1, mnq)
+      {
+         real_t u = 0;
+         for (int dy = 0; dy < NDOF_O; ++dy)
+         {
+            u += DDQ[1][tidz][dz][dy][qx] * Boo(qy, dy);
+         }
+         DQQ[1][tidz][dz][qy][qx] = u;
+      }
+      MFEM_SYNC_THREAD;
+
+      // threads assigned to mitigate bank conflicts
+      MFEM_FOREACH_THREAD_DIRECT_3D_OFFSET(qx, qy, qz, x, NQUAD_C, NQUAD_O,
+                                           NQUAD_O, mnq, mnq, mnq - 1)
+      {
+         real_t u = 0;
+         for (int dz = 0; dz < NDOF_O; ++dz)
+         {
+            u += DQQ[0][tidz][dz][qy][qx] * Boo(qz, dz);
+         }
+         QQQ[0][tidz][qz][qy][qx] = u;
+      }
+      MFEM_SYNC_THREAD;
+      // threads assigned to mitigate bank conflicts
+      MFEM_FOREACH_THREAD_DIRECT_3D_OFFSET(qx, qy, qz, x, NQUAD_C, NQUAD_O,
+                                           NQUAD_O, mnq, mnq, mnq - 1)
+      {
+         real_t u = 0;
+         for (int dz = 0; dz < NDOF_C; ++dz)
+         {
+            u += DQQ[1][tidz][dz][qy][qx] * Gco(qz, dz);
+         }
+         Y(qx + (qy + qz * NQUAD_O) * NQUAD_C, e) =
+            QQQ[0][tidz][qz][qy][qx] - u;
+      }
+      MFEM_SYNC_THREAD;
+
+      // y: Vx Boo Bcc Gco - Vz Gco Bcc Boo
+      // threads assigned to mitigate bank conflicts
+      MFEM_FOREACH_THREAD_DIRECT_3D_OFFSET(qx, dy, dz, x, NQUAD_O, NDOF_C,
+                                           NDOF_C, mnq - 1, mnq, mnq)
+      {
+         real_t u = 0;
+         for (int dx = 0; dx < NDOF_O; ++dx)
+         {
+            u += X[0][tidz][dx + (dy + dz * NDOF_C) * NDOF_O] * Boo(qx, dx);
+         }
+         DDQ[0][tidz][dz][dy][qx] = u;
+      }
+      // threads assigned to mitigate bank conflicts
+      MFEM_FOREACH_THREAD_DIRECT_3D_OFFSET(qx, dy, dz, x, NQUAD_O, NDOF_C,
+                                           NDOF_O, mnq, mnq, mnq - 1)
+      {
+         real_t u = 0;
+         for (int dx = 0; dx < NDOF_C; ++dx)
+         {
+            u += X[2][tidz][dx + (dy + dz * NDOF_C) * NDOF_C] * Gco(qx, dx);
+         }
+         DDQ[1][tidz][dz][dy][qx] = u;
+      }
+      MFEM_SYNC_THREAD;
+
+      // threads assigned to mitigate bank conflicts
+      MFEM_FOREACH_THREAD_DIRECT_3D_OFFSET(qx, qy, dz, x, NQUAD_O, NQUAD_C,
+                                           NDOF_C, mnq - 1, mnq, mnq)
+      {
+         real_t u = 0;
+         for (int dy = 0; dy < NDOF_C; ++dy)
+         {
+            u += DDQ[0][tidz][dz][dy][qx] * Bcc(qy, dy);
+         }
+         DQQ[0][tidz][dz][qy][qx] = u;
+      }
+      // threads assigned to mitigate bank conflicts
+      MFEM_FOREACH_THREAD_DIRECT_3D_OFFSET(qx, qy, dz, x, NQUAD_O, NQUAD_C,
+                                           NDOF_O, mnq - 1, mnq, mnq)
+      {
+         real_t u = 0;
+         for (int dy = 0; dy < NDOF_C; ++dy)
+         {
+            u += DDQ[1][tidz][dz][dy][qx] * Bcc(qy, dy);
+         }
+         DQQ[1][tidz][dz][qy][qx] = u;
+      }
+      MFEM_SYNC_THREAD;
+
+      // threads assigned to mitigate bank conflicts
+      MFEM_FOREACH_THREAD_DIRECT_3D_OFFSET(qx, qy, qz, x, NQUAD_O, NQUAD_C,
+                                           NQUAD_O, mnq, mnq, mnq - 1)
+      {
+         real_t u = 0;
+         for (int dz = 0; dz < NDOF_C; ++dz)
+         {
+            u += DQQ[0][tidz][dz][qy][qx] * Gco(qz, dz);
+         }
+         QQQ[0][tidz][qz][qy][qx] = u;
+      }
+      MFEM_SYNC_THREAD;
+      // threads assigned to mitigate bank conflicts
+      MFEM_FOREACH_THREAD_DIRECT_3D_OFFSET(qx, qy, qz, x, NQUAD_O, NQUAD_C,
+                                           NQUAD_O, mnq, mnq, mnq - 1)
+      {
+         real_t u = 0;
+         for (int dz = 0; dz < NDOF_O; ++dz)
+         {
+            u += DQQ[1][tidz][dz][qy][qx] * Boo(qz, dz);
+         }
+         Y(qx + (qy + qz * NQUAD_C) * NQUAD_O + offsetq, e) =
+            QQQ[0][tidz][qz][qy][qx] - u;
+      }
+      MFEM_SYNC_THREAD;
+
+      // z: Vy Gco Boo Bcc - Vx Boo Gco Bcc
+      // threads assigned to mitigate bank conflicts
+      MFEM_FOREACH_THREAD_DIRECT_3D_OFFSET(qx, dy, dz, x, NQUAD_O, NDOF_O,
+                                           NDOF_C, mnq, mnq - 1, mnq)
+      {
+         real_t u = 0;
+         for (int dx = 0; dx < NDOF_C; ++dx)
+         {
+            u += X[1][tidz][dx + (dy + dz * NDOF_O) * NDOF_C] * Gco(qx, dx);
+         }
+         DDQ[0][tidz][dz][dy][qx] = u;
+      }
+      // threads assigned to mitigate bank conflicts
+      MFEM_FOREACH_THREAD_DIRECT_3D_OFFSET(qx, dy, dz, x, NQUAD_O, NDOF_C,
+                                           NDOF_C, mnq - 1, mnq, mnq)
+      {
+         real_t u = 0;
+         for (int dx = 0; dx < NDOF_O; ++dx)
+         {
+            u += X[0][tidz][dx + (dy + dz * NDOF_C) * NDOF_O] * Boo(qx, dx);
+         }
+         DDQ[1][tidz][dz][dy][qx] = u;
+      }
+      MFEM_SYNC_THREAD;
+
+      // threads assigned to mitigate bank conflicts
+      MFEM_FOREACH_THREAD_DIRECT_3D_OFFSET(qx, qy, dz, x, NQUAD_O, NQUAD_O,
+                                           NDOF_C, mnq, mnq - 1, mnq)
+      {
+         real_t u = 0;
+         for (int dy = 0; dy < NDOF_O; ++dy)
+         {
+            u += DDQ[0][tidz][dz][dy][qx] * Boo(qy, dy);
+         }
+         DQQ[0][tidz][dz][qy][qx] = u;
+      }
+      // threads assigned to mitigate bank conflicts
+      MFEM_FOREACH_THREAD_DIRECT_3D_OFFSET(qx, qy, dz, x, NQUAD_O, NQUAD_O,
+                                           NDOF_C, mnq, mnq - 1, mnq)
+      {
+         real_t u = 0;
+         for (int dy = 0; dy < NDOF_C; ++dy)
+         {
+            u += DDQ[1][tidz][dz][dy][qx] * Gco(qy, dy);
+         }
+         DQQ[1][tidz][dz][qy][qx] = u;
+      }
+      MFEM_SYNC_THREAD;
+
+      // threads assigned to mitigate bank conflicts
+      MFEM_FOREACH_THREAD_DIRECT_3D_OFFSET(qx, qy, qz, x, NQUAD_O, NQUAD_O,
+                                           NQUAD_C, mnq, mnq - 1, mnq)
+      {
+         real_t u = 0;
+         for (int dz = 0; dz < NDOF_C; ++dz)
+         {
+            u += DQQ[0][tidz][dz][qy][qx] * Bcc(qz, dz);
+         }
+         QQQ[0][tidz][qz][qy][qx] = u;
+      }
+      MFEM_SYNC_THREAD;
+      // threads assigned to mitigate bank conflicts
+      MFEM_FOREACH_THREAD_DIRECT_3D_OFFSET(qx, qy, qz, x, NQUAD_O, NQUAD_O,
+                                           NQUAD_C, mnq, mnq - 1, mnq)
+      {
+         real_t u = 0;
+         for (int dz = 0; dz < NDOF_C; ++dz)
+         {
+            u += DQQ[1][tidz][dz][qy][qx] * Bcc(qz, dz);
+         }
+         Y(qx + (qy + qz * NQUAD_O) * NQUAD_O + 2 * offsetq, e) =
+            QQQ[0][tidz][qz][qy][qx] - u;
+      }
+      MFEM_SYNC_THREAD;
+   });
+}
+
+template <int T_NDOF_O, int T_NQUAD_O>
+void CurlInterpolatorTApply3DSmem(const int ne, const int ndof_o,
+                                  const int nquad_o, const Vector &pa,
+                                  const Vector &x_, Vector &y_)
+{
+   constexpr int mnd_o = T_NDOF_O ? T_NDOF_O : DofQuadLimits::HCURL_MAX_D1D - 1;
+   constexpr int mnq_o =
+      T_NQUAD_O ? T_NQUAD_O : DofQuadLimits::HDIV_MAX_D1D - 1;
+   constexpr int mndq = std::max(mnd_o + 1, mnq_o + 1);
+   constexpr int tbatch = curlinterp::NBZ3D(T_NDOF_O, T_NQUAD_O, mndq);
+   MFEM_VERIFY(ndof_o <= mnd_o, "Error: H(curl) order larger than supported");
+   MFEM_VERIFY(nquad_o <= mnq_o, "Error: H(div) order larger than supported");
+   int mnq = std::max(ndof_o + 1, nquad_o + 1);
+   auto pa_data = pa.Read();
+   auto x_d = x_.Read();
+   auto y_d = y_.ReadWrite();
+   mfem::forall_2D_batch<mndq * mndq * (mndq - 1) * tbatch>(
+      ne, mnq * mnq * (mnq - 1), 1, tbatch, [=] MFEM_HOST_DEVICE(int e)
+   {
+      constexpr int MND_O =
+         T_NDOF_O ? T_NDOF_O : DofQuadLimits::HCURL_MAX_D1D - 1;
+      constexpr int MNQ_O =
+         T_NQUAD_O ? T_NQUAD_O : DofQuadLimits::HDIV_MAX_D1D - 1;
+      constexpr int MNDQ = std::max(MND_O + 1, MNQ_O + 1);
+#if defined(__CUDA_ARCH__) || defined(__HIP_DEVICE_COMPILE__)
+      constexpr int nbz = curlinterp::NBZ3D(T_NDOF_O, T_NQUAD_O, MNDQ);
+      int tidz = MFEM_THREAD_ID(z);
+      // Make mnq a local variable since capturing would result in different
+      // captures between host/device versions, and spuriously fails
+      int mnq = std::max(ndof_o + 1, nquad_o + 1);
+#else
+      constexpr int nbz = 1;
+      constexpr int tidz = 0;
+#endif
+      const int NDOF_O = T_NDOF_O ? T_NDOF_O : ndof_o;
+      const int NQUAD_O = T_NQUAD_O ? T_NQUAD_O : nquad_o;
+      const int NDOF_C = NDOF_O + 1;
+      const int NQUAD_C = NQUAD_O + 1;
+      MFEM_SHARED real_t
+      sBG[(MND_O + 1) * MNQ_O + (MND_O + 1) * (MNQ_O + 1) + MND_O * MNQ_O];
+      auto X_ = Reshape(x_d, 3 * NQUAD_C * NQUAD_O * NQUAD_O, ne);
+      auto Y = Reshape(y_d, 3 * NDOF_C * NDOF_C * NDOF_O, ne);
+      auto Gco = Reshape(sBG, NQUAD_O, NDOF_C);
+      auto Bcc = Reshape(sBG + NDOF_C * NQUAD_O, NQUAD_C, NDOF_C);
+      auto Boo =
+         Reshape(sBG + NDOF_C * NQUAD_O + NDOF_C * NQUAD_C, NQUAD_O, NDOF_O);
+      MFEM_SHARED real_t X[3][nbz][MNQ_O * MNQ_O * (MNQ_O + 1)];
+      MFEM_SHARED real_t sm0[nbz * 2 * MNDQ * MNDQ * MNDQ];
+      MFEM_SHARED real_t sm1[nbz * 2 * MNDQ * MNDQ * MNDQ];
+
+      // shapes of buffers always use MNDQ to mitigate shared memory bank
+      // conflicts
+      real_t(*QQD)[nbz][MNDQ][MNDQ][MNDQ] =
+         (real_t(*)[nbz][MNDQ][MNDQ][MNDQ])(sm0);
+      real_t(*QDD)[nbz][MNDQ][MNDQ][MNDQ] =
+         (real_t(*)[nbz][MNDQ][MNDQ][MNDQ])(sm1);
+      real_t(*DDD)[nbz][MNDQ][MNDQ][MNDQ] =
+         (real_t(*)[nbz][MNDQ][MNDQ][MNDQ])(sm0);
+      const int offset = NDOF_O * NDOF_C * NDOF_C;
+      const int offsetq = NQUAD_C * NQUAD_O * NQUAD_O;
+      MFEM_FOREACH_THREAD_DIRECT(ix, x, offsetq)
+      {
+         for (int dim = 0; dim < 3; ++dim)
+         {
+            X[dim][tidz][ix] = X_(ix + dim * offsetq, e);
+         }
+      }
+      // load basis functions data
+      if (tidz == 0)
+      {
+         auto npts = NDOF_C * NQUAD_O + NDOF_C * NQUAD_C + NDOF_O * NQUAD_O;
+         MFEM_FOREACH_THREAD(ix, x, npts) { sBG[ix] = pa_data[ix]; }
+      }
+      MFEM_SYNC_THREAD;
+
+      // x: Vy Boo Bcc Gco - Vz Boo Gco Bcc
+      // threads assigned to mitigate bank conflicts
+      MFEM_FOREACH_THREAD_DIRECT_3D_OFFSET(dz, qx, qy, x, NDOF_C, NQUAD_O,
+                                           NQUAD_C, mnq, mnq - 1, mnq)
+      {
+         real_t u = 0;
+         for (int qz = 0; qz < NQUAD_O; ++qz)
+         {
+            u += X[1][tidz][qx + (qy + qz * NQUAD_C) * NQUAD_O] * Gco(qz, dz);
+         }
+         QQD[0][tidz][qy][qx][dz] = u;
+      }
+      // threads assigned to mitigate bank conflicts
+      MFEM_FOREACH_THREAD_DIRECT_3D_OFFSET(dz, qx, qy, x, NDOF_C, NQUAD_O,
+                                           NQUAD_O, mnq, mnq, mnq - 1)
+      {
+         real_t u = 0;
+         for (int qz = 0; qz < NQUAD_C; ++qz)
+         {
+            u += X[2][tidz][qx + (qy + qz * NQUAD_O) * NQUAD_O] * Bcc(qz, dz);
+         }
+         QQD[1][tidz][qy][qx][dz] = u;
+      }
+      MFEM_SYNC_THREAD;
+
+      // threads assigned to mitigate bank conflicts
+      MFEM_FOREACH_THREAD_DIRECT_3D_OFFSET(dy, dz, qx, x, NDOF_C, NDOF_C,
+                                           NQUAD_O, mnq, mnq, mnq - 1)
+      {
+         real_t u = 0;
+         for (int qy = 0; qy < NQUAD_C; ++qy)
+         {
+            u += QQD[0][tidz][qy][qx][dz] * Bcc(qy, dy);
+         }
+         QDD[0][tidz][qx][dz][dy] = u;
+      }
+      // threads assigned to mitigate bank conflicts
+      MFEM_FOREACH_THREAD_DIRECT_3D_OFFSET(dy, dz, qx, x, NDOF_C, NDOF_C,
+                                           NQUAD_O, mnq, mnq, mnq - 1)
+      {
+         real_t u = 0;
+         for (int qy = 0; qy < NQUAD_O; ++qy)
+         {
+            u += QQD[1][tidz][qy][qx][dz] * Gco(qy, dy);
+         }
+         QDD[1][tidz][qx][dz][dy] = u;
+      }
+      MFEM_SYNC_THREAD;
+
+      // threads assigned to mitigate bank conflicts
+      MFEM_FOREACH_THREAD_DIRECT_3D_OFFSET(dx, dy, dz, x, NDOF_O, NDOF_C,
+                                           NDOF_C, mnq - 1, mnq, mnq)
+      {
+         real_t u = 0;
+         for (int qx = 0; qx < NQUAD_O; ++qx)
+         {
+            u += QDD[0][tidz][qx][dz][dy] * Boo(qx, dx);
+         }
+         DDD[0][tidz][dz][dy][dx] = u;
+      }
+      MFEM_SYNC_THREAD;
+      // threads assigned to mitigate bank conflicts
+      MFEM_FOREACH_THREAD_DIRECT_3D_OFFSET(dx, dy, dz, x, NDOF_O, NDOF_C,
+                                           NDOF_C, mnq - 1, mnq, mnq)
+      {
+         real_t u = 0;
+         for (int qx = 0; qx < NQUAD_O; ++qx)
+         {
+            u += QDD[1][tidz][qx][dz][dy] * Boo(qx, dx);
+         }
+         Y(dx + (dy + dz * NDOF_C) * NDOF_O, e) =
+            DDD[0][tidz][dz][dy][dx] - u;
+      }
+      MFEM_SYNC_THREAD;
+
+      // y: Vz Gco Boo Bcc - Vx Bcc Boo Gco
+      // threads assigned to mitigate bank conflicts
+      MFEM_FOREACH_THREAD_DIRECT_3D_OFFSET(dz, qx, qy, x, NDOF_C, NQUAD_O,
+                                           NQUAD_O, mnq, mnq, mnq - 1)
+      {
+         real_t u = 0;
+         for (int qz = 0; qz < NQUAD_C; ++qz)
+         {
+            u += X[2][tidz][qx + (qy + qz * NQUAD_O) * NQUAD_O] * Bcc(qz, dz);
+         }
+         QQD[0][tidz][qy][qx][dz] = u;
+      }
+      // threads assigned to mitigate bank conflicts
+      MFEM_FOREACH_THREAD_DIRECT_3D_OFFSET(dz, qx, qy, x, NDOF_C, NQUAD_C,
+                                           NQUAD_O, mnq, mnq, mnq - 1)
+      {
+         real_t u = 0;
+         for (int qz = 0; qz < NQUAD_O; ++qz)
+         {
+            u += X[0][tidz][qx + (qy + qz * NQUAD_O) * NQUAD_C] * Gco(qz, dz);
+         }
+         QQD[1][tidz][qy][qx][dz] = u;
+      }
+      MFEM_SYNC_THREAD;
+
+      // threads assigned to mitigate bank conflicts
+      MFEM_FOREACH_THREAD_DIRECT_3D_OFFSET(dy, dz, qx, x, NDOF_O, NDOF_C,
+                                           NQUAD_O, mnq, mnq, mnq - 1)
+      {
+         real_t u = 0;
+         for (int qy = 0; qy < NQUAD_O; ++qy)
+         {
+            u += QQD[0][tidz][qy][qx][dz] * Boo(qy, dy);
+         }
+         QDD[0][tidz][qx][dz][dy] = u;
+      }
+      // threads assigned to mitigate bank conflicts
+      MFEM_FOREACH_THREAD_DIRECT_3D_OFFSET(dy, dz, qx, x, NDOF_O, NDOF_C,
+                                           NQUAD_C, mnq - 1, mnq, mnq)
+      {
+         real_t u = 0;
+         for (int qy = 0; qy < NQUAD_O; ++qy)
+         {
+            u += QQD[1][tidz][qy][qx][dz] * Boo(qy, dy);
+         }
+         QDD[1][tidz][qx][dz][dy] = u;
+      }
+      MFEM_SYNC_THREAD;
+
+      // threads assigned to mitigate bank conflicts
+      MFEM_FOREACH_THREAD_DIRECT_3D_OFFSET(dx, dy, dz, x, NDOF_C, NDOF_O,
+                                           NDOF_C, mnq, mnq - 1, mnq)
+      {
+         real_t u = 0;
+         for (int qx = 0; qx < NQUAD_O; ++qx)
+         {
+            u += QDD[0][tidz][qx][dz][dy] * Gco(qx, dx);
+         }
+         DDD[0][tidz][dz][dy][dx] = u;
+      }
+      MFEM_SYNC_THREAD;
+      // threads assigned to mitigate bank conflicts
+      MFEM_FOREACH_THREAD_DIRECT_3D_OFFSET(dx, dy, dz, x, NDOF_C, NDOF_O,
+                                           NDOF_C, mnq, mnq - 1, mnq)
+      {
+         real_t u = 0;
+         for (int qx = 0; qx < NQUAD_C; ++qx)
+         {
+            u += QDD[1][tidz][qx][dz][dy] * Bcc(qx, dx);
+         }
+         Y(dx + (dy + dz * NDOF_O) * NDOF_C + offset, e) =
+            DDD[0][tidz][dz][dy][dx] - u;
+      }
+      MFEM_SYNC_THREAD;
+
+      // z: Vx Bcc Gco Boo - Vy Gco Bcc Boo
+      // threads assigned to mitigate bank conflicts
+      MFEM_FOREACH_THREAD_DIRECT_3D_OFFSET(dz, qx, qy, x, NDOF_O, NQUAD_C,
+                                           NQUAD_O, mnq, mnq, mnq - 1)
+      {
+         real_t u = 0;
+         for (int qz = 0; qz < NQUAD_O; ++qz)
+         {
+            u += X[0][tidz][qx + (qy + qz * NQUAD_O) * NQUAD_C] * Boo(qz, dz);
+         }
+         QQD[0][tidz][qy][qx][dz] = u;
+      }
+      // threads assigned to mitigate bank conflicts
+      MFEM_FOREACH_THREAD_DIRECT_3D_OFFSET(dz, qx, qy, x, NDOF_O, NQUAD_O,
+                                           NQUAD_C, mnq, mnq - 1, mnq)
+      {
+         real_t u = 0;
+         for (int qz = 0; qz < NQUAD_O; ++qz)
+         {
+            u += X[1][tidz][qx + (qy + qz * NQUAD_C) * NQUAD_O] * Boo(qz, dz);
+         }
+         QQD[1][tidz][qy][qx][dz] = u;
+      }
+      MFEM_SYNC_THREAD;
+
+      // threads assigned to mitigate bank conflicts
+      MFEM_FOREACH_THREAD_DIRECT_3D_OFFSET(dy, dz, qx, x, NDOF_C, NDOF_O,
+                                           NQUAD_C, mnq, mnq - 1, mnq)
+      {
+         real_t u = 0;
+         for (int qy = 0; qy < NQUAD_O; ++qy)
+         {
+            u += QQD[0][tidz][qy][qx][dz] * Gco(qy, dy);
+         }
+         QDD[0][tidz][qx][dz][dy] = u;
+      }
+      // threads assigned to mitigate bank conflicts
+      MFEM_FOREACH_THREAD_DIRECT_3D_OFFSET(dy, dz, qx, x, NDOF_C, NDOF_O,
+                                           NQUAD_O, mnq, mnq, mnq - 1)
+      {
+         real_t u = 0;
+         for (int qy = 0; qy < NQUAD_C; ++qy)
+         {
+            u += QQD[1][tidz][qy][qx][dz] * Bcc(qy, dy);
+         }
+         QDD[1][tidz][qx][dz][dy] = u;
+      }
+      MFEM_SYNC_THREAD;
+
+      // threads assigned to mitigate bank conflicts
+      MFEM_FOREACH_THREAD_DIRECT_3D_OFFSET(dx, dy, dz, x, NDOF_C, NDOF_C,
+                                           NDOF_O, mnq, mnq, mnq - 1)
+      {
+         real_t u = 0;
+         for (int qx = 0; qx < NQUAD_C; ++qx)
+         {
+            u += QDD[0][tidz][qx][dz][dy] * Bcc(qx, dx);
+         }
+         DDD[0][tidz][dz][dy][dx] = u;
+      }
+      MFEM_SYNC_THREAD;
+      // threads assigned to mitigate bank conflicts
+      MFEM_FOREACH_THREAD_DIRECT_3D_OFFSET(dx, dy, dz, x, NDOF_C, NDOF_C,
+                                           NDOF_O, mnq, mnq, mnq - 1)
+      {
+         real_t u = 0;
+         for (int qx = 0; qx < NQUAD_O; ++qx)
+         {
+            u += QDD[1][tidz][qx][dz][dy] * Gco(qx, dx);
+         }
+         Y(dx + (dy + dz * NDOF_C) * NDOF_C + 2 * offset, e) =
+            DDD[0][tidz][dz][dy][dx] - u;
+      }
+      MFEM_SYNC_THREAD;
+   });
+}
+
 } // namespace internal
 
+template <int DIM, int NDOF_O, int NQUAD_O>
+CurlInterpolator::ApplyKernelType
+CurlInterpolator::ApplyPAKernels::Kernel()
+{
+   if constexpr (DIM == 3)
+   {
+      return internal::CurlInterpolatorApply3DSmem<NDOF_O, NQUAD_O>;
+   }
+   MFEM_ABORT("Bad dimension!");
+}
+
+template <int DIM, int NDOF_O, int NQUAD_O>
+CurlInterpolator::ApplyKernelType
+CurlInterpolator::ApplyTPAKernels::Kernel()
+{
+   if constexpr (DIM == 3)
+   {
+      return internal::CurlInterpolatorTApply3DSmem<NDOF_O, NQUAD_O>;
+   }
+   MFEM_ABORT("Bad dimension!");
+}
+
 } // namespace mfem
 
 /// \endcond DO_NOT_DOCUMENT

fem/integ/bilininteg_interp_pa.cpp

@@ -14,9 +14,218 @@
 #include "../gridfunc.hpp"
 #include "../qfunction.hpp"
 
+#include "bilininteg_hcurlhdiv_kernels.hpp"
+
 namespace mfem
 {
 
+namespace
+{
+
+void PAHcurlApplyCurl2D(const int c_dofs1D,
+                        const int o_dofs1D,
+                        const int NE,
+                        const Array<real_t> &Bo_,
+                        const Array<real_t> &Gc_,
+                        const Vector &x_,
+                        Vector &y_)
+{
+   auto Bo = Reshape(Bo_.Read(), o_dofs1D, o_dofs1D);
+   auto Gc = Reshape(Gc_.Read(), o_dofs1D, c_dofs1D);
+   auto X = Reshape(x_.Read(), 2 * c_dofs1D * o_dofs1D, NE);
+   auto Y = Reshape(y_.ReadWrite(), o_dofs1D, o_dofs1D, NE);
+
+   mfem::forall(NE, [=] MFEM_HOST_DEVICE (int e)
+   {
+      for (int iy = 0; iy < c_dofs1D; ++iy)
+      {
+         for (int ix = 0; ix < o_dofs1D; ++ix)
+         {
+            const real_t xv = X(ix + iy * o_dofs1D, e);
+            for (int oy = 0; oy < o_dofs1D; ++oy)
+            {
+               const real_t gy = Gc(oy, iy);
+               for (int ox = 0; ox < o_dofs1D; ++ox)
+               {
+                  Y(ox, oy, e) -= Bo(ox, ix) * gy * xv;
+               }
+            }
+         }
+      }
+
+      const int y_nd = c_dofs1D * o_dofs1D;
+      for (int iy = 0; iy < o_dofs1D; ++iy)
+      {
+         for (int ix = 0; ix < c_dofs1D; ++ix)
+         {
+            const real_t xv = X(y_nd + ix + iy * c_dofs1D, e);
+            for (int oy = 0; oy < o_dofs1D; ++oy)
+            {
+               const real_t by = Bo(oy, iy);
+               for (int ox = 0; ox < o_dofs1D; ++ox)
+               {
+                  Y(ox, oy, e) += Gc(ox, ix) * by * xv;
+               }
+            }
+         }
+      }
+   });
+}
+
+void PAHcurlApplyCurlTranspose2D(const int c_dofs1D,
+                                 const int o_dofs1D,
+                                 const int NE,
+                                 const Array<real_t> &Bo_,
+                                 const Array<real_t> &Gc_,
+                                 const Vector &x_,
+                                 Vector &y_)
+{
+   auto Bo = Reshape(Bo_.Read(), o_dofs1D, o_dofs1D);
+   auto Gc = Reshape(Gc_.Read(), o_dofs1D, c_dofs1D);
+   auto X = Reshape(x_.Read(), o_dofs1D, o_dofs1D, NE);
+   auto Y = Reshape(y_.ReadWrite(), 2 * c_dofs1D * o_dofs1D, NE);
+
+   mfem::forall(NE, [=] MFEM_HOST_DEVICE (int e)
+   {
+      for (int dy = 0; dy < c_dofs1D; ++dy)
+      {
+         for (int dx = 0; dx < o_dofs1D; ++dx)
+         {
+            real_t sum = 0.0;
+            for (int oy = 0; oy < o_dofs1D; ++oy)
+            {
+               const real_t gy = Gc(oy, dy);
+               for (int ox = 0; ox < o_dofs1D; ++ox)
+               {
+                  sum -= Bo(ox, dx) * gy * X(ox, oy, e);
+               }
+            }
+            Y(dx + dy * o_dofs1D, e) += sum;
+         }
+      }
+
+      const int y_nd = c_dofs1D * o_dofs1D;
+      for (int dy = 0; dy < o_dofs1D; ++dy)
+      {
+         for (int dx = 0; dx < c_dofs1D; ++dx)
+         {
+            real_t sum = 0.0;
+            for (int oy = 0; oy < o_dofs1D; ++oy)
+            {
+               const real_t by = Bo(oy, dy);
+               for (int ox = 0; ox < o_dofs1D; ++ox)
+               {
+                  sum += Gc(ox, dx) * by * X(ox, oy, e);
+               }
+            }
+            Y(y_nd + dx + dy * c_dofs1D, e) += sum;
+         }
+      }
+   });
+}
+
+void PAHdivApplyCurl2D(const int c_dofs1D,
+                       const int o_dofs1D,
+                       const int NE,
+                       const Array<real_t> &Bc_,
+                       const Array<real_t> &Gc_,
+                       const Vector &x_,
+                       Vector &y_)
+{
+   auto Bc = Reshape(Bc_.Read(), c_dofs1D, c_dofs1D);
+   auto Gc = Reshape(Gc_.Read(), o_dofs1D, c_dofs1D);
+   auto X = Reshape(x_.Read(), c_dofs1D, c_dofs1D, NE);
+   auto Y = Reshape(y_.ReadWrite(), 2 * c_dofs1D * o_dofs1D, NE);
+
+   mfem::forall(NE, [=] MFEM_HOST_DEVICE (int e)
+   {
+      for (int iy = 0; iy < c_dofs1D; ++iy)
+      {
+         for (int ix = 0; ix < c_dofs1D; ++ix)
+         {
+            const real_t xv = X(ix, iy, e);
+            for (int oy = 0; oy < o_dofs1D; ++oy)
+            {
+               const real_t gy = Gc(oy, iy);
+               for (int ox = 0; ox < c_dofs1D; ++ox)
+               {
+                  Y(ox + oy * c_dofs1D, e) += Bc(ox, ix) * gy * xv;
+               }
+            }
+         }
+      }
+
+      const int y_nd = c_dofs1D * o_dofs1D;
+      for (int iy = 0; iy < c_dofs1D; ++iy)
+      {
+         for (int ix = 0; ix < c_dofs1D; ++ix)
+         {
+            const real_t xv = X(ix, iy, e);
+            for (int oy = 0; oy < c_dofs1D; ++oy)
+            {
+               const real_t by = Bc(oy, iy);
+               for (int ox = 0; ox < o_dofs1D; ++ox)
+               {
+                  Y(y_nd + ox + oy * o_dofs1D, e) -= Gc(ox, ix) * by * xv;
+               }
+            }
+         }
+      }
+   });
+}
+
+void PAHdivApplyCurlTranspose2D(const int c_dofs1D,
+                                const int o_dofs1D,
+                                const int NE,
+                                const Array<real_t> &Bc_,
+                                const Array<real_t> &Gc_,
+                                const Vector &x_,
+                                Vector &y_)
+{
+   auto Bc = Reshape(Bc_.Read(), c_dofs1D, c_dofs1D);
+   auto Gc = Reshape(Gc_.Read(), o_dofs1D, c_dofs1D);
+   auto X = Reshape(x_.Read(), 2 * c_dofs1D * o_dofs1D, NE);
+   auto Y = Reshape(y_.ReadWrite(), c_dofs1D, c_dofs1D, NE);
+
+   mfem::forall(NE, [=] MFEM_HOST_DEVICE (int e)
+   {
+      for (int dy = 0; dy < o_dofs1D; ++dy)
+      {
+         for (int dx = 0; dx < c_dofs1D; ++dx)
+         {
+            const real_t xv = X(dx + dy * c_dofs1D, e);
+            for (int iy = 0; iy < c_dofs1D; ++iy)
+            {
+               const real_t gy = Gc(dy, iy);
+               for (int ix = 0; ix < c_dofs1D; ++ix)
+               {
+                  Y(ix, iy, e) += Bc(dx, ix) * gy * xv;
+               }
+            }
+         }
+      }
+
+      const int y_nd = c_dofs1D * o_dofs1D;
+      for (int dy = 0; dy < c_dofs1D; ++dy)
+      {
+         for (int dx = 0; dx < o_dofs1D; ++dx)
+         {
+            const real_t xv = X(y_nd + dx + dy * o_dofs1D, e);
+            for (int iy = 0; iy < c_dofs1D; ++iy)
+            {
+               const real_t by = Bc(dy, iy);
+               for (int ix = 0; ix < c_dofs1D; ++ix)
+               {
+                  Y(ix, iy, e) -= Gc(dx, ix) * by * xv;
+               }
+            }
+         }
+      }
+   });
+}
+
+}
+
 // Apply to x corresponding to DOFs in H^1 (domain) the (topological) gradient
 // to get a dof in H(curl) (range). You can think of the range as the "test" space
 // and the domain as the "trial" space, but there's no integration.
@@ -1950,4 +2159,266 @@ void IdentityInterpolator::AddMultTransposePA(const Vector &x, Vector &y) const
    }
 }
 
+void CurlInterpolator::AssemblePA(const FiniteElementSpace &dom_fes,
+                                  const FiniteElementSpace &ran_fes)
+{
+   Mesh *mesh = dom_fes.GetMesh();
+   dim = mesh->Dimension();
+   ne = dom_fes.GetNE();
+   pa_mode_2d = 0;
+   MFEM_VERIFY(ne == ran_fes.GetNE(),
+               "Different meshes for domain and range spaces");
+
+   if (dim == 2)
+   {
+      pa_data.SetSize(0);
+      const FiniteElement *dom_fel = dom_fes.GetTypicalFE();
+      const FiniteElement *ran_fel = ran_fes.GetTypicalFE();
+      const bool hcurl_to_scalar =
+         dynamic_cast<const VectorTensorFiniteElement*>(dom_fel) != NULL &&
+         dom_fel->GetDerivType() == FiniteElement::CURL &&
+         dynamic_cast<const TensorBasisElement*>(ran_fel) != NULL &&
+         ran_fel->GetRangeType() == FiniteElement::SCALAR;
+      const bool scalar_to_hdiv =
+         dynamic_cast<const TensorBasisElement*>(dom_fel) != NULL &&
+         dom_fel->GetRangeType() == FiniteElement::SCALAR &&
+         dynamic_cast<const VectorTensorFiniteElement*>(ran_fel) != NULL &&
+         ran_fel->GetDerivType() == FiniteElement::DIV;
+
+      MFEM_VERIFY(hcurl_to_scalar || scalar_to_hdiv,
+                  "2D CurlInterpolator PA supports H(curl)->scalar and scalar->H(div) only.");
+
+      int closed_basis_type = -1;
+      int open_basis_type = -1;
+      if (hcurl_to_scalar)
+      {
+         const auto *trial_fec = dynamic_cast<const ND_FECollection*>(dom_fes.FEColl());
+         const auto *range_fec = dynamic_cast<const L2_FECollection*>(ran_fes.FEColl());
+         MFEM_VERIFY(trial_fec != NULL, "H(curl) domain must use ND_FECollection.");
+         MFEM_VERIFY(range_fec != NULL, "Scalar range must use L2_FECollection.");
+         MFEM_VERIFY(ran_fel->GetMapType() == FiniteElement::INTEGRAL,
+                     "2D H(curl)->scalar CurlInterpolator PA supports integral-map scalar range spaces only.");
+         closed_basis_type = trial_fec->GetClosedBasisType();
+         open_basis_type = trial_fec->GetOpenBasisType();
+         MFEM_VERIFY(range_fec->GetBasisType() == open_basis_type,
+                     "Domain/range open basis types do not match.");
+         pa_mode_2d = 1;
+      }
+      else
+      {
+         const auto *trial_fec = dynamic_cast<const H1_FECollection*>(dom_fes.FEColl());
+         const auto *range_fec = dynamic_cast<const RT_FECollection*>(ran_fes.FEColl());
+         MFEM_VERIFY(trial_fec != NULL, "Scalar domain must use H1_FECollection.");
+         MFEM_VERIFY(range_fec != NULL, "H(div) range must use RT_FECollection.");
+         closed_basis_type = trial_fec->GetBasisType();
+         open_basis_type = range_fec->GetOpenBasisType();
+         MFEM_VERIFY(range_fec->GetClosedBasisType() == closed_basis_type,
+                     "Domain/range closed basis types do not match.");
+         pa_mode_2d = 2;
+      }
+
+      const int order = hcurl_to_scalar
+                        ? dynamic_cast<const VectorTensorFiniteElement*>(dom_fel)->GetOrder()
+                        : dynamic_cast<const NodalTensorFiniteElement*>(dom_fel)->GetOrder();
+      c_dofs1D = order + 1;
+      o_dofs1D = order;
+
+      closed_dofquad_fe.reset(new H1_SegmentElement(order, closed_basis_type));
+      open_dofquad_fe.reset(new L2_SegmentElement(order - 1, open_basis_type));
+
+      mfem::QuadratureFunctions1D qf1d;
+      mfem::IntegrationRule closed_ir;
+      closed_ir.SetSize(c_dofs1D);
+      qf1d.GaussLobatto(c_dofs1D, &closed_ir);
+
+      mfem::IntegrationRule open_ir;
+      open_ir.SetSize(o_dofs1D);
+      qf1d.GaussLegendre(o_dofs1D, &open_ir);
+
+      maps_C_C = &closed_dofquad_fe->GetDofToQuad(closed_ir, DofToQuad::TENSOR);
+      maps_O_C = &closed_dofquad_fe->GetDofToQuad(open_ir, DofToQuad::TENSOR);
+      maps_O_O = &open_dofquad_fe->GetDofToQuad(open_ir, DofToQuad::TENSOR);
+
+      MFEM_VERIFY(maps_C_C->ndof == c_dofs1D && maps_C_C->nqpt == c_dofs1D, "");
+      MFEM_VERIFY(maps_O_C->ndof == c_dofs1D && maps_O_C->nqpt == o_dofs1D, "");
+      MFEM_VERIFY(maps_O_O->ndof == o_dofs1D && maps_O_O->nqpt == o_dofs1D, "");
+      return;
+   }
+
+   closed_dofquad_fe.reset();
+   open_dofquad_fe.reset();
+   maps_C_C = nullptr;
+   maps_O_C = nullptr;
+   maps_O_O = nullptr;
+
+   const VectorTensorFiniteElement *dom_el =
+      dynamic_cast<const VectorTensorFiniteElement *>(dom_fes.GetTypicalFE());
+   const VectorTensorFiniteElement *ran_el =
+      dynamic_cast<const VectorTensorFiniteElement *>(ran_fes.GetTypicalFE());
+   MFEM_VERIFY(dom_el != NULL, "Only VectorTensorFiniteElement is supported!");
+   MFEM_VERIFY(ran_el != NULL, "Only VectorTensorFiniteElement is supported!");
+   MFEM_VERIFY(dom_el->GetDerivType() == FiniteElement::CURL,
+               "Domain space must be H(curl)");
+   MFEM_VERIFY(ran_el->GetDerivType() == FiniteElement::DIV,
+               "Range space must be H(div)");
+
+   const int dims = dom_el->GetDim();
+   MFEM_VERIFY(dims == 3, "");
+
+   ndof_o = dom_el->GetOrder();
+   int ndof_c = ndof_o + 1;
+   nquad_o = ran_el->GetOrder();
+   int nquad_c = nquad_o + 1;
+
+   // extract the tensor product range dof locations
+   std::vector<real_t> qc(nquad_c);
+   std::vector<real_t> qo(nquad_o);
+   {
+      const IntegrationRule &ran_nodes = ran_el->GetNodes();
+      const Array<int> &quad_map = ran_el->GetDofMap();
+      for (int i = 0; i < nquad_c; ++i)
+      {
+         int idx = UnsignIndex(quad_map[i]);
+         qc[i] = ran_nodes.IntPoint(idx).x;
+      }
+      int offset = ndof_c * ndof_o * ndof_o;
+      for (int i = 0; i < nquad_o; ++i)
+      {
+         int idx = UnsignIndex(quad_map[i + offset]);
+         qo[i] = ran_nodes.IntPoint(idx).x;
+      }
+   }
+
+   // evaluate closed/open 1D basis (and their derivatives) at closed and
+   // open quads
+   // storage order: GCO, BCC, BOO
+   pa_data.SetSize(ndof_c * nquad_o + ndof_c * nquad_c + ndof_o * nquad_o);
+   auto ptr = pa_data.HostWrite();
+   auto &cbasis1d = dom_el->GetBasis1D();
+   auto &obasis1d = dom_el->GetOpenBasis1D();
+   Vector b, g;
+   b.SetSize(ndof_c);
+   g.SetSize(ndof_c);
+   for (int j = 0; j < nquad_o; ++j)
+   {
+      cbasis1d.Eval(qo[j], b, g);
+      for (int i = 0; i < ndof_c; ++i)
+      {
+         ptr[j + i * nquad_o] = g[i];
+      }
+   }
+   ptr += nquad_o * ndof_c;
+
+   for (int j = 0; j < nquad_c; ++j)
+   {
+      cbasis1d.Eval(qc[j], b);
+      for (int i = 0; i < ndof_c; ++i)
+      {
+         ptr[j + i * nquad_c] = b[i];
+      }
+   }
+   ptr += ndof_c * nquad_c;
+
+   b.SetSize(ndof_o);
+   for (int j = 0; j < nquad_o; ++j)
+   {
+      obasis1d.Eval(qo[j], b);
+      for (int i = 0; i < ndof_o; ++i)
+      {
+         ptr[j + i * nquad_o] = b[i];
+      }
+   }
+}
+
+CurlInterpolator::Kernels::Kernels()
+{
+   CurlInterpolator::AddSpecialization<3, 1, 1>();
+   CurlInterpolator::AddSpecialization<3, 2, 2>();
+   CurlInterpolator::AddSpecialization<3, 3, 3>();
+   CurlInterpolator::AddSpecialization<3, 4, 4>();
+   CurlInterpolator::AddSpecialization<3, 5, 5>();
+}
+
+CurlInterpolator::CurlInterpolator() { static Kernels kernels{}; }
+
+void CurlInterpolator::AddMultPA(const Vector &x, Vector &y) const
+{
+   if (dim == 2)
+   {
+      MFEM_VERIFY(maps_C_C != nullptr && maps_O_C != nullptr,
+                  "2D CurlInterpolator PA data is not assembled.");
+      if (pa_mode_2d == 1)
+      {
+         MFEM_VERIFY(maps_O_O != nullptr,
+                     "2D CurlInterpolator scalar curl map is not assembled.");
+         PAHcurlApplyCurl2D(c_dofs1D, o_dofs1D, ne, maps_O_O->B, maps_O_C->G,
+                            x, y);
+      }
+      else if (pa_mode_2d == 2)
+      {
+         PAHdivApplyCurl2D(c_dofs1D, o_dofs1D, ne, maps_C_C->B, maps_O_C->G,
+                           x, y);
+      }
+      else
+      {
+         MFEM_ABORT("Unsupported 2D CurlInterpolator mode.");
+      }
+      return;
+   }
+
+   ApplyPAKernels::Run(dim, ndof_o, nquad_o, ne, ndof_o, nquad_o, pa_data, x, y);
+}
+
+void CurlInterpolator::AddMultTransposePA(const Vector &x, Vector &y) const
+{
+   if (dim == 2)
+   {
+      MFEM_VERIFY(maps_C_C != nullptr && maps_O_C != nullptr,
+                  "2D CurlInterpolator PA data is not assembled.");
+      if (pa_mode_2d == 1)
+      {
+         MFEM_VERIFY(maps_O_O != nullptr,
+                     "2D CurlInterpolator scalar curl map is not assembled.");
+         PAHcurlApplyCurlTranspose2D(c_dofs1D, o_dofs1D, ne, maps_O_O->B,
+                                     maps_O_C->G, x, y);
+      }
+      else if (pa_mode_2d == 2)
+      {
+         PAHdivApplyCurlTranspose2D(c_dofs1D, o_dofs1D, ne, maps_C_C->B,
+                                    maps_O_C->G, x, y);
+      }
+      else
+      {
+         MFEM_ABORT("Unsupported 2D CurlInterpolator mode.");
+      }
+      return;
+   }
+
+   ApplyTPAKernels::Run(dim, ndof_o, nquad_o, ne, ndof_o, nquad_o, pa_data, x, y);
+}
+
+/// \cond DO_NOT_DOCUMENT
+
+CurlInterpolator::ApplyKernelType
+CurlInterpolator::ApplyPAKernels::Fallback(int DIM, int, int)
+{
+   if (DIM == 3)
+   {
+      return internal::CurlInterpolatorApply3DSmem<0, 0>;
+   }
+   MFEM_ABORT("Bad dimension!");
+}
+
+CurlInterpolator::ApplyKernelType
+CurlInterpolator::ApplyTPAKernels::Fallback(int DIM, int, int)
+{
+   if (DIM == 3)
+   {
+      return internal::CurlInterpolatorTApply3DSmem<0, 0>;
+   }
+   MFEM_ABORT("Bad dimension!");
+}
+
+/// \endcond DO_NOT_DOCUMENT
+
 } // namespace mfem

fem/integ/bilininteg_vecmass_pa.cpp

@@ -22,6 +22,8 @@ void VectorMassIntegrator::AssemblePA(const FiniteElementSpace &fes)
 {
    Mesh *mesh = fes.GetMesh();
    const FiniteElement &el = *fes.GetTypicalFE();
+   MFEM_VERIFY(el.GetMapType() == FiniteElement::VALUE,
+               "Only value map type supported");
    ElementTransformation &Trans = *mesh->GetTypicalElementTransformation();
    const auto *ir = IntRule ? IntRule : &MassIntegrator::GetRule(el, el, Trans);
 

fem/lor/lor_ams.cpp

@@ -94,10 +94,10 @@ void BatchedLOR_AMS::Form2DEdgeToVertex_RT(Array<int> &edge2vert)
          const int iv0 = ix + iy*op1;
          const int iv1 = ix1 + iy1*op1;
 
-         // Rotated gradient in 2D (-dy, dx), so flip the sign for the first
-         // component (c == 0).
-         e2v(0, iedge) = (c == 1) ? iv0 : iv1;
-         e2v(1, iedge) = (c == 1) ? iv1 : iv0;
+         // 2D curl (dy, -dx), so flip the sign for the second
+         // component (c == 1).
+         e2v(0, iedge) = (c == 0) ? iv0 : iv1;
+         e2v(1, iedge) = (c == 0) ? iv1 : iv0;
       }
    }
 }

fem/lor/lor_batched.cpp

@@ -142,8 +142,6 @@ static MFEM_HOST_DEVICE int GetAndIncrementNnzIndex(const int i_L, int* I)
 
 int BatchedLORAssembly::FillI(SparseMatrix &A) const
 {
-   static constexpr int Max = 16;
-
    const int nvdof = fes_ho.GetVSize();
 
    const int ndof_per_el = fes_ho.GetTypicalFE()->GetDof();
@@ -165,6 +163,8 @@ int BatchedLORAssembly::FillI(SparseMatrix &A) const
    const auto K = dof_glob2loc_offsets_.Read();
    const auto map = Reshape(sparse_mapping.Read(), nnz_per_row, ndof_per_el);
 
+   Array<int> ij_elts(dof_glob2loc_.Size() * 2);
+   auto d_ij_elts = Reshape(ij_elts.Write(), dof_glob2loc_.Size(), 2);
 
    auto I = A.WriteI();
 
@@ -176,10 +176,10 @@ int BatchedLORAssembly::FillI(SparseMatrix &A) const
       const int sii = el_dof_lex(ii_el, iel_ho);
       const int ii = (sii >= 0) ? sii : -1 -sii;
       // Get number and list of elements containing this DOF
-      int i_elts[Max];
       const int i_offset = K[ii];
       const int i_next_offset = K[ii+1];
       const int i_ne = i_next_offset - i_offset;
+      int *i_elts = &d_ij_elts(i_offset, 0);
       for (int e_i = 0; e_i < i_ne; ++e_i)
       {
          const int si_E = dof_glob2loc[i_offset+e_i]; // signed
@@ -202,7 +202,7 @@ int BatchedLORAssembly::FillI(SparseMatrix &A) const
          }
          else // assembly required
          {
-            int j_elts[Max];
+            int *j_elts = &d_ij_elts(j_offset, 1);
             for (int e_j = 0; e_j < j_ne; ++e_j)
             {
                const int sj_E = dof_glob2loc[j_offset+e_j]; // signed
@@ -269,7 +269,8 @@ void BatchedLORAssembly::FillJAndData(SparseMatrix &A) const
       mfem::forall(nvdof + 1, [=] MFEM_HOST_DEVICE (int i) { I[i] = I2[i]; });
    }
 
-   static constexpr int Max = 16;
+   Array<int> ij_B_el(dof_glob2loc_.Size() * 4);
+   auto d_ij_B_el = Reshape(ij_B_el.Write(), dof_glob2loc_.Size(), 4);
 
    mfem::forall(ndof_per_el*nel_ho, [=] MFEM_HOST_DEVICE (int i)
    {
@@ -279,11 +280,13 @@ void BatchedLORAssembly::FillJAndData(SparseMatrix &A) const
       const int sii = el_dof_lex(ii_el, iel_ho); // signed
       const int ii = (sii >= 0) ? sii : -1 - sii;
       // Get number and list of elements containing this DOF
-      int i_elts[Max];
-      int i_B[Max];
       const int i_offset = K[ii];
       const int i_next_offset = K[ii+1];
       const int i_ne = i_next_offset - i_offset;
+
+      int *i_elts = &d_ij_B_el(i_offset, 0);
+      int *i_B = &d_ij_B_el(i_offset, 1);
+
       for (int e_i = 0; e_i < i_ne; ++e_i)
       {
          const int si_E = dof_glob2loc[i_offset+e_i]; // signed
@@ -312,8 +315,8 @@ void BatchedLORAssembly::FillJAndData(SparseMatrix &A) const
          }
          else // assembly required
          {
-            int j_elts[Max];
-            int j_B[Max];
+            int *j_elts = &d_ij_B_el(j_offset, 2);
+            int *j_B = &d_ij_B_el(j_offset, 3);
             for (int e_j = 0; e_j < j_ne; ++e_j)
             {
                const int sj_E = dof_glob2loc[j_offset+e_j]; // signed

fem/pfespace.cpp

@@ -26,6 +26,8 @@
 
 #include <limits>
 #include <list>
+#include <unordered_map>
+#include <unordered_set>
 
 namespace mfem
 {
@@ -1285,6 +1287,342 @@ void ParFiniteElementSpace::GetExteriorVDofs(Array<int> &ext_dofs,
    Synchronize(ext_dofs);
 }
 
+void ParFiniteElementSpace::GetBoundaryLoopEdgeDofs(
+   const Array<int> &boundary_element_indices,
+   Array<int> &ess_tdof_list,
+   Array<int> &boundary_edge_dofs_out,
+   Array<int> *ldof_marker,
+   Array<int> *dof_edges,
+   Array<int> *dof_boundary_elements,
+   Array<int> *ess_edge_list)
+{
+   MFEM_VERIFY(!pmesh->Nonconforming(),
+               "GetBoundaryLoopEdgeDofs does not support nonconforming meshes");
+   MFEM_VERIFY(pmesh->Dimension() >= 2,
+               "GetBoundaryLoopEdgeDofs requires 2D or 3D meshes to find 1D edge objects");
+
+   // Call the serial version, then rebuild scratch maps/set from the returned
+   // arrays for the O(1) lookups the parallel reconciliation below needs.
+   Array<int> loc_dofs, loc_edges, loc_belems;
+   FiniteElementSpace::GetBoundaryLoopEdgeDofs(boundary_element_indices, loc_dofs,
+                                               &loc_edges, &loc_belems);
+
+   std::unordered_set<int> boundary_edge_dofs;
+   std::unordered_map<int, int> dof_to_edge_map;
+   std::unordered_map<int, int> dof_to_boundary_element;
+   boundary_edge_dofs.reserve(loc_dofs.Size());
+   dof_to_edge_map.reserve(loc_dofs.Size());
+   dof_to_boundary_element.reserve(loc_dofs.Size());
+   for (int i = 0; i < loc_dofs.Size(); i++)
+   {
+      const int dof = loc_dofs[i];
+      boundary_edge_dofs.insert(dof);
+      dof_to_edge_map[dof] = loc_edges[i];
+      dof_to_boundary_element[dof] = loc_belems[i];
+   }
+
+   // Parallel processing: Build edge sharing lookup table
+   std::unordered_map<int, int> edge_to_group_size;
+   int num_groups = pmesh->GetNGroups();
+
+   int total_shared_edges = 0;
+   for (int group = 1; group < num_groups; group++)
+   {
+      total_shared_edges += pmesh->GroupNEdges(group);
+   }
+   edge_to_group_size.reserve(total_shared_edges);
+
+   for (int group = 1; group < num_groups; group++)
+   {
+      int group_size = pmesh->gtopo.GetGroupSize(group);
+      int num_edges_in_group = pmesh->GroupNEdges(group);
+
+      for (int i = 0; i < num_edges_in_group; i++)
+      {
+         edge_to_group_size.emplace(pmesh->GroupEdge(group, i), group_size);
+      }
+   }
+
+   // Get global indices
+   Array<HYPRE_BigInt> global_edge_indices;
+   pmesh->GetGlobalEdgeIndices(global_edge_indices);
+
+   // Handle dimension-specific boundary element relationships
+   Array<HYPRE_BigInt> global_face_indices;
+   std::unordered_map<int, int> boundary_element_to_companion;
+   std::unordered_set<int> dofs_to_remove;
+
+   const int dim = pmesh->Dimension();
+   if (dim == 3)
+   {
+      // In 3D: boundary elements are faces, we track which face each boundary element is
+      pmesh->GetGlobalFaceIndices(global_face_indices);
+      for (int boundary_element_idx : boundary_element_indices)
+      {
+         int face_index, face_orientation;
+         pmesh->GetBdrElementFace(boundary_element_idx, &face_index, &face_orientation);
+         boundary_element_to_companion[boundary_element_idx] = face_index;
+      }
+
+      std::vector<HYPRE_BigInt> local_data;
+      local_data.reserve(boundary_edge_dofs.size() * 2);
+
+      std::unordered_set<int> processed_edges;
+      processed_edges.reserve(boundary_edge_dofs.size());
+
+      for (const auto& [dof, local_edge] : dof_to_edge_map)
+      {
+         // Skip if already processed this edge
+         if (!processed_edges.insert(local_edge).second) { continue; }
+
+         // Check if edge is shared (fast lookup)
+         auto it = edge_to_group_size.find(local_edge);
+         if (it != edge_to_group_size.end() && it->second > 1)
+         {
+            // Get boundary element and companion index directly from pre-computed map
+            int boundary_element_idx = dof_to_boundary_element[dof];
+            int companion_index = boundary_element_to_companion[boundary_element_idx];
+
+            // Store edge-face pair for 3D artificial boundary detection
+            local_data.push_back(global_edge_indices[local_edge]);
+            local_data.push_back(global_face_indices[companion_index]);
+         }
+      }
+
+      // MPI communication for 3D artificial boundary detection
+      int num_procs = pmesh->GetNRanks();
+      int local_size = local_data.size();
+
+      std::vector<int> mpi_arrays(num_procs * 4);
+      int* all_sizes = mpi_arrays.data();
+      int* displs = all_sizes + num_procs;
+      int* byte_sizes = displs + num_procs;
+      int* byte_displs = byte_sizes + num_procs;
+
+      MPI_Allgather(&local_size, 1, MPI_INT, all_sizes, 1, MPI_INT, pmesh->GetComm());
+
+      int total_size = 0;
+      constexpr int hypre_size = sizeof(HYPRE_BigInt);
+      for (int i = 0; i < num_procs; i++)
+      {
+         displs[i] = total_size;
+         byte_displs[i] = total_size * hypre_size;
+         total_size += all_sizes[i];
+         byte_sizes[i] = all_sizes[i] * hypre_size;
+      }
+
+      if (total_size > 0)
+      {
+         std::vector<HYPRE_BigInt> all_data(total_size);
+         MPI_Allgatherv(local_data.data(), local_size * hypre_size, MPI_BYTE,
+                        all_data.data(), byte_sizes, byte_displs, MPI_BYTE, pmesh->GetComm());
+
+         // Build global-to-local edge mapping
+         std::unordered_map<HYPRE_BigInt, int> global_to_local_edge;
+         global_to_local_edge.reserve(global_edge_indices.Size());
+         for (int i = 0; i < global_edge_indices.Size(); ++i)
+         {
+            global_to_local_edge[global_edge_indices[i]] = i;
+         }
+
+         // Process collected data to find edges in multiple faces (artificial boundaries)
+         std::unordered_map<HYPRE_BigInt, std::unordered_set<HYPRE_BigInt>>edge_to_faces;
+         edge_to_faces.reserve(total_size / 2);
+
+         for (size_t i = 0; i < all_data.size(); i += 2)
+         {
+            edge_to_faces[all_data[i]].insert(all_data[i + 1]);
+         }
+
+         // Mark DOFs from artificial edges for removal
+         dofs_to_remove.reserve(local_data.size() / 4);
+
+         for (size_t i = 0; i < local_data.size(); i += 2)
+         {
+            HYPRE_BigInt global_edge_id = local_data[i];
+
+            // If this edge appears in 2+ distinct faces, it's artificial
+            if (edge_to_faces[global_edge_id].size() >= 2)
+            {
+               int local_edge = global_to_local_edge[global_edge_id];
+               Array<int> local_edge_dofs;
+               GetEdgeDofs(local_edge, local_edge_dofs);
+
+               // Mark boundary DOFs of this edge for removal
+               for (int k = 0; k < local_edge_dofs.Size(); ++k)
+               {
+                  int dof = local_edge_dofs[k];
+                  if (boundary_edge_dofs.count(dof))
+                  {
+                     dofs_to_remove.insert(dof);
+                  }
+               }
+            }
+         }
+      }
+   }
+   else if (dim == 2)
+   {
+      // In 2D the boundary elements are themselves the edges, so there are no
+      // artificial boundary edges to detect. However, for collections with
+      // vertex DOFs (e.g. ND_R2D), a vertex shared by two boundary segments is
+      // interior to the boundary curve and must be dropped. The serial code
+      // does this by erasing a DOF on its second occurrence, which only sees
+      // the occurrences local to this rank. When the two segments meeting at a
+      // vertex live on different ranks, each rank sees a single occurrence and
+      // wrongly keeps the DOF. Reconcile the occurrence parity across each
+      // sharing group: membership in boundary_edge_dofs is the local parity,
+      // and the parities sum (mod 2) to the global occurrence parity.
+      Array<int> boundary_dof_count(GetVSize());
+      boundary_dof_count = 0;
+      for (const int dof : boundary_edge_dofs)
+      {
+         boundary_dof_count[dof] = 1;
+      }
+
+      // implement allreduce(+) as reduce(+) + broadcast
+      gcomm->Reduce<int>(boundary_dof_count, GroupCommunicator::Sum);
+      gcomm->Bcast(boundary_dof_count);
+
+      for (const int dof : boundary_edge_dofs)
+      {
+         if (boundary_dof_count[dof] % 2 == 0)
+         {
+            dofs_to_remove.insert(dof);
+         }
+      }
+   }
+
+   // Remove artificial DOFs
+   for (int dof : dofs_to_remove)
+   {
+      boundary_edge_dofs.erase(dof);
+      dof_to_edge_map.erase(dof);
+      dof_to_boundary_element.erase(dof);
+   }
+
+   // Convert to true DOFs and output
+   ess_tdof_list.SetSize(0);
+   ess_tdof_list.Reserve(boundary_edge_dofs.size());
+   if (ess_edge_list)
+   {
+      // Reset as well, so that it stays in correspondence with ess_tdof_list
+      // when the same output array is reused across calls.
+      ess_edge_list->SetSize(0);
+      ess_edge_list->Reserve(boundary_edge_dofs.size());
+   }
+   // Marker of the boundary edge DOFs. Always computed locally because the
+   // parallel reconciliation below needs it; only copied to the caller's output
+   // if requested (see the ldof_marker parameter).
+   Array<int> local_ldof_marker(GetVSize());
+   local_ldof_marker = 0;
+
+   for (int dof : boundary_edge_dofs)
+   {
+      local_ldof_marker[dof] = 1; // Mark all boundary edge dofs
+   }
+
+   // Make sure that a selected shared DOF is marked on every rank of its
+   // sharing group, including ranks holding none of the selected boundary
+   // elements. Only the group master owns the corresponding true DOF, so
+   // without this the true DOF would be emitted by no rank at all: the
+   // non-master ranks get -1 from GetLocalTDofNumber(), while the master may
+   // not have selected the DOF locally.
+   Synchronize(local_ldof_marker);
+
+   // A DOF marked only through the synchronization above has no local
+   // dof_to_edge_map entry, but the shared edge carrying it is still present in
+   // the local mesh. Build the missing DOF -> edge entries from the shared
+   // edges of the groups, so that ess_edge_list stays in correspondence with
+   // ess_tdof_list. Note that a vertex DOF is not associated with a unique
+   // edge, so it is only resolved when it is an interior DOF of an edge.
+   std::unordered_map<int, int> shared_dof_to_edge;
+   Array<int> shared_edge_dofs;
+   for (int group = 1; group < num_groups; group++)
+   {
+      const int num_edges_in_group = pmesh->GroupNEdges(group);
+      for (int i = 0; i < num_edges_in_group; i++)
+      {
+         const int edge = pmesh->GroupEdge(group, i);
+         GetEdgeInteriorDofs(edge, shared_edge_dofs);
+         for (int k = 0; k < shared_edge_dofs.Size(); k++)
+         {
+            shared_dof_to_edge.emplace(shared_edge_dofs[k], edge);
+         }
+      }
+   }
+
+   // Build parallel arrays for DOFs and corresponding edges
+   std::vector<std::pair<int, int>> tdof_edge_pairs;
+   tdof_edge_pairs.reserve(boundary_edge_dofs.size());
+
+   for (int dof = 0; dof < local_ldof_marker.Size(); dof++)
+   {
+      if (!local_ldof_marker[dof]) { continue; }
+
+      const int tdof = GetLocalTDofNumber(dof);
+      if (tdof < 0) { continue; } // tdof == -1 means not owned by this rank
+
+      int edge = -1;
+      auto it = dof_to_edge_map.find(dof);
+      if (it != dof_to_edge_map.end())
+      {
+         edge = it->second;
+      }
+      else
+      {
+         auto shared_it = shared_dof_to_edge.find(dof);
+         if (shared_it != shared_dof_to_edge.end())
+         {
+            edge = shared_it->second;
+         }
+      }
+      tdof_edge_pairs.push_back({tdof, edge});
+   }
+
+   // Sort by true DOF index to maintain consistent ordering
+   std::sort(tdof_edge_pairs.begin(), tdof_edge_pairs.end());
+
+   // Extract sorted true DOFs and edges
+   for (const auto& pair : tdof_edge_pairs)
+   {
+      ess_tdof_list.Append(pair.first);
+      if (ess_edge_list)
+      {
+         ess_edge_list->Append(pair.second);
+      }
+   }
+
+   // Emit the local boundary-loop DOFs in a deterministic (increasing DOF
+   // index) order shared by all output arrays.
+   std::vector<int> kept(boundary_edge_dofs.begin(), boundary_edge_dofs.end());
+   std::sort(kept.begin(), kept.end());
+
+   boundary_edge_dofs_out.SetSize(0);
+   boundary_edge_dofs_out.Reserve(static_cast<int>(kept.size()));
+   if (dof_edges)
+   {
+      dof_edges->SetSize(0);
+      dof_edges->Reserve(static_cast<int>(kept.size()));
+   }
+   if (dof_boundary_elements)
+   {
+      dof_boundary_elements->SetSize(0);
+      dof_boundary_elements->Reserve(static_cast<int>(kept.size()));
+   }
+   for (int dof : kept)
+   {
+      boundary_edge_dofs_out.Append(dof);
+      if (dof_edges) { dof_edges->Append(dof_to_edge_map[dof]); }
+      if (dof_boundary_elements)
+      {
+         dof_boundary_elements->Append(dof_to_boundary_element[dof]);
+      }
+   }
+
+   if (ldof_marker) { ldof_marker->Swap(local_ldof_marker); }
+}
+
 void ParFiniteElementSpace::GetExteriorTrueDofs(Array<int> &ext_tdof_list,
                                                 int component) const
 {

fem/pfespace.hpp

@@ -460,6 +460,41 @@ public:
    void GetExteriorTrueDofs(Array<int> &ext_tdof_list,
                             int component = -1) const override;
 
+   /** @brief Extract the edge degrees of freedom of a boundary "loop" on a
+       parallel mesh (see the serial FiniteElementSpace::GetBoundaryLoopEdgeDofs
+       for the definition of a loop). This version removes the artificial
+       boundary edges that appear at processor boundaries, so the selected DOFs
+       are independent of the mesh partitioning.
+
+       As in the serial version, the @a boundary_edge_dofs_out, @a dof_edges and
+       @a dof_boundary_elements outputs share a single indexing describing the
+       same local DOF at each position.
+
+       Requirements:
+       - Mesh must be conforming (no hanging nodes)
+       - Mesh dimension must be >= 2
+       @param[in]  boundary_element_indices Array of boundary element indices.
+       @param[out] ess_tdof_list Essential true DOF indices, sorted ascending.
+       @param[out] boundary_edge_dofs_out Local boundary-loop DOF indices.
+       @param[out] ldof_marker Optional; marker of the boundary edge DOFs,
+                   derivable from @a boundary_edge_dofs_out via ListToMarker().
+       @param[out] dof_edges Optional; local edge index of each DOF.
+       @param[out] dof_boundary_elements Optional; a boundary element containing
+                   each DOF.
+       @param[out] ess_edge_list Optional array of edge indices, in one-to-one
+                                 correspondence with @a ess_tdof_list. An entry
+                                 is -1 when the true DOF is owned by this rank
+                                 but no local edge can be associated with it,
+                                 which can happen for a shared vertex DOF whose
+                                 boundary elements are all on other ranks. */
+   void GetBoundaryLoopEdgeDofs(const Array<int> &boundary_element_indices,
+                                Array<int> &ess_tdof_list,
+                                Array<int> &boundary_edge_dofs_out,
+                                Array<int> *ldof_marker = nullptr,
+                                Array<int> *dof_edges = nullptr,
+                                Array<int> *dof_boundary_elements = nullptr,
+                                Array<int> *ess_edge_list = nullptr);
+
    /** If the given ldof is owned by the current processor, return its local
        tdof number, otherwise return -1 */
    int GetLocalTDofNumber(int ldof) const;

fem/transfer.cpp

@@ -231,9 +231,11 @@ const Operator &InterpolationGridTransfer::BackwardOperator()
 
 L2ProjectionGridTransfer::L2Projection::L2Projection(
    const FiniteElementSpace &fes_ho_, const FiniteElementSpace &fes_lor_,
+   CoefficientWithOrder coeff_ho_, CoefficientWithOrder coeff_lor_,
    MemoryType d_mt_)
    : Operator(fes_lor_.GetVSize(), fes_ho_.GetVSize()),
-     fes_ho(fes_ho_), fes_lor(fes_lor_), d_mt(d_mt_)
+     fes_ho(fes_ho_), fes_lor(fes_lor_), coeff_ho(coeff_ho_),
+     coeff_lor(coeff_lor_), d_mt(d_mt_)
 { }
 
 void L2ProjectionGridTransfer::L2Projection::BuildHo2Lor(
@@ -263,12 +265,13 @@ void L2ProjectionGridTransfer::L2Projection::ElemMixedMass(
    IntegrationPointTransformation& ip_tr,
    DenseMatrix& M_mixed_el) const
 {
-   int order = fe_lor.GetOrder() + fe_ho.GetOrder() + tr_lor->OrderW();
-   const IntegrationRule* ir = &IntRules.Get(geom, order);
+   int order = fe_lor.GetOrder() + fe_ho.GetOrder() + tr_lor->OrderW() +
+               coeff_ho.order;
+   const IntegrationRule &ir = IntRules.Get(geom, order);
    M_mixed_el = 0.0;
-   for (int i = 0; i < ir->GetNPoints(); i++)
+   for (int i = 0; i < ir.GetNPoints(); i++)
    {
-      const IntegrationPoint& ip_lor = ir->IntPoint(i);
+      const IntegrationPoint& ip_lor = ir.IntPoint(i);
       IntegrationPoint ip_ho;
       ip_tr.Transform(ip_lor, ip_ho);
       Vector shape_lor(fe_lor.GetDof());
@@ -284,23 +287,23 @@ void L2ProjectionGridTransfer::L2Projection::ElemMixedMass(
       {
          w *= tr_lor->Weight();
       }
+      if (coeff_ho)
+      {
+         w *= coeff_ho.coeff->Eval(*tr_ho, ip_ho);
+      }
       shape_lor *= w;
       AddMultVWt(shape_lor, shape_ho, M_mixed_el);
    }
 }
 
-void L2ProjectionGridTransfer::L2Projection::ElemMixedMass(
-   Geometry::Type geom, const FiniteElement& fe_ho,
-   const FiniteElement& fe_lor, ElementTransformation* el_tr,
-   IntegrationPointTransformation& ip_tr,
+void L2ProjectionGridTransfer::L2Projection::ElemMixedEvaluation(
+   Geometry::Type geom, const FiniteElement& fe_ho, const FiniteElement& fe_lor,
+   IntegrationPointTransformation& ip_tr, const IntegrationRule& ir,
    DenseMatrix& B_L, DenseMatrix& B_H) const
 {
-   int order = fe_lor.GetOrder() + fe_ho.GetOrder() + el_tr->OrderW();
-   const IntegrationRule* ir = &IntRules.Get(geom, order);
-
-   for (int i = 0; i < ir->GetNPoints(); i++)
+   for (int i = 0; i < ir.GetNPoints(); i++)
    {
-      const IntegrationPoint& ip_lor = ir->IntPoint(i);
+      const IntegrationPoint& ip_lor = ir.IntPoint(i);
       IntegrationPoint ip_ho;
 
       // maps integration point ip_lor -> ip_ho
@@ -320,7 +323,6 @@ void L2ProjectionGridTransfer::L2Projection::ElemMixedMass(
          B_H(i, j) = shape_ho(j);
       }
    }
-
 }
 
 void L2ProjectionGridTransfer::L2Projection::MixedMassEA(
@@ -328,10 +330,11 @@ void L2ProjectionGridTransfer::L2Projection::MixedMassEA(
    const FiniteElementSpace& fes_lor_ea,
    Vector &M_LH, MemoryType d_mt_)
 {
-   Mesh* mesh_ho = fes_ho_ea.GetMesh();
-   Mesh* mesh_lor = fes_lor_ea.GetMesh();
-   int nel_ho = mesh_ho->GetNE();
-   int nel_lor = mesh_lor->GetNE();
+   Mesh &mesh_ho = *fes_ho_ea.GetMesh();
+   Mesh &mesh_lor = *fes_lor_ea.GetMesh();
+
+   const int nel_ho = mesh_ho.GetNE();
+   const int nel_lor = mesh_lor.GetNE();
 
    if (nel_ho == 0)
    {
@@ -339,11 +342,11 @@ void L2ProjectionGridTransfer::L2Projection::MixedMassEA(
       return;
    }
 
-   const CoarseFineTransformations& cf_tr = mesh_lor->GetRefinementTransforms();
+   const CoarseFineTransformations& cf_tr = mesh_lor.GetRefinementTransforms();
 
    int nref_max = 0;
    Array<Geometry::Type> geoms;
-   mesh_ho->GetGeometries(mesh_ho->Dimension(), geoms);
+   mesh_ho.GetGeometries(mesh_ho.Dimension(), geoms);
    for (int ig = 0; ig < geoms.Size(); ++ig)
    {
       Geometry::Type geom = geoms[ig];
@@ -360,130 +363,226 @@ void L2ProjectionGridTransfer::L2Projection::MixedMassEA(
    {
       // Assume all HO elements are LOR in the same way
       const int iho = 0;
-      {
-         Array<int> lor_els;
-         ho2lor.GetRow(iho, lor_els);
-         int nref = ho2lor.RowSize(iho);
-
-         Geometry::Type geom = mesh_ho->GetElementBaseGeometry(iho);
-         const FiniteElement &fe_ho = *fes_ho_ea.GetFE(iho);
-         const FiniteElement &fe_lor = *fes_lor_ea.GetFE(lor_els[0]);
-
-         // Allocate space for DenseTensors
-         ElementTransformation *el_tr = fes_lor_ea.GetElementTransformation(0);
-         int order = fe_lor.GetOrder() + fe_ho.GetOrder() + el_tr->OrderW();
-         const IntegrationRule* ir_ea = &IntRules.Get(geom, order);
-         int qPts = ir_ea->GetNPoints();
-
-         // Containers for the basis functions sampled at quadrature points
-         B_L.SetSize(qPts, fe_lor.GetDof(), nref, d_mt);
-         B_H.SetSize(qPts, fe_ho.GetDof(), nref, d_mt);
-         D.SetSize(qPts, nref, nel_ho, d_mt);
-
-         const GeometricFactors *geo_facts =
-            mesh_lor->GetGeometricFactors(*ir_ea, GeometricFactors::DETERMINANTS);
-
-         MFEM_ASSERT(nel_ho*nref == nel_lor, "we expect nel_ho*nref == nel_lor");
-
-         // Setup data at quadrature points
-         // TODO add support for user coefficient
-         const auto W = Reshape(ir_ea->GetWeights().Read(), qPts);
-         const auto J = Reshape(geo_facts->detJ.Read(), qPts, nel_lor);
-         const auto d_D = Reshape(D.Write(), qPts, nref, nel_ho);
-
-         mfem::forall(qPts * nref * nel_ho, [=] MFEM_HOST_DEVICE (int tid)
-         {
-            const int q    = tid % qPts;
-            const int iref = (tid / qPts) % nref;
-            const int iho  = (tid / (qPts * nref)) % nel_ho;
-
-            const int lo_el_id = iref + nref*iho;
-            const real_t detJ = J(q, lo_el_id);
-
-            d_D(q, iref, iho) = W(q) * detJ;
-
-         });
-
-         emb_tr.SetIdentityTransformation(geom);
-         const DenseTensor &pmats = cf_tr.point_matrices[geom];
-
-         // Collect the basis functions
-         for (int iref = 0; iref < nref; ++iref)
-         {
-            int ilor = lor_els[iref];
-            // Now assemble the block-row of the mixed mass matrix associated
-            // with integrating HO functions against LOR functions on the LOR
-            // sub-element.
-
-            // Create the transformation that embeds the fine low-order element
-            // within the coarse high-order element in reference space
-            emb_tr.SetPointMat(pmats(cf_tr.embeddings[ilor].matrix));
-
-            DenseMatrix &b_lo = B_L(ilor);
-            DenseMatrix &b_ho = B_H(ilor);
-
-            ElemMixedMass(geom, fe_ho, fe_lor, el_tr, ip_tr, b_lo, b_ho);
-
-         } // loop over subcells of ho element
-         // end of quadrature point setup
-      }
-
-   } // completed setup of basis function and quadrature point
-
-   // Assemble mixed mass matrix
-   {
-      int iho = 0;
       Array<int> lor_els;
       ho2lor.GetRow(iho, lor_els);
-      int nref = ho2lor.RowSize(iho);
+      const int nref = ho2lor.RowSize(iho);
+      MFEM_VERIFY(nel_ho*nref == nel_lor, "we expect nel_ho*nref == nel_lor");
+
+      Geometry::Type geom = mesh_ho.GetElementBaseGeometry(iho);
+
+      emb_tr.SetIdentityTransformation(geom);
+      const DenseTensor &pmats = cf_tr.point_matrices[geom];
 
       const FiniteElement &fe_ho = *fes_ho_ea.GetFE(iho);
       const FiniteElement &fe_lor = *fes_lor_ea.GetFE(lor_els[0]);
-      const int ndof_ho = fe_ho.GetDof();
-      const int ndof_lor = fe_lor.GetDof();
 
-      const int qPts = D.SizeI();
+      // Allocate space for DenseTensors
+      ElementTransformation &el_tr = *mesh_lor.GetTypicalElementTransformation();
+      const int order = fe_lor.GetOrder() + fe_ho.GetOrder() + el_tr.OrderW()
+                        + coeff_ho.order;
+      const IntegrationRule &ir_ea = IntRules.Get(geom, order);
+      const int qPts = ir_ea.GetNPoints();
 
-      M_LH.SetSize(ndof_lor*ndof_ho*nref*nel_ho, d_mt);
+      // Containers for the basis functions sampled at quadrature points
+      B_L.SetSize(qPts, fe_lor.GetDof(), nref, d_mt);
+      B_H.SetSize(qPts, fe_ho.GetDof(), nref, d_mt);
+      D.SetSize(qPts, nref, nel_ho, d_mt);
 
-      // Rows x columns
-      // Recall MFEM is column major
-      // rows x columns is inverted  - matrix is ndof_lor x ndof_ho
-      auto v_M_LH = Reshape(M_LH.Write(), ndof_lor, ndof_ho, nref,
-                            nel_ho);
+      const GeometricFactors *geo_facts =
+         mesh_lor.GetGeometricFactors(ir_ea, GeometricFactors::DETERMINANTS);
 
-      const int fe_ho_ndof = fe_ho.GetDof();
-      const int fe_lor_ndof = fe_lor.GetDof();
+      Vector coeff_vec(qPts*nel_lor);
+      coeff_vec.UseDevice(true);
 
-      auto d_B_L = Reshape(B_L.Read(), qPts, fe_lor_ndof, nref);
-      auto d_B_H = Reshape(B_H.Read(), qPts, fe_ho_ndof, nref);
-      auto d_D   = Reshape(D.Read(), qPts, nref, nel_ho);
+      const int dim = mesh_ho.Dimension();
+      const int nq1d = (int)floor(pow(ir_ea.Size(), 1.0/dim) + 0.5);
+      const int nref_1d = (int)floor(pow(nref, 1.0/dim) + 0.5);
 
-      mfem::forall(fe_ho_ndof*nref*nel_ho, [=] MFEM_HOST_DEVICE (int idx)
+      if (!coeff_ho)
       {
-         const int bh = idx % fe_ho_ndof;
-         const int iref = (idx / fe_ho_ndof) % nref;
-         const int iho = idx / fe_ho_ndof / nref;
-         // (B_lo_dofs x Q) x (Q x B_ho_dofs)
-         for (int bl = 0; bl < fe_lor_ndof; ++bl)
+         coeff_vec = 1.0;
+      }
+      else if (UsesTensorBasis(fes_ho) &&
+               nq1d*nref_1d <= DeviceDofQuadLimits::Get().MAX_Q1D)
+      {
+         // Fast coefficient evaluation for tensor-product case. We create a
+         // "composite" quadrature rule in the high-order element that is the
+         // union of the quadrature rules within each of the low-order-refined
+         // subelements.
+         //
+         // NOTE: if the integration rule order is high and there are many LOR
+         // subelements, this can create a very big quadrature rule. That is
+         // why we need to check that we do not exceed MAX_Q1D. If we do, then
+         // we fall back on the slower "legacy" evaluation.
+
+         // Construct the composite rule as a tensor-product of the 1D LOR rule.
+         IntegrationRule ir_ho = [&]()
          {
-            real_t dot = 0.0;
-            for (int qi=0; qi<qPts; ++qi)
+            IntegrationRule ir_ho_1d(nq1d * nref_1d);
+            for (int iref = 0; iref < nref_1d; ++iref)
             {
-               dot += d_B_L(qi, bl, iref) *  d_D(qi, iref, iho) * d_B_H(qi, bh, iref);
+               const real_t a = pmats(cf_tr.embeddings[iref].matrix)(0,0);
+               const real_t b = pmats(cf_tr.embeddings[iref].matrix)(0,1);
+               for (int iq = 0; iq < nq1d; ++iq)
+               {
+                  ir_ho_1d[iq + iref*nq1d].x = a + ir_ea[iq].x*(b - a);
+               }
+            }
+            if (dim == 1) { return ir_ho_1d; }
+            else if (dim == 2) { return IntegrationRule(ir_ho_1d, ir_ho_1d); }
+            else { return IntegrationRule(ir_ho_1d, ir_ho_1d, ir_ho_1d); }
+         }();
+
+         // Project the high-order coefficient on the high-order composite rule.
+         QuadratureSpace qs(mesh_ho, ir_ho);
+         CoefficientVector coeff_vec_ho(*coeff_ho.coeff, qs);
+
+         // Permute the coefficient values to the expected LOR ordering.
+         const int nq_ho = ir_ho.Size();
+         const auto d_Q_ho = Reshape(coeff_vec_ho.Read(), nq_ho, nel_ho);
+         const auto d_Q = Reshape(coeff_vec.Write(), qPts, nel_lor);
+
+         mfem::forall(nq_ho * nel_ho, [=] MFEM_HOST_DEVICE (int ii)
+         {
+            const int e_ho = ii / nq_ho;
+            const int iq_ho = ii % nq_ho;
+
+            int iq_tensor = iq_ho;
+            int iq_lor = 0;
+            int iref = 0;
+            int iq_stride = 1;
+            int iref_stride = 1;
+            const int nq_ho_1d = nq1d*nref_1d;
+            for (int d = 0; d < dim; ++d)
+            {
+               const int iq_ho_1d = iq_tensor % nq_ho_1d;
+               iq_tensor /= nq_ho_1d;
+
+               iq_lor += (iq_ho_1d % nq1d)*iq_stride;
+               iref += (iq_ho_1d / nq1d)*iref_stride;
+               iq_stride *= nq1d;
+               iref_stride *= nref_1d;
+            }
+            const int e_lor = iref + e_ho*nref;
+
+            d_Q(iq_lor, e_lor) = d_Q_ho(iq_ho, e_ho);
+         });
+      }
+      else
+      {
+         // Legacy/fallback coefficient evaluation for non-tensor-product cases
+         // or when the number of quadrature points is too large for the device
+         // kernels.
+         IntegrationPoint ip_ho;
+         for (int e_ho = 0; e_ho < nel_ho; ++e_ho)
+         {
+            ElementTransformation &ho_tr = *mesh_ho.GetElementTransformation(e_ho);
+            for (int iref = 0; iref < nref; ++iref)
+            {
+               const int e_lor = iref + e_ho*nref;
+               emb_tr.SetPointMat(pmats(cf_tr.embeddings[e_lor].matrix));
+
+               for (int iq = 0; iq < qPts; ++iq)
+               {
+                  const IntegrationPoint &ip_lor = ir_ea[iq];
+                  ip_tr.Transform(ip_lor, ip_ho);
+                  ho_tr.SetIntPoint(&ip_ho);
+                  coeff_vec[iq + e_lor*qPts] = coeff_ho.coeff->Eval(ho_tr, ip_ho);
+               }
             }
-            // column major storage
-            v_M_LH(bl, bh, iref, iho) = dot;
          }
+      }
+
+      // Setup data at quadrature points
+      const auto W = Reshape(ir_ea.GetWeights().Read(), qPts);
+      const auto J = Reshape(geo_facts->detJ.Read(), qPts, nel_lor);
+      const auto d_D = Reshape(D.Write(), qPts, nref, nel_ho);
+      const auto d_Q = Reshape(coeff_vec.Read(), qPts, nel_lor);
+
+      mfem::forall(qPts * nref * nel_ho, [=] MFEM_HOST_DEVICE (int tid)
+      {
+         const int q    = tid % qPts;
+         const int iref = (tid / qPts) % nref;
+         const int iho  = (tid / (qPts * nref)) % nel_ho;
+
+         const int lo_el_id = iref + nref*iho;
+         const real_t detJ = J(q, lo_el_id);
+
+         d_D(q, iref, iho) = W(q) * d_Q(q, lo_el_id) * detJ;
       });
-   } // end of mixed assembly mass matrix
+
+      // Collect the basis functions
+      for (int iref = 0; iref < nref; ++iref)
+      {
+         int ilor = lor_els[iref];
+         // Now assemble the block-row of the mixed mass matrix associated
+         // with integrating HO functions against LOR functions on the LOR
+         // sub-element.
+
+         // Create the transformation that embeds the fine low-order element
+         // within the coarse high-order element in reference space
+         emb_tr.SetPointMat(pmats(cf_tr.embeddings[ilor].matrix));
+
+         DenseMatrix &b_lo = B_L(ilor);
+         DenseMatrix &b_ho = B_H(ilor);
+
+         ElemMixedEvaluation(geom, fe_ho, fe_lor, ip_tr, ir_ea, b_lo, b_ho);
+      } // loop over subcells of ho element
+      // end of quadrature point setup
+   } // completed setup of basis function and quadrature point
+
+   // Assemble mixed mass matrix
+   int iho = 0;
+   Array<int> lor_els;
+   ho2lor.GetRow(iho, lor_els);
+   int nref = ho2lor.RowSize(iho);
+
+   const FiniteElement &fe_ho = *fes_ho_ea.GetFE(iho);
+   const FiniteElement &fe_lor = *fes_lor_ea.GetFE(lor_els[0]);
+   const int ndof_ho = fe_ho.GetDof();
+   const int ndof_lor = fe_lor.GetDof();
+
+   const int qPts = D.SizeI();
+
+   M_LH.SetSize(ndof_lor*ndof_ho*nref*nel_ho, d_mt);
+
+   // Rows x columns
+   // Recall MFEM is column major
+   // rows x columns is inverted  - matrix is ndof_lor x ndof_ho
+   auto v_M_LH = Reshape(M_LH.Write(), ndof_lor, ndof_ho, nref,
+                         nel_ho);
+
+   const int fe_ho_ndof = fe_ho.GetDof();
+   const int fe_lor_ndof = fe_lor.GetDof();
+
+   auto d_B_L = Reshape(B_L.Read(), qPts, fe_lor_ndof, nref);
+   auto d_B_H = Reshape(B_H.Read(), qPts, fe_ho_ndof, nref);
+   auto d_D   = Reshape(D.Read(), qPts, nref, nel_ho);
+
+   mfem::forall(fe_ho_ndof*nref*nel_ho, [=] MFEM_HOST_DEVICE (int idx)
+   {
+      const int bh = idx % fe_ho_ndof;
+      const int iref = (idx / fe_ho_ndof) % nref;
+      const int iho = idx / fe_ho_ndof / nref;
+      // (B_lo_dofs x Q) x (Q x B_ho_dofs)
+      for (int bl = 0; bl < fe_lor_ndof; ++bl)
+      {
+         real_t dot = 0.0;
+         for (int qi=0; qi<qPts; ++qi)
+         {
+            dot += d_B_L(qi, bl, iref) *  d_D(qi, iref, iho) * d_B_H(qi, bh, iref);
+         }
+         // column major storage
+         v_M_LH(bl, bh, iref, iho) = dot;
+      }
+   });
 }
 
 L2ProjectionGridTransfer::L2ProjectionL2Space::L2ProjectionL2Space
 (const FiniteElementSpace &fes_ho_, const FiniteElementSpace &fes_lor_,
+ CoefficientWithOrder coeff_ho_, CoefficientWithOrder coeff_lor_,
  const bool use_ea_, MemoryType d_mt_)
-   : L2Projection(fes_ho_, fes_lor_, d_mt_),
-     use_ea(use_ea_)
+   : L2Projection(fes_ho_, fes_lor_, coeff_ho_, coeff_lor_, d_mt_), use_ea(use_ea_)
 {
    if (use_ea)
    {
@@ -559,7 +658,11 @@ L2ProjectionGridTransfer::L2ProjectionL2Space::L2ProjectionL2Space
       DenseMatrix Minv_lor(ndof_lor*nref, ndof_lor*nref);
       DenseMatrix M_mixed(ndof_lor*nref, ndof_ho);
 
-      MassIntegrator mi;
+      MassIntegrator mi = [&]()
+      {
+         return coeff_lor ? MassIntegrator(*coeff_lor.coeff) : MassIntegrator();
+      }();
+
       DenseMatrix M_lor_el(ndof_lor, ndof_lor);
       DenseMatrixInverse Minv_lor_el(&M_lor_el);
       DenseMatrix M_lor(ndof_lor*nref, ndof_lor*nref);
@@ -577,6 +680,10 @@ L2ProjectionGridTransfer::L2ProjectionL2Space::L2ProjectionL2Space
          // Assemble the low-order refined mass matrix and invert locally
          int ilor = lor_els[iref];
          ElementTransformation *tr_lor = fes_lor.GetElementTransformation(ilor);
+
+         const int order = 2*fe_lor.GetOrder() + tr_lor->OrderW() + coeff_lor.order;
+         mi.SetIntegrationRule(IntRules.Get(geom, order));
+
          mi.AssembleElementMatrix(fe_lor, *tr_lor, M_lor_el);
          M_lor.CopyMN(M_lor_el, iref*ndof_lor, iref*ndof_lor);
          Minv_lor_el.Factor();
@@ -668,25 +775,22 @@ void L2ProjectionGridTransfer::L2ProjectionL2Space::EAL2ProjectionL2Space()
    // Need to compute M_L
    // Note: Using user-inputted M_LH IntegrationRule ir
    // (higher order than needed) in order to re-use coeff
-   MassIntegrator mi;
+   MassIntegrator mi = [&]()
+   {
+      return coeff_lor ? MassIntegrator(*coeff_lor.coeff) : MassIntegrator();
+   }();
+
+   const int order = 2*fes_lor.GetMaxElementOrder()
+                     + mesh_lor->GetTypicalElementTransformation()->OrderW()
+                     + coeff_lor.order;
+   mi.SetIntegrationRule(
+      IntRules.Get(mesh_lor->GetTypicalElementGeometry(), order));
 
    Vector M_ea_lor;
-   int ndof_lor;
-   int ndof_ho;
-   int nref;
-   {
-      int iho = 0;
-      Array<int> lor_els;
-      ho2lor.GetRow(iho, lor_els);
-      nref = ho2lor.RowSize(iho);
-
-      const FiniteElement &fe_ho = *fes_ho.GetFE(iho);
-      const FiniteElement &fe_lor = *fes_lor.GetFE(lor_els[0]);
-      ndof_ho = fe_ho.GetDof();
-      ndof_lor = fe_lor.GetDof();
-
-      M_ea_lor.SetSize(ndof_lor*ndof_lor*nel_lor, d_mt);
-   }
+   const int ndof_lor = fes_lor.GetTypicalFE()->GetDof();
+   const int ndof_ho = fes_ho.GetTypicalFE()->GetDof();
+   const int nref = ho2lor.RowSize(0);
+   M_ea_lor.SetSize(ndof_lor*ndof_lor*nel_lor, d_mt);
 
    const bool add = false;
    mi.AssembleEA(fes_lor, M_ea_lor, add);
@@ -1032,8 +1136,9 @@ void L2ProjectionGridTransfer::L2ProjectionL2Space::EAProlongateTranspose(
 
 L2ProjectionGridTransfer::L2ProjectionH1Space::L2ProjectionH1Space(
    const FiniteElementSpace& fes_ho_, const FiniteElementSpace& fes_lor_,
+   CoefficientWithOrder coeff_ho_, CoefficientWithOrder coeff_lor_,
    const bool use_ea_, MemoryType d_mt_)
-   : L2Projection(fes_ho_, fes_lor_, d_mt_),
+   : L2Projection(fes_ho_, fes_lor_, coeff_ho_, coeff_lor_, d_mt_),
      use_ea(use_ea_)
 {
 
@@ -1092,8 +1197,9 @@ L2ProjectionGridTransfer::L2ProjectionH1Space::L2ProjectionH1Space(
 
 L2ProjectionGridTransfer::L2ProjectionH1Space::L2ProjectionH1Space(
    const ParFiniteElementSpace& pfes_ho, const ParFiniteElementSpace& pfes_lor,
+   CoefficientWithOrder coeff_ho_, CoefficientWithOrder coeff_lor_,
    const bool use_ea_, MemoryType d_mt_)
-   : L2Projection(pfes_ho, pfes_lor, d_mt_),
+   : L2Projection(pfes_ho, pfes_lor, coeff_ho_, coeff_lor_, d_mt_),
      use_ea(use_ea_), pcg(pfes_ho.GetComm())
 {
 
@@ -1165,12 +1271,12 @@ void L2ProjectionGridTransfer::L2ProjectionH1Space::SetupPCG()
 
 void L2ProjectionGridTransfer::L2ProjectionH1Space::EAL2ProjectionH1Space()
 {
-   Mesh* mesh_ho = fes_ho.GetMesh();
-   Mesh* mesh_lor = fes_lor.GetMesh();
-   int nel_ho = mesh_ho->GetNE();
-   int nel_lor = mesh_lor->GetNE();
-   int ndof_ho = fes_ho.GetNDofs();
-   int ndof_lor = fes_lor.GetNDofs();
+   Mesh &mesh_ho = *fes_ho.GetMesh();
+   Mesh &mesh_lor = *fes_lor.GetMesh();
+   const int nel_ho = mesh_ho.GetNE();
+   const int nel_lor = mesh_lor.GetNE();
+   const int ndof_ho = fes_ho.GetNDofs();
+   const int ndof_lor = fes_lor.GetNDofs();
 
    // If the local mesh is empty, skip all computations
    if (nel_ho == 0)
@@ -1178,11 +1284,11 @@ void L2ProjectionGridTransfer::L2ProjectionH1Space::EAL2ProjectionH1Space()
       return;
    }
 
-   const CoarseFineTransformations& cf_tr = mesh_lor->GetRefinementTransforms();
+   const CoarseFineTransformations& cf_tr = mesh_lor.GetRefinementTransforms();
 
    int nref_max = 0;
    Array<Geometry::Type> geoms;
-   mesh_ho->GetGeometries(mesh_ho->Dimension(), geoms);
+   mesh_ho.GetGeometries(mesh_ho.Dimension(), geoms);
    for (int ig = 0; ig < geoms.Size(); ++ig)
    {
       Geometry::Type geom = geoms[ig];
@@ -1205,7 +1311,8 @@ void L2ProjectionGridTransfer::L2ProjectionH1Space::EAL2ProjectionH1Space()
 
    BilinearForm Mho(fes_ho_scalar.get());
    Mho.SetAssemblyLevel(AssemblyLevel::PARTIAL);
-   Mho.AddDomainIntegrator(new MassIntegrator);
+   Mho.AddDomainIntegrator(coeff_ho ? new MassIntegrator(*coeff_ho.coeff)
+                           : new MassIntegrator);
    Mho.Assemble();
 
    // Processor local lumped Mass
@@ -1215,7 +1322,16 @@ void L2ProjectionGridTransfer::L2ProjectionH1Space::EAL2ProjectionH1Space()
 
    BilinearForm Mlor(fes_lor_scalar.get());
    Mlor.SetAssemblyLevel(AssemblyLevel::PARTIAL);
-   Mlor.AddDomainIntegrator(new MassIntegrator);
+   {
+      MassIntegrator *mi = coeff_lor ? new MassIntegrator(*coeff_lor.coeff)
+                           : new MassIntegrator;
+      const int order = 2*fes_lor.GetMaxElementOrder()
+                        + mesh_lor.GetTypicalElementTransformation()->OrderW()
+                        + coeff_lor.order;
+      mi->SetIntegrationRule(
+         IntRules.Get(mesh_lor.GetTypicalElementGeometry(), order));
+      Mlor.AddDomainIntegrator(mi);
+   }
    Mlor.Assemble();
 
    Vector ones_lor(Mlor.Width()); ones_lor = 1.0;
@@ -1228,15 +1344,14 @@ void L2ProjectionGridTransfer::L2ProjectionH1Space::EAL2ProjectionH1Space()
    MixedMassEA(fes_ho, fes_lor, M_LH_ea, d_mt);
 
    // Set ownership
-   M_LH_local_op = new H1SpaceMixedMassOperator(fes_ho_scalar.get(),
-                                                fes_lor_scalar.get(),
-                                                &ho2lor,
-                                                &M_LH_ea);
+   M_LH.reset(new H1SpaceMixedMassOperator(fes_ho_scalar.get(),
+                                           fes_lor_scalar.get(),
+                                           &ho2lor,
+                                           &M_LH_ea));
 
    ML_inv_vea.reset(new H1SpaceLumpedMassOperator(fes_ho_scalar.get(),
                                                   fes_lor_scalar.get(),
                                                   ML_inv_ea));
-   M_LH.reset(M_LH_local_op);
    R.reset(new ProductOperator(ML_inv_vea.get(), M_LH.get(), false,
                                false));
 
@@ -1253,18 +1368,18 @@ void L2ProjectionGridTransfer::L2ProjectionH1Space::EAL2ProjectionH1Space()
 void L2ProjectionGridTransfer::L2ProjectionH1Space::EAL2ProjectionH1Space
 (const ParFiniteElementSpace& pfes_ho, const ParFiniteElementSpace& pfes_lor)
 {
-   Mesh* mesh_ho = pfes_ho.GetParMesh();
-   Mesh* mesh_lor = pfes_lor.GetParMesh();
-   int nel_ho = mesh_ho->GetNE();
-   int nel_lor = mesh_lor->GetNE();
+   Mesh &mesh_ho = *pfes_ho.GetParMesh();
+   Mesh &mesh_lor = *pfes_lor.GetParMesh();
+   int nel_ho = mesh_ho.GetNE();
+   int nel_lor = mesh_lor.GetNE();
    int ndof_ho = pfes_ho.GetNDofs();
    int ndof_lor = pfes_lor.GetNDofs();
 
-   const CoarseFineTransformations& cf_tr = mesh_lor->GetRefinementTransforms();
+   const CoarseFineTransformations& cf_tr = mesh_lor.GetRefinementTransforms();
 
    int nref_max = 0;
    Array<Geometry::Type> geoms;
-   mesh_ho->GetGeometries(mesh_ho->Dimension(), geoms);
+   mesh_ho.GetGeometries(mesh_ho.Dimension(), geoms);
    for (int ig = 0; ig < geoms.Size(); ++ig)
    {
       Geometry::Type geom = geoms[ig];
@@ -1287,7 +1402,8 @@ void L2ProjectionGridTransfer::L2ProjectionH1Space::EAL2ProjectionH1Space
 
    ParBilinearForm pMho(pfes_ho_scalar.get());
    pMho.SetAssemblyLevel(AssemblyLevel::PARTIAL);
-   pMho.AddDomainIntegrator(new MassIntegrator);
+   pMho.AddDomainIntegrator(coeff_ho ? new MassIntegrator(*coeff_ho.coeff)
+                            : new MassIntegrator);
    pMho.Assemble();
 
    // Processor local lumped Mass
@@ -1297,7 +1413,16 @@ void L2ProjectionGridTransfer::L2ProjectionH1Space::EAL2ProjectionH1Space
 
    ParBilinearForm pMlor(pfes_lor_scalar.get());
    pMlor.SetAssemblyLevel(AssemblyLevel::PARTIAL);
-   pMlor.AddDomainIntegrator(new MassIntegrator);
+   {
+      MassIntegrator *mi = coeff_lor ? new MassIntegrator(*coeff_lor.coeff)
+                           : new MassIntegrator;
+      const int order = 2*fes_lor.GetMaxElementOrder()
+                        + mesh_lor.GetTypicalElementTransformation()->OrderW()
+                        + coeff_lor.order;
+      mi->SetIntegrationRule(
+         IntRules.Get(mesh_lor.GetTypicalElementGeometry(), order));
+      pMlor.AddDomainIntegrator(mi);
+   }
    pMlor.Assemble();
 
    Vector ones_lor(pMlor.Width()); ones_lor = 1.0;
@@ -1570,7 +1695,7 @@ std::unique_ptr<SparseMatrix>>
          int ilor = lor_els[iref];
          ElementTransformation* el_tr = fes_lor.GetElementTransformation(ilor);
 
-         int order = 2 * fe_lor.GetOrder() + el_tr->OrderW();
+         int order = 2 * fe_lor.GetOrder() + el_tr->OrderW() + coeff_lor.order;
          const IntegrationRule* ir = &IntRules.Get(geom, order);
          ML_el = 0.0;
          for (int i = 0; i < ir->GetNPoints(); ++i)
@@ -1578,7 +1703,13 @@ std::unique_ptr<SparseMatrix>>
             const IntegrationPoint& ip_lor = ir->IntPoint(i);
             fe_lor.CalcShape(ip_lor, shape_lor);
             el_tr->SetIntPoint(&ip_lor);
-            ML_el += (shape_lor *= (el_tr->Weight() * ip_lor.weight));
+            real_t w = ip_lor.weight;
+            if (coeff_lor)
+            {
+               w *= coeff_lor.coeff->Eval(*el_tr, ip_lor);
+            }
+            shape_lor *= el_tr->Weight() * w;
+            ML_el += shape_lor;
          }
          fes_lor.GetElementDofs(ilor, dofs_lor);
          ML_inv.AddElementVector(dofs_lor, ML_el);
@@ -2024,8 +2155,8 @@ void L2ProjectionGridTransfer::BuildF()
    {
       if (!Parallel())
       {
-         F = new L2ProjectionH1Space(dom_fes, ran_fes,
-                                     use_ea, d_mt);
+         F = new L2ProjectionH1Space(
+            dom_fes, ran_fes, coeff_ho, coeff_lor, use_ea, d_mt);
       }
       else
       {
@@ -2034,15 +2165,15 @@ void L2ProjectionGridTransfer::BuildF()
             static_cast<mfem::ParFiniteElementSpace&>(dom_fes);
          const mfem::ParFiniteElementSpace& ran_pfes =
             static_cast<mfem::ParFiniteElementSpace&>(ran_fes);
-         F = new L2ProjectionH1Space(dom_pfes, ran_pfes,
-                                     use_ea, d_mt);
+         F = new L2ProjectionH1Space(
+            dom_pfes, ran_pfes, coeff_ho, coeff_lor, use_ea, d_mt);
 #endif
       }
    }
    else
    {
-      F = new L2ProjectionL2Space(dom_fes, ran_fes,
-                                  use_ea, d_mt);
+      F = new L2ProjectionL2Space(
+         dom_fes, ran_fes, coeff_ho, coeff_lor, use_ea, d_mt);
    }
 }
 

fem/transfer.hpp

@@ -19,6 +19,8 @@
 #include "pfespace.hpp"
 #endif
 
+#include <cstddef>
+
 namespace mfem
 {
 
@@ -162,6 +164,18 @@ public:
 };
 
 
+struct CoefficientWithOrder
+{
+   Coefficient *coeff;
+   int order;
+   CoefficientWithOrder() : coeff(nullptr), order(0) { }
+   CoefficientWithOrder(std::nullptr_t) : coeff(nullptr), order(0) { }
+   CoefficientWithOrder(Coefficient &coeff_) : coeff(&coeff_), order(1) { }
+   CoefficientWithOrder(Coefficient &coeff_, int order_)
+      : coeff(&coeff_), order(order_) { }
+   operator bool() const { return coeff != nullptr; }
+};
+
 /** @brief Transfer data in L2 and H1 finite element spaces between a coarse
     mesh and an embedded refined mesh using L2 projection. */
 /** The forward, coarse-to-fine, transfer uses L2 projection. The backward,
@@ -207,6 +221,8 @@ public:
    protected:
       const FiniteElementSpace& fes_ho;
       const FiniteElementSpace& fes_lor;
+      CoefficientWithOrder coeff_ho;
+      CoefficientWithOrder coeff_lor;
 
       MemoryType d_mt;
       Array<int> offsets;
@@ -214,8 +230,15 @@ public:
 
       L2Projection(const FiniteElementSpace& fes_ho_,
                    const FiniteElementSpace& fes_lor_,
+                   CoefficientWithOrder coeff_ho_,
+                   CoefficientWithOrder coeff_lor_,
                    MemoryType d_mt_ = Device::GetHostMemoryType());
 
+      L2Projection(const FiniteElementSpace& fes_ho_,
+                   const FiniteElementSpace& fes_lor_,
+                   MemoryType d_mt_ = Device::GetHostMemoryType())
+         : L2Projection(fes_ho_, fes_lor_, nullptr, nullptr, d_mt_) { }
+
       void BuildHo2Lor(int nel_ho, int nel_lor,
                        const CoarseFineTransformations& cf_tr);
 
@@ -225,11 +248,11 @@ public:
                          IntegrationPointTransformation& ip_tr,
                          DenseMatrix& M_mixed_el) const;
 
-      void ElemMixedMass(Geometry::Type geom, const FiniteElement& fe_ho,
-                         const FiniteElement& fe_lor,
-                         ElementTransformation* el_tr,
-                         IntegrationPointTransformation& ip_tr,
-                         DenseMatrix& B_L, DenseMatrix& B_H) const;
+      void ElemMixedEvaluation(Geometry::Type geom, const FiniteElement& fe_ho,
+                               const FiniteElement& fe_lor,
+                               IntegrationPointTransformation& ip_tr,
+                               const IntegrationRule& ir,
+                               DenseMatrix& B_L, DenseMatrix& B_H) const;
    public:
       /* Returns the Mixed Mass M_LH via device element assembly by building the
       basis functions and data at the quadrature points. */
@@ -287,9 +310,17 @@ public:
    public:
       L2ProjectionL2Space(const FiniteElementSpace& fes_ho_,
                           const FiniteElementSpace& fes_lor_,
+                          CoefficientWithOrder coeff_ho_,
+                          CoefficientWithOrder coeff_lor_,
                           const bool use_ea_,
                           MemoryType d_mt_ = Device::GetHostMemoryType());
 
+      L2ProjectionL2Space(const FiniteElementSpace& fes_ho_,
+                          const FiniteElementSpace& fes_lor_,
+                          const bool use_ea_,
+                          MemoryType d_mt_ = Device::GetHostMemoryType())
+         : L2ProjectionL2Space(fes_ho_, fes_lor_, nullptr, nullptr, use_ea_, d_mt_) { }
+
       /*Same as above but assembles and stores R_ea, P_ea */
       void EAL2ProjectionL2Space();
 
@@ -356,13 +387,30 @@ public:
    public:
       L2ProjectionH1Space(const FiniteElementSpace &fes_ho_,
                           const FiniteElementSpace &fes_lor_,
+                          CoefficientWithOrder coeff_ho_,
+                          CoefficientWithOrder coeff_lor_,
                           const bool use_ea_,
                           MemoryType d_mt_ = Device::GetHostMemoryType());
+
+      L2ProjectionH1Space(const FiniteElementSpace& fes_ho_,
+                          const FiniteElementSpace& fes_lor_,
+                          const bool use_ea_,
+                          MemoryType d_mt_ = Device::GetHostMemoryType())
+         : L2ProjectionH1Space(fes_ho_, fes_lor_, nullptr, nullptr, use_ea_, d_mt_) { }
+
 #ifdef MFEM_USE_MPI
       L2ProjectionH1Space(const ParFiniteElementSpace &pfes_ho_,
                           const ParFiniteElementSpace &pfes_lor_,
+                          CoefficientWithOrder coeff_ho_,
+                          CoefficientWithOrder coeff_lor_,
                           const bool use_ea_,
                           MemoryType d_mt_ = Device::GetHostMemoryType());
+
+      L2ProjectionH1Space(const ParFiniteElementSpace& fes_ho_,
+                          const ParFiniteElementSpace& fes_lor_,
+                          const bool use_ea_,
+                          MemoryType d_mt_ = Device::GetHostMemoryType())
+         : L2ProjectionH1Space(fes_ho_, fes_lor_, nullptr, nullptr, use_ea_, d_mt_) { }
 #endif
       /// Same as above but assembles action of R through 4 parts:
       ///   ( )  inv( lumped(M_L) ), which is a diagonal matrix (essentially a vector)
@@ -508,18 +556,38 @@ public:
       virtual ~L2Prolongation() { }
    };
 
+   /// Coefficient for the mixed L2 inner product.
+   CoefficientWithOrder coeff_ho;
+   /// Coefficient for the low-order L2 inner product.
+   CoefficientWithOrder coeff_lor;
    L2Projection   *F; ///< Forward, coarse-to-fine, operator
    L2Prolongation *B; ///< Backward, fine-to-coarse, operator
    bool force_l2_space;
 
 public:
+   /// Construct the unweighted L2 projection grid transfer.
    L2ProjectionGridTransfer(FiniteElementSpace &coarse_fes_,
                             FiniteElementSpace &fine_fes_,
                             bool force_l2_space_ = false,
                             MemoryType d_mt_ = Device::GetHostMemoryType()) // move to method
       : GridTransfer(coarse_fes_, fine_fes_),
-        F(NULL), B(NULL), force_l2_space(force_l2_space_)
-   { }
+        coeff_ho(nullptr), coeff_lor(nullptr), F(nullptr), B(nullptr),
+        force_l2_space(force_l2_space_) { }
+
+   /// @brief Construct the weighted L2 projection grid transfer.
+   ///
+   /// The low-order inner product is weighted by @a coeff_lor, and the mixed
+   /// inner product is weighted by @a coeff_ho.
+   L2ProjectionGridTransfer(FiniteElementSpace &coarse_fes_,
+                            FiniteElementSpace &fine_fes_,
+                            CoefficientWithOrder coeff_ho_,
+                            CoefficientWithOrder coeff_lor_,
+                            bool force_l2_space_ = false,
+                            MemoryType d_mt_ = Device::GetHostMemoryType()) // move to method
+      : GridTransfer(coarse_fes_, fine_fes_),
+        coeff_ho(coeff_ho_), coeff_lor(coeff_lor_), F(nullptr), B(nullptr),
+        force_l2_space(force_l2_space_) { }
+
    virtual ~L2ProjectionGridTransfer();
 
    const Operator &ForwardOperator() override;
@@ -527,6 +595,7 @@ public:
    const Operator &BackwardOperator() override;
 
    bool SupportsBackwardsOperator() const override;
+
 private:
    void BuildF();
 };

general/backends.hpp

@@ -14,7 +14,7 @@
 
 #include "../config/config.hpp"
 
-#if defined(MFEM_USE_CUDA) && defined(__CUDACC__)
+#if defined(MFEM_USE_CUDA)
 #include <cusparse.h>
 #include <library_types.h>
 #include <cuda_runtime.h>
@@ -22,7 +22,7 @@
 #endif
 #include "cuda.hpp"
 
-#if defined(MFEM_USE_HIP) && defined(__HIP__)
+#if defined(MFEM_USE_HIP)
 #include <hip/hip_runtime.h>
 #endif
 #include "hip.hpp"
@@ -45,15 +45,17 @@
 #endif
 
 #if !defined(MFEM_USE_CUDA_OR_HIP)
-constexpr bool mfem_use_gpu = false;
-#define MFEM_DEVICE
-#define MFEM_HOST
-#define MFEM_LAMBDA
-// #define MFEM_HOST_DEVICE // defined in config/config.hpp
 // MFEM_DEVICE_SYNC is made available for debugging purposes
 #define MFEM_DEVICE_SYNC
 // MFEM_STREAM_SYNC is used for UVM and MPI GPU-Aware kernels
 #define MFEM_STREAM_SYNC
+#endif
+
+#if !defined(MFEM_USE_CUDA_OR_HIP_LANG)
+#define MFEM_DEVICE
+#define MFEM_HOST
+#define MFEM_LAMBDA
+// #define MFEM_HOST_DEVICE // defined in config/config.hpp
 #define MFEM_LAUNCH_BOUNDS(...)
 #endif
 
@@ -126,4 +128,23 @@ MFEM_HOST_DEVICE T AtomicAdd(T &add, const T val)
 #endif
 }
 
+namespace mfem::internal
+{
+
+#if defined(MFEM_USE_CUDA_OR_HIP) && !defined(MFEM_USE_CUDA_OR_HIP_LANG)
+static constexpr bool can_compile_kernels = false;
+#else
+static constexpr bool can_compile_kernels = true;
+#endif
+
+template <bool can_compile_kernels = can_compile_kernels>
+void RequireKernelCompilation()
+{
+   static_assert(
+      can_compile_kernels,
+      "The calling function needs to be compiled with CUDA/HIP language!");
+}
+
+}
+
 #endif // MFEM_BACKENDS_HPP

general/communication.cpp

@@ -1108,6 +1108,126 @@ void GroupCommunicator::ReduceEnd(T *ldata, int layout,
    num_requests = 0;
 }
 
+template <class T>
+void GroupCommunicator::ReduceMarked(T *ldata, const Array<int> &marker,
+                                     int layout,
+                                     void (*Op)(OpData<T>)) const
+{
+   if (comm_lock == 0) { return; }
+   // The above also handles the case (group_buf_size == 0).
+   MFEM_VERIFY(comm_lock == 2, "object is NOT locked for Reduce");
+
+   switch (mode)
+   {
+      case byGroup: // ***** Communication by groups *****
+      {
+         OpData<T> opd;
+         opd.ldata = ldata;
+         Array<int> group_num_req(group_ldof.Size());
+         for (int gr = 1; gr < group_ldof.Size(); gr++)
+         {
+            group_num_req[gr] =
+               gtopo.IAmMaster(gr) ? gtopo.GetGroupSize(gr)-1 : 0;
+         }
+         int idx;
+         while (MPI_Waitany(num_requests, requests, &idx, MPI_STATUS_IGNORE),
+                idx != MPI_UNDEFINED)
+         {
+            int gr = request_marker[idx];
+            if (gr == -1) { continue; } // skip send requests
+
+            // Delay the processing of a group until all receive requests, for
+            // that group, are done:
+            if ((--group_num_req[gr]) != 0) { continue; }
+
+            opd.nldofs = group_ldof.RowSize(gr);
+            // groups without dofs are skipped, so here nldofs > 0.
+
+            opd.buf = (T *)group_buf.GetData() + buf_offsets[gr];
+            opd.ldofs = (layout == 0) ?
+                        group_ldof.GetRow(gr) : group_ltdof.GetRow(gr);
+            opd.nb = gtopo.GetGroupSize(gr)-1;
+
+            // Apply operation only to marked DOFs. The receive buffer is
+            // neighbor-major with stride opd.nldofs, i.e. the contributions to
+            // DOF i are buf[j*opd.nldofs + i] for j = 0 ... opd.nb-1. Setting
+            // nldofs = 1 for a single DOF changes that stride to 1, so the
+            // strided values must first be gathered into a contiguous buffer.
+            Array<T> single_buf(opd.nb);
+            for (int i = 0; i < opd.nldofs; i++)
+            {
+               if (marker[opd.ldofs[i]])
+               {
+                  for (int j = 0; j < opd.nb; j++)
+                  {
+                     single_buf[j] = opd.buf[j*opd.nldofs + i];
+                  }
+
+                  // Create a temporary OpData with just this one DOF
+                  OpData<T> single_opd;
+                  single_opd.ldata = ldata;
+                  single_opd.buf = single_buf.GetData();
+                  single_opd.ldofs = opd.ldofs + i;
+                  single_opd.nldofs = 1;
+                  single_opd.nb = opd.nb;
+
+                  // Apply the operation
+                  Op(single_opd);
+               }
+            }
+         }
+         break;
+      }
+
+      case byNeighbor: // ***** Communication by neighbors *****
+      {
+         MPI_Waitall(num_requests, requests, MPI_STATUSES_IGNORE);
+
+         for (int nbr = 1; nbr < nbr_send_groups.Size(); nbr++)
+         {
+            // In Reduce operation: send_groups <--> recv_groups
+            const int num_recv_groups = nbr_send_groups.RowSize(nbr);
+            if (num_recv_groups > 0)
+            {
+               const int *grp_list = nbr_send_groups.GetRow(nbr);
+               const T *buf = (T*)group_buf.GetData() + buf_offsets[nbr];
+               for (int i = 0; i < num_recv_groups; i++)
+               {
+                  // Custom version of ReduceGroupFromBuffer that checks marker
+                  int gr = grp_list[i];
+                  const int *ldofs = (layout == 0) ?
+                                     group_ldof.GetRow(gr) : group_ltdof.GetRow(gr);
+                  const int nldofs = group_ldof.RowSize(gr);
+
+                  for (int j = 0; j < nldofs; j++)
+                  {
+                     if (marker[ldofs[j]])
+                     {
+                        // Create a temporary OpData with just this one DOF
+                        OpData<T> opd;
+                        opd.ldata = ldata;
+                        opd.buf = const_cast<T*>(buf) + j;
+                        opd.ldofs = ldofs + j;
+                        opd.nldofs = 1;
+                        opd.nb = 1;
+
+                        // Apply the operation
+                        Op(opd);
+                     }
+                  }
+
+                  buf += nldofs;
+               }
+            }
+         }
+         break;
+      }
+   }
+
+   comm_lock = 0; // 0 - no lock
+   num_requests = 0;
+}
+
 template <class T>
 void GroupCommunicator::Sum(OpData<T> opd)
 {
@@ -1171,6 +1291,8 @@ void GroupCommunicator::Max(OpData<T> opd)
 template <class T>
 void GroupCommunicator::BitOR(OpData<T> opd)
 {
+   static_assert(std::is_integral<T>::value,
+                 "BitOR reduction requires an integral type.");
    for (int i = 0; i < opd.nldofs; i++)
    {
       T data = opd.ldata[opd.ldofs[i]];
@@ -1182,6 +1304,33 @@ void GroupCommunicator::BitOR(OpData<T> opd)
    }
 }
 
+template <class T>
+void GroupCommunicator::MaxAbs(OpData<T> opd)
+{
+   for (int i = 0; i < opd.nldofs; i++)
+   {
+      T data = opd.ldata[opd.ldofs[i]];
+      T abs_data = std::abs(data);
+
+      for (int j = 0; j < opd.nb; j++)
+      {
+         T b = opd.buf[j*opd.nldofs+i];
+         T abs_b = std::abs(b);
+
+         // On an equal-magnitude tie keep the more positive value, so
+         // opposite-sign ties resolve deterministically to the positive one.
+         if (abs_data < abs_b || (abs_data == abs_b && data < b))
+         {
+            data = b;
+            abs_data = abs_b;
+         }
+      }
+
+      opd.ldata[opd.ldofs[i]] = data;
+   }
+}
+
+
 void GroupCommunicator::PrintInfo(std::ostream &os) const
 {
    char c = '\0';
@@ -1318,18 +1467,24 @@ template void GroupCommunicator::BcastEnd<int>(int *, int) const;
 template void GroupCommunicator::ReduceBegin<int>(const int *) const;
 template void GroupCommunicator::ReduceEnd<int>(
    int *, int, void (*)(OpData<int>)) const;
+template void GroupCommunicator::ReduceMarked<int>(
+   int*, const Array<int>&, int, void (*)(OpData<int>)) const;
 
 template void GroupCommunicator::BcastBegin<double>(double *, int) const;
 template void GroupCommunicator::BcastEnd<double>(double *, int) const;
 template void GroupCommunicator::ReduceBegin<double>(const double *) const;
 template void GroupCommunicator::ReduceEnd<double>(
    double *, int, void (*)(OpData<double>)) const;
+template void GroupCommunicator::ReduceMarked<double>(
+   double*, const Array<int>&, int, void (*)(OpData<double>)) const;
 
 template void GroupCommunicator::BcastBegin<float>(float *, int) const;
 template void GroupCommunicator::BcastEnd<float>(float *, int) const;
 template void GroupCommunicator::ReduceBegin<float>(const float *) const;
 template void GroupCommunicator::ReduceEnd<float>(
    float *, int, void (*)(OpData<float>)) const;
+template void GroupCommunicator::ReduceMarked<float>(
+   float*, const Array<int>&, int, void (*)(OpData<float>)) const;
 
 // @endcond
 
@@ -1338,14 +1493,17 @@ template void GroupCommunicator::Sum<int>(OpData<int>);
 template void GroupCommunicator::Min<int>(OpData<int>);
 template void GroupCommunicator::Max<int>(OpData<int>);
 template void GroupCommunicator::BitOR<int>(OpData<int>);
+template void GroupCommunicator::MaxAbs<int>(OpData<int>);
 
 template void GroupCommunicator::Sum<double>(OpData<double>);
 template void GroupCommunicator::Min<double>(OpData<double>);
 template void GroupCommunicator::Max<double>(OpData<double>);
+template void GroupCommunicator::MaxAbs<double>(OpData<double>);
 
 template void GroupCommunicator::Sum<float>(OpData<float>);
 template void GroupCommunicator::Min<float>(OpData<float>);
 template void GroupCommunicator::Max<float>(OpData<float>);
+template void GroupCommunicator::MaxAbs<float>(OpData<float>);
 
 
 #ifdef __bgq__

general/communication.hpp

@@ -22,6 +22,7 @@
 #include "globals.hpp"
 #include <mpi.h>
 #include <cstdint>
+#include <type_traits>
 
 // can't directly use MPI_CXX_BOOL because Microsoft's MPI implementation
 // doesn't include MPI_CXX_BOOL. Fallback to MPI_C_BOOL if unavailable.
@@ -408,14 +409,38 @@ public:
    template <class T> void Reduce(Array<T> &ldata, void (*Op)(OpData<T>)) const
    { Reduce<T>((T *)ldata, Op); }
 
-   /// Reduce operation Sum, instantiated for int and double
+   /// Reduce operation Sum, instantiated for int, double and float
    template <class T> static void Sum(OpData<T>);
-   /// Reduce operation Min, instantiated for int and double
+   /// Reduce operation Min, instantiated for int, double and float
    template <class T> static void Min(OpData<T>);
-   /// Reduce operation Max, instantiated for int and double
+   /// Reduce operation Max, instantiated for int, double and float
    template <class T> static void Max(OpData<T>);
    /// Reduce operation bitwise OR, instantiated for int only
    template <class T> static void BitOR(OpData<T>);
+   /// Reduce operation selecting the signed value with the largest absolute
+   /// value, instantiated for int, double and float. The result keeps its sign;
+   /// it is not the non-negative absolute value. Equal-magnitude ties are
+   /// broken deterministically toward the more positive value, so opposite-sign
+   /// ties resolve to the positive one regardless of accumulation order.
+   template <class T> static void MaxAbs(OpData<T>);
+
+   /** @brief Finalize reduction operation started with ReduceBegin(), but only apply
+       the reduction to DOFs marked in the marker array.
+
+       @note The reduction is carried out in the signed type @a T, so the result
+       is signed even for bitwise operations.
+   */
+   template <class T>
+   void ReduceMarked(T *ldata, const Array<int> &marker, int layout,
+                     void (*Op)(OpData<T>)) const;
+
+   /** @brief Reduce within each group where the master is the root, but only for marked DOFs. */
+   template <class T>
+   void Reduce(T *ldata, const Array<int> &marker, void (*Op)(OpData<T>)) const
+   {
+      ReduceBegin(ldata);
+      ReduceMarked(ldata, marker, 0, Op);
+   }
 
    /// Print information about the GroupCommunicator from all MPI ranks.
    void PrintInfo(std::ostream &out = mfem::out) const;

general/cuda.hpp

@@ -18,14 +18,8 @@
 // CUDA block size used by MFEM.
 #define MFEM_CUDA_BLOCKS 256
 
-#if defined(MFEM_USE_CUDA) && defined(__CUDACC__)
+#if defined(MFEM_USE_CUDA)
 #define MFEM_USE_CUDA_OR_HIP
-constexpr bool mfem_use_gpu = true;
-#define MFEM_DEVICE __device__
-#define MFEM_HOST __host__
-#define MFEM_LAMBDA __host__
-#define MFEM_LAUNCH_BOUNDS __launch_bounds__
-// #define MFEM_HOST_DEVICE __host__ __device__ // defined in config/config.hpp
 #define MFEM_DEVICE_SYNC MFEM_GPU_CHECK(cudaDeviceSynchronize())
 #define MFEM_STREAM_SYNC MFEM_GPU_CHECK(cudaStreamSynchronize(0))
 // Define a CUDA error check macro, MFEM_GPU_CHECK(x), where x returns/is of
@@ -40,6 +34,15 @@ constexpr bool mfem_use_gpu = true;
     }                                                                          \
   } while (0)
 
+// Macros defined only when compiling with CUDA language
+#if defined(__CUDACC__)
+#define MFEM_USE_CUDA_OR_HIP_LANG
+#define MFEM_DEVICE __device__
+#define MFEM_HOST __host__
+#define MFEM_LAMBDA __host__
+#define MFEM_LAUNCH_BOUNDS __launch_bounds__
+// #define MFEM_HOST_DEVICE __host__ __device__ // defined in config/config.hpp
+
 // Define the MFEM inner threading macros
 #if defined(__CUDA_ARCH__)
 #define MFEM_SHARED __shared__
@@ -67,12 +70,13 @@ constexpr bool mfem_use_gpu = true;
    if (int ix = threadIdx.k % (OX), iy = threadIdx.k / (OX), iz = iy / (OY);   \
        (ix < (SX)) && ((iy %= (OY)) < (SY)) && (iz < (SZ)))
 #endif // defined(__CUDA_ARCH__)
-#endif // defined(MFEM_USE_CUDA) && defined(__CUDACC__)
+#endif // defined(__CUDACC__)
+#endif // defined(MFEM_USE_CUDA)
 
 namespace mfem
 {
 
-#if defined(MFEM_USE_CUDA) && defined(__CUDACC__)
+#if defined(MFEM_USE_CUDA)
 // Function used by the macro MFEM_GPU_CHECK.
 void mfem_cuda_error(cudaError_t err, const char *expr, const char *func,
                      const char *file, int line);

general/error.cpp

@@ -171,7 +171,7 @@ void mfem_error(const char *msg)
 #ifdef MFEM_USE_EXCEPTIONS
    if (mfem_error_action == MFEM_ERROR_THROW)
    {
-      throw ErrorException(msg);
+      throw ErrorException(msg ? msg : "");
    }
 #endif
 

general/error.hpp

@@ -15,7 +15,7 @@
 #include "../config/config.hpp"
 #include <iomanip>
 #include <sstream>
-#ifdef MFEM_USE_HIP
+#if defined(MFEM_USE_HIP)
 #include <hip/hip_runtime.h>
 #endif
 
@@ -153,21 +153,13 @@ void mfem_warning(const char *msg = NULL);
 
 
 // Additional abort functions for HIP
-#if defined(MFEM_USE_HIP)
-#ifndef __HIP_DEVICE_COMPILE__
-template<typename T>
-__host__ void abort_msg(T & msg)
-{
-   MFEM_ABORT(msg);
-}
-#else
+#if defined(__HIP_DEVICE_COMPILE__)
 template<typename T>
 __device__ void abort_msg(T & msg)
 {
    abort();
 }
 #endif
-#endif
 
 // Abort inside a device kernel
 #if defined(__CUDA_ARCH__)

general/forall.hpp

@@ -1044,6 +1044,8 @@ inline void ForallWrap(const bool use_dev, const int N,
                        const int X=0, const int Y=0, const int Z=0,
                        const int G=0)
 {
+   internal::RequireKernelCompilation();
+
    MFEM_CONTRACT_VAR(X);
    MFEM_CONTRACT_VAR(Y);
    MFEM_CONTRACT_VAR(Z);
@@ -1276,6 +1278,9 @@ inline void hypre_forall_cpu(int N, lambda &&body)
 template<typename lambda>
 inline void hypre_forall_gpu(int N, lambda &&body)
 {
+   internal::RequireKernelCompilation();
+
+#if defined(MFEM_USE_CUDA_OR_HIP_LANG)
 #if defined(HYPRE_USING_CUDA)
    CuWrap1D(N, body);
 #elif defined(HYPRE_USING_HIP)
@@ -1283,6 +1288,7 @@ inline void hypre_forall_gpu(int N, lambda &&body)
 #else
 #error Unknown HYPRE GPU backend!
 #endif
+#endif
 }
 #endif
 

general/hip.hpp

@@ -18,14 +18,8 @@
 // HIP block size used by MFEM.
 #define MFEM_HIP_BLOCKS 256
 
-#if defined(MFEM_USE_HIP) && defined(__HIP__)
+#if defined(MFEM_USE_HIP)
 #define MFEM_USE_CUDA_OR_HIP
-constexpr bool mfem_use_gpu = true;
-#define MFEM_DEVICE __device__
-#define MFEM_HOST __host__
-#define MFEM_LAMBDA __host__ __device__
-#define MFEM_LAUNCH_BOUNDS __launch_bounds__
-// #define MFEM_HOST_DEVICE __host__ __device__ // defined in config/config.hpp
 #define MFEM_DEVICE_SYNC MFEM_GPU_CHECK(hipDeviceSynchronize())
 #define MFEM_STREAM_SYNC MFEM_GPU_CHECK(hipStreamSynchronize(0))
 // Define a HIP error check macro, MFEM_GPU_CHECK(x), where x returns/is of
@@ -40,6 +34,15 @@ constexpr bool mfem_use_gpu = true;
     }                                                                          \
   } while (0)
 
+// Macros defined only when compiling with HIP language
+#if defined(__HIP__)
+#define MFEM_USE_CUDA_OR_HIP_LANG
+#define MFEM_DEVICE __device__
+#define MFEM_HOST __host__
+#define MFEM_LAMBDA __host__ __device__
+#define MFEM_LAUNCH_BOUNDS __launch_bounds__
+// #define MFEM_HOST_DEVICE __host__ __device__ // defined in config/config.hpp
+
 // Define the MFEM inner threading macros
 #if defined(__HIP_DEVICE_COMPILE__)
 #define MFEM_SHARED __shared__
@@ -71,7 +74,8 @@ constexpr bool mfem_use_gpu = true;
        iz = iy / (OY);                                                         \
        (ix < (SX)) && ((iy %= (OY)) < (SY)) && (iz < (SZ)))
 #endif // defined(__HIP_DEVICE_COMPILE__)
-#endif // defined(MFEM_USE_HIP) && defined(__HIP__)
+#endif // defined(__HIP__)
+#endif // defined(MFEM_USE_HIP)
 
 namespace mfem
 {

general/reducers.hpp

@@ -550,10 +550,10 @@ void reduce(int N, T &res, B &&body, const R &reducer, bool use_dev,
 
       int num_mp = Device::NumMultiprocessors(Device::GetId());
 #if defined(MFEM_USE_CUDA)
-      // good value of mp_sat found experimentally on Lassen
+      // good value of mp_sat found experimentally on Lassen (V100)
       constexpr int mp_sat = 8;
 #elif defined(MFEM_USE_HIP)
-      // good value of mp_sat found experimentally on Tuolumne
+      // good value of mp_sat found experimentally on Tuolumne (MI300A)
       constexpr int mp_sat = 4;
 #else
       num_mp = 1;

general/scan.hpp

@@ -15,6 +15,10 @@
 #include "backends.hpp"
 #include "forall.hpp"
 
+#if defined(MFEM_USE_CUDA_OR_HIP) && !defined(MFEM_USE_CUDA_OR_HIP_LANG)
+#error "This header requires compilation with CUDA/HIP language!"
+#else
+
 #ifdef MFEM_USE_CUDA
 #include <cub/device/device_scan.cuh>
 #include <cub/device/device_select.cuh>
@@ -406,4 +410,6 @@ void CopyUnique(bool use_dev, InputIt d_in, OutputIt d_out,
 
 #undef MFEM_CUB_NAMESPACE
 
-#endif
+#endif // defined(MFEM_USE_CUDA_OR_HIP) && !defined(MFEM_USE_CUDA_OR_HIP_LANG)
+
+#endif // MFEM_SCAN_HPP

linalg/CMakeLists.txt

@@ -27,6 +27,7 @@ list(APPEND SRCS
   handle.cpp
   matrix.cpp
   mma.cpp
+  multivector.cpp
   ode.cpp
   operator.cpp
   ordering.cpp
@@ -63,6 +64,7 @@ list(APPEND HDRS
   linalg.hpp
   matrix.hpp
   mma.hpp
+  multivector.hpp
   ode.hpp
   operator.hpp
   ordering.hpp

linalg/densemat.hpp

@@ -1136,6 +1136,17 @@ private:
 public:
    DenseTensor() : ni(0), nj(0), nk(0) { }
 
+   DenseTensor(const DenseTensor &other)
+      : tdata(other.tdata), ni(other.ni), nj(other.nj), nk(other.nk) { }
+
+   DenseTensor(DenseTensor &&other)
+      : tdata(std::move(other.tdata)), ni(other.ni), nj(other.nj), nk(other.nk)
+   {
+      // Reset other; other.tdata is reset in Array<T> move constructror.
+      other.Mk.ClearExternalData();
+      other.ni = other.nj = other.nk = 0;
+   }
+
    DenseTensor(int i, int j, int k) : tdata(i*j*k), ni(i), nj(j), nk(k) { }
 
    DenseTensor(real_t *d, int i, int j, int k)
@@ -1144,6 +1155,33 @@ public:
    DenseTensor(int i, int j, int k, MemoryType mt)
       : tdata(i*j*k, mt), ni(i), nj(j), nk(k) { }
 
+   DenseTensor &operator=(const DenseTensor &other)
+   {
+      if (this == &other) { return *this; }
+      Mk.ClearExternalData();
+      tdata = other.tdata;
+      ni = other.ni;
+      nj = other.nj;
+      nk = other.nk;
+      return *this;
+   }
+
+   DenseTensor &operator=(DenseTensor &&other)
+   {
+      if (this == &other) { return *this; }
+      Mk.ClearExternalData();
+      tdata = std::move(other.tdata);
+      ni = other.ni;
+      nj = other.nj;
+      nk = other.nk;
+
+      // Reset other; other.tdata is reset in Array<T> move assignment.
+      other.Mk.ClearExternalData();
+      other.ni = other.nj = other.nk = 0;
+
+      return *this;
+   }
+
    int SizeI() const { return ni; }
    int SizeJ() const { return nj; }
    int SizeK() const { return nk; }

linalg/hypre.cpp

@@ -5842,6 +5842,10 @@ void HypreAMS::MakeGradientAndInterpolation(
    {
       grad->AddTraceFaceInterpolator(new GradientInterpolator);
    }
+   else if (dynamic_cast<const RT_FECollection *>(edge_fec))
+   {
+      grad->AddDomainInterpolator(new CurlInterpolator);
+   }
    else
    {
       grad->AddDomainInterpolator(new GradientInterpolator);

linalg/linalg.hpp

@@ -15,6 +15,7 @@
 // Linear algebra header file
 
 #include "vector.hpp"
+#include "multivector.hpp"
 #include "operator.hpp"
 #include "matrix.hpp"
 #include "sparsemat.hpp"

linalg/multivector.cpp

@@ -0,0 +1,60 @@
+// Copyright (c) 2010-2025, 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 "multivector.hpp"
+
+namespace mfem
+{
+
+MultiVector::MultiVector(const Array<int> &vector_sizes)
+{
+   SetSizes(vector_sizes);
+}
+
+MultiVector::MultiVector(const Array<int> &vector_sizes, MemoryType mt)
+{
+   SetSizes(vector_sizes, mt);
+}
+
+MultiVector::MultiVector(Vector &base, const Array<int> &vector_sizes)
+{
+   MakeRef(base, vector_sizes);
+}
+
+void MultiVector::SetSizes(const Array<int> &vector_sizes)
+{
+   blocks.resize(vector_sizes.Size());
+   for (int i = 0; i < vector_sizes.Size(); i++)
+   {
+      operator[](i).SetSize(vector_sizes[i]);
+   }
+}
+
+void MultiVector::SetSizes(const Array<int> &vector_sizes, MemoryType mt)
+{
+   blocks.resize(vector_sizes.Size());
+   for (int i = 0; i < vector_sizes.Size(); i++)
+   {
+      operator[](i).SetSize(vector_sizes[i], mt);
+   }
+}
+
+void MultiVector::MakeRef(Vector &base, const Array<int> &vector_sizes)
+{
+   blocks.resize(vector_sizes.Size());
+   for (int offset = 0, i = 0; i < vector_sizes.Size(); i++)
+   {
+      blocks[i].emplace<0>(base, offset, vector_sizes[i]);
+      offset += vector_sizes[i];
+   }
+}
+
+} // namespace mfem

linalg/multivector.hpp

@@ -0,0 +1,251 @@
+// Copyright (c) 2010-2025, 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_MULTIVECTOR_HPP
+#define MFEM_MULTIVECTOR_HPP
+
+#include "../general/array.hpp"
+#include "vector.hpp"
+#include <vector>
+#include <array>
+#include <variant>
+
+namespace mfem
+{
+
+/// Class representing an array of Vectors with generally different sizes.
+/** This class is similar to BlockVector with the following two main
+    differences:
+    - the data for the individual Vector blocks does not need to be part of one
+      big contiguous memory allocation;
+    - this class does not inherit from class Vector (as a consequence of the
+      first bullet).
+
+    Internally, each Vector block is represented as one of the following
+    three options:
+    - (default) a Vector object constructed and owned by this class; this
+      object, in turn, as any Vector object, can own its Memory allocation or
+      refer to a sub-Memory of another Memory object; or
+    - a pointer to an externally allocated Vector or classes derived from
+      Vector.
+    - a pointer to an externally allocated const Vector or classes derived from
+      Vector. This option is helpful for wrapping const Vector objects as a
+      MultiVector that will be then used as a const MultiVector. */
+class MultiVector
+{
+private:
+   std::vector<std::variant<Vector,Vector*,const Vector*>> blocks;
+
+public:
+   /// Create an empty MultiVector with zero blocks.
+   MultiVector() = default;
+
+   /** @brief Create a MultiVector with @a num_blocks blocks. The individual
+       Vector blocks are default initialized, i.e. they all have size zero. */
+   MultiVector(int num_blocks)
+      : blocks(num_blocks) { }
+
+   /** @brief Construct a MultiVector with number of blocks and individual block
+       Vector sizes given by @a vector_sizes.
+
+       @note The memory of the individual Vector blocks is NOT initialized. */
+   MultiVector(const Array<int> &vector_sizes);
+
+   /** @brief Construct a MultiVector with number of blocks and individual block
+       Vector sizes given by @a vector_sizes. All Vector blocks use the
+       MemoryType @a mt.
+
+       @note The memory of the individual Vector blocks is NOT initialized. */
+   MultiVector(const Array<int> &vector_sizes, MemoryType mt);
+
+   /** @brief Construct a MultiVector referencing data within a given monolithic
+       Vector @a base.
+
+       With this constructor, the Memory flags of @a base and of the individual
+       Vector blocks may need to be explicitly synchronized when data is moved
+       between host and device. */
+   MultiVector(Vector &base, const Array<int> &vector_sizes);
+
+   /** @brief Construct a MultiVector referencing multiple Vectors given as
+       arguments.
+
+       The VectorTypes reference arguments are expected to be static_cast-able
+       to (Vector &) which is the case if the types are derived from Vector,
+       e.g. HypreParVector, GridFunction, etc.
+
+       With this constructor, operations on individual Vector blocks are
+       performed directly on the objects @a vs. In particular, there is no need
+       to synchronize the Memory flags of @a vs and the ones of the individual
+       Vector blocks when data is moved between host and device. */
+   template <typename... VectorTypes,
+             std::enable_if_t<
+                std::conjunction_v<
+                   std::is_convertible<VectorTypes&,Vector&>...>, bool> = true>
+   MultiVector(VectorTypes &...vs) { MakeRef(vs...); }
+
+   /** @brief Construct a MultiVector referencing multiple const Vectors given
+       as arguments. Individual blocks are read-only; non-const operator[]
+       will generate an error. */
+   template <typename... VectorTypes,
+             std::enable_if_t<
+                std::conjunction_v<
+                   std::is_convertible<const VectorTypes&,const Vector&>...>,
+                bool> = true>
+   MultiVector(const VectorTypes &...vs) { MakeRef(vs...); }
+
+   /// Return the number of Vectors in the MultiVector.
+   int NumBlocks() const { return blocks.size(); }
+
+   /** @brief Set the number of Vectors in the MultiVector. Existing Vector
+       blocks will remain unmodified. New Vector blocks will be default
+       initialized, i.e. they all have size zero. */
+   void SetNumBlocks(int num_blocks) { blocks.resize(num_blocks); }
+
+   /** @brief Read-write access to the i-th Vector. Generates an error if the
+       i-th block is read-only, i.e. it is a pointer to a const Vector. */
+   inline Vector &operator[](int i);
+
+   /// Read-only access to the i-th Vector.
+   inline const Vector &operator[](int i) const;
+
+   /** @brief Update the MultiVector according to the given @a vector_sizes.
+
+       This method can be used to add or remove blocks. The individual Vector
+       sizes are updated using the method Vector::SetSize(int). */
+   void SetSizes(const Array<int> &vector_sizes);
+
+   /** @brief Update the MultiVector according to the given @a vector_sizes and
+       MemoryType @a mt.
+
+       This method can be used to add or remove blocks. The individual Vector
+       sizes and MemoryType are updated using the method
+       Vector::SetSize(int, MemoryType). */
+   void SetSizes(const Array<int> &vector_sizes, MemoryType mt);
+
+   /** @brief Update the MultiVector to reference data within a given monolithic
+       Vector @a base.
+
+       After calling this method, the Memory flags of @a base and of the
+       individual Vector blocks may need to be explicitly synchronized when data
+       is moved between host and device.*/
+   void MakeRef(Vector &base, const Array<int> &vector_sizes);
+
+   /** @brief Update the @a i-th MultiVector block to reference data within the
+       given monolithic Vector @a base at the given @a offset and with the given
+       @a size.
+
+       After calling this method, the Memory flags of @a base and of the @a i-th
+       Vector block may need to be explicitly synchronized when data is moved
+       between host and device.*/
+   inline void MakeRef(int i, Vector &base, int offset, int size)
+   {
+      blocks[i].emplace<0>(base, offset, size);
+   }
+
+   /** @brief Update the MultiVector to reference multiple Vectors given as
+       arguments.
+
+       The VectorTypes reference arguments are expected to be static_cast-able
+       to (Vector &) which is the case if the types are derived from Vector,
+       e.g. HypreParVector, GridFunction, etc.
+
+       After calling this method, operations on individual Vector blocks are
+       performed directly on the objects @a vs. In particular, there is no need
+       to synchronize the Memory flags of @a vs and the ones of the individual
+       Vector blocks when data is moved between host and device. */
+   template <typename... VectorTypes,
+             std::enable_if_t<
+                std::conjunction_v<
+                   std::is_convertible<VectorTypes&,Vector&>...>, bool> = true>
+   inline void MakeRef(VectorTypes &...vs);
+
+   /** @brief Update the MultiVector to reference multiple const Vectors given
+       as arguments. Individual blocks are read-only; non-const operator[]
+       will generate an error. */
+   template <typename... VectorTypes,
+             std::enable_if_t<
+                std::conjunction_v<
+                   std::is_convertible<const VectorTypes&,const Vector&>...>,
+                bool> = true>
+   inline void MakeRef(const VectorTypes &...vs);
+
+   /** @brief Update the @a i-th MultiVector block to reference the given
+       Vector @a v.
+
+       After calling this method, operations on the @a i-th Vector block are
+       performed directly on the Vector @a v. In particular, there is no need
+       to synchronize the Memory flags of @a v and the ones of the @a i-th
+       Vector blocks when data is moved between host and device. */
+   inline void MakeRef(int i, Vector &v) { blocks[i] = &v; }
+
+   /** @brief Update the @a i-th MultiVector block to reference the given
+       const Vector @a v. The block becomes read-only. */
+   inline void MakeRef(int i, const Vector &v) { blocks[i] = &v; }
+};
+
+// Inline and template methods
+
+inline Vector &MultiVector::operator[](int i)
+{
+   auto &bi = blocks[i];
+   const auto idx = bi.index();
+   if (idx == 0) { return std::get<0>(bi); }
+   if (idx == 1) { return *std::get<1>(bi); }
+   MFEM_ABORT("Non-const access to a const Vector block!");
+}
+
+inline const Vector &MultiVector::operator[](int i) const
+{
+   auto &bi = blocks[i];
+   const auto idx = bi.index();
+   return (idx == 0) ? std::get<0>(bi) :
+          (idx == 1) ? *std::get<1>(bi) :
+          /**/         *std::get<2>(bi);
+}
+
+template <typename... VectorTypes,
+          std::enable_if_t<
+             std::conjunction_v<
+                std::is_convertible<VectorTypes&,Vector&>...>, bool>>
+inline void MultiVector::MakeRef(VectorTypes &...vs)
+{
+   blocks.resize(sizeof...(vs));
+   if constexpr (sizeof...(vs) > 0)
+   {
+      const std::array vs_p{&static_cast<Vector&>(vs)...};
+      for (std::size_t i = 0; i < sizeof...(vs); i++)
+      {
+         blocks[i] = vs_p[i];
+      }
+   }
+}
+
+template <typename... VectorTypes,
+          std::enable_if_t<
+             std::conjunction_v<
+                std::is_convertible<const VectorTypes&,const Vector&>...>,
+             bool>>
+inline void MultiVector::MakeRef(const VectorTypes &...vs)
+{
+   blocks.resize(sizeof...(vs));
+   if constexpr (sizeof...(vs) > 0)
+   {
+      const std::array vs_p{&static_cast<const Vector&>(vs)...};
+      for (std::size_t i = 0; i < sizeof...(vs); i++)
+      {
+         blocks[i] = vs_p[i];
+      }
+   }
+}
+
+} // namespace mfem
+
+#endif // MFEM_MULTIVECTOR_HPP

linalg/operator.cpp

@@ -111,6 +111,21 @@ void Operator::ArrayAddMultTranspose(const Array<const Vector *> &X,
    }
 }
 
+void Operator::MultMV(const MultiVector &, MultiVector &) const
+{
+   MFEM_ABORT("this method is not overridden for this class!");
+}
+
+void Operator::MultTransposeMV(const MultiVector &x, MultiVector &y) const
+{
+   MFEM_ABORT("this method is not overridden for this class!");
+}
+
+Operator &Operator::GetGradientMV(const MultiVector &) const
+{
+   MFEM_ABORT("this method is not overridden for this class!");
+}
+
 void Operator::FormLinearSystem(const Array<int> &ess_tdof_list,
                                 Vector &x, Vector &b,
                                 Operator* &Aout, Vector &X, Vector &B,

linalg/operator.hpp

@@ -13,6 +13,7 @@
 #define MFEM_OPERATOR
 
 #include "vector.hpp"
+#include "multivector.hpp"
 
 namespace mfem
 {
@@ -129,6 +130,20 @@ public:
    virtual void ArrayAddMultTranspose(const Array<const Vector *> &X,
                                       Array<Vector *> &Y, const real_t a = 1.0) const;
 
+   /** @brief Operator application, y = A(x), where the input @a x and the
+       output @a y are MultiVector objects, i.e. they generally use
+       non-contiguous memory representation.
+
+       The base class implementation for the method is to generate an error. */
+   virtual void MultMV(const MultiVector &x, MultiVector &y) const;
+
+   /** @brief Action of the transpose operator, y = A^t(x), where the input @a x
+       and the output @a y are MultiVector objects, i.e. they generally use
+       non-contiguous memory representation.
+
+       The base class implementation for this method is to generate an error. */
+   virtual void MultTransposeMV(const MultiVector &x, MultiVector &y) const;
+
    /** @brief Evaluate the gradient operator at the point @a x. The default
        behavior in class Operator is to generate an error. */
    virtual Operator &GetGradient(const Vector &x) const
@@ -137,6 +152,13 @@ public:
       return const_cast<Operator &>(*this);
    }
 
+   /** @brief Evaluate the gradient operator at the point @a x. The input @a x
+       is provided as a MultiVector, i.e. it generally uses non-contiguous
+       memory representation.
+
+       The base class implementation for the method is to generate an error. */
+   virtual Operator &GetGradientMV(const MultiVector &x) const;
+
    /** @brief Computes the diagonal entries into @a diag. Typically, this
        operation only makes sense for linear Operator%s. In some cases, only an
        approximation of the diagonal is computed. */

mesh/pncmesh.cpp

@@ -2624,7 +2624,8 @@ void ParNCMesh::RedistributeElements(Array<int> &new_ranks, int target_elements,
          for (int i = 0; i < rank_neighbors.Size(); i++)
          {
             int elem = rank_neighbors[i];
-            msg.AddElementRank(elem, new_ranks[elements[elem].index]);
+            const Element &el = elements[elem];
+            msg.AddElement(elem, new_ranks[el.index], el.attribute);
          }
 
          msg.Isend(rank, MyComm);
@@ -2647,7 +2648,9 @@ void ParNCMesh::RedistributeElements(Array<int> &new_ranks, int target_elements,
       {
          int ghost_index = elements[msg.elements[i]].index;
          MFEM_ASSERT(element_type[ghost_index] == 2, "");
-         new_ranks[ghost_index] = msg.values[i];
+         const ElementRankAndAttribute &value = msg.values[i];
+         new_ranks[ghost_index] = value.rank;
+         elements[msg.elements[i]].attribute = value.attribute;
       }
    }
 
@@ -2718,7 +2721,7 @@ void ParNCMesh::RedistributeElements(Array<int> &new_ranks, int target_elements,
 
                if ((element_type[el.index] & 1) || el.rank != rank)
                {
-                  msg.AddElementRank(elem, el.rank);
+                  msg.AddElement(elem, el.rank, el.attribute);
                }
                // NOTE: we skip 'ghosts' that are of the receiver's rank because
                // they are not really ghosts and would get sent multiple times,
@@ -2770,10 +2773,12 @@ void ParNCMesh::RedistributeElements(Array<int> &new_ranks, int target_elements,
 
          for (int i = 0; i < msg.Size(); i++)
          {
-            int elem_rank = msg.values[i];
-            elements[msg.elements[i]].rank = elem_rank;
+            const ElementRankAndAttribute &value = msg.values[i];
+            Element &el = elements[msg.elements[i]];
+            el.rank = value.rank;
+            el.attribute = value.attribute;
 
-            if (elem_rank == MyRank) { received_elements++; }
+            if (value.rank == MyRank) { received_elements++; }
          }
 
          // save the ranks we received from, for later use in RecvRebalanceDofs
@@ -2809,7 +2814,10 @@ void ParNCMesh::RedistributeElements(Array<int> &new_ranks, int target_elements,
 
             for (int i = 0; i < msg.Size(); i++)
             {
-               elements[msg.elements[i]].rank = msg.values[i];
+               const ElementRankAndAttribute &value = msg.values[i];
+               Element &el = elements[msg.elements[i]];
+               el.rank = value.rank;
+               el.attribute = value.attribute;
             }
 
             // save the ranks we received from, for later use in RecvRebalanceDofs

mesh/pncmesh.hpp

@@ -531,26 +531,36 @@ protected: // implementation
       typedef std::map<int, NeighborDerefinementMessage> Map;
    };
 
-   /** Used in Step 2 of Rebalance() to synchronize new rank assignments in
-    *  the ghost layer.
+   struct ElementRankAndAttribute
+   {
+      int rank;
+      int attribute;
+   };
+
+   /** Used in RedistributeElements() to synchronize new rank assignments and
+    *  element attributes in the ghost layer.
     */
-   class NeighborElementRankMessage : public ElementValueMessage<int, false,
+   class NeighborElementRankMessage :
+      public ElementValueMessage<ElementRankAndAttribute, false,
       VarMessageTag::NEIGHBOR_ELEMENT_RANK_VM>
    {
    public:
-      void AddElementRank(int elem, int rank) { Add(elem, rank); }
+      void AddElement(int elem, int rank, int attribute)
+      { Add(elem, {rank, attribute}); }
       typedef std::map<int, NeighborElementRankMessage> Map;
    };
 
-   /** Used by Rebalance() to send elements and their ranks. Note that
+   /** Used by Rebalance() to send elements, ranks, and attributes. Note that
     *  RefTypes == true which means the refinement hierarchy will be recreated
     *  on the receiving side.
     */
-   class RebalanceMessage : public ElementValueMessage<int, true,
+   class RebalanceMessage :
+      public ElementValueMessage<ElementRankAndAttribute, true,
       VarMessageTag::REBALANCE_VM>
    {
    public:
-      void AddElementRank(int elem, int rank) { Add(elem, rank); }
+      void AddElement(int elem, int rank, int attribute)
+      { Add(elem, {rank, attribute}); }
       typedef std::map<int, RebalanceMessage> Map;
    };
 

miniapps/fluids/navier/makefile

@@ -52,6 +52,8 @@ endif
 .SUFFIXES:
 .SUFFIXES: .o .cpp .mk
 .PHONY: all lib-common clean clean-build clean-exec
+# Keeping the *.o files fixes an issue with the MacOS version of 'make'.
+.PRECIOUS: %.o
 
 # Remove built-in rules
 %: %.cpp

miniapps/multidomain/makefile

@@ -68,7 +68,7 @@ multidomain-test-par: multidomain
 multidomain_nd-test-par: multidomain_nd
 	@$(call mfem-test,$<, $(RUN_MPI), Multidomain ND miniapp,-tf 0.001)
 multidomain_rt-test-par: multidomain_rt
-	@$(call mfem-test,$<, $(RUN_MPI), Multidomain RT iniapp,-tf 0.001)
+	@$(call mfem-test,$<, $(RUN_MPI), Multidomain RT miniapp,-tf 0.001)
 
 # Generate an error message if the MFEM library is not built and exit
 $(MFEM_LIB_FILE):

miniapps/parelag/MultilevelHcurlHdivSolver.cpp

@@ -761,7 +761,7 @@ int main(int argc, char *argv[])
       if (visualize)
       {
          hcurlhdiv_dofTrueDof.Distribute(X, x);
-         MultiVector tmp(x.GetData(), 1, x.Size());
+         parelag::MultiVector tmp(x.GetData(), 1, x.Size());
          sequence[0]->show(jform, tmp);
       }
       post_timer.Stop();

miniapps/tools/lor-transfer.cpp

@@ -33,6 +33,7 @@
 //
 // Sample runs:  lor-transfer
 //               lor-transfer -h1
+//               lor-transfer -ea -w
 //               lor-transfer -t
 //               lor-transfer -m ../../data/star-q2.mesh -lref 5 -p 4
 //               lor-transfer -m ../../data/star-mixed.mesh -lref 3 -p 2
@@ -59,11 +60,12 @@ string direction;
 
 // Exact functions to project
 real_t RHO_exact(const Vector &x);
+real_t W_exact(const Vector &x);
+real_t weight(const Vector &x);
 
 // Helper functions
 void visualize(VisItDataCollection &, string, int, int, int visport = 19916);
-real_t compute_mass(FiniteElementSpace *, real_t, VisItDataCollection &,
-                    string);
+real_t compute_mass(GridFunction &, real_t, string, CoefficientWithOrder);
 
 int main(int argc, char *argv[])
 {
@@ -76,6 +78,7 @@ int main(int argc, char *argv[])
    bool useH1 = false;
    int visport = 19916;
    bool use_pointwise_transfer = false;
+   bool use_weighted_transfer = false;
    const char *device_config = "cpu";
    bool use_ea       = false;
 
@@ -98,6 +101,9 @@ int main(int argc, char *argv[])
    args.AddOption(&use_pointwise_transfer, "-t", "--use-pointwise-transfer",
                   "-no-t", "--dont-use-pointwise-transfer",
                   "Use pointwise transfer operators instead of L2 projection.");
+   args.AddOption(&use_weighted_transfer, "-w", "--use-weighted-transfer",
+                  "-no-w", "--dont-use-weighted-transfer",
+                  "Use coefficient-weighted L2 projection.");
    args.AddOption(&device_config, "-d", "--device",
                   "Device configuration string, see Device::Configure().");
    args.AddOption(&use_ea, "-ea", "--ea-version", "-no-ea",
@@ -107,6 +113,15 @@ int main(int argc, char *argv[])
    // Configure device
    Device device(device_config);
 
+   if (use_weighted_transfer && !use_pointwise_transfer)
+   {
+      if (problem != 5)
+      {
+         cout << "Switching to positive problem = 5 for weighted transfer.\n";
+      }
+      problem = 5;
+   }
+
    // Read the mesh from the given mesh file.
    Mesh mesh(mesh_file, 1, 1);
    int dim = mesh.Dimension();
@@ -138,6 +153,14 @@ int main(int argc, char *argv[])
    FiniteElementSpace fespace(&mesh, fec);
    FiniteElementSpace fespace_lor(&mesh_lor, fec_lor);
 
+   FunctionCoefficient weight_fn_coeff(weight);
+   CoefficientWithOrder weight_coeff;
+   if (use_weighted_transfer)
+   {
+      weight_coeff.coeff = &weight_fn_coeff;
+      weight_coeff.order = 2;
+   }
+
    GridFunction rho(&fespace);
    GridFunction rho_lor(&fespace_lor);
 
@@ -165,7 +188,7 @@ int main(int argc, char *argv[])
    rho.SetTrueVector();
    rho.SetFromTrueVector();
 
-   real_t ho_mass = compute_mass(&fespace, -1.0, HO_dc, "HO       ");
+   real_t ho_mass = compute_mass(rho, -1.0, "HO       ", weight_coeff);
    if (vis) { visualize(HO_dc, "HO", Wx, Wy, visport); Wx += offx; }
 
    GridTransfer *gt;
@@ -175,7 +198,8 @@ int main(int argc, char *argv[])
    }
    else
    {
-      gt = new L2ProjectionGridTransfer(fespace, fespace_lor);
+      gt = new L2ProjectionGridTransfer(fespace, fespace_lor, weight_coeff,
+                                        weight_coeff);
    }
 
    // Configure element assembly for device acceleration
@@ -186,9 +210,44 @@ int main(int argc, char *argv[])
    // HO->LOR restriction
    direction = "HO -> LOR @ LOR";
    R.Mult(rho, rho_lor);
-   compute_mass(&fespace_lor, ho_mass, LOR_dc, "R(HO)    ");
+   compute_mass(rho_lor, ho_mass, "R(HO)    ", weight_coeff);
    if (vis) { visualize(LOR_dc, "R(HO)", Wx, Wy, visport); Wx += offx; }
 
+   if (use_weighted_transfer && !use_pointwise_transfer)
+   {
+      // Transfer velocity while conserving rho-weighted momentum.
+      GridFunctionCoefficient rho_coeff(&rho);
+      GridFunctionCoefficient rho_lor_coeff(&rho_lor);
+      ProductCoefficient prod_coeff(weight_fn_coeff, rho_coeff);
+      ProductCoefficient prod_lor_coeff(weight_fn_coeff, rho_lor_coeff);
+      CoefficientWithOrder prod_weight(prod_coeff, order + 2);
+      CoefficientWithOrder prod_lor_weight(prod_lor_coeff, lorder + 2);
+
+      GridFunction w(&fespace), w_lor(&fespace_lor);
+      FunctionCoefficient W(W_exact);
+      w.ProjectCoefficient(W);
+
+      cout << '\n';
+      const real_t ho_momentum = compute_mass(w, -1.0, "rho w HO ", prod_weight);
+
+      L2ProjectionGridTransfer vel_gt(fespace, fespace_lor, prod_weight,
+                                      prod_lor_weight);
+      vel_gt.UseEA(use_ea);
+      vel_gt.ForwardOperator().Mult(w, w_lor);
+      compute_mass(w_lor, ho_momentum, "rho w LOR", prod_lor_weight);
+
+      if (vel_gt.SupportsBackwardsOperator())
+      {
+         GridFunction w_prev = w;
+         vel_gt.BackwardOperator().Mult(w_lor, w);
+         compute_mass(w, ho_momentum, "P(rho w) ", prod_weight);
+
+         w_prev -= w;
+         cout.precision(12);
+         cout << "|w - P(R(w))|_∞     = " << w_prev.Normlinf() << "\n\n";
+      }
+   }
+
    if (gt->SupportsBackwardsOperator())
    {
       const Operator &P = gt->BackwardOperator();
@@ -196,7 +255,7 @@ int main(int argc, char *argv[])
       direction = "HO -> LOR @ HO";
       GridFunction rho_prev = rho;
       P.Mult(rho_lor, rho);
-      compute_mass(&fespace, ho_mass, HO_dc, "P(R(HO)) ");
+      compute_mass(rho, ho_mass, "P(R(HO)) ", weight_coeff);
       if (vis) { visualize(HO_dc, "P(R(HO))", Wx, Wy, visport); Wx = 0; Wy += offy; }
 
       rho_prev -= rho;
@@ -218,7 +277,7 @@ int main(int argc, char *argv[])
    direction = "LOR -> HO @ LOR";
    rho_lor.ProjectCoefficient(RHO);
    GridFunction rho_lor_prev = rho_lor;
-   real_t lor_mass = compute_mass(&fespace_lor, -1.0, LOR_dc, "LOR      ");
+   real_t lor_mass = compute_mass(rho_lor, -1.0, "LOR      ", weight_coeff);
    if (vis) { visualize(LOR_dc, "LOR", Wx, Wy, visport); Wx += offx; }
 
    if (gt->SupportsBackwardsOperator())
@@ -227,14 +286,14 @@ int main(int argc, char *argv[])
       // Prolongate to HO space
       direction = "LOR -> HO @ HO";
       P.Mult(rho_lor, rho);
-      compute_mass(&fespace, lor_mass, HO_dc, "P(LOR)   ");
+      compute_mass(rho, lor_mass, "P(LOR)   ", weight_coeff);
       if (vis) { visualize(HO_dc, "P(LOR)", Wx, Wy, visport); Wx += offx; }
 
       // Restrict back to LOR space. This won't give the original function because
       // the rho_lor doesn't necessarily live in the range of R.
       direction = "LOR -> HO @ LOR";
       R.Mult(rho, rho_lor);
-      compute_mass(&fespace_lor, lor_mass, LOR_dc, "R(P(LOR))");
+      compute_mass(rho_lor, lor_mass, "R(P(LOR))", weight_coeff);
       if (vis) { visualize(LOR_dc, "R(P(LOR))", Wx, Wy, visport); }
 
       rho_lor_prev -= rho_lor;
@@ -270,12 +329,26 @@ real_t RHO_exact(const Vector &x)
          return M_PI/2-atan(5*(2*x.Norml2()-1));
       case 4: // basis function
          return (x.Norml2() < 0.1) ? 1 : 0;
+      case 5: // positive function
+         return 2.0 + 2*x(0)*x(0) + 3*x(1)*x(1) - x(0)*x(1) + 0.1*sin(x.Norml2());
       default:
          return 1.0;
    }
 }
 
 
+real_t W_exact(const Vector &x)
+{
+   return x(1) + 0.25*cos(2*M_PI*x.Norml2());
+}
+
+
+real_t weight(const Vector &x)
+{
+   return x(0)*x(0) + x(1)*x(1) + 1.0;
+}
+
+
 void visualize(VisItDataCollection &dc, string prefix, int x, int y,
                int visport)
 {
@@ -292,21 +365,32 @@ void visualize(VisItDataCollection &dc, string prefix, int x, int y,
 }
 
 
-real_t compute_mass(FiniteElementSpace *L2, real_t massL2,
-                    VisItDataCollection &dc, string prefix)
+real_t compute_mass(GridFunction &gf, real_t oldmass, string prefix,
+                    CoefficientWithOrder mass_coeff)
 {
+   FiniteElementSpace &fes = *gf.FESpace();
+   Mesh &mesh = *fes.GetMesh();
+
+   // Integration order is a * (element order) + b.
+   const int a = 2;
+   const int b = mesh.GetTypicalElementTransformation()->OrderW() +
+                 mass_coeff.order;
+
    ConstantCoefficient one(1.0);
-   LinearForm lf(L2);
-   lf.AddDomainIntegrator(new DomainLFIntegrator(one));
+   Coefficient &coeff = mass_coeff ? *mass_coeff.coeff : one;
+   DomainLFIntegrator *integ = new DomainLFIntegrator(coeff, a, b);
+
+   LinearForm lf(&fes);
+   lf.AddDomainIntegrator(integ);
    lf.Assemble();
 
-   real_t newmass = lf(*dc.GetField("density"));
+   const real_t newmass = lf(gf);
    cout.precision(18);
    cout << space << " " << prefix << " mass   = " << newmass;
-   if (massL2 >= 0)
+   if (oldmass >= 0)
    {
       cout.precision(4);
-      cout << " ("  << fabs(newmass-massL2)*100/massL2 << "%)";
+      cout << " ("  << fabs(newmass-oldmass)*100/oldmass << "%)";
    }
    cout << endl;
    return newmass;

miniapps/tools/plor-transfer.cpp

@@ -33,6 +33,7 @@
 //
 // Sample runs:  plor-transfer
 //               plor-transfer -h1
+//               plor-transfer -ea -w
 //               plor-transfer -t
 //               plor-transfer -m ../../data/star-q2.mesh -lref 5 -p 4
 //               plor-transfer -m ../../data/star-mixed.mesh -lref 3 -p 2
@@ -59,11 +60,12 @@ string direction;
 
 // Exact functions to project
 real_t RHO_exact(const Vector &x);
+real_t W_exact(const Vector &x);
+real_t weight(const Vector &x);
 
 // Helper functions
 void visualize(VisItDataCollection &, string, int, int, int /* visport */);
-real_t compute_mass(ParFiniteElementSpace *, real_t, VisItDataCollection &,
-                    string);
+real_t compute_mass(ParGridFunction &, real_t, string, CoefficientWithOrder);
 
 int main(int argc, char *argv[])
 {
@@ -80,6 +82,7 @@ int main(int argc, char *argv[])
    bool useH1 = false;
    int visport = 19916;
    bool use_pointwise_transfer = false;
+   bool use_weighted_transfer = false;
    const char *device_config = "cpu";
    bool use_ea       = false;
 
@@ -102,6 +105,9 @@ int main(int argc, char *argv[])
    args.AddOption(&use_pointwise_transfer, "-t", "--use-pointwise-transfer",
                   "-no-t", "--dont-use-pointwise-transfer",
                   "Use pointwise transfer operators instead of L2 projection.");
+   args.AddOption(&use_weighted_transfer, "-w", "--use-weighted-transfer",
+                  "-no-w", "--dont-use-weighted-transfer",
+                  "Use coefficient-weighted L2 projection.");
    args.AddOption(&device_config, "-d", "--device",
                   "Device configuration string, see Device::Configure().");
    args.AddOption(&use_ea, "-ea", "--ea-version", "-no-ea",
@@ -112,6 +118,15 @@ int main(int argc, char *argv[])
    Device device(device_config);
    if (Mpi::Root()) { device.Print(); }
 
+   if (use_weighted_transfer && !use_pointwise_transfer)
+   {
+      if (problem != 5 && Mpi::Root())
+      {
+         cout << "Switching to positive problem = 5 for weighted transfer.\n";
+      }
+      problem = 5;
+   }
+
    // Read the mesh from the given mesh file.
    Mesh serial_mesh(mesh_file, 1, 1);
    ParMesh mesh(MPI_COMM_WORLD, serial_mesh);
@@ -154,6 +169,14 @@ int main(int argc, char *argv[])
    ParFiniteElementSpace fespace(&mesh, fec);
    ParFiniteElementSpace fespace_lor(&mesh_lor, fec_lor);
 
+   FunctionCoefficient weight_fn_coeff(weight);
+   CoefficientWithOrder weight_coeff;
+   if (use_weighted_transfer)
+   {
+      weight_coeff.coeff = &weight_fn_coeff;
+      weight_coeff.order = 2;
+   }
+
    ParGridFunction rho(&fespace);
    ParGridFunction rho_lor(&fespace_lor);
 
@@ -183,7 +206,7 @@ int main(int argc, char *argv[])
    rho.SetTrueVector();
    rho.SetFromTrueVector();
 
-   real_t ho_mass = compute_mass(&fespace, -1.0, HO_dc, "HO       ");
+   real_t ho_mass = compute_mass(rho, -1.0, "HO       ", weight_coeff);
    if (vis) { visualize(HO_dc, "HO", Wx, Wy, visport); Wx += offx; }
 
    GridTransfer *gt;
@@ -193,7 +216,8 @@ int main(int argc, char *argv[])
    }
    else
    {
-      gt = new L2ProjectionGridTransfer(fespace, fespace_lor);
+      gt = new L2ProjectionGridTransfer(fespace, fespace_lor, weight_coeff,
+                                        weight_coeff);
    }
 
    // Configure element assembly for device acceleration
@@ -204,7 +228,7 @@ int main(int argc, char *argv[])
    // HO->LOR restriction
    direction = "HO -> LOR @ LOR";
    R.Mult(rho, rho_lor);
-   compute_mass(&fespace_lor, ho_mass, LOR_dc, "R(HO)    ");
+   compute_mass(rho_lor, ho_mass, "R(HO)    ", weight_coeff);
    if (vis) { visualize(LOR_dc, "R(HO)", Wx, Wy, visport); Wx += offx; }
    auto global_max = [](const Vector& v)
    {
@@ -214,6 +238,47 @@ int main(int argc, char *argv[])
       return max;
    };
 
+   if (use_weighted_transfer && !use_pointwise_transfer)
+   {
+      // Transfer velocity while conserving rho-weighted momentum.
+      GridFunctionCoefficient rho_coeff(&rho);
+      GridFunctionCoefficient rho_lor_coeff(&rho_lor);
+      ProductCoefficient prod_coeff(weight_fn_coeff, rho_coeff);
+      ProductCoefficient prod_lor_coeff(weight_fn_coeff, rho_lor_coeff);
+      CoefficientWithOrder prod_weight(prod_coeff, order + 2);
+      CoefficientWithOrder prod_lor_weight(prod_lor_coeff, lorder + 2);
+
+      ParGridFunction w(&fespace), w_lor(&fespace_lor);
+      FunctionCoefficient W(W_exact);
+      w.ProjectCoefficient(W);
+
+      if (Mpi::Root()) { cout << '\n'; }
+      const real_t ho_momentum = compute_mass(w, -1.0, "rho w HO ", prod_weight);
+
+      L2ProjectionGridTransfer vel_gt(fespace, fespace_lor, prod_weight,
+                                      prod_lor_weight);
+      vel_gt.UseEA(use_ea);
+      vel_gt.ForwardOperator().Mult(w, w_lor);
+      compute_mass(w_lor, ho_momentum, "rho w LOR", prod_lor_weight);
+
+      if (vel_gt.SupportsBackwardsOperator())
+      {
+         ParGridFunction w_prev = w;
+         vel_gt.BackwardOperator().Mult(w_lor, w);
+         compute_mass(w, ho_momentum, "P(rho w) ", prod_weight);
+
+         w_prev -= w;
+         Vector w_prev_true(fespace.GetTrueVSize());
+         w_prev.GetTrueDofs(w_prev_true);
+         const real_t l_inf = global_max(w_prev_true);
+         if (Mpi::Root())
+         {
+            cout.precision(12);
+            cout << "|w - P(R(w))|_∞     = " << l_inf << "\n\n";
+         }
+      }
+   }
+
    if (gt->SupportsBackwardsOperator())
    {
       const Operator &P = gt->BackwardOperator();
@@ -221,7 +286,7 @@ int main(int argc, char *argv[])
       direction = "HO -> LOR @ HO";
       ParGridFunction rho_prev = rho;
       P.Mult(rho_lor, rho);
-      compute_mass(&fespace, ho_mass, HO_dc, "P(R(HO)) ");
+      compute_mass(rho, ho_mass, "P(R(HO)) ", weight_coeff);
       if (vis) { visualize(HO_dc, "P(R(HO))", Wx, Wy, visport); Wx = 0; Wy += offy; }
 
       rho_prev -= rho;
@@ -263,7 +328,7 @@ int main(int argc, char *argv[])
    direction = "LOR -> HO @ LOR";
    rho_lor.ProjectCoefficient(RHO);
    ParGridFunction rho_lor_prev = rho_lor;
-   real_t lor_mass = compute_mass(&fespace_lor, -1.0, LOR_dc, "LOR      ");
+   real_t lor_mass = compute_mass(rho_lor, -1.0, "LOR      ", weight_coeff);
    if (vis) { visualize(LOR_dc, "LOR", Wx, Wy, visport); Wx += offx; }
 
    if (gt->SupportsBackwardsOperator())
@@ -272,14 +337,14 @@ int main(int argc, char *argv[])
       // Prolongate to HO space
       direction = "LOR -> HO @ HO";
       P.Mult(rho_lor, rho);
-      compute_mass(&fespace, lor_mass, HO_dc, "P(LOR)   ");
+      compute_mass(rho, lor_mass, "P(LOR)   ", weight_coeff);
       if (vis) { visualize(HO_dc, "P(LOR)", Wx, Wy, visport); Wx += offx; }
 
       // Restrict back to LOR space. This won't give the original function because
       // the rho_lor doesn't necessarily live in the range of R.
       direction = "LOR -> HO @ LOR";
       R.Mult(rho, rho_lor);
-      compute_mass(&fespace_lor, lor_mass, LOR_dc, "R(P(LOR))");
+      compute_mass(rho_lor, lor_mass, "R(P(LOR))", weight_coeff);
       if (vis) { visualize(LOR_dc, "R(P(LOR))", Wx, Wy, visport); }
 
       rho_lor_prev -= rho_lor;
@@ -334,12 +399,26 @@ real_t RHO_exact(const Vector &x)
          return M_PI/2-atan(5*(2*x.Norml2()-1));
       case 4: // basis function
          return (x.Norml2() < 0.1) ? 1 : 0;
+      case 5: // positive function
+         return 2.0 + 2*x(0)*x(0) + 3*x(1)*x(1) - x(0)*x(1) + 0.1*sin(x.Norml2());
       default:
          return 1.0;
    }
 }
 
 
+real_t W_exact(const Vector &x)
+{
+   return x(1) + 0.25*cos(2*M_PI*x.Norml2());
+}
+
+
+real_t weight(const Vector &x)
+{
+   return x(0)*x(0) + x(1)*x(1) + 1.0;
+}
+
+
 void visualize(VisItDataCollection &dc, string prefix, int x, int y,
                int visport)
 {
@@ -358,23 +437,34 @@ void visualize(VisItDataCollection &dc, string prefix, int x, int y,
 }
 
 
-real_t compute_mass(ParFiniteElementSpace *L2, real_t massL2,
-                    VisItDataCollection &dc, string prefix)
+real_t compute_mass(ParGridFunction &gf, real_t oldmass, string prefix,
+                    CoefficientWithOrder mass_coeff)
 {
+   ParFiniteElementSpace &fes = *gf.ParFESpace();
+   Mesh &mesh = *fes.GetMesh();
+
+   // Integration order is a * (element order) + b.
+   const int a = 2;
+   const int b = mesh.GetTypicalElementTransformation()->OrderW() +
+                 mass_coeff.order;
+
    ConstantCoefficient one(1.0);
-   ParLinearForm lf(L2);
-   lf.AddDomainIntegrator(new DomainLFIntegrator(one));
+   Coefficient &coeff = mass_coeff ? *mass_coeff.coeff : one;
+   DomainLFIntegrator *integ = new DomainLFIntegrator(coeff, a, b);
+
+   ParLinearForm lf(&fes);
+   lf.AddDomainIntegrator(integ);
    lf.Assemble();
 
-   real_t newmass = lf(*dc.GetParField("density"));
+   const real_t newmass = lf(gf);
    if (Mpi::Root())
    {
       cout.precision(18);
       cout << space << " " << prefix << " mass   = " << newmass;
-      if (massL2 >= 0)
+      if (oldmass >= 0)
       {
          cout.precision(4);
-         cout << " ("  << fabs(newmass-massL2)*100/massL2 << "%)";
+         cout << " ("  << fabs(newmass-oldmass)*100/oldmass << "%)";
       }
       cout << endl;
    }

tests/benchmarks/bench_ceed.cpp

@@ -32,7 +32,11 @@
 // Custom benchmark arguments generator
 static void CustomArguments(bm::Benchmark *b) noexcept
 {
-   constexpr int MAX_NDOFS = 16 * 1024 * (mfem_use_gpu ? 1024 : 8);
+#if defined(MFEM_USE_CUDA_OR_HIP_LANG)
+   constexpr int MAX_NDOFS = 16 * 1024 * 1024;
+#else
+   constexpr int MAX_NDOFS = 16 * 1024 * 8;
+#endif
 
    const auto orders = { 7, 6, 5, 4, 3, 2, 1 };
 

tests/unit/CMakeLists.txt

@@ -39,6 +39,7 @@ set(UNIT_TESTS_SRCS
   dfem/test_divergence.cpp
   dfem/test_lvector_interface.cpp
   dfem/test_mass.cpp
+  dfem/test_tuple.cpp
   general/test_array.cpp
   general/test_scan.cpp
   general/test_arrays_by_name.cpp
@@ -96,6 +97,7 @@ set(UNIT_TESTS_SRCS
   fem/test_2d_bilininteg.cpp
   fem/test_3d_bilininteg.cpp
   fem/test_assembly_levels.cpp
+  fem/test_bdr_edgedof.cpp
   fem/test_bilinearform.cpp
   fem/test_block_operators.cpp
   fem/test_blocknonlinearform.cpp

tests/unit/dfem/test_tuple.cpp

@@ -0,0 +1,274 @@
+// Copyright (c) 2010-2025, 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 "../unit_tests.hpp"
+#include "mfem.hpp"
+#ifndef MFEM_USE_MPI
+#include "../../../fem/dfem/tuple.hpp"
+#endif
+
+using namespace mfem;
+using namespace mfem::future;
+
+namespace tuple_test
+{
+
+// A payload that is not a scalar, mimicking what dFEM kernels actually store.
+using vec3 = tensor<real_t, 3>;
+using tuple3 = tuple<real_t, int, vec3>;
+
+// mfem::future::tuple is no longer an aggregate: it derives from tuple_leaf
+// bases so that it can be defined for an arbitrary number of elements. These
+// checks pin down the properties that the aggregate used to provide for free
+// and that device kernels (which capture tuples by value) depend on.
+static_assert(std::is_trivially_copyable<tuple3>::value,
+              "tuple must be trivially copyable to be captured by value in device kernels");
+static_assert(std::is_trivially_destructible<tuple3>::value,
+              "tuple must be trivially destructible");
+static_assert(std::is_trivially_default_constructible<tuple3>::value,
+              "tuple must be trivially default constructible");
+static_assert(std::is_trivially_copy_assignable<tuple3>::value,
+              "tuple must be trivially copy assignable");
+static_assert(sizeof(tuple3) == sizeof(real_t) + sizeof(int) + sizeof(vec3) +
+              (alignof(real_t) - sizeof(int)),
+              "tuple must not be larger than the sum of its (padded) members");
+
+// Size and element types, both through mfem::future and through the std
+// specializations that drive structured bindings.
+static_assert(tuple_size<tuple3>::value == 3, "");
+static_assert(std::tuple_size<tuple3>::value == 3, "");
+static_assert(std::is_same<tuple_element<0, tuple3>::type, real_t>::value, "");
+static_assert(std::is_same<tuple_element<1, tuple3>::type, int>::value, "");
+static_assert(std::is_same<tuple_element<2, tuple3>::type, vec3>::value, "");
+static_assert(std::is_same<std::tuple_element_t<0, tuple3>, real_t>::value, "");
+static_assert(std::is_same<std::tuple_element_t<2, tuple3>, vec3>::value, "");
+
+// get must preserve the value category and constness of its argument.
+static_assert(std::is_same<decltype(get<1>(std::declval<tuple3&>())),
+              int&>::value, "get on an lvalue must return an lvalue reference");
+static_assert(std::is_same<decltype(get<1>(std::declval<const tuple3&>())),
+              const int&>::value,
+              "get on a const lvalue must return a const lvalue reference");
+static_assert(std::is_same<decltype(get<1>(std::declval<tuple3&&>())),
+              int&&>::value, "get on an rvalue must return an rvalue reference");
+static_assert(std::is_same<decltype(get<1>(std::declval<const tuple3&&>())),
+              const int&&>::value,
+              "get on a const rvalue must return a const rvalue reference");
+
+// += and -= must return a reference, not a copy of the whole tuple.
+using tuple2 = tuple<real_t, vec3>;
+static_assert(std::is_same<decltype(std::declval<tuple2&>() +=
+                                       std::declval<const tuple2&>()), tuple2&>::value,
+              "operator+= must return a reference");
+static_assert(std::is_same<decltype(std::declval<tuple2&>() -=
+                                       std::declval<const tuple2&>()), tuple2&>::value,
+              "operator-= must return a reference");
+
+// The element-wise constructor must stay implicit, so that the
+// copy-list-initialization forms that worked with the aggregate keep working.
+static_assert(std::is_convertible<int, tuple<int>>::value,
+              "tuple's element-wise constructor must not be explicit");
+
+// Constructing from an incompatible type must SFINAE out rather than hard-error,
+// so that the constructor does not poison type traits.
+struct not_a_number { };
+static_assert(!std::is_constructible<tuple<int, int>, int, not_a_number>::value,
+              "");
+static_assert(!std::is_constructible<tuple<int, int>, int>::value,
+              "arity mismatch must not be constructible");
+
+// Usable at compile time.
+constexpr tuple<int, real_t> const_tuple {2, 3.0};
+static_assert(get<0>(const_tuple) == 2, "");
+
+// Copy-list-initialization in a return statement (broken by an explicit ctor).
+tuple<int, real_t> returns_braced_init_list() { return {7, 8.0}; }
+
+} // namespace tuple_test
+
+using namespace tuple_test;
+
+TEST_CASE("dFEM tuple structured bindings", "[dFEM]")
+{
+   tuple3 t {1.0, 2, vec3{{3.0, 4.0, 5.0}}};
+
+   SECTION("binding by reference writes through")
+   {
+      auto &[a, b, c] = t;
+      a = 10.0;
+      b = 20;
+      c(0) = 30.0;
+      REQUIRE(get<0>(t) == 10.0_r);
+      REQUIRE(get<1>(t) == 20);
+      REQUIRE(get<2>(t)(0) == 30.0_r);
+   }
+
+   SECTION("binding by value copies")
+   {
+      auto [a, b, c] = t;
+      a = 10.0;
+      b = 20;
+      c(0) = 30.0;
+      REQUIRE(get<0>(t) == 1.0_r);
+      REQUIRE(get<1>(t) == 2);
+      REQUIRE(get<2>(t)(0) == 3.0_r);
+   }
+
+   SECTION("binding to const")
+   {
+      const auto &[a, b, c] = t;
+      REQUIRE(a == 1.0_r);
+      REQUIRE(b == 2);
+      REQUIRE(c(2) == 5.0_r);
+      static_assert(std::is_same<decltype(a), const real_t>::value, "");
+      static_assert(std::is_same<decltype(c), const vec3>::value, "");
+   }
+
+   SECTION("the bindings alias the tuple storage")
+   {
+      auto &[a, b, c] = t;
+      REQUIRE(&a == &get<0>(t));
+      REQUIRE(&b == &get<1>(t));
+      REQUIRE(&c == &get<2>(t));
+   }
+}
+
+TEST_CASE("dFEM tuple construction", "[dFEM]")
+{
+   SECTION("copy-list-initialization")
+   {
+      tuple<int, real_t> a = {1, 2.0};
+      REQUIRE(get<0>(a) == 1);
+      REQUIRE(get<1>(a) == 2.0_r);
+
+      const auto b = returns_braced_init_list();
+      REQUIRE(get<0>(b) == 7);
+      REQUIRE(get<1>(b) == 8.0_r);
+   }
+
+   SECTION("direct initialization and CTAD")
+   {
+      tuple c {1, 2.0_r, vec3{{1.0, 2.0, 3.0}}};
+      static_assert(std::is_same<decltype(c), tuple<int, real_t, vec3>>::value,
+                    "CTAD must decay the arguments");
+      REQUIRE(get<1>(c) == 2.0_r);
+   }
+
+   SECTION("make_tuple")
+   {
+      const auto d = make_tuple(1, 2.0_r);
+      static_assert(std::is_same<decltype(d), const tuple<int, real_t>>::value, "");
+      REQUIRE(get<0>(d) == 1);
+   }
+
+   SECTION("copy and move construction preserve values")
+   {
+      tuple3 t {1.0, 2, vec3{{3.0, 4.0, 5.0}}};
+      tuple3 copy(t);
+      tuple3 moved(std::move(t));
+      REQUIRE(get<1>(copy) == 2);
+      REQUIRE(get<2>(moved)(1) == 4.0_r);
+   }
+
+   SECTION("value initialization zeroes trivial members")
+   {
+      tuple<int, real_t> z {};
+      REQUIRE(get<0>(z) == 0);
+      REQUIRE(get<1>(z) == 0.0_r);
+   }
+}
+
+TEST_CASE("dFEM tuple arithmetic", "[dFEM]")
+{
+   const tuple2 x {1.0, vec3{{1.0, 2.0, 3.0}}};
+   const tuple2 y {2.0, vec3{{4.0, 5.0, 6.0}}};
+
+   SECTION("element-wise binary operators")
+   {
+      const auto sum = x + y;
+      REQUIRE(get<0>(sum) == 3.0_r);
+      REQUIRE(get<1>(sum)(2) == 9.0_r);
+
+      const auto diff = y - x;
+      REQUIRE(get<0>(diff) == 1.0_r);
+      REQUIRE(get<1>(diff)(0) == 3.0_r);
+   }
+
+   SECTION("compound assignment mutates in place and returns a reference")
+   {
+      tuple2 z = x;
+      auto &ref = (z += y);
+      REQUIRE(&ref == &z);
+      REQUIRE(get<0>(z) == 3.0_r);
+      REQUIRE(get<1>(z)(1) == 7.0_r);
+
+      auto &ref2 = (z -= y);
+      REQUIRE(&ref2 == &z);
+      REQUIRE(get<0>(z) == 1.0_r);
+      REQUIRE(get<1>(z)(1) == 2.0_r);
+   }
+
+   SECTION("scalar operators and unary minus")
+   {
+      const auto scaled = 2.0_r * x;
+      REQUIRE(get<0>(scaled) == 2.0_r);
+      REQUIRE(get<1>(scaled)(2) == 6.0_r);
+
+      const auto halved = x / 2.0_r;
+      REQUIRE(get<0>(halved) == 0.5_r);
+
+      const auto negated = -x;
+      REQUIRE(get<0>(negated) == -1.0_r);
+      REQUIRE(get<1>(negated)(0) == -1.0_r);
+   }
+
+   SECTION("apply")
+   {
+      const auto s = apply([](const real_t &a, const vec3 &b) { return a + b(0); },
+      x);
+      REQUIRE(s == 2.0_r);
+   }
+}
+
+// The tuples are captured by value in device kernels, so exercise a round trip
+// through device memory: construct, mutate through structured bindings and read
+// back on the device.
+TEST_CASE("dFEM tuple on device", "[dFEM][GPU]")
+{
+   Vector res(4);
+   auto d_res = res.Write();
+
+   forall(1, [=] MFEM_HOST_DEVICE (int)
+   {
+      tuple3 t {1.0, 2, vec3{{3.0, 4.0, 5.0}}};
+      auto &[a, b, c] = t;
+      a += static_cast<real_t>(b);
+      c(0) = a;
+
+      tuple2 u {get<0>(t), get<2>(t)};
+      u += tuple2 {1.0, vec3{{1.0, 1.0, 1.0}}};
+
+      d_res[0] = get<0>(u);
+      d_res[1] = get<1>(u)(0);
+      d_res[2] = get<1>(u)(1);
+      d_res[3] = static_cast<real_t>(get<1>(t));
+
+      tuple2 v1{0_r, vec3{0_r, 0_r, 0_r}};
+      tuple2 v2{0_r, vec3{0_r, 0_r, 0_r}};
+      [[maybe_unused]] auto v = v1 + v2;
+   });
+
+   res.HostRead();
+   REQUIRE(std::as_const(res)(0) == 4.0_r);
+   REQUIRE(std::as_const(res)(1) == 4.0_r);
+   REQUIRE(std::as_const(res)(2) == 5.0_r);
+   REQUIRE(std::as_const(res)(3) == 2.0_r);
+}

tests/unit/fem/test_2d_bilininteg.cpp

@@ -3451,4 +3451,81 @@ TEST_CASE("2D Bilinear Scalar Weak Curl Cross Integrators",
    }
 }
 
+TEST_CASE("2D Bilinear Scalar Curl Integrator PartialAssembly",
+          "[MixedScalarCurlIntegrator]"
+          "[BilinearFormIntegrator]"
+          "[NonlinearFormIntegrator]"
+          "[GPU]")
+{
+   int order = 2, n = 1, dim = 2;
+   double tol = 1e-9;
+
+   Mesh mesh = Mesh::MakeCartesian2D(n, n, Element::QUADRILATERAL, 1, 2.0, 3.0);
+
+   VectorFunctionCoefficient   F2_coef(dim, F2);
+   FunctionCoefficient         q2_coef(q2);
+
+   SECTION("Operators on ND")
+   {
+      ND_FECollection    fec_nd(order, dim);
+      FiniteElementSpace fespace_nd(&mesh, &fec_nd);
+
+      GridFunction f_nd(&fespace_nd); f_nd.ProjectCoefficient(F2_coef);
+
+      for (int map_type = (int)FiniteElement::VALUE;
+           map_type <= (int)FiniteElement::INTEGRAL; map_type++)
+      {
+         SECTION("Mapping ND to L2 (" +
+                 MapTypeName((FiniteElement::MapType)map_type) + ")")
+         {
+            L2_FECollection    fec_l2(order - 1, dim,
+                                      BasisType::GaussLegendre,
+                                      (FiniteElement::MapType)map_type);
+            FiniteElementSpace fespace_l2(&mesh, &fec_l2);
+
+            Vector tmp_l2(fespace_l2.GetNDofs());
+            Vector tmp_l2_pa(fespace_l2.GetNDofs());
+
+            SECTION("Without Coefficient")
+            {
+               MixedBilinearForm blf_fa(&fespace_nd, &fespace_l2);
+               blf_fa.AddDomainIntegrator(new MixedScalarCurlIntegrator());
+               blf_fa.Assemble();
+               blf_fa.Finalize();
+
+               blf_fa.Mult(f_nd, tmp_l2);
+
+               MixedBilinearForm blf_pa(&fespace_nd, &fespace_l2);
+               blf_pa.SetAssemblyLevel(mfem::AssemblyLevel::PARTIAL);
+               blf_pa.AddDomainIntegrator(new MixedScalarCurlIntegrator());
+               blf_pa.Assemble();
+
+               blf_pa.Mult(f_nd, tmp_l2_pa);
+               tmp_l2_pa -= tmp_l2;
+               REQUIRE(tmp_l2_pa.Normlinf() < tol);
+            }
+            SECTION("With Scalar Coefficient")
+            {
+               MixedBilinearForm blf_fa(&fespace_nd, &fespace_l2);
+               blf_fa.AddDomainIntegrator(
+                  new MixedScalarCurlIntegrator(q2_coef));
+               blf_fa.Assemble();
+               blf_fa.Finalize();
+
+               blf_fa.Mult(f_nd, tmp_l2);
+
+               MixedBilinearForm blf_pa(&fespace_nd, &fespace_l2);
+               blf_pa.SetAssemblyLevel(mfem::AssemblyLevel::PARTIAL);
+               blf_pa.AddDomainIntegrator(new MixedScalarCurlIntegrator(q2_coef));
+               blf_pa.Assemble();
+
+               blf_pa.Mult(f_nd, tmp_l2_pa);
+               tmp_l2_pa -= tmp_l2;
+               REQUIRE(tmp_l2_pa.Normlinf() < tol);
+            }
+         }
+      }
+   }
+}
+
 } // namespace bilininteg_2d

tests/unit/fem/test_bdr_edgedof.cpp

@@ -0,0 +1,705 @@
+// Copyright (c) 2010-2025, 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 "unit_tests.hpp"
+#include "mfem.hpp"
+#include "../mesh/mesh_test_utils.hpp"
+
+#include <set>
+#include <unordered_set>
+#include <vector>
+
+using namespace mfem;
+
+#ifdef MFEM_USE_MPI
+
+TEST_CASE("BoundaryEdgeDOFsPartitionInvariant",
+          "[Parallel][ParMesh][BoundaryEdgeDOFs]")
+{
+   constexpr int orientation = 3;
+   constexpr int order = 1;
+
+   // Use all available MPI processes for partitioning
+   const int test_num_procs = Mpi::WorldSize();
+
+   // Create base mesh
+   Mesh base_mesh = OrientedTriFaceMesh(orientation, true);
+   base_mesh.UniformRefinement();
+   const int n_elements = base_mesh.GetNE();
+
+   // Use a small set of representative partitionings
+   std::vector<std::vector<int>> all_partitionings;
+   // 1. All elements on rank 0
+   all_partitionings.push_back(std::vector<int>(n_elements, 0));
+   if (test_num_procs > 1)
+   {
+      // 2. Block partition: first half on rank 0, second half on last rank
+      std::vector<int> &block = all_partitionings.emplace_back(n_elements);
+      for (int i = 0; i < n_elements; i++)
+      {
+         block[i] = (i < n_elements/2) ? 0 : test_num_procs-1;
+      }
+
+      // 3. Round-robin partition: elements assigned cyclically to all ranks
+      std::vector<int> &round_robin = all_partitionings.emplace_back(n_elements);
+      for (int i = 0; i < n_elements; i++)
+      {
+         round_robin[i] = i % test_num_procs;
+      }
+   }
+
+   // Create reusable FEC
+   ND_FECollection fec(order, 3);
+
+   std::vector<int> all_results;
+   all_results.reserve(all_partitionings.size());
+
+   // Test each partitioning
+   for (const auto& partition : all_partitionings)
+   {
+      // Create parallel mesh with current partitioning
+      Mesh test_mesh = OrientedTriFaceMesh(orientation, true);
+      test_mesh.UniformRefinement();
+      // For single process, use default partitioning; for multiple, use custom partition
+      ParMesh pmesh = (test_num_procs == 1) ?
+                      ParMesh(MPI_COMM_WORLD, test_mesh) :
+                      ParMesh(MPI_COMM_WORLD, test_mesh, partition.data());
+
+      // Create finite element space
+      ParFiniteElementSpace fespace(&pmesh, &fec);
+
+      // Extract boundary edge DOFs
+      Array<int> ess_tdof_list;
+      Array<int> boundary_edge_ldofs;
+      std::vector<Array<int>> attr_to_elements;
+
+      // Select the shared face to be the tested boundary
+      int bdr_attr = pmesh.bdr_attributes.Max();
+      Array<int> bdr_attrs(1);
+      bdr_attrs[0] = bdr_attr;
+
+      fespace.GetBoundaryElementsByAttribute(bdr_attrs, attr_to_elements);
+      Array<int> boundary_elements = attr_to_elements[0];
+
+      Array<int> dof_edges, dof_boundary_elements, ess_edge_list;
+
+      fespace.GetBoundaryLoopEdgeDofs(boundary_elements, ess_tdof_list,
+                                      boundary_edge_ldofs, nullptr, &dof_edges,
+                                      &dof_boundary_elements, &ess_edge_list);
+
+      // Collect total boundary edge DOFs
+      int local_dofs = boundary_edge_ldofs.Size();
+      int total_dofs;
+      MPI_Allreduce(&local_dofs, &total_dofs, 1, MPI_INT, MPI_SUM, MPI_COMM_WORLD);
+
+      all_results.push_back(total_dofs);
+   }
+
+   // The set of boundary edge DOFs is a property of the mesh geometry and must
+   // not depend on how the elements are distributed across ranks. Each result
+   // is the global count of selected boundary edge DOFs for one partitioning, so
+   // if the method correctly removes the artificial edges introduced at
+   // processor boundaries, every partitioning yields the same total. A mismatch
+   // means some partition kept or dropped a DOF that another did not.
+   REQUIRE(!all_results.empty());
+   // One refinement splits the triangular face into four sub-triangles. Its
+   // perimeter has six loop edges (order-1 ND: one DOF per edge); the three
+   // interior edges of the middle sub-triangle are shared and correctly dropped.
+   constexpr int expected = 6;
+   for (int result : all_results)
+   {
+      REQUIRE(result == expected);
+   }
+}
+
+TEST_CASE("BoundaryEdgeDOFsBasicFunctionality",
+          "[Parallel][ParMesh][BoundaryEdgeDOFs]")
+{
+   const int orientation = GENERATE(1, 3, 5);
+   const int order = GENERATE(1, 2);
+
+   CAPTURE(orientation, order);
+
+   // Create test mesh
+   Mesh mesh = OrientedTriFaceMesh(orientation, true);
+   mesh.UniformRefinement();
+   ParMesh pmesh(MPI_COMM_WORLD, mesh);
+
+   // Create finite element space
+   ND_FECollection fec(order, 3);
+   ParFiniteElementSpace fespace(&pmesh, &fec);
+
+   // Test boundary edge DOF extraction
+   Array<int> ess_tdof_list;
+   Array<int> boundary_edge_ldofs;
+   Array<int> ldof_marker;
+   std::vector<Array<int>> attr_to_elements;
+
+   // Get boundary elements for the shared face
+   int bdr_attr = pmesh.bdr_attributes.Max();
+   Array<int> bdr_attrs(1);
+   bdr_attrs[0] = bdr_attr;
+
+   fespace.GetBoundaryElementsByAttribute(bdr_attrs, attr_to_elements);
+   Array<int> boundary_elements = attr_to_elements[0];
+
+   Array<int> dof_edges, dof_boundary_elements, ess_edge_list;
+
+   fespace.GetBoundaryLoopEdgeDofs(boundary_elements, ess_tdof_list,
+                                   boundary_edge_ldofs, &ldof_marker, &dof_edges,
+                                   &dof_boundary_elements, &ess_edge_list);
+
+   // Basic validation
+   REQUIRE(ldof_marker.Size() == fespace.GetVSize());
+   REQUIRE(ess_tdof_list.Size() >= 0);
+
+   // The output arrays share a single indexing, so they must have equal size.
+   REQUIRE(boundary_edge_ldofs.Size() == dof_edges.Size());
+   REQUIRE(dof_edges.Size() == dof_boundary_elements.Size());
+
+   // Verify all boundary edge DOFs are marked in ldof_marker
+   for (int dof : boundary_edge_ldofs)
+   {
+      REQUIRE(ldof_marker[dof] == 1);
+   }
+}
+
+// Helper function to compute boundary loop length
+real_t ComputeBoundaryLoopLength(ParMesh* pmesh, const Array<int>& dof_edges)
+{
+   real_t local_length = 0.0;
+   std::unordered_set<int> processed_edges;
+
+   for (int i = 0; i < dof_edges.Size(); i++)
+   {
+      int edge_id = dof_edges[i];
+      if (!processed_edges.insert(edge_id).second) { continue; }
+
+      Array<int> edge_verts;
+      pmesh->GetEdgeVertices(edge_id, edge_verts);
+      const real_t* v0 = pmesh->GetVertex(edge_verts[0]);
+      const real_t* v1 = pmesh->GetVertex(edge_verts[1]);
+
+      real_t edge_length = 0.0;
+      for (int d = 0; d < pmesh->SpaceDimension(); d++)
+      {
+         real_t diff = v1[d] - v0[d];
+         edge_length += diff * diff;
+      }
+      local_length += sqrt(edge_length);
+   }
+   return local_length;
+}
+
+TEST_CASE("BoundaryEdgeDOFsNestedCubes",
+          "[Parallel][ParMesh][BoundaryEdgeDOFs]")
+{
+   const int order = GENERATE(1, 2);
+
+   // Expected processor-invariant results for nested cubes mesh (1 refinement)
+   // order=1: 16 tdofs, sum=16.0, length=2.0
+   // order=2: 32 tdofs, sum=32.0, length=2.0
+   int exp_tdofs = (order == 1) ? 16 : 32;
+   real_t exp_sum = (order == 1) ? real_t(16.0) : real_t(32.0);
+   real_t exp_length = real_t(2.0);
+
+   struct BoundaryTest
+   {
+      int attr_value;
+      Vector normal;
+      std::string name;
+   };
+
+   std::vector<BoundaryTest> boundary_tests =
+   {
+      {7, Vector({0, 0, -1}), "-z"},
+      {8, Vector({0, 0, 1}), "+z"},
+      {9, Vector({0, -1, 0}), "-y"},
+      {10, Vector({1, 0, 0}), "+x"},
+      {11, Vector({0, 1, 0}), "+y"},
+      {12, Vector({-1, 0, 0}), "-x"}
+   };
+
+   const char* mesh_file = "../../data/nested_cubes.msh";
+   Mesh mesh(mesh_file, 1, 1);
+   mesh.UniformRefinement();
+   ParMesh pmesh(MPI_COMM_WORLD, mesh);
+
+   ND_FECollection fec(order, 3);
+   ParFiniteElementSpace fespace(&pmesh, &fec);
+
+   int num_procs;
+   MPI_Comm_size(MPI_COMM_WORLD, &num_procs);
+
+   for (const auto& test : boundary_tests)
+   {
+      CAPTURE(test.name, test.attr_value, order, num_procs);
+
+      std::vector<Array<int>> attr_to_elements;
+      Array<int> bdr_attrs(1);
+      bdr_attrs[0] = test.attr_value;
+
+      fespace.GetBoundaryElementsByAttribute(bdr_attrs, attr_to_elements);
+      Array<int> boundary_elements = attr_to_elements[0];
+
+      Array<int> ess_tdof_list;
+      Array<int> ldof_marker;
+      Array<int> boundary_edge_ldofs;
+      Array<int> dof_edges, dof_boundary_elements, ess_edge_list;
+
+      fespace.GetBoundaryLoopEdgeDofs(boundary_elements, ess_tdof_list,
+                                      boundary_edge_ldofs, &ldof_marker, &dof_edges,
+                                      &dof_boundary_elements, &ess_edge_list);
+
+      Array<int> dof_orientations;
+      fespace.ComputeLoopEdgeOrientations(dof_edges, dof_boundary_elements,
+                                          test.normal, dof_orientations);
+
+      ParGridFunction x(&fespace);
+      x = real_t(0.0);
+      for (int i = 0; i < boundary_edge_ldofs.Size(); i++)
+      {
+         x(boundary_edge_ldofs[i]) = real_t(1.0) * dof_orientations[i];
+      }
+
+      GroupCommunicator *gc = fespace.ScalarGroupComm();
+      Array<int> global_marker(ldof_marker);
+      gc->Reduce<int>(global_marker.GetData(), GroupCommunicator::BitOR<int>);
+      gc->Bcast(global_marker);
+
+      Array<real_t> values(x.GetData(), x.Size());
+      gc->ReduceBegin(values.GetData());
+      gc->ReduceMarked<real_t>(values.GetData(), global_marker, 0,
+                               GroupCommunicator::MaxAbs<real_t>);
+      gc->Bcast(values.GetData());
+      delete gc;
+
+      Vector x_true;
+      x.GetTrueDofs(x_true);
+
+      int local_nonzero_tdofs = 0;
+      real_t local_tdof_sum = 0.0;
+      for (int tdof = 0; tdof < x_true.Size(); tdof++)
+      {
+         real_t tdof_value = x_true(tdof);
+         if (abs(tdof_value) > 1e-12)
+         {
+            local_nonzero_tdofs++;
+            local_tdof_sum += abs(tdof_value);
+         }
+      }
+
+      real_t local_length = ComputeBoundaryLoopLength(&pmesh, dof_edges);
+
+      int global_nonzero_tdofs;
+      real_t global_tdof_sum, total_length;
+      MPI_Allreduce(&local_nonzero_tdofs, &global_nonzero_tdofs, 1, MPI_INT, MPI_SUM,
+                    MPI_COMM_WORLD);
+      MPI_Allreduce(&local_tdof_sum, &global_tdof_sum, 1,
+                    MPITypeMap<real_t>::mpi_type, MPI_SUM,
+                    MPI_COMM_WORLD);
+      MPI_Allreduce(&local_length, &total_length, 1,
+                    MPITypeMap<real_t>::mpi_type, MPI_SUM,
+                    MPI_COMM_WORLD);
+
+      // Verify processor-invariant results match expected values
+      REQUIRE(global_nonzero_tdofs == exp_tdofs);
+      REQUIRE(abs(global_tdof_sum - exp_sum) < real_t(1e-12));
+      REQUIRE(abs(total_length - exp_length) < real_t(1e-12));
+   }
+}
+
+TEST_CASE("BoundaryEdgeDOFs2DSquareInSquare",
+          "[Parallel][ParMesh][BoundaryEdgeDOFs]")
+{
+   // Test 2D boundary edge DOF extraction using square-in-square mesh
+   constexpr int order = 2;
+
+   // Test multiple inner boundary attributes
+   std::vector<int> inner_attrs_to_test = {5, 6, 7, 8};
+
+   // Load 2D square-in-square mesh from file
+   const char* mesh_file = "../../data/square_in_square.msh";
+   Mesh serial_mesh(mesh_file, 1, 1);
+   serial_mesh.UniformRefinement();
+
+   int num_procs = Mpi::WorldSize();
+
+   // Test each boundary attribute
+   for (int inner_attr : inner_attrs_to_test)
+   {
+      CAPTURE(inner_attr); // Capture the attribute being tested for better test output
+
+      // Test that results are consistent across different mesh partitionings
+      const int n_elements = serial_mesh.GetNE();
+
+      // Generate multiple different partitionings
+      std::vector<std::vector<int>> all_partitionings;
+
+      // 1. All elements on rank 0
+      all_partitionings.push_back(std::vector<int>(n_elements, 0));
+
+      if (num_procs > 1)
+      {
+         // 2. Block partition: first half on rank 0, second half on last rank
+         std::vector<int> block(n_elements);
+         for (int i = 0; i < n_elements; i++)
+         {
+            block[i] = (i < n_elements/2) ? 0 : num_procs-1;
+         }
+         all_partitionings.push_back(block);
+
+         // 3. Round-robin partition: elements assigned cyclically to all ranks
+         std::vector<int> round_robin(n_elements);
+         for (int i = 0; i < n_elements; i++)
+         {
+            round_robin[i] = i % num_procs;
+         }
+         all_partitionings.push_back(round_robin);
+      }
+
+      ND_FECollection fec(order, 2);
+      std::vector<int> all_dof_results;
+      all_dof_results.reserve(all_partitionings.size());
+
+      // Test each partitioning
+      for (const auto& partition : all_partitionings)
+      {
+         // Create parallel mesh with current partitioning
+         Mesh test_mesh(mesh_file, 1, 1);
+         test_mesh.UniformRefinement();
+         ParMesh pmesh = (num_procs == 1) ?
+                         ParMesh(MPI_COMM_WORLD, test_mesh) :
+                         ParMesh(MPI_COMM_WORLD, test_mesh, partition.data());
+
+         ParFiniteElementSpace fespace(&pmesh, &fec);
+
+         // Find boundary elements with the inner attribute
+         std::vector<Array<int>> attr_to_elements;
+         Array<int> inner_attrs(1);
+         inner_attrs[0] = inner_attr;
+         fespace.GetBoundaryElementsByAttribute(inner_attrs, attr_to_elements);
+
+         Array<int> inner_boundary_elements = attr_to_elements[0];
+
+         Array<int> ess_tdofs, ess_edges;
+         Array<int> boundary_dofs;
+         Array<int> dof_edges, dof_boundary_elements;
+
+         fespace.GetBoundaryLoopEdgeDofs(inner_boundary_elements, ess_tdofs,
+                                         boundary_dofs, nullptr, &dof_edges,
+                                         &dof_boundary_elements, &ess_edges);
+
+         // The output arrays share one indexing, so their sizes must match.
+         REQUIRE(boundary_dofs.Size() == dof_edges.Size());
+         REQUIRE(dof_edges.Size() == dof_boundary_elements.Size());
+
+         // Gather global counts for this partitioning
+         int local_dof_count = boundary_dofs.Size();
+         int global_dof_count;
+         MPI_Allreduce(&local_dof_count, &global_dof_count, 1, MPI_INT, MPI_SUM,
+                       MPI_COMM_WORLD);
+
+         all_dof_results.push_back(global_dof_count);
+      }
+
+      // Verify all partitionings give identical results
+      REQUIRE(!all_dof_results.empty());
+
+      int expected_dofs = all_dof_results[0];
+      for (int result : all_dof_results)
+      {
+         REQUIRE(result == expected_dofs);
+      }
+   } // End of inner_attr loop
+}
+
+TEST_CASE("BoundaryEdgeDOFsSharedDOFsAreOwnedBySomeRank",
+          "[Parallel][ParMesh][BoundaryEdgeDOFs]")
+{
+   // Every selected shared DOF must appear in exactly one rank's ess_tdof_list.
+   // Only the group master owns the corresponding true DOF and returns a
+   // non-negative value from GetLocalTDofNumber(), so if the master holds none
+   // of the selected boundary elements the DOF would be emitted by no rank at
+   // all unless the local marker is synchronized across the sharing group.
+   const int nranks = Mpi::WorldSize();
+   if (nranks < 2) { return; }
+
+   constexpr int order = 1;
+   ND_FECollection fec(order, 3);
+
+   for (int orientation : {1, 3, 5})
+   {
+      Mesh probe = OrientedTriFaceMesh(orientation, true);
+      probe.UniformRefinement();
+      const int ne = probe.GetNE();
+
+      // Several partitionings, to vary which rank masters each shared group
+      std::vector<std::vector<int>> partitionings;
+      {
+         std::vector<int> round_robin(ne), block(ne), strided(ne);
+         for (int i = 0; i < ne; i++)
+         {
+            round_robin[i] = i % nranks;
+            block[i] = (i < ne/2) ? 0 : nranks-1;
+            strided[i] = (i * 7 + 3) % nranks;
+         }
+         partitionings = {round_robin, block, strided};
+      }
+
+      for (const auto &partition : partitionings)
+      {
+         Mesh mesh = OrientedTriFaceMesh(orientation, true);
+         mesh.UniformRefinement();
+         ParMesh pmesh(MPI_COMM_WORLD, mesh, partition.data());
+         ParFiniteElementSpace fes(&pmesh, &fec);
+
+         const int bdr_attr = pmesh.bdr_attributes.Max();
+         Array<int> bdr_attrs(1);
+         bdr_attrs[0] = bdr_attr;
+         std::vector<Array<int>> attr_to_elements;
+         fes.GetBoundaryElementsByAttribute(bdr_attrs, attr_to_elements);
+         Array<int> bdr_elements = attr_to_elements[0];
+
+         Array<int> ess_tdofs;
+         Array<int> boundary_dofs;
+         fes.GetBoundaryLoopEdgeDofs(bdr_elements, ess_tdofs, boundary_dofs);
+
+         // Identify DOFs by global true DOF number, which is agreed upon by all
+         // ranks sharing the DOF, then compare the set selected anywhere with
+         // the set actually emitted in ess_tdof_list.
+         std::set<HYPRE_BigInt> selected, emitted;
+         for (int dof : boundary_dofs)
+         {
+            selected.insert(fes.GetGlobalTDofNumber(dof));
+         }
+         for (int i = 0; i < ess_tdofs.Size(); i++)
+         {
+            emitted.insert(fes.GetMyTDofOffset() + ess_tdofs[i]);
+         }
+
+         auto all_gather = [nranks](const std::set<HYPRE_BigInt> &s)
+         {
+            std::vector<HYPRE_BigInt> local(s.begin(), s.end());
+            int n = static_cast<int>(local.size()), total = 0;
+            std::vector<int> counts(nranks), bytes(nranks), displs(nranks);
+            MPI_Allgather(&n, 1, MPI_INT, counts.data(), 1, MPI_INT,
+                          MPI_COMM_WORLD);
+            constexpr int sz = sizeof(HYPRE_BigInt);
+            for (int r = 0; r < nranks; r++)
+            {
+               displs[r] = total * sz;
+               total += counts[r];
+               bytes[r] = counts[r] * sz;
+            }
+            std::vector<HYPRE_BigInt> all(total);
+            MPI_Allgatherv(local.data(), n * sz, MPI_BYTE, all.data(),
+                           bytes.data(), displs.data(), MPI_BYTE,
+                           MPI_COMM_WORLD);
+            return std::set<HYPRE_BigInt>(all.begin(), all.end());
+         };
+
+         // Gather both sets across all ranks. global_selected is every shared
+         // boundary DOF chosen on any rank; global_emitted is every true DOF
+         // actually placed in some rank's ess_tdof_list. A selected DOF missing
+         // from global_emitted is one that no rank owns and outputs, which is
+         // exactly the synchronization bug this test guards against.
+         const std::set<HYPRE_BigInt> global_selected = all_gather(selected);
+         const std::set<HYPRE_BigInt> global_emitted = all_gather(emitted);
+
+         int num_missing = 0;
+         for (auto gtdof : global_selected)
+         {
+            if (!global_emitted.count(gtdof)) { num_missing++; }
+         }
+
+         CAPTURE(orientation, nranks, global_selected.size(),
+                 global_emitted.size(), num_missing);
+         REQUIRE(num_missing == 0);
+      }
+   }
+}
+
+TEST_CASE("BoundaryEdgeDOFs2DLoopVertexDOFsPartitionInvariant",
+          "[Parallel][ParMesh][BoundaryEdgeDOFs]")
+{
+   // A closed boundary loop split between ranks must give the same result as
+   // the serial code. With a collection carrying vertex DOFs (ND_R2D), a vertex
+   // shared by two boundary segments is interior to the loop and must be
+   // dropped. When the two segments live on different ranks, each rank sees the
+   // vertex only once locally, so the occurrence parity has to be reconciled
+   // across the sharing group.
+   if (Mpi::WorldSize() < 2) { return; }
+
+   constexpr int order = 1;
+   ND_R2D_FECollection fec(order, 2);
+
+   // Serial reference result
+   Mesh serial_mesh = Mesh::MakeCartesian2D(2, 2, Element::QUADRILATERAL, false,
+                                            1.0, 1.0);
+   FiniteElementSpace serial_fes(&serial_mesh, &fec);
+
+   Array<int> serial_bdr_elements(serial_mesh.GetNBE());
+   for (int i = 0; i < serial_bdr_elements.Size(); i++)
+   {
+      serial_bdr_elements[i] = i;
+   }
+
+   Array<int> serial_boundary_dofs;
+
+   serial_fes.GetBoundaryLoopEdgeDofs(serial_bdr_elements, serial_boundary_dofs);
+
+   const int serial_count = serial_boundary_dofs.Size();
+
+   // Compare against several partitionings of the same mesh
+   const int num_procs = Mpi::WorldSize();
+   std::vector<std::vector<int>> partitionings;
+   {
+      Mesh probe = Mesh::MakeCartesian2D(2, 2, Element::QUADRILATERAL, false,
+                                         1.0, 1.0);
+      const int ne = probe.GetNE();
+
+      std::vector<int> block(ne), round_robin(ne);
+      for (int i = 0; i < ne; i++)
+      {
+         block[i] = (i < ne/2) ? 0 : num_procs-1;
+         round_robin[i] = i % num_procs;
+      }
+      partitionings.push_back(block);
+      partitionings.push_back(round_robin);
+   }
+
+   for (const auto &partition : partitionings)
+   {
+      Mesh mesh = Mesh::MakeCartesian2D(2, 2, Element::QUADRILATERAL, false,
+                                        1.0, 1.0);
+      ParMesh pmesh(MPI_COMM_WORLD, mesh, partition.data());
+      ParFiniteElementSpace pfes(&pmesh, &fec);
+
+      Array<int> local_bdr_elements(pmesh.GetNBE());
+      for (int i = 0; i < local_bdr_elements.Size(); i++)
+      {
+         local_bdr_elements[i] = i;
+      }
+
+      Array<int> ess_tdofs;
+      Array<int> local_boundary_dofs;
+      pfes.GetBoundaryLoopEdgeDofs(local_bdr_elements, ess_tdofs,
+                                   local_boundary_dofs);
+
+      // The true DOFs are owned by exactly one rank each, so summing the local
+      // counts gives a partition-independent global count.
+      int local_tdofs = ess_tdofs.Size();
+      int global_tdofs = 0;
+      MPI_Allreduce(&local_tdofs, &global_tdofs, 1, MPI_INT, MPI_SUM,
+                    MPI_COMM_WORLD);
+
+      CAPTURE(num_procs, serial_count, global_tdofs);
+      REQUIRE(global_tdofs == serial_count);
+   }
+}
+
+TEST_CASE("GroupCommunicatorReduceMarkedByGroupStride",
+          "[Parallel][GroupCommunicator]")
+{
+   // Regression test for the neighbor-major stride of the byGroup receive
+   // buffer: with more than one DOF in a group, the contributions to DOF i are
+   // at buf[j*nldofs + i], so reducing a single marked DOF must gather the
+   // strided values rather than reading a contiguous run.
+   const int rank = Mpi::WorldRank();
+   const int nranks = Mpi::WorldSize();
+   if (nranks < 3) { return; }
+
+   ListOfIntegerSets groups;
+
+   IntegerSet local_group(1);
+   local_group[0] = rank;
+   groups.Insert(local_group);
+
+   IntegerSet shared_group(nranks);
+   for (int r = 0; r < nranks; r++)
+   {
+      shared_group[r] = r;
+   }
+   groups.Insert(shared_group);
+
+   GroupTopology topology(MPI_COMM_WORLD);
+   topology.Create(groups, 4983);
+
+   GroupCommunicator comm(topology, GroupCommunicator::byGroup);
+   // Two DOFs in the same shared group, so the buffer stride is 2.
+   Array<int> ldof_group(2);
+   ldof_group = 1;
+   comm.Create(ldof_group);
+
+   Array<real_t> values(2);
+   values[0] = real_t(10.0) * rank + real_t(1.0);
+   values[1] = real_t(100.0) * rank + real_t(2.0);
+
+   Array<int> marker(2);
+   marker = 1;
+
+   comm.ReduceBegin(values.GetData());
+   comm.ReduceMarked<real_t>(values.GetData(), marker, 0,
+                             GroupCommunicator::Sum<real_t>);
+   comm.Bcast(values);
+
+   const real_t rank_sum = real_t(nranks) * real_t(nranks - 1) / real_t(2.0);
+   REQUIRE(values[0] == MFEM_Approx(real_t(10.0) * rank_sum + real_t(nranks)));
+   REQUIRE(values[1] == MFEM_Approx(real_t(100.0) * rank_sum +
+                                    real_t(2.0) * real_t(nranks)));
+}
+
+TEST_CASE("GroupCommunicatorMaxAbs", "[Parallel][GroupCommunicator]")
+{
+   const int rank = Mpi::WorldRank();
+   const int nranks = Mpi::WorldSize();
+   if (nranks < 2) { return; }
+
+   ListOfIntegerSets groups;
+
+   IntegerSet local_group(1);
+   local_group[0] = rank;
+   groups.Insert(local_group);
+
+   IntegerSet shared_group(nranks);
+   for (int r = 0; r < nranks; r++)
+   {
+      shared_group[r] = r;
+   }
+   groups.Insert(shared_group);
+
+   GroupTopology topology(MPI_COMM_WORLD);
+   topology.Create(groups, 4983);
+
+   GroupCommunicator comm(topology, GroupCommunicator::byGroup);
+   Array<int> ldof_group(2);
+   ldof_group = 1;
+   comm.Create(ldof_group);
+
+   // The group master (rank 0) reduces the peers' contributions into its own.
+   Array<real_t> values(2);
+   // DOF 0: equal magnitude across ranks with opposite signs, with the negative
+   // value held by the master, so the opposite-sign tie must still resolve
+   // deterministically to the positive value.
+   values[0] = (rank == 0) ? real_t(-5.0) : real_t(5.0);
+   // DOF 1: the largest magnitude is negative and held by a peer, so it must
+   // win over the master's smaller positive value and keep its sign.
+   values[1] = (rank == nranks - 1) ? real_t(-10.0) : real_t(5.0);
+
+   comm.Reduce<real_t>(values.GetData(), GroupCommunicator::MaxAbs<real_t>);
+   comm.Bcast(values);
+
+   REQUIRE(values[0] == MFEM_Approx(real_t(5.0)));
+   REQUIRE(values[1] == MFEM_Approx(real_t(-10.0)));
+}
+
+#endif // MFEM_USE_MPI

tests/unit/fem/test_lin_interp.cpp

@@ -1069,4 +1069,238 @@ TEST_CASE("Exact Sequence Properties: d(df)=0",
    }
 }
 
+template <class A, class B>
+static void TestCurl(FiniteElementSpace &dom_fes, FiniteElementSpace &ran_fes,
+                     A coeff, B dcoeff)
+{
+   real_t tol = 1e-10;
+   DiscreteLinearOperator CurlFA(&dom_fes, &ran_fes);
+   CurlFA.AddDomainInterpolator(new CurlInterpolator());
+   CurlFA.Assemble();
+   CurlFA.Finalize();
+
+   SparseMatrix &Curl = CurlFA.SpMat();
+   GridFunction x(&dom_fes), y_fa(&ran_fes), y(&ran_fes);
+   x.ProjectCoefficient(coeff);
+   y.ProjectCoefficient(dcoeff);
+   REQUIRE(x.Size() == Curl.Width());
+   REQUIRE(y_fa.Size() == Curl.Height());
+   Curl.Mult(x, y_fa);
+   y_fa -= y;
+   REQUIRE(y_fa.Normlinf() < tol);
+}
+
+template<class Coeff, class TCoeff>
+static void CompareCurlPA(FiniteElementSpace& dom_fes,
+                          FiniteElementSpace &ran_fes,
+                          Coeff coeff, TCoeff tcoeff)
+{
+   real_t tol = 1e-10;
+   DiscreteLinearOperator CurlFA(&dom_fes, &ran_fes);
+   CurlFA.AddDomainInterpolator(new CurlInterpolator());
+   CurlFA.Assemble();
+   CurlFA.Finalize();
+   DiscreteLinearOperator CurlPA(&dom_fes, &ran_fes);
+   CurlPA.AddDomainInterpolator(new CurlInterpolator());
+   CurlPA.SetAssemblyLevel(AssemblyLevel::PARTIAL);
+   CurlPA.Assemble();
+
+   SparseMatrix &Curl = CurlFA.SpMat();
+   GridFunction x(&dom_fes), y_fa(&ran_fes), y_pa(&ran_fes);
+   x.ProjectCoefficient(coeff);
+   REQUIRE(x.Size() == Curl.Width());
+   REQUIRE(y_fa.Size() == Curl.Height());
+   REQUIRE(x.Size() == CurlPA.Width());
+   REQUIRE(y_pa.Size() == CurlPA.Height());
+   Curl.Mult(x, y_fa);
+   CurlPA.Mult(x, y_pa);
+   y_pa -= y_fa;
+   REQUIRE(y_pa.Normlinf() < tol);
+   // transpose
+   y_fa.ProjectCoefficient(tcoeff);
+   GridFunction x_fa(&dom_fes), x_pa(&dom_fes);
+   Curl.MultTranspose(y_fa, x_fa);
+   CurlPA.MultTranspose(y_fa, x_pa);
+   x_pa -= x_fa;
+   REQUIRE(x_pa.Normlinf() < tol);
+}
+
+TEST_CASE("Partial Assemble Linear Interpolator",
+          "[CurlInterpolator]"
+          "[GPU]")
+{
+   constexpr int maxOrder = 3;
+   auto order = GENERATE_COPY(range(1, maxOrder + 1));
+   CAPTURE(order);
+
+   auto dim = GENERATE(2, 3);
+   CAPTURE(dim);
+
+   int n = 3;
+
+   Mesh mesh;
+
+   switch (dim)
+   {
+      case 2:
+         mesh =
+            Mesh::MakeCartesian2D(n, n, Element::QUADRILATERAL, true, 2.0, 3.0);
+         break;
+      case 3:
+         mesh = Mesh::MakeCartesian3D(n, n, n, Element::HEXAHEDRON, 2.0, 3.0, 5.0);
+         break;
+   }
+
+   // domain spaces
+   H1_FECollection fec_h1(order, dim);
+   FiniteElementSpace fespace_h1(&mesh, &fec_h1);
+   ND_FECollection fec_nd(order, dim);
+   FiniteElementSpace fespace_nd(&mesh, &fec_nd);
+
+   // range spaces
+   RT_FECollection fec_rt(order - 1, dim);
+   FiniteElementSpace fespace_rt(&mesh, &fec_rt);
+   L2_FECollection fec_l2(order - 1, dim, BasisType::GaussLegendre,
+                          FiniteElement::INTEGRAL);
+   FiniteElementSpace fespace_l2(&mesh, &fec_l2);
+
+   switch (dim)
+   {
+      case 2:
+      {
+         FunctionCoefficient coeff([](const Vector &x)
+         { return sin(2 * M_PI * x[1] / 3) - cos(2 * M_PI * x[0] / 2); });
+         VectorFunctionCoefficient vcoeff(2, [](const Vector &x, Vector &y)
+         {
+            y.SetSize(2);
+            y[0] = -cos(2 * M_PI * x[1] / 3);
+            y[1] = sin(2 * M_PI * x[0] / 2);
+         });
+         // out of plane H1 -> in-plane RT
+         SECTION("H1 to RT")
+         {
+            CompareCurlPA(fespace_h1, fespace_rt, coeff, vcoeff);
+         }
+         // in-plane ND -> out of plane L2
+         SECTION("ND to L2")
+         {
+            CompareCurlPA(fespace_nd, fespace_l2, vcoeff, coeff);
+         }
+         break;
+      }
+      case 3:
+      {
+         VectorFunctionCoefficient coeff(3, [](const Vector &x, Vector &y)
+         {
+            y.SetSize(3);
+            y[0] = sin(2 * M_PI * x[2] / 5) - cos(2 * M_PI * x[1] / 3);
+            y[1] = sin(2 * M_PI * x[0] / 2) - cos(2 * M_PI * x[2] / 5);
+            y[2] = sin(2 * M_PI * x[1] / 3) - cos(2 * M_PI * x[0] / 2);
+         });
+         CompareCurlPA(fespace_nd, fespace_rt, coeff, coeff);
+         break;
+      }
+   }
+}
+
+TEST_CASE("Curl Linear Interpolator",
+          "[CurlInterpolator]"
+          "[GPU]")
+{
+   int order = 2;
+
+   auto type = (Element::Type)GENERATE(range((int)Element::TRIANGLE,
+                                             (int)Element::PYRAMID + 1));
+   CAPTURE(type);
+
+   int n = 3;
+
+   Mesh mesh;
+
+   int dim;
+
+   if (type < (int)Element::TETRAHEDRON)
+   {
+      dim = 2;
+      mesh = Mesh::MakeCartesian2D(n, n, (Element::Type)type, 1, 2.0, 3.0);
+   }
+   else
+   {
+      dim = 3;
+      mesh = Mesh::MakeCartesian3D(n, n, n, (Element::Type)type,
+                                   2.0, 3.0, 5.0);
+   }
+
+   // domain spaces
+   H1_FECollection fec_h1(order, dim);
+   FiniteElementSpace fespace_h1(&mesh, &fec_h1);
+   ND_FECollection fec_nd(order, dim);
+   FiniteElementSpace fespace_nd(&mesh, &fec_nd);
+
+   // range spaces
+   RT_FECollection fec_rt(order - 1, dim);
+   FiniteElementSpace fespace_rt(&mesh, &fec_rt);
+   L2_FECollection fec_l2(order - 1, dim, BasisType::GaussLegendre,
+                          FiniteElement::INTEGRAL);
+   FiniteElementSpace fespace_l2(&mesh, &fec_l2);
+
+   switch (dim)
+   {
+      case 2:
+      {
+         // out of plane H1 -> in-plane RT
+         SECTION("H1 to RT")
+         {
+            FunctionCoefficient coeff([](const Vector &x)
+            {
+               return 1 - 2 * x[0] + 3 * x[1];
+            });
+            VectorFunctionCoefficient dcoeff(2, [](const Vector &x, Vector &y)
+            {
+               y.SetSize(2);
+               // d Ez/dy
+               y[0] = 3;
+               // -d Ez/dx
+               y[1] = 2;
+            });
+
+            TestCurl(fespace_h1, fespace_rt, coeff, dcoeff);
+         }
+         // in-plane ND -> out of plane L2
+         SECTION("ND to L2")
+         {
+            VectorFunctionCoefficient coeff(2, [](const Vector &x, Vector &y)
+            {
+               y.SetSize(2);
+               y[0] = 1 - 2 * x[0] + 3 * x[1];
+               y[1] = 2 * (1 - 2 * x[0] + 3 * x[1]);
+            });
+            FunctionCoefficient dcoeff([](const Vector &x)
+            { return 2 * (-2) - 3; });
+            TestCurl(fespace_nd, fespace_l2, coeff, dcoeff);
+         }
+         break;
+      }
+      case 3:
+      {
+         VectorFunctionCoefficient coeff(3, [](const Vector &x, Vector &y)
+         {
+            y.SetSize(3);
+            y[0] = 1 + 2 * x[0] - 3 * x[1] + 4 * x[2];
+            y[1] = 4 + 3 * x[0] - 2 * x[1] + 1 * x[2];
+            y[2] = 2 - 1 * x[0] + 4 * x[1] - 3 * x[2];
+         });
+         VectorFunctionCoefficient dcoeff(3, [](const Vector &x, Vector &y)
+         {
+            y.SetSize(3);
+            y[0] = 4 - 1;
+            y[1] = 4 + 1;
+            y[2] = 3 + 3;
+         });
+         TestCurl(fespace_nd, fespace_rt, coeff, dcoeff);
+         break;
+      }
+   }
+}
+
 } // namespace lin_interp

tests/unit/fem/test_lor_batched.cpp

@@ -214,7 +214,14 @@ TEST_CASE("LOR AMS", "[LOR][BatchedLOR][AMS][Parallel][GPU]")
    ParFiniteElementSpace vert_fespace(edge_fespace.GetParMesh(), &vert_fec);
 
    ParDiscreteLinearOperator grad(&vert_fespace, &edge_fespace);
-   grad.AddDomainInterpolator(new GradientInterpolator);
+   if (space_type == RT)
+   {
+      grad.AddDomainInterpolator(new CurlInterpolator);
+   }
+   else
+   {
+      grad.AddDomainInterpolator(new GradientInterpolator);
+   }
    grad.Assemble();
    grad.Finalize();
    std::unique_ptr<HypreParMatrix> G(grad.ParallelAssemble());

tests/unit/fem/test_pa_coeff.cpp

@@ -750,6 +750,89 @@ TEST_CASE("Hcurl/Hdiv Mixed PA Coefficient",
    }
 }
 
+TEST_CASE("Hcurl/Hdiv MixedVectorGradientPA",
+          "[GPU][PartialAssembly][Coefficient]")
+{
+   constexpr real_t tol = 4e-12;
+   dimension = GENERATE(2, 3);
+   // no coeff, scalar coeff, diagonal matrix coeff, full matrix coeff
+   auto coeffType = GENERATE(0, 1, 2, 3);
+   auto order = GENERATE(1, 2, 3);
+   // RT, ND
+   auto vFEType = GENERATE(0, 1);
+   CAPTURE(dimension, coeffType, order, vFEType);
+
+   const int ne = 3;
+   Mesh mesh = MakeCartesianNonaligned(dimension, ne);
+
+   H1_FECollection scalar_fec(order, dimension);
+   FiniteElementSpace s_fespace(&mesh, &scalar_fec);
+
+   std::unique_ptr<FiniteElementCollection> vector_fec;
+
+   switch (vFEType)
+   {
+      case 0:
+         vector_fec.reset(new RT_FECollection(order - 1, dimension));
+         break;
+      case 1:
+         vector_fec.reset(new ND_FECollection(order, dimension));
+         break;
+   }
+   FiniteElementSpace v_fespace(&mesh, vector_fec.get());
+
+   MixedBilinearForm pa_form(&s_fespace, &v_fespace);
+   pa_form.SetAssemblyLevel(AssemblyLevel::PARTIAL);
+   MixedBilinearForm fa_form(&s_fespace, &v_fespace);
+
+   std::unique_ptr<Coefficient> coeff;
+   std::unique_ptr<DiagonalMatrixCoefficient> dq_coeff;
+   std::unique_ptr<MatrixCoefficient> mq_coeff;
+   switch (coeffType)
+   {
+      case 0:
+         pa_form.AddDomainIntegrator(new MixedVectorGradientIntegrator);
+         fa_form.AddDomainIntegrator(new MixedVectorGradientIntegrator);
+         break;
+      case 1:
+         coeff.reset(new FunctionCoefficient(&coeffFunction));
+         pa_form.AddDomainIntegrator(new MixedVectorGradientIntegrator(*coeff));
+         fa_form.AddDomainIntegrator(new MixedVectorGradientIntegrator(*coeff));
+         break;
+      case 2:
+         dq_coeff.reset(new VectorFunctionCoefficient(dimension, &vectorCoeffFunction));
+         pa_form.AddDomainIntegrator(new MixedVectorGradientIntegrator(*dq_coeff));
+         fa_form.AddDomainIntegrator(new MixedVectorGradientIntegrator(*dq_coeff));
+         break;
+      case 3:
+         mq_coeff.reset(new MatrixFunctionCoefficient(
+                           dimension, &asymmetricMatrixCoeffFunction));
+         pa_form.AddDomainIntegrator(new MixedVectorGradientIntegrator(*mq_coeff));
+         fa_form.AddDomainIntegrator(new MixedVectorGradientIntegrator(*mq_coeff));
+         break;
+   }
+   pa_form.Assemble();
+   fa_form.Assemble();
+
+   GridFunction x_fa(&s_fespace), y_fa(&v_fespace), y_pa(&v_fespace);
+   x_fa.Randomize(1234);
+   REQUIRE(x_fa.Size() == pa_form.Width());
+   REQUIRE(x_fa.Size() == fa_form.Width());
+   REQUIRE(y_fa.Size() == fa_form.Height());
+   REQUIRE(y_pa.Size() == pa_form.Height());
+   pa_form.Mult(x_fa, y_pa);
+   fa_form.Mult(x_fa, y_fa);
+   y_pa -= y_fa;
+   REQUIRE(y_pa.Normlinf() <= tol);
+
+   GridFunction x_pa(&s_fespace);
+   y_fa.Randomize(1234);
+   pa_form.MultTranspose(y_fa, x_pa);
+   fa_form.MultTranspose(y_fa, x_fa);
+   x_pa -= x_fa;
+   REQUIRE(x_pa.Normlinf() <= tol);
+}
+
 TEST_CASE("3D Bilinear VectorFE Integrators PartialAssembly",
           "[BilinearFormIntegrator]"
           "[PartialAssembly]"
@@ -1059,4 +1142,111 @@ TEST_CASE("3D Bilinear VectorFE Integrators PartialAssembly",
    }
 }
 
+TEST_CASE("3D Bilinear Weak Curl Integrators Partial Assembly",
+          "[MixedVectorWeakCurlIntegrator]"
+          "[BilinearFormIntegrator]"
+          "[PartialAssembly]"
+          "[GPU]")
+{
+   auto order = GENERATE(1, 2);
+   CAPTURE(order);
+   int dim = 3;
+
+   FunctionCoefficient q3_coeff(coeffFunction);
+   VectorFunctionCoefficient F3_coeff(dim, vectorCoeffFunction);
+
+   auto mesh_fname =
+      GENERATE("../../data/fichera-amr.mesh", "../../data/ball-nurbs.mesh");
+   CAPTURE(mesh_fname);
+   Mesh mesh(mesh_fname);
+   REQUIRE(mesh.Dimension() == dim);
+   REQUIRE(mesh.SpaceDimension() == dim);
+
+   // convert nurbs into piecewise-quadratic curved mesh
+   if (mesh.NURBSext)
+   {
+      mesh.UniformRefinement();
+      mesh.SetCurvature(2);
+   }
+
+   SECTION("RT to ND No Coeff")
+   {
+      ND_FECollection fec_nd(order, dim);
+      FiniteElementSpace fespace_nd(&mesh, &fec_nd);
+      RT_FECollection fec_rt(order - 1, dim);
+      FiniteElementSpace fespace_rt(&mesh, &fec_rt);
+
+      MixedBilinearForm bfa(&fespace_rt, &fespace_nd);
+      bfa.AddDomainIntegrator(new MixedVectorWeakCurlIntegrator);
+      bfa.Assemble();
+      bfa.Finalize();
+
+      MixedBilinearForm bpa(&fespace_rt, &fespace_nd);
+      bpa.SetAssemblyLevel(AssemblyLevel::PARTIAL);
+      bpa.AddDomainIntegrator(new MixedVectorWeakCurlIntegrator);
+      bpa.Assemble();
+
+      GridFunction x(&fespace_rt), y_fa(&fespace_nd), y_pa(&fespace_nd);
+      x.Randomize(1234);
+      REQUIRE(bfa.Height() == y_fa.Size());
+      REQUIRE(bfa.Width() == x.Size());
+      REQUIRE(bpa.Height() == y_fa.Size());
+      REQUIRE(bpa.Width() == x.Size());
+      bfa.Mult(x, y_fa);
+      bpa.Mult(x, y_pa);
+      y_pa -= y_fa;
+      REQUIRE( y_pa.Normlinf() == MFEM_Approx(0_r) );
+   }
+
+   SECTION("RT to ND Scalar Coeff")
+   {
+      ND_FECollection fec_nd(order, dim);
+      FiniteElementSpace fespace_nd(&mesh, &fec_nd);
+      RT_FECollection fec_rt(order - 1, dim);
+      FiniteElementSpace fespace_rt(&mesh, &fec_rt);
+
+      MixedBilinearForm bfa(&fespace_rt, &fespace_nd);
+      bfa.AddDomainIntegrator(new MixedVectorWeakCurlIntegrator(q3_coeff));
+      bfa.Assemble();
+      bfa.Finalize();
+
+      MixedBilinearForm bpa(&fespace_rt, &fespace_nd);
+      bpa.SetAssemblyLevel(AssemblyLevel::PARTIAL);
+      bpa.AddDomainIntegrator(new MixedVectorWeakCurlIntegrator(q3_coeff));
+      bpa.Assemble();
+
+      GridFunction x(&fespace_rt), y_fa(&fespace_nd), y_pa(&fespace_nd);
+      x.Randomize(1234);
+      bfa.Mult(x, y_fa);
+      bpa.Mult(x, y_pa);
+      y_pa -= y_fa;
+      REQUIRE( y_pa.Normlinf() == MFEM_Approx(0_r) );
+   }
+
+   SECTION("RT to ND Diagonal Matrix Coeff")
+   {
+      ND_FECollection fec_nd(order, dim);
+      FiniteElementSpace fespace_nd(&mesh, &fec_nd);
+      RT_FECollection fec_rt(order - 1, dim);
+      FiniteElementSpace fespace_rt(&mesh, &fec_rt);
+
+      MixedBilinearForm bfa(&fespace_rt, &fespace_nd);
+      bfa.AddDomainIntegrator(new MixedVectorWeakCurlIntegrator(F3_coeff));
+      bfa.Assemble();
+      bfa.Finalize();
+
+      MixedBilinearForm bpa(&fespace_rt, &fespace_nd);
+      bpa.SetAssemblyLevel(AssemblyLevel::PARTIAL);
+      bpa.AddDomainIntegrator(new MixedVectorWeakCurlIntegrator(F3_coeff));
+      bpa.Assemble();
+
+      GridFunction x(&fespace_rt), y_fa(&fespace_nd), y_pa(&fespace_nd);
+      x.Randomize(1234);
+      bfa.Mult(x, y_fa);
+      bpa.Mult(x, y_pa);
+      y_pa -= y_fa;
+      REQUIRE( y_pa.Normlinf() == MFEM_Approx(0_r) );
+   }
+}
+
 } // namespace pa_coeff

tests/unit/linalg/test_complex_operator.cpp

@@ -164,12 +164,29 @@ TEST_CASE("ComplexHypreParMatrix GetSystemMatrix",
    a.AddDomainIntegrator(new VectorFEMassIntegrator(one),
                          new VectorFEMassIntegrator(one));
    a.Assemble();
+
+   // 2. Test ParSesquilinearForm::FormSystemMatrix directly and verify that
+   //    essential entries on the imaginary diagonal are zero.
    OperatorPtr Ah;
+   a.FormSystemMatrix(ess_tdof_list, Ah);
+   ComplexHypreParMatrix *A_complex = Ah.Is<ComplexHypreParMatrix>();
+   REQUIRE(A_complex != nullptr);
+   Vector diag;
+   A_complex->imag().GetDiag(diag);
+   const Array<int> &ess_tdofs = ess_tdof_list;
+   const Vector &diag_h = diag;
+   ess_tdofs.HostRead();
+   diag_h.HostRead();
+   for (const int tdof : ess_tdofs)
+   {
+      REQUIRE(diag_h[tdof] == 0.0);
+   }
+
+   // 3. Test the call to ComplexHypreParMatrix::GetSystemMatrix and destroying
+   //    the returned matrix.
    Vector B, X;
    a.FormLinearSystem(ess_tdof_list, x, b, Ah, X, B);
 
-   // 2. Test the call to ComplexHypreParMatrix::GetSystemMatrix and destroying
-   //    the returned matrix.
    HypreParMatrix *A = Ah.As<ComplexHypreParMatrix>()->GetSystemMatrix();
    delete A;
 }

tests/unit/mesh/test_bb_grid_map.cpp

@@ -152,7 +152,7 @@ TEST_CASE("GlobalBBoxTensorGridMap Parallel",
    std::map<int, std::vector<int>> pt_to_procs;
    map.MapPointsToProcs(centers, 1, pt_to_procs);
 
-   REQUIRE(pt_to_procs.size() == nel + 1);
+   REQUIRE(pt_to_procs.size() == (unsigned)nel + 1);
    for (int i = 0; i < nel; i++)
    {
       std::vector<int> procs = pt_to_procs[i];

tests/unit/mesh/test_ncmesh.cpp

@@ -304,6 +304,66 @@ TEST_CASE("pNCMesh PA diagonal",  "[Parallel], [NCMesh]")
    }
 } // test case
 
+TEST_CASE("ParNCMesh Rebalance preserves element attributes",
+          "[Parallel], [NCMesh]")
+{
+   const int rank = Mpi::WorldRank();
+   const int nranks = Mpi::WorldSize();
+   if (nranks < 2) { return; }
+
+   auto mesh_fname = GENERATE("../../data/star.mesh",
+                              "../../data/fichera.mesh");
+   CAPTURE(mesh_fname);
+
+   auto CheckRebalance = [rank, nranks, mesh_fname](bool refine,
+                                                    bool custom_partition)
+   {
+      Mesh mesh(mesh_fname);
+      mesh.EnsureNCMesh();
+      ParMesh pmesh(MPI_COMM_WORLD, mesh);
+
+      const int attribute = 1234 + (custom_partition ? rank : 0);
+      for (int i = 0; i < pmesh.GetNE(); i++)
+      {
+         pmesh.SetAttribute(i, attribute);
+      }
+      pmesh.SetAttributes();
+
+      if (refine)
+      {
+         Array<int> refinements;
+         if (pmesh.GetNE() && (custom_partition || rank == 0))
+         {
+            refinements.Append(0);
+         }
+         pmesh.GeneralRefinement(refinements);
+      }
+
+      int expected_attribute = attribute;
+      if (custom_partition)
+      {
+         // Move every element to the next rank, as in GitHub issue #4009.
+         Array<int> partition(pmesh.GetNE());
+         partition = (rank + 1) % nranks;
+         pmesh.Rebalance(partition);
+         expected_attribute = 1234 + (rank + nranks - 1) % nranks;
+      }
+      else
+      {
+         pmesh.Rebalance();
+      }
+
+      for (int i = 0; i < pmesh.GetNE(); i++)
+      {
+         CHECK(pmesh.GetAttribute(i) == expected_attribute);
+      }
+   };
+
+   SECTION("Custom partition, unrefined") { CheckRebalance(false, true); }
+   SECTION("Custom partition, refined") { CheckRebalance(true, true); }
+   SECTION("Default partition, refined") { CheckRebalance(true, false); }
+}
+
 TEST_CASE("EdgeFaceConstraint", "[Parallel], [NCMesh]")
 {
    auto exact_soln = [](const Vector& x)