minor bug

remove comments
fixing bugs
2023-05-31 11:00:58 -07:00 · 2023-04-07 18:43:48 -07:00 · 2023-04-07 18:43:09 -07:00 · 2023-04-04 16:40:28 -07:00 · 2023-04-04 12:14:57 -07:00 · 2023-04-04 00:09:53 -07:00
210 changed files with 11310 additions and 3543 deletions
@@ -62,7 +62,7 @@ jobs:

      - name: GHCR Login
        if: (github.event_name != 'pull_request')
-        uses: docker/login-action@v1
+        uses: docker/login-action@v2
        with:
          registry: ghcr.io
          username: ${{ github.actor }}
@@ -94,7 +94,7 @@ jobs:
    # This external action allows to interrupt a workflow already running on
    # the same branch to save resource
    - name: Cancel Previous Runs
-      uses: styfle/cancel-workflow-action@0.9.0
+      uses: styfle/cancel-workflow-action@0.11.0
      with:
        access_token: ${{ github.token }}

@@ -102,7 +102,7 @@ jobs:
    # /home/runner/work/mfem/mfem/mfem
    # Note: Done now to access "install-hypre" and "install-metis" actions.
    - name: checkout mfem
-      uses: actions/checkout@v2
+      uses: actions/checkout@v3
      with:
        path: ${{ env.MFEM_TOP_DIR }}
        # Fetch the complete history for codecov to access commits ID
@@ -115,25 +115,25 @@ jobs:
      if: matrix.mpi == 'par' && matrix.os == 'ubuntu-latest'
      run: |
        sudo apt-get install mpich libmpich-dev
-        export MAKE_CXX_FLAG="MPICXX=mpic++"

    - name: get lcov (Linux)
      if: matrix.codecov == 'YES' && matrix.os == 'ubuntu-latest'
      run: |
        sudo apt-get install lcov

-    - name: Set up Homebrew
-      if: ( matrix.mpi == 'par' || matrix.codecov == 'YES' ) && matrix.os == 'macos-latest'
-      uses: Homebrew/actions/setup-homebrew@c4aafe8c4620bf08883dd4679c374f11e73329d3
+    # Keep the following section in case we need it again in the future,
+    # see: https://github.com/mfem/mfem/pull/3385#discussion_r1058013032
+    # - name: Set up Homebrew
+    #   if: ( matrix.mpi == 'par' || matrix.codecov == 'YES' ) && matrix.os == 'macos-latest'
+    #   uses: Homebrew/actions/setup-homebrew@master

    - name: get MPI (MacOS)
      if: matrix.mpi == 'par' && matrix.os == 'macos-latest'
      run: |
        export HOMEBREW_NO_INSTALL_CLEANUP=1
        brew install openmpi
-        export MAKE_CXX_FLAG="MPICXX=mpic++"

-    - name: get MPI (MacOS)
+    - name: get lcov (MacOS)
      if: matrix.codecov == 'YES' && matrix.os == 'macos-latest'
      run: |
        export HOMEBREW_NO_INSTALL_CLEANUP=1
@@ -141,14 +141,14 @@ jobs:

    - name: get MPI (Windows)
      if: matrix.mpi == 'par' && matrix.os == 'windows-latest'
-      uses: mpi4py/setup-mpi@v1.0.3
+      uses: mpi4py/setup-mpi@v1.1.4

    # Get Hypre through cache, or build it.
    # Install will only run on cache miss.
    - name: cache hypre
      id: hypre-cache
      if: matrix.mpi == 'par'
-      uses: actions/cache@v2
+      uses: actions/cache@v3
      with:
        path: ${{ env.HYPRE_TOP_DIR }}
        key: ${{ runner.os }}-build-${{ env.HYPRE_TOP_DIR }}-${{ matrix.hypre-target }}-v2.2
@@ -176,7 +176,7 @@ jobs:
    - name: cache metis
      id: metis-cache
      if: matrix.mpi == 'par' && matrix.os != 'windows-latest'
-      uses: actions/cache@v2
+      uses: actions/cache@v3
      with:
        path: ${{ env.METIS_TOP_DIR }}
        key: ${{ runner.os }}-build-${{ env.METIS_TOP_DIR }}-v2.2
@@ -190,12 +190,13 @@ jobs:

    - name: cache vcpkg (Windows)
      id: vcpkg-cache
+      if: matrix.os == 'windows-latest'
      uses: actions/cache@v3
      with:
        path: vcpkg_cache
        key: ${{ runner.os }}-${{ matrix.mpi }}-vcpkg-v1

-    - name: prepare binary cache location
+    - name: prepare vcpkg binary cache location (Windows)
      if: matrix.os == 'windows-latest' && steps.vcpkg-cache.outputs.cache-hit != 'true'
      run: |
        mkdir -p vcpkg_cache
@@ -230,11 +231,7 @@ jobs:
      run: |
        cd ${{ env.MFEM_TOP_DIR }}  && make check

-    - name: unit tests
-      if: matrix.build-system == 'make' && matrix.target == 'opt'
-      run: |
-        cd ${{ env.MFEM_TOP_DIR }} && make unittest
-
+    # Note: 'tests' include the unit tests
    - name: tests
      if: matrix.build-system == 'make' && matrix.target == 'opt'
      run: |
@@ -247,7 +244,7 @@ jobs:
        cd ${{ env.MFEM_TOP_DIR }} && cmake --build build --target check --config ${CTEST_CONFIG}
      shell: bash

-    - name: cmake unit tests (Ubuntu 20.04)
+    - name: cmake unit tests (Ubuntu)
      if: matrix.build-system == 'cmake' && matrix.target == 'opt' && matrix.os == 'ubuntu-latest'
      run: |
        CTEST_CONFIG="Release"
@@ -265,7 +262,7 @@ jobs:
    # Code coverage (process and upload reports)
    - name: codecov
      if: matrix.codecov == 'YES'
-      uses: mfem/github-actions/upload-coverage@v2.0
+      uses: mfem/github-actions/upload-coverage@v2.2
      with:
        name: ${{ matrix.os }}-${{ matrix.build-system }}-${{ matrix.target }}-${{ matrix.mpi }}-${{ matrix.hypre-target }}
        project_dir: ${{ env.MFEM_TOP_DIR }}
@@ -35,23 +35,22 @@ jobs:

    steps:
    - name: Cancel Previous Runs
-      uses: styfle/cancel-workflow-action@0.9.0
+      uses: styfle/cancel-workflow-action@0.11.0
      with:
        access_token: ${{ github.token }}

    - name: checkout MFEM
-      uses: actions/checkout@v2
+      uses: actions/checkout@v3
      with:
        path: mfem

    - name: Get MPI (Linux)
      run: |
        sudo apt-get install mpich libmpich-dev
-        export MAKE_CXX_FLAG="MPICXX=mpic++"

    - name: Cache Hypre Install
      id: hypre-cache
-      uses: actions/cache@v2
+      uses: actions/cache@v3
      with:
        path: ${{ env.HYPRE_TOP_DIR }}
        key: ${{ runner.os }}-build-${{ env.HYPRE_TOP_DIR }}-v2.2
@@ -66,7 +65,7 @@ jobs:

    - name: Cache Metis Install
      id: metis-cache
-      uses: actions/cache@v2
+      uses: actions/cache@v3
      with:
        path: ${{ env.METIS_TOP_DIR }}
        key: ${{ runner.os }}-build-${{ env.METIS_TOP_DIR }}-v2.2
@@ -34,12 +34,12 @@ jobs:
       github.event.pull_request.head.repo.full_name != github.repository)
    steps:
    - name: Cancel Previous Runs
-      uses: styfle/cancel-workflow-action@0.9.0
+      uses: styfle/cancel-workflow-action@0.11.0
      with:
        access_token: ${{ github.token }}

    - name: checkout mfem
-      uses: actions/checkout@v2
+      uses: actions/checkout@v3

    - name: copyright check
      id: copyright
@@ -84,7 +84,7 @@ jobs:
       github.event.pull_request.head.repo.full_name != github.repository)
    steps:
    - name: checkout mfem
-      uses: actions/checkout@v2
+      uses: actions/checkout@v3

    - name: get astyle
      run: |
@@ -101,11 +101,14 @@ jobs:
       github.event.pull_request.head.repo.full_name != github.repository)
    steps:
    - name: checkout mfem
-      uses: actions/checkout@v2
+      uses: actions/checkout@v3

    - name: get doxygen and graphviz
      run: |
        sudo apt-get install doxygen graphviz
+
+    - name: update doxygen config file
+      run: |
        cd doc
        doxygen -u CodeDocumentation.conf.in

@@ -123,7 +126,7 @@ jobs:
    runs-on: ubuntu-latest
    steps:
    - name: checkout mfem
-      uses: actions/checkout@v2
+      uses: actions/checkout@v3
      with:
        fetch-depth: 0

@@ -131,6 +131,12 @@ examples/hiop/ex9-mesh.*
 examples/hiop/ex9-init.*
 examples/hiop/ex9-final.*

+examples/ipopt/exContactBlockTL
+examples/ipopt/exContactBlockTL.mesh
+examples/ipopt/exContactBlockTL-mesh.*
+examples/ipopt/exContactBlockTL-init.*
+examples/ipopt/exContactBlockTL-final.*
+
 examples/petsc/ex[1-69]p
 examples/petsc/ex1[0-1]p
 examples/petsc/mesh.*
@@ -203,6 +209,7 @@ miniapps/meshing/mesh-explorer
 miniapps/meshing/shaper
 miniapps/meshing/extruder
 miniapps/meshing/trimmer
+miniapps/meshing/reflector
 miniapps/meshing/mesh-optimizer
 miniapps/meshing/pmesh-optimizer
 miniapps/meshing/minimal-surface
@@ -214,9 +221,12 @@ miniapps/meshing/toroid-*.mesh
 miniapps/meshing/twist-*.mesh
 miniapps/meshing/mesh-explorer.mesh
 miniapps/meshing/partitioning.txt
+miniapps/meshing/mesh-explorer-visit*
+miniapps/meshing/mesh-explorer-paraview/
 miniapps/meshing/shaper.mesh
 miniapps/meshing/extruder.mesh
 miniapps/meshing/trimmer.mesh
+miniapps/meshing/reflected.mesh
 miniapps/meshing/optimized*
 miniapps/meshing/perturbed*
 miniapps/meshing/polar-nc.mesh
@@ -276,6 +286,7 @@ miniapps/tools/convert-dc
 miniapps/tools/lor-transfer
 miniapps/tools/get-values
 miniapps/tools/check-tmop-metric
+miniapps/tools/tmop-metric-magnitude

 miniapps/toys/automata
 miniapps/toys/life
@@ -93,7 +93,7 @@ report_baseline:
          git pull && \
          git add ${rundir} && \
          git commit -m "${msg}" && \
-          git push origin master
+          ${CI_PROJECT_DIR}/.gitlab/scripts/git_try_to_push
        else
          for file in ${rundir}/*; do
            echo "------------------------------"
@@ -0,0 +1,41 @@
+#!/bin/bash
+
+# Copyright (c) 2010-2022, Lawrence Livermore National Security, LLC. Produced
+# at the Lawrence Livermore National Laboratory. All Rights reserved. See files
+# LICENSE and NOTICE for details. LLNL-CODE-806117.
+#
+# This file is part of the MFEM library. For more information and source code
+# availability visit https://mfem.org.
+#
+# MFEM is free software; you can redistribute it and/or modify it under the
+# terms of the BSD-3 license. We welcome feedback and contributions, see file
+# CONTRIBUTING.md for details.
+
+# Try to push to the remote 5 times. If the push fails, and the local and remote
+# have diverged, then pull from the remote to merge changes, and try pushing
+# again. If some other failure happens
+for i in {1..5}; do
+   git push origin master && exit 0
+   # Wait for 20 seconds in case someone else is pushing to the remote
+   # concurrently
+   sleep 20
+   # Fetch any updates from the remote
+   git remote update
+   # Get the latest commit on the local branch
+   LOCAL=$(git rev-parse @)
+   # Get the latest commit on the remote
+   REMOTE=$(git rev-parse @{u})
+   # Get the common ancestor
+   BASE=$(git merge-base @ @{u})
+   # Have the local and remote diverged?
+   if [[ $LOCAL != $REMOTE && $LOCAL != $BASE && $REMOTE != $BASE ]]; then
+      git pull
+      if [[ $? == 0 ]]; then
+         continue
+      else
+         exit 1 # Something else went wrong trying to pull
+      fi
+   fi
+done
+
+exit 1 # Did not succeed in 5 attempts
@@ -32,7 +32,7 @@ if [[ "$AUTOTEST_COMMIT" != "NO" ]]; then
  git pull && \
  git add ${rundir} && \
  git commit -m "${msg}" && \
-  git push origin master
+  ${CI_PROJECT_DIR}/.gitlab/scripts/git_try_to_push
 else
  for file in ${rundir}/*; do
    echo "------------------------------"
@@ -29,7 +29,7 @@ if [[ "$AUTOTEST_COMMIT" != "NO" ]]; then
  git pull && \
  git add ${rundir} && \
  git commit -m "${msg}" && \
-  git push origin master
+  ${CI_PROJECT_DIR}/.gitlab/scripts/git_try_to_push
 else
  for file in ${rundir}/*; do
    echo "------------------------------"
@@ -8,6 +8,71 @@
                               https://mfem.org


+Version 4.5.1 (development)
+===========================
+- When using discontinuous (L2) spaces, use local (element-wise) L2 projection
+  as the coarsening operator for non-conforming AMR meshes.
+
+Meshing improvements
+--------------------
+- Added support for pyramids in non-conforming meshes. Currently only isotropic
+  refinement is supported in this case.
+
+- Updated logic in FindPointsGSLIB to ignore points found near (but outside) the
+  domain boundary.
+
+- Added support for pyramids in Gmsh meshes.
+
+- Fixed a bug in TMOP metric 301.
+
+- Added an option to auto-balance compound TMOP metrics.
+
+Discretization improvements
+---------------------------
+- TBD
+
+Linear and nonlinear solvers
+----------------------------
+- Added a fast normalization-based distance solver, see the Distance miniapp
+  in the miniapps/shifted/ directory.
+
+New and updated examples and miniapps
+-------------------------------------
+- Added a new meshing miniapp, reflector, which reflects a high-order or NURBS
+  hexahedral mesh about a plane.
+
+- The mesh-explorer miniapp can now save mesh files in the VisIt or ParaView
+  formats using the corresponding DataCollection objects. See option 'D' in the
+  main menu.
+
+Integrations, testing and documentation
+---------------------------------------
+- Removed the support for the Mesquite toolkit. We recommend using MFEM's TMOP
+  functionality instead for mesh optimization. See the mesh-optimizer miniapp.
+
+- The following integrations have updated minimum version requirements:
+  * RAJA >= 2022.10.3
+
+Miscellaneous
+-------------
+- VisItDataCollection now correctly handles collection names containing
+  underscores.
+
+- VisItDataCollection::SetPadDigits() no longer alters the number of digits
+  used to represent the MPI rank because VisIt seems to require 6 digits.
+  This parameter can still be explicitly overridden with
+  VisItDataCollection::SetPadDigitsRank().
+
+API changes
+-----------
+- The implicit cast methods of class Vector to 'double *' and 'const double *'
+  have been deprecated and generate deprecation warnings if used. They will be
+  removed in a future release.
+
+- The methods Mesh::GetFaceBaseGeometry and Mesh::GetFaceGeometryType have been
+  deprecated, and Mesh::GetFaceGeometry (which provides identical functionality)
+  should be used instead.
+
 Version 4.5, released on October 22, 2022
 =========================================

@@ -51,6 +116,9 @@ Discretization improvements
 - Added a class CoefficientVector for efficient access of variable coefficient
  values at quadrature points (in particular for GPU/device kernels).

+- Added support for GridFunction::GetGradients() and
+  GriFunction::GetVectorGradient() on face-neighbor elements.
+
 - Added WhiteGaussianNoiseDomainLFIntegrator: a LinearFormIntegrator class for
  spatial Gaussian white noise.

@@ -10,7 +10,9 @@
 # CONTRIBUTING.md for details.

 # The variable CMAKE_CXX_STANDARD and related were introduced in CMake v3.1
-cmake_minimum_required(VERSION 3.1)
+# Version 3.8 fixes the handling of CMAKE_CXX_STANDARD for try_compile.
+# Version 3.8 or newer is required for direct CUDA support.
+cmake_minimum_required(VERSION 3.8)
 set(USER_CONFIG "${CMAKE_CURRENT_SOURCE_DIR}/config/user.cmake" CACHE PATH
  "Path to optional user configuration file.")

@@ -51,7 +53,7 @@ project(mfem NONE)
 # Current version of MFEM, see also `makefile`.
 #   mfem_VERSION = (string)
 #   MFEM_VERSION = (int)   [automatically derived from mfem_VERSION]
-set(${PROJECT_NAME}_VERSION 4.5.0)
+set(${PROJECT_NAME}_VERSION 4.5.1)

 # Prohibit in-source build
 if (${PROJECT_SOURCE_DIR} STREQUAL ${PROJECT_BINARY_DIR})
@@ -66,8 +68,7 @@ set(XSDK_ENABLE_C OFF)
 set(XSDK_ENABLE_Fortran OFF)

 # Check if we need to enable C or Fortran.
-if (CMAKE_VERSION VERSION_LESS 3.2 OR
-    MFEM_USE_CONDUIT OR
+if (MFEM_USE_CONDUIT OR
    MFEM_USE_SIDRE OR
    MFEM_USE_PETSC)
   # This seems to be needed by:
@@ -81,11 +82,13 @@ if (MFEM_USE_STRUMPACK)
  # Just needed to find the MPI_Fortran libraries to link with
  set(XSDK_ENABLE_Fortran ON)
 endif()
-# SUNDIALS >= 6.4.0 requires C++14:
-if (MFEM_USE_SUNDIALS AND ("${CMAKE_CXX_STANDARD}" LESS "14"))
-  set(CMAKE_CXX_STANDARD 14)
-endif()
-if (MFEM_USE_GINKGO AND ("${CMAKE_CXX_STANDARD}" LESS "14"))
+# SUNDIALS, STRUMPACK, Ginkgo, RAJA and Umpire require C++14:
+if ((MFEM_USE_SUNDIALS OR
+     MFEM_USE_STRUMPACK OR
+     MFEM_USE_GINKGO OR
+     MFEM_USE_RAJA OR
+     MFEM_USE_UMPIRE) AND
+    ("${CMAKE_CXX_STANDARD}" LESS "14"))
  set(CMAKE_CXX_STANDARD 14)
 endif()

@@ -103,17 +106,12 @@ if (MFEM_USE_CUDA)
   if (MFEM_USE_HIP)
      message(FATAL_ERROR " *** MFEM_USE_HIP cannot be combined with MFEM_USE_CUDA.")
   endif()
-   # MFEM_USE_CUDA requires CMake 3.8 or newer (for direct CUDA support)
-   cmake_minimum_required(VERSION 3.8 FATAL_ERROR)
   # Use ${CMAKE_CXX_COMPILER} as the cuda host compiler.
   if (NOT CMAKE_CUDA_HOST_COMPILER)
      set(CMAKE_CUDA_HOST_COMPILER ${CMAKE_CXX_COMPILER})
   endif()
   enable_language(CUDA)
-   set(CMAKE_CUDA_STANDARD 11)
-   if (MFEM_USE_GINKGO)
-     set(CMAKE_CUDA_STANDARD 14)
-   endif()
+   set(CMAKE_CUDA_STANDARD ${CMAKE_CXX_STANDARD})
   set(CMAKE_CUDA_STANDARD_REQUIRED ON)
   set(CMAKE_CUDA_EXTENSIONS OFF)
   set(CUDA_FLAGS "--expt-extended-lambda")
@@ -136,8 +134,7 @@ if (MFEM_USE_CUDA)
      set(CUDA_FLAGS "-ccbin=${CMAKE_CXX_COMPILER} ${CUDA_FLAGS}")
      set(CMAKE_CUDA_HOST_LINK_LAUNCHER ${CMAKE_CXX_COMPILER})
   endif()
-   set(CMAKE_CUDA_FLAGS "${CUDA_FLAGS}" CACHE STRING
-       "CUDA flags set for MFEM" FORCE)
+   set(CMAKE_CUDA_FLAGS ${CMAKE_CUDA_FLAGS} ${CUDA_FLAGS})
   set(CUSPARSE_FOUND TRUE)
   set(CUSPARSE_LIBRARIES "cusparse")
   set(CUBLAS_FOUND TRUE)
@@ -199,6 +196,26 @@ if (MFEM_USE_HIP)
  find_package(HIPSPARSE REQUIRED)
 endif()

+# OpenMP
+if (MFEM_USE_OPENMP OR MFEM_USE_LEGACY_OPENMP)
+  if (NOT MFEM_THREAD_SAFE AND MFEM_USE_LEGACY_OPENMP)
+    message(FATAL_ERROR " *** MFEM_USE_LEGACY_OPENMP requires MFEM_THREAD_SAFE=ON.")
+  endif()
+  find_package(OpenMP REQUIRED)
+  set(OPENMP_LIBRARIES ${OpenMP_CXX_LIBRARIES})
+  if(APPLE)
+    # On macOS, the compiler needs additional help to find the <omp.h> header.
+    # See issue #2642 for more information.
+    set(OPENMP_INCLUDE_DIRS ${OpenMP_CXX_INCLUDE_DIRS})
+  endif(APPLE)
+  if (OPENMP_FOUND)
+    set(CMAKE_CXX_FLAGS "${CMAKE_CXX_FLAGS} ${OpenMP_CXX_FLAGS}")
+    if (MFEM_USE_CUDA)
+      set(CMAKE_CUDA_FLAGS "${CMAKE_CUDA_FLAGS} -Xcompiler=${OpenMP_CXX_FLAGS}")
+    endif()
+  endif()
+endif()
+
 # MPI -> hypre; PETSc (optional)
 if (MFEM_USE_MPI)
  find_package(MPI REQUIRED)
@@ -264,20 +281,6 @@ if (MFEM_USE_LAPACK)
  find_package(LAPACK REQUIRED)
 endif()

-# OpenMP
-if (MFEM_USE_OPENMP OR MFEM_USE_LEGACY_OPENMP)
-  if (NOT MFEM_THREAD_SAFE AND MFEM_USE_LEGACY_OPENMP)
-    message(FATAL_ERROR " *** MFEM_USE_LEGACY_OPENMP requires MFEM_THREAD_SAFE=ON.")
-  endif()
-  find_package(OpenMP REQUIRED)
-  set(OPENMP_LIBRARIES ${OpenMP_CXX_LIBRARIES})
-  if(APPLE)
-    # On macOS, the compiler needs additional help to find the <omp.h> header.
-    # See issue #2642 for more information.
-    set(OPENMP_INCLUDE_DIRS ${OpenMP_CXX_INCLUDE_DIRS})
-  endif(APPLE)
-endif()
-
 # SuiteSparse (before SUNDIALS which may depend on KLU)
 if (MFEM_USE_SUITESPARSE)
  find_package(SuiteSparse REQUIRED
@@ -296,11 +299,6 @@ if (MFEM_USE_SUNDIALS)
  find_package(SUNDIALS REQUIRED ${SUNDIALS_COMPONENTS})
 endif()

-# Mesquite
-if (MFEM_USE_MESQUITE)
-  find_package(Mesquite REQUIRED)
-endif()
-
 # SuperLU_DIST can only be enabled in parallel
 if (MFEM_USE_SUPERLU)
  if (MFEM_USE_MPI)
@@ -406,6 +404,15 @@ if (MFEM_USE_HIOP)
  # find_package updates HIOP_FOUND, HIOP_INCLUDE_DIRS, HIOP_LIBRARIES
 endif()

+# IpOpt optimizer
+if (MFEM_USE_IPOPT)
+  find_package(IPOPT REQUIRED)
+  message(
+      STATUS
+      "IPOPT_INCLUDE_DIRS=${IPOPT_INCLUDE_DIRS}, IPOPT_LIBRARIES=${IPOPT_LIBRARIES}, IPOPT_DIR=${IPOPT_DIR}")
+  # find_package updates IPOPT_FOUND, IPOPT_INCLUDE_DIRS, IPOPT_LIBRARIES
+endif()
+
 # CoDiPack package
 if (MFEM_USE_CODIPACK)
  find_package(CODIPACK REQUIRED)
@@ -488,6 +495,10 @@ endif()
 #   an ALIAS target is missing?
 # Call Stack (most recent call first):
 #   CMakeLists.txt:474 (mfem_add_library)
+#
+# NOTE: We need to figure out which TPL library adds the dependency on
+#       "Threads::Threads" and call the next line only when that TPL library is
+#       enabled.  -V. Dobrev
 find_package(Threads REQUIRED)

 # List all possible libraries in order of dependencies.
@@ -495,9 +506,9 @@ find_package(Threads REQUIRED)
 #    With newer versions of SuiteSparse which include METIS header using 64-bit
 #    integers, the METIS header (with 32-bit indices, as used by mfem) needs to
 #    be before SuiteSparse.
-set(MFEM_TPLS OPENMP HYPRE LAPACK BLAS SuperLUDist METIS SuiteSparse SUNDIALS
-    PETSC SLEPC MESQUITE MUMPS STRUMPACK AXOM FMS CONDUIT Ginkgo GNUTLS GSLIB
-    NETCDF MPFR PUMI HIOP POSIXCLOCKS MFEMBacktrace ZLIB OCCA CEED RAJA UMPIRE
+set(MFEM_TPLS OPENMP HYPRE LAPACK BLAS SuperLUDist STRUMPACK METIS SuiteSparse
+    SUNDIALS PETSC SLEPC MUMPS AXOM FMS CONDUIT Ginkgo GNUTLS GSLIB
+    NETCDF MPFR PUMI HIOP IPOPT POSIXCLOCKS MFEMBacktrace ZLIB OCCA CEED RAJA UMPIRE
    ADIOS2 CUBLAS CUSPARSE MKL_CPARDISO AMGX CALIPER CODIPACK BENCHMARK PARELAG
    MPI_CXX HIP HIPSPARSE MOONOLITH BLITZ ALGOIM ENZYME)

@@ -511,18 +522,12 @@ foreach(TPL IN LISTS MFEM_TPLS)
    list(APPEND TPL_INCLUDE_DIRS ${${TPL}_INCLUDE_DIRS})
  endif()
 endforeach(TPL)
+list(REVERSE TPL_LIBRARIES)
 list(REMOVE_DUPLICATES TPL_LIBRARIES)
+list(REVERSE TPL_LIBRARIES)
 list(REMOVE_DUPLICATES TPL_INCLUDE_DIRS)
 # message(STATUS "TPL_INCLUDE_DIRS = ${TPL_INCLUDE_DIRS}")

-if (OPENMP_FOUND)
-  message(STATUS "MFEM: using package OpenMP")
-  set(CMAKE_CXX_FLAGS "${CMAKE_CXX_FLAGS} ${OpenMP_CXX_FLAGS}")
-  if (MFEM_USE_CUDA)
-    set(CMAKE_CUDA_FLAGS "${CMAKE_CUDA_FLAGS} -Xcompiler=${OpenMP_CXX_FLAGS}")
-  endif()
-endif()
-
 message(STATUS "MFEM build type: CMAKE_BUILD_TYPE = ${CMAKE_BUILD_TYPE}")
 message(STATUS "MFEM version: v${MFEM_VERSION_STRING}")
 message(STATUS "MFEM git string: ${MFEM_GIT_STRING}")
@@ -701,6 +706,8 @@ add_subdirectory(doc)
 message(STATUS "CMAKE_INSTALL_PREFIX = ${CMAKE_INSTALL_PREFIX}")
 set(INSTALL_INCLUDE_DIR include
  CACHE PATH "Relative path for installing header files.")
+set(INSTALL_BIN_DIR bin
+  CACHE PATH "Relative path for installing the binaries.")
 set(INSTALL_LIB_DIR lib
  CACHE PATH "Relative path for installing the library.")
 # other options: "share/mfem/cmake", "lib/mfem/cmake"
@@ -719,7 +726,9 @@ set(CMAKE_INSTALL_DEFAULT_COMPONENT_NAME Development)
 # Install the library
 install(TARGETS ${PROJECT_NAME}
  EXPORT ${PROJECT_NAME_UC}Targets
-  DESTINATION ${INSTALL_LIB_DIR})
+  RUNTIME DESTINATION ${INSTALL_BIN_DIR}
+  LIBRARY DESTINATION ${INSTALL_LIB_DIR}
+  ARCHIVE DESTINATION ${INSTALL_LIB_DIR})

 # Install the master headers
 foreach(Header mfem.hpp mfem-performance.hpp)
@@ -112,6 +112,7 @@ The MFEM source code has the following structure:
  │   ├── caliper
  │   ├── ginkgo
  │   ├── hiop
+  │   ├── ipopt
  │   ├── jupyter
  │   ├── moonolith
  │   ├── petsc
@@ -337,10 +337,6 @@ MFEM_USE_SUNDIALS = YES/NO
   library. When enabled, this option uses the SUNDIALS_* library options,
   see below.

-MFEM_USE_MESQUITE = YES/NO
-   Enable MFEM functionality based on the Mesquite library. When enabled, this
-   option uses the MESQUITE_* library options, see below.
-
 MFEM_USE_SUITESPARSE = YES/NO
   Enable MFEM functionality based on the SuiteSparse library. Currently, this
   option adds the classes UMFPackSolver and KLUSolver (both sparse serial
@@ -475,6 +471,9 @@ MFEM_USE_HIOP = YES/NO
   Enable the usage of HiOp (https://github.com/LLNL/hiop) in MFEM. HiOp is an
   HPC solver for nonlinear optimization problems.

+MFEM_USE_IPOPT = YES/NO
+   Enable the usage of Ipopt in MFEM.
+
 MFEM_USE_CODIPACK = YES/NO
  Enable automatic differentiation using the CoDiPack library.
  www.scicomp.uni-kl.de/codi/
@@ -636,11 +635,6 @@ The specific libraries and their options are:
  Options: SUNDIALS_OPT, SUNDIALS_LIB.
  Versions: SUNDIALS >= 5.0.0, SUNDIALS >= 5.4.0 for CUDA support.

- Mesquite (optional), used when MFEM_USE_MESQUITE = YES.
-  URL: http://trilinos.org/oldsite/packages/mesquite
-  Options: MESQUITE_OPT, MESQUITE_LIB.
-  The Mesquite support is deprecated and will be removed in the future.
-
 - SuiteSparse (optional), used when MFEM_USE_SUITESPARSE = YES.
  URL: http://faculty.cse.tamu.edu/davis/suitesparse.html
  Options: SUITESPARSE_OPT, SUITESPARSE_LIB.
@@ -747,6 +741,11 @@ The specific libraries and their options are:
  Options: HIOP_OPT, HIOP_LIB.
  Versions: HIOP >= 0.4.6.

+- Ipopt (optional), used when MFEM_USE_IPOPT = YES.
+  URL: https://github.com/coin-or/Ipopt
+  Options: IPOPT_OPT, IPOPT_LIB.
+  Versions: IPOPT >= 3.14
+
 - CoDiPack (optional), used with MFEM_USE_CODIPACK = YES
  URL: https://www.scicomp.uni-kl.de/codi/
  Options: CODIPACK_OPT
@@ -802,10 +801,10 @@ The specific libraries and their options are:
  Versions: libCEED >= 0.10.

 - RAJA (optional), used when MFEM_USE_RAJA = YES.
-  Beginning with MFEM v4.3, only RAJA v0.14.0+ is supported.
+  Beginning with MFEM v4.5.1, only RAJA v2022.10.3+ is supported.
  URL: https://github.com/LLNL/RAJA
  Options: RAJA_DIR, RAJA_OPT, RAJA_LIB.
-  Versions: RAJA >= 0.14.0.
+  Versions: RAJA >= 2022.10.3.

 - Moonolith (optional), use when MFEM_USE_MOONOLITH = YES.
  URL: https://bitbucket.org/zulianp/par_moonolith
@@ -969,7 +968,6 @@ MFEM_USE_LEGACY_OPENMP
 MFEM_USE_OPENMP
 MFEM_USE_MEMALLOC
 MFEM_TIMER_TYPE - Set automatically, can be overwritten.
-MFEM_USE_MESQUITE
 MFEM_USE_SUITESPARSE
 MFEM_USE_SUPERLU
 MFEM_USE_MUMPS
@@ -982,6 +980,7 @@ MFEM_USE_MPFR
 MFEM_USE_ZLIB
 MFEM_USE_PUMI
 MFEM_USE_HIOP
+MFEM_USE_IPOPT
 MFEM_USE_CODIPACK
 MFEM_USE_ADFORWARD
 MFEM_USE_CUDA
@@ -1034,7 +1033,6 @@ The CMake build system adds auto-detection for the following packages/libraries:
 - HYPRE
 - METIS - The option MFEM_USE_METIS_5 is auto-detected.
 - ParMETIS
- - MESQUITE
 - SuiteSparse
 - SuperLUDist, STRUMPACK
 - Ginkgo
@@ -1046,6 +1044,7 @@ The CMake build system adds auto-detection for the following packages/libraries:
 - POSIXCLOCKS
 - PUMI
 - HIOP
+ - IPOPT
 - CoDiPack
 - OCCA
 - RAJA
@@ -212,10 +212,6 @@ IF (DEFINED TPL_ENABLE_SUNDIALS)
  SET(MFEM_USE_SUNDIALS ${TPL_ENABLE_SUNDIALS} CACHE BOOL "Enable SUNDIALS usage" FORCE)
 ENDIF()

-IF (DEFINED TPL_ENABLE_MESQUITE)
-  SET(MFEM_USE_MESQUITE ${TPL_ENABLE_MESQUITE} CACHE BOOL "Enable MESQUITE usage" FORCE)
-ENDIF()
-
 IF (DEFINED TPL_ENABLE_SUITESPARSE)
  SET(MFEM_USE_SUITESPARSE ${TPL_ENABLE_SUITESPARSE} CACHE BOOL "Enable SuiteSparse usage" FORCE)
 ENDIF()
@@ -29,7 +29,6 @@ set(MFEM_USE_LEGACY_OPENMP @MFEM_USE_LEGACY_OPENMP@)
 set(MFEM_USE_MEMALLOC @MFEM_USE_MEMALLOC@)
 set(MFEM_TIMER_TYPE @MFEM_TIMER_TYPE@)
 set(MFEM_USE_SUNDIALS @MFEM_USE_SUNDIALS@)
-set(MFEM_USE_MESQUITE @MFEM_USE_MESQUITE@)
 set(MFEM_USE_SUITESPARSE @MFEM_USE_SUITESPARSE@)
 set(MFEM_USE_SUPERLU @MFEM_USE_SUPERLU@)
 set(MFEM_USE_MUMPS @MFEM_USE_MUMPS@)
@@ -37,6 +36,7 @@ set(MFEM_USE_STRUMPACK @MFEM_USE_STRUMPACK@)
 set(MFEM_USE_GINKGO @MFEM_USE_GINKGO@)
 set(MFEM_USE_AMGX @MFEM_USE_AMGX@)
 set(MFEM_USE_HIOP @MFEM_USE_HIOP@)
+set(MFEM_USE_IPOPT @MFEM_USE_IPOPT@)
 set(MFEM_USE_GNUTLS @MFEM_USE_GNUTLS@)
 set(MFEM_USE_GSLIB @MFEM_USE_GSLIB@)
 set(MFEM_USE_NETCDF @MFEM_USE_NETCDF@)
@@ -77,9 +77,6 @@
 // Internal MFEM option: enable group/batch allocation for some small objects.
 #cmakedefine MFEM_USE_MEMALLOC

-// Enable MFEM functionality based on the Mesquite library.
-#cmakedefine MFEM_USE_MESQUITE
-
 // Enable MFEM functionality based on the SuiteSparse library.
 #cmakedefine MFEM_USE_SUITESPARSE

@@ -134,6 +131,9 @@
 // Enable MFEM functionality based on the HiOp library
 #cmakedefine MFEM_USE_HIOP

+// Enable MFEM functionality based on the Ipopt library
+#cmakedefine MFEM_USE_IPOPT
+
 // Build the GPU/CUDA-enabled version of the MFEM library.
 // Requires a CUDA compiler (nvcc).
 #cmakedefine MFEM_USE_CUDA
@@ -16,7 +16,7 @@

 include(MfemCmakeUtilities)
 mfem_find_package(Algoim ALGOIM ALGOIM_DIR
-       	"include" "algoim_quad.hpp"
+        "include;src" "algoim_quad.hpp"
       	"" ""
-	"Paths to headers required by Algoim."
-	"Libraries required by Algoim.")
+        "Paths to headers required by Algoim."
+        "Libraries required by Algoim.")
@@ -16,7 +16,22 @@

 include(MfemCmakeUtilities)
 mfem_find_package(Caliper CALIPER CALIPER_DIR
-       	"include" "caliper/cali.h"
-       	"lib" "caliper"
-  	"Paths to headers required by Caliper."
-       	"Libraries required by Caliper.")
+      "include" "caliper/cali.h"
+      "lib" "caliper"
+      "Paths to headers required by Caliper."
+      "Libraries required by Caliper.")
+
+# Append adiak path/lib if the user provided ADIAK_DIR
+if(ADIAK_DIR AND EXISTS ${ADIAK_DIR})
+    find_package(adiak NO_DEFAULT_PATH REQUIRED PATHS ${ADIAK_DIR}/lib/cmake/adiak ${ADIAK_DIR})
+    list(APPEND CALIPER_INCLUDE_DIRS ${adiak_INCLUDE_DIRS})
+    list(APPEND CALIPER_LIBRARIES ${adiak_LIBRARIES})
+endif()
+
+# Append gotcha path/lib if the user provided GOTCHA_DIR
+if(GOTCHA_DIR AND EXISTS ${GOTCHA_DIR})
+    find_package(gotcha NO_DEFAULT_PATH REQUIRED PATHS ${GOTCHA_DIR}/lib/cmake/gotcha ${GOTCHA_DIR})
+    list(APPEND CALIPER_INCLUDE_DIRS ${gotcha_INCLUDE_DIRS})
+    list(APPEND CALIPER_LIBRARIES ${gotcha_LIBRARIES})
+endif()
+
@@ -14,9 +14,21 @@
 #   - HDF5_LIBRARIES - The HDF5 libraries
 #   - HDF5_INCLUDE_DIRS - The HDF5 include directories

+# NOTE: Using this FindHDF5.cmake instead of the CMake provided version may lead
+#       to issues with some TPL libraries that depend (or may depend) on HDF5.
+#       For this reason, we should consider removing this file, or at least
+#       making it use the CMake provided version by default and apply the logic
+#       below only when specifically requested by a user.  -V. Dobrev
+
 # First Check for HDF5_DIR
 if(NOT HDF5_DIR)
-    MESSAGE(FATAL_ERROR "Could not find HDF5. HDF5 support needs explicit HDF5_DIR")
+  message(FATAL_ERROR
+    "Could not find HDF5. HDF5 support needs explicit HDF5_DIR")
+endif()
+
+if (NOT HDF5_FIND_QUIETLY)
+  message(STATUS "Looking for HDF5 ...")
+  message(STATUS "   in HDF5_DIR = ${HDF5_DIR}")
 endif()

 # Find includes
@@ -50,8 +62,9 @@ include(FindPackageHandleStandardArgs)

 # Handle the QUIETLY and REQUIRED arguments and set HDF5_FOUND to TRUE if all
 # listed variables are TRUE
-find_package_handle_standard_args(HDF5  DEFAULT_MSG
+find_package_handle_standard_args(HDF5
+                                  " *** HDF5 not found. Please set HDF5_DIR."
+                                  HDF5_LIBRARIES
                                  HDF5_INCLUDE_DIRS
                                  __HDF5_LIBRARY
-                                  __HDF5_HL_LIBRARY
-                                  HDF5_LIBRARIES )
+                                  __HDF5_HL_LIBRARY)
@@ -9,12 +9,15 @@
 # terms of the BSD-3 license. We welcome feedback and contributions, see file
 # CONTRIBUTING.md for details.

-# Defines the following variables:
-#   - MESQUITE_FOUND
-#   - MESQUITE_LIBRARIES
-#   - MESQUITE_INCLUDE_DIRS
+# Sets the following variables:
+#   - IPOPT_FOUND
+#   - IPOPT_INCLUDE_DIRS
+#   - IPOPT_LIBRARIES

 include(MfemCmakeUtilities)
-mfem_find_package(Mesquite MESQUITE MESQUITE_DIR
-  "include" "Mesquite_all_headers.hpp" "lib" "mesquite"
-  "Paths to headers required by Mesquite." "Libraries required by Mesquite.")
+mfem_find_package(IPOPT IPOPT IPOPT_DIR
+  "include" "IpTNLP.hpp"
+  "lib" "ipopt"
+  "Paths to headers required by IPOPT."
+  "Libraries required by IPOPT.")
+
@@ -17,18 +17,24 @@
 include(MfemCmakeUtilities)

 # FindHDF5.cmake uses HDF5_ROOT, so we "translate" from the MFEM convention
-set(HDF5_ROOT ${HDF5_DIR} CACHE PATH "")
+# (MFEM's FindHDF5.cmake does not need HDF5_ROOT)
+# set(HDF5_ROOT ${HDF5_DIR} CACHE PATH "")
+
 # We need to guard against the case where HDF5 was already found but without
 # the HL extensions (in which case mfem_find_package will treat the package
 # as already having been found), so we reset the variable to force FindHDF5.cmake
 # to be called for a second time
 set(HDF5_FOUND OFF)
 enable_language(C) # FindHDF5.cmake uses the C compiler
+
 mfem_find_package(NetCDF NETCDF NETCDF_DIR "include" netcdf.h "lib" netcdf
  "Paths to headers required by NetCDF." "Libraries required by NetCDF.")
+
 # The HL extension libraries are in a separate variable and must precede
 # the "regular" hdf5 library, as hdf5_hl depends on hdf5
 # The netcdf library will always be the first element of NETCDF_LIBRARIES
 # and we need to insert after that library but before the hdf5 library, so
 # position 1 is used
-list(INSERT NETCDF_LIBRARIES 1 ${HDF5_C_LIBRARY_hdf5_hl})
+# (MFEM's FindHDF5.cmake does not set HDF5_C_LIBRARY_hdf5_hl and the HL library
+# is already added to NETCDF_LIBRARIES)
+# list(INSERT NETCDF_LIBRARIES 1 ${HDF5_C_LIBRARY_hdf5_hl})
@@ -14,17 +14,27 @@
 #   - RAJA_LIBRARIES
 #   - RAJA_INCLUDE_DIRS

-include(MfemCmakeUtilities)
-mfem_find_package(RAJA RAJA RAJA_DIR "include" "RAJA/RAJA.hpp" "lib" "RAJA"
-  "Paths to headers required by RAJA." "Libraries required by RAJA.")
-
-if (NOT RAJA_CONFIG_CMAKE)
-   set(RAJA_CONFIG_CMAKE "${RAJA_DIR}/share/raja/cmake/raja-config.cmake")
+if (RAJA_FOUND)
+   return()
 endif()
-if (EXISTS "${RAJA_CONFIG_CMAKE}")
-   include("${RAJA_CONFIG_CMAKE}")
-   if (ENABLE_CUDA AND NOT MFEM_USE_CUDA)
-      message(FATAL_ERROR
-         "RAJA is built with CUDA: MFEM_USE_CUDA=YES is required")
+message(STATUS "Looking for RAJA ...")
+if (RAJA_DIR)
+   message(STATUS "   in RAJA_DIR = ${RAJA_DIR}")
+   find_package(RAJA CONFIG NO_DEFAULT_PATH PATHS "${RAJA_DIR}")
+endif()
+if (NOT RAJA_FOUND)
+   message(STATUS "   in standard CMake locations")
+   find_package(RAJA CONFIG)
+endif()
+if (RAJA_FOUND)
+   set(RAJA_LIBRARIES "RAJA" CACHE STRING "RAJA imported target." FORCE)
+   set(RAJA_INCLUDE_DIRS "" CACHE STRING "RAJA include dirs (not used)" FORCE)
+   message(STATUS
+      "Found RAJA target: ${RAJA_LIBRARIES} (version: ${RAJA_VERSION})")
+else()
+   set(msg STATUS)
+   if (RAJA_FIND_REQUIRED)
+      set(msg FATAL_ERROR)
   endif()
+   message(${msg} "RAJA not found. Please set RAJA_DIR to the RAJA prefix.")
 endif()
@@ -14,6 +14,23 @@
 #   - UMPIRE_LIBRARIES
 #   - UMPIRE_INCLUDE_DIRS

-find_package(umpire REQUIRED CONFIG)
+if (NOT umpire_DIR AND UMPIRE_DIR)
+  set(umpire_DIR ${UMPIRE_DIR}/lib/cmake/umpire)
+endif()
+message(STATUS "Looking for UMPIRE ...")
+message(STATUS "   in UMPIRE_DIR = ${UMPIRE_DIR}")
+message(STATUS "      umpire_DIR = ${umpire_DIR}")
+find_package(umpire CONFIG)
 set(UMPIRE_FOUND ${umpire_FOUND})
 set(UMPIRE_LIBRARIES "umpire")
+if (UMPIRE_FOUND)
+  message(STATUS
+    "Found UMPIRE target: ${UMPIRE_LIBRARIES} (version: ${umpire_VERSION})")
+else()
+  set(msg STATUS)
+  if (UMPIRE_FIND_REQUIRED)
+    set(msg FATAL_ERROR)
+  endif()
+  message(${msg}
+    "UMPIRE not found. Please set UMPIRE_DIR to the install prefix.")
+endif()
@@ -46,6 +46,10 @@ endfunction()
 # Wrapper for add_executable
 macro(mfem_add_executable NAME)
  add_executable(${NAME} ${ARGN})
+  if (MFEM_USE_CUDA)
+    set_target_properties(${NAME} PROPERTIES
+      CUDA_RESOLVE_DEVICE_SYMBOLS ON)
+  endif()
 endmacro()

 # Wrapper for add_library
@@ -158,27 +162,12 @@ macro(add_mfem_miniapp MFEM_EXE_NAME)

  # Append the additional libraries and options
  if (LIBRARIES_LIST)
-    if(CMAKE_VERSION VERSION_GREATER 2.8.11)
-      target_link_libraries(${MFEM_EXE_NAME} PRIVATE ${LIBRARIES_LIST})
-    else()
-      target_link_libraries(${MFEM_EXE_NAME} ${LIBRARIES_LIST})
-    endif()
+    target_link_libraries(${MFEM_EXE_NAME} PRIVATE ${LIBRARIES_LIST})
  endif()
  if (EXTRA_OPTIONS_LIST)
    string(REPLACE ";" " " EXTRA_OPTIONS_STRING "${EXTRA_OPTIONS_LIST}")
    message(STATUS "${MFEM_EXE_NAME}: add flags \"${EXTRA_OPTIONS_STRING}\"")
-    if(CMAKE_VERSION VERSION_GREATER 2.8.11)
-      target_compile_options(${MFEM_EXE_NAME} PRIVATE ${EXTRA_OPTIONS_LIST})
-    else()
-      get_target_property(THIS_COMPILE_FLAGS ${MFEM_EXE_NAME} COMPILE_FLAGS)
-      if (THIS_COMPILE_FLAGS)
-        set(THIS_COMPILE_FLAGS "${THIS_COMPILE_FLAGS} ${EXTRA_OPTIONS_STRING}")
-      else()
-        set(THIS_COMPILE_FLAGS "${EXTRA_OPTIONS_STRING}")
-      endif()
-      set_target_properties(${MFEM_EXE_NAME}
-        PROPERTIES COMPILE_FLAGS ${THIS_COMPILE_FLAGS})
-    endif()
+    target_compile_options(${MFEM_EXE_NAME} PRIVATE ${EXTRA_OPTIONS_LIST})
  endif()
  if (EXTRA_DEFINES_LIST)
    target_compile_definitions(${MFEM_EXE_NAME} PRIVATE ${EXTRA_DEFINES_LIST})
@@ -187,17 +176,15 @@ macro(add_mfem_miniapp MFEM_EXE_NAME)
  # Handle the MPI separately
  if (MFEM_USE_MPI)
    # Add MPI_CXX_LIBRARIES, in case this target does not link with mfem.
-    if(CMAKE_VERSION VERSION_GREATER 2.8.11)
-      target_link_libraries(${MFEM_EXE_NAME} PRIVATE ${MPI_CXX_LIBRARIES})
-    else()
-      target_link_libraries(${MFEM_EXE_NAME} ${MPI_CXX_LIBRARIES})
-    endif()
+    target_link_libraries(${MFEM_EXE_NAME} PRIVATE ${MPI_CXX_LIBRARIES})

    if (MPI_CXX_INCLUDE_PATH)
      target_include_directories(${MFEM_EXE_NAME} PRIVATE ${MPI_CXX_INCLUDE_PATH})
    endif()
    if (MPI_CXX_COMPILE_FLAGS)
-      target_compile_options(${MFEM_EXE_NAME} PRIVATE ${MPI_CXX_COMPILE_FLAGS})
+      separate_arguments(MPI_CXX_COMPILE_ARGS UNIX_COMMAND
+        "${MPI_CXX_COMPILE_FLAGS}")
+      target_compile_options(${MFEM_EXE_NAME} PRIVATE ${MPI_CXX_COMPILE_ARGS})
    endif()

    if (MPI_CXX_LINK_FLAGS)
@@ -878,11 +865,11 @@ function(mfem_export_mk_files)
  set(CONFIG_MK_BOOL_VARS MFEM_USE_MPI MFEM_USE_METIS MFEM_USE_METIS_5
      MFEM_DEBUG MFEM_USE_EXCEPTIONS MFEM_USE_ZLIB MFEM_USE_LIBUNWIND
      MFEM_USE_LAPACK MFEM_THREAD_SAFE MFEM_USE_LEGACY_OPENMP MFEM_USE_OPENMP
-      MFEM_USE_MEMALLOC MFEM_USE_SUNDIALS MFEM_USE_MESQUITE MFEM_USE_SUITESPARSE
+      MFEM_USE_MEMALLOC MFEM_USE_SUNDIALS MFEM_USE_SUITESPARSE
      MFEM_USE_SUPERLU MFEM_USE_SUPERLU5 MFEM_USE_MUMPS MFEM_USE_STRUMPACK
      MFEM_USE_GINKGO MFEM_USE_AMGX MFEM_USE_GNUTLS MFEM_USE_NETCDF
      MFEM_USE_PETSC MFEM_USE_SLEPC MFEM_USE_MPFR MFEM_USE_SIDRE MFEM_USE_FMS
-      MFEM_USE_CONDUIT MFEM_USE_PUMI MFEM_USE_HIOP MFEM_USE_GSLIB MFEM_USE_CUDA
+      MFEM_USE_CONDUIT MFEM_USE_PUMI MFEM_USE_HIOP MFEM_USE_IPOPT MFEM_USE_GSLIB MFEM_USE_CUDA
      MFEM_USE_HIP MFEM_USE_RAJA MFEM_USE_OCCA MFEM_USE_CEED MFEM_USE_CALIPER
      MFEM_USE_UMPIRE MFEM_USE_SIMD MFEM_USE_ADIOS2 MFEM_USE_MKL_CPARDISO
      MFEM_USE_ADFORWARD MFEM_USE_CODIPACK MFEM_USE_BENCHMARK MFEM_USE_PARELAG
@@ -85,9 +85,6 @@
 // Enable MFEM functionality based on the SUNDIALS libraries.
 // #define MFEM_USE_SUNDIALS

-// Enable MFEM functionality based on the Mesquite library.
-// #define MFEM_USE_MESQUITE
-
 // Enable MFEM functionality based on the SuiteSparse library.
 // #define MFEM_USE_SUITESPARSE

@@ -144,6 +141,9 @@
 // Enable MFEM functionality based on the HIOP library.
 // #define MFEM_USE_HIOP

+// Enable MFEM functionality based on the IPOPT library.
+// #define MFEM_USE_IPOPT
+
 // Enable MFEM functionality based on the GSLIB library
 // #define MFEM_USE_GSLIB

@@ -29,7 +29,6 @@ MFEM_USE_OPENMP        = @MFEM_USE_OPENMP@
 MFEM_USE_MEMALLOC      = @MFEM_USE_MEMALLOC@
 MFEM_TIMER_TYPE        = @MFEM_TIMER_TYPE@
 MFEM_USE_SUNDIALS      = @MFEM_USE_SUNDIALS@
-MFEM_USE_MESQUITE      = @MFEM_USE_MESQUITE@
 MFEM_USE_SUITESPARSE   = @MFEM_USE_SUITESPARSE@
 MFEM_USE_SUPERLU       = @MFEM_USE_SUPERLU@
 MFEM_USE_SUPERLU5      = @MFEM_USE_SUPERLU5@
@@ -47,6 +46,7 @@ MFEM_USE_FMS           = @MFEM_USE_FMS@
 MFEM_USE_CONDUIT       = @MFEM_USE_CONDUIT@
 MFEM_USE_PUMI          = @MFEM_USE_PUMI@
 MFEM_USE_HIOP          = @MFEM_USE_HIOP@
+MFEM_USE_IPOPT          = @MFEM_USE_IPOPT@
 MFEM_USE_GSLIB         = @MFEM_USE_GSLIB@
 MFEM_USE_CUDA          = @MFEM_USE_CUDA@
 MFEM_USE_HIP           = @MFEM_USE_HIP@
@@ -30,7 +30,6 @@ option(MFEM_USE_OPENMP "Enable the OpenMP backend" OFF)
 option(MFEM_USE_LEGACY_OPENMP "Enable legacy OpenMP usage" OFF)
 option(MFEM_USE_MEMALLOC "Enable the internal MEMALLOC option." ON)
 option(MFEM_USE_SUNDIALS "Enable SUNDIALS usage" OFF)
-option(MFEM_USE_MESQUITE "Enable MESQUITE usage" OFF)
 option(MFEM_USE_SUITESPARSE "Enable SuiteSparse usage" OFF)
 option(MFEM_USE_SUPERLU "Enable SuperLU_DIST usage" OFF)
 option(MFEM_USE_SUPERLU5 "Use the old SuperLU_DIST 5.1 version" OFF)
@@ -49,6 +48,7 @@ option(MFEM_USE_FMS "Enable FMS usage" OFF)
 option(MFEM_USE_CONDUIT "Enable Conduit usage" OFF)
 option(MFEM_USE_PUMI "Enable PUMI" OFF)
 option(MFEM_USE_HIOP "Enable HiOp" OFF)
+option(MFEM_USE_IPOPT "Enable Ipopt" OFF)
 option(MFEM_USE_CUDA "Enable CUDA" OFF)
 option(MFEM_USE_HIP "Enable HIP" OFF)
 option(MFEM_USE_OCCA "Enable OCCA" OFF)
@@ -124,9 +124,6 @@ set(SUNDIALS_DIR "${MFEM_DIR}/../sundials-5.0.0/instdir" CACHE PATH
 # set(SUNDIALS_REQUIRED_PACKAGES "SuiteSparse/KLU/AMD/BTF/COLAMD/config"
 #     CACHE STRING "Additional packages required by SUNDIALS.")

-set(MESQUITE_DIR "${MFEM_DIR}/../mesquite-2.99" CACHE PATH
-    "Path to the Mesquite library.")
-
 set(SuiteSparse_DIR "${MFEM_DIR}/../SuiteSparse" CACHE PATH
    "Path to the SuiteSparse library.")
 set(SuiteSparse_REQUIRED_PACKAGES "BLAS" "METIS"
@@ -188,6 +185,7 @@ set(GNUTLS_DIR "" CACHE PATH "Path to the GnuTLS library.")

 set(GSLIB_DIR "" CACHE PATH "Path to the GSLIB library.")

+set(HDF5_DIR "/usr" CACHE PATH "Path to the HDF5 library.")
 set(NETCDF_DIR "" CACHE PATH "Path to the NetCDF library.")
 set(NetCDF_REQUIRED_PACKAGES "HDF5/C/HL" CACHE STRING
    "Additional packages required by NetCDF.")
@@ -223,6 +221,10 @@ set(HIOP_DIR "${MFEM_DIR}/../hiop/install" CACHE STRING
    "Directory where HiOp is installed")
 set(HIOP_REQUIRED_PACKAGES "BLAS" "LAPACK" CACHE STRING
    "Packages that HiOp depends on.")
+set(IPOPT_DIR "${MFEM_DIR}/../ipopt/install" CACHE STRING
+    "Directory where IpOpt is installed")
+set(IPOPT_REQUIRED_PACKAGES "BLAS" "LAPACK" CACHE STRING
+    "Packages that IpOpt depends on.")

 set(MKL_CPARDISO_DIR "" CACHE STRING "MKL installation path.")
 set(MKL_MPI_WRAPPER_LIB "mkl_blacs_mpich_lp64" CACHE STRING "MKL MPI wrapper library")
@@ -230,17 +232,12 @@ set(MKL_LIBRARY_DIR "" CACHE STRING "Custom library subdirectory")

 set(OCCA_DIR "${MFEM_DIR}/../occa" CACHE PATH "Path to OCCA")
 set(RAJA_DIR "${MFEM_DIR}/../raja" CACHE PATH "Path to RAJA")
-# If RAJA is built with external CAMP:
-# set(RAJA_REQUIRED_PACKAGES "camp"
-#     CACHE STRING "Packages that RAJA depends on.")
-# set(camp_DIR "${MFEM_DIR}/../camp/lib/cmake/camp"
-#     CACHE PATH "Path to CAMP CMake files.")
 set(CEED_DIR "${MFEM_DIR}/../libCEED" CACHE PATH "Path to libCEED")
 set(UMPIRE_DIR "${MFEM_DIR}/../umpire" CACHE PATH "Path to Umpire")
 set(CALIPER_DIR "${MFEM_DIR}/../caliper" CACHE PATH "Path to Caliper")
 set(BLITZ_DIR "${MFEM_DIR}/../blitz" CACHE PATH "Path to Blitz")
 set(ALGOIM_DIR "${MFEM_DIR}/../algoim" CACHE PATH "Path to Algoim")
-set(ALGOIM_REQUIRED_PACKAGES "BLITZ" CACHE STRING
+set(Algoim_REQUIRED_PACKAGES "Blitz" CACHE STRING
    "Packages that ALGOIM depends on.")

 set(BENCHMARK_DIR "${MFEM_DIR}/../google-benchmark" CACHE PATH
@@ -131,7 +131,6 @@ MFEM_USE_LEGACY_OPENMP = NO
 MFEM_USE_MEMALLOC      = YES
 MFEM_TIMER_TYPE        = $(if $(NOTMAC),2,4)
 MFEM_USE_SUNDIALS      = NO
-MFEM_USE_MESQUITE      = NO
 MFEM_USE_SUITESPARSE   = NO
 MFEM_USE_SUPERLU       = NO
 MFEM_USE_SUPERLU5      = NO
@@ -149,6 +148,7 @@ MFEM_USE_FMS           = NO
 MFEM_USE_CONDUIT       = NO
 MFEM_USE_PUMI          = NO
 MFEM_USE_HIOP          = NO
+MFEM_USE_IPOPT         = NO
 MFEM_USE_GSLIB         = NO
 MFEM_USE_CUDA          = NO
 MFEM_USE_HIP           = NO
@@ -270,11 +270,6 @@ endif
 # If SUNDIALS was built with KLU:
 # MFEM_USE_SUITESPARSE = YES

-# MESQUITE library configuration
-MESQUITE_DIR = @MFEM_DIR@/../mesquite-2.99
-MESQUITE_OPT = -I$(MESQUITE_DIR)/include
-MESQUITE_LIB = -L$(MESQUITE_DIR)/lib -lmesquite
-
 # SuiteSparse library configuration
 LIB_RT = $(if $(NOTMAC),-lrt,)
 SUITESPARSE_DIR = @MFEM_DIR@/../SuiteSparse
@@ -324,6 +319,9 @@ MUMPS_LIB = $(XLINKER)-rpath,$(MUMPS_DIR)/lib -L$(MUMPS_DIR)/lib -ldmumps\

 # STRUMPACK library configuration
 STRUMPACK_DIR = @MFEM_DIR@/../STRUMPACK-build
+ifeq ($(MFEM_USE_STRUMPACK),YES)
+   BASE_FLAGS = -std=c++14
+endif
 STRUMPACK_OPT = -I$(STRUMPACK_DIR)/include $(SCOTCH_OPT)
 # If STRUMPACK was build with OpenMP support, the following may be need:
 # STRUMPACK_OPT += $(OPENMP_OPT)
@@ -450,6 +448,11 @@ HIOP_DIR = @MFEM_DIR@/../hiop/install
 HIOP_OPT = -I$(HIOP_DIR)/include
 HIOP_LIB = -L$(HIOP_DIR)/lib -lhiop $(LAPACK_LIB)

+# IPOPT
+IPOPT_DIR = @MFEM_DIR@/../ipopt/install
+IPOPT_OPT = -I$(IPOPT_DIR)/include
+IPOPT_LIB = -L$(IPOPT_DIR)/lib -lipopt $(LAPACK_LIB)
+
 # CoDiPack
 CODIPACK_DIR = @MFEM_DIR@/../CoDiPack
 CODIPACK_OPT = -I$(CODIPACK_DIR)
@@ -476,7 +479,17 @@ OCCA_LIB = $(XLINKER)-rpath,$(OCCA_DIR)/lib -L$(OCCA_DIR)/lib -locca
 # CALIPER library configuration
 CALIPER_DIR = @MFEM_DIR@/../caliper
 CALIPER_OPT = -I$(CALIPER_DIR)/include
-CALIPER_LIB = $(XLINKER)-rpath,$(CALIPER_DIR)/lib64 -L$(CALIPER_DIR)/lib64 -lcaliper
+CALIPER_LIB = $(XLINKER)-rpath,$(CALIPER_DIR)/lib64 $(XLINKER)-rpath,$(CALIPER_DIR)/lib -L$(CALIPER_DIR)/lib64 -L$(CALIPER_DIR)/lib -lcaliper
+
+ifdef ADIAK_DIR
+   CALIPER_OPT += -I$(ADIAK_DIR)/include
+   CALIPER_LIB += $(XLINKER)-rpath,$(ADIAK_DIR)/lib64 $(XLINKER)-rpath,$(ADIAK_DIR)/lib -L$(ADIAK_DIR)/lib64 -L$(ADIAK_DIR)/lib -ladiak
+endif
+ifdef GOTCHA_DIR
+   CALIPER_OPT += -I$(GOTCHA_DIR)/include
+   CALIPER_LIB += $(XLINKER)-rpath,$(GOTCHA_DIR)/lib64 $(XLINKER)-rpath,$(GOTCHA_DIR)/lib -L$(GOTCHA_DIR)/lib64 -L$(GOTCHA_DIR)/lib -lgotcha
+endif
+

 # BLITZ library configuration
 BLITZ_DIR = @MFEM_DIR@/../blitz
@@ -499,20 +512,29 @@ CEED_OPT = -I$(CEED_DIR)/include
 CEED_LIB = $(XLINKER)-rpath,$(CEED_DIR)/lib -L$(CEED_DIR)/lib -lceed

 # RAJA library configuration
+ifeq ($(MFEM_USE_RAJA),YES)
+   BASE_FLAGS = -std=c++14
+endif
 RAJA_DIR = @MFEM_DIR@/../raja
 RAJA_OPT = -I$(RAJA_DIR)/include
 ifdef CUB_DIR
   RAJA_OPT += -I$(CUB_DIR)
 endif
+
+CAMP_LIB = -lcamp
 ifdef CAMP_DIR
   RAJA_OPT += -I$(CAMP_DIR)/include
+   CAMP_LIB = $(XLINKER)-rpath,$(CAMP_DIR)/lib -L$(CAMP_DIR)/lib -lcamp
 endif
-RAJA_LIB = $(XLINKER)-rpath,$(RAJA_DIR)/lib -L$(RAJA_DIR)/lib -lRAJA
+RAJA_LIB = $(XLINKER)-rpath,$(RAJA_DIR)/lib -L$(RAJA_DIR)/lib -lRAJA $(CAMP_LIB)

 # UMPIRE library configuration
+ifeq ($(MFEM_USE_UMPIRE),YES)
+   BASE_FLAGS = -std=c++14
+endif
 UMPIRE_DIR = @MFEM_DIR@/../umpire
 UMPIRE_OPT = -I$(UMPIRE_DIR)/include $(if $(CAMP_DIR), -I$(CAMP_DIR)/include)
-UMPIRE_LIB = -L$(UMPIRE_DIR)/lib -lumpire
+UMPIRE_LIB = -L$(UMPIRE_DIR)/lib -lumpire $(CAMP_LIB)

 # MKL CPardiso library configuration
 MKL_CPARDISO_DIR ?=
@@ -58,6 +58,10 @@ groups_serial=(
   "HiOp examples:"
   "examples/hiop"
   "ex9.cpp"'
+'"ipopt"
+   "IpOpt examples:"
+   "examples/ipopt"
+   "ex10.cpp"'
 '"pumi"
   "PUMI examples:"
   "examples/pumi"
@@ -215,6 +219,10 @@ groups_all=(
   "HiOp examples:"
   "examples/hiop"
   "ex9.cpp ex9p.cpp"'
+'"ipopt"
+   "IpOpt examples:"
+   "examples/ipopt"
+   "ex10.cpp"'
 '"pumi"
   "PUMI examples:"
   "examples/pumi"
@@ -38,7 +38,7 @@ PROJECT_NAME           = "MFEM"
 # could be handy for archiving the generated documentation or if some version
 # control system is used.

-PROJECT_NUMBER         = v4.5.0
+PROJECT_NUMBER         = v4.5.1

 # Using the PROJECT_BRIEF tag one can provide an optional one line description
 # for a project that appears at the top of each page and should give viewer a
@@ -763,36 +763,55 @@ INPUT                  = @MFEM_SOURCE_DIR@/doc/CodeDocumentation.dox \
                         @MFEM_SOURCE_DIR@/config \
                         @MFEM_SOURCE_DIR@/general \
                         @MFEM_SOURCE_DIR@/linalg \
+                         @MFEM_SOURCE_DIR@/linalg/simd \
                         @MFEM_SOURCE_DIR@/mesh \
+                         @MFEM_SOURCE_DIR@/mesh/submesh \
                         @MFEM_SOURCE_DIR@/fem \
-                         @MFEM_SOURCE_DIR@/fem/moonolith \
+                         @MFEM_SOURCE_DIR@/fem/ceed \
+                         @MFEM_SOURCE_DIR@/fem/ceed/integrators \
+                         @MFEM_SOURCE_DIR@/fem/ceed/integrators/convection \
+                         @MFEM_SOURCE_DIR@/fem/ceed/integrators/diffusion \
+                         @MFEM_SOURCE_DIR@/fem/ceed/integrators/mass \
+                         @MFEM_SOURCE_DIR@/fem/ceed/integrators/nlconvection \
+                         @MFEM_SOURCE_DIR@/fem/ceed/interface \
+                         @MFEM_SOURCE_DIR@/fem/ceed/solvers \
                         @MFEM_SOURCE_DIR@/fem/fe \
                         @MFEM_SOURCE_DIR@/fem/lor \
+                         @MFEM_SOURCE_DIR@/fem/moonolith \
+                         @MFEM_SOURCE_DIR@/fem/qinterp \
+                         @MFEM_SOURCE_DIR@/fem/tmop \
                         @MFEM_SOURCE_DIR@/examples \
-                         @MFEM_SOURCE_DIR@/examples/caliper \
                         @MFEM_SOURCE_DIR@/examples/amgx \
+                         @MFEM_SOURCE_DIR@/examples/caliper \
                         @MFEM_SOURCE_DIR@/examples/ginkgo \
-                         @MFEM_SOURCE_DIR@/examples/moonolith \
                         @MFEM_SOURCE_DIR@/examples/hiop \
+                         @MFEM_SOURCE_DIR@/examples/ipopt \
+                         @MFEM_SOURCE_DIR@/examples/moonolith \
                         @MFEM_SOURCE_DIR@/examples/petsc \
                         @MFEM_SOURCE_DIR@/examples/pumi \
                         @MFEM_SOURCE_DIR@/examples/sundials \
                         @MFEM_SOURCE_DIR@/examples/superlu \
                         @MFEM_SOURCE_DIR@/miniapps/adjoint \
+                         @MFEM_SOURCE_DIR@/miniapps/autodiff \
                         @MFEM_SOURCE_DIR@/miniapps/common \
                         @MFEM_SOURCE_DIR@/miniapps/electromagnetics \
                         @MFEM_SOURCE_DIR@/miniapps/gslib \
+                         @MFEM_SOURCE_DIR@/miniapps/hooke \
+                         @MFEM_SOURCE_DIR@/miniapps/hooke/kernels \
+                         @MFEM_SOURCE_DIR@/miniapps/hooke/materials \
+                         @MFEM_SOURCE_DIR@/miniapps/hooke/operators \
+                         @MFEM_SOURCE_DIR@/miniapps/hooke/preconditioners \
                         @MFEM_SOURCE_DIR@/miniapps/meshing \
                         @MFEM_SOURCE_DIR@/miniapps/mtop \
-                         @MFEM_SOURCE_DIR@/miniapps/autodiff \
+                         @MFEM_SOURCE_DIR@/miniapps/multidomain \
                         @MFEM_SOURCE_DIR@/miniapps/navier \
                         @MFEM_SOURCE_DIR@/miniapps/nurbs \
+                         @MFEM_SOURCE_DIR@/miniapps/parelag \
                         @MFEM_SOURCE_DIR@/miniapps/performance \
                         @MFEM_SOURCE_DIR@/miniapps/shifted \
                         @MFEM_SOURCE_DIR@/miniapps/solvers \
                         @MFEM_SOURCE_DIR@/miniapps/tools \
-                         @MFEM_SOURCE_DIR@/miniapps/toys \
-                         @MFEM_SOURCE_DIR@/miniapps/parelag
+                         @MFEM_SOURCE_DIR@/miniapps/toys

 # This tag can be used to specify the character encoding of the source files
 # that doxygen parses. Internally doxygen uses the UTF-8 encoding. Doxygen uses
@@ -178,6 +178,11 @@ if (MFEM_USE_HIOP)
  add_subdirectory(hiop)
 endif()

+# Include the examples/ipopt directory if IpOpt is enabled
+if (MFEM_USE_IPOPT)
+  add_subdirectory(ipopt)
+endif()
+
 # Include the examples/petsc directory if PETSc is enabled.
 if (MFEM_USE_PETSC)
  add_subdirectory(petsc)
@@ -33,13 +33,13 @@ add_mfem_examples(CALIPER_EXE_SRCS ${PREFIX})
 if (MFEM_ENABLE_TESTING)
  foreach(SRC_FILE ${CALIPER_EXE_SRCS})
    get_filename_component(SRC_FILENAME ${SRC_FILE} NAME)
-    string(REPLACE ".cpp" "" TEST_NAME ${SRC_FILENAME})
+    string(REPLACE ".cpp" "" TEST_NAME ${PREFIX}${SRC_FILENAME})

    set(THIS_TEST_OPTIONS "-no-vis")

    if (NOT (${TEST_NAME} MATCHES ".*p$"))
      add_test(NAME ${TEST_NAME}_ser
-        COMMAND ${TEST_NAME} ${THIS_TEST_OPTIONS})
+        COMMAND $<TARGET_FILE:${TEST_NAME}> ${THIS_TEST_OPTIONS})
    else()
      add_test(NAME ${TEST_NAME}_np=${MFEM_MPI_NP}
        COMMAND ${MPIEXEC} ${MPIEXEC_NUMPROC_FLAG} ${MFEM_MPI_NP}
@@ -0,0 +1,810 @@
+//                                Contact example
+//
+// Compile with: make contact
+//
+// Sample runs:  ./contact -m1 block1.mesh -m2 block2.mesh -at "5 6 7 8"
+// Sample runs:  ./contact -m1 block1_d.mesh -m2 block2_d.mesh -at "5 6 7 8"
+
+#include "mfem.hpp"
+#include <fstream>
+#include <iostream>
+#include "nodepair.hpp"
+
+using namespace std;
+using namespace mfem;
+
+bool ifequalarray(const Array<int> a1, const Array<int> a2)
+{
+   if (a1.Size()!=a2.Size())
+   {
+      return false;
+   }
+   for (int i=0; i<a1.Size(); i++)
+   {
+      if (a1[i] != a2[i])
+      {
+         return false;
+      }
+   }
+   return true;
+}
+
+void FindSurfaceToProject(Mesh& mesh, const int elem, int& cbdrface)
+{
+   Array<int> attr;
+   attr.Append(2);
+   Array<int> faces;
+   Array<int> ori;
+   std::vector<Array<int> > facesVertices;
+   std::vector<int > faceid;
+   mesh.GetElementFaces(elem, faces, ori);
+   int face = -1;
+   for (int i=0; i<faces.Size(); i++)
+   {
+      face = faces[i];
+      Array<int> faceVert;
+      if (!mesh.FaceIsInterior(face)) // if on the boundary
+      {
+         mesh.GetFaceVertices(face, faceVert);
+         faceVert.Sort();
+         facesVertices.push_back(faceVert);
+         faceid.push_back(face);
+      }
+   }
+   int bdrface = facesVertices.size();
+
+   Array<int> bdryFaces;
+   // This shoulnd't need to be rebuilt
+   std::vector<Array<int> > bdryVerts;
+   for (int b=0; b<mesh.GetNBE(); ++b)
+   {
+      if (attr.FindSorted(mesh.GetBdrAttribute(b)) >= 0)  // found the contact surface
+      {
+         bdryFaces.Append(b);
+         Array<int> vert;
+         mesh.GetBdrElementVertices(b, vert);
+         vert.Sort();
+         bdryVerts.push_back(vert);
+      }
+   }
+
+   int bdrvert = bdryVerts.size();
+   cbdrface = -1;  // the face number of the contact surface element
+   int count_cbdrface = 0;  // the number of matching surfaces, used for checks
+
+   for (int i=0; i<bdrface; i++)
+   {
+      for (int j=0; j<bdrvert; j++)
+      {
+         if (ifequalarray(facesVertices[i], bdryVerts[j]))
+         {
+            cbdrface = faceid[i];
+            count_cbdrface += 1;
+         }
+      }
+   }
+   MFEM_VERIFY(count_cbdrface == 1,"projection surface not found");
+
+};
+
+Vector GetNormalVector(Mesh & mesh, const int elem, const double *ref,
+                       int & refFace, int & refNormal, bool & interior)
+{
+   ElementTransformation *trans = mesh.GetElementTransformation(elem);
+   const int dim = mesh.Dimension();
+   const int spaceDim = trans->GetSpaceDim();
+
+   MFEM_VERIFY(spaceDim == 3, "");
+
+   Vector n(spaceDim);
+
+   IntegrationPoint ip;
+   ip.Set(ref, dim);
+
+   trans->SetIntPoint(&ip);
+   //CalcOrtho(trans->Jacobian(), n);  // Works only for face transformations
+   const DenseMatrix jac = trans->Jacobian();
+
+   int dimNormal = -1;
+   int normalSide = -1;
+
+   const double tol = 1.0e-8;
+   for (int i=0; i<dim; ++i)
+   {
+      const double d0 = std::abs(ref[i]);
+      const double d1 = std::abs(ref[i] - 1.0);
+
+      const double d = std::min(d0, d1);
+      // TODO: this works only for hexahedral meshes!
+
+      if (d < tol)
+      {
+         MFEM_VERIFY(dimNormal == -1, "");
+         dimNormal = i;
+
+         if (d0 < tol)
+         {
+            normalSide = 0;
+         }
+         else
+         {
+            normalSide = 1;
+         }
+      }
+   }
+   // closest point on the boundary
+   if (dimNormal < 0 || normalSide < 0) // node is inside the element
+   {
+      interior = 1;
+      Vector n(3);
+      n = 0.0;
+      return n;
+   }
+
+   MFEM_VERIFY(dimNormal >= 0 && normalSide >= 0, "");
+   refNormal = dimNormal;
+
+   MFEM_VERIFY(dim == 3, "");
+
+   {
+      // Find the reference face
+      if (dimNormal == 0)
+      {
+         refFace = (normalSide == 1) ? 2 : 4;
+      }
+      else if (dimNormal == 1)
+      {
+         refFace = (normalSide == 1) ? 3 : 1;
+      }
+      else
+      {
+         refFace = (normalSide == 1) ? 5 : 0;
+      }
+   }
+
+   std::vector<Vector> tang(2);
+
+   int tangDir[2] = {-1, -1};
+   {
+      int t = 0;
+      for (int i=0; i<dim; ++i)
+      {
+         if (i != dimNormal)
+         {
+            tangDir[t] = i;
+            t++;
+         }
+      }
+
+      MFEM_VERIFY(t == 2, "");
+   }
+
+   for (int i=0; i<2; ++i)
+   {
+      tang[i].SetSize(3);
+
+      Vector tangRef(3);
+      tangRef = 0.0;
+      tangRef[tangDir[i]] = 1.0;
+
+      jac.Mult(tangRef, tang[i]);
+   }
+
+   Vector c(3);  // Cross product
+
+   c[0] = (tang[0][1] * tang[1][2]) - (tang[0][2] * tang[1][1]);
+   c[1] = (tang[0][2] * tang[1][0]) - (tang[0][0] * tang[1][2]);
+   c[2] = (tang[0][0] * tang[1][1]) - (tang[0][1] * tang[1][0]);
+
+   c /= c.Norml2();
+
+   Vector nref(3);
+   nref = 0.0;
+   nref[dimNormal] = 1.0;
+
+   Vector ndir(3);
+   jac.Mult(nref, ndir);
+
+   ndir /= ndir.Norml2();
+
+   const double dp = ndir * c;
+
+   // TODO: eliminate c?
+   n = c;
+   if (dp < 0.0)
+   {
+      n *= -1.0;
+   }
+   interior = 0;
+   return n;
+}
+
+// WARNING: global variable, just for this little example.
+std::array<std::array<int, 3>, 8> HEX_VERT =
+{
+   {  {0,0,0},
+      {1,0,0},
+      {1,1,0},
+      {0,1,0},
+      {0,0,1},
+      {1,0,1},
+      {1,1,1},
+      {0,1,1}
+   }
+};
+
+int GetHexVertex(int cdim, int c, int fa, int fb, Vector & refCrd)
+{
+   int ref[3];
+   ref[cdim] = c;
+   ref[cdim == 0 ? 1 : 0] = fa;
+   ref[cdim == 2 ? 1 : 2] = fb;
+
+   for (int i=0; i<3; ++i) { refCrd[i] = ref[i]; }
+
+   int refv = -1;
+
+   for (int i=0; i<8; ++i)
+   {
+      bool match = true;
+      for (int j=0; j<3; ++j)
+      {
+         if (ref[j] != HEX_VERT[i][j]) { match = false; }
+      }
+
+      if (match) { refv = i; }
+   }
+
+   MFEM_VERIFY(refv >= 0, "");
+
+   return refv;
+}
+
+// Coordinates in xyz are assumed to be ordered as [X, Y, Z]
+// where X is the list of x-coordinates for all points and so on.
+// conn: connectivity of the target surface elements
+// xi: surface reference cooridnates for the cloest point, involves a linear transformation from [0,1] to [-1,1]
+void FindPointsInMesh(Mesh & mesh, Vector const& xyz, Array<int>& conn,
+                      Vector& xi)
+{
+   const int dim = mesh.Dimension();
+   const int np = xyz.Size() / dim;
+
+   MFEM_VERIFY(np * dim == xyz.Size(), "");
+
+   mesh.EnsureNodes();
+
+   //FindPointsGSLIB finder(MPI_COMM_WORLD);
+   FindPointsGSLIB finder;
+
+   finder.SetDistanceToleranceForPointsFoundOnBoundary(0.5);
+
+   const double bb_t = 0.5;
+   finder.Setup(mesh, bb_t);
+
+   finder.FindPoints(xyz);
+
+   /// Return code for each point searched by FindPoints: inside element (0), on
+   /// element boundary (1), or not found (2).
+   Array<unsigned int> codes = finder.GetCode();
+
+   /// Return element number for each point found by FindPoints.
+   Array<unsigned int> elems = finder.GetElem();
+
+   /// Return reference coordinates for each point found by FindPoints.
+   Vector refcrd = finder.GetReferencePosition();
+
+   /// Return distance between the sought and the found point in physical space,
+   /// for each point found by FindPoints.
+   Vector dist = finder.GetDist();
+
+   MFEM_VERIFY(dist.Size() == np, "");
+   MFEM_VERIFY(refcrd.Size() == np * dim, "");
+   MFEM_VERIFY(elems.Size() == np, "");
+   MFEM_VERIFY(codes.Size() == np, "");
+
+   bool allfound = true;
+   for (auto code : codes)
+      if (code == 2) { allfound = false; }
+
+   MFEM_VERIFY(allfound, "A point was not found");
+
+   cout << "Maximum distance of projected points: " << dist.Max() << endl;
+
+   // extract information
+   for (int i=0; i<np; ++i)
+   {
+      /*cout << "Point " << i << ": (";
+      for (int j=0; j<dim; ++j)
+      {
+         cout << xyz[i + (j*np)];
+         if (j == dim-1) {cout << ")" << endl;}
+         else{cout << ", ";}
+      }*/
+      //cout << "  element: " << elems[i] << endl;
+      //cout << "  element " << elems[i] << " vertices:" << endl;
+      //Array<int> vert;
+      //mesh.GetElementVertices(elems[i], vert);
+      //for (auto v : vert)
+      //{
+      //   cout << "    " << v << endl;
+      //}
+
+      /*cout << "  reference coordinates: (";
+      for (int j=0; j<dim; ++j)
+      {
+         cout << refcrd[(i*dim) + j];
+         if (j == dim-1)
+         {
+            cout << ")" << endl;
+         }
+         else
+         {
+            cout << ", ";
+         }
+      }*/
+
+      int refFace, refNormal, refNormalSide;
+      bool is_interior = -1;
+      Vector normal = GetNormalVector(mesh, elems[i], refcrd.GetData() + (i*dim),
+                                      refFace, refNormal, is_interior);
+      int phyFace;
+      if (is_interior)
+      {
+         phyFace = -1; // the id of the face that has the closest point
+         FindSurfaceToProject(mesh, elems[i], phyFace);
+
+         Array<int> cbdrVert;
+         mesh.GetFaceVertices(phyFace, cbdrVert);
+         Vector xs(dim);
+         xs[0] = xyz[i + 0*np];
+         xs[1] = xyz[i + 1*np];
+         xs[2] = xyz[i + 2*np];
+         Vector xi_tmp(dim-1);
+         // get nodes!
+
+         GridFunction *nodes = mesh.GetNodes();
+         DenseMatrix coords(4,3);
+         for (int i=0; i<4; i++)
+         {
+            for (int j=0; j<3; j++)
+            {
+               coords(i,j) = (*nodes)[cbdrVert[i]*3+j];
+            }
+         }
+         SlaveToMaster(coords, xs, xi_tmp);
+
+         for (int j=0; j<dim-1; ++j)
+         {
+            xi[i*(dim-1)+j] = xi_tmp[j];
+         }
+         // now get get the projection to the surface
+      }
+      else
+      {
+         Vector faceRefCrd(dim-1);
+         {
+            int fd = 0;
+            for (int j=0; j<dim; ++j)
+            {
+               if (j == refNormal)
+               {
+                  refNormalSide = (refcrd[(i*dim) + j] > 0.5);
+               }
+               else
+               {
+                  faceRefCrd[fd] = refcrd[(i*dim) + j];
+                  fd++;
+               }
+            }
+
+            MFEM_VERIFY(fd == dim-1, "");
+         }
+
+         for (int j=0; j<dim-1; ++j)
+         {
+            xi[i*(dim-1)+j] = faceRefCrd[j]*2.0 - 1.0;
+         }
+         //cout << "  face reference coordinates: (";
+         /*for (int j=0; j<dim-1; ++j)
+         {
+            cout << faceRefCrd[j];
+            if (j == dim-2){cout << ")" << endl;}
+            else{cout << ", ";}
+         }*/
+      }
+      //cout << "  normal vector: ";
+      //normal.Print();
+
+      // ask, does this do anything?
+      /*
+      IntegrationPoint ip;
+      ip.Set(refcrd.GetData() + (i*dim), dim);
+      ElementTransformation *trans = mesh.GetElementTransformation(elems[i]);
+      Vector phys(trans->GetSpaceDim());
+      trans->Transform(ip, phys);
+      cout << "  physical coordinates: ";
+      phys.Print();
+      */
+
+      // Get the element face
+      Array<int> faces;
+      Array<int> ori;
+      int face;
+
+      if (is_interior)
+      {
+         face = phyFace;
+      }
+      else
+      {
+         mesh.GetElementFaces(elems[i], faces, ori);
+         face = faces[refFace];
+      }
+
+      Array<int> faceVert;
+      mesh.GetFaceVertices(face, faceVert);
+
+      //cout << "  face " << face << " vertices:" << endl;
+      //for (auto v : faceVert){ cout << "    " << v << endl;}
+
+      for (int p=0; p<4; p++)
+      {
+         conn[4*i+p] = faceVert[p];
+      }
+      /*
+      Vector ref(dim);
+
+      for (int p=0; p<2; ++p)
+         for (int q=0; q<2; ++q)
+         {
+            const int refv = GetHexVertex(refNormal, refNormalSide, p, q, ref);
+            cout << "  face reference vertex (" << p << "," << q
+                 << ") is global vertex " << vert[refv] << endl;
+
+            {
+               // Sanity check
+               ip.Set(ref.GetData(), dim);
+               trans->Transform(ip, phys);
+               for (int j=0; j<dim; ++j)
+               {
+                  phys[j] -= mesh.GetVertex(vert[refv])[j];
+               }
+          phys.Print();
+          cout<<vert[refv]<<endl;
+               cout<<mesh.GetVertex(vert[refv])[0]<<endl;
+               cout<<mesh.GetVertex(vert[refv])[1]<<endl;
+               cout<<mesh.GetVertex(vert[refv])[2]<<endl;
+               MFEM_VERIFY(phys.Norml2() < 1.0e-12, "Sanity check failed");
+            }
+         }*/
+   }
+}
+
+int main(int argc, char *argv[])
+{
+   // 1. Parse command-line options.
+   const char *mesh_file1 = "block1.mesh";
+   const char *mesh_file2 = "block2.mesh";
+
+   Array<int> attr;
+   Array<int> m_attr;
+
+   OptionsParser args(argc, argv);
+   args.AddOption(&mesh_file1, "-m1", "--mesh1",
+                  "First mesh file to use.");
+   args.AddOption(&mesh_file2, "-m2", "--mesh2",
+                  "Second mesh file to use.");
+   args.AddOption(&attr, "-at", "--attributes-surf",
+                  "Attributes of boundary faces on contact surface for mesh 2.");
+   args.Parse();
+   if (!args.Good())
+   {
+      args.PrintUsage(cout);
+      return 1;
+   }
+   args.PrintOptions(cout);
+
+   Mesh mesh1(mesh_file1, 1, 1);
+   Mesh mesh2(mesh_file2, 1, 1);
+
+   {
+      char vishost[] = "localhost";
+      int  visport   = 19916;
+      socketstream mesh1a_sock(vishost, visport);
+      mesh1a_sock.precision(8);
+      mesh1a_sock << "mesh\n" << mesh1 << flush;
+      socketstream mesh2a_sock(vishost, visport);
+      mesh2a_sock.precision(8);
+      mesh2a_sock << "mesh\n" << mesh2 << flush;
+   }
+
+   const int dim = mesh1.Dimension();
+   MFEM_VERIFY(dim == mesh2.Dimension(), "");
+
+   // boundary attribute 2 is the potential contact surface of nodes
+   attr.Append(2);
+   // boundary attribute 2 is the potential contact surface for master surface
+   m_attr.Append(2);
+
+   //  Define a finite element space on the mesh. Here we use vector finite
+   //  elements, i.e. dim copies of a scalar finite element space. The vector
+   //  dimension is specified by the last argument of the FiniteElementSpace
+   //  constructor.
+   FiniteElementCollection *fec1;
+   FiniteElementSpace *fespace1;
+   fec1 = new H1_FECollection(1, dim);
+   fespace1 = new FiniteElementSpace(&mesh1, fec1, dim, Ordering::byVDIM);
+   cout << "Number of finite element unknowns for mesh1: "
+        << fespace1->GetTrueVSize() << endl;
+   mesh1.SetNodalFESpace(fespace1);
+   GridFunction nodes0 = *mesh1.GetNodes(); // undeformed mesh1 nodal grid function
+   GridFunction *nodes1 = mesh1.GetNodes();
+
+   FiniteElementCollection *fec2 = new H1_FECollection(1, dim);
+   FiniteElementSpace *fespace2 = new FiniteElementSpace(&mesh2, fec2, dim,
+                                                         Ordering::byVDIM);
+   cout << "Number of finite element unknowns for mesh2: "
+        << fespace2->GetTrueVSize() << endl;
+
+   // degrees of freedom of both meshes
+   int ndof_1 = fespace1->GetTrueVSize();
+   int ndof_2 = fespace2->GetTrueVSize();
+   int ndofs = ndof_1 + ndof_2;
+   // number of nodes for each mesh
+   int nnd_1 = mesh1.GetNV();
+   int nnd_2 = mesh2.GetNV();
+   int nnd = nnd_1 + nnd_2;
+   // Determine the list of true (i.e. conforming) essential boundary dofs.
+   // In this example, the boundary conditions are defined by marking only
+   // boundary attribute 1 from the mesh as essential and converting it to a
+   // list of true dofs.
+   Array<int> ess_tdof_list1, ess_bdr1(mesh1.bdr_attributes.Max());
+   ess_bdr1 = 0;
+   //ess_bdr1[0] = 1;
+   // Not ready to be passed on yet
+   // fespace->GetEssentialTrueDofs(ess_bdr, ess_tdof_list);
+
+   Array<int> ess_tdof_list2, ess_bdr2(mesh2.bdr_attributes.Max());
+   ess_bdr2 = 0;
+   //ess_bdr2[0] = 1;
+
+   // Define the displacement vector x as a finite element grid function
+   // corresponding to fespace. GridFunction is a derived class of Vector.
+   GridFunction x1(fespace1);
+   x1 = 0.0;
+   GridFunction x2(fespace2);
+   x2 = 0.0;
+
+   // Generate force
+   LinearForm *b1 = new LinearForm(fespace1);
+   b1->Assemble();
+
+   LinearForm *b2 = new LinearForm(fespace2);
+   b2->Assemble();
+
+   // Set up the bilinear form a(.,.) on the finite element space
+   //  corresponding to the linear elasticity integrator with piece-wise
+   //  constants coefficient lambda and mu.
+   Vector lambda1(mesh1.attributes.Max());
+   lambda1 = 57.6923076923;
+   PWConstCoefficient lambda1_func(lambda1);
+   Vector mu1(mesh1.attributes.Max());
+   mu1 = 38.4615384615;
+   PWConstCoefficient mu1_func(mu1);
+
+   BilinearForm *a1 = new BilinearForm(fespace1);
+   a1->AddDomainIntegrator(new ElasticityIntegrator(lambda1_func,mu1_func));
+
+   Vector lambda2(mesh2.attributes.Max());
+   lambda2 = 57.6923076923;
+   PWConstCoefficient lambda2_func(lambda2);
+   Vector mu2(mesh2.attributes.Max());
+   mu2 = 38.4615384615;
+   PWConstCoefficient mu2_func(mu2);
+
+   BilinearForm *a2 = new BilinearForm(fespace2);
+   a2->AddDomainIntegrator(new ElasticityIntegrator(lambda2_func,mu2_func));
+
+   a1->Assemble();
+   SparseMatrix A1;
+   Vector B1, X1;
+   a1->FormLinearSystem(ess_tdof_list1, x1, *b1, A1, X1, B1);
+
+   a2->Assemble();
+   SparseMatrix A2;
+   Vector B2, X2;
+   a2->FormLinearSystem(ess_tdof_list2, x2, *b2, A2, X2, B2);
+
+   // Combine elasticity operator for two meshes into one.
+   // Block Matrix
+   SparseMatrix K(ndofs,ndofs);
+   for (int i=0; i<A1.Height(); i++)
+   {
+      Array<int> col_tmp;
+      Vector v_tmp;
+      col_tmp = 0;
+      v_tmp = 0.0;
+      A1.GetRow(i, col_tmp, v_tmp);
+      K.SetRow(i, col_tmp, v_tmp);
+   }
+   for (int i=0; i<A2.Height(); i++)
+   {
+      Array<int> col_tmp;
+      Vector v_tmp;
+      col_tmp = 0;
+      v_tmp = 0.0;
+      A2.GetRow(i, col_tmp, v_tmp);
+      for (int j=0; j<col_tmp.Size(); j++)
+      {
+         col_tmp[j] += ndof_1;
+      }
+      K.SetRow(i+ndof_1, col_tmp, v_tmp);  // mesh1 top left corner
+   }
+
+   // Construct node to segment contact constraint.
+
+   attr.Sort();
+   cout << "Boundary attributes for contact surface faces in mesh 2" << endl;
+   for (auto a : attr) { cout << a << endl; }
+
+   Array<int> bdryFaces2;  // TODO: remove this?
+
+   std::set<int> bdryVerts2;
+   for (int b=0; b<mesh2.GetNBE(); ++b)
+   {
+      if (attr.FindSorted(mesh2.GetBdrAttribute(b)) >= 0)
+      {
+         bdryFaces2.Append(b);
+         Array<int> vert;
+         mesh2.GetBdrElementVertices(b, vert);
+         for (auto v : vert)
+         {
+            bdryVerts2.insert(v);
+         }
+      }
+   }
+
+   int npoints = bdryVerts2.size();
+   Array<int> s_conn(npoints); // connectivity of the second/slave mesh
+   Vector xyz(dim * npoints);
+   xyz = 0.0;
+
+   cout << "Boundary vertices for contact surface vertices in mesh 2" << endl;
+
+   // construct the nodal coordinates on mesh2 to be projected, including displacement
+   int count = 0;
+   for (auto v : bdryVerts2)
+   {
+      cout << v << ": " << mesh2.GetVertex(v)[0] << ", "
+           << mesh2.GetVertex(v)[1] << ", "
+           << mesh2.GetVertex(v)[2] << endl;
+
+      for (int i=0; i<dim; ++i)
+      {
+         xyz[count + (i * npoints)] = mesh2.GetVertex(v)[i] + x2[v*dim+i];
+      }
+
+      s_conn[count] = v + nnd_1; // dof1 is the master
+      count++;
+   }
+
+   MFEM_VERIFY(count == npoints, "");
+
+   // gap function
+   Vector g(npoints*dim);
+   g = -1.0;
+   // segment reference coordinates of the closest point
+   Vector m_xi(npoints*(dim-1));
+   m_xi = -1.0;
+   Vector xs(dim*npoints);
+   xs = 0.0;
+   for (int i=0; i<npoints; i++)
+   {
+      for (int j=0; j<dim; j++)
+      {
+         xs[i*dim+j] = xyz[i + (j*npoints)];
+      }
+   }
+
+   Array<int> m_conn(
+      npoints*4); // only works for linear elements that have 4 vertices!
+   DenseMatrix coordsm(npoints*4, dim);
+
+   // adding displacement to mesh1 using a fixed grid function from mesh1
+   x1 = 1e-4; // x1 order: [xyz xyz... xyz]
+   add(nodes0, x1, *nodes1);
+
+   FindPointsInMesh(mesh1, xyz, m_conn, m_xi);
+
+   for (int i=0; i<npoints; i++)
+   {
+      for (int j=0; j<4; j++)
+      {
+         for (int k=0; k<dim; k++)
+         {
+            coordsm(i*4+j,k) = mesh1.GetVertex(m_conn[i*4+j])[k]+x1[dim*m_conn[i*4+j]+k];
+         }
+      }
+   }
+   //coordsm.Print();
+   SparseMatrix M(nnd,ndofs);
+   std::vector<SparseMatrix> dM(nnd, SparseMatrix(ndofs,ndofs));
+
+   Assemble_Contact(nnd, npoints, ndofs, xs, m_xi, coordsm,
+                    s_conn, m_conn, g, M, dM);
+
+   std::set<int> dirbdryv2;
+   for (int b=0; b<mesh2.GetNBE(); ++b)
+   {
+      if (mesh2.GetBdrAttribute(b) == 1)
+      {
+         Array<int> vert;
+         mesh2.GetBdrElementVertices(b, vert);
+         for (auto v : vert)
+         {
+            dirbdryv2.insert(v);
+         }
+      }
+   }
+   std::set<int> dirbdryv1;
+   for (int b=0; b<mesh1.GetNBE(); ++b)
+   {
+      if (mesh1.GetBdrAttribute(b) == 1)
+      {
+         Array<int> vert;
+         mesh1.GetBdrElementVertices(b, vert);
+         for (auto v : vert)
+         {
+            dirbdryv1.insert(v);
+         }
+      }
+   }
+
+   Array<int> Dirichlet_dof;
+   Array<double> Dirichlet_val;
+
+   for (auto v : dirbdryv2)
+   {
+      for (int i=0; i<dim; ++i)
+      {
+         Dirichlet_dof.Append(v*dim + i + ndof_1);
+         Dirichlet_val.Append(0.);
+      }
+   }
+   double delta = 0.1;
+   for (auto v : dirbdryv1)
+   {
+      Dirichlet_dof.Append(v*dim + 0);
+      Dirichlet_val.Append(delta);
+      Dirichlet_dof.Append(v*dim + 1);
+      Dirichlet_val.Append(0.);
+      Dirichlet_dof.Append(v*dim + 2);
+      Dirichlet_val.Append(0.);
+   }
+
+
+   {
+      char vishost[] = "localhost";
+      int  visport   = 19916;
+      socketstream mesh1_sock(vishost, visport);
+      mesh1_sock.precision(8);
+      mesh1_sock << "mesh\n" << mesh1 << flush;
+      socketstream mesh2_sock(vishost, visport);
+      mesh2_sock.precision(8);
+      mesh2_sock << "mesh\n" << mesh2 << flush;
+   }
+
+   //M.Print();
+   /*Vector eps(ndofs);
+   Vector sol(ndofs); sol = 0.;
+   for(int i=0;i<ndofs;i++) eps[i] = 1e-5 * i ;
+   for(int i=0;i<9;i++)
+   {
+     cout<<i<<endl;
+     dM[s_conn[i]].Mult(eps,sol);
+     sol.Print();
+   }
+   */
+   return 0;
+}
@@ -30,7 +30,7 @@
 //
 // Device sample runs:
 //               ex1 -pa -d cuda
-//             * ex1 -fa -d cuda
+//               ex1 -fa -d cuda
 //               ex1 -pa -d raja-cuda
 //             * ex1 -pa -d raja-hip
 //               ex1 -pa -d occa-cuda
@@ -192,7 +192,14 @@ int main(int argc, char *argv[])
   //    domain integrator.
   BilinearForm a(&fespace);
   if (pa) { a.SetAssemblyLevel(AssemblyLevel::PARTIAL); }
-   if (fa) { a.SetAssemblyLevel(AssemblyLevel::FULL); }
+   if (fa)
+   {
+      a.SetAssemblyLevel(AssemblyLevel::FULL);
+      // Sort the matrix column indices when running on GPU or with OpenMP (i.e.
+      // when Device::IsEnabled() returns true). This makes the results
+      // bit-for-bit deterministic at the cost of somewhat longer run time.
+      a.EnableSparseMatrixSorting(Device::IsEnabled());
+   }
   a.AddDomainIntegrator(new DiffusionIntegrator(one));

   // 10. Assemble the bilinear form and the corresponding linear system,
@@ -30,7 +30,7 @@
 //
 // Device sample runs:
 //               mpirun -np 4 ex1p -pa -d cuda
-//             * mpirun -np 4 ex1p -fa -d cuda
+//               mpirun -np 4 ex1p -fa -d cuda
 //               mpirun -np 4 ex1p -pa -d occa-cuda
 //               mpirun -np 4 ex1p -pa -d raja-omp
 //               mpirun -np 4 ex1p -pa -d ceed-cpu
@@ -219,7 +219,14 @@ int main(int argc, char *argv[])
   //     Diffusion domain integrator.
   ParBilinearForm a(&fespace);
   if (pa) { a.SetAssemblyLevel(AssemblyLevel::PARTIAL); }
-   if (fa) { a.SetAssemblyLevel(AssemblyLevel::FULL); }
+   if (fa)
+   {
+      a.SetAssemblyLevel(AssemblyLevel::FULL);
+      // Sort the matrix column indices when running on GPU or with OpenMP (i.e.
+      // when Device::IsEnabled() returns true). This makes the results
+      // bit-for-bit deterministic at the cost of somewhat longer run time.
+      a.EnableSparseMatrixSorting(Device::IsEnabled());
+   }
   a.AddDomainIntegrator(new DiffusionIntegrator(one));

   // 12. Assemble the parallel bilinear form and the corresponding linear
@@ -0,0 +1,60 @@
+# Copyright (c) 2010-2022, 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.
+
+set(IPOPT_EXAMPLES_SRCS)
+list(APPEND IPOPT_EXAMPLES_SRCS exContactBlockTL.cpp)
+
+# Include the source directory where mfem.hpp and mfem-performance.hpp are.
+include_directories(BEFORE ${PROJECT_BINARY_DIR})
+
+# Add "test_ipopt" target, see below.
+add_custom_target(test_ipopt
+  ${CMAKE_CTEST_COMMAND} -R ipopt USES_TERMINAL)
+
+# Add one executable per cpp file, adding "ipopt_" as prefix. Sets
+# "test_ipopt" as a target that depends on the given examples.
+set(PFX ipopt_)
+add_mfem_examples(IPOPT_EXAMPLES_SRCS ${PFX} "" test_ipopt)
+
+# Testing.
+# The IPOPT tests can be run separately using the target "test_ipopt"
+# which builds the examples and runs:
+#   ctest -R ipopt
+
+if (MFEM_ENABLE_TESTING)
+  # Command line options for the tests.
+  # Example 9:
+  set(EXCONTACTBTL_COMMON_OPTS -m ../../data/periodic-segment.mesh -p 0 -dt 0.005)
+  set(EXCONTACTBTL_TEST_OPTS ${EXCONTACTBTL_COMMON_OPTS}  -r  2 )
+
+  # Add the tests: one test per source file.
+  foreach(SRC_FILE ${IPOPT_EXAMPLES_SRCS})
+    get_filename_component(SRC_FILENAME ${SRC_FILE} NAME)
+    string(REPLACE ".cpp" "" TEST_NAME ${SRC_FILENAME})
+    string(TOUPPER ${TEST_NAME} UP_TEST_NAME)
+    set(TEST_NAME ${PFX}${TEST_NAME})
+
+    set(THIS_TEST_OPTIONS "-no-vis")
+    list(APPEND THIS_TEST_OPTIONS ${${UP_TEST_NAME}_TEST_OPTS})
+    # message(STATUS "Test ${TEST_NAME} options: ${THIS_TEST_OPTIONS}")
+
+    if (NOT (${TEST_NAME} MATCHES ".*p$"))
+      add_test(NAME ${TEST_NAME}_ser
+        COMMAND ${TEST_NAME} ${THIS_TEST_OPTIONS})
+    else()
+      add_test(NAME ${TEST_NAME}_np=4
+        COMMAND ${MPIEXEC} ${MPIEXEC_NUMPROC_FLAG} 4
+        ${MPIEXEC_PREFLAGS}
+        $<TARGET_FILE:${TEST_NAME}> ${THIS_TEST_OPTIONS}
+        ${MPIEXEC_POSTFLAGS})
+    endif()
+  endforeach()
+endif()
@@ -0,0 +1,19 @@
+                    Finite Element Discretization Library
+                                   __
+                       _ __ ___   / _|  ___  _ __ ___
+                      | '_ ` _ \ | |_  / _ \| '_ ` _ \
+                      | | | | | ||  _||  __/| | | | | |
+                      |_| |_| |_||_|   \___||_| |_| |_|
+
+                               https://mfem.org
+
+This directory contains modifications of the example codes that illustrate the
+use of MFEM for solving nonlinear constrained optimization problems, including
+features based on the IpOpt, a lightweight HPC solver for nonlinear optimization
+problems.
+
+To use the Ipopt features, make sure that MFEM is configured with the option
+"MFEM_USE_IPOPT = YES", see the top-level INSTALL file for details.
+
+We recommend comparing the original example codes with the corresponding files
+in the current directory.
@@ -0,0 +1,103 @@
+MFEM mesh v1.0
+
+#
+# MFEM Geometry Types (see mesh/geom.hpp):
+#
+# POINT       = 0
+# SEGMENT     = 1
+# TRIANGLE    = 2
+# SQUARE      = 3
+# TETRAHEDRON = 4
+# CUBE        = 5
+# PRISM       = 6
+#
+
+dimension
+3
+
+elements
+9
+1   5     0      1     3     2     8    9     11    10
+1   5     2      3     5     4    10    11    13    12
+1   5     4      5     7     6    12    13    15    14
+1   5     8      9    11    10    16    17    19    18
+1   5     10    11    13    12    18    19    21    20
+1   5     12    13    15    14    20    21    23    22
+1   5     16    17    19    18    24    25    27    26
+1   5     18    19    21    20    26    27    29    28
+1   5     20    21    23    22    28    29    31    30
+
+
+
+# 0 nothing
+# 1 dirichlet bc
+# 2 contact
+boundary
+30
+0   3    1     0     2     3
+0   3    3     2     4     5
+0   3    5     4     6     7
+0   3    24    25    27    26
+0   3    26    27    29    28
+0   3    28    29    31    30
+1   3    2     0     8    10
+1   3    4     2    10    12
+1   3    6     4    12    14
+1  3    10    8    16    18
+1  3    12    10    18    20
+1  3    14    12    20    22
+1  3    18    16    24    26
+1  3    20    18    26    28
+1  3    22    20    28    30
+2  3    1     3    11    9
+2  3     3     5    13    11
+2  3     5     7    15    13
+2  3    9    11    19    17
+2  3    11    13    21    19
+2  3    13    15    23    21
+2  3    17    19    27    25
+2  3    19    21    29    27
+2  3    21    23    31    29
+0  3     8     0     1    9
+0  3    16     8    9    17
+0  3    24    16    17    25
+0  3     6    14    15     7
+0  3    14    22    23    15
+0  3    22    30    31    23
+
+
+vertices
+32
+3
+   -1.0000         0         0
+         0         0         0
+   -1.0000    0.3333         0
+         0    0.3333         0
+   -1.0000    0.6667         0
+         0    0.6667         0
+   -1.0000    1.0000         0
+         0    1.0000         0
+   -1.0000         0    0.3333
+         0         0    0.3333
+   -1.0000    0.3333    0.3333
+         0    0.3333    0.3333
+   -1.0000    0.6667    0.3333
+         0    0.6667    0.3333
+   -1.0000    1.0000    0.3333
+         0    1.0000    0.3333
+   -1.0000         0    0.6667
+         0         0    0.6667
+   -1.0000    0.3333    0.6667
+         0    0.3333    0.6667
+   -1.0000    0.6667    0.6667
+         0    0.6667    0.6667
+   -1.0000    1.0000    0.6667
+         0    1.0000    0.6667
+   -1.0000         0    1.0000
+         0         0    1.0000
+   -1.0000    0.3333    1.0000
+         0    0.3333    1.0000
+   -1.0000    0.6667    1.0000
+         0    0.6667    1.0000
+   -1.0000    1.0000    1.0000
+         0    1.0000    1.0000
@@ -0,0 +1,68 @@
+MFEM mesh v1.0
+
+#
+# MFEM Geometry Types (see mesh/geom.hpp):
+#
+# POINT       = 0
+# SEGMENT     = 1
+# TRIANGLE    = 2
+# SQUARE      = 3
+# TETRAHEDRON = 4
+# CUBE        = 5
+# PRISM       = 6
+#
+
+dimension
+3
+
+# 1 nothing
+elements
+4
+1   5     0     1     3     2     6     7    9     8
+1   5     2     3     5     4     8     9    11    10
+1   5     6     7     9     8    12    13    15    14
+1   5     8     9    11    10    14    15    17    16
+
+# 0 nothing
+# 1 dirichlet bc
+# 2 contact
+boundary
+16
+0   3     1     0     2     3
+0   3     3     2     4     5
+0   3    12    13    15    14
+0   3    14    15    17    16
+2   3     2     0     6     8
+2   3     4     2     8    10
+2   3     8     6    12    14
+2   3    10     8    14    16
+1   3     1     3     9     7
+1   3     3     5    11    9
+1   3     7    9    15    13
+1   3     9    11    17    15
+0   3     6     0     1     7
+0   3    12     6     7    13
+0   3     4    10    11     5
+0   3    10    16    17    11
+
+vertices
+18
+3
+         0    0.2464    0.2464
+    0.5071    0.2464    0.2464
+         0    0.5000    0.2464
+    0.5071    0.5000    0.2464
+         0    0.7536    0.2464
+    0.5071    0.7536    0.2464
+         0    0.2464    0.5000
+    0.5071    0.2464    0.5000
+         0    0.5000    0.5000
+    0.5071    0.5000    0.5000
+         0    0.7536    0.5000
+    0.5071    0.7536    0.5000
+         0    0.2464    0.7536
+    0.5071    0.2464    0.7536
+         0    0.5000    0.7536
+    0.5071    0.5000    0.7536
+         0    0.7536    0.7536
+    0.5071    0.7536    0.7536
@@ -0,0 +1,742 @@
+//                                Contact example
+//
+// Compile with: make contact
+//
+// Sample runs:  ./contact -m1 block1.mesh -m2 block2.mesh -at "5 6 7 8"
+// Sample runs:  ./contact -m1 block1_d.mesh -m2 block2_d.mesh -at "5 6 7 8"
+
+#include "mfem.hpp"
+#include <fstream>
+#include <iostream>
+#include "nodepair.hpp"
+
+using namespace std;
+using namespace mfem;
+
+bool ifequalarray(const Array<int> a1, const Array<int> a2)
+{
+   if (a1.Size()!=a2.Size())
+   {
+      return false;
+   }
+   for (int i=0; i<a1.Size(); i++)
+   {
+      if (a1[i] != a2[i])
+      {
+         return false;
+      }
+   }
+   return true;
+}
+
+void FindSurfaceToProject(Mesh& mesh, const int elem, int& cbdrface)
+{
+   Array<int> attr;
+   attr.Append(2);
+   Array<int> faces;
+   Array<int> ori;
+   std::vector<Array<int> > facesVertices;
+   std::vector<int > faceid;
+   mesh.GetElementFaces(elem, faces, ori);
+   int face = -1;
+   for (int i=0; i<faces.Size(); i++)
+   {
+      face = faces[i];
+      Array<int> faceVert;
+      if (!mesh.FaceIsInterior(face)) // if on the boundary
+      {
+         mesh.GetFaceVertices(face, faceVert);
+         faceVert.Sort();
+         facesVertices.push_back(faceVert);
+         faceid.push_back(face);
+      }
+   }
+   int bdrface = facesVertices.size();
+
+   Array<int> bdryFaces;
+   // This shoulnd't need to be rebuilt
+   std::vector<Array<int> > bdryVerts;
+   for (int b=0; b<mesh.GetNBE(); ++b)
+   {
+      if (attr.FindSorted(mesh.GetBdrAttribute(b)) >= 0)  // found the contact surface
+      {
+         bdryFaces.Append(b);
+         Array<int> vert;
+         mesh.GetBdrElementVertices(b, vert);
+         vert.Sort();
+         bdryVerts.push_back(vert);
+      }
+   }
+
+   int bdrvert = bdryVerts.size();
+   cbdrface = -1;  // the face number of the contact surface element
+   int count_cbdrface = 0;  // the number of matching surfaces, used for checks
+
+   for (int i=0; i<bdrface; i++)
+   {
+      for (int j=0; j<bdrvert; j++)
+      {
+         if (ifequalarray(facesVertices[i], bdryVerts[j]))
+         {
+            cbdrface = faceid[i];
+            count_cbdrface += 1;
+         }
+      }
+   }
+   MFEM_VERIFY(count_cbdrface == 1,"projection surface not found");
+
+};
+
+Vector GetNormalVector(Mesh & mesh, const int elem, const double *ref,
+                       int & refFace, int & refNormal, bool & interior)
+{
+   ElementTransformation *trans = mesh.GetElementTransformation(elem);
+   const int dim = mesh.Dimension();
+   const int spaceDim = trans->GetSpaceDim();
+
+   MFEM_VERIFY(spaceDim == 3, "");
+
+   Vector n(spaceDim);
+
+   IntegrationPoint ip;
+   ip.Set(ref, dim);
+
+   trans->SetIntPoint(&ip);
+   //CalcOrtho(trans->Jacobian(), n);  // Works only for face transformations
+   const DenseMatrix jac = trans->Jacobian();
+
+   int dimNormal = -1;
+   int normalSide = -1;
+
+   const double tol = 1.0e-8;
+   for (int i=0; i<dim; ++i)
+   {
+      const double d0 = std::abs(ref[i]);
+      const double d1 = std::abs(ref[i] - 1.0);
+
+      const double d = std::min(d0, d1);
+      // TODO: this works only for hexahedral meshes!
+
+      if (d < tol)
+      {
+         MFEM_VERIFY(dimNormal == -1, "");
+         dimNormal = i;
+
+         if (d0 < tol)
+         {
+            normalSide = 0;
+         }
+         else
+         {
+            normalSide = 1;
+         }
+      }
+   }
+   // closest point on the boundary
+   if (dimNormal < 0 || normalSide < 0) // node is inside the element
+   {
+      interior = 1;
+      Vector n(3);
+      n = 0.0;
+      return n;
+   }
+
+   MFEM_VERIFY(dimNormal >= 0 && normalSide >= 0, "");
+   refNormal = dimNormal;
+
+   MFEM_VERIFY(dim == 3, "");
+
+   {
+      // Find the reference face
+      if (dimNormal == 0)
+      {
+         refFace = (normalSide == 1) ? 2 : 4;
+      }
+      else if (dimNormal == 1)
+      {
+         refFace = (normalSide == 1) ? 3 : 1;
+      }
+      else
+      {
+         refFace = (normalSide == 1) ? 5 : 0;
+      }
+   }
+
+   std::vector<Vector> tang(2);
+
+   int tangDir[2] = {-1, -1};
+   {
+      int t = 0;
+      for (int i=0; i<dim; ++i)
+      {
+         if (i != dimNormal)
+         {
+            tangDir[t] = i;
+            t++;
+         }
+      }
+
+      MFEM_VERIFY(t == 2, "");
+   }
+
+   for (int i=0; i<2; ++i)
+   {
+      tang[i].SetSize(3);
+
+      Vector tangRef(3);
+      tangRef = 0.0;
+      tangRef[tangDir[i]] = 1.0;
+
+      jac.Mult(tangRef, tang[i]);
+   }
+
+   Vector c(3);  // Cross product
+
+   c[0] = (tang[0][1] * tang[1][2]) - (tang[0][2] * tang[1][1]);
+   c[1] = (tang[0][2] * tang[1][0]) - (tang[0][0] * tang[1][2]);
+   c[2] = (tang[0][0] * tang[1][1]) - (tang[0][1] * tang[1][0]);
+
+   c /= c.Norml2();
+
+   Vector nref(3);
+   nref = 0.0;
+   nref[dimNormal] = 1.0;
+
+   Vector ndir(3);
+   jac.Mult(nref, ndir);
+
+   ndir /= ndir.Norml2();
+
+   const double dp = ndir * c;
+
+   // TODO: eliminate c?
+   n = c;
+   if (dp < 0.0)
+   {
+      n *= -1.0;
+   }
+   interior = 0;
+   return n;
+}
+
+// WARNING: global variable, just for this little example.
+std::array<std::array<int, 3>, 8> HEX_VERT =
+{
+   {  {0,0,0},
+      {1,0,0},
+      {1,1,0},
+      {0,1,0},
+      {0,0,1},
+      {1,0,1},
+      {1,1,1},
+      {0,1,1}
+   }
+};
+
+int GetHexVertex(int cdim, int c, int fa, int fb, Vector & refCrd)
+{
+   int ref[3];
+   ref[cdim] = c;
+   ref[cdim == 0 ? 1 : 0] = fa;
+   ref[cdim == 2 ? 1 : 2] = fb;
+
+   for (int i=0; i<3; ++i) { refCrd[i] = ref[i]; }
+
+   int refv = -1;
+
+   for (int i=0; i<8; ++i)
+   {
+      bool match = true;
+      for (int j=0; j<3; ++j)
+      {
+         if (ref[j] != HEX_VERT[i][j]) { match = false; }
+      }
+
+      if (match) { refv = i; }
+   }
+
+   MFEM_VERIFY(refv >= 0, "");
+
+   return refv;
+}
+
+// Coordinates in xyz are assumed to be ordered as [X, Y, Z]
+// where X is the list of x-coordinates for all points and so on.
+// conn: connectivity of the target surface elements
+// xi: surface reference cooridnates for the cloest point, involves a linear transformation from [0,1] to [-1,1]
+void FindPointsInMesh(Mesh & mesh, Vector const& xyz, Array<int>& conn,
+                      Vector& xi)
+{
+   const int dim = mesh.Dimension();
+   const int np = xyz.Size() / dim;
+
+   MFEM_VERIFY(np * dim == xyz.Size(), "");
+
+   mesh.EnsureNodes();
+
+   //FindPointsGSLIB finder(MPI_COMM_WORLD);
+   FindPointsGSLIB finder;
+
+   finder.SetDistanceToleranceForPointsFoundOnBoundary(0.5);
+
+   const double bb_t = 0.5;
+   finder.Setup(mesh, bb_t);
+
+   finder.FindPoints(xyz);
+
+   /// Return code for each point searched by FindPoints: inside element (0), on
+   /// element boundary (1), or not found (2).
+   Array<unsigned int> codes = finder.GetCode();
+
+   /// Return element number for each point found by FindPoints.
+   Array<unsigned int> elems = finder.GetElem();
+
+   /// Return reference coordinates for each point found by FindPoints.
+   Vector refcrd = finder.GetReferencePosition();
+
+   /// Return distance between the sought and the found point in physical space,
+   /// for each point found by FindPoints.
+   Vector dist = finder.GetDist();
+
+   MFEM_VERIFY(dist.Size() == np, "");
+   MFEM_VERIFY(refcrd.Size() == np * dim, "");
+   MFEM_VERIFY(elems.Size() == np, "");
+   MFEM_VERIFY(codes.Size() == np, "");
+
+   bool allfound = true;
+   for (auto code : codes)
+      if (code == 2) { allfound = false; }
+
+   MFEM_VERIFY(allfound, "A point was not found");
+
+   cout << "Maximum distance of projected points: " << dist.Max() << endl;
+
+   // extract information
+   for (int i=0; i<np; ++i)
+   {
+      cout << "Point " << i << ": (";
+      for (int j=0; j<dim; ++j)
+      {
+         cout << xyz[i + (j*np)];
+         if (j == dim-1) {cout << ")" << endl;}
+         else {cout << ", ";}
+      }
+      //cout << "  element: " << elems[i] << endl;
+      //cout << "  element " << elems[i] << " vertices:" << endl;
+      //Array<int> vert;
+      //mesh.GetElementVertices(elems[i], vert);
+      //for (auto v : vert)
+      //{
+      //   cout << "    " << v << endl;
+      //}
+
+      /*cout << "  reference coordinates: (";
+      for (int j=0; j<dim; ++j)
+      {
+         cout << refcrd[(i*dim) + j];
+         if (j == dim-1)
+         {
+            cout << ")" << endl;
+         }
+         else
+         {
+            cout << ", ";
+         }
+      }*/
+
+      int refFace, refNormal, refNormalSide;
+      bool is_interior = -1;
+      Vector normal = GetNormalVector(mesh, elems[i], refcrd.GetData() + (i*dim),
+                                      refFace, refNormal, is_interior);
+      int phyFace;
+      if (is_interior)
+      {
+         phyFace = -1; // the id of the face that has the closest point
+         FindSurfaceToProject(mesh, elems[i], phyFace);
+
+         Array<int> cbdrVert;
+         mesh.GetFaceVertices(phyFace, cbdrVert);
+         Vector xs(dim);
+         xs[0] = xyz[i + 0*np];
+         xs[1] = xyz[i + 1*np];
+         xs[2] = xyz[i + 2*np];
+         Vector xi_tmp(dim-1);
+         // get nodes!
+
+         GridFunction *nodes = mesh.GetNodes();
+         DenseMatrix coords(4,3);
+         for (int i=0; i<4; i++)
+         {
+            for (int j=0; j<3; j++)
+            {
+               coords(i,j) = (*nodes)[cbdrVert[i]*3+j];
+            }
+         }
+         SlaveToMaster(coords, xs, xi_tmp);
+
+         for (int j=0; j<dim-1; ++j)
+         {
+            xi[i*(dim-1)+j] = xi_tmp[j];
+         }
+         // now get get the projection to the surface
+      }
+      else
+      {
+         Vector faceRefCrd(dim-1);
+         {
+            int fd = 0;
+            for (int j=0; j<dim; ++j)
+            {
+               if (j == refNormal)
+               {
+                  refNormalSide = (refcrd[(i*dim) + j] > 0.5);
+               }
+               else
+               {
+                  faceRefCrd[fd] = refcrd[(i*dim) + j];
+                  fd++;
+               }
+            }
+
+            MFEM_VERIFY(fd == dim-1, "");
+         }
+
+         for (int j=0; j<dim-1; ++j)
+         {
+            xi[i*(dim-1)+j] = faceRefCrd[j]*2.0 - 1.0;
+         }
+         //cout << "  face reference coordinates: (";
+         for (int j=0; j<dim-1; ++j)
+         {
+            cout << faceRefCrd[j];
+            if (j == dim-2) {cout << ")" << endl;}
+            else {cout << ", ";}
+         }
+      }
+      //cout << "  normal vector: ";
+      //normal.Print();
+
+      // ask, does this do anything?
+      /*
+      IntegrationPoint ip;
+      ip.Set(refcrd.GetData() + (i*dim), dim);
+      ElementTransformation *trans = mesh.GetElementTransformation(elems[i]);
+      Vector phys(trans->GetSpaceDim());
+      trans->Transform(ip, phys);
+      cout << "  physical coordinates: ";
+      phys.Print();
+      */
+
+      // Get the element face
+      Array<int> faces;
+      Array<int> ori;
+      int face;
+
+      if (is_interior)
+      {
+         face = phyFace;
+      }
+      else
+      {
+         mesh.GetElementFaces(elems[i], faces, ori);
+         face = faces[refFace];
+      }
+
+      Array<int> faceVert;
+      mesh.GetFaceVertices(face, faceVert);
+
+      //cout << "  face " << face << " vertices:" << endl;
+      //for (auto v : faceVert){ cout << "    " << v << endl;}
+
+      for (int p=0; p<4; p++)
+      {
+         conn[4*i+p] = faceVert[p];
+      }
+      /*
+      Vector ref(dim);
+
+      for (int p=0; p<2; ++p)
+         for (int q=0; q<2; ++q)
+         {
+            const int refv = GetHexVertex(refNormal, refNormalSide, p, q, ref);
+            cout << "  face reference vertex (" << p << "," << q
+                 << ") is global vertex " << vert[refv] << endl;
+
+            {
+               // Sanity check
+               ip.Set(ref.GetData(), dim);
+               trans->Transform(ip, phys);
+               for (int j=0; j<dim; ++j)
+               {
+                  phys[j] -= mesh.GetVertex(vert[refv])[j];
+               }
+          phys.Print();
+          cout<<vert[refv]<<endl;
+               cout<<mesh.GetVertex(vert[refv])[0]<<endl;
+               cout<<mesh.GetVertex(vert[refv])[1]<<endl;
+               cout<<mesh.GetVertex(vert[refv])[2]<<endl;
+               MFEM_VERIFY(phys.Norml2() < 1.0e-12, "Sanity check failed");
+            }
+         }*/
+   }
+}
+
+int main(int argc, char *argv[])
+{
+   // 1. Parse command-line options.
+   const char *mesh_file1 = "block1.mesh";
+   const char *mesh_file2 = "block2.mesh";
+
+   Array<int> attr;
+   Array<int> m_attr;
+
+   OptionsParser args(argc, argv);
+   args.AddOption(&mesh_file1, "-m1", "--mesh1",
+                  "First mesh file to use.");
+   args.AddOption(&mesh_file2, "-m2", "--mesh2",
+                  "Second mesh file to use.");
+   args.AddOption(&attr, "-at", "--attributes-surf",
+                  "Attributes of boundary faces on contact surface for mesh 2.");
+   args.Parse();
+   if (!args.Good())
+   {
+      args.PrintUsage(cout);
+      return 1;
+   }
+   args.PrintOptions(cout);
+
+   Mesh mesh1(mesh_file1, 1, 1);
+   Mesh mesh2(mesh_file2, 1, 1);
+
+   const int dim = mesh1.Dimension();
+   MFEM_VERIFY(dim == mesh2.Dimension(), "");
+
+   // boundary attribute 2 is the potential contact surface of nodes
+   attr.Append(2);
+   // boundary attribute 2 is the potential contact surface for master surface
+   m_attr.Append(2);
+
+   //  Define a finite element space on the mesh. Here we use vector finite
+   //  elements, i.e. dim copies of a scalar finite element space. The vector
+   //  dimension is specified by the last argument of the FiniteElementSpace
+   //  constructor.
+   FiniteElementCollection *fec1;
+   FiniteElementSpace *fespace1;
+   fec1 = new H1_FECollection(1, dim);
+   fespace1 = new FiniteElementSpace(&mesh1, fec1, dim, Ordering::byVDIM);
+   cout << "Number of finite element unknowns for mesh1: "
+        << fespace1->GetTrueVSize() << endl;
+   mesh1.SetNodalFESpace(fespace1);
+   GridFunction nodes0 = *mesh1.GetNodes(); // undeformed mesh1 nodal grid function
+   GridFunction *nodes1 = mesh1.GetNodes();
+
+   FiniteElementCollection *fec2 = new H1_FECollection(1, dim);
+   FiniteElementSpace *fespace2 = new FiniteElementSpace(&mesh2, fec2, dim,
+                                                         Ordering::byVDIM);
+   cout << "Number of finite element unknowns for mesh2: "
+        << fespace2->GetTrueVSize() << endl;
+
+   // degrees of freedom of both meshes
+   int ndof_1 = fespace1->GetTrueVSize();
+   int ndof_2 = fespace2->GetTrueVSize();
+   int ndofs = ndof_1 + ndof_2;
+   // number of nodes for each mesh
+   int nnd_1 = mesh1.GetNV();
+   int nnd_2 = mesh2.GetNV();
+   int nnd = nnd_1 + nnd_2;
+   // Determine the list of true (i.e. conforming) essential boundary dofs.
+   // In this example, the boundary conditions are defined by marking only
+   // boundary attribute 1 from the mesh as essential and converting it to a
+   // list of true dofs.
+   Array<int> ess_tdof_list1, ess_bdr1(mesh1.bdr_attributes.Max());
+   cout<<mesh1.bdr_attributes.Max()<<endl;
+   ess_bdr1 = 0;
+   //ess_bdr1[0] = 1;
+   // Not ready to be passed on yet
+   // fespace->GetEssentialTrueDofs(ess_bdr, ess_tdof_list);
+
+   Array<int> ess_tdof_list2, ess_bdr2(mesh2.bdr_attributes.Max());
+   ess_bdr2 = 0;
+   //ess_bdr2[0] = 1;
+
+   // Define the displacement vector x as a finite element grid function
+   // corresponding to fespace. GridFunction is a derived class of Vector.
+   GridFunction x1(fespace1);
+   x1 = 0.0;
+   GridFunction x2(fespace2);
+   x2 = 0.0;
+
+   // Generate force
+   LinearForm *b1 = new LinearForm(fespace1);
+   b1->Assemble();
+
+   LinearForm *b2 = new LinearForm(fespace2);
+   b2->Assemble();
+
+   // Set up the bilinear form a(.,.) on the finite element space
+   //  corresponding to the linear elasticity integrator with piece-wise
+   //  constants coefficient lambda and mu.
+   Vector lambda1(mesh1.attributes.Max());
+   lambda1 = 57.6923076923;
+   PWConstCoefficient lambda1_func(lambda1);
+   Vector mu1(mesh1.attributes.Max());
+   mu1 = 38.4615384615;
+   PWConstCoefficient mu1_func(mu1);
+
+   BilinearForm *a1 = new BilinearForm(fespace1);
+   a1->AddDomainIntegrator(new ElasticityIntegrator(lambda1_func,mu1_func));
+
+   Vector lambda2(mesh2.attributes.Max());
+   lambda2 = 57.6923076923;
+   PWConstCoefficient lambda2_func(lambda2);
+   Vector mu2(mesh2.attributes.Max());
+   mu2 = 38.4615384615;
+   PWConstCoefficient mu2_func(mu2);
+
+   BilinearForm *a2 = new BilinearForm(fespace2);
+   a2->AddDomainIntegrator(new ElasticityIntegrator(lambda2_func,mu2_func));
+
+   a1->Assemble();
+   SparseMatrix A1;
+   Vector B1, X1;
+   a1->FormLinearSystem(ess_tdof_list1, x1, *b1, A1, X1, B1);
+
+   a2->Assemble();
+   SparseMatrix A2;
+   Vector B2, X2;
+   a2->FormLinearSystem(ess_tdof_list2, x2, *b2, A2, X2, B2);
+
+   // Combine elasticity operator for two meshes into one.
+   // Block Matrix
+   SparseMatrix K(ndofs,ndofs);
+   for (int i=0; i<A1.Height(); i++)
+   {
+      Array<int> col_tmp;
+      Vector v_tmp;
+      col_tmp = 0;
+      v_tmp = 0.0;
+      A1.GetRow(i, col_tmp, v_tmp);
+      K.SetRow(i, col_tmp, v_tmp);
+   }
+   for (int i=0; i<A2.Height(); i++)
+   {
+      Array<int> col_tmp;
+      Vector v_tmp;
+      col_tmp = 0;
+      v_tmp = 0.0;
+      A2.GetRow(i, col_tmp, v_tmp);
+      for (int j=0; j<col_tmp.Size(); j++)
+      {
+         col_tmp[j] += ndof_1;
+      }
+      K.SetRow(i+ndof_1, col_tmp, v_tmp);  // mesh1 top left corner
+   }
+
+   // Construct node to segment contact constraint.
+
+   attr.Sort();
+   cout << "Boundary attributes for contact surface faces in mesh 2" << endl;
+   for (auto a : attr)
+   {
+      cout << a << endl;
+   }
+
+   Array<int> bdryFaces2;  // TODO: remove this?
+
+   std::set<int> bdryVerts2;
+   for (int b=0; b<mesh2.GetNBE(); ++b)
+   {
+      if (attr.FindSorted(mesh2.GetBdrAttribute(b)) >= 0)
+      {
+         bdryFaces2.Append(b);
+         Array<int> vert;
+         mesh2.GetBdrElementVertices(b, vert);
+         for (auto v : vert)
+         {
+            bdryVerts2.insert(v);
+         }
+      }
+   }
+
+   int npoints = bdryVerts2.size();
+   Array<int> s_conn(npoints); // connectivity of the second/slave mesh
+   Vector xyz(dim * npoints);
+   xyz = 0.0;
+
+   cout << "Boundary vertices for contact surface vertices in mesh 2" << endl;
+
+   // construct the nodal coordinates on mesh2 to be projected, including displacement
+   int count = 0;
+   for (auto v : bdryVerts2)
+   {
+      cout << v << ": " << mesh2.GetVertex(v)[0] << ", "
+           << mesh2.GetVertex(v)[1] << ", "
+           << mesh2.GetVertex(v)[2] << endl;
+
+      for (int i=0; i<dim; ++i)
+      {
+         xyz[count + (i * npoints)] = mesh2.GetVertex(v)[i] + x2[v*dim+i];
+      }
+
+      s_conn[count] = v + nnd_1; // dof1 is the master
+      count++;
+   }
+
+   MFEM_VERIFY(count == npoints, "");
+
+   // gap function
+   Vector g(npoints*dim);
+   g = -1.0;
+   // segment reference coordinates of the closest point
+   Vector m_xi(npoints*(dim-1));
+   m_xi = -1.0;
+   Vector xs(dim*npoints);
+   xs = 0.0;
+   for (int i=0; i<npoints; i++)
+   {
+      for (int j=0; j<dim; j++)
+      {
+         xs[i*dim+j] = xyz[i + (j*npoints)];
+      }
+   }
+
+   Array<int> m_conn(
+      npoints*4); // only works for linear elements that have 4 vertices!
+   DenseMatrix coordsm(npoints*4, dim);
+
+   // adding displacement to mesh1 using a fixed grid function from mesh1
+   x1 = 1e-4; // x1 order: [xyz xyz... xyz]
+   add(nodes0, x1, *nodes1);
+
+   FindPointsInMesh(mesh1, xyz, m_conn, m_xi);
+
+   for (int i=0; i<npoints; i++)
+   {
+      for (int j=0; j<4; j++)
+      {
+         for (int k=0; k<dim; k++)
+         {
+            coordsm(i*4+j,k) = mesh1.GetVertex(m_conn[i*4+j])[k]+x1[dim*m_conn[i*4+j]+k];
+         }
+      }
+   }
+   //coordsm.Print();
+   SparseMatrix M(nnd,ndofs);
+   std::vector<SparseMatrix> dM(nnd, SparseMatrix(ndofs,ndofs));
+
+   Assemble_Contact(nnd, npoints, ndofs, xs, m_xi, coordsm,
+                    s_conn, m_conn, g, M, dM);
+
+   //M.Print();
+   /*Vector eps(ndofs);
+   Vector sol(ndofs); sol = 0.;
+   for(int i=0;i<ndofs;i++) eps[i] = 1e-5 * i ;
+   for(int i=0;i<9;i++)
+   {
+     cout<<i<<endl;
+     dM[s_conn[i]].Mult(eps,sol);
+     sol.Print();
+   }
+   */
+   return 0;
+}
@@ -0,0 +1,230 @@
+//                                Contact example
+//
+// Compile with: make exContactBlockTL
+//
+// Sample runs:  ./exContactBlockTL -m1 block1.mesh -m2 block2.mesh -at "5 6 7 8"
+// Sample runs:  ./exContactBlockTL -m1 block1_d.mesh -m2 block2_d.mesh -at "5 6 7 8"
+
+#ifndef EXCONTACTBLOCKTL_HPP
+#define EXCONTACTBLOCKTL_HPP
+
+#include "mfem.hpp"
+#include "IpTNLP.hpp"
+
+using namespace std;
+using namespace mfem;
+using namespace Ipopt;
+
+
+class ExContactBlockTL: public TNLP
+{
+public:
+   /** default constructor */
+   ExContactBlockTL(int argc, char *argv[]);
+
+   /** default destructor */
+   virtual ~ExContactBlockTL();
+
+   /**@name Overloaded from TNLP */
+   /** Method to return some info about the nlp */
+   virtual bool get_nlp_info(
+      Index&          n,
+      Index&          m,
+      Index&          nnz_jac_g,
+      Index&          nnz_h_lag,
+      IndexStyleEnum& index_style
+   );
+
+   /** Method to return the bounds for my problem */
+   virtual bool get_bounds_info(
+      Index   n,
+      Number* x_l,
+      Number* x_u,
+      Index   m,
+      Number* g_l,
+      Number* g_u
+   );
+
+   /** Method to return the starting point for the algorithm */
+   virtual bool get_starting_point(
+      Index   n,
+      bool    init_x,
+      Number* x,
+      bool    init_z,
+      Number* z_L,
+      Number* z_U,
+      Index   m,
+      bool    init_lambda,
+      Number* lambda
+   );
+
+   /** Method to return the objective value */
+   virtual bool eval_f(
+      Index         n,
+      const Number* x,
+      bool          new_x,
+      Number&       obj_value
+   );
+
+   /** Method to return the gradient of the objective */
+   virtual bool eval_grad_f(
+      Index         n,
+      const Number* x,
+      bool          new_x,
+      Number*       grad_f
+   );
+
+   /** Method to return the constraint residuals */
+   virtual bool eval_g(
+      Index         n,
+      const Number* x,
+      bool          new_x,
+      Index         m,
+      Number*       cons
+   );
+
+   /** Method to return:
+    *   1) The structure of the Jacobian (if "values" is NULL)
+    *   2) The values of the Jacobian (if "values" is not NULL)
+    */
+   virtual bool eval_jac_g(
+      Index         n,
+      const Number* x,
+      bool          new_x,
+      Index         m,
+      Index         nele_jac,
+      Index*        iRow,
+      Index*        jCol,
+      Number*       values
+   );
+
+   /** Method to return:
+    *   1) The structure of the Hessian of the Lagrangian (if "values" is NULL)
+    *   2) The values of the Hessian of the Lagrangian (if "values" is not NULL)
+    */
+   virtual bool eval_h(
+      Index         n,
+      const Number* x,
+      bool          new_x,
+      Number        obj_factor,
+      Index         m,
+      const Number* lambda,
+      bool          new_lambda,
+      Index         nele_hess,
+      Index*        iRow,
+      Index*        jCol,
+      Number*       values
+   );
+
+   /** This method is called when the algorithm is complete so the TNLP can store/write the solution */
+   virtual void finalize_solution(
+      SolverReturn               status,
+      Index                      n,
+      const Number*              x,
+      const Number*              z_L,
+      const Number*              z_U,
+      Index                      m,
+      const Number*              g,
+      const Number*              lambda,
+      Number                     obj_value,
+      const IpoptData*           ip_data,
+      IpoptCalculatedQuantities* ip_cq
+   );
+
+private:
+   void update_g();
+   void update_jac();
+   void update_hess();
+
+private:
+   /**@name Methods to block default compiler methods.
+    *
+    * The compiler automatically generates the following three methods.
+    *  Since the default compiler implementation is generally not what
+    *  you want (for all but the most simple classes), we usually
+    *  put the declarations of these methods in the private section
+    *  and never implement them. This prevents the compiler from
+    *  implementing an incorrect "default" behavior without us
+    *  knowing. (See Scott Meyers book, "Effective C++")
+    */
+   ExContactBlockTL(
+      const ExContactBlockTL&
+   );
+
+   ExContactBlockTL& operator=(
+      const ExContactBlockTL&
+   );
+
+   Array<int> attr;
+   Array<int> m_attr;
+   Array<int> s_conn; // connectivity of the second/slave mesh
+   std::string mesh_file1;
+   std::string mesh_file2;
+   Mesh* mesh1;
+   Mesh* mesh2;
+   FiniteElementCollection* fec1;
+   FiniteElementCollection* fec2;
+   FiniteElementSpace* fespace1;
+   FiniteElementSpace* fespace2;
+   Array<int> ess_tdof_list1;
+   Array<int> ess_tdof_list2;
+   GridFunction nodes0;
+   GridFunction* nodes1;
+   GridFunction* nodes2;
+   GridFunction* x1;
+   GridFunction* x2;
+   LinearForm* b1;
+   LinearForm* b2;
+   PWConstCoefficient* lambda1_func;
+   PWConstCoefficient* lambda2_func;
+   PWConstCoefficient* mu1_func;
+   PWConstCoefficient* mu2_func;
+   BilinearForm* a1;
+   BilinearForm* a2;
+
+   mfem::Vector lambda1;
+   mfem::Vector lambda2;
+   mfem::Vector mu1;
+   mfem::Vector mu2;
+   mfem::Vector xyz;
+
+   std::set<int> bdryVerts2;
+
+   int dim;
+   // degrees of freedom of both meshes
+   int ndof_1;
+   int ndof_2;
+   int ndofs;
+   // number of nodes for each mesh
+   int nnd_1;
+   int nnd_2;
+   int nnd;
+
+   int npoints;
+
+   SparseMatrix A1;
+   mfem::Vector B1, X1;
+   SparseMatrix A2;
+   mfem::Vector B2, X2;
+
+   SparseMatrix* K;
+   mfem::Vector g;
+   mfem::Vector m_xi;
+   mfem::Vector xs;
+
+   Array<int> m_conn; // only works for linear elements that have 4 vertices!
+   DenseMatrix* coordsm;
+   SparseMatrix* M;
+
+   std::vector<SparseMatrix>* dM;
+
+   Array<int> Dirichlet_dof;
+   Array<double> Dirichlet_val;
+
+public: 
+   Mesh * GetMesh1() {return mesh1;}
+   Mesh * GetMesh2() {return mesh2;}
+
+};
+
+#endif
@@ -0,0 +1,68 @@
+# Copyright (c) 2010-2022, 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.
+
+# Use the MFEM build directory
+MFEM_DIR ?= ../..
+MFEM_BUILD_DIR ?= ../..
+SRC = $(if $(MFEM_DIR:../..=),$(MFEM_DIR)/examples/ipopt/,)
+CONFIG_MK = $(MFEM_BUILD_DIR)/config/config.mk
+# Use the MFEM install directory
+# MFEM_INSTALL_DIR = ../../mfem
+# CONFIG_MK = $(MFEM_INSTALL_DIR)/share/mfem/config.mk
+
+MFEM_LIB_FILE = mfem_is_not_built
+-include $(CONFIG_MK)
+
+SEQ_EXAMPLES = exContactBlockTL
+EXAMPLES = $(SEQ_EXAMPLES)
+
+.SUFFIXES:
+.SUFFIXES: .o .cpp .mk
+.PHONY: all clean clean-build clean-exec
+
+# Remove built-in rule
+%: %.cpp
+
+# Replace the default implicit rule for *.cpp files
+%: $(SRC)%.cpp $(MFEM_LIB_FILE) $(CONFIG_MK)
+	$(MFEM_CXX) $(MFEM_FLAGS) $< -o $@ $(MFEM_LIBS)
+
+all: $(EXAMPLES)
+
+ifeq ($(MFEM_USE_IPOPT),NO)
+$(EXAMPLES):
+	$(error MFEM is not configured with IPOPT)
+endif
+
+MFEM_TESTS = EXAMPLES
+include $(MFEM_TEST_MK)
+
+# Testing: Parallel vs. serial runs
+RUN_MPI = $(MFEM_MPIEXEC) $(MFEM_MPIEXEC_NP) $(MFEM_MPI_NP)
+%-test-par: %
+	@$(call mfem-test,$<, $(RUN_MPI), Parallel example)
+%-test-seq: %
+	@$(call mfem-test,$<,, Serial example)
+
+# Testing: "test" target and mfem-test* variables are defined in config/test.mk
+
+# Generate an error message if the MFEM library is not built and exit
+$(MFEM_LIB_FILE):
+	$(error The MFEM library is not built)
+
+clean: clean-build clean-exec
+
+clean-build:
+	rm -f *.o *~ $(SEQ_EXAMPLES)
+	rm -rf *.dSYM *.TVD.*breakpoints
+
+clean-exec:
+	@rm -f exContactBlockTL.mesh exContactBlockTL-mesh.* exContactBlockTL-init.* exContactBlockTL-final.* ExampleContactBlockTL*
@@ -0,0 +1,888 @@
+
+using namespace std;
+using namespace mfem;
+
+void BasisEval(const Vector xi, Vector &N, DenseMatrix &dNdxi) // dNdxi is 2*4
+{
+   N[0] = 0.25*(1-xi[0])*(1-xi[1]);
+   N[1] = 0.25*(1+xi[0])*(1-xi[1]);
+   N[2] = 0.25*(1+xi[0])*(1+xi[1]);
+   N[3] = 0.25*(1-xi[0])*(1+xi[1]);
+
+   dNdxi(0,0) = 0.25*(-1+xi[1]);
+   dNdxi(0,1) = 0.25*(1-xi[1]);
+   dNdxi(0,2) = 0.25*(1+xi[1]);
+   dNdxi(0,3) = 0.25*(-1-xi[1]);
+   dNdxi(1,0) = 0.25*(-1+xi[0]);
+   dNdxi(1,1) = 0.25*(-1-xi[0]);
+   dNdxi(1,2) = 0.25*(1+xi[0]);
+   dNdxi(1,3) = 0.25*(1-xi[0]);
+}
+
+
+void BasisEvalDerivs(const Vector xi, Vector& N, DenseMatrix& dNdxi,
+                     DenseMatrix& dN2dxi)
+{
+   N[0] = 0.25*(1-xi[0])*(1-xi[1]);
+   N[1] = 0.25*(1+xi[0])*(1-xi[1]);
+   N[2] = 0.25*(1+xi[0])*(1+xi[1]);
+   N[3] = 0.25*(1-xi[0])*(1+xi[1]);
+
+   dNdxi.SetSize(2,4); dNdxi = 0.0;
+   dN2dxi.SetSize(3,4);
+   dN2dxi = 0.0;  // first row dxi2, second detadxi, third deta2
+
+   dNdxi(0,0) = 0.25*(-1+xi[1]); dNdxi(0,1) = 0.25*(1-xi[1]);
+   dNdxi(0,2) = 0.25*(1+xi[1]);  dNdxi(0,3) = 0.25*(-1-xi[1]);
+   dNdxi(1,0) = 0.25*(-1+xi[0]); dNdxi(1,1) = 0.25*(-1-xi[0]);
+   dNdxi(1,2) = 0.25*(1+xi[0]);  dNdxi(1,3) = 0.25*(1-xi[0]);
+
+   dN2dxi(1,0) = 0.25; dN2dxi(1,1) = -0.25; dN2dxi(1,2) = 0.25;
+   dN2dxi(1,3) = -0.25;
+}
+
+// returns the vector and matrix form of the shape functions and its derivative
+void BasisVectorDerivs(const Vector xi, DenseMatrix& N, DenseMatrix& dNdxi,
+                       DenseMatrix& ddNdxi)
+{
+   N.SetSize(3,12); N = 0.0;
+   N(0,0) = 0.25*(1-xi[0])*(1-xi[1]); N(0,3) = 0.25*(1+xi[0])*(1-xi[1]);
+   N(0,6) = 0.25*(1+xi[0])*(1+xi[1]); N(0,9) = 0.25*(1-xi[0])*(1+xi[1]);
+
+   N(1,1) = 0.25*(1-xi[0])*(1-xi[1]); N(1,4) = 0.25*(1+xi[0])*(1-xi[1]);
+   N(1,7) = 0.25*(1+xi[0])*(1+xi[1]); N(1,10) = 0.25*(1-xi[0])*(1+xi[1]);
+
+   N(2,2) = 0.25*(1-xi[0])*(1-xi[1]); N(2,5) = 0.25*(1+xi[0])*(1-xi[1]);
+   N(2,8) = 0.25*(1+xi[0])*(1+xi[1]); N(2,11) = 0.25*(1-xi[0])*(1+xi[1]);
+
+   dNdxi.SetSize(3*2, 3*4);  dNdxi = 0.0;
+   dNdxi(0,0) = 0.25*(-1+xi[1]);  dNdxi(0,3) = 0.25*(1-xi[1]);
+   dNdxi(0,6) = 0.25*(1+xi[1]);  dNdxi(0,9) = 0.25*(-1-xi[1]);
+   dNdxi(1,1) = 0.25*(-1+xi[1]);  dNdxi(1,4) = 0.25*(1-xi[1]);
+   dNdxi(1,7) = 0.25*(1+xi[1]);  dNdxi(1,10) = 0.25*(-1-xi[1]);
+   dNdxi(2,2) = 0.25*(-1+xi[1]);  dNdxi(2,5) = 0.25*(1-xi[1]);
+   dNdxi(2,8) = 0.25*(1+xi[1]);  dNdxi(2,11) = 0.25*(-1-xi[1]);
+
+   dNdxi(3,0) = 0.25*(-1+xi[0]);  dNdxi(3,3) = 0.25*(-1-xi[0]);
+   dNdxi(3,6) = 0.25*(1+xi[0]);  dNdxi(3,9) = 0.25*(1-xi[0]);
+   dNdxi(4,1) = 0.25*(-1+xi[0]);  dNdxi(4,4) = 0.25*(-1-xi[0]);
+   dNdxi(4,7) = 0.25*(1+xi[0]);  dNdxi(4,10) = 0.25*(1-xi[0]);
+   dNdxi(5,2) = 0.25*(-1+xi[0]);  dNdxi(5,5) = 0.25*(-1-xi[0]);
+   dNdxi(5,8) = 0.25*(1+xi[0]);  dNdxi(5,11) = 0.25*(1-xi[0]);
+
+   ddNdxi.SetSize(3*4, 3*4); ddNdxi = 0.0;
+   ddNdxi(3,0) = 0.25; ddNdxi(3,3) = -0.25;
+   ddNdxi(3,6) = 0.25; ddNdxi(3,9) = -0.25;
+   ddNdxi(4,1) = 0.25; ddNdxi(4,4) = -0.25;
+   ddNdxi(4,7) = 0.25; ddNdxi(4,10) = -0.25;
+   ddNdxi(5,2) = 0.25; ddNdxi(5,5) = -0.25;
+   ddNdxi(5,8) = 0.25; ddNdxi(5,11) = -0.25;
+
+   ddNdxi(6,0) = 0.25; ddNdxi(6,3) = -0.25;
+   ddNdxi(6,6) = 0.25; ddNdxi(6,9) = -0.25;
+   ddNdxi(7,1) = 0.25; ddNdxi(7,4) = -0.25;
+   ddNdxi(7,7) = 0.25; ddNdxi(7,10) = -0.25;
+   ddNdxi(8,2) = 0.25; ddNdxi(8,5) = -0.25;
+   ddNdxi(8,8) = 0.25; ddNdxi(8,11) = -0.25;
+}
+
+
+void cross(const Vector a, const Vector b, Vector& c)
+{
+   assert(a.Size()==3);
+   c.SetSize(3);
+   c[0] = a[1]*b[2] - a[2]*b[1];
+   c[1] = -a[0]*b[2] + b[0]*a[2];
+   c[2] = a[0]*b[1] - a[1]*b[0];
+
+}
+// a outer b
+void outer(const Vector a, const Vector b, DenseMatrix& c)
+{
+   int m = a.Size();
+   int n = b.Size();
+   assert(c.Height()==m);
+   assert(c.Width() ==n);
+   for (int i=0; i<m; i++)
+   {
+      for (int j=0; j<n; j++)
+      {
+         c(i,j) = a[i]*b[j];
+      }
+   }
+}
+// dphidxi 2*4
+// coords 4*3
+void ComputeNormal(const DenseMatrix& dphidxi, const DenseMatrix& coords,
+                   Vector& normal, double& nnorm)
+{
+
+   DenseMatrix dxdxi(2,3);
+   Mult(dphidxi, coords, dxdxi);
+   Vector dxdxi1(3);
+   Vector dxdxi2(3);
+
+   dxdxi.GetRow(0,dxdxi1);
+   dxdxi.GetRow(1,dxdxi2);
+
+   cross(dxdxi1, dxdxi2, normal);  // is there a cross product?  no
+   // VectorCrossProductCoefficient::Eval has hard-coded cross product
+   nnorm = normal.Norml2( );
+   normal /=  nnorm;
+}
+
+void SlaveToMaster(const DenseMatrix& m_coords, const Vector& s_x, Vector& xi)
+{
+   bool converged = false;
+   bool pt_on_elem = false;
+   int dim = 3;
+   xi.SetSize(dim-1);
+   xi = 0.0;
+   double r = 1e10;
+   int max_iter = 15;
+   double off_el_xi = 1e-2;
+   double proj_newton_tol = 1e-13;
+   double proj_max_gap = 0.5;
+   Vector gap_v(dim);
+   // warm start from linear solution
+
+   for (int it=0; it<max_iter; it++)
+   {
+      //cout<<it<<endl;
+      Vector m_N(4);
+      m_N = 0.;
+      DenseMatrix m_dN(2,4);
+      m_dN = 0.;
+      DenseMatrix m_dN2(3,4);
+      m_dN2 = 0.;
+      BasisEvalDerivs(xi, m_N, m_dN, m_dN2);
+
+      Vector x_c(dim);
+      m_coords.MultTranspose(m_N, x_c);
+
+      gap_v = s_x;
+      gap_v -= x_c;
+
+      DenseMatrix m_dx(2,3);
+      m_dx = 0.;
+      Mult(m_dN, m_coords, m_dx);
+
+      Vector r(dim-1);
+      r = 0.0;
+      m_dx.Mult(gap_v, r);
+
+      if (r.Normlinf() < proj_newton_tol)
+      {
+         converged = true;
+         break;
+      }
+
+      DenseMatrix drdxi(dim-1,dim-1);
+      drdxi = 0.;
+      MultABt(m_dx, m_dx, drdxi); // m_dx * m_dx.T
+      drdxi *= -1.0;
+
+      DenseMatrix m_dx2(3,3); m_dx2 = 0.0;
+      Mult(m_dN2,m_coords, m_dx2);
+
+      //m_d2x = m_dN(:,:,2) * m_elem_coords(1:4,:);  //m_dN(:,:,2) is 3*4
+      for (int d=0; d<3; d++)
+      {
+         DenseMatrix Mtemp(2,2); Mtemp = 0.0;
+         Mtemp(0,0) = m_dx2(0,d); Mtemp(0,1) = m_dx2(1,d);
+         Mtemp(1,0) = m_dx2(1,d); Mtemp(1,1) = m_dx2(2,d);
+
+         drdxi.Add(gap_v[d], Mtemp);
+      }
+
+      //cond_num = rcond(drdxi);  condition number?
+      //drdxi.TestInversion();
+      DenseMatrixInverse drdxi_inv(drdxi);
+      Vector xi_tmp(dim-1);
+
+      drdxi_inv.Mult(r,xi_tmp);
+      xi -= xi_tmp;
+   }
+   if (!converged)
+   {
+      xi = 0.0;
+   }
+   off_el_xi += 1 ; // tolerance of offset of xi outside [-1,1]
+
+   //cout<<gap_v.Norml2()<<" " <<xi.Normlinf()<<endl;
+   if (gap_v.Norml2() < proj_max_gap && xi.Normlinf() <= off_el_xi)
+   {
+      pt_on_elem = true;
+   }
+
+   MFEM_VERIFY(pt_on_elem == true, "xi went out of bounds");
+   MFEM_VERIFY(converged == true, "projection didn't converge");
+}
+
+
+
+// m_coords is expected to be 4 * 3
+void  ComputeGapJacobian(const Vector x_s, const Vector xi,
+                         const DenseMatrix m_coords,
+                         double& gap, Vector& normal, Vector& dgdxm, Vector& dgdxs)
+{
+   Vector m_N(4);
+   DenseMatrix m_dN(2,4);
+   DenseMatrix m_dN2(3,4);
+   BasisEvalDerivs(xi, m_N, m_dN, m_dN2);
+
+   Vector x_c(3);
+   m_coords.MultTranspose(m_N, x_c);
+
+   Vector gap_v(3); gap_v = 0.0;
+   gap_v  = x_s;
+   gap_v -= x_c;
+
+   DenseMatrix m_dx(2,3);
+   Mult(m_dN, m_coords, m_dx);
+
+   double nnorm = 0;
+   ComputeNormal(m_dN, m_coords, normal, nnorm);
+
+   gap = gap_v * normal; // gap function value, dot product between vectors
+
+   //dr_dx = zeros(2,4,3); % nsegment, nodes in quad, ndim
+
+   DenseMatrix dr_dx_res1(4,3); dr_dx_res1 = 0.;
+   DenseMatrix dr_dx_res2(4,3); dr_dx_res2 = 0.;
+
+   Vector m_dxrow1(3);
+   m_dx.GetRow(0, m_dxrow1);
+   outer(m_N, m_dxrow1, dr_dx_res1);// 4*1 times 1*3
+   dr_dx_res1 *= -1.0;
+
+   Vector m_dxrow2(3);
+   m_dx.GetRow(1, m_dxrow2);
+   outer(m_N, m_dxrow2, dr_dx_res2);// 4*1 times 1*3
+   dr_dx_res2 *= -1.0;
+
+   Vector m_dNrow1(4); m_dN.GetRow(0, m_dNrow1);
+   Vector m_dNrow2(4); m_dN.GetRow(1, m_dNrow2);
+
+   DenseMatrix dr_dx_res1_tmp(4,3); dr_dx_res1_tmp = 0.;
+   DenseMatrix dr_dx_res2_tmp(4,3); dr_dx_res2_tmp = 0.;
+   outer(m_dNrow1, gap_v, dr_dx_res1_tmp);// 4*1 times 1*3
+   outer(m_dNrow2, gap_v, dr_dx_res2_tmp);// 4*1 times 1*3
+
+   dr_dx_res1 += dr_dx_res1_tmp; // outer product in vector?
+   dr_dx_res2 += dr_dx_res2_tmp;
+
+
+   DenseMatrix K_dxidx1(2,2); // 2*2
+   K_dxidx1 = 0.;
+   MultABt(m_dx, m_dx, K_dxidx1); // m_dx * m_dx.T
+
+   Vector v_dxidx2(4);
+   m_coords.Mult(gap_v, v_dxidx2); // m_coords * gap_v;  // 4*3 * 3 = 4
+
+   DenseMatrix K_dxidx2(2,2); K_dxidx2 = 0.0;
+
+   Vector m_dN2row1(4); m_dN2.GetRow(0, m_dN2row1);
+   Vector m_dN2row2(4); m_dN2.GetRow(1, m_dN2row2);
+   Vector m_dN2row3(4); m_dN2.GetRow(2, m_dN2row3);
+   // how to get 2nd order? multidimensional matrix?
+   K_dxidx2(0,0) = m_dN2row1 * v_dxidx2; // how would 4*1 * 1*4 be computed?
+   K_dxidx2(0,1) = m_dN2row2 * v_dxidx2;
+   K_dxidx2(1,0) = m_dN2row2 * v_dxidx2;
+   K_dxidx2(1,1) = m_dN2row3 * v_dxidx2;
+
+   DenseMatrix K_dxidx(2,2);
+   K_dxidx -= K_dxidx1;
+   K_dxidx += K_dxidx2;
+
+   // resize the vectors and matrices
+   Vector dxidx(24); dxidx = 0.0;
+   Vector drdx_r(24); drdx_r = 0.0;
+
+   for (int i=0; i<4; i++)
+   {
+      for (int j=0; j<3; j++)
+      {
+         drdx_r[4*j+i] = dr_dx_res1(i,j);
+         drdx_r[4*j+i+12] = dr_dx_res2(i,j);
+
+      }
+   }
+   //drdx_r(1:4*3,1) = reshape(dr_dx_res(:,:,1),4*3,1);
+   //drdx_r(4*3+1:2*4*3,1) = reshape(dr_dx_res(:,:,2),4*3,1);
+   DenseMatrix drdx_K(24,24); drdx_K = 0.;
+   for (int i =0; i<12; i++)
+   {
+      drdx_K(i,i) = K_dxidx(0,0);
+      drdx_K(i,12+i) = K_dxidx(0,1);
+      drdx_K(12+i,i) = K_dxidx(1,0);
+      drdx_K(12+i,12+i) = K_dxidx(1,1);
+   }
+
+   DenseMatrixInverse drdxK_inv(drdx_K);
+   drdxK_inv.Mult(drdx_r,dxidx);
+   // LinearSolve (drdx_K,drdx_r, dxidx) ; //???
+   dxidx *= -1.0;
+
+
+
+   Vector drdxs_r(6);
+   drdxs_r[0] = m_dx(0,0); drdxs_r[1] = m_dx(0,1); drdxs_r[2] = m_dx(0,2);
+   drdxs_r[3] = m_dx(1,0); drdxs_r[4] = m_dx(1,1); drdxs_r[5] = m_dx(1,2);
+
+   DenseMatrix drdxs_K(6,6); drdxs_K = 0.;
+   for (int i=0; i<3; i++)
+   {
+      drdxs_K(i,i) = K_dxidx(0,0);
+      drdxs_K(i,3+i) = K_dxidx(0,1);
+      drdxs_K(i+3,i) = K_dxidx(1,0);
+      drdxs_K(i+3,i+3) = K_dxidx(1,1);
+   }
+
+   Vector dxidxs(6); dxidxs = 0.0;
+   DenseMatrixInverse drdxsK_inv(drdxs_K);
+   drdxsK_inv.Mult(drdxs_r,dxidxs);
+   dxidxs *= -1.0;
+   //dxidxs = -drdxs_K\drdxs_r;
+
+   //dxidx = reshape(dxidx, 4,3,2); dxidxs = reshape(dxidxs, 1,3,2);
+
+   dgdxm.SetSize(12); dgdxm = 0.;
+   DenseMatrix dgdxm_tmp(4,3);
+   outer(m_N, normal,dgdxm_tmp);
+   for (int i=0; i<4; i++)
+   {
+      for (int j=0; j<3; j++)
+      {
+         dgdxm[3*i+j] = -dgdxm_tmp(i,j);
+      }
+   }
+   //dxidx_M = -m_dN(1:2,:,1) * (m_coords(1:4,:)*normal'); % this turns out to be 0
+
+   dgdxs.SetSize(3);
+   dgdxs += normal;
+   //dgdxs = dgdxs + dxidx_M(1) * dxidxs(:,:,1) + dxidx_M(2) * dxidxs(:,:,2);
+};
+
+void ComputeGapHessian(const Vector x_s, const Vector xi,
+                       const DenseMatrix m_coords,
+                       DenseMatrix& dg2dx)
+{
+   Vector m_N(4);
+   DenseMatrix m_dN(2,4);
+   DenseMatrix m_dN2(3,4);
+   BasisEvalDerivs(xi, m_N, m_dN, m_dN2);
+
+   int dim = 3;
+   int num_dofs1 = dim;
+   int num_dofs2 = 4*dim;
+   int num_dofs = num_dofs1 + num_dofs2;
+   dg2dx.SetSize(num_dofs,num_dofs); dg2dx = 0.0;
+
+   Vector x_c(3);
+   m_coords.MultTranspose(m_N,x_c);
+
+   Vector gap_v(3); gap_v = 0.0;
+   gap_v  = x_s;
+   gap_v -= x_c;
+
+   DenseMatrix m_dx(2,3);
+   Mult(m_dN, m_coords, m_dx);
+
+   DenseMatrix m_dx2(3,3); m_dx2 = 0.0;
+   Mult(m_dN2,m_coords, m_dx2);
+   double nnorm = 0.0;
+   Vector normal(3); normal = 0.0;
+   ComputeNormal(m_dN, m_coords, normal, nnorm);
+
+   double gap = gap_v * normal; // gap function value, dot product between vectors
+
+   DenseMatrix M(2,2); M = 0.0;
+   MultABt(m_dx, m_dx, M);
+
+   DenseMatrix f(2, num_dofs2); f = 0.0;
+
+   for (int d=0; d<3; d++)
+   {
+      DenseMatrix Mtemp(2,2); Mtemp = 0.0;
+      Mtemp(0,0) = m_dx2(0,d); Mtemp(0,1) = m_dx2(1,d);
+      Mtemp(1,0) = m_dx2(1,d); Mtemp(1,1) = m_dx2(2,d);
+
+      M.Add(-gap_v[d], Mtemp);
+
+      Vector m_dxcol(2); m_dx.GetColumn(d, m_dxcol);
+      DenseMatrix ftmp(2,4);
+      outer(m_dxcol, m_N, ftmp);
+      ftmp *= -1;
+      ftmp.Add( gap_v[d], m_dN);   // 2*4
+
+      for (int j=0; j<4; j++)
+      {
+         assert(d+3*j<num_dofs2);
+         f(0,d+j*3) = ftmp(0,j);
+         f(1,d+j*3) = ftmp(1,j);
+      }
+   }
+   //fprintf('hess dxidxm\n');
+   DenseMatrixInverse Minv(M);
+   DenseMatrix dxidxm(2,num_dofs2); dxidxm = 0.0;
+   Minv.Mult(f, dxidxm);
+   //LinearSolve??
+   //dxidxm = M\f;
+
+   DenseMatrix nde2(2,2); nde2 = 0.0;
+   DenseMatrix Nndx2(2,num_dofs2); Nndx2 = 0.0;
+
+   for (int d=0; d<3; d++)
+   {
+      DenseMatrix ndetmp(2,2); ndetmp = 0.0;
+      ndetmp(0,0) = normal(d)*m_dx2(0,d); ndetmp(0,1) = normal(d)*m_dx2(1,d);
+      ndetmp(1,0) = normal(d)*m_dx2(1,d); ndetmp(1,1) = normal(d)*m_dx2(2,d);
+
+      nde2 += ndetmp;
+
+      for (int j=0; j<4; j++)
+      {
+         assert(d+3*j<num_dofs2);
+         Nndx2(0,d+j*3) = normal[d]*m_dN(0,j);
+         Nndx2(1,d+j*3) = normal[d]*m_dN(1,j);
+      }
+
+   }
+
+   DenseMatrix Ndn(2,num_dofs2); Ndn = 0.0;
+   Ndn += Nndx2;
+   AddMult(nde2, dxidxm, Ndn);
+
+
+   DenseMatrix M2(2,2); M2 = 0.0;
+   MultABt(m_dx, m_dx, M2);
+   DenseMatrixInverse M2inv(M2);
+   DenseMatrix diag2(2,2); diag2(0,0) = 1.0; diag2(1,1) = 1.0;
+   DenseMatrix m_con(2,2); m_con = 0.0;
+
+   M2inv.Mult(diag2, m_con);
+
+   DenseMatrix dg2dxm(num_dofs2, num_dofs2); dg2dxm = 0.0;
+
+   DenseMatrix dg2dxm_tmp(num_dofs2,2); dg2dxm_tmp = 0.0;
+   MultAtB(Ndn, m_con, dg2dxm_tmp);
+   Mult(dg2dxm_tmp, Ndn, dg2dxm);
+   dg2dxm *= gap;
+
+   DenseMatrix dg2dxm_tmp2(num_dofs2,num_dofs2); dg2dxm_tmp2 = 0.0;
+   MultAtB(Nndx2, dxidxm, dg2dxm_tmp2);
+   dg2dxm.Add(-1.0, dg2dxm_tmp2);
+
+   dg2dxm_tmp = 0.0;
+   MultAtB(dxidxm, nde2, dg2dxm_tmp);
+
+   AddMult_a(-1.0, dg2dxm_tmp, dxidxm, dg2dxm);
+
+   dg2dxm_tmp2 = 0.0;
+   MultAtB(dxidxm, Nndx2, dg2dxm_tmp2);
+   dg2dxm.Add(-1.0, dg2dxm_tmp2);
+
+   Vector v_dxidx2(4);
+   m_coords.Mult(gap_v, v_dxidx2); // m_coords * gap_v;  // 4*3 * 3 = 4
+
+   DenseMatrix K_dxidx2(2,2); K_dxidx2 = 0.0;
+
+   Vector m_dN2row1(4); m_dN2.GetRow(0, m_dN2row1);
+   Vector m_dN2row2(4); m_dN2.GetRow(1, m_dN2row2);
+   Vector m_dN2row3(4); m_dN2.GetRow(2, m_dN2row3);
+   K_dxidx2(0,0) = m_dN2row1 * v_dxidx2; // how would 4*1 * 1*4 be computed?
+   K_dxidx2(0,1) = m_dN2row2 * v_dxidx2;
+   K_dxidx2(1,0) = m_dN2row2 * v_dxidx2;
+   K_dxidx2(1,1) = m_dN2row3 * v_dxidx2;
+
+   DenseMatrix K_dxidx(2,2);
+   K_dxidx -= M2;
+   K_dxidx += K_dxidx2;
+
+   Vector drdxs_r(6);
+   drdxs_r[0] = m_dx(0,0); drdxs_r[1] = m_dx(0,1); drdxs_r[2] = m_dx(0,2);
+   drdxs_r[3] = m_dx(1,0); drdxs_r[4] = m_dx(1,1); drdxs_r[5] = m_dx(1,2);
+
+   DenseMatrix drdxs_K(6,6); drdxs_K = 0.;
+   for (int i=0; i<3; i++)
+   {
+      drdxs_K(i,i) = K_dxidx(0,0);
+      drdxs_K(i,3+i) = K_dxidx(0,1);
+      drdxs_K(i+3,i) = K_dxidx(1,0);
+      drdxs_K(i+3,i+3) = K_dxidx(1,1);
+   }
+   Vector dxidxs(6);
+
+   DenseMatrixInverse drdxsK_inv(drdxs_K);
+   drdxsK_inv.Mult(drdxs_r,dxidxs);
+   dxidxs *= -1.0;
+   //dxidxs = -drdxs_K\drdxs_r;
+
+   DenseMatrix dxidxs_m(2,3); dxidxs_m = 0.0;
+   dxidxs_m(0,0) = dxidxs[0]; dxidxs_m(0,1) = dxidxs[1]; dxidxs_m(0,2) = dxidxs[2];
+   dxidxs_m(1,0) = dxidxs[3]; dxidxs_m(1,1) = dxidxs[4]; dxidxs_m(1,2) = dxidxs[5];
+
+   DenseMatrix dtao1dxs(3,3); dtao1dxs = 0.0;
+   DenseMatrix dtao2dxs(3,3); dtao2dxs = 0.0;
+
+   Vector dxidxs_row1(3); dxidxs_row1 = 0.0; Vector dxidxs_row2(3);
+   dxidxs_row2 = 0.0;
+   Vector mdx2_row1(3); mdx2_row1 = 0.0; Vector mdx2_row2(3); mdx2_row2 = 0.0;
+   Vector mdx2_row3(3); mdx2_row3 = 0.0;
+   dxidxs_m.GetRow(0,dxidxs_row1);
+   dxidxs_m.GetRow(1,dxidxs_row2);
+   m_dx2.GetRow(0,mdx2_row1);
+   m_dx2.GetRow(1,mdx2_row2);
+   m_dx2.GetRow(2,mdx2_row3);
+
+   DenseMatrix dtaotmp(3,3); dtaotmp = 0.0;
+   outer(mdx2_row1, dxidxs_row1,dtaotmp);
+   dtao1dxs += dtaotmp; dtaotmp = 0.0;
+   outer(mdx2_row2, dxidxs_row1,dtaotmp);
+   dtao1dxs += dtaotmp; dtaotmp = 0.0;
+
+   outer(mdx2_row2, dxidxs_row2, dtaotmp);
+   dtao2dxs += dtaotmp; dtaotmp = 0.0;
+   outer(mdx2_row3, dxidxs_row2, dtaotmp);
+   dtao2dxs += dtaotmp; dtaotmp = 0.0;
+
+   DenseMatrix dtaodxs(3,3); dtaodxs = 0.0; //tao = tao1 cross tao2
+
+   for (int d=0; d<3; d++)
+   {
+      Vector dtao1dxs_tmp(3); dtao1dxs_tmp = 0.0;
+      dtao1dxs.GetColumn(d,dtao1dxs_tmp);
+      Vector m_dxrow(3); m_dx.GetRow(1, m_dxrow);
+
+      Vector dtaodxs_tmp(3); dtaodxs_tmp = 0.0;
+      cross(dtao1dxs_tmp, m_dxrow, dtaodxs_tmp);
+
+      Vector dtaodxs_tmp2(3); dtaodxs_tmp2 = 0.0;
+      m_dx.GetRow(0, m_dxrow);
+      dtao1dxs_tmp = 0.0;  // reuse the same vector for dtao2
+      dtao2dxs.GetColumn(d,dtao1dxs_tmp);
+      cross(m_dxrow, dtao1dxs_tmp, dtaodxs_tmp2);
+
+      dtaodxs_tmp2 += dtaodxs_tmp;
+      dtaodxs.SetCol(d, dtaodxs_tmp2);
+   }
+
+   DenseMatrix dndxs(3,3); dndxs = 0.0; dndxs += dtaodxs; dndxs *= 1.0/nnorm;
+   DenseMatrix dndxs_tmp(3,3); dndxs_tmp = 0.0;
+   outer(normal, normal, dndxs_tmp);
+   AddMult_a(-1/nnorm, dndxs_tmp, dtaodxs, dndxs);
+
+   DenseMatrix dgvdxs(3,3); dgvdxs = 0.0;
+   MultAtB(m_dx, dxidxs_m, dgvdxs);
+   dgvdxs *= -1;
+   for (int d=0; d<3; d++)
+   {
+      dgvdxs(d,d) += 1.0;
+   }
+   //dxidxs: 2*3
+
+   DenseMatrix dg2dxs(3,3); dg2dxs = 0.0;
+   DenseMatrix dg2dxs_tmp(3,2); dg2dxs_tmp = 0.0;
+   MultAtB(dxidxs_m, nde2, dg2dxs_tmp);
+   AddMult_a(-1.0, dg2dxs_tmp, dxidxs_m, dg2dxs);
+   DenseMatrix dg2dxs_tmp2(3,3); dg2dxs_tmp2 = 0.0;
+   MultAtB(dgvdxs, dndxs, dg2dxs_tmp2);
+   dg2dxs += dg2dxs_tmp2;
+   dg2dxs_tmp2 = 0.0;
+   MultAtB(dndxs, dndxs_tmp, dg2dxs_tmp2);
+   AddMult(dg2dxs_tmp2, dgvdxs, dg2dxs);
+
+   DenseMatrix Ne(3,12), Be(6,12), dBe(12,12);
+   BasisVectorDerivs(xi, Ne, Be, dBe);
+
+   DenseMatrix dtao1dxm(3,12); dtao1dxm.CopyRows(Be, 0, 2);
+   DenseMatrix dtao2dxm(3,12); dtao2dxm.CopyRows(Be, 3, 5);
+
+   Vector m_coords_v(12);
+   for (int i=0; i<4; i++)
+   {
+      for (int j=0; j<3; j++)
+      {
+         m_coords_v[i*3+j] = m_coords(i,j);
+      }
+   }
+
+   for (int i=0; i<2; i++)
+   {
+      Vector dxidxm_tmp(num_dofs2); dxidxm_tmp = 0.0;
+      dxidxm.GetRow(i,dxidxm_tmp);
+
+      DenseMatrix dBe_tmp(3,12);
+      dBe_tmp.CopyRows(dBe,i*3,(i+1)*3-1);
+
+      DenseMatrix dtaodxm_tmp(12,12); dtaodxm_tmp = 0.0;
+      outer(m_coords_v, dxidxm_tmp, dtaodxm_tmp);
+      AddMult(dBe_tmp, dtaodxm_tmp, dtao1dxm);
+
+      //dtao1dxm += dBe(:,:,i)*reshape(m_coords(1:4,:)',12,1)*reshape(dxidxm(i,:),1,12); % 3*12
+      dBe_tmp = 0.0;
+      dBe_tmp.CopyRows(dBe,(i+2)*3,(i+3)*3-1);
+      AddMult(dBe_tmp, dtaodxm_tmp, dtao2dxm);
+
+   }
+
+   DenseMatrix dtaodxm(3,12); dtaodxm = 0.0;//tao = tao1 cross tao2
+
+   for (int d=0; d<12; d++)
+   {
+      Vector dtaodxm_tmp(3); dtaodxm_tmp = 0.0;
+      Vector dtaodxm_tmp2(3); dtaodxm_tmp2 = 0.0;
+      Vector tmp1(3); tmp1 = 0.0;  dtao1dxm.GetColumn(d,tmp1);
+      Vector m_dxrow2(3); m_dx.GetRow(1, m_dxrow2);
+      Vector m_dxrow1(3); m_dx.GetRow(0, m_dxrow1);
+      Vector tmp2(3); tmp2 = 0.0;  dtao2dxm.GetColumn(d,tmp2);
+
+      cross(tmp1, m_dxrow2, dtaodxm_tmp);
+      cross(m_dxrow1,tmp2, dtaodxm_tmp2);
+      dtaodxm_tmp += dtaodxm_tmp2;
+
+      dtaodxm.SetCol(d, dtaodxm_tmp);
+   }
+
+   DenseMatrix dndxm(3,12); dndxm = 0.0;
+   dndxm += dtaodxm;
+   dndxm *= 1.0/nnorm;
+   AddMult_a(-1/nnorm, dndxs_tmp, dtaodxm, dndxm); //dndxs_tmp = normal'*normal
+
+   DenseMatrix dgvdxm(3,12); dgvdxm = 0.0;
+   dgvdxm -= Ne;
+
+   for (int i=0; i<2; i++)
+   {
+      Vector dxidxm_tmp(num_dofs2); dxidxm_tmp = 0.0;
+      dxidxm.GetRow(i,dxidxm_tmp);
+
+      DenseMatrix Be_tmp(3,12);
+      Be_tmp.CopyRows(Be,i*3,(i+1)*3-1);
+
+      DenseMatrix dgvdxm_tmp(12,12); dgvdxm_tmp = 0.0;
+      outer(m_coords_v, dxidxm_tmp, dgvdxm_tmp);
+      AddMult_a(-1.0, Be_tmp, dgvdxm_tmp, dgvdxm);
+
+   }
+
+   DenseMatrix dg2dxsxm(3,12); dg2dxsxm = 0.0;
+   DenseMatrix dg2dxsxm_tmp(3,3); dg2dxsxm_tmp = 0.0;
+   MultAtB(dgvdxs, dndxm, dg2dxsxm);
+
+   MultAtB(dndxs, dndxs_tmp, dg2dxsxm_tmp);
+   AddMult(dg2dxsxm_tmp, dgvdxm, dg2dxsxm); // += dndxs'*normal'*normal*dgvdxm;
+
+   DenseMatrix dgvdxsxmn(3,12); dgvdxsxmn = 0.0;
+   DenseMatrix dgvdxsxmn_tmp(3,2); dgvdxsxmn_tmp = 0.0;
+   MultAtB(dxidxs_m, nde2, dgvdxsxmn_tmp);   //dxidxs_m: 2*3
+
+   AddMult_a(-1.0, dgvdxsxmn_tmp, dxidxm, dgvdxsxmn);
+
+
+   for (int i =0; i<2; i++)
+   {
+      DenseMatrix Be_tmp(3,12);
+      Be_tmp.CopyRows(Be,i*3,(i+1)*3-1);
+
+      Vector dxidxs_row(3); dxidxs_row = 0.0; dxidxs_m.GetRow(i,dxidxs_row);
+      DenseMatrix dgvdxsxmn_tmp2(3,3); dgvdxsxmn_tmp2 = 0.0;
+      outer(dxidxs_row, normal, dgvdxsxmn_tmp2);
+      AddMult_a(-1.0, dgvdxsxmn_tmp2, Be_tmp, dgvdxsxmn);
+   }
+
+   dg2dxsxm += dgvdxsxmn;
+
+   DenseMatrix dg2dxmxs(12,3); dg2dxmxs = 0.0;
+   DenseMatrix dg2dxmxs_tmp(12,3); dg2dxmxs_tmp = 0.0;
+   MultAtB(dgvdxm, dndxs, dg2dxmxs);
+   MultAtB(dndxm, dndxs_tmp, dg2dxmxs_tmp);
+   AddMult(dg2dxmxs_tmp, dgvdxs, dg2dxmxs);
+
+   DenseMatrix dgvdxmxsn(12,3); dgvdxmxsn = 0.0;
+   DenseMatrix dgvdxmxsn_tmp(12,2); dgvdxmxsn_tmp = 0.0;
+
+   MultAtB(dxidxm, nde2, dgvdxmxsn_tmp);
+   dgvdxmxsn_tmp *= -1.0;
+   AddMult(dgvdxmxsn_tmp, dxidxs_m, dgvdxmxsn);
+
+   for (int i =0; i<2; i++)
+   {
+      DenseMatrix Be_tmp(3,12);
+      Be_tmp.CopyRows(Be,i*3,(i+1)*3-1);
+      Be_tmp.Transpose(); // Be is now 12*3
+
+      Vector dxidxs_row(3); dxidxs_row = 0.0; dxidxs_m.GetRow(i,dxidxs_row);
+      DenseMatrix dgvdxmxsn_tmp2(3,3); dgvdxmxsn_tmp2 = 0.0;
+      outer(normal, dxidxs_row, dgvdxmxsn_tmp2);
+      AddMult_a(-1.0, Be_tmp, dgvdxmxsn_tmp2, dgvdxmxsn);
+
+   }
+
+   dg2dxmxs += dgvdxmxsn;
+
+   dg2dx.CopyMN(dg2dxs, 0, 0);
+   dg2dx.CopyMN(dg2dxm, 3, 3);
+   dg2dx.CopyMN(dg2dxsxm, 0, 3);
+   dg2dx.CopyMN(dg2dxmxs, 3, 0);
+
+};
+
+
+
+void NodeSegConPairs(const Vector x1, const Vector xi2,
+                     const DenseMatrix coords2,
+                     double& node_g, Vector& node_dg, DenseMatrix& node_dg2)
+{
+   double gap = 0.0;
+   Vector normal(3); normal = 0.0;
+   Vector dgdxm(12); dgdxm = 0.0;
+   Vector dgdxs(3); dgdxs = 0.0;
+
+   ComputeGapJacobian(x1, xi2, coords2, gap, normal, dgdxm, dgdxs);
+   node_g = gap;
+
+   node_dg.SetSize(12+3);
+   for (int i=0; i<3; i++) { node_dg[i] = dgdxs[i]; }
+   for (int i=0; i<12; i++) { node_dg[i+3] = dgdxm[i]; }
+
+   DenseMatrix dg2dx(15,15); dg2dx = 0.0;
+   DenseMatrix dgvdxmxsn(12,3); dgvdxmxsn = 0.0;
+   ComputeGapHessian(x1, xi2, coords2, dg2dx);
+
+   node_dg2.SetSize(15,15);
+   node_dg2 = dg2dx;
+
+   /*
+   if(obj.space1.conns{e1}(i)==150)  % for debugging purpose
+
+   v1 = 1:3;
+   v2 = 1:12;
+   %v1 = ones(1,3)
+   %v2 = ones(1,12)
+   v2 = reshape(v2,4,3);
+   x1n1 = x1 + 0.01*v1;
+   coords2n1 = coords2 + 0.001*v2;
+      [xi2n1, gapv1, ~, ~] = SlaveToMaster(obj, coords2n1, x1n1);
+      [gapn1, n1,dgdxmn1, dgdxsn1] = ComputeGapJacobian(obj, x1n1, xi2n1, coords2n1);
+   x1n2 = x1 - 0.01*v1;
+   coords2n2 = coords2 - 0.001*v2;
+      [xi2n2, gapv2, ~, ~] = SlaveToMaster(obj, coords2n2, x1n2);
+      [gapn2, n2,dgdxmn2, dgdxsn2] = ComputeGapJacobian(obj, x1n2, xi2n2, coords2n2);
+   fprintf('fd\n');
+      %gapv1-gapv2
+   [dgdxsn1(:)',dgdxmn1(:)'] - [dgdxsn2(:)',dgdxmn2(:)']
+
+   %dgdxsn1-dgdxsn2
+   fprintf('code\n');
+   v2n = v2';
+   %dg2dx(1:3,1:3)*0.04*ones(3,1)
+   temp = zeros(12,3);
+   for i = 1:4
+   temp1 = dg2dx(3+(i-1)*3+1:3+i*3,1:3);
+   temp((i-1)*3+1:i*3,:) = temp1';
+   end
+   temp2 = zeros(3,12);
+   for i = 1:4
+   temp3 = dg2dx(1:3,3+(i-1)*3+1:3+i*3);
+   temp2(:,(i-1)*3+1:i*3) = temp3';
+   end
+   %dg2dx
+   %dg2dx(4:end,1:3) = temp;
+   %dg2dx(1:3,4:end) = temp2;
+   %dgvdxm * 0.002*v2n(:)
+   (dg2dx*[0.02*v1(:)',0.002*v2n(:)']')'
+   %dg2dx(4:end,1:3)
+   end*/
+
+};
+
+
+// coordsm : (npoints*4, 3) use what class?
+// m_conn: (npoints*4)
+void Assemble_Contact(const int m, const int npoints, const int ndofs,
+                      const Vector x_s,
+                      const Vector xi, const DenseMatrix coordsm, const Array<int> s_conn,
+                      const Array<int> m_conn, Vector& g, SparseMatrix& M,
+                      std::vector<SparseMatrix>& dM)
+{
+   int n = ndofs;
+   int ndim = 3;
+
+   g.SetSize(m);
+   g = 0.0;
+
+   //SparseMatrix M(m, n);  // M needs to be the correct size
+
+   //dM.resize(m); // needs to clear?
+
+   double g_tmp = 0.;
+   Vector dg(4*ndim+ndim);
+   dg = 0.;
+   DenseMatrix dg2(4*ndim+ndim,4*ndim+ndim);
+   dg2 = 0.;
+
+   for (int i=0; i<npoints; i++)
+   {
+      Vector x1(ndim);
+      x1[0] = x_s[i*ndim];
+      x1[1] = x_s[i*ndim+1];
+      x1[2] = x_s[i*ndim+2];
+
+      Vector xi2(ndim-1);
+      xi2[0] = xi[i*(ndim-1)];
+      xi2[1] = xi[i*(ndim-1)+1];
+
+      DenseMatrix coords2(4,3);
+      coords2.CopyRows(coordsm, i*4,(i+1)*4-1);
+
+      //how to get coords2?
+      dg = 0.0;
+      dg2 = 0.;
+      NodeSegConPairs(x1, xi2, coords2, g_tmp, dg, dg2);
+      //x1.Print();
+      //xi2.Print();
+      //coords2.Print();
+      g[s_conn[i]] = g_tmp; // should be unique
+      Array<int> m_conn_i(4);
+      m_conn.GetSubArray(4*i, 4, m_conn_i);
+
+      Array<int> node_conn(5);
+      node_conn[0] = s_conn[i];
+      for (int j=0; j<4; j++)
+      {
+         node_conn[j+1] = m_conn_i[j];
+      }
+
+      Array<int>  M_i_tmp(1);
+      M_i_tmp[0] = s_conn[i];
+
+      //j_idx = (node_conn-1)*obj.disp_field.num_components +repmat((1:obj.disp_field.num_components)', 1, length(node_conn{i}));
+      Array<int> j_idx(5*ndim); j_idx = 0;
+      for (int j=0; j< 5; j++)
+      {
+         for (int k=0; k<ndim; k++)
+         {
+            j_idx[j*ndim+k] = node_conn[j]*ndim+k;
+         }
+      }
+      DenseMatrix M_v_tmp(1, ndim*(4+1));   // SetData now?
+      M_v_tmp.SetRow(0, dg);
+
+      M.AddSubMatrix(M_i_tmp, j_idx, M_v_tmp);
+
+      Array<int> dM_i(ndim*(4+1));
+      Array<int> dM_j(ndim*(4+1));
+
+      for (int j=0; j< ndim*(4+1); j++)
+      {
+         dM_i[j] = j_idx[j];
+         dM_j[j] = j_idx[j];
+      }
+      //dg2.Print();
+      //dM[s_conn[i]].Print();
+      dM[s_conn[i]].AddSubMatrix(dM_i,dM_j, dg2);
+   }
+};
+
@@ -46,6 +46,9 @@ endif
 ifeq ($(MFEM_USE_HIOP),YES)
   SUBDIRS += hiop
 endif
+ifeq ($(MFEM_USE_IPOPT),YES)
+   SUBDIRS += ipopt
+endif
 ifeq ($(MFEM_USE_PETSC),YES)
   SUBDIRS += petsc
 endif
@@ -0,0 +1,888 @@
+
+using namespace std;
+using namespace mfem;
+
+void BasisEval(const Vector xi, Vector &N, DenseMatrix &dNdxi) // dNdxi is 2*4
+{
+   N[0] = 0.25*(1-xi[0])*(1-xi[1]);
+   N[1] = 0.25*(1+xi[0])*(1-xi[1]);
+   N[2] = 0.25*(1+xi[0])*(1+xi[1]);
+   N[3] = 0.25*(1-xi[0])*(1+xi[1]);
+
+   dNdxi(0,0) = 0.25*(-1+xi[1]);
+   dNdxi(0,1) = 0.25*(1-xi[1]);
+   dNdxi(0,2) = 0.25*(1+xi[1]);
+   dNdxi(0,3) = 0.25*(-1-xi[1]);
+   dNdxi(1,0) = 0.25*(-1+xi[0]);
+   dNdxi(1,1) = 0.25*(-1-xi[0]);
+   dNdxi(1,2) = 0.25*(1+xi[0]);
+   dNdxi(1,3) = 0.25*(1-xi[0]);
+}
+
+
+void BasisEvalDerivs(const Vector xi, Vector& N, DenseMatrix& dNdxi,
+                     DenseMatrix& dN2dxi)
+{
+   N[0] = 0.25*(1-xi[0])*(1-xi[1]);
+   N[1] = 0.25*(1+xi[0])*(1-xi[1]);
+   N[2] = 0.25*(1+xi[0])*(1+xi[1]);
+   N[3] = 0.25*(1-xi[0])*(1+xi[1]);
+
+   dNdxi.SetSize(2,4); dNdxi = 0.0;
+   dN2dxi.SetSize(3,4);
+   dN2dxi = 0.0;  // first row dxi2, second detadxi, third deta2
+
+   dNdxi(0,0) = 0.25*(-1+xi[1]); dNdxi(0,1) = 0.25*(1-xi[1]);
+   dNdxi(0,2) = 0.25*(1+xi[1]);  dNdxi(0,3) = 0.25*(-1-xi[1]);
+   dNdxi(1,0) = 0.25*(-1+xi[0]); dNdxi(1,1) = 0.25*(-1-xi[0]);
+   dNdxi(1,2) = 0.25*(1+xi[0]);  dNdxi(1,3) = 0.25*(1-xi[0]);
+
+   dN2dxi(1,0) = 0.25; dN2dxi(1,1) = -0.25; dN2dxi(1,2) = 0.25;
+   dN2dxi(1,3) = -0.25;
+}
+
+// returns the vector and matrix form of the shape functions and its derivative
+void BasisVectorDerivs(const Vector xi, DenseMatrix& N, DenseMatrix& dNdxi,
+                       DenseMatrix& ddNdxi)
+{
+   N.SetSize(3,12); N = 0.0;
+   N(0,0) = 0.25*(1-xi[0])*(1-xi[1]); N(0,3) = 0.25*(1+xi[0])*(1-xi[1]);
+   N(0,6) = 0.25*(1+xi[0])*(1+xi[1]); N(0,9) = 0.25*(1-xi[0])*(1+xi[1]);
+
+   N(1,1) = 0.25*(1-xi[0])*(1-xi[1]); N(1,4) = 0.25*(1+xi[0])*(1-xi[1]);
+   N(1,7) = 0.25*(1+xi[0])*(1+xi[1]); N(1,10) = 0.25*(1-xi[0])*(1+xi[1]);
+
+   N(2,2) = 0.25*(1-xi[0])*(1-xi[1]); N(2,5) = 0.25*(1+xi[0])*(1-xi[1]);
+   N(2,8) = 0.25*(1+xi[0])*(1+xi[1]); N(2,11) = 0.25*(1-xi[0])*(1+xi[1]);
+
+   dNdxi.SetSize(3*2, 3*4);  dNdxi = 0.0;
+   dNdxi(0,0) = 0.25*(-1+xi[1]);  dNdxi(0,3) = 0.25*(1-xi[1]);
+   dNdxi(0,6) = 0.25*(1+xi[1]);  dNdxi(0,9) = 0.25*(-1-xi[1]);
+   dNdxi(1,1) = 0.25*(-1+xi[1]);  dNdxi(1,4) = 0.25*(1-xi[1]);
+   dNdxi(1,7) = 0.25*(1+xi[1]);  dNdxi(1,10) = 0.25*(-1-xi[1]);
+   dNdxi(2,2) = 0.25*(-1+xi[1]);  dNdxi(2,5) = 0.25*(1-xi[1]);
+   dNdxi(2,8) = 0.25*(1+xi[1]);  dNdxi(2,11) = 0.25*(-1-xi[1]);
+
+   dNdxi(3,0) = 0.25*(-1+xi[0]);  dNdxi(3,3) = 0.25*(-1-xi[0]);
+   dNdxi(3,6) = 0.25*(1+xi[0]);  dNdxi(3,9) = 0.25*(1-xi[0]);
+   dNdxi(4,1) = 0.25*(-1+xi[0]);  dNdxi(4,4) = 0.25*(-1-xi[0]);
+   dNdxi(4,7) = 0.25*(1+xi[0]);  dNdxi(4,10) = 0.25*(1-xi[0]);
+   dNdxi(5,2) = 0.25*(-1+xi[0]);  dNdxi(5,5) = 0.25*(-1-xi[0]);
+   dNdxi(5,8) = 0.25*(1+xi[0]);  dNdxi(5,11) = 0.25*(1-xi[0]);
+
+   ddNdxi.SetSize(3*4, 3*4); ddNdxi = 0.0;
+   ddNdxi(3,0) = 0.25; ddNdxi(3,3) = -0.25;
+   ddNdxi(3,6) = 0.25; ddNdxi(3,9) = -0.25;
+   ddNdxi(4,1) = 0.25; ddNdxi(4,4) = -0.25;
+   ddNdxi(4,7) = 0.25; ddNdxi(4,10) = -0.25;
+   ddNdxi(5,2) = 0.25; ddNdxi(5,5) = -0.25;
+   ddNdxi(5,8) = 0.25; ddNdxi(5,11) = -0.25;
+
+   ddNdxi(6,0) = 0.25; ddNdxi(6,3) = -0.25;
+   ddNdxi(6,6) = 0.25; ddNdxi(6,9) = -0.25;
+   ddNdxi(7,1) = 0.25; ddNdxi(7,4) = -0.25;
+   ddNdxi(7,7) = 0.25; ddNdxi(7,10) = -0.25;
+   ddNdxi(8,2) = 0.25; ddNdxi(8,5) = -0.25;
+   ddNdxi(8,8) = 0.25; ddNdxi(8,11) = -0.25;
+}
+
+
+void cross(const Vector a, const Vector b, Vector& c)
+{
+   assert(a.Size()==3);
+   c.SetSize(3);
+   c[0] = a[1]*b[2] - a[2]*b[1];
+   c[1] = -a[0]*b[2] + b[0]*a[2];
+   c[2] = a[0]*b[1] - a[1]*b[0];
+
+}
+// a outer b
+void outer(const Vector a, const Vector b, DenseMatrix& c)
+{
+   int m = a.Size();
+   int n = b.Size();
+   assert(c.Height()==m);
+   assert(c.Width() ==n);
+   for (int i=0; i<m; i++)
+   {
+      for (int j=0; j<n; j++)
+      {
+         c(i,j) = a[i]*b[j];
+      }
+   }
+}
+// dphidxi 2*4
+// coords 4*3
+void ComputeNormal(const DenseMatrix& dphidxi, const DenseMatrix& coords,
+                   Vector& normal, double& nnorm)
+{
+
+   DenseMatrix dxdxi(2,3);
+   Mult(dphidxi, coords, dxdxi);
+   Vector dxdxi1(3);
+   Vector dxdxi2(3);
+
+   dxdxi.GetRow(0,dxdxi1);
+   dxdxi.GetRow(1,dxdxi2);
+
+   cross(dxdxi1, dxdxi2, normal);  // is there a cross product?  no
+   // VectorCrossProductCoefficient::Eval has hard-coded cross product
+   nnorm = normal.Norml2( );
+   normal /=  nnorm;
+}
+
+void SlaveToMaster(const DenseMatrix& m_coords, const Vector& s_x, Vector& xi)
+{
+   bool converged = false;
+   bool pt_on_elem = false;
+   int dim = 3;
+   xi.SetSize(dim-1);
+   xi = 0.0;
+   double r = 1e10;
+   int max_iter = 15;
+   double off_el_xi = 1e-2;
+   double proj_newton_tol = 1e-13;
+   double proj_max_gap = 0.5;
+   Vector gap_v(dim);
+   // warm start from linear solution
+
+   for (int it=0; it<max_iter; it++)
+   {
+      //cout<<it<<endl;
+      Vector m_N(4);
+      m_N = 0.;
+      DenseMatrix m_dN(2,4);
+      m_dN = 0.;
+      DenseMatrix m_dN2(3,4);
+      m_dN2 = 0.;
+      BasisEvalDerivs(xi, m_N, m_dN, m_dN2);
+
+      Vector x_c(dim);
+      m_coords.MultTranspose(m_N, x_c);
+
+      gap_v = s_x;
+      gap_v -= x_c;
+
+      DenseMatrix m_dx(2,3);
+      m_dx = 0.;
+      Mult(m_dN, m_coords, m_dx);
+
+      Vector r(dim-1);
+      r = 0.0;
+      m_dx.Mult(gap_v, r);
+
+      if (r.Normlinf() < proj_newton_tol)
+      {
+         converged = true;
+         break;
+      }
+
+      DenseMatrix drdxi(dim-1,dim-1);
+      drdxi = 0.;
+      MultABt(m_dx, m_dx, drdxi); // m_dx * m_dx.T
+      drdxi *= -1.0;
+
+      DenseMatrix m_dx2(3,3); m_dx2 = 0.0;
+      Mult(m_dN2,m_coords, m_dx2);
+
+      //m_d2x = m_dN(:,:,2) * m_elem_coords(1:4,:);  //m_dN(:,:,2) is 3*4
+      for (int d=0; d<3; d++)
+      {
+         DenseMatrix Mtemp(2,2); Mtemp = 0.0;
+         Mtemp(0,0) = m_dx2(0,d); Mtemp(0,1) = m_dx2(1,d);
+         Mtemp(1,0) = m_dx2(1,d); Mtemp(1,1) = m_dx2(2,d);
+
+         drdxi.Add(gap_v[d], Mtemp);
+      }
+
+      //cond_num = rcond(drdxi);  condition number?
+      //drdxi.TestInversion();
+      DenseMatrixInverse drdxi_inv(drdxi);
+      Vector xi_tmp(dim-1);
+
+      drdxi_inv.Mult(r,xi_tmp);
+      xi -= xi_tmp;
+   }
+   if (!converged)
+   {
+      xi = 0.0;
+   }
+   off_el_xi += 1 ; // tolerance of offset of xi outside [-1,1]
+
+   //cout<<gap_v.Norml2()<<" " <<xi.Normlinf()<<endl;
+   if (gap_v.Norml2() < proj_max_gap && xi.Normlinf() <= off_el_xi)
+   {
+      pt_on_elem = true;
+   }
+
+   MFEM_VERIFY(pt_on_elem == true, "xi went out of bounds");
+   MFEM_VERIFY(converged == true, "projection didn't converge");
+}
+
+
+
+// m_coords is expected to be 4 * 3
+void  ComputeGapJacobian(const Vector x_s, const Vector xi,
+                         const DenseMatrix m_coords,
+                         double& gap, Vector& normal, Vector& dgdxm, Vector& dgdxs)
+{
+   Vector m_N(4);
+   DenseMatrix m_dN(2,4);
+   DenseMatrix m_dN2(3,4);
+   BasisEvalDerivs(xi, m_N, m_dN, m_dN2);
+
+   Vector x_c(3);
+   m_coords.MultTranspose(m_N, x_c);
+
+   Vector gap_v(3); gap_v = 0.0;
+   gap_v  = x_s;
+   gap_v -= x_c;
+
+   DenseMatrix m_dx(2,3);
+   Mult(m_dN, m_coords, m_dx);
+
+   double nnorm = 0;
+   ComputeNormal(m_dN, m_coords, normal, nnorm);
+
+   gap = gap_v * normal; // gap function value, dot product between vectors
+
+   //dr_dx = zeros(2,4,3); % nsegment, nodes in quad, ndim
+
+   DenseMatrix dr_dx_res1(4,3); dr_dx_res1 = 0.;
+   DenseMatrix dr_dx_res2(4,3); dr_dx_res2 = 0.;
+
+   Vector m_dxrow1(3);
+   m_dx.GetRow(0, m_dxrow1);
+   outer(m_N, m_dxrow1, dr_dx_res1);// 4*1 times 1*3
+   dr_dx_res1 *= -1.0;
+
+   Vector m_dxrow2(3);
+   m_dx.GetRow(1, m_dxrow2);
+   outer(m_N, m_dxrow2, dr_dx_res2);// 4*1 times 1*3
+   dr_dx_res2 *= -1.0;
+
+   Vector m_dNrow1(4); m_dN.GetRow(0, m_dNrow1);
+   Vector m_dNrow2(4); m_dN.GetRow(1, m_dNrow2);
+
+   DenseMatrix dr_dx_res1_tmp(4,3); dr_dx_res1_tmp = 0.;
+   DenseMatrix dr_dx_res2_tmp(4,3); dr_dx_res2_tmp = 0.;
+   outer(m_dNrow1, gap_v, dr_dx_res1_tmp);// 4*1 times 1*3
+   outer(m_dNrow2, gap_v, dr_dx_res2_tmp);// 4*1 times 1*3
+
+   dr_dx_res1 += dr_dx_res1_tmp; // outer product in vector?
+   dr_dx_res2 += dr_dx_res2_tmp;
+
+
+   DenseMatrix K_dxidx1(2,2); // 2*2
+   K_dxidx1 = 0.;
+   MultABt(m_dx, m_dx, K_dxidx1); // m_dx * m_dx.T
+
+   Vector v_dxidx2(4);
+   m_coords.Mult(gap_v, v_dxidx2); // m_coords * gap_v;  // 4*3 * 3 = 4
+
+   DenseMatrix K_dxidx2(2,2); K_dxidx2 = 0.0;
+
+   Vector m_dN2row1(4); m_dN2.GetRow(0, m_dN2row1);
+   Vector m_dN2row2(4); m_dN2.GetRow(1, m_dN2row2);
+   Vector m_dN2row3(4); m_dN2.GetRow(2, m_dN2row3);
+   // how to get 2nd order? multidimensional matrix?
+   K_dxidx2(0,0) = m_dN2row1 * v_dxidx2; // how would 4*1 * 1*4 be computed?
+   K_dxidx2(0,1) = m_dN2row2 * v_dxidx2;
+   K_dxidx2(1,0) = m_dN2row2 * v_dxidx2;
+   K_dxidx2(1,1) = m_dN2row3 * v_dxidx2;
+
+   DenseMatrix K_dxidx(2,2);
+   K_dxidx -= K_dxidx1;
+   K_dxidx += K_dxidx2;
+
+   // resize the vectors and matrices
+   Vector dxidx(24); dxidx = 0.0;
+   Vector drdx_r(24); drdx_r = 0.0;
+
+   for (int i=0; i<4; i++)
+   {
+      for (int j=0; j<3; j++)
+      {
+         drdx_r[4*j+i] = dr_dx_res1(i,j);
+         drdx_r[4*j+i+12] = dr_dx_res2(i,j);
+
+      }
+   }
+   //drdx_r(1:4*3,1) = reshape(dr_dx_res(:,:,1),4*3,1);
+   //drdx_r(4*3+1:2*4*3,1) = reshape(dr_dx_res(:,:,2),4*3,1);
+   DenseMatrix drdx_K(24,24); drdx_K = 0.;
+   for (int i =0; i<12; i++)
+   {
+      drdx_K(i,i) = K_dxidx(0,0);
+      drdx_K(i,12+i) = K_dxidx(0,1);
+      drdx_K(12+i,i) = K_dxidx(1,0);
+      drdx_K(12+i,12+i) = K_dxidx(1,1);
+   }
+
+   DenseMatrixInverse drdxK_inv(drdx_K);
+   drdxK_inv.Mult(drdx_r,dxidx);
+   // LinearSolve (drdx_K,drdx_r, dxidx) ; //???
+   dxidx *= -1.0;
+
+
+
+   Vector drdxs_r(6);
+   drdxs_r[0] = m_dx(0,0); drdxs_r[1] = m_dx(0,1); drdxs_r[2] = m_dx(0,2);
+   drdxs_r[3] = m_dx(1,0); drdxs_r[4] = m_dx(1,1); drdxs_r[5] = m_dx(1,2);
+
+   DenseMatrix drdxs_K(6,6); drdxs_K = 0.;
+   for (int i=0; i<3; i++)
+   {
+      drdxs_K(i,i) = K_dxidx(0,0);
+      drdxs_K(i,3+i) = K_dxidx(0,1);
+      drdxs_K(i+3,i) = K_dxidx(1,0);
+      drdxs_K(i+3,i+3) = K_dxidx(1,1);
+   }
+
+   Vector dxidxs(6); dxidxs = 0.0;
+   DenseMatrixInverse drdxsK_inv(drdxs_K);
+   drdxsK_inv.Mult(drdxs_r,dxidxs);
+   dxidxs *= -1.0;
+   //dxidxs = -drdxs_K\drdxs_r;
+
+   //dxidx = reshape(dxidx, 4,3,2); dxidxs = reshape(dxidxs, 1,3,2);
+
+   dgdxm.SetSize(12); dgdxm = 0.;
+   DenseMatrix dgdxm_tmp(4,3);
+   outer(m_N, normal,dgdxm_tmp);
+   for (int i=0; i<4; i++)
+   {
+      for (int j=0; j<3; j++)
+      {
+         dgdxm[3*i+j] = -dgdxm_tmp(i,j);
+      }
+   }
+   //dxidx_M = -m_dN(1:2,:,1) * (m_coords(1:4,:)*normal'); % this turns out to be 0
+
+   dgdxs.SetSize(3);
+   dgdxs += normal;
+   //dgdxs = dgdxs + dxidx_M(1) * dxidxs(:,:,1) + dxidx_M(2) * dxidxs(:,:,2);
+};
+
+void ComputeGapHessian(const Vector x_s, const Vector xi,
+                       const DenseMatrix m_coords,
+                       DenseMatrix& dg2dx)
+{
+   Vector m_N(4);
+   DenseMatrix m_dN(2,4);
+   DenseMatrix m_dN2(3,4);
+   BasisEvalDerivs(xi, m_N, m_dN, m_dN2);
+
+   int dim = 3;
+   int num_dofs1 = dim;
+   int num_dofs2 = 4*dim;
+   int num_dofs = num_dofs1 + num_dofs2;
+   dg2dx.SetSize(num_dofs,num_dofs); dg2dx = 0.0;
+
+   Vector x_c(3);
+   m_coords.MultTranspose(m_N,x_c);
+
+   Vector gap_v(3); gap_v = 0.0;
+   gap_v  = x_s;
+   gap_v -= x_c;
+
+   DenseMatrix m_dx(2,3);
+   Mult(m_dN, m_coords, m_dx);
+
+   DenseMatrix m_dx2(3,3); m_dx2 = 0.0;
+   Mult(m_dN2,m_coords, m_dx2);
+   double nnorm = 0.0;
+   Vector normal(3); normal = 0.0;
+   ComputeNormal(m_dN, m_coords, normal, nnorm);
+
+   double gap = gap_v * normal; // gap function value, dot product between vectors
+
+   DenseMatrix M(2,2); M = 0.0;
+   MultABt(m_dx, m_dx, M);
+
+   DenseMatrix f(2, num_dofs2); f = 0.0;
+
+   for (int d=0; d<3; d++)
+   {
+      DenseMatrix Mtemp(2,2); Mtemp = 0.0;
+      Mtemp(0,0) = m_dx2(0,d); Mtemp(0,1) = m_dx2(1,d);
+      Mtemp(1,0) = m_dx2(1,d); Mtemp(1,1) = m_dx2(2,d);
+
+      M.Add(-gap_v[d], Mtemp);
+
+      Vector m_dxcol(2); m_dx.GetColumn(d, m_dxcol);
+      DenseMatrix ftmp(2,4);
+      outer(m_dxcol, m_N, ftmp);
+      ftmp *= -1;
+      ftmp.Add( gap_v[d], m_dN);   // 2*4
+
+      for (int j=0; j<4; j++)
+      {
+         assert(d+3*j<num_dofs2);
+         f(0,d+j*3) = ftmp(0,j);
+         f(1,d+j*3) = ftmp(1,j);
+      }
+   }
+   //fprintf('hess dxidxm\n');
+   DenseMatrixInverse Minv(M);
+   DenseMatrix dxidxm(2,num_dofs2); dxidxm = 0.0;
+   Minv.Mult(f, dxidxm);
+   //LinearSolve??
+   //dxidxm = M\f;
+
+   DenseMatrix nde2(2,2); nde2 = 0.0;
+   DenseMatrix Nndx2(2,num_dofs2); Nndx2 = 0.0;
+
+   for (int d=0; d<3; d++)
+   {
+      DenseMatrix ndetmp(2,2); ndetmp = 0.0;
+      ndetmp(0,0) = normal(d)*m_dx2(0,d); ndetmp(0,1) = normal(d)*m_dx2(1,d);
+      ndetmp(1,0) = normal(d)*m_dx2(1,d); ndetmp(1,1) = normal(d)*m_dx2(2,d);
+
+      nde2 += ndetmp;
+
+      for (int j=0; j<4; j++)
+      {
+         assert(d+3*j<num_dofs2);
+         Nndx2(0,d+j*3) = normal[d]*m_dN(0,j);
+         Nndx2(1,d+j*3) = normal[d]*m_dN(1,j);
+      }
+
+   }
+
+   DenseMatrix Ndn(2,num_dofs2); Ndn = 0.0;
+   Ndn += Nndx2;
+   AddMult(nde2, dxidxm, Ndn);
+
+
+   DenseMatrix M2(2,2); M2 = 0.0;
+   MultABt(m_dx, m_dx, M2);
+   DenseMatrixInverse M2inv(M2);
+   DenseMatrix diag2(2,2); diag2(0,0) = 1.0; diag2(1,1) = 1.0;
+   DenseMatrix m_con(2,2); m_con = 0.0;
+
+   M2inv.Mult(diag2, m_con);
+
+   DenseMatrix dg2dxm(num_dofs2, num_dofs2); dg2dxm = 0.0;
+
+   DenseMatrix dg2dxm_tmp(num_dofs2,2); dg2dxm_tmp = 0.0;
+   MultAtB(Ndn, m_con, dg2dxm_tmp);
+   Mult(dg2dxm_tmp, Ndn, dg2dxm);
+   dg2dxm *= gap;
+
+   DenseMatrix dg2dxm_tmp2(num_dofs2,num_dofs2); dg2dxm_tmp2 = 0.0;
+   MultAtB(Nndx2, dxidxm, dg2dxm_tmp2);
+   dg2dxm.Add(-1.0, dg2dxm_tmp2);
+
+   dg2dxm_tmp = 0.0;
+   MultAtB(dxidxm, nde2, dg2dxm_tmp);
+
+   AddMult_a(-1.0, dg2dxm_tmp, dxidxm, dg2dxm);
+
+   dg2dxm_tmp2 = 0.0;
+   MultAtB(dxidxm, Nndx2, dg2dxm_tmp2);
+   dg2dxm.Add(-1.0, dg2dxm_tmp2);
+
+   Vector v_dxidx2(4);
+   m_coords.Mult(gap_v, v_dxidx2); // m_coords * gap_v;  // 4*3 * 3 = 4
+
+   DenseMatrix K_dxidx2(2,2); K_dxidx2 = 0.0;
+
+   Vector m_dN2row1(4); m_dN2.GetRow(0, m_dN2row1);
+   Vector m_dN2row2(4); m_dN2.GetRow(1, m_dN2row2);
+   Vector m_dN2row3(4); m_dN2.GetRow(2, m_dN2row3);
+   K_dxidx2(0,0) = m_dN2row1 * v_dxidx2; // how would 4*1 * 1*4 be computed?
+   K_dxidx2(0,1) = m_dN2row2 * v_dxidx2;
+   K_dxidx2(1,0) = m_dN2row2 * v_dxidx2;
+   K_dxidx2(1,1) = m_dN2row3 * v_dxidx2;
+
+   DenseMatrix K_dxidx(2,2);
+   K_dxidx -= M2;
+   K_dxidx += K_dxidx2;
+
+   Vector drdxs_r(6);
+   drdxs_r[0] = m_dx(0,0); drdxs_r[1] = m_dx(0,1); drdxs_r[2] = m_dx(0,2);
+   drdxs_r[3] = m_dx(1,0); drdxs_r[4] = m_dx(1,1); drdxs_r[5] = m_dx(1,2);
+
+   DenseMatrix drdxs_K(6,6); drdxs_K = 0.;
+   for (int i=0; i<3; i++)
+   {
+      drdxs_K(i,i) = K_dxidx(0,0);
+      drdxs_K(i,3+i) = K_dxidx(0,1);
+      drdxs_K(i+3,i) = K_dxidx(1,0);
+      drdxs_K(i+3,i+3) = K_dxidx(1,1);
+   }
+   Vector dxidxs(6);
+
+   DenseMatrixInverse drdxsK_inv(drdxs_K);
+   drdxsK_inv.Mult(drdxs_r,dxidxs);
+   dxidxs *= -1.0;
+   //dxidxs = -drdxs_K\drdxs_r;
+
+   DenseMatrix dxidxs_m(2,3); dxidxs_m = 0.0;
+   dxidxs_m(0,0) = dxidxs[0]; dxidxs_m(0,1) = dxidxs[1]; dxidxs_m(0,2) = dxidxs[2];
+   dxidxs_m(1,0) = dxidxs[3]; dxidxs_m(1,1) = dxidxs[4]; dxidxs_m(1,2) = dxidxs[5];
+
+   DenseMatrix dtao1dxs(3,3); dtao1dxs = 0.0;
+   DenseMatrix dtao2dxs(3,3); dtao2dxs = 0.0;
+
+   Vector dxidxs_row1(3); dxidxs_row1 = 0.0; Vector dxidxs_row2(3);
+   dxidxs_row2 = 0.0;
+   Vector mdx2_row1(3); mdx2_row1 = 0.0; Vector mdx2_row2(3); mdx2_row2 = 0.0;
+   Vector mdx2_row3(3); mdx2_row3 = 0.0;
+   dxidxs_m.GetRow(0,dxidxs_row1);
+   dxidxs_m.GetRow(1,dxidxs_row2);
+   m_dx2.GetRow(0,mdx2_row1);
+   m_dx2.GetRow(1,mdx2_row2);
+   m_dx2.GetRow(2,mdx2_row3);
+
+   DenseMatrix dtaotmp(3,3); dtaotmp = 0.0;
+   outer(mdx2_row1, dxidxs_row1,dtaotmp);
+   dtao1dxs += dtaotmp; dtaotmp = 0.0;
+   outer(mdx2_row2, dxidxs_row1,dtaotmp);
+   dtao1dxs += dtaotmp; dtaotmp = 0.0;
+
+   outer(mdx2_row2, dxidxs_row2, dtaotmp);
+   dtao2dxs += dtaotmp; dtaotmp = 0.0;
+   outer(mdx2_row3, dxidxs_row2, dtaotmp);
+   dtao2dxs += dtaotmp; dtaotmp = 0.0;
+
+   DenseMatrix dtaodxs(3,3); dtaodxs = 0.0; //tao = tao1 cross tao2
+
+   for (int d=0; d<3; d++)
+   {
+      Vector dtao1dxs_tmp(3); dtao1dxs_tmp = 0.0;
+      dtao1dxs.GetColumn(d,dtao1dxs_tmp);
+      Vector m_dxrow(3); m_dx.GetRow(1, m_dxrow);
+
+      Vector dtaodxs_tmp(3); dtaodxs_tmp = 0.0;
+      cross(dtao1dxs_tmp, m_dxrow, dtaodxs_tmp);
+
+      Vector dtaodxs_tmp2(3); dtaodxs_tmp2 = 0.0;
+      m_dx.GetRow(0, m_dxrow);
+      dtao1dxs_tmp = 0.0;  // reuse the same vector for dtao2
+      dtao2dxs.GetColumn(d,dtao1dxs_tmp);
+      cross(m_dxrow, dtao1dxs_tmp, dtaodxs_tmp2);
+
+      dtaodxs_tmp2 += dtaodxs_tmp;
+      dtaodxs.SetCol(d, dtaodxs_tmp2);
+   }
+
+   DenseMatrix dndxs(3,3); dndxs = 0.0; dndxs += dtaodxs; dndxs *= 1.0/nnorm;
+   DenseMatrix dndxs_tmp(3,3); dndxs_tmp = 0.0;
+   outer(normal, normal, dndxs_tmp);
+   AddMult_a(-1/nnorm, dndxs_tmp, dtaodxs, dndxs);
+
+   DenseMatrix dgvdxs(3,3); dgvdxs = 0.0;
+   MultAtB(m_dx, dxidxs_m, dgvdxs);
+   dgvdxs *= -1;
+   for (int d=0; d<3; d++)
+   {
+      dgvdxs(d,d) += 1.0;
+   }
+   //dxidxs: 2*3
+
+   DenseMatrix dg2dxs(3,3); dg2dxs = 0.0;
+   DenseMatrix dg2dxs_tmp(3,2); dg2dxs_tmp = 0.0;
+   MultAtB(dxidxs_m, nde2, dg2dxs_tmp);
+   AddMult_a(-1.0, dg2dxs_tmp, dxidxs_m, dg2dxs);
+   DenseMatrix dg2dxs_tmp2(3,3); dg2dxs_tmp2 = 0.0;
+   MultAtB(dgvdxs, dndxs, dg2dxs_tmp2);
+   dg2dxs += dg2dxs_tmp2;
+   dg2dxs_tmp2 = 0.0;
+   MultAtB(dndxs, dndxs_tmp, dg2dxs_tmp2);
+   AddMult(dg2dxs_tmp2, dgvdxs, dg2dxs);
+
+   DenseMatrix Ne(3,12), Be(6,12), dBe(12,12);
+   BasisVectorDerivs(xi, Ne, Be, dBe);
+
+   DenseMatrix dtao1dxm(3,12); dtao1dxm.CopyRows(Be, 0, 2);
+   DenseMatrix dtao2dxm(3,12); dtao2dxm.CopyRows(Be, 3, 5);
+
+   Vector m_coords_v(12);
+   for (int i=0; i<4; i++)
+   {
+      for (int j=0; j<3; j++)
+      {
+         m_coords_v[i*3+j] = m_coords(i,j);
+      }
+   }
+
+   for (int i=0; i<2; i++)
+   {
+      Vector dxidxm_tmp(num_dofs2); dxidxm_tmp = 0.0;
+      dxidxm.GetRow(i,dxidxm_tmp);
+
+      DenseMatrix dBe_tmp(3,12);
+      dBe_tmp.CopyRows(dBe,i*3,(i+1)*3-1);
+
+      DenseMatrix dtaodxm_tmp(12,12); dtaodxm_tmp = 0.0;
+      outer(m_coords_v, dxidxm_tmp, dtaodxm_tmp);
+      AddMult(dBe_tmp, dtaodxm_tmp, dtao1dxm);
+
+      //dtao1dxm += dBe(:,:,i)*reshape(m_coords(1:4,:)',12,1)*reshape(dxidxm(i,:),1,12); % 3*12
+      dBe_tmp = 0.0;
+      dBe_tmp.CopyRows(dBe,(i+2)*3,(i+3)*3-1);
+      AddMult(dBe_tmp, dtaodxm_tmp, dtao2dxm);
+
+   }
+
+   DenseMatrix dtaodxm(3,12); dtaodxm = 0.0;//tao = tao1 cross tao2
+
+   for (int d=0; d<12; d++)
+   {
+      Vector dtaodxm_tmp(3); dtaodxm_tmp = 0.0;
+      Vector dtaodxm_tmp2(3); dtaodxm_tmp2 = 0.0;
+      Vector tmp1(3); tmp1 = 0.0;  dtao1dxm.GetColumn(d,tmp1);
+      Vector m_dxrow2(3); m_dx.GetRow(1, m_dxrow2);
+      Vector m_dxrow1(3); m_dx.GetRow(0, m_dxrow1);
+      Vector tmp2(3); tmp2 = 0.0;  dtao2dxm.GetColumn(d,tmp2);
+
+      cross(tmp1, m_dxrow2, dtaodxm_tmp);
+      cross(m_dxrow1,tmp2, dtaodxm_tmp2);
+      dtaodxm_tmp += dtaodxm_tmp2;
+
+      dtaodxm.SetCol(d, dtaodxm_tmp);
+   }
+
+   DenseMatrix dndxm(3,12); dndxm = 0.0;
+   dndxm += dtaodxm;
+   dndxm *= 1.0/nnorm;
+   AddMult_a(-1/nnorm, dndxs_tmp, dtaodxm, dndxm); //dndxs_tmp = normal'*normal
+
+   DenseMatrix dgvdxm(3,12); dgvdxm = 0.0;
+   dgvdxm -= Ne;
+
+   for (int i=0; i<2; i++)
+   {
+      Vector dxidxm_tmp(num_dofs2); dxidxm_tmp = 0.0;
+      dxidxm.GetRow(i,dxidxm_tmp);
+
+      DenseMatrix Be_tmp(3,12);
+      Be_tmp.CopyRows(Be,i*3,(i+1)*3-1);
+
+      DenseMatrix dgvdxm_tmp(12,12); dgvdxm_tmp = 0.0;
+      outer(m_coords_v, dxidxm_tmp, dgvdxm_tmp);
+      AddMult_a(-1.0, Be_tmp, dgvdxm_tmp, dgvdxm);
+
+   }
+
+   DenseMatrix dg2dxsxm(3,12); dg2dxsxm = 0.0;
+   DenseMatrix dg2dxsxm_tmp(3,3); dg2dxsxm_tmp = 0.0;
+   MultAtB(dgvdxs, dndxm, dg2dxsxm);
+
+   MultAtB(dndxs, dndxs_tmp, dg2dxsxm_tmp);
+   AddMult(dg2dxsxm_tmp, dgvdxm, dg2dxsxm); // += dndxs'*normal'*normal*dgvdxm;
+
+   DenseMatrix dgvdxsxmn(3,12); dgvdxsxmn = 0.0;
+   DenseMatrix dgvdxsxmn_tmp(3,2); dgvdxsxmn_tmp = 0.0;
+   MultAtB(dxidxs_m, nde2, dgvdxsxmn_tmp);   //dxidxs_m: 2*3
+
+   AddMult_a(-1.0, dgvdxsxmn_tmp, dxidxm, dgvdxsxmn);
+
+
+   for (int i =0; i<2; i++)
+   {
+      DenseMatrix Be_tmp(3,12);
+      Be_tmp.CopyRows(Be,i*3,(i+1)*3-1);
+
+      Vector dxidxs_row(3); dxidxs_row = 0.0; dxidxs_m.GetRow(i,dxidxs_row);
+      DenseMatrix dgvdxsxmn_tmp2(3,3); dgvdxsxmn_tmp2 = 0.0;
+      outer(dxidxs_row, normal, dgvdxsxmn_tmp2);
+      AddMult_a(-1.0, dgvdxsxmn_tmp2, Be_tmp, dgvdxsxmn);
+   }
+
+   dg2dxsxm += dgvdxsxmn;
+
+   DenseMatrix dg2dxmxs(12,3); dg2dxmxs = 0.0;
+   DenseMatrix dg2dxmxs_tmp(12,3); dg2dxmxs_tmp = 0.0;
+   MultAtB(dgvdxm, dndxs, dg2dxmxs);
+   MultAtB(dndxm, dndxs_tmp, dg2dxmxs_tmp);
+   AddMult(dg2dxmxs_tmp, dgvdxs, dg2dxmxs);
+
+   DenseMatrix dgvdxmxsn(12,3); dgvdxmxsn = 0.0;
+   DenseMatrix dgvdxmxsn_tmp(12,2); dgvdxmxsn_tmp = 0.0;
+
+   MultAtB(dxidxm, nde2, dgvdxmxsn_tmp);
+   dgvdxmxsn_tmp *= -1.0;
+   AddMult(dgvdxmxsn_tmp, dxidxs_m, dgvdxmxsn);
+
+   for (int i =0; i<2; i++)
+   {
+      DenseMatrix Be_tmp(3,12);
+      Be_tmp.CopyRows(Be,i*3,(i+1)*3-1);
+      Be_tmp.Transpose(); // Be is now 12*3
+
+      Vector dxidxs_row(3); dxidxs_row = 0.0; dxidxs_m.GetRow(i,dxidxs_row);
+      DenseMatrix dgvdxmxsn_tmp2(3,3); dgvdxmxsn_tmp2 = 0.0;
+      outer(normal, dxidxs_row, dgvdxmxsn_tmp2);
+      AddMult_a(-1.0, Be_tmp, dgvdxmxsn_tmp2, dgvdxmxsn);
+
+   }
+
+   dg2dxmxs += dgvdxmxsn;
+
+   dg2dx.CopyMN(dg2dxs, 0, 0);
+   dg2dx.CopyMN(dg2dxm, 3, 3);
+   dg2dx.CopyMN(dg2dxsxm, 0, 3);
+   dg2dx.CopyMN(dg2dxmxs, 3, 0);
+
+};
+
+
+
+void NodeSegConPairs(const Vector x1, const Vector xi2,
+                     const DenseMatrix coords2,
+                     double& node_g, Vector& node_dg, DenseMatrix& node_dg2)
+{
+   double gap = 0.0;
+   Vector normal(3); normal = 0.0;
+   Vector dgdxm(12); dgdxm = 0.0;
+   Vector dgdxs(3); dgdxs = 0.0;
+
+   ComputeGapJacobian(x1, xi2, coords2, gap, normal, dgdxm, dgdxs);
+   node_g = gap;
+
+   node_dg.SetSize(12+3);
+   for (int i=0; i<3; i++) { node_dg[i] = dgdxs[i]; }
+   for (int i=0; i<12; i++) { node_dg[i+3] = dgdxm[i]; }
+
+   DenseMatrix dg2dx(15,15); dg2dx = 0.0;
+   DenseMatrix dgvdxmxsn(12,3); dgvdxmxsn = 0.0;
+   ComputeGapHessian(x1, xi2, coords2, dg2dx);
+
+   node_dg2.SetSize(15,15);
+   node_dg2 = dg2dx;
+
+   /*
+   if(obj.space1.conns{e1}(i)==150)  % for debugging purpose
+
+   v1 = 1:3;
+   v2 = 1:12;
+   %v1 = ones(1,3)
+   %v2 = ones(1,12)
+   v2 = reshape(v2,4,3);
+   x1n1 = x1 + 0.01*v1;
+   coords2n1 = coords2 + 0.001*v2;
+      [xi2n1, gapv1, ~, ~] = SlaveToMaster(obj, coords2n1, x1n1);
+      [gapn1, n1,dgdxmn1, dgdxsn1] = ComputeGapJacobian(obj, x1n1, xi2n1, coords2n1);
+   x1n2 = x1 - 0.01*v1;
+   coords2n2 = coords2 - 0.001*v2;
+      [xi2n2, gapv2, ~, ~] = SlaveToMaster(obj, coords2n2, x1n2);
+      [gapn2, n2,dgdxmn2, dgdxsn2] = ComputeGapJacobian(obj, x1n2, xi2n2, coords2n2);
+   fprintf('fd\n');
+      %gapv1-gapv2
+   [dgdxsn1(:)',dgdxmn1(:)'] - [dgdxsn2(:)',dgdxmn2(:)']
+
+   %dgdxsn1-dgdxsn2
+   fprintf('code\n');
+   v2n = v2';
+   %dg2dx(1:3,1:3)*0.04*ones(3,1)
+   temp = zeros(12,3);
+   for i = 1:4
+   temp1 = dg2dx(3+(i-1)*3+1:3+i*3,1:3);
+   temp((i-1)*3+1:i*3,:) = temp1';
+   end
+   temp2 = zeros(3,12);
+   for i = 1:4
+   temp3 = dg2dx(1:3,3+(i-1)*3+1:3+i*3);
+   temp2(:,(i-1)*3+1:i*3) = temp3';
+   end
+   %dg2dx
+   %dg2dx(4:end,1:3) = temp;
+   %dg2dx(1:3,4:end) = temp2;
+   %dgvdxm * 0.002*v2n(:)
+   (dg2dx*[0.02*v1(:)',0.002*v2n(:)']')'
+   %dg2dx(4:end,1:3)
+   end*/
+
+};
+
+
+// coordsm : (npoints*4, 3) use what class?
+// m_conn: (npoints*4)
+void Assemble_Contact(const int m, const int npoints, const int ndofs,
+                      const Vector x_s,
+                      const Vector xi, const DenseMatrix coordsm, const Array<int> s_conn,
+                      const Array<int> m_conn, Vector& g, SparseMatrix& M,
+                      std::vector<SparseMatrix>& dM)
+{
+   int n = ndofs;
+   int ndim = 3;
+
+   g.SetSize(m);
+   g = 0.0;
+
+   //SparseMatrix M(m, n);  // M needs to be the correct size
+
+   //dM.resize(m); // needs to clear?
+
+   double g_tmp = 0.;
+   Vector dg(4*ndim+ndim);
+   dg = 0.;
+   DenseMatrix dg2(4*ndim+ndim,4*ndim+ndim);
+   dg2 = 0.;
+
+   for (int i=0; i<npoints; i++)
+   {
+      Vector x1(ndim);
+      x1[0] = x_s[i*ndim];
+      x1[1] = x_s[i*ndim+1];
+      x1[2] = x_s[i*ndim+2];
+
+      Vector xi2(ndim-1);
+      xi2[0] = xi[i*(ndim-1)];
+      xi2[1] = xi[i*(ndim-1)+1];
+
+      DenseMatrix coords2(4,3);
+      coords2.CopyRows(coordsm, i*4,(i+1)*4-1);
+
+      //how to get coords2?
+      dg = 0.0;
+      dg2 = 0.;
+      NodeSegConPairs(x1, xi2, coords2, g_tmp, dg, dg2);
+      //x1.Print();
+      //xi2.Print();
+      //coords2.Print();
+      g[s_conn[i]] = g_tmp; // should be unique
+      Array<int> m_conn_i(4);
+      m_conn.GetSubArray(4*i, 4, m_conn_i);
+
+      Array<int> node_conn(5);
+      node_conn[0] = s_conn[i];
+      for (int j=0; j<4; j++)
+      {
+         node_conn[j+1] = m_conn_i[j];
+      }
+
+      Array<int>  M_i_tmp(1);
+      M_i_tmp[0] = s_conn[i];
+
+      //j_idx = (node_conn-1)*obj.disp_field.num_components +repmat((1:obj.disp_field.num_components)', 1, length(node_conn{i}));
+      Array<int> j_idx(5*ndim); j_idx = 0;
+      for (int j=0; j< 5; j++)
+      {
+         for (int k=0; k<ndim; k++)
+         {
+            j_idx[j*ndim+k] = node_conn[j]*ndim+k;
+         }
+      }
+      DenseMatrix M_v_tmp(1, ndim*(4+1));   // SetData now?
+      M_v_tmp.SetRow(0, dg);
+
+      M.AddSubMatrix(M_i_tmp, j_idx, M_v_tmp);
+
+      Array<int> dM_i(ndim*(4+1));
+      Array<int> dM_j(ndim*(4+1));
+
+      for (int j=0; j< ndim*(4+1); j++)
+      {
+         dM_i[j] = j_idx[j];
+         dM_j[j] = j_idx[j];
+      }
+      //dg2.Print();
+      //dM[s_conn[i]].Print();
+      dM[s_conn[i]].AddSubMatrix(dM_i,dM_j, dg2);
+   }
+};
+
@@ -14,4 +14,5 @@
 -mg_levels_esteig_ksp_type cg
 -mg_levels_esteig_ksp_max_it 10
 -mg_levels_ksp_chebyshev_esteig 0,0.05,0,1.05
+-pc_gamg_use_sa_esteig 0
 -mg_levels_pc_type sor
@@ -124,6 +124,7 @@ void BilinearForm::SetAssemblyLevel(AssemblyLevel assembly_level)
      case AssemblyLevel::LEGACY:
         break;
      case AssemblyLevel::FULL:
+         SetDiagonalPolicy( DIAG_ONE ); // Only diagonal policy supported on device
         ext = new FABilinearFormExtension(this);
         break;
      case AssemblyLevel::ELEMENT:
@@ -80,6 +80,9 @@ protected:
   /** @brief Extension for supporting Full Assembly (FA), Element Assembly (EA),
       Partial Assembly (PA), or Matrix Free assembly (MF). */
   BilinearFormExtension *ext;
+   /** Indicates if the sparse matrix is sorted after assembly when using
+       Full Assembly (FA). */
+   bool sort_sparse_matrix = false;

   /** @brief Indicates the Mesh::sequence corresponding to the current state of
       the BilinearForm. */
@@ -181,6 +184,21 @@ public:
       If used, this method must be called before assembly. */
   void SetAssemblyLevel(AssemblyLevel assembly_level);

+   /** @brief Force the sparse matrix column indices to be sorted when using
+       AssemblyLevel::FULL.
+
+       When assembling on device the assembly algorithm uses atomic operations
+       to insert values in the sparse matrix, which can result in different
+       column index orderings across runs. Calling this method with @a enable_it
+       set to @a true forces a sorting algorithm to be called at the end of the
+       assembly procedure to ensure sorted column indices (and therefore
+       deterministic results).
+   */
+   void EnableSparseMatrixSorting(bool enable_it)
+   {
+      sort_sparse_matrix = enable_it;
+   }
+
   /// Returns the assembly level
   AssemblyLevel GetAssemblyLevel() const { return assembly; }

@@ -529,7 +529,7 @@ void EABilinearFormExtension::Assemble()
   }

   faceDofs = trial_fes ->
-              GetTraceElement(0, trial_fes->GetMesh()->GetFaceBaseGeometry(0)) ->
+              GetTraceElement(0, trial_fes->GetMesh()->GetFaceGeometry(0)) ->
              GetDof();

   MFEM_VERIFY(a->GetBBFI()->Size() == 0,
@@ -955,6 +955,10 @@ void FABilinearFormExtension::Assemble()
      }
      a->mat = mat;
   }
+   if ( a->sort_sparse_matrix )
+   {
+      a->mat->SortColumnIndices();
+   }
 }


@@ -208,10 +208,6 @@ public:
                                            Vector &x, Vector &b,
                                            OperatorHandle &A, Vector &X, Vector &B) = 0;

-   virtual void AddMult(const Vector &x, Vector &y, const double c=1.0) const = 0;
-   virtual void AddMultTranspose(const Vector &x, Vector &y,
-                                 const double c=1.0) const = 0;
-
   virtual void AssembleDiagonal_ADAt(const Vector &D, Vector &diag) const = 0;

   virtual void Update() = 0;
@@ -287,7 +283,7 @@ public:
   /// Partial assembly of all internal integrators
   void Assemble();

-   void AddMult(const Vector &x, Vector &y, const double c) const;
+   void AddMult(const Vector &x, Vector &y, const double c=1.0) const;

   void AddMultTranspose(const Vector &x, Vector &y, const double c=1.0) const;

@@ -2338,7 +2338,7 @@ void MixedCurlIntegrator::AssembleElementMatrix2(
   if (spaceH1)
   {
      dshape.SetSize(trial_dof,dim);
-      curlshape.SetSize(dim*trial_dof,1);
+      curlshape.SetSize(trial_dof,dim);
      dimc = dim;
   }
   else
@@ -2367,7 +2367,7 @@ void MixedCurlIntegrator::AssembleElementMatrix2(
      if (spaceH1)
      {
         trial_fe.CalcPhysDShape(Trans, dshape);
-         dshape.GradToCurl(curlshape);
+         dshape.GradToVectorCurl2D(curlshape);
      }
      else
      {
@@ -3059,8 +3059,8 @@ public:

 /** Integrator for the DG form:

-    - < {(Q grad(u)).n}, [v] > + sigma < [u], {(Q grad(v)).n} >
-    + kappa < {h^{-1} Q} [u], [v] >,
+        - < {(Q grad(u)).n}, [v] > + sigma < [u], {(Q grad(v)).n} >
+        + kappa < {h^{-1} Q} [u], [v] >

    where Q is a scalar or matrix diffusion coefficient and u, v are the trial
    and test spaces, respectively. The parameters sigma and kappa determine the
@@ -145,7 +145,7 @@ void DGTraceIntegrator::SetupPA(const FiniteElementSpace &fes, FaceType type)
   // Assumes tensor-product elements
   Mesh *mesh = fes.GetMesh();
   const FiniteElement &el =
-      *fes.GetTraceElement(0, fes.GetMesh()->GetFaceBaseGeometry(0));
+      *fes.GetTraceElement(0, fes.GetMesh()->GetFaceGeometry(0));
   FaceElementTransformations &T0 =
      *fes.GetMesh()->GetFaceElementTransformations(0);
   const IntegrationRule *ir = IntRule?
@@ -56,8 +56,7 @@ void MassIntegrator::AssemblePA(const FiniteElementSpace &fes)
   dim = mesh->Dimension();
   ne = fes.GetMesh()->GetNE();
   nq = ir->GetNPoints();
-   geom = mesh->GetGeometricFactors(*ir, GeometricFactors::COORDINATES |
-                                    GeometricFactors::JACOBIANS, mt);
+   geom = mesh->GetGeometricFactors(*ir, GeometricFactors::DETERMINANTS, mt);
   maps = &el.GetDofToQuad(*ir, DofToQuad::TENSOR);
   dofs1D = maps->ndof;
   quad1D = maps->nqpt;
@@ -74,7 +73,7 @@ void MassIntegrator::AssemblePA(const FiniteElementSpace &fes)
      const bool const_c = coeff.Size() == 1;
      const bool by_val = map_type == FiniteElement::VALUE;
      const auto W = Reshape(ir->GetWeights().Read(), Q1D,Q1D);
-      const auto J = Reshape(geom->J.Read(), Q1D,Q1D,2,2,NE);
+      const auto J = Reshape(geom->detJ.Read(), Q1D,Q1D,NE);
      const auto C = const_c ? Reshape(coeff.Read(), 1,1,1) :
                     Reshape(coeff.Read(), Q1D,Q1D,NE);
      auto v = Reshape(pa_data.Write(), Q1D,Q1D, NE);
@@ -84,11 +83,7 @@ void MassIntegrator::AssemblePA(const FiniteElementSpace &fes)
         {
            MFEM_FOREACH_THREAD(qy,y,Q1D)
            {
-               const double J11 = J(qx,qy,0,0,e);
-               const double J12 = J(qx,qy,1,0,e);
-               const double J21 = J(qx,qy,0,1,e);
-               const double J22 = J(qx,qy,1,1,e);
-               const double detJ = (J11*J22)-(J21*J12);
+               const double detJ = J(qx,qy,e);
               const double coeff = const_c ? C(0,0,0) : C(qx,qy,e);
               v(qx,qy,e) =  W(qx,qy) * coeff * (by_val ? detJ : 1.0/detJ);
            }
@@ -102,7 +97,7 @@ void MassIntegrator::AssemblePA(const FiniteElementSpace &fes)
      const bool const_c = coeff.Size() == 1;
      const bool by_val = map_type == FiniteElement::VALUE;
      const auto W = Reshape(ir->GetWeights().Read(), Q1D,Q1D,Q1D);
-      const auto J = Reshape(geom->J.Read(), Q1D,Q1D,Q1D,3,3,NE);
+      const auto J = Reshape(geom->detJ.Read(), Q1D,Q1D,Q1D,NE);
      const auto C = const_c ? Reshape(coeff.Read(), 1,1,1,1) :
                     Reshape(coeff.Read(), Q1D,Q1D,Q1D,NE);
      auto v = Reshape(pa_data.Write(), Q1D,Q1D,Q1D,NE);
@@ -114,18 +109,7 @@ void MassIntegrator::AssemblePA(const FiniteElementSpace &fes)
            {
               MFEM_FOREACH_THREAD(qz,z,Q1D)
               {
-                  const double J11 = J(qx,qy,qz,0,0,e);
-                  const double J21 = J(qx,qy,qz,1,0,e);
-                  const double J31 = J(qx,qy,qz,2,0,e);
-                  const double J12 = J(qx,qy,qz,0,1,e);
-                  const double J22 = J(qx,qy,qz,1,1,e);
-                  const double J32 = J(qx,qy,qz,2,1,e);
-                  const double J13 = J(qx,qy,qz,0,2,e);
-                  const double J23 = J(qx,qy,qz,1,2,e);
-                  const double J33 = J(qx,qy,qz,2,2,e);
-                  const double detJ = J11 * (J22 * J33 - J32 * J23) -
-                  /* */               J21 * (J12 * J33 - J32 * J13) +
-                  /* */               J31 * (J12 * J23 - J22 * J13);
+                  const double detJ = J(qx,qy,qz,e);
                  const double coeff = const_c ? C(0,0,0,0) : C(qx,qy,qz,e);
                  v(qx,qy,qz,e) = W(qx,qy,qz) * coeff * (by_val ? detJ : 1.0/detJ);
               }
@@ -75,7 +75,7 @@ void TransposeIntegrator::AssembleEAInteriorFaces(const FiniteElementSpace& fes,
      Vector ea_data_ext_tmp(ea_data_ext.Size());
      bfi->AssembleEAInteriorFaces(fes, ea_data_int_tmp, ea_data_ext_tmp, false);
      const int faceDofs = fes.GetTraceElement(0,
-                                               fes.GetMesh()->GetFaceBaseGeometry(0))->GetDof();
+                                               fes.GetMesh()->GetFaceGeometry(0))->GetDof();
      auto A_int = Reshape(ea_data_int_tmp.Read(), faceDofs, faceDofs, 2, nf);
      auto A_ext = Reshape(ea_data_ext_tmp.Read(), faceDofs, faceDofs, 2, nf);
      auto AT_int = Reshape(ea_data_int.ReadWrite(), faceDofs, faceDofs, 2, nf);
@@ -102,7 +102,7 @@ void TransposeIntegrator::AssembleEAInteriorFaces(const FiniteElementSpace& fes,
   {
      bfi->AssembleEAInteriorFaces(fes, ea_data_int, ea_data_ext, false);
      const int faceDofs = fes.GetTraceElement(0,
-                                               fes.GetMesh()->GetFaceBaseGeometry(0))->GetDof();
+                                               fes.GetMesh()->GetFaceGeometry(0))->GetDof();
      auto A_int = Reshape(ea_data_int.ReadWrite(), faceDofs, faceDofs, 2, nf);
      auto A_ext = Reshape(ea_data_ext.ReadWrite(), faceDofs, faceDofs, 2, nf);
      MFEM_FORALL(f, nf,
@@ -146,7 +146,7 @@ void TransposeIntegrator::AssembleEABoundaryFaces(const FiniteElementSpace& fes,
      Vector ea_data_bdr_tmp(ea_data_bdr.Size());
      bfi->AssembleEABoundaryFaces(fes, ea_data_bdr_tmp, false);
      const int faceDofs = fes.GetTraceElement(0,
-                                               fes.GetMesh()->GetFaceBaseGeometry(0))->GetDof();
+                                               fes.GetMesh()->GetFaceGeometry(0))->GetDof();
      auto A_bdr = Reshape(ea_data_bdr_tmp.Read(), faceDofs, faceDofs, nf);
      auto AT_bdr = Reshape(ea_data_bdr.ReadWrite(), faceDofs, faceDofs, nf);
      MFEM_FORALL(f, nf,
@@ -165,7 +165,7 @@ void TransposeIntegrator::AssembleEABoundaryFaces(const FiniteElementSpace& fes,
   {
      bfi->AssembleEABoundaryFaces(fes, ea_data_bdr, false);
      const int faceDofs = fes.GetTraceElement(0,
-                                               fes.GetMesh()->GetFaceBaseGeometry(0))->GetDof();
+                                               fes.GetMesh()->GetFaceGeometry(0))->GetDof();
      auto A_bdr = Reshape(ea_data_bdr.ReadWrite(), faceDofs, faceDofs, nf);
      MFEM_FORALL(f, nf,
      {
@@ -69,9 +69,11 @@ void Operator::Mult(const mfem::Vector &x, mfem::Vector &y) const
 #endif
 }

-void Operator::AddMult(const mfem::Vector &x, mfem::Vector &y) const
+void Operator::AddMult(const mfem::Vector &x, mfem::Vector &y,
+                       const double a) const
 {
 #ifdef MFEM_USE_CEED
+   MFEM_VERIFY(a == 1.0, "General coefficient case is not yet supported!");
   const CeedScalar *x_ptr;
   CeedScalar *y_ptr;
   CeedMemType mem;
@@ -38,7 +38,8 @@ public:
   Operator(CeedOperator op);
 #endif
   void Mult(const mfem::Vector &x, mfem::Vector &y) const override;
-   void AddMult(const mfem::Vector &x, mfem::Vector &y) const;
+   void AddMult(const mfem::Vector &x, mfem::Vector &y,
+                const double a = 1.0) const override;
   void GetDiagonal(mfem::Vector &diag) const;
   using mfem::Operator::SetupRAP;
   virtual ~Operator()
@@ -47,6 +47,7 @@ void InitRestrictionWithIndices(const FiniteElementSpace &fes,

 /** @brief Initialize a strided CeedElemRestriction

+    @param[in] fes Input finite element space.
    @param[in] nelem is the number of elements.
    @param[in] nqpts is the total number of quadrature points.
    @param[in] qdatasize is the number of data per quadrature point.
@@ -140,7 +140,11 @@ int CeedOperatorGetActiveField(CeedOperator oper, CeedOperatorField *field)
   CeedOperator *subops;
   if (isComposite)
   {
+#if CEED_VERSION_GE(0, 10, 2)
+      ierr = CeedCompositeOperatorGetSubList(oper, &subops); CeedChk(ierr);
+#else
      ierr = CeedOperatorGetSubList(oper, &subops); CeedChk(ierr);
+#endif
      ierr = CeedOperatorGetQFunction(subops[0], &qf); CeedChk(ierr);
   }
   else
@@ -275,9 +275,14 @@ CeedOperator CoarsenCeedCompositeOperator(
                                      &op_coarse); PCeedChk(ierr);

   int nsub;
-   ierr = CeedOperatorGetNumSub(op, &nsub); PCeedChk(ierr);
   CeedOperator *subops;
+#if CEED_VERSION_GE(0, 10, 2)
+   ierr = CeedCompositeOperatorGetNumSub(op, &nsub); PCeedChk(ierr);
+   ierr = CeedCompositeOperatorGetSubList(op, &subops); PCeedChk(ierr);
+#else
+   ierr = CeedOperatorGetNumSub(op, &nsub); PCeedChk(ierr);
   ierr = CeedOperatorGetSubList(op, &subops); PCeedChk(ierr);
+#endif
   for (int isub=0; isub<nsub; ++isub)
   {
      CeedOperator subop = subops[isub];
@@ -310,8 +310,13 @@ int CeedOperatorFullAssemble(CeedOperator op, SparseMatrix **mat)
   {
      CeedInt numsub;
      CeedOperator *subops;
+#if CEED_VERSION_GE(0, 10, 2)
+      CeedCompositeOperatorGetNumSub(op, &numsub);
+      ierr = CeedCompositeOperatorGetSubList(op, &subops); CeedChk(ierr);
+#else
      CeedOperatorGetNumSub(op, &numsub);
      ierr = CeedOperatorGetSubList(op, &subops); CeedChk(ierr);
+#endif
      for (int i = 0; i < numsub; ++i)
      {
         ierr = CeedSingleOperatorFullAssemble(subops[i], out); CeedChk(ierr);
@@ -66,6 +66,7 @@ int CeedBasisATPMGCoarsen(CeedBasis basisin, CeedBasis* basisout,
    @param[in] coarse_er         CeedElemRestriction for coarse operator
                                 (see CeedATPMGElemRestriction)
    @param[out] coarse_basis_out CeedBasis for coarser operator
+    @param[out] basis_ctof_out   CeedBasis describing interpolation from coarse to fine
    @param[out] out              coarsened CeedOperator
 */
 int CeedATPMGOperator(CeedOperator oper, int order_reduction,
@@ -1319,7 +1319,6 @@ public:
   virtual void Eval(DenseSymmetricMatrix &K, ElementTransformation &T,
                     const IntegrationPoint &ip) = 0;

-   using MatrixCoefficient::Eval;
   /** @brief Evaluate the matrix coefficient in the element described by @a T
       at the point @a ip, storing the result as a dense matrix @a K. */
   /** This function allows the use of SymmetricMatrixCoefficient in situations
@@ -1348,7 +1347,6 @@ public:
   ///Construct using matrix @a m for the constant.
   SymmetricMatrixConstantCoefficient(const DenseSymmetricMatrix &m)
      : SymmetricMatrixCoefficient(m.Height()), mat(m) { }
-   using MatrixCoefficient::Eval;
   using SymmetricMatrixCoefficient::Eval;
   /// Evaluate the matrix coefficient at @a ip.
   virtual void Eval(DenseSymmetricMatrix &M, ElementTransformation &T,
@@ -1400,7 +1398,6 @@ public:
   /// Set the time for internally stored coefficients
   void SetTime(double t);

-   using MatrixCoefficient::Eval;
   using SymmetricMatrixCoefficient::Eval;
   /// Evaluate the matrix coefficient at @a ip.
   virtual void Eval(DenseSymmetricMatrix &K, ElementTransformation &T,
@@ -110,8 +110,8 @@ DataCollection::DataCollection(const std::string& collection_name, Mesh *mesh_)
   precision = precision_default;
   pad_digits_cycle = pad_digits_rank = pad_digits_default;
   format = SERIAL_FORMAT; // use serial mesh format
-   compression = false;
-   error = NO_ERROR;
+   compression = 0;
+   error = No_Error;
 }

 void DataCollection::SetMesh(Mesh *new_mesh)
@@ -494,7 +494,7 @@ void VisItDataCollection::Load(int cycle_)
 {
   DeleteAll();
   time_step = 0.0;
-   error = NO_ERROR;
+   error = No_Error;
   cycle = cycle_;
   std::string root_name = prefix_path + name + "_" +
                           to_padded_string(cycle, pad_digits_cycle) +
@@ -724,7 +724,7 @@ void VisItDataCollection::ParseVisItRootString(const std::string& json)

   // Set the DataCollection::name using the mesh path
   std::string path = mesh.get("path").get<std::string>();
-   size_t right_sep = path.find('_');
+   size_t right_sep = path.rfind('_');
   if (right_sep == std::string::npos)
   {
      error = READ_ERROR;
@@ -767,10 +767,13 @@ ParaViewDataCollection::ParaViewDataCollection(const std::string&
     high_order_output(false),
     restart_mode(false)
 {
+   cycle = 0; // always include a valid cycle index in file names
+
+   compression_level = -1;  // default zlib compression level, equivalent to 6
 #ifdef MFEM_USE_ZLIB
-   compression = -1; // default zlib compression level, equivalent to 6
+   compression = true; // if we have zlib, enable compression
 #else
-   compression = 0;
+   compression = false; // otherwise, disable compression
 #endif
 }

@@ -919,7 +922,7 @@ void ParaViewDataCollection::Save()
   {
      const std::string &field_name = qfield.first;
      std::ofstream os(vtu_prefix + GenerateVTUFileName(field_name, myid));
-      qfield.second->SaveVTU(os, pv_data_format, compression);
+      qfield.second->SaveVTU(os, pv_data_format, GetCompressionLevel());
   }

   // MPI rank 0 also creates a "PVTU" file that points to all of the separately
@@ -1033,13 +1036,13 @@ void ParaViewDataCollection::WritePVTUFooter(std::ostream &os,
 void ParaViewDataCollection::SaveDataVTU(std::ostream &os, int ref)
 {
   os << "<VTKFile type=\"UnstructuredGrid\"";
-   if (compression != 0)
+   if (GetCompressionLevel() != 0)
   {
      os << " compressor=\"vtkZLibDataCompressor\"";
   }
   os << " version=\"0.1\" byte_order=\"" << VTKByteOrder() << "\">\n";
   os << "<UnstructuredGrid>\n";
-   mesh->PrintVTU(os,ref,pv_data_format,high_order_output,compression);
+   mesh->PrintVTU(os,ref,pv_data_format,high_order_output,GetCompressionLevel());

   // dump out the grid functions as point data
   os << "<PointData >\n";
@@ -1103,7 +1106,7 @@ void ParaViewDataCollection::SaveGFieldVTU(std::ostream &os, int ref_,

   if (IsBinaryFormat())
   {
-      WriteVTKEncodedCompressed(os,buf.data(),buf.size(),compression);
+      WriteVTKEncodedCompressed(os,buf.data(),buf.size(),GetCompressionLevel());
      os << '\n';
   }
   os << "</DataArray>" << std::endl;
@@ -1128,18 +1131,13 @@ void ParaViewDataCollection::SetCompressionLevel(int compression_level_)
 {
   MFEM_ASSERT(compression_level_ >= -1 && compression_level_ <= 9,
               "Compression level must be between -1 and 9 (inclusive).");
-   compression = compression_level_;
+   compression_level = compression_level_;
+   compression = compression_level_ != 0;
 }

 void ParaViewDataCollection::SetCompression(bool compression_)
 {
-   // If we are enabling compression, and it was disabled previously, use the
-   // default compression level. Otherwise, leave the compression level
-   // unchanged.
-   if (compression_ && compression == 0)
-   {
-      SetCompressionLevel(-1);
-   }
+   compression = compression_;
 }

 void ParaViewDataCollection::UseRestartMode(bool restart_mode_)
@@ -1171,4 +1169,9 @@ const char *ParaViewDataCollection::GetDataTypeString() const
   }
 }

+int ParaViewDataCollection::GetCompressionLevel() const
+{
+   return compression ? compression_level : 0;
+}
+
 }  // end namespace MFEM
@@ -338,11 +338,12 @@ public:
   /// Set the precision (number of digits) used for the text output of doubles
   void SetPrecision(int prec) { precision = prec; }
   /// Set the number of digits used for both the cycle and the MPI rank
-   void SetPadDigits(int digits) { pad_digits_cycle=pad_digits_rank = digits; }
+   virtual void SetPadDigits(int digits)
+   { pad_digits_cycle=pad_digits_rank = digits; }
   /// Set the number of digits used for the cycle
-   void SetPadDigitsCycle(int digits) { pad_digits_cycle = digits; }
+   virtual void SetPadDigitsCycle(int digits) { pad_digits_cycle = digits; }
   /// Set the number of digits used for the MPI rank in filenames
-   void SetPadDigitsRank(int digits) { pad_digits_rank = digits; }
+   virtual void SetPadDigitsRank(int digits) { pad_digits_rank = digits; }
   /// Set the desired output mesh and data format.
   /** See the enumeration #Format for valid options. Derived classes can define
       their own format enumerations and override this method to perform input
@@ -377,12 +378,24 @@ public:
   virtual ~DataCollection();

   /// Errors returned by Error()
-   enum { NO_ERROR = 0, READ_ERROR = 1, WRITE_ERROR = 2 };
+   enum
+   {
+      // Workaround for use with headers that define NO_ERROR as a macro,
+      // e.g. winerror.h (which is included by Windows.h):
+#ifndef NO_ERROR
+      NO_ERROR = 0,
+#endif
+      // Use the following identifier if NO_ERROR is defined as a macro,
+      // e.g. winerror.h (which is included by Windows.h):
+      No_Error    = 0,
+      READ_ERROR  = 1,
+      WRITE_ERROR = 2
+   };

   /// Get the current error state
   int Error() const { return error; }
   /// Reset the error state
-   void ResetError(int err_state = NO_ERROR) { error = err_state; }
+   void ResetError(int err_state = No_Error) { error = err_state; }

 #ifdef MFEM_USE_MPI
   friend class ParMesh;
@@ -441,21 +454,29 @@ public:
 #endif

   /// Set/change the mesh associated with the collection
-   virtual void SetMesh(Mesh *new_mesh);
+   virtual void SetMesh(Mesh *new_mesh) override;

 #ifdef MFEM_USE_MPI
   /// Set/change the mesh associated with the collection.
-   virtual void SetMesh(MPI_Comm comm, Mesh *new_mesh);
+   virtual void SetMesh(MPI_Comm comm, Mesh *new_mesh) override;
 #endif

   /// Add a grid function to the collection and update the root file
-   virtual void RegisterField(const std::string& field_name, GridFunction *gf);
+   virtual void RegisterField(const std::string& field_name,
+                              GridFunction *gf) override;

   /// Add a quadrature function to the collection and update the root file.
   /** Visualization of quadrature function is not supported in VisIt(3.12).
       A patch has been sent to VisIt developers in June 2020. */
   virtual void RegisterQField(const std::string& q_field_name,
-                               QuadratureFunction *qf);
+                               QuadratureFunction *qf) override;
+
+   /// Set the number of digits used for both the cycle and the MPI rank
+   /// @note VisIt seems to require 6 pad digits for the MPI rank. Therefore,
+   /// this function uses this default value. This behavior can be overridden
+   /// by calling SetPadDigitsCycle() and SetPadDigitsRank() instead.
+   virtual void SetPadDigits(int digits) override
+   { pad_digits_cycle=digits; pad_digits_rank=6; }

   /// Set VisIt parameter: default levels of detail for the MultiresControl
   void SetLevelsOfDetail(int levels_of_detail);
@@ -468,13 +489,13 @@ public:
   void DeleteAll();

   /// Save the collection and a VisIt root file
-   virtual void Save();
+   virtual void Save() override;

   /// Save a VisIt root file for the collection
   void SaveRootFile();

   /// Load the collection based on its VisIt data (described in its root file)
-   virtual void Load(int cycle_ = 0);
+   virtual void Load(int cycle_ = 0) override;

   /// We will delete the mesh and fields if we own them
   virtual ~VisItDataCollection() {}
@@ -486,6 +507,7 @@ class ParaViewDataCollection : public DataCollection
 {
 private:
   int levels_of_detail;
+   int compression_level;
   std::fstream pvd_stream;
   VTKFormat pv_data_format;
   bool high_order_output;
@@ -498,6 +520,9 @@ protected:
   void SaveGFieldVTU(std::ostream& out, int ref_, const FieldMapIterator& it);
   const char *GetDataFormatString() const;
   const char *GetDataTypeString() const;
+   /// @brief If compression is enabled, return the compression level, otherwise
+   /// return 0.
+   int GetCompressionLevel() const;

   std::string GenerateCollectionPath();
   std::string GenerateVTUFileName(const std::string &prefix, int rank);
@@ -516,7 +541,7 @@ public:
                          mfem::Mesh *mesh_ = NULL);

   /// Set refinement levels - every element is uniformly split based on
-   /// levels_of_detail_
+   /// levels_of_detail_. The initial value is 1.
   void SetLevelsOfDetail(int levels_of_detail_);

   /// Save the collection - the directory name is constructed based on the
@@ -527,18 +552,27 @@ public:
   /// VTKFormat::ASCII, VTKFormat::BINARY, and VTKFormat::BINARY32.
   /// The ASCII and BINARY options output double precision data, whereas the
   /// BINARY32 option outputs single precision data.
+   ///
+   /// The initial format is VTKFormat::BINARY.
   void SetDataFormat(VTKFormat fmt);

-   /// Set the zlib compression level. 0 indicates no compression, -1 indicates
-   /// the default compression level. Otherwise, specify a number between 1 and
-   /// 9, 1 being the fastest, and 9 being the best compression. Compression
-   /// only takes effect if the output format is BINARY or BINARY32. MFEM must
-   /// be compiled with MFEM_USE_ZLIB = YES.
+   /// @brief Set the zlib compression level.
+   ///
+   /// 0 indicates no compression, -1 indicates the default compression level.
+   /// Otherwise, specify a number between 1 and 9, 1 being the fastest, and 9
+   /// being the best compression. Compression only takes effect if the output
+   /// format is BINARY or BINARY32. MFEM must be compiled with MFEM_USE_ZLIB =
+   /// YES.
+   ///
+   /// The initial compression level is 0 if MFEM is compiled with MFEM_USE_ZLIB
+   /// turned off, and -1 otherwise.
+   ///
+   /// Any nonzero compression level will enable compression.
   void SetCompressionLevel(int compression_level_);

   /// Enable or disable zlib compression. If the input is true, use the default
   /// zlib compression level (unless the compression level has previously been
-   /// set by calling SetCompressionLevel).
+   /// set by calling SetCompressionLevel()).
   void SetCompression(bool compression_) override;

   /// Returns true if the output format is BINARY or BINARY32, false if ASCII.
@@ -551,6 +585,8 @@ public:
   /// Enable or disable restart mode. If restart is enabled, new writes will
   /// preserve timestep metadata for any solutions prior to the currently
   /// defined time.
+   ///
+   /// Initially, restart mode is disabled.
   void UseRestartMode(bool restart_mode_);

   /// Load the collection - not implemented in the ParaView writer
@@ -339,8 +339,11 @@ ND_TriDofTransformation::TransformDual(double *v) const
 void
 ND_TriDofTransformation::InvTransformDual(double *v) const
 {
-   int nedofs = order; // number of DoFs per edge
-   int nfdofs = order*(order-1); // number of DoFs per face
+   // Return immediately when no face DoFs are present
+   if (nfdofs < 2) { return; }
+
+   MFEM_VERIFY(Fo.Size() >= 1,
+               "Face orientations are unset in ND_TriDofTransformation");

   double data[2];
   Vector v2(data, 2);
@@ -432,8 +435,11 @@ ND_TetDofTransformation::TransformDual(double *v) const
 void
 ND_TetDofTransformation::InvTransformDual(double *v) const
 {
-   int nedofs = order; // number of DoFs per edge
-   int nfdofs = order*(order-1); // number of DoFs per face
+   // Return immediately when no face DoFs are present
+   if (nfdofs < 2) { return; }
+
+   MFEM_VERIFY(Fo.Size() >= 4,
+               "Face orientations are unset in ND_TetDofTransformation");

   double data[2];
   Vector v2(data, 2);
@@ -36,19 +36,19 @@ FiniteElement::FiniteElement(int D, Geometry::Type G,
 #endif
 }

-void FiniteElement::CalcVShape (
+void FiniteElement::CalcVShape(
   const IntegrationPoint &ip, DenseMatrix &shape) const
 {
   MFEM_ABORT("method is not implemented for this class");
 }

-void FiniteElement::CalcVShape (
+void FiniteElement::CalcVShape(
   ElementTransformation &Trans, DenseMatrix &shape) const
 {
   MFEM_ABORT("method is not implemented for this class");
 }

-void FiniteElement::CalcDivShape (
+void FiniteElement::CalcDivShape(
   const IntegrationPoint &ip, Vector &divshape) const
 {
   MFEM_ABORT("method is not implemented for this class");
@@ -97,14 +97,14 @@ void FiniteElement::GetFaceDofs(int face, int **dofs, int *ndofs) const
   MFEM_ABORT("method is not overloaded");
 }

-void FiniteElement::CalcHessian (const IntegrationPoint &ip,
-                                 DenseMatrix &h) const
+void FiniteElement::CalcHessian(const IntegrationPoint &ip,
+                                DenseMatrix &h) const
 {
   MFEM_ABORT("method is not overloaded");
 }

-void FiniteElement::GetLocalInterpolation (ElementTransformation &Trans,
-                                           DenseMatrix &I) const
+void FiniteElement::GetLocalInterpolation(ElementTransformation &Trans,
+                                          DenseMatrix &I) const
 {
   MFEM_ABORT("method is not overloaded");
 }
@@ -122,13 +122,13 @@ void FiniteElement::GetTransferMatrix(const FiniteElement &fe,
   MFEM_ABORT("method is not overloaded");
 }

-void FiniteElement::Project (
+void FiniteElement::Project(
   Coefficient &coeff, ElementTransformation &Trans, Vector &dofs) const
 {
   MFEM_ABORT("method is not overloaded");
 }

-void FiniteElement::Project (
+void FiniteElement::Project(
   VectorCoefficient &vc, ElementTransformation &Trans, Vector &dofs) const
 {
   MFEM_ABORT("method is not overloaded");
@@ -137,7 +137,7 @@ void FiniteElement::Project (
 void FiniteElement::ProjectFromNodes(Vector &vc, ElementTransformation &Trans,
                                     Vector &dofs) const
 {
-   mfem_error ("FiniteElement::ProjectFromNodes() (vector) is not overloaded!");
+   mfem_error("FiniteElement::ProjectFromNodes() (vector) is not overloaded!");
 }

 void FiniteElement::ProjectMatrixCoefficient(
@@ -239,7 +239,6 @@ void FiniteElement::CalcPhysLaplacian(ElementTransformation &Trans,
   }
 }

-
 // Assume a linear mapping
 void FiniteElement::CalcPhysLinLaplacian(ElementTransformation &Trans,
                                         Vector &Laplacian) const
@@ -250,7 +249,7 @@ void FiniteElement::CalcPhysLinLaplacian(ElementTransformation &Trans,
   DenseMatrix Gij(dim,dim);
   Vector scale(size);

-   CalcHessian (Trans.GetIntPoint(), hess);
+   CalcHessian(Trans.GetIntPoint(), hess);
   MultAAt(Trans.InverseJacobian(), Gij);

   if (dim == 3)
@@ -283,7 +282,6 @@ void FiniteElement::CalcPhysLinLaplacian(ElementTransformation &Trans,
         Laplacian[nd] += hess(nd,ii)*scale[ii];
      }
   }
-
 }

 void  FiniteElement::CalcPhysHessian(ElementTransformation &Trans,
@@ -363,11 +361,128 @@ void  FiniteElement::CalcPhysHessian(ElementTransformation &Trans,
   Mult( hess, lhm, Hessian);
 }

-const DofToQuad &FiniteElement::GetDofToQuad(const IntegrationRule &,
-                                             DofToQuad::Mode) const
+const DofToQuad &FiniteElement::GetDofToQuad(const IntegrationRule &ir,
+                                             DofToQuad::Mode mode) const
 {
-   MFEM_ABORT("method is not implemented for this element");
-   return *dof2quad_array[0]; // suppress a warning
+   MFEM_VERIFY(mode == DofToQuad::FULL, "invalid mode requested");
+
+   for (int i = 0; i < dof2quad_array.Size(); i++)
+   {
+      const DofToQuad &d2q = *dof2quad_array[i];
+      if (d2q.IntRule == &ir && d2q.mode == mode) { return d2q; }
+   }
+
+#ifdef MFEM_THREAD_SAFE
+   DenseMatrix vshape(dof, dim);
+#endif
+
+   DofToQuad *d2q = new DofToQuad;
+   const int nqpt = ir.GetNPoints();
+   d2q->FE = this;
+   d2q->IntRule = &ir;
+   d2q->mode = mode;
+   d2q->ndof = dof;
+   d2q->nqpt = nqpt;
+   if (range_type == SCALAR)
+   {
+      d2q->B.SetSize(nqpt*dof);
+      d2q->Bt.SetSize(dof*nqpt);
+
+      Vector shape;
+      vshape.GetColumnReference(0, shape);
+      for (int i = 0; i < nqpt; i++)
+      {
+         const IntegrationPoint &ip = ir.IntPoint(i);
+         CalcShape(ip, shape);
+         for (int j = 0; j < dof; j++)
+         {
+            d2q->B[i+nqpt*j] = d2q->Bt[j+dof*i] = shape(j);
+         }
+      }
+   }
+   else
+   {
+      d2q->B.SetSize(nqpt*dim*dof);
+      d2q->Bt.SetSize(dof*nqpt*dim);
+
+      for (int i = 0; i < nqpt; i++)
+      {
+         const IntegrationPoint &ip = ir.IntPoint(i);
+         CalcVShape(ip, vshape);
+         for (int d = 0; d < dim; d++)
+         {
+            for (int j = 0; j < dof; j++)
+            {
+               d2q->B[i+nqpt*(d+dim*j)] = d2q->Bt[j+dof*(i+nqpt*d)] = vshape(j, d);
+            }
+         }
+      }
+   }
+   switch (deriv_type)
+   {
+      case GRAD:
+      {
+         d2q->G.SetSize(nqpt*dim*dof);
+         d2q->Gt.SetSize(dof*nqpt*dim);
+
+         for (int i = 0; i < nqpt; i++)
+         {
+            const IntegrationPoint &ip = ir.IntPoint(i);
+            CalcDShape(ip, vshape);
+            for (int d = 0; d < dim; d++)
+            {
+               for (int j = 0; j < dof; j++)
+               {
+                  d2q->G[i+nqpt*(d+dim*j)] = d2q->Gt[j+dof*(i+nqpt*d)] = vshape(j, d);
+               }
+            }
+         }
+         break;
+      }
+      case DIV:
+      {
+         d2q->G.SetSize(nqpt*dof);
+         d2q->Gt.SetSize(dof*nqpt);
+
+         Vector divshape;
+         vshape.GetColumnReference(0, divshape);
+         for (int i = 0; i < nqpt; i++)
+         {
+            const IntegrationPoint &ip = ir.IntPoint(i);
+            CalcDivShape(ip, divshape);
+            for (int j = 0; j < dof; j++)
+            {
+               d2q->G[i+nqpt*j] = d2q->Gt[j+dof*i] = divshape(j);
+            }
+         }
+         break;
+      }
+      case CURL:
+      {
+         d2q->G.SetSize(nqpt*cdim*dof);
+         d2q->Gt.SetSize(dof*nqpt*cdim);
+
+         DenseMatrix curlshape(vshape.GetData(), dof, cdim);  // cdim <= dim
+         for (int i = 0; i < nqpt; i++)
+         {
+            const IntegrationPoint &ip = ir.IntPoint(i);
+            CalcCurlShape(ip, curlshape);
+            for (int d = 0; d < cdim; d++)
+            {
+               for (int j = 0; j < dof; j++)
+               {
+                  d2q->G[i+nqpt*(d+dim*j)] = d2q->Gt[j+dof*(i+nqpt*d)] = curlshape(j, d);
+               }
+            }
+         }
+         break;
+      }
+      case NONE:
+      default:
+         MFEM_ABORT("invalid finite element derivative type");
+   }
+   dof2quad_array.Append(d2q);
+   return *d2q;
 }

 FiniteElement::~FiniteElement()
@@ -379,16 +494,19 @@ FiniteElement::~FiniteElement()
 }


-void ScalarFiniteElement::NodalLocalInterpolation (
+void ScalarFiniteElement::NodalLocalInterpolation(
   ElementTransformation &Trans, DenseMatrix &I,
   const ScalarFiniteElement &fine_fe) const
 {
   double v[Geometry::MaxDim];
-   Vector vv (v, dim);
+   Vector vv(v, dim);
   IntegrationPoint f_ip;

 #ifdef MFEM_THREAD_SAFE
-   Vector c_shape(dof);
+   Vector shape(dof);
+#else
+   Vector shape;
+   vshape.GetColumnReference(0, shape);
 #endif

   MFEM_ASSERT(map_type == fine_fe.GetMapType(), "");
@@ -398,10 +516,10 @@ void ScalarFiniteElement::NodalLocalInterpolation (
   {
      Trans.Transform(fine_fe.Nodes.IntPoint(i), vv);
      f_ip.Set(v, dim);
-      CalcShape(f_ip, c_shape);
+      CalcShape(f_ip, shape);
      for (int j = 0; j < dof; j++)
      {
-         if (fabs(I(i,j) = c_shape(j)) < 1.0e-12)
+         if (fabs(I(i,j) = shape(j)) < 1.0e-12)
         {
            I(i,j) = 0.0;
         }
@@ -422,7 +540,7 @@ void ScalarFiniteElement::ScalarLocalInterpolation(
   // General "interpolation", defined by L2 projection

   double v[Geometry::MaxDim];
-   Vector vv (v, dim);
+   Vector vv(v, dim);
   IntegrationPoint f_ip;

   const int fs = fine_fe.GetDof(), cs = this->GetDof();
@@ -456,14 +574,13 @@ void ScalarFiniteElement::ScalarLocalInterpolation(
   }
 }

-void ScalarFiniteElement::ScalarLocalRestriction(
+void ScalarFiniteElement::ScalarLocalL2Restriction(
   ElementTransformation &Trans, DenseMatrix &R,
   const ScalarFiniteElement &coarse_fe) const
 {
   // General "restriction", defined by L2 projection
   double v[Geometry::MaxDim];
-   Vector vv (v, dim);
-   IntegrationPoint f_ip;
+   Vector vv(v, dim);

   const int cs = coarse_fe.GetDof(), fs = this->GetDof();
   R.SetSize(cs, fs);
@@ -472,16 +589,27 @@ void ScalarFiniteElement::ScalarLocalRestriction(
   const int ir_order = GetOrder() + coarse_fe.GetOrder();
   const IntegrationRule &ir = IntRules.Get(coarse_fe.GetGeomType(), ir_order);

+   // integrate coarse_mass in the coarse space
   for (int i = 0; i < ir.GetNPoints(); i++)
   {
-      const IntegrationPoint &ip = ir.IntPoint(i);
-      this->CalcShape(ip, fine_shape);
-      Trans.Transform(ip, vv);
-      f_ip.Set(v, dim);
-      coarse_fe.CalcShape(f_ip, coarse_shape);
+      const IntegrationPoint &c_ip = ir.IntPoint(i);
+      coarse_fe.CalcShape(c_ip, coarse_shape);
+      AddMult_a_VVt(c_ip.weight, coarse_shape, coarse_mass);
+   }

-      AddMult_a_VVt(ip.weight, coarse_shape, coarse_mass);
-      AddMult_a_VWt(ip.weight, coarse_shape, fine_shape, coarse_fine_mass);
+   // integrate coarse_fine_mass in the fine space
+   Trans.SetIntPoint(&Geometries.GetCenter(geom_type));
+   for (int i = 0; i < ir.GetNPoints(); i++)
+   {
+      const IntegrationPoint &f_ip = ir.IntPoint(i);
+      this->CalcShape(f_ip, fine_shape);
+      Trans.Transform(f_ip, vv);
+
+      IntegrationPoint c_ip;
+      c_ip.Set(v, dim);
+      coarse_fe.CalcShape(c_ip, coarse_shape);
+      AddMult_a_VWt(f_ip.weight*Trans.Weight(), coarse_shape, fine_shape,
+                    coarse_fine_mass);
   }

   DenseMatrixInverse coarse_mass_inv(coarse_mass);
@@ -494,95 +622,6 @@ void ScalarFiniteElement::ScalarLocalRestriction(
      R *= 1.0 / Trans.Weight();
   }
 }
-const DofToQuad &ScalarFiniteElement::GetDofToQuad(const IntegrationRule &ir,
-                                                   DofToQuad::Mode mode) const
-{
-   MFEM_VERIFY(mode == DofToQuad::FULL, "invalid mode requested");
-
-   for (int i = 0; i < dof2quad_array.Size(); i++)
-   {
-      const DofToQuad &d2q = *dof2quad_array[i];
-      if (d2q.IntRule == &ir && d2q.mode == mode) { return d2q; }
-   }
-
-   DofToQuad *d2q = new DofToQuad;
-   const int nqpt = ir.GetNPoints();
-   d2q->FE = this;
-   d2q->IntRule = &ir;
-   d2q->mode = mode;
-   d2q->ndof = dof;
-   d2q->nqpt = nqpt;
-   d2q->B.SetSize(nqpt*dof);
-   d2q->Bt.SetSize(dof*nqpt);
-   d2q->G.SetSize(nqpt*dim*dof);
-   d2q->Gt.SetSize(dof*nqpt*dim);
-#ifdef MFEM_THREAD_SAFE
-   Vector c_shape(dof);
-   DenseMatrix vshape(dof, dim);
-#endif
-   for (int i = 0; i < nqpt; i++)
-   {
-      const IntegrationPoint &ip = ir.IntPoint(i);
-      CalcShape(ip, c_shape);
-      for (int j = 0; j < dof; j++)
-      {
-         d2q->B[i+nqpt*j] = d2q->Bt[j+dof*i] = c_shape(j);
-      }
-      CalcDShape(ip, vshape);
-      for (int d = 0; d < dim; d++)
-      {
-         for (int j = 0; j < dof; j++)
-         {
-            d2q->G[i+nqpt*(d+dim*j)] = d2q->Gt[j+dof*(i+nqpt*d)] = vshape(j,d);
-         }
-      }
-   }
-   dof2quad_array.Append(d2q);
-   return *d2q;
-}
-
-// protected method
-const DofToQuad &ScalarFiniteElement::GetTensorDofToQuad(
-   const TensorBasisElement &tb,
-   const IntegrationRule &ir, DofToQuad::Mode mode) const
-{
-   MFEM_VERIFY(mode == DofToQuad::TENSOR, "invalid mode requested");
-
-   for (int i = 0; i < dof2quad_array.Size(); i++)
-   {
-      const DofToQuad &d2q = *dof2quad_array[i];
-      if (d2q.IntRule == &ir && d2q.mode == mode) { return d2q; }
-   }
-
-   DofToQuad *d2q = new DofToQuad;
-   const Poly_1D::Basis &basis_1d = tb.GetBasis1D();
-   const int ndof = order + 1;
-   const int nqpt = (int)floor(pow(ir.GetNPoints(), 1.0/dim) + 0.5);
-   d2q->FE = this;
-   d2q->IntRule = &ir;
-   d2q->mode = mode;
-   d2q->ndof = ndof;
-   d2q->nqpt = nqpt;
-   d2q->B.SetSize(nqpt*ndof);
-   d2q->Bt.SetSize(ndof*nqpt);
-   d2q->G.SetSize(nqpt*ndof);
-   d2q->Gt.SetSize(ndof*nqpt);
-   Vector val(ndof), grad(ndof);
-   for (int i = 0; i < nqpt; i++)
-   {
-      // The first 'nqpt' points in 'ir' have the same x-coordinates as those
-      // of the 1D rule.
-      basis_1d.Eval(ir.IntPoint(i).x, val, grad);
-      for (int j = 0; j < ndof; j++)
-      {
-         d2q->B[i+nqpt*j] = d2q->Bt[j+ndof*i] = val(j);
-         d2q->G[i+nqpt*j] = d2q->Gt[j+ndof*i] = grad(j);
-      }
-   }
-   dof2quad_array.Append(d2q);
-   return *d2q;
-}
-

 void NodalFiniteElement::ProjectCurl_2D(
   const FiniteElement &fe, ElementTransformation &Trans,
@@ -618,7 +657,7 @@ void InvertLinearTrans(ElementTransformation &trans,

   double store[3];
   Vector v(store, x.Size());
-   pt.Get(v, x.Size());
+   pt.Get(store, x.Size());
   v -= x;

   trans.InverseJacobian().Mult(v, x);
@@ -631,7 +670,10 @@ void NodalFiniteElement::GetLocalRestriction(ElementTransformation &Trans,
   Vector pt(&ipt.x, dim);

 #ifdef MFEM_THREAD_SAFE
-   Vector c_shape(dof);
+   Vector shape(dof);
+#else
+   Vector shape;
+   vshape.GetColumnReference(0, shape);
 #endif

   Trans.SetIntPoint(&Nodes[0]);
@@ -641,8 +683,8 @@ void NodalFiniteElement::GetLocalRestriction(ElementTransformation &Trans,
      InvertLinearTrans(Trans, Nodes[j], pt);
      if (Geometries.CheckPoint(geom_type, ipt)) // do we need an epsilon here?
      {
-         CalcShape(ipt, c_shape);
-         R.SetRow(j, c_shape);
+         CalcShape(ipt, shape);
+         R.SetRow(j, shape);
      }
      else
      {
@@ -653,7 +695,7 @@ void NodalFiniteElement::GetLocalRestriction(ElementTransformation &Trans,
   R.Threshold(1e-12);
 }

-void NodalFiniteElement::Project (
+void NodalFiniteElement::Project(
   Coefficient &coeff, ElementTransformation &Trans, Vector &dofs) const
 {
   for (int i = 0; i < dof; i++)
@@ -662,7 +704,7 @@ void NodalFiniteElement::Project (
      // some coefficients expect that Trans.IntPoint is the same
      // as the second argument of Eval
      Trans.SetIntPoint(&ip);
-      dofs(i) = coeff.Eval (Trans, ip);
+      dofs(i) = coeff.Eval(Trans, ip);
      if (map_type == INTEGRAL)
      {
         dofs(i) *= Trans.Weight();
@@ -670,7 +712,7 @@ void NodalFiniteElement::Project (
   }
 }

-void NodalFiniteElement::Project (
+void NodalFiniteElement::Project(
   VectorCoefficient &vc, ElementTransformation &Trans, Vector &dofs) const
 {
   MFEM_ASSERT(dofs.Size() == vc.GetVDim()*dof, "");
@@ -849,18 +891,18 @@ VectorFiniteElement::VectorFiniteElement(int D, Geometry::Type G,
   }
 }

-void VectorFiniteElement::CalcShape (
+void VectorFiniteElement::CalcShape(
   const IntegrationPoint &ip, Vector &shape ) const
 {
-   mfem_error ("Error: Cannot use scalar CalcShape(...) function with\n"
-               "   VectorFiniteElements!");
+   mfem_error("Error: Cannot use scalar CalcShape(...) function with\n"
+              "   VectorFiniteElements!");
 }

-void VectorFiniteElement::CalcDShape (
+void VectorFiniteElement::CalcDShape(
   const IntegrationPoint &ip, DenseMatrix &dshape ) const
 {
-   mfem_error ("Error: Cannot use scalar CalcDShape(...) function with\n"
-               "   VectorFiniteElements!");
+   mfem_error("Error: Cannot use scalar CalcDShape(...) function with\n"
+              "   VectorFiniteElements!");
 }

 void VectorFiniteElement::SetDerivMembers()
@@ -900,7 +942,7 @@ void VectorFiniteElement::SetDerivMembers()
   }
 }

-void VectorFiniteElement::CalcVShape_RT (
+void VectorFiniteElement::CalcVShape_RT(
   ElementTransformation &Trans, DenseMatrix &shape) const
 {
   MFEM_ASSERT(map_type == H_DIV, "");
@@ -912,7 +954,7 @@ void VectorFiniteElement::CalcVShape_RT (
   shape *= (1.0 / Trans.Weight());
 }

-void VectorFiniteElement::CalcVShape_ND (
+void VectorFiniteElement::CalcVShape_ND(
   ElementTransformation &Trans, DenseMatrix &shape) const
 {
   MFEM_ASSERT(map_type == H_CURL, "");
@@ -954,8 +996,8 @@ void VectorFiniteElement::Project_RT(
   {
      Trans.SetIntPoint(&Nodes.IntPoint(k));
      // dof_k = nk^t adj(J) xk
-      Vector vk(vc.GetData()+k*sdim, sdim);
-      dofs(k) = Trans.AdjugateJacobian().InnerProduct(vk, nk + d2n[k]*dim);
+      dofs(k) = Trans.AdjugateJacobian().InnerProduct(
+                   &vc[k*sdim], nk + d2n[k]*dim);
      if (!square_J) { dofs(k) /= Trans.Weight(); }
   }
 }
@@ -1171,9 +1213,8 @@ void VectorFiniteElement::Project_ND(
   for (int k = 0; k < dof; k++)
   {
      Trans.SetIntPoint(&Nodes.IntPoint(k));
-      Vector vk(vc.GetData()+k*dim, dim);
      // dof_k = xk^t J tk
-      dofs(k) = Trans.Jacobian().InnerProduct(tk + d2t[k]*dim, vk);
+      dofs(k) = Trans.Jacobian().InnerProduct(tk + d2t[k]*dim, &vc[k*dim]);
   }
 }

@@ -1320,7 +1361,7 @@ void VectorFiniteElement::LocalL2Projection_RT(
      double w = ip.weight;
      this->CalcVShape(ip, fine_shape);
      Trans.Transform(ip, v);
-      tr_ip.Set(v, dim);
+      tr_ip.Set(v.GetData(), dim);
      cfe.CalcVShape(tr_ip, coarse_shape);

      AddMult_a_AAt(w, fine_shape, fine_mass);
@@ -1407,7 +1448,7 @@ void VectorFiniteElement::LocalL2Projection_ND(
      const IntegrationPoint &ip = ir.IntPoint(i);
      this->CalcVShape(ip, fine_shape);
      Trans.Transform(ip, v);
-      tr_ip.Set(v, dim);
+      tr_ip.Set(v.GetData(), dim);
      cfe.CalcVShape(tr_ip, coarse_shape);

      AddMult_a_AAt(ip.weight, fine_shape, fine_mass);
@@ -2403,6 +2444,46 @@ TensorBasisElement::TensorBasisElement(const int dims, const int p,
   }
 }

+const DofToQuad &TensorBasisElement::GetTensorDofToQuad(
+   const FiniteElement &fe, const IntegrationRule &ir,
+   DofToQuad::Mode mode, const Poly_1D::Basis &basis, bool closed,
+   Array<DofToQuad*> &dof2quad_array)
+{
+   MFEM_VERIFY(mode == DofToQuad::TENSOR, "invalid mode requested");
+
+   for (int i = 0; i < dof2quad_array.Size(); i++)
+   {
+      const DofToQuad &d2q = *dof2quad_array[i];
+      if (d2q.IntRule == &ir && d2q.mode == mode) { return d2q; }
+   }
+
+   DofToQuad *d2q = new DofToQuad;
+   const int ndof = closed ? fe.GetOrder() + 1 : fe.GetOrder();
+   const int nqpt = (int)floor(pow(ir.GetNPoints(), 1.0/fe.GetDim()) + 0.5);
+   d2q->FE = &fe;
+   d2q->IntRule = &ir;
+   d2q->mode = mode;
+   d2q->ndof = ndof;
+   d2q->nqpt = nqpt;
+   d2q->B.SetSize(nqpt*ndof);
+   d2q->Bt.SetSize(ndof*nqpt);
+   d2q->G.SetSize(nqpt*ndof);
+   d2q->Gt.SetSize(ndof*nqpt);
+   Vector val(ndof), grad(ndof);
+   for (int i = 0; i < nqpt; i++)
+   {
+      // The first 'nqpt' points in 'ir' have the same x-coordinates as those
+      // of the 1D rule.
+      basis.Eval(ir.IntPoint(i).x, val, grad);
+      for (int j = 0; j < ndof; j++)
+      {
+         d2q->B[i+nqpt*j] = d2q->Bt[j+ndof*i] = val(j);
+         d2q->G[i+nqpt*j] = d2q->Gt[j+ndof*i] = grad(j);
+      }
+   }
+   dof2quad_array.Append(d2q);
+   return *d2q;
+}

 NodalTensorFiniteElement::NodalTensorFiniteElement(const int dims,
                                                   const int p,
@@ -2437,8 +2518,7 @@ VectorTensorFiniteElement::VectorTensorFiniteElement(const int dims,
                                                     const DofMapType dmtype)
   : VectorFiniteElement(dims, GetTensorProductGeometry(dims), d,
                         p, M, FunctionSpace::Qk),
-     TensorBasisElement(dims, p, VerifyNodal(cbtype), dmtype),
-     cbasis1d(poly1d.GetBasis(p, VerifyClosed(cbtype))),
+     TensorBasisElement(dims, p, VerifyNodal(VerifyClosed(cbtype)), dmtype),
     obasis1d(poly1d.GetBasis(p - 1, VerifyOpen(obtype)))
 {
   MFEM_VERIFY(dims > 1, "Constructor for VectorTensorFiniteElement with both "
@@ -2453,93 +2533,13 @@ VectorTensorFiniteElement::VectorTensorFiniteElement(const int dims,
                                                     const DofMapType dmtype)
   : VectorFiniteElement(dims, GetTensorProductGeometry(dims), d,
                         p, M, FunctionSpace::Pk),
-     TensorBasisElement(dims, p, obtype, dmtype),
-     cbasis1d(poly1d.GetBasis(p, VerifyOpen(obtype))),
+     TensorBasisElement(dims, p, VerifyOpen(obtype), dmtype),
     obasis1d(poly1d.GetBasis(p, VerifyOpen(obtype)))
 {
   MFEM_VERIFY(dims == 1, "Constructor for VectorTensorFiniteElement without "
               "closed basis is only valid for 1D elements.");
 }

-const DofToQuad &VectorTensorFiniteElement::GetDofToQuad(
-   const IntegrationRule &ir,
-   DofToQuad::Mode mode) const
-{
-   MFEM_VERIFY(mode != DofToQuad::FULL, "invalid mode requested");
-
-   return GetTensorDofToQuad(ir, mode, true);
-}
-
-const DofToQuad &VectorTensorFiniteElement::GetDofToQuadOpen(
-   const IntegrationRule &ir,
-   DofToQuad::Mode mode) const
-{
-   MFEM_VERIFY(mode != DofToQuad::FULL, "invalid mode requested");
-
-   return GetTensorDofToQuad(ir, mode, false);
-}
-
-const DofToQuad &VectorTensorFiniteElement::GetTensorDofToQuad(
-   const IntegrationRule &ir,
-   DofToQuad::Mode mode,
-   const bool closed) const
-{
-   MFEM_VERIFY(mode == DofToQuad::TENSOR, "invalid mode requested");
-
-   for (int i = 0;
-        i < (closed ? dof2quad_array.Size() : dof2quad_array_open.Size());
-        i++)
-   {
-      const DofToQuad &d2q = closed ? *dof2quad_array[i] : *dof2quad_array_open[i];
-      if (d2q.IntRule == &ir && d2q.mode == mode) { return d2q; }
-   }
-
-   DofToQuad *d2q = new DofToQuad;
-   const int ndof = closed ? order + 1 : order;
-   const int nqpt = (int)floor(pow(ir.GetNPoints(), 1.0/dim) + 0.5);
-   d2q->FE = this;
-   d2q->IntRule = &ir;
-   d2q->mode = mode;
-   d2q->ndof = ndof;
-   d2q->nqpt = nqpt;
-   d2q->B.SetSize(nqpt*ndof);
-   d2q->Bt.SetSize(ndof*nqpt);
-   d2q->G.SetSize(nqpt*ndof);
-   d2q->Gt.SetSize(ndof*nqpt);
-   Vector val(ndof), grad(ndof);
-   for (int i = 0; i < nqpt; i++)
-   {
-      // The first 'nqpt' points in 'ir' have the same x-coordinates as those
-      // of the 1D rule.
-
-      if (closed)
-      {
-         cbasis1d.Eval(ir.IntPoint(i).x, val, grad);
-      }
-      else
-      {
-         obasis1d.Eval(ir.IntPoint(i).x, val, grad);
-      }
-
-      for (int j = 0; j < ndof; j++)
-      {
-         d2q->B[i+nqpt*j] = d2q->Bt[j+ndof*i] = val(j);
-         d2q->G[i+nqpt*j] = d2q->Gt[j+ndof*i] = grad(j);
-      }
-   }
-
-   if (closed)
-   {
-      dof2quad_array.Append(d2q);
-   }
-   else
-   {
-      dof2quad_array_open.Append(d2q);
-   }
-
-   return *d2q;
-}
-
 VectorTensorFiniteElement::~VectorTensorFiniteElement()
 {
   for (int i = 0; i < dof2quad_array_open.Size(); i++)
@@ -127,7 +127,6 @@ public:
   }
 };

-
 /** @brief Structure representing the matrices/tensors needed to evaluate (in
    reference space) the values, gradients, divergences, or curls of a
    FiniteElement at a the quadrature points of a given IntegrationRule. */
@@ -157,8 +156,7 @@ public:
          dimensions using 1D number of quadrature points and degrees of
          freedom. */
      /** When representing a vector-valued FiniteElement, two DofToQuad objects
-          are used to describe the "closed" and "open" 1D basis functions
-          (TODO). */
+          are used to describe the "closed" and "open" 1D basis functions. */
      TENSOR
   };

@@ -176,7 +174,7 @@ public:
   /// Basis functions evaluated at quadrature points.
   /** The storage layout is column-major with dimensions:
       - #nqpt x #ndof, for scalar elements, or
-       - #nqpt x dim x #ndof, for vector elements, (TODO)
+       - #nqpt x dim x #ndof, for vector elements,

       where

@@ -187,15 +185,15 @@ public:
   /// Transpose of #B.
   /** The storage layout is column-major with dimensions:
       - #ndof x #nqpt, for scalar elements, or
-       - #ndof x #nqpt x dim, for vector elements (TODO). */
+       - #ndof x #nqpt x dim, for vector elements. */
   Array<double> Bt;

   /** @brief Gradients/divergences/curls of basis functions evaluated at
       quadrature points. */
   /** The storage layout is column-major with dimensions:
       - #nqpt x dim x #ndof, for scalar elements, or
-       - #nqpt x #ndof, for H(div) vector elements (TODO), or
-       - #nqpt x cdim x #ndof, for H(curl) vector elements (TODO),
+       - #nqpt x #ndof, for H(div) vector elements, or
+       - #nqpt x cdim x #ndof, for H(curl) vector elements,

       where

@@ -208,12 +206,11 @@ public:
   /// Transpose of #G.
   /** The storage layout is column-major with dimensions:
       - #ndof x #nqpt x dim, for scalar elements, or
-       - #ndof x #nqpt, for H(div) vector elements (TODO), or
-       - #ndof x #nqpt x cdim, for H(curl) vector elements (TODO). */
+       - #ndof x #nqpt, for H(div) vector elements, or
+       - #ndof x #nqpt x cdim, for H(curl) vector elements. */
   Array<double> Gt;
 };

-
 /// Describes the function space on each element
 class FunctionSpace
 {
@@ -247,7 +244,7 @@ protected:
   mutable int orders[Geometry::MaxDim]; ///< Anisotropic orders
   IntegrationRule Nodes;
 #ifndef MFEM_THREAD_SAFE
-   mutable DenseMatrix vshape; // Dof x VDim
+   mutable DenseMatrix vshape; // Dof x Dim
 #endif
   /// Container for all DofToQuad objects created by the FiniteElement.
   /** Multiple DofToQuad objects may be needed when different quadrature rules
@@ -256,7 +253,7 @@ protected:

 public:
   /// Enumeration for range_type and deriv_range_type
-   enum RangeType { SCALAR, VECTOR };
+   enum RangeType { UNKNOWN_RANGE_TYPE = -1, SCALAR, VECTOR };

   /** @brief Enumeration for MapType: defines how reference functions are
       mapped to physical space.
@@ -270,6 +267,8 @@ public:
   */
   enum MapType
   {
+      UNKNOWN_MAP_TYPE = -1, /**< Used to distinguish an unset MapType variable
+                                  from the known values below. */
      VALUE,     /**< For scalar fields; preserves point values
                          \f$ u(x) = \hat u(\hat x) \f$ */
      INTEGRAL,  /**< For scalar fields; preserves volume integrals
@@ -348,7 +347,6 @@ public:
       H_DIV, H_CURL}. */
   int GetMapType() const { return map_type; }

-
   /** @brief Returns the FiniteElement::DerivType of the element describing the
       spatial derivative method implemented, one of {NONE, GRAD,
       DIV, CURL}. */
@@ -455,8 +453,8 @@ public:
       part of the Hessian of one shape function.
       The order in 2D is {u_xx, u_xy, u_yy}.
       The size (#dof x (#dim (#dim+1)/2) of @a Hessian must be set in advance.*/
-   virtual void CalcHessian (const IntegrationPoint &ip,
-                             DenseMatrix &Hessian) const;
+   virtual void CalcHessian(const IntegrationPoint &ip,
+                            DenseMatrix &Hessian) const;

   /** @brief Evaluate the Hessian of all shape functions of a scalar finite
       element in reference space at the given point @a ip. */
@@ -577,6 +575,7 @@ public:
   /** See the documentation for DofToQuad for more details. */
   virtual const DofToQuad &GetDofToQuad(const IntegrationRule &ir,
                                         DofToQuad::Mode mode) const;
+
   /// Deconstruct the FiniteElement
   virtual ~FiniteElement();

@@ -623,16 +622,11 @@ public:
   }
 };

-
 /** @brief Class for finite elements with basis functions
    that return scalar values. */
 class ScalarFiniteElement : public FiniteElement
 {
 protected:
-#ifndef MFEM_THREAD_SAFE
-   mutable Vector c_shape;
-#endif
-
   static const ScalarFiniteElement &CheckScalarFE(const FiniteElement &fe)
   {
      MFEM_VERIFY(fe.GetRangeType() == SCALAR,
@@ -640,10 +634,6 @@ protected:
      return static_cast<const ScalarFiniteElement &>(fe);
   }

-   const DofToQuad &GetTensorDofToQuad(const class TensorBasisElement &tb,
-                                       const IntegrationRule &ir,
-                                       DofToQuad::Mode mode) const;
-
 public:
   /** @brief Construct ScalarFiniteElement with given
       @param D    Reference space dimension
@@ -654,13 +644,8 @@ public:
    */
   ScalarFiniteElement(int D, Geometry::Type G, int Do, int O,
                       int F = FunctionSpace::Pk)
-#ifdef MFEM_THREAD_SAFE
      : FiniteElement(D, G, Do, O, F)
   { deriv_type = GRAD; deriv_range_type = VECTOR; deriv_map_type = H_CURL; }
-#else
-      : FiniteElement(D, G, Do, O, F), c_shape(dof)
-   { deriv_type = GRAD; deriv_range_type = VECTOR; deriv_map_type = H_CURL; }
-#endif

   /** @brief Set the FiniteElement::MapType of the element to either VALUE or
       INTEGRAL. Also sets the FiniteElement::DerivType to GRAD if the
@@ -672,7 +657,6 @@ public:
      deriv_type = (M == VALUE) ? GRAD : NONE;
   }

-
   /** @brief Get the matrix @a I that defines nodal interpolation
       @a between this element and the refined element @a fine_fe. */
   void NodalLocalInterpolation(ElementTransformation &Trans,
@@ -693,15 +677,11 @@ public:
   /** If the "fine" elements cannot represent all basis functions of the
       "coarse" element, then boundary values from different sub-elements are
       generally different. */
-   void ScalarLocalRestriction(ElementTransformation &Trans,
-                               DenseMatrix &R,
-                               const ScalarFiniteElement &coarse_fe) const;
-
-   virtual const DofToQuad &GetDofToQuad(const IntegrationRule &ir,
-                                         DofToQuad::Mode mode) const;
+   void ScalarLocalL2Restriction(ElementTransformation &Trans,
+                                 DenseMatrix &R,
+                                 const ScalarFiniteElement &coarse_fe) const;
 };

-
 /// Class for standard nodal finite elements.
 class NodalFiniteElement : public ScalarFiniteElement
 {
@@ -723,38 +703,38 @@ public:
                      int F = FunctionSpace::Pk)
      : ScalarFiniteElement(D, G, Do, O, F) { }

-   virtual void GetLocalInterpolation(ElementTransformation &Trans,
-                                      DenseMatrix &I) const
+   void GetLocalInterpolation(ElementTransformation &Trans,
+                              DenseMatrix &I) const override
   { NodalLocalInterpolation(Trans, I, *this); }

-   virtual void GetLocalRestriction(ElementTransformation &Trans,
-                                    DenseMatrix &R) const;
+   void GetLocalRestriction(ElementTransformation &Trans,
+                            DenseMatrix &R) const override;

-   virtual void GetTransferMatrix(const FiniteElement &fe,
-                                  ElementTransformation &Trans,
-                                  DenseMatrix &I) const
+   void GetTransferMatrix(const FiniteElement &fe,
+                          ElementTransformation &Trans,
+                          DenseMatrix &I) const override
   { CheckScalarFE(fe).NodalLocalInterpolation(Trans, I, *this); }

-   virtual void Project (Coefficient &coeff,
-                         ElementTransformation &Trans, Vector &dofs) const;
+   void Project(Coefficient &coeff,
+                ElementTransformation &Trans, Vector &dofs) const override;

-   virtual void Project (VectorCoefficient &vc,
-                         ElementTransformation &Trans, Vector &dofs) const;
+   void Project(VectorCoefficient &vc,
+                ElementTransformation &Trans, Vector &dofs) const override;

   // (mc.height x mc.width) @ DOFs -> (Dof x mc.width x mc.height) in dofs
-   virtual void ProjectMatrixCoefficient(
-      MatrixCoefficient &mc, ElementTransformation &T, Vector &dofs) const;
+   void ProjectMatrixCoefficient(
+      MatrixCoefficient &mc, ElementTransformation &T, Vector &dofs) const override;

-   virtual void Project(const FiniteElement &fe, ElementTransformation &Trans,
-                        DenseMatrix &I) const;
+   void Project(const FiniteElement &fe, ElementTransformation &Trans,
+                DenseMatrix &I) const override;

-   virtual void ProjectGrad(const FiniteElement &fe,
-                            ElementTransformation &Trans,
-                            DenseMatrix &grad) const;
+   void ProjectGrad(const FiniteElement &fe,
+                    ElementTransformation &Trans,
+                    DenseMatrix &grad) const override;

-   virtual void ProjectDiv(const FiniteElement &fe,
-                           ElementTransformation &Trans,
-                           DenseMatrix &div) const;
+   void ProjectDiv(const FiniteElement &fe,
+                   ElementTransformation &Trans,
+                   DenseMatrix &div) const override;

   /** @brief Get an Array<int> that maps lexicographically ordered indices to
       the indices of the respective nodes/dofs/basis functions.
@@ -788,12 +768,12 @@ class VectorFiniteElement : public FiniteElement
   // Hide the scalar functions CalcShape and CalcDShape.
 private:
   /// Overrides the scalar CalcShape function to print an error.
-   virtual void CalcShape(const IntegrationPoint &ip,
-                          Vector &shape) const;
+   void CalcShape(const IntegrationPoint &ip,
+                  Vector &shape) const override;

   /// Overrides the scalar CalcDShape function to print an error.
-   virtual void CalcDShape(const IntegrationPoint &ip,
-                           DenseMatrix &dshape) const;
+   void CalcDShape(const IntegrationPoint &ip,
+                   DenseMatrix &dshape) const override;

 protected:
   bool is_nodal;
@@ -952,11 +932,10 @@ protected:
   }

 public:
-   VectorFiniteElement (int D, Geometry::Type G, int Do, int O, int M,
-                        int F = FunctionSpace::Pk);
+   VectorFiniteElement(int D, Geometry::Type G, int Do, int O, int M,
+                       int F = FunctionSpace::Pk);
 };

-
 /// @brief Class for computing 1D special polynomials and their associated basis
 /// functions
 class Poly_1D
@@ -1091,6 +1070,11 @@ public:
   // { CalcLegendre(p, x, u); }
   { CalcChebyshev(p, x, u); }

+   /** @brief Evaluate the values of a hierarchical 1D basis at point x
+       hierarchical = k-th basis function is degree k polynomial */
+   static void CalcBasis(const int p, const double x, Vector &u)
+   { CalcBasis(p, x, u.GetData()); }
+
   /// Evaluate the values and derivatives of a hierarchical 1D basis at point @a x
   static void CalcBasis(const int p, const double x, double *u, double *d)
   // { CalcMono(p, x, u, d); }
@@ -1098,6 +1082,11 @@ public:
   // { CalcLegendre(p, x, u, d); }
   { CalcChebyshev(p, x, u, d); }

+   /** @brief Evaluate the values and derivatives of a hierarchical 1D basis at
+       point @a x. */
+   static void CalcBasis(const int p, const double x, Vector &u, Vector &d)
+   { CalcBasis(p, x, u.GetData(), d.GetData()); }
+
   /// Evaluate the values, derivatives and second derivatives of a hierarchical 1D basis at point x
   static void CalcBasis(const int p, const double x, double *u, double *d,
                         double *dd)
@@ -1106,6 +1095,12 @@ public:
   // { CalcLegendre(p, x, u, d); }
   { CalcChebyshev(p, x, u, d, dd); }

+   /** @brief Evaluate the values, derivatives and second derivatives of a
+       hierarchical 1D basis at point @a x. */
+   static void CalcBasis(const int p, const double x, Vector &u, Vector &d,
+                         Vector &dd)
+   { CalcBasis(p, x, u.GetData(), d.GetData(), dd.GetData()); }
+
   /// Evaluate a representation of a Delta function at point x
   static double CalcDelta(const int p, const double x)
   { return pow(x, (double) p); }
@@ -1135,12 +1130,24 @@ public:
   static void CalcBernstein(const int p, const double x, double *u)
   { CalcBinomTerms(p, x, 1. - x, u); }

+   /** @brief Compute the values of the Bernstein basis functions of order
+       @a p at coordinate @a x and store the results in the already allocated
+       @a u array. */
+   static void CalcBernstein(const int p, const double x, Vector &u)
+   { CalcBernstein(p, x, u.GetData()); }
+
   /** @brief Compute the values and derivatives of the Bernstein basis functions
       of order @a p at coordinate @a x and store the results in the already allocated
       @a u and @a d arrays. */
   static void CalcBernstein(const int p, const double x, double *u, double *d)
   { CalcBinomTerms(p, x, 1. - x, u, d); }

+   /** @brief Compute the values and derivatives of the Bernstein basis
+       functions of order @a p at coordinate @a x and store the results in the
+       already allocated @a u and @a d arrays. */
+   static void CalcBernstein(const int p, const double x, Vector &u, Vector &d)
+   { CalcBernstein(p, x, u.GetData(), d.GetData()); }
+
   static void CalcLegendre(const int p, const double x, double *u);
   static void CalcLegendre(const int p, const double x, double *u, double *d);

@@ -1149,7 +1156,6 @@ public:

 extern Poly_1D poly1d;

-
 /// An element defined as an ND tensor product of 1D elements on a segment,
 /// square, or cube
 class TensorBasisElement
@@ -1173,7 +1179,7 @@ public:

   int GetBasisType() const { return b_type; }

-   const Poly_1D::Basis& GetBasis1D() const { return basis1d; }
+   const Poly_1D::Basis &GetBasis1D() const { return basis1d; }

   /** @brief Get an Array<int> that maps lexicographically ordered indices to
       the indices of the respective nodes/dofs/basis functions. If the dofs are
@@ -1205,6 +1211,11 @@ public:
         default: MFEM_ABORT("invalid dimension: " << dim); return -1;
      }
   }
+
+   static const DofToQuad &GetTensorDofToQuad(
+      const FiniteElement &fe, const IntegrationRule &ir,
+      DofToQuad::Mode mode, const Poly_1D::Basis &basis, bool closed,
+      Array<DofToQuad*> &dof2quad_array);
 };

 class NodalTensorFiniteElement : public NodalFiniteElement,
@@ -1215,18 +1226,18 @@ public:
                            const DofMapType dmtype);

   const DofToQuad &GetDofToQuad(const IntegrationRule &ir,
-                                 DofToQuad::Mode mode) const
+                                 DofToQuad::Mode mode) const override
   {
      return (mode == DofToQuad::FULL) ?
-             ScalarFiniteElement::GetDofToQuad(ir, mode) :
-             ScalarFiniteElement::GetTensorDofToQuad(*this, ir, mode);
+             FiniteElement::GetDofToQuad(ir, mode) :
+             GetTensorDofToQuad(*this, ir, mode, basis1d, true, dof2quad_array);
   }

-   virtual void SetMapType(const int map_type_);
+   void SetMapType(const int map_type_) override;

-   virtual void GetTransferMatrix(const FiniteElement &fe,
-                                  ElementTransformation &Trans,
-                                  DenseMatrix &I) const
+   void GetTransferMatrix(const FiniteElement &fe,
+                          ElementTransformation &Trans,
+                          DenseMatrix &I) const override
   {
      if (basis1d.IsIntegratedType())
      {
@@ -1246,7 +1257,7 @@ private:
   mutable Array<DofToQuad*> dof2quad_array_open;

 protected:
-   Poly_1D::Basis &cbasis1d, &obasis1d;
+   Poly_1D::Basis &obasis1d;

 public:
   VectorTensorFiniteElement(const int dims, const int d, const int p,
@@ -1259,16 +1270,22 @@ public:
                             const DofMapType dmtype);

   const DofToQuad &GetDofToQuad(const IntegrationRule &ir,
-                                 DofToQuad::Mode mode) const;
+                                 DofToQuad::Mode mode) const override
+   {
+      MFEM_VERIFY(mode != DofToQuad::FULL, "invalid mode requested");
+      return GetTensorDofToQuad(*this, ir, mode, basis1d, true,
+                                dof2quad_array);
+   }

   const DofToQuad &GetDofToQuadOpen(const IntegrationRule &ir,
-                                     DofToQuad::Mode mode) const;
+                                     DofToQuad::Mode mode) const
+   {
+      MFEM_VERIFY(mode != DofToQuad::FULL, "invalid mode requested");
+      return GetTensorDofToQuad(*this, ir, mode, obasis1d, false,
+                                dof2quad_array_open);
+   }

-   const DofToQuad &GetTensorDofToQuad(const IntegrationRule &ir,
-                                       DofToQuad::Mode mode,
-                                       const bool closed) const;
-
-   ~VectorTensorFiniteElement();
+   virtual ~VectorTensorFiniteElement();
 };

 void InvertLinearTrans(ElementTransformation &trans,
@@ -32,6 +32,11 @@ public:
   virtual void CalcDShape(const IntegrationPoint &ip,
                           DenseMatrix &dshape) const;
   virtual void ProjectDelta(int vertex, Vector &dofs) const;
+
+   virtual void GetLocalRestriction(ElementTransformation &Trans,
+                                    DenseMatrix &R) const
+   { ScalarLocalL2Restriction(Trans, R, *this); }
+
 };


@@ -55,6 +60,11 @@ public:
                            ElementTransformation &Trans,
                            DenseMatrix &curl) const
   { ProjectCurl_2D(fe, Trans, curl); }
+
+   virtual void GetLocalRestriction(ElementTransformation &Trans,
+                                    DenseMatrix &R) const
+   { ScalarLocalL2Restriction(Trans, R, *this); }
+
   using FiniteElement::Project;
   virtual void ProjectDiv(const FiniteElement &fe,
                           ElementTransformation &Trans,
@@ -80,6 +90,11 @@ public:
   virtual void CalcDShape(const IntegrationPoint &ip,
                           DenseMatrix &dshape) const;
   virtual void ProjectDelta(int vertex, Vector &dofs) const;
+
+   virtual void GetLocalRestriction(ElementTransformation &Trans,
+                                    DenseMatrix &R) const
+   { ScalarLocalL2Restriction(Trans, R, *this); }
+
   using FiniteElement::Project;
   virtual void ProjectDiv(const FiniteElement &fe,
                           ElementTransformation &Trans,
@@ -111,6 +126,11 @@ public:
                            ElementTransformation &Trans,
                            DenseMatrix &curl) const
   { ProjectCurl_2D(fe, Trans, curl); }
+
+   virtual void GetLocalRestriction(ElementTransformation &Trans,
+                                    DenseMatrix &R) const
+   { ScalarLocalL2Restriction(Trans, R, *this); }
+
 };


@@ -133,6 +153,11 @@ public:
   virtual void CalcDShape(const IntegrationPoint &ip,
                           DenseMatrix &dshape) const;
   virtual void ProjectDelta(int vertex, Vector &dofs) const;
+
+   virtual void GetLocalRestriction(ElementTransformation &Trans,
+                                    DenseMatrix &R) const
+   { ScalarLocalL2Restriction(Trans, R, *this); }
+
 };


@@ -314,14 +314,16 @@ void ND_HexahedronElement::CalcVShape(const IntegrationPoint &ip,
 #ifdef MFEM_THREAD_SAFE
   Vector shape_cx(p + 1), shape_ox(p), shape_cy(p + 1), shape_oy(p);
   Vector shape_cz(p + 1), shape_oz(p);
-   Vector dshape_cx(p + 1), dshape_cy(p + 1), dshape_cz(p + 1);
 #endif

   if (obasis1d.IsIntegratedType())
   {
-      cbasis1d.Eval(ip.x, shape_cx, dshape_cx);
-      cbasis1d.Eval(ip.y, shape_cy, dshape_cy);
-      cbasis1d.Eval(ip.z, shape_cz, dshape_cz);
+#ifdef MFEM_THREAD_SAFE
+      Vector dshape_cx(p + 1), dshape_cy(p + 1), dshape_cz(p + 1);
+#endif
+      basis1d.Eval(ip.x, shape_cx, dshape_cx);
+      basis1d.Eval(ip.y, shape_cy, dshape_cy);
+      basis1d.Eval(ip.z, shape_cz, dshape_cz);
      obasis1d.ScaleIntegrated(false);
      obasis1d.EvalIntegrated(dshape_cx, shape_ox);
      obasis1d.EvalIntegrated(dshape_cy, shape_oy);
@@ -329,9 +331,9 @@ void ND_HexahedronElement::CalcVShape(const IntegrationPoint &ip,
   }
   else
   {
-      cbasis1d.Eval(ip.x, shape_cx);
-      cbasis1d.Eval(ip.y, shape_cy);
-      cbasis1d.Eval(ip.z, shape_cz);
+      basis1d.Eval(ip.x, shape_cx);
+      basis1d.Eval(ip.y, shape_cy);
+      basis1d.Eval(ip.z, shape_cz);
      obasis1d.Eval(ip.x, shape_ox);
      obasis1d.Eval(ip.y, shape_oy);
      obasis1d.Eval(ip.z, shape_oz);
@@ -405,9 +407,9 @@ void ND_HexahedronElement::CalcCurlShape(const IntegrationPoint &ip,
   Vector dshape_cx(p + 1), dshape_cy(p + 1), dshape_cz(p + 1);
 #endif

-   cbasis1d.Eval(ip.x, shape_cx, dshape_cx);
-   cbasis1d.Eval(ip.y, shape_cy, dshape_cy);
-   cbasis1d.Eval(ip.z, shape_cz, dshape_cz);
+   basis1d.Eval(ip.x, shape_cx, dshape_cx);
+   basis1d.Eval(ip.y, shape_cy, dshape_cy);
+   basis1d.Eval(ip.z, shape_cz, dshape_cz);
   if (obasis1d.IsIntegratedType())
   {
      obasis1d.ScaleIntegrated(false);
@@ -656,21 +658,23 @@ void ND_QuadrilateralElement::CalcVShape(const IntegrationPoint &ip,

 #ifdef MFEM_THREAD_SAFE
   Vector shape_cx(p + 1), shape_ox(p), shape_cy(p + 1), shape_oy(p);
-   Vector dshape_cx(p + 1), dshape_cy(p + 1);
 #endif

   if (obasis1d.IsIntegratedType())
   {
-      cbasis1d.Eval(ip.x, shape_cx, dshape_cx);
-      cbasis1d.Eval(ip.y, shape_cy, dshape_cy);
+#ifdef MFEM_THREAD_SAFE
+      Vector dshape_cx(p + 1), dshape_cy(p + 1);
+#endif
+      basis1d.Eval(ip.x, shape_cx, dshape_cx);
+      basis1d.Eval(ip.y, shape_cy, dshape_cy);
      obasis1d.ScaleIntegrated(false);
      obasis1d.EvalIntegrated(dshape_cx, shape_ox);
      obasis1d.EvalIntegrated(dshape_cy, shape_oy);
   }
   else
   {
-      cbasis1d.Eval(ip.x, shape_cx);
-      cbasis1d.Eval(ip.y, shape_cy);
+      basis1d.Eval(ip.x, shape_cx);
+      basis1d.Eval(ip.y, shape_cy);
      obasis1d.Eval(ip.x, shape_ox);
      obasis1d.Eval(ip.y, shape_oy);
   }
@@ -720,8 +724,8 @@ void ND_QuadrilateralElement::CalcCurlShape(const IntegrationPoint &ip,
   Vector dshape_cx(p + 1), dshape_cy(p + 1);
 #endif

-   cbasis1d.Eval(ip.x, shape_cx, dshape_cx);
-   cbasis1d.Eval(ip.y, shape_cy, dshape_cy);
+   basis1d.Eval(ip.x, shape_cx, dshape_cx);
+   basis1d.Eval(ip.y, shape_cy, dshape_cy);
   if (obasis1d.IsIntegratedType())
   {
      obasis1d.ScaleIntegrated(false);
@@ -13,6 +13,7 @@

 #include "fe_pos.hpp"
 #include "../bilininteg.hpp"
+#include "../lininteg.hpp"
 #include "../coefficient.hpp"

 namespace mfem
@@ -212,7 +213,7 @@ void BiQuadPos2DFiniteElement::GetLocalInterpolation(
 void BiQuadPos2DFiniteElement::Project(
   Coefficient &coeff, ElementTransformation &Trans, Vector &dofs) const
 {
-   double *d = dofs;
+   double *d = dofs.GetData();

   for (int i = 0; i < 9; i++)
   {
@@ -382,8 +383,8 @@ void H1Pos_QuadrilateralElement::CalcShape(const IntegrationPoint &ip,
   Vector shape_x(p+1), shape_y(p+1);
 #endif

-   Poly_1D::CalcBernstein(p, ip.x, shape_x.GetData() );
-   Poly_1D::CalcBernstein(p, ip.y, shape_y.GetData() );
+   Poly_1D::CalcBernstein(p, ip.x, shape_x);
+   Poly_1D::CalcBernstein(p, ip.y, shape_y);

   // Reorder so that vertices are at the beginning of the list
   for (int o = 0, j = 0; j <= p; j++)
@@ -402,8 +403,8 @@ void H1Pos_QuadrilateralElement::CalcDShape(const IntegrationPoint &ip,
   Vector shape_x(p+1), shape_y(p+1), dshape_x(p+1), dshape_y(p+1);
 #endif

-   Poly_1D::CalcBernstein(p, ip.x, shape_x.GetData(), dshape_x.GetData() );
-   Poly_1D::CalcBernstein(p, ip.y, shape_y.GetData(), dshape_y.GetData() );
+   Poly_1D::CalcBernstein(p, ip.x, shape_x, dshape_x);
+   Poly_1D::CalcBernstein(p, ip.y, shape_y, dshape_y);

   // Reorder so that vertices are at the beginning of the list
   for (int o = 0, j = 0; j <= p; j++)
@@ -40,7 +40,7 @@ public:

   virtual void GetLocalRestriction(ElementTransformation &Trans,
                                    DenseMatrix &R) const
-   { ScalarLocalRestriction(Trans, R, *this); }
+   { ScalarLocalL2Restriction(Trans, R, *this); }

   virtual void GetTransferMatrix(const FiniteElement &fe,
                                  ElementTransformation &Trans,
@@ -73,8 +73,8 @@ public:
                                 DofToQuad::Mode mode) const
   {
      return (mode == DofToQuad::FULL) ?
-             ScalarFiniteElement::GetDofToQuad(ir, mode) :
-             ScalarFiniteElement::GetTensorDofToQuad(*this, ir, mode);
+             FiniteElement::GetDofToQuad(ir, mode) :
+             GetTensorDofToQuad(*this, ir, mode, basis1d, true, dof2quad_array);
   }
 };

@@ -145,21 +145,23 @@ void RT_QuadrilateralElement::CalcVShape(const IntegrationPoint &ip,

 #ifdef MFEM_THREAD_SAFE
   Vector shape_cx(pp1 + 1), shape_ox(pp1), shape_cy(pp1 + 1), shape_oy(pp1);
-   Vector dshape_cx(pp1 + 1), dshape_cy(pp1 + 1);
 #endif

   if (obasis1d.IsIntegratedType())
   {
-      cbasis1d.Eval(ip.x, shape_cx, dshape_cx);
-      cbasis1d.Eval(ip.y, shape_cy, dshape_cy);
+#ifdef MFEM_THREAD_SAFE
+      Vector dshape_cx(pp1 + 1), dshape_cy(pp1 + 1);
+#endif
+      basis1d.Eval(ip.x, shape_cx, dshape_cx);
+      basis1d.Eval(ip.y, shape_cy, dshape_cy);
      obasis1d.ScaleIntegrated(false);
      obasis1d.EvalIntegrated(dshape_cx, shape_ox);
      obasis1d.EvalIntegrated(dshape_cy, shape_oy);
   }
   else
   {
-      cbasis1d.Eval(ip.x, shape_cx);
-      cbasis1d.Eval(ip.y, shape_cy);
+      basis1d.Eval(ip.x, shape_cx);
+      basis1d.Eval(ip.y, shape_cy);
      obasis1d.Eval(ip.x, shape_ox);
      obasis1d.Eval(ip.y, shape_oy);
   }
@@ -207,8 +209,8 @@ void RT_QuadrilateralElement::CalcDivShape(const IntegrationPoint &ip,
   Vector dshape_cx(pp1 + 1), dshape_cy(pp1 + 1);
 #endif

-   cbasis1d.Eval(ip.x, shape_cx, dshape_cx);
-   cbasis1d.Eval(ip.y, shape_cy, dshape_cy);
+   basis1d.Eval(ip.x, shape_cx, dshape_cx);
+   basis1d.Eval(ip.y, shape_cy, dshape_cy);
   if (obasis1d.IsIntegratedType())
   {
      obasis1d.ScaleIntegrated(false);
@@ -473,14 +475,16 @@ void RT_HexahedronElement::CalcVShape(const IntegrationPoint &ip,
 #ifdef MFEM_THREAD_SAFE
   Vector shape_cx(pp1 + 1), shape_ox(pp1), shape_cy(pp1 + 1), shape_oy(pp1);
   Vector shape_cz(pp1 + 1), shape_oz(pp1);
-   Vector dshape_cx(pp1 + 1), dshape_cy(pp1 + 1), dshape_cz(pp1 + 1);
 #endif

   if (obasis1d.IsIntegratedType())
   {
-      cbasis1d.Eval(ip.x, shape_cx, dshape_cx);
-      cbasis1d.Eval(ip.y, shape_cy, dshape_cy);
-      cbasis1d.Eval(ip.z, shape_cz, dshape_cz);
+#ifdef MFEM_THREAD_SAFE
+      Vector dshape_cx(pp1 + 1), dshape_cy(pp1 + 1), dshape_cz(pp1 + 1);
+#endif
+      basis1d.Eval(ip.x, shape_cx, dshape_cx);
+      basis1d.Eval(ip.y, shape_cy, dshape_cy);
+      basis1d.Eval(ip.z, shape_cz, dshape_cz);
      obasis1d.ScaleIntegrated(false);
      obasis1d.EvalIntegrated(dshape_cx, shape_ox);
      obasis1d.EvalIntegrated(dshape_cy, shape_oy);
@@ -488,9 +492,9 @@ void RT_HexahedronElement::CalcVShape(const IntegrationPoint &ip,
   }
   else
   {
-      cbasis1d.Eval(ip.x, shape_cx);
-      cbasis1d.Eval(ip.y, shape_cy);
-      cbasis1d.Eval(ip.z, shape_cz);
+      basis1d.Eval(ip.x, shape_cx);
+      basis1d.Eval(ip.y, shape_cy);
+      basis1d.Eval(ip.z, shape_cz);
      obasis1d.Eval(ip.x, shape_ox);
      obasis1d.Eval(ip.y, shape_oy);
      obasis1d.Eval(ip.z, shape_oz);
@@ -564,9 +568,9 @@ void RT_HexahedronElement::CalcDivShape(const IntegrationPoint &ip,
   Vector dshape_cx(pp1 + 1), dshape_cy(pp1 + 1), dshape_cz(pp1 + 1);
 #endif

-   cbasis1d.Eval(ip.x, shape_cx, dshape_cx);
-   cbasis1d.Eval(ip.y, shape_cy, dshape_cy);
-   cbasis1d.Eval(ip.z, shape_cz, dshape_cz);
+   basis1d.Eval(ip.x, shape_cx, dshape_cx);
+   basis1d.Eval(ip.y, shape_cy, dshape_cy);
+   basis1d.Eval(ip.z, shape_cz, dshape_cz);
   if (obasis1d.IsIntegratedType())
   {
      obasis1d.ScaleIntegrated(false);
@@ -22,6 +22,71 @@ namespace mfem

 using namespace std;

+const FiniteElement *
+FiniteElementCollection::FiniteElementForDim(int dim) const
+{
+   ErrorMode save_error_mode = error_mode;
+   error_mode = RETURN_NULL;
+   const FiniteElement *fe = nullptr;
+   for (int g = Geometry::DimStart[dim]; g < Geometry::DimStart[dim+1]; g++)
+   {
+      fe = FiniteElementForGeometry((Geometry::Type)g);
+      if (fe != nullptr) { break; }
+   }
+   error_mode = save_error_mode;
+   return fe;
+}
+
+int FiniteElementCollection::GetRangeType(int dim) const
+{
+   const FiniteElement *fe = FiniteElementForDim(dim);
+   if (fe)
+   {
+      return fe->GetRangeType();
+   }
+   return FiniteElement::UNKNOWN_RANGE_TYPE;
+}
+
+int FiniteElementCollection::GetDerivRangeType(int dim) const
+{
+   const FiniteElement *fe = FiniteElementForDim(dim);
+   if (fe)
+   {
+      return fe->GetDerivRangeType();
+   }
+   return FiniteElement::UNKNOWN_RANGE_TYPE;
+}
+
+int FiniteElementCollection::GetMapType(int dim) const
+{
+   const FiniteElement *fe = FiniteElementForDim(dim);
+   if (fe)
+   {
+      return fe->GetMapType();
+   }
+   return FiniteElement::UNKNOWN_MAP_TYPE;
+}
+
+int FiniteElementCollection::GetDerivType(int dim) const
+{
+   const FiniteElement *fe = FiniteElementForDim(dim);
+   if (fe)
+   {
+      return fe->GetDerivType();
+   }
+   return FiniteElement::NONE;
+}
+
+int FiniteElementCollection::GetDerivMapType(int dim) const
+{
+   const FiniteElement *fe = FiniteElementForDim(dim);
+   if (fe)
+   {
+      return fe->GetDerivMapType();
+   }
+   return FiniteElement::UNKNOWN_MAP_TYPE;
+}
+
 int FiniteElementCollection::HasFaceDofs(Geometry::Type geom, int p) const
 {
   switch (geom)
@@ -579,6 +644,7 @@ LinearFECollection::FiniteElementForGeometry(Geometry::Type GeomType) const
      case Geometry::PRISM:       return &WedgeFE;
      case Geometry::PYRAMID:     return &PyramidFE;
      default:
+         if (error_mode == RETURN_NULL) { return nullptr; }
         mfem_error ("LinearFECollection: unknown geometry type.");
   }
   return &SegmentFE; // Make some compilers happy
@@ -622,6 +688,7 @@ QuadraticFECollection::FiniteElementForGeometry(Geometry::Type GeomType) const
      case Geometry::CUBE:        return &ParallelepipedFE;
      case Geometry::PRISM:       return &WedgeFE;
      default:
+         if (error_mode == RETURN_NULL) { return nullptr; }
         mfem_error ("QuadraticFECollection: unknown geometry type.");
   }
   return &SegmentFE; // Make some compilers happy
@@ -662,6 +729,7 @@ QuadraticPosFECollection::FiniteElementForGeometry(
      case Geometry::SEGMENT:     return &SegmentFE;
      case Geometry::SQUARE:      return &QuadrilateralFE;
      default:
+         if (error_mode == RETURN_NULL) { return nullptr; }
         mfem_error ("QuadraticPosFECollection: unknown geometry type.");
   }
   return NULL; // Make some compilers happy
@@ -702,6 +770,7 @@ CubicFECollection::FiniteElementForGeometry(Geometry::Type GeomType) const
      case Geometry::CUBE:        return &ParallelepipedFE;
      case Geometry::PRISM:       return &WedgeFE;
      default:
+         if (error_mode == RETURN_NULL) { return nullptr; }
         mfem_error ("CubicFECollection: unknown geometry type.");
   }
   return &SegmentFE; // Make some compilers happy
@@ -768,6 +837,7 @@ CrouzeixRaviartFECollection::FiniteElementForGeometry(
      case Geometry::TRIANGLE:    return &TriangleFE;
      case Geometry::SQUARE:      return &QuadrilateralFE;
      default:
+         if (error_mode == RETURN_NULL) { return nullptr; }
         mfem_error ("CrouzeixRaviartFECollection: unknown geometry type.");
   }
   return &SegmentFE; // Make some compilers happy
@@ -805,6 +875,7 @@ RT0_2DFECollection::FiniteElementForGeometry(Geometry::Type GeomType) const
      case Geometry::TRIANGLE:    return &TriangleFE;
      case Geometry::SQUARE:      return &QuadrilateralFE;
      default:
+         if (error_mode == RETURN_NULL) { return nullptr; }
         mfem_error ("RT0_2DFECollection: unknown geometry type.");
   }
   return &SegmentFE; // Make some compilers happy
@@ -847,6 +918,7 @@ RT1_2DFECollection::FiniteElementForGeometry(Geometry::Type GeomType) const
      case Geometry::TRIANGLE:    return &TriangleFE;
      case Geometry::SQUARE:      return &QuadrilateralFE;
      default:
+         if (error_mode == RETURN_NULL) { return nullptr; }
         mfem_error ("RT1_2DFECollection: unknown geometry type.");
   }
   return &SegmentFE; // Make some compilers happy
@@ -888,6 +960,7 @@ RT2_2DFECollection::FiniteElementForGeometry(Geometry::Type GeomType) const
      case Geometry::TRIANGLE:    return &TriangleFE;
      case Geometry::SQUARE:      return &QuadrilateralFE;
      default:
+         if (error_mode == RETURN_NULL) { return nullptr; }
         mfem_error ("RT2_2DFECollection: unknown geometry type.");
   }
   return &SegmentFE; // Make some compilers happy
@@ -929,6 +1002,7 @@ Const2DFECollection::FiniteElementForGeometry(Geometry::Type GeomType) const
      case Geometry::TRIANGLE:    return &TriangleFE;
      case Geometry::SQUARE:      return &QuadrilateralFE;
      default:
+         if (error_mode == RETURN_NULL) { return nullptr; }
         mfem_error ("Const2DFECollection: unknown geometry type.");
   }
   return &TriangleFE; // Make some compilers happy
@@ -964,6 +1038,7 @@ LinearDiscont2DFECollection::FiniteElementForGeometry(
      case Geometry::TRIANGLE:    return &TriangleFE;
      case Geometry::SQUARE:      return &QuadrilateralFE;
      default:
+         if (error_mode == RETURN_NULL) { return nullptr; }
         mfem_error ("LinearDiscont2DFECollection: unknown geometry type.");
   }
   return &TriangleFE; // Make some compilers happy
@@ -999,6 +1074,7 @@ GaussLinearDiscont2DFECollection::FiniteElementForGeometry(
      case Geometry::TRIANGLE:    return &TriangleFE;
      case Geometry::SQUARE:      return &QuadrilateralFE;
      default:
+         if (error_mode == RETURN_NULL) { return nullptr; }
         mfem_error ("GaussLinearDiscont2DFECollection:"
                     " unknown geometry type.");
   }
@@ -1033,6 +1109,7 @@ P1OnQuadFECollection::FiniteElementForGeometry(Geometry::Type GeomType) const
 {
   if (GeomType != Geometry::SQUARE)
   {
+      if (error_mode == RETURN_NULL) { return nullptr; }
      mfem_error ("P1OnQuadFECollection: unknown geometry type.");
   }
   return &QuadrilateralFE;
@@ -1067,6 +1144,7 @@ QuadraticDiscont2DFECollection::FiniteElementForGeometry(
      case Geometry::TRIANGLE:    return &TriangleFE;
      case Geometry::SQUARE:      return &QuadrilateralFE;
      default:
+         if (error_mode == RETURN_NULL) { return nullptr; }
         mfem_error ("QuadraticDiscont2DFECollection: unknown geometry type.");
   }
   return &TriangleFE; // Make some compilers happy
@@ -1102,6 +1180,7 @@ QuadraticPosDiscont2DFECollection::FiniteElementForGeometry(
   {
      case Geometry::SQUARE:  return &QuadrilateralFE;
      default:
+         if (error_mode == RETURN_NULL) { return nullptr; }
         mfem_error ("QuadraticPosDiscont2DFECollection: unknown geometry type.");
   }
   return NULL; // Make some compilers happy
@@ -1132,6 +1211,7 @@ const
      case Geometry::TRIANGLE:    return &TriangleFE;
      case Geometry::SQUARE:      return &QuadrilateralFE;
      default:
+         if (error_mode == RETURN_NULL) { return nullptr; }
         mfem_error ("GaussQuadraticDiscont2DFECollection:"
                     " unknown geometry type.");
   }
@@ -1170,6 +1250,7 @@ CubicDiscont2DFECollection::FiniteElementForGeometry(
      case Geometry::TRIANGLE:    return &TriangleFE;
      case Geometry::SQUARE:      return &QuadrilateralFE;
      default:
+         if (error_mode == RETURN_NULL) { return nullptr; }
         mfem_error ("CubicDiscont2DFECollection: unknown geometry type.");
   }
   return &TriangleFE; // Make some compilers happy
@@ -1207,6 +1288,7 @@ LinearNonConf3DFECollection::FiniteElementForGeometry(
      case Geometry::TETRAHEDRON: return &TetrahedronFE;
      case Geometry::CUBE:        return &ParallelepipedFE;
      default:
+         if (error_mode == RETURN_NULL) { return nullptr; }
         mfem_error ("LinearNonConf3DFECollection: unknown geometry type.");
   }
   return &TriangleFE; // Make some compilers happy
@@ -1247,6 +1329,7 @@ Const3DFECollection::FiniteElementForGeometry(Geometry::Type GeomType) const
      case Geometry::PRISM:       return &WedgeFE;
      case Geometry::PYRAMID:     return &PyramidFE;
      default:
+         if (error_mode == RETURN_NULL) { return nullptr; }
         mfem_error ("Const3DFECollection: unknown geometry type.");
   }
   return &TetrahedronFE; // Make some compilers happy
@@ -1288,6 +1371,7 @@ LinearDiscont3DFECollection::FiniteElementForGeometry(
      case Geometry::PRISM:       return &WedgeFE;
      case Geometry::CUBE:        return &ParallelepipedFE;
      default:
+         if (error_mode == RETURN_NULL) { return nullptr; }
         mfem_error ("LinearDiscont3DFECollection: unknown geometry type.");
   }
   return &TetrahedronFE; // Make some compilers happy
@@ -1327,6 +1411,7 @@ QuadraticDiscont3DFECollection::FiniteElementForGeometry(
      case Geometry::TETRAHEDRON: return &TetrahedronFE;
      case Geometry::CUBE:        return &ParallelepipedFE;
      default:
+         if (error_mode == RETURN_NULL) { return nullptr; }
         mfem_error ("QuadraticDiscont3DFECollection: unknown geometry type.");
   }
   return &TetrahedronFE; // Make some compilers happy
@@ -1368,6 +1453,7 @@ RefinedLinearFECollection::FiniteElementForGeometry(
      case Geometry::TETRAHEDRON: return &TetrahedronFE;
      case Geometry::CUBE:        return &ParallelepipedFE;
      default:
+         if (error_mode == RETURN_NULL) { return nullptr; }
         mfem_error ("RefinedLinearFECollection: unknown geometry type.");
   }
   return &SegmentFE; // Make some compilers happy
@@ -1408,6 +1494,7 @@ ND1_3DFECollection::FiniteElementForGeometry(Geometry::Type GeomType) const
      case Geometry::PRISM:       return &WedgeFE;
      case Geometry::PYRAMID:     return &PyramidFE;
      default:
+         if (error_mode == RETURN_NULL) { return nullptr; }
         mfem_error ("ND1_3DFECollection: unknown geometry type.");
   }
   return &HexahedronFE; // Make some compilers happy
@@ -1457,6 +1544,7 @@ RT0_3DFECollection::FiniteElementForGeometry(Geometry::Type GeomType) const
      case Geometry::PRISM:       return &WedgeFE;
      case Geometry::PYRAMID:     return &PyramidFE;
      default:
+         if (error_mode == RETURN_NULL) { return nullptr; }
         mfem_error ("RT0_3DFECollection: unknown geometry type.");
   }
   return &HexahedronFE; // Make some compilers happy
@@ -1506,6 +1594,7 @@ RT1_3DFECollection::FiniteElementForGeometry(Geometry::Type GeomType) const
      case Geometry::SQUARE:      return &QuadrilateralFE;
      case Geometry::CUBE:        return &HexahedronFE;
      default:
+         if (error_mode == RETURN_NULL) { return nullptr; }
         mfem_error ("RT1_3DFECollection: unknown geometry type.");
   }
   return &HexahedronFE; // Make some compilers happy
@@ -1867,6 +1956,7 @@ H1_FECollection::FiniteElementForGeometry(Geometry::Type GeomType) const
   }
   else
   {
+      if (error_mode == RETURN_NULL) { return nullptr; }
      MFEM_ABORT("H1 Pyramid basis functions are not yet supported "
                 "for order > 1.");
      return NULL;
@@ -2247,6 +2337,7 @@ L2_FECollection::FiniteElementForGeometry(Geometry::Type GeomType) const
   }
   else
   {
+      if (error_mode == RETURN_NULL) { return nullptr; }
      MFEM_ABORT("L2 Pyramid basis functions are not yet supported "
                 "for order > 0.");
      return NULL;
@@ -2502,6 +2593,7 @@ RT_FECollection::FiniteElementForGeometry(Geometry::Type GeomType) const
   }
   else
   {
+      if (error_mode == RETURN_NULL) { return nullptr; }
      MFEM_ABORT("RT Pyramid basis functions are not yet supported "
                 "for order > 0.");
      return NULL;
@@ -2787,6 +2879,7 @@ ND_FECollection::FiniteElementForGeometry(Geometry::Type GeomType) const
   }
   else
   {
+      if (error_mode == RETURN_NULL) { return nullptr; }
      MFEM_ABORT("ND Pyramid basis functions are not yet supported "
                 "for order > 1.");
      return NULL;
@@ -3389,6 +3482,7 @@ NURBSFECollection::FiniteElementForGeometry(Geometry::Type GeomType) const
      case Geometry::SQUARE:      return QuadrilateralFE;
      case Geometry::CUBE:        return ParallelepipedFE;
      default:
+         if (error_mode == RETURN_NULL) { return nullptr; }
         mfem_error ("NURBSFECollection: unknown geometry type.");
   }
   return SegmentFE; // Make some compilers happy
@@ -51,6 +51,14 @@ public:
   virtual const FiniteElement *
   FiniteElementForGeometry(Geometry::Type GeomType) const = 0;

+   /** @brief Returns the first non-NULL FiniteElement for the given dimension
+
+       @note Repeatedly calls FiniteElementForGeometry in the order defined in
+       the Geometry::Type enumeration.
+   */
+   virtual const FiniteElement *
+   FiniteElementForDim(int dim) const;
+
   virtual int DofForGeometry(Geometry::Type GeomType) const = 0;

   /** @brief Returns an array, say p, that maps a local permuted index i to a
@@ -66,6 +74,17 @@ public:

   virtual int GetContType() const = 0;

+   /** @note The following methods provide the same information as the
+       corresponding methods of the FiniteElement base class.
+       @{
+   */
+   virtual int GetRangeType(int dim) const;
+   virtual int GetDerivRangeType(int dim) const;
+   virtual int GetMapType(int dim) const;
+   virtual int GetDerivType(int dim) const;
+   virtual int GetDerivMapType(int dim) const;
+   /** @} */
+
   int HasFaceDofs(Geometry::Type geom, int p) const;

   virtual const FiniteElement *TraceFiniteElementForGeometry(
@@ -214,6 +233,19 @@ protected:
   void InitVarOrder(int p) const;

   mutable Array<FiniteElementCollection*> var_orders;
+
+   /// How to treat errors in FiniteElementForGeometry() calls.
+   enum ErrorMode
+   {
+      RETURN_NULL,      ///< Return NULL on errors
+      RAISE_MFEM_ERROR  /**< Raise an MFEM error (default in base class).
+                             Sub-classes can ignore this and return NULL. */
+   };
+
+   /// How to treat errors in FiniteElementForGeometry() calls.
+   /** The typical error in derived classes is that no FiniteElement is defined
+       for the given Geometry, or the input is not a valid Geometry. */
+   mutable ErrorMode error_mode = RAISE_MFEM_ERROR;
 };

 /// Arbitrary order H1-conforming (continuous) finite elements.
@@ -1958,7 +1958,7 @@ FiniteElementSpace::DerefinementOperator::DerefinementOperator(
         DenseMatrix &lM = localM[g](mi[s]);
         DenseMatrix &lR = localR[g](lR_offset+s);
         MultAtB(lP, lM, lR); // lR = lP^T lM
-         AddMult(lR, lP, lPtMP); // lPtMP += lP^T lM lP
+         mfem::AddMult(lR, lP, lPtMP); // lPtMP += lP^T lM lP
      }
      DenseMatrixInverse lPtMP_inv(lPtMP);
      for (int s = 0; s < nm; s++)
@@ -2005,7 +2005,7 @@ void FiniteElementSpace::DerefinementOperator
         x.GetSubVector(f_vdofs, loc_x);
         loc_x_mat.UseExternalData(loc_x.GetData(), f_vdofs.Size()/fine_vdim,
                                   fine_vdim);
-         AddMult(lR, loc_x_mat, loc_y_mat);
+         mfem::AddMult(lR, loc_x_mat, loc_y_mat);
      }
      y.SetSubVector(c_vdofs, loc_y);
   }
@@ -2056,10 +2056,7 @@ SparseMatrix* FiniteElementSpace::DerefinementMatrix(int old_ndofs,
      GetLocalDerefinementMatrices(elem_geoms[i], localR[elem_geoms[i]]);
   }

-   SparseMatrix *R = (elem_geoms.Size() != 1)
-                     ? new SparseMatrix(ndofs*vdim, old_ndofs*vdim) // variable row size
-                     : new SparseMatrix(ndofs*vdim, old_ndofs*vdim,
-                                        localR[elem_geoms[0]].SizeI());
+   SparseMatrix *R = new SparseMatrix(ndofs*vdim, old_ndofs*vdim);

   Array<int> mark(R->Height());
   mark = 0;
@@ -2069,6 +2066,7 @@ SparseMatrix* FiniteElementSpace::DerefinementMatrix(int old_ndofs,

   MFEM_ASSERT(dtrans.embeddings.Size() == old_elem_dof->Size(), "");

+   bool is_dg = FEColl()->GetContType() == FiniteElementCollection::DISCONTINUOUS;
   int num_marked = 0;
   for (int k = 0; k < dtrans.embeddings.Size(); k++)
   {
@@ -2091,10 +2089,11 @@ SparseMatrix* FiniteElementSpace::DerefinementMatrix(int old_ndofs,
            int r = DofToVDof(dofs[i], vd);
            int m = (r >= 0) ? r : (-1 - r);

-            if (!mark[m])
+            if (is_dg || !mark[m])
            {
               lR.GetRow(i, row);
               R->SetRow(r, old_vdofs, row);
+
               mark[m] = 1;
               num_marked++;
            }
@@ -2102,8 +2101,11 @@ SparseMatrix* FiniteElementSpace::DerefinementMatrix(int old_ndofs,
      }
   }

-   MFEM_VERIFY(num_marked == R->Height(),
-               "internal error: not all rows of R were set.");
+   if (!is_dg)
+   {
+      MFEM_VERIFY(num_marked == R->Height(),
+                  "internal error: not all rows of R were set.");
+   }

   R->Finalize(); // no-op if fixed width
   return R;
@@ -3145,7 +3147,7 @@ const FiniteElement *FiniteElementSpace::GetFaceElement(int i) const
         break;
      case 3:
      default:
-         fe = fec->FiniteElementForGeometry(mesh->GetFaceBaseGeometry(i));
+         fe = fec->FiniteElementForGeometry(mesh->GetFaceGeometry(i));
   }

   if (NURBSext)
@@ -38,7 +38,7 @@ public:
   /// Construct an empty finite element space hierarchy. This is useful if the
   /// hierarchy is constructed by coarsening a fine space, rather than refining
   /// a coarse space.
-   FiniteElementSpaceHierarchy() { }
+   FiniteElementSpaceHierarchy() = default;

   /// @brief Constructs a space hierarchy with the given mesh and space on the
   /// coarsest level.
@@ -91,6 +91,7 @@ public:
 class ParFiniteElementSpaceHierarchy : public FiniteElementSpaceHierarchy
 {
 public:
+   ParFiniteElementSpaceHierarchy() = default;
   /// @brief Constructs a parallel space hierarchy with the given mesh and spaces
   /// on level zero.
   /** The ownership of the mesh and space may be transferred to the
@@ -1920,7 +1920,7 @@ DataCollectionToFmsDataCollection(DataCollection *mfem_dc,
                                             FMS_NODAL_GAUSS_CLOSED, 1);
      err |= FmsDataCollectionAddField(*dc, "Coords", &fcoords);
      err |= FmsFieldSet(fcoords, fdcoords, mmesh->SpaceDimension(), FMS_BY_NODES,
-                         FMS_DOUBLE, mverts);
+                         FMS_DOUBLE, mverts.HostRead());
      err |= FmsComponentSetCoordinates(volume, fcoords);
   }

@@ -416,13 +416,24 @@ void GridFunction::GetNodalValues(int i, Array<double> &nval, int vdim) const

   if (FElem->GetRangeType() == FiniteElement::SCALAR)
   {
-      MFEM_ASSERT(FElem->GetMapType() == FiniteElement::VALUE,
-                  "invalid FE map type");
      Vector shape(dof);
-      for (k = 0; k < n; k++)
+      if (FElem->GetMapType() == FiniteElement::VALUE)
      {
-         FElem->CalcShape(ElemVert->IntPoint(k), shape);
-         nval[k] = shape * ((const double *)loc_data + dof * vdim);
+         for (k = 0; k < n; k++)
+         {
+            FElem->CalcShape(ElemVert->IntPoint(k), shape);
+            nval[k] = shape * (&loc_data[dof * vdim]);
+         }
+      }
+      else
+      {
+         ElementTransformation *Tr = fes->GetElementTransformation(i);
+         for (k = 0; k < n; k++)
+         {
+            Tr->SetIntPoint(&ElemVert->IntPoint(k));
+            FElem->CalcPhysShape(*Tr, shape);
+            nval[k] = shape * (&loc_data[dof * vdim]);
+         }
      }
   }
   else
@@ -495,7 +506,7 @@ void GridFunction::GetVectorValue(int i, const IntegrationPoint &ip,
      val.SetSize(vdim);
      for (int k = 0; k < vdim; k++)
      {
-         val(k) = shape * ((const double *)loc_data + dof * k);
+         val(k) = shape * (&loc_data[dof * k]);
      }
   }
   else
@@ -1027,7 +1038,7 @@ void GridFunction::GetVectorValue(ElementTransformation &T,
      val.SetSize(vdim);
      for (int k = 0; k < vdim; k++)
      {
-         val(k) = shape * ((const double *)loc_data + dof * k);
+         val(k) = shape * (&loc_data[dof * k]);
      }
   }
   else
@@ -1078,7 +1089,7 @@ void GridFunction::GetVectorValues(ElementTransformation &T,

         for (int k = 0; k < vdim; k++)
         {
-            vals(k,j) = shape * ((const double *)loc_data + dof * k);
+            vals(k,j) = shape * (&loc_data[dof * k]);
         }
      }
   }
@@ -1181,8 +1192,7 @@ void GridFunction::GetValuesFrom(const GridFunction &orig_func)
         orig_fe->CalcShape(ip, shape);
         for (d = 0; d < vdim; d++)
         {
-            loc_values(d*dof+j) =
-               shape * ((const double *)orig_loc_values + d * odof) ;
+            loc_values(d*dof+j) = shape * (&orig_loc_values[d * odof]);
         }
      }
      if (doftrans)
@@ -1224,8 +1234,7 @@ void GridFunction::GetBdrValuesFrom(const GridFunction &orig_func)
         orig_fe->CalcShape(ip, shape);
         for (d = 0; d < vdim; d++)
         {
-            loc_values(d*dof+j) =
-               shape * ((const double *)orig_loc_values + d * odof);
+            loc_values(d*dof+j) = shape * (&orig_loc_values[d * odof]);
         }
      }
      SetSubVector(vdofs, loc_values);
@@ -1432,21 +1441,13 @@ void GridFunction::GetDerivative(int comp, int der_comp, GridFunction &der)
   }
 }

-
 void GridFunction::GetVectorGradientHat(
   ElementTransformation &T, DenseMatrix &gh) const
 {
-   int elNo = T.ElementNo;
-   const FiniteElement *FElem = fes->GetFE(elNo);
+   const FiniteElement *FElem = fes->GetFE(T.ElementNo);
   int dim = FElem->GetDim(), dof = FElem->GetDof();
-   Array<int> vdofs;
-   DofTransformation * doftrans = fes->GetElementVDofs(elNo, vdofs);
   Vector loc_data;
-   GetSubVector(vdofs, loc_data);
-   if (doftrans)
-   {
-      doftrans->InvTransformPrimal(loc_data);
-   }
+   GetElementDofValues(T.ElementNo, loc_data);
   // assuming scalar FE
   int vdim = fes->GetVDim();
   DenseMatrix dshape(dof, dim);
@@ -1651,6 +1652,7 @@ void GridFunction::GetGradient(ElementTransformation &T, Vector &grad) const
         const FiniteElement *fe = fes->GetFE(T.ElementNo);
         MFEM_ASSERT(fe->GetMapType() == FiniteElement::VALUE,
                     "invalid FE map type");
+         MFEM_ASSERT(fes->GetVDim() == 1, "Defined for scalar functions.");
         int spaceDim = fes->GetMesh()->SpaceDimension();
         int dim = fe->GetDim(), dof = fe->GetDof();
         DenseMatrix dshape(dof, dim);
@@ -1719,13 +1721,8 @@ void GridFunction::GetGradients(ElementTransformation &tr,
   MFEM_ASSERT(fe->GetMapType() == FiniteElement::VALUE, "invalid FE map type");
   DenseMatrix dshape(fe->GetDof(), fe->GetDim());
   Vector lval, gh(fe->GetDim()), gcol;
-   Array<int> dofs;
-   DofTransformation * doftrans = fes->GetElementDofs(elNo, dofs);
-   GetSubVector(dofs, lval);
-   if (doftrans)
-   {
-      doftrans->InvTransformPrimal(lval);
-   }
+
+   GetElementDofValues(tr.ElementNo, lval);
   grad.SetSize(fe->GetDim(), ir.GetNPoints());
   for (int i = 0; i < ir.GetNPoints(); i++)
   {
@@ -4099,16 +4096,16 @@ void TensorProductLegendre(int dim,                // input
   // Map x to [0, 1] to use CalcLegendre since it uses shifted Legendre Polynomials.
   double x1 = (x(0) - xmin(0))/(xmax(0)-xmin(0)), x2, x3;
   Vector poly_x(order+1), poly_y(order+1), poly_z(order+1);
-   poly1d.CalcLegendre(order, x1, poly_x);
+   poly1d.CalcLegendre(order, x1, poly_x.GetData());
   if (dim > 1)
   {
      x2 = (x(1)-xmin(1))/(xmax(1)-xmin(1));
-      poly1d.CalcLegendre(order, x2, poly_y);
+      poly1d.CalcLegendre(order, x2, poly_y.GetData());
   }
   if (dim == 3)
   {
      x3 = (x(2)-xmin(2))/(xmax(2)-xmin(2));
-      poly1d.CalcLegendre(order, x3, poly_z);
+      poly1d.CalcLegendre(order, x3, poly_z.GetData());
   }

   int basis_dimension = static_cast<int>(pow(order+1,dim));
@@ -48,8 +48,6 @@ protected:

   void SaveSTLTri(std::ostream &out, double p1[], double p2[], double p3[]);

-   void GetVectorGradientHat(ElementTransformation &T, DenseMatrix &gh) const;
-
   // Project the delta coefficient without scaling and return the (local)
   // integral of the projection.
   void ProjectDeltaCoefficient(DeltaCoefficient &delta_coeff,
@@ -329,17 +327,34 @@ public:

   void GetCurl(ElementTransformation &tr, Vector &curl) const;

+   /** @brief Gradient of a scalar function at a quadrature point.
+
+       @note It is assumed that the IntegrationPoint of interest has been
+       specified by ElementTransformation::SetIntPoint() before calling
+       GetGradient().
+
+       @note Can be used from a ParGridFunction when @a tr is an
+       ElementTransformation of a face-neighbor element and face-neighbor data
+       has been exchanged. */
   void GetGradient(ElementTransformation &tr, Vector &grad) const;

+   /// Extension of GetGradient(...) for a collection of IntegrationPoints.
   void GetGradients(ElementTransformation &tr, const IntegrationRule &ir,
                     DenseMatrix &grad) const;

+   /// Extension of GetGradient(...) for a collection of IntegrationPoints.
   void GetGradients(const int elem, const IntegrationRule &ir,
                     DenseMatrix &grad) const
   { GetGradients(*fes->GetElementTransformation(elem), ir, grad); }

+   /** @brief Compute the vector gradient with respect to the physical element
+       variable. */
   void GetVectorGradient(ElementTransformation &tr, DenseMatrix &grad) const;

+   /** @brief Compute the vector gradient with respect to the reference element
+       variable. */
+   void GetVectorGradientHat(ElementTransformation &T, DenseMatrix &gh) const;
+
   /** Compute \f$ (\int_{\Omega} (*this) \psi_i)/(\int_{\Omega} \psi_i) \f$,
       where \f$ \psi_i \f$ are the basis functions for the FE space of avgs.
       Both FE spaces should be scalar and on the same mesh. */
@@ -37,7 +37,7 @@ FindPointsGSLIB::FindPointsGSLIB()
     fec_map_lin(NULL),
     fdata2D(NULL), fdata3D(NULL), cr(NULL), gsl_comm(NULL),
     dim(-1), points_cnt(0), setupflag(false), default_interp_value(0),
-     avgtype(AvgType::ARITHMETIC)
+     avgtype(AvgType::ARITHMETIC), bdr_tol(1e-8)
 {
   mesh_split.SetSize(4);
   ir_split.SetSize(4);
@@ -84,7 +84,7 @@ FindPointsGSLIB::FindPointsGSLIB(MPI_Comm comm_)
     fec_map_lin(NULL),
     fdata2D(NULL), fdata3D(NULL), cr(NULL), gsl_comm(NULL),
     dim(-1), points_cnt(0), setupflag(false), default_interp_value(0),
-     avgtype(AvgType::ARITHMETIC)
+     avgtype(AvgType::ARITHMETIC), bdr_tol(1e-8)
 {
   mesh_split.SetSize(4);
   ir_split.SetSize(4);
@@ -179,24 +179,27 @@ void FindPointsGSLIB::FindPoints(const Vector &point_pos,
   gsl_ref.SetSize(points_cnt * dim);
   gsl_dist.SetSize(points_cnt);

-   const double *xv_base[dim];
-   unsigned xv_stride[dim];
-   for (int d = 0; d < dim; d++)
+   auto xvFill = [&](const double *xv_base[], unsigned xv_stride[], int dim)
   {
-      if (point_pos_ordering == Ordering::byNODES)
+      for (int d = 0; d < dim; d++)
      {
-         xv_base[d] = point_pos.GetData() + d*points_cnt;
-         xv_stride[d] = sizeof(double);
+         if (point_pos_ordering == Ordering::byNODES)
+         {
+            xv_base[d] = point_pos.GetData() + d*points_cnt;
+            xv_stride[d] = sizeof(double);
+         }
+         else
+         {
+            xv_base[d] = point_pos.GetData() + d;
+            xv_stride[d] = dim*sizeof(double);
+         }
      }
-      else
-      {
-         xv_base[d] = point_pos.GetData() + d;
-         xv_stride[d] = dim*sizeof(double);
-      }
-   }
-
+   };
   if (dim == 2)
   {
+      const double *xv_base[2];
+      unsigned xv_stride[2];
+      xvFill(xv_base, xv_stride, dim);
      findpts_2(gsl_code.GetData(), sizeof(unsigned int),
                gsl_proc.GetData(), sizeof(unsigned int),
                gsl_elem.GetData(), sizeof(unsigned int),
@@ -204,8 +207,11 @@ void FindPointsGSLIB::FindPoints(const Vector &point_pos,
                gsl_dist.GetData(), sizeof(double),
                xv_base, xv_stride, points_cnt, fdata2D);
   }
-   else
+   else  // dim == 3
   {
+      const double *xv_base[3];
+      unsigned xv_stride[3];
+      xvFill(xv_base, xv_stride, dim);
      findpts_3(gsl_code.GetData(), sizeof(unsigned int),
                gsl_proc.GetData(), sizeof(unsigned int),
                gsl_elem.GetData(), sizeof(unsigned int),
@@ -217,10 +223,12 @@ void FindPointsGSLIB::FindPoints(const Vector &point_pos,
   // Set the element number and reference position to 0 for points not found
   for (int i = 0; i < points_cnt; i++)
   {
-      if (gsl_code[i] == 2)
+      if (gsl_code[i] == 2 ||
+          (gsl_code[i] == 1 && gsl_dist(i) > bdr_tol))
      {
         gsl_elem[i] = 0;
         for (int d = 0; d < dim; d++) { gsl_ref(i*dim + d) = -1.; }
+         gsl_code[i] = 2;
      }
   }

@@ -565,7 +573,7 @@ void FindPointsGSLIB::GetNodalValues(const GridFunction *gf_in,
   const int pts_el = std::pow(dof_1D, dim);
   const int pts_cnt = NE_split_total * pts_el;
   node_vals.SetSize(vdim * pts_cnt);
-   node_vals *= 0;
+   node_vals = 0.0;

   int gsl_mesh_pt_index = 0;

@@ -1080,7 +1088,7 @@ void FindPointsGSLIB::InterpolateGeneral(const GridFunction &field_in,

            sarray_transfer(struct send_pt, sendpt, proc, 1, cr);
            sdpt = (struct send_pt *)sendpt->ptr;
-            for (int index = 0; index < sendpt->n; index++)
+            for (int index = 0; index < static_cast<int>(sendpt->n); index++)
            {
               int idx = field_in.FESpace()->GetOrdering() == Ordering::byNODES ?
                         sdpt->index + j*nptorig :
@@ -1147,7 +1155,7 @@ void OversetFindPointsGSLIB::Setup(Mesh &m, const int meshid,
   distfint.SetSize(pts_cnt);
   if (!gfmax)
   {
-      distfint = 0.;
+      distfint = 0.0;
   }
   else
   {
@@ -1194,24 +1202,27 @@ void OversetFindPointsGSLIB::FindPoints(const Vector &point_pos,
   gsl_ref.SetSize(points_cnt * dim);
   gsl_dist.SetSize(points_cnt);

-   const double *xv_base[dim];
-   unsigned xv_stride[dim];
-   for (int d = 0; d < dim; d++)
+   auto xvFill = [&](const double *xv_base[], unsigned xv_stride[], int dim)
   {
-      if (point_pos_ordering == Ordering::byNODES)
+      for (int d = 0; d < dim; d++)
      {
-         xv_base[d] = point_pos.GetData() + d*points_cnt;
-         xv_stride[d] = sizeof(double);
+         if (point_pos_ordering == Ordering::byNODES)
+         {
+            xv_base[d] = point_pos.GetData() + d*points_cnt;
+            xv_stride[d] = sizeof(double);
+         }
+         else
+         {
+            xv_base[d] = point_pos.GetData() + d;
+            xv_stride[d] = dim*sizeof(double);
+         }
      }
-      else
-      {
-         xv_base[d] = point_pos.GetData() + d;
-         xv_stride[d] = dim*sizeof(double);
-      }
-   }
-
+   };
   if (dim == 2)
   {
+      const double *xv_base[2];
+      unsigned xv_stride[2];
+      xvFill(xv_base, xv_stride, dim);
      findptsms_2(gsl_code.GetData(), sizeof(unsigned int),
                  gsl_proc.GetData(), sizeof(unsigned int),
                  gsl_elem.GetData(), sizeof(unsigned int),
@@ -1221,8 +1232,11 @@ void OversetFindPointsGSLIB::FindPoints(const Vector &point_pos,
                  point_id.GetData(), sizeof(unsigned int), &match,
                  points_cnt, fdata2D);
   }
-   else
+   else  // dim == 3
   {
+      const double *xv_base[3];
+      unsigned xv_stride[3];
+      xvFill(xv_base, xv_stride, dim);
      findptsms_3(gsl_code.GetData(), sizeof(unsigned int),
                  gsl_proc.GetData(), sizeof(unsigned int),
                  gsl_elem.GetData(), sizeof(unsigned int),
@@ -1236,10 +1250,12 @@ void OversetFindPointsGSLIB::FindPoints(const Vector &point_pos,
   // Set the element number and reference position to 0 for points not found
   for (int i = 0; i < points_cnt; i++)
   {
-      if (gsl_code[i] == 2)
+      if (gsl_code[i] == 2 ||
+          (gsl_code[i] == 1 && gsl_dist(i) > bdr_tol))
      {
         gsl_elem[i] = 0;
         for (int d = 0; d < dim; d++) { gsl_ref(i*dim + d) = -1.; }
+         gsl_code[i] = 2;
      }
   }

@@ -38,6 +38,11 @@ namespace mfem
 *     coordinates inside the element that each point is located in. gslib also
 *     returns a code that indicates whether the point was found inside an
 *     element, on element border, or not found in the domain.
+ *     For points returned as found on `element border`, the point is either
+ *     on an element edge/face or near the domain boundary, and gslib also
+ *     returns a distance to the border. Points near (but outside) the domain
+ *     boundary must then be marked as not found using the distance returned
+ *     by gslib.
 *
 *  3. Interpolate - Interpolates any grid function at the points found using 2.
 *
@@ -70,6 +75,8 @@ protected:
   Array<int> split_element_map;
   Array<int> split_element_index;
   int        NE_split_total;
+   // Tolerance to ignore points just outside elements at the boundary.
+   double     bdr_tol;

   /// Use GSLIB for communication and interpolation
   virtual void InterpolateH1(const GridFunction &field_in, Vector &field_out);
@@ -181,6 +188,14 @@ public:
      default_interp_value = interp_value_;
   }

+   /// Set the tolerance for detecting points outside the 'curvilinear' boundary
+   /// that gslib may return as found on the boundary. Points found on boundary
+   /// with distance greater than @ bdr_tol are marked as not found.
+   virtual void SetDistanceToleranceForPointsFoundOnBoundary(double bdr_tol_)
+   {
+      bdr_tol = bdr_tol_;
+   }
+
   /** Cleans up memory allocated internally by gslib.
       Note that in parallel, this must be called before MPI_Finalize(), as it
       calls MPI_Comm_free() for internal gslib communicators. */
@@ -1032,12 +1032,12 @@ void WhiteGaussianNoiseDomainLFIntegrator::AssembleRHSElementVect
      massinteg.AssembleElementMatrix(el, Tr, *M);
      CholeskyFactors chol(M->Data());
      chol.Factor(M->Height());
-      chol.LMult(n,1,elvect);
+      chol.LMult(n,1,elvect.GetData());
   }
   else
   {
      CholeskyFactors chol(L[iel]->Data());
-      chol.LMult(n,1,elvect);
+      chol.LMult(n,1,elvect.GetData());
   }
 }

@@ -14,41 +14,26 @@
 namespace mfem
 {

-Multigrid::Multigrid()
-   : cycleType(CycleType::VCYCLE), preSmoothingSteps(1), postSmoothingSteps(1)
+MultigridBase::MultigridBase()
+   : cycleType(CycleType::VCYCLE), preSmoothingSteps(1), postSmoothingSteps(1),
+     nrhs(0)
 {}

-Multigrid::Multigrid(const Array<Operator*>& operators_,
-                     const Array<Solver*>& smoothers_,
-                     const Array<Operator*>& prolongations_,
-                     const Array<bool>& ownedOperators_,
-                     const Array<bool>& ownedSmoothers_,
-                     const Array<bool>& ownedProlongations_)
-   : Solver(operators_.Last()->NumRows()), cycleType(CycleType::VCYCLE),
-     preSmoothingSteps(1), postSmoothingSteps(1),
-     X(operators_.Size()), Y(X.Size()), R(X.Size()), Z(X.Size())
+MultigridBase::MultigridBase(const Array<Operator*>& operators_,
+                             const Array<Solver*>& smoothers_,
+                             const Array<bool>& ownedOperators_,
+                             const Array<bool>& ownedSmoothers_)
+   : Solver(operators_.Last()->Height(), operators_.Last()->Width()),
+     cycleType(CycleType::VCYCLE), preSmoothingSteps(1), postSmoothingSteps(1),
+     nrhs(0)
 {
   operators_.Copy(operators);
   smoothers_.Copy(smoothers);
-   prolongations_.Copy(prolongations);
   ownedOperators_.Copy(ownedOperators);
   ownedSmoothers_.Copy(ownedSmoothers);
-   ownedProlongations_.Copy(ownedProlongations);
-
-   for (int level = 0; level < operators.Size(); ++level)
-   {
-      X[level] = new Vector(operators[level]->NumRows());
-      *X[level] = 0.0;
-      Y[level] = new Vector(operators[level]->NumRows());
-      *Y[level] = 0.0;
-      R[level] = new Vector(operators[level]->NumRows());
-      *R[level] = 0.0;
-      Z[level] = new Vector(operators[level]->NumRows());
-      *Z[level] = 0.0;
-   }
 }

-Multigrid::~Multigrid()
+MultigridBase::~MultigridBase()
 {
   for (int i = 0; i < operators.Size(); ++i)
   {
@@ -60,12 +45,210 @@ Multigrid::~Multigrid()
      {
         delete smoothers[i];
      }
-      delete X[i];
-      delete Y[i];
-      delete R[i];
-      delete Z[i];
+   }
+   EraseVectors();
+}
+
+void MultigridBase::InitVectors() const
+{
+   if (X.NumRows() > 0 && X.NumCols() > 0) { EraseVectors(); }
+   const int M = NumLevels();
+   X.SetSize(M, nrhs);
+   Y.SetSize(M, nrhs);
+   R.SetSize(M, nrhs);
+   Z.SetSize(M, nrhs);
+   for (int i = 0; i < X.NumRows(); ++i)
+   {
+      const int n = operators[i]->Height();
+      for (int j = 0; j < X.NumCols(); ++j)
+      {
+         X(i, j) = new Vector(n);
+         Y(i, j) = new Vector(n);
+         R(i, j) = new Vector(n);
+         Z(i, j) = new Vector(n);
+      }
+   }
+}
+
+void MultigridBase::EraseVectors() const
+{
+   for (int i = 0; i < X.NumRows(); ++i)
+   {
+      for (int j = 0; j < X.NumCols(); ++j)
+      {
+         delete X(i, j);
+         delete Y(i, j);
+         delete R(i, j);
+         delete Z(i, j);
+      }
+   }
+}
+
+void MultigridBase::AddLevel(Operator* op, Solver* smoother,
+                             bool ownOperator, bool ownSmoother)
+{
+   height = op->Height();
+   width = op->Width();
+   operators.Append(op);
+   smoothers.Append(smoother);
+   ownedOperators.Append(ownOperator);
+   ownedSmoothers.Append(ownSmoother);
+}
+
+void MultigridBase::SetCycleType(CycleType cycleType_, int preSmoothingSteps_,
+                                 int postSmoothingSteps_)
+{
+   cycleType = cycleType_;
+   preSmoothingSteps = preSmoothingSteps_;
+   postSmoothingSteps = postSmoothingSteps_;
+}
+
+void MultigridBase::Mult(const Vector& x, Vector& y) const
+{
+   Array<const Vector*> X_(1);
+   Array<Vector*> Y_(1);
+   X_[0] = &x;
+   Y_[0] = &y;
+   ArrayMult(X_, Y_);
+}
+
+void MultigridBase::ArrayMult(const Array<const Vector*>& X_,
+                              Array<Vector*>& Y_) const
+{
+   MFEM_ASSERT(operators.Size() > 0,
+               "Multigrid solver does not have operators set!");
+   MFEM_ASSERT(X_.Size() == Y_.Size(),
+               "Number of columns mismatch in MultigridBase::Mult!");
+   if (iterative_mode)
+   {
+      MFEM_WARNING("Multigrid solver does not use iterative_mode and ignores "
+                   "the initial guess!");
   }

+   // Add capacity as necessary
+   nrhs = X_.Size();
+   if (X.NumCols() < nrhs) { InitVectors(); }
+
+   // Perform a single cycle
+   const int M = NumLevels();
+   for (int j = 0; j < nrhs; ++j)
+   {
+      MFEM_ASSERT(X_[j] && Y_[j], "Missing Vector in MultigridBase::Mult!");
+      *X(M - 1, j) = *X_[j];
+      *Y(M - 1, j) = 0.0;
+   }
+   Cycle(M - 1);
+   for (int j = 0; j < nrhs; ++j)
+   {
+      *Y_[j] = *Y(M - 1, j);
+   }
+}
+
+void MultigridBase::SmoothingStep(int level, bool zero, bool transpose) const
+{
+   // y = y + S (x - A y) or y = y + S^T (x - A y)
+   if (zero)
+   {
+      Array<Vector *> X_(X[level], nrhs), Y_(Y[level], nrhs);
+      GetSmootherAtLevel(level)->ArrayMult(X_, Y_);
+   }
+   else
+   {
+      Array<Vector *> Y_(Y[level], nrhs), R_(R[level], nrhs),
+            Z_(Z[level], nrhs);
+      for (int j = 0; j < nrhs; ++j)
+      {
+         *R_[j] = *X(level, j);
+      }
+      GetOperatorAtLevel(level)->ArrayAddMult(Y_, R_, -1.0);
+      if (transpose)
+      {
+         GetSmootherAtLevel(level)->ArrayMultTranspose(R_, Z_);
+      }
+      else
+      {
+         GetSmootherAtLevel(level)->ArrayMult(R_, Z_);
+      }
+      for (int j = 0; j < nrhs; ++j)
+      {
+         *Y_[j] += *Z_[j];
+      }
+   }
+}
+
+void MultigridBase::Cycle(int level) const
+{
+   // Coarse solve
+   if (level == 0)
+   {
+      SmoothingStep(0, true, false);
+      return;
+   }
+
+   // Pre-smooth
+   for (int i = 0; i < preSmoothingSteps; ++i)
+   {
+      SmoothingStep(level, (cycleType == CycleType::VCYCLE && i == 0), false);
+   }
+
+   // Compute residual and restrict
+   {
+      Array<Vector *> Y_(Y[level], nrhs), R_(R[level], nrhs),
+            X_(X[level - 1], nrhs);
+      for (int j = 0; j < nrhs; ++j)
+      {
+         *R_[j] = *X(level, j);
+      }
+      GetOperatorAtLevel(level)->ArrayAddMult(Y_, R_, -1.0);
+      GetProlongationAtLevel(level - 1)->ArrayMultTranspose(R_, X_);
+      for (int j = 0; j < nrhs; ++j)
+      {
+         *Y(level - 1, j) = 0.0;
+      }
+   }
+
+   // Corrections
+   Cycle(level - 1);
+   if (cycleType == CycleType::WCYCLE)
+   {
+      Cycle(level - 1);
+   }
+
+   // Prolongate and add
+   {
+      Array<Vector *> Y_(Y[level - 1], nrhs), Z_(Z[level], nrhs);
+      GetProlongationAtLevel(level - 1)->ArrayMult(Y_, Z_);
+      for (int j = 0; j < nrhs; ++j)
+      {
+         *Y(level, j) += *Z_[j];
+      }
+   }
+
+   // Post-smooth
+   for (int i = 0; i < postSmoothingSteps; ++i)
+   {
+      SmoothingStep(level, false, true);
+   }
+}
+
+Multigrid::Multigrid()
+   : MultigridBase()
+{}
+
+Multigrid::Multigrid(const Array<Operator*>& operators_,
+                     const Array<Solver*>& smoothers_,
+                     const Array<Operator*>& prolongations_,
+                     const Array<bool>& ownedOperators_,
+                     const Array<bool>& ownedSmoothers_,
+                     const Array<bool>& ownedProlongations_)
+   : MultigridBase(operators_, smoothers_, ownedOperators_, ownedSmoothers_)
+{
+   prolongations_.Copy(prolongations);
+   ownedProlongations_.Copy(ownedProlongations);
+}
+
+Multigrid::~Multigrid()
+{
   for (int i = 0; i < prolongations.Size(); ++i)
   {
      if (ownedProlongations[i])
@@ -73,158 +256,12 @@ Multigrid::~Multigrid()
         delete prolongations[i];
      }
   }
-
-   operators.DeleteAll();
-   smoothers.DeleteAll();
-   prolongations.DeleteAll();
-   X.DeleteAll();
-   Y.DeleteAll();
-   R.DeleteAll();
-   Z.DeleteAll();
 }

-void Multigrid::AddLevel(Operator* opr, Solver* smoother, bool ownOperator,
-                         bool ownSmoother)
-{
-   operators.Append(opr);
-   smoothers.Append(smoother);
-   ownedOperators.Append(ownOperator);
-   ownedSmoothers.Append(ownSmoother);
-   width = opr->Width();
-   height = opr->Height();
-
-   X.Append(new Vector(height));
-   *X.Last() = 0.0;
-   Y.Append(new Vector(height));
-   *Y.Last() = 0.0;
-   R.Append(new Vector(height));
-   *R.Last() = 0.0;
-   Z.Append(new Vector(height));
-   *Z.Last() = 0.0;
-}
-
-int Multigrid::NumLevels() const { return operators.Size(); }
-
-int Multigrid::GetFinestLevelIndex() const { return NumLevels() - 1; }
-
-const Operator* Multigrid::GetOperatorAtLevel(int level) const
-{
-   return operators[level];
-}
-
-Operator* Multigrid::GetOperatorAtLevel(int level)
-{
-   return operators[level];
-}
-
-const Operator* Multigrid::GetOperatorAtFinestLevel() const
-{
-   return GetOperatorAtLevel(operators.Size() - 1);
-}
-
-Operator* Multigrid::GetOperatorAtFinestLevel()
-{
-   return GetOperatorAtLevel(operators.Size() - 1);
-}
-
-Solver* Multigrid::GetSmootherAtLevel(int level) const
-{
-   return smoothers[level];
-}
-
-Solver* Multigrid::GetSmootherAtLevel(int level)
-{
-   return smoothers[level];
-}
-
-void Multigrid::SetCycleType(CycleType cycleType_, int preSmoothingSteps_,
-                             int postSmoothingSteps_)
-{
-   cycleType = cycleType_;
-   preSmoothingSteps = preSmoothingSteps_;
-   postSmoothingSteps = postSmoothingSteps_;
-}
-
-void Multigrid::Mult(const Vector& x, Vector& y) const
-{
-   MFEM_ASSERT(NumLevels() > 0, "");
-   *X.Last() = x;
-   *Y.Last() = 0.0;
-   Cycle(GetFinestLevelIndex());
-   y = *Y.Last();
-}
-
-void Multigrid::SetOperator(const Operator& op)
-{
-   MFEM_ABORT("SetOperator not supported in Multigrid");
-}
-
-void Multigrid::SmoothingStep(int level, bool transpose) const
-{
-   GetOperatorAtLevel(level)->Mult(*Y[level], *R[level]); // r = A x
-   subtract(*X[level], *R[level], *R[level]);             // r = b - A x
-   if (transpose)
-   {
-      GetSmootherAtLevel(level)->MultTranspose(*R[level], *Z[level]); // z = S r
-   }
-   else
-   {
-      GetSmootherAtLevel(level)->Mult(*R[level], *Z[level]); // z = S r
-   }
-   add(*Y[level], 1.0, *Z[level], *Y[level]);             // x = x + S (b - A x)
-}
-
-void Multigrid::Cycle(int level) const
-{
-   if (level == 0)
-   {
-      GetSmootherAtLevel(level)->Mult(*X[level], *Y[level]);
-      return;
-   }
-
-   for (int i = 0; i < preSmoothingSteps; i++)
-   {
-      SmoothingStep(level, false);
-   }
-
-   // Compute residual
-   GetOperatorAtLevel(level)->Mult(*Y[level], *R[level]);
-   subtract(*X[level], *R[level], *R[level]);
-
-   // Restrict residual
-   GetProlongationAtLevel(level - 1)->MultTranspose(*R[level], *X[level - 1]);
-
-   // Init zeros
-   *Y[level - 1] = 0.0;
-
-   // Corrections
-   int corrections = 1;
-   if (cycleType == CycleType::WCYCLE)
-   {
-      corrections = 2;
-   }
-   for (int correction = 0; correction < corrections; ++correction)
-   {
-      Cycle(level - 1);
-   }
-
-   // Prolongate
-   GetProlongationAtLevel(level - 1)->Mult(*Y[level - 1], *R[level]);
-
-   // Add update
-   *Y[level] += *R[level];
-
-   // Post-smooth
-   for (int i = 0; i < postSmoothingSteps; i++)
-   {
-      SmoothingStep(level, true);
-   }
-}
-
-const Operator* Multigrid::GetProlongationAtLevel(int level) const
-{
-   return prolongations[level];
-}
+GeometricMultigrid::
+GeometricMultigrid(const FiniteElementSpaceHierarchy& fespaces_)
+   : MultigridBase(), fespaces(fespaces_)
+{}

 GeometricMultigrid::~GeometricMultigrid()
 {
@@ -232,15 +269,10 @@ GeometricMultigrid::~GeometricMultigrid()
   {
      delete bfs[i];
   }
-
-   bfs.DeleteAll();
-
   for (int i = 0; i < essentialTrueDofs.Size(); ++i)
   {
      delete essentialTrueDofs[i];
   }
-
-   essentialTrueDofs.DeleteAll();
 }

 void GeometricMultigrid::FormFineLinearSystem(Vector& x, Vector& b,
@@ -256,9 +288,4 @@ void GeometricMultigrid::RecoverFineFEMSolution(const Vector& X,
   bfs.Last()->RecoverFEMSolution(X, b, x);
 }

-const Operator* GeometricMultigrid::GetProlongationAtLevel(int level) const
-{
-   return fespaces.GetProlongationAtLevel(level);
-}
-
 } // namespace mfem
@@ -21,8 +21,8 @@
 namespace mfem
 {

-/// Multigrid solver class
-class Multigrid : public Solver
+/// Abstract base class for Multigrid solvers
+class MultigridBase : public Solver
 {
 public:
   enum class CycleType
@@ -34,65 +34,72 @@ public:
 protected:
   Array<Operator*> operators;
   Array<Solver*> smoothers;
-   Array<Operator*> prolongations;
-
   Array<bool> ownedOperators;
   Array<bool> ownedSmoothers;
-   Array<bool> ownedProlongations;

   CycleType cycleType;
   int preSmoothingSteps;
   int postSmoothingSteps;

-   mutable Array<Vector*> X;
-   mutable Array<Vector*> Y;
-   mutable Array<Vector*> R;
-   mutable Array<Vector*> Z;
+   mutable Array2D<Vector*> X, Y, R, Z;
+   mutable int nrhs;

 public:
-   /// Constructs an empty multigrid hierarchy.
-   Multigrid();
+   /// Constructs an empty multigrid hierarchy
+   MultigridBase();

-   /// Constructs a multigrid hierarchy from the given inputs.
-   /** Inputs include operators and smoothers on all levels, prolongation
-       operators that go from coarser to finer levels, and ownership of the
-       given operators, smoothers, and prolongations. */
-   Multigrid(const Array<Operator*>& operators_, const Array<Solver*>& smoothers_,
-             const Array<Operator*>& prolongations_, const Array<bool>& ownedOperators_,
-             const Array<bool>& ownedSmoothers_, const Array<bool>& ownedProlongations_);
+   /// Constructs a multigrid hierarchy from the given inputs
+   /** Inputs include operators and smoothers on all levels, and ownership of
+       the given operators and smoothers */
+   MultigridBase(const Array<Operator*>& operators_,
+                 const Array<Solver*>& smoothers_,
+                 const Array<bool>& ownedOperators_,
+                 const Array<bool>& ownedSmoothers_);

   /// Destructor
-   virtual ~Multigrid();
+   virtual ~MultigridBase();

-   /// Adds a level to the multigrid operator hierarchy.
+   /// Adds a level to the multigrid operator hierarchy
   /** The ownership of the operators and solvers/smoothers may be transferred
-       to the Multigrid by setting the according boolean variables. */
-   void AddLevel(Operator* opr, Solver* smoother, bool ownOperator,
+       to the Multigrid by setting the according boolean variables */
+   void AddLevel(Operator* op, Solver* smoother, bool ownOperator,
                 bool ownSmoother);

   /// Returns the number of levels
-   int NumLevels() const;
+   int NumLevels() const { return operators.Size(); }

   /// Returns the index of the finest level
-   int GetFinestLevelIndex() const;
+   int GetFinestLevelIndex() const { return NumLevels() - 1; }

   /// Returns operator at given level
-   const Operator* GetOperatorAtLevel(int level) const;
-
-   /// Returns operator at given level
-   Operator* GetOperatorAtLevel(int level);
+   const Operator* GetOperatorAtLevel(int level) const
+   {
+      return operators[level];
+   }
+   Operator* GetOperatorAtLevel(int level)
+   {
+      return operators[level];
+   }

   /// Returns operator at finest level
-   const Operator* GetOperatorAtFinestLevel() const;
-
-   /// Returns operator at finest level
-   Operator* GetOperatorAtFinestLevel();
+   const Operator* GetOperatorAtFinestLevel() const
+   {
+      return GetOperatorAtLevel(GetFinestLevelIndex());
+   }
+   Operator* GetOperatorAtFinestLevel()
+   {
+      return GetOperatorAtLevel(GetFinestLevelIndex());
+   }

   /// Returns smoother at given level
-   Solver* GetSmootherAtLevel(int level) const;
-
-   /// Returns smoother at given level
-   Solver* GetSmootherAtLevel(int level);
+   const Solver* GetSmootherAtLevel(int level) const
+   {
+      return smoothers[level];
+   }
+   Solver* GetSmootherAtLevel(int level)
+   {
+      return smoothers[level];
+   }

   /// Set cycle type and number of pre- and post-smoothing steps used by Mult
   void SetCycleType(CycleType cycleType_, int preSmoothingSteps_,
@@ -100,23 +107,62 @@ public:

   /// Application of the multigrid as a preconditioner
   virtual void Mult(const Vector& x, Vector& y) const override;
+   virtual void ArrayMult(const Array<const Vector*>& X_,
+                          Array<Vector*>& Y_) const override;

   /// Not supported for multigrid
-   virtual void SetOperator(const Operator& op) override;
+   virtual void SetOperator(const Operator& op) override
+   {
+      MFEM_ABORT("SetOperator is not supported in Multigrid!");
+   }

 private:
-   /// Application of a smoothing step at particular level
-   void SmoothingStep(int level, bool transpose) const;
-
   /// Application of a multigrid cycle at particular level
   void Cycle(int level) const;

+   /// Application of a pre-/post-smoothing step at particular level
+   void SmoothingStep(int level, bool zero, bool transpose) const;
+
+   /// Allocate or destroy temporary storage
+   void InitVectors() const;
+   void EraseVectors() const;
+
   /// Returns prolongation operator at given level
-   virtual const Operator* GetProlongationAtLevel(int level) const;
+   virtual const Operator* GetProlongationAtLevel(int level) const = 0;
+};
+
+/// Multigrid solver class
+class Multigrid : public MultigridBase
+{
+protected:
+   Array<Operator*> prolongations;
+   Array<bool> ownedProlongations;
+
+public:
+   /// Constructs an empty multigrid hierarchy
+   Multigrid();
+
+   /// Constructs a multigrid hierarchy from the given inputs
+   /** Inputs include operators and smoothers on all levels, prolongation
+       operators that go from coarser to finer levels, and ownership of the
+       given operators, smoothers, and prolongations */
+   Multigrid(const Array<Operator*>& operators_, const Array<Solver*>& smoothers_,
+             const Array<Operator*>& prolongations_, const Array<bool>& ownedOperators_,
+             const Array<bool>& ownedSmoothers_, const Array<bool>& ownedProlongations_);
+
+   /// Destructor
+   virtual ~Multigrid();
+
+private:
+   /// Returns prolongation operator at given level
+   virtual const Operator* GetProlongationAtLevel(int level) const override
+   {
+      return prolongations[level];
+   }
 };

 /// Geometric multigrid associated with a hierarchy of finite element spaces
-class GeometricMultigrid : public Multigrid
+class GeometricMultigrid : public MultigridBase
 {
 protected:
   const FiniteElementSpaceHierarchy& fespaces;
@@ -126,8 +172,7 @@ protected:
 public:
   /** Construct an empty multigrid object for the given finite element space
       hierarchy @a fespaces_ */
-   GeometricMultigrid(const FiniteElementSpaceHierarchy& fespaces_)
-      : Multigrid(), fespaces(fespaces_) { }
+   GeometricMultigrid(const FiniteElementSpaceHierarchy& fespaces_);

   /// Destructor
   virtual ~GeometricMultigrid();
@@ -142,7 +187,10 @@ public:

 private:
   /// Returns prolongation operator at given level
-   virtual const Operator* GetProlongationAtLevel(int level) const override;
+   virtual const Operator* GetProlongationAtLevel(int level) const override
+   {
+      return fespaces.GetProlongationAtLevel(level);
+   }
 };

 } // namespace mfem
@@ -347,14 +347,12 @@ ParallelEliminateEssentialBC(const Array<int> &bdr_attr_is_ess,
 void ParBilinearForm::TrueAddMult(const Vector &x, Vector &y, const double a)
 const
 {
-   if (Xaux.ParFESpace() != pfes)
-   {
-      Xaux.SetSpace(pfes);
-      Yaux.SetSpace(pfes);
-      Ytmp.SetSize(pfes->GetTrueVSize());
-   }
+   const Operator *P = pfes->GetProlongationMatrix();
+   Xaux.SetSize(P->Height());
+   Yaux.SetSize(P->Height());
+   Ytmp.SetSize(P->Width());

-   Xaux.Distribute(&x);
+   P->Mult(x, Xaux);
   if (ext)
   {
      ext->Mult(Xaux, Yaux);
@@ -366,8 +364,8 @@ const
                  " implemented");
      mat->Mult(Xaux, Yaux);
   }
-   pfes->GetProlongationMatrix()->MultTranspose(Yaux, Ytmp);
-   y.Add(a,Ytmp);
+   P->MultTranspose(Yaux, Ytmp);
+   y.Add(a, Ytmp);
 }

 void ParBilinearForm::FormLinearSystem(
@@ -31,9 +31,8 @@ class ParBilinearForm : public BilinearForm
 protected:
   ParFiniteElementSpace *pfes; ///< Points to the same object as #fes

-   /// Auxiliary objects used in TrueAddMult().
-   mutable ParGridFunction Xaux, Yaux;
-   mutable Vector Ytmp;
+   /// Auxiliary vectors used in TrueAddMult(): L-, L-, and T-vector, resp.
+   mutable Vector Xaux, Yaux, Ytmp;

   OperatorHandle p_mat, p_mat_e;

@@ -1536,7 +1536,7 @@ const FiniteElement *ParFiniteElementSpace::GetFaceNbrFaceFE(int i) const
   // Works in tandem with GetFaceNbrFaceVDofs() defined above.

   MFEM_ASSERT(Nonconforming() && !NURBSext, "");
-   Geometry::Type face_geom = pmesh->GetFaceGeometryType(i);
+   Geometry::Type face_geom = pmesh->GetFaceGeometry(i);
   return fec->FiniteElementForGeometry(face_geom);
 }

@@ -2606,7 +2606,7 @@ int ParFiniteElementSpace
      if (dump < 10)
      {
         char fname[100];
-         sprintf(fname, "dofs%02d.txt", MyRank);
+         snprintf(fname, 100, "dofs%02d.txt", MyRank);
         std::ofstream f(fname);
         DebugDumpDOFs(f, deps, dof_group, dof_owner, finalized);
         dump++;
@@ -3025,6 +3025,8 @@ ParFiniteElementSpace::ParallelDerefinementMatrix(int old_ndofs,
   Array<char> mark(diag->Height());
   mark = 0;

+   bool is_dg = FEColl()->GetContType() == FiniteElementCollection::DISCONTINUOUS;
+
   for (int k = 0; k < dtrans.embeddings.Size(); k++)
   {
      const Embedding &emb = dtrans.embeddings[k];
@@ -3053,7 +3055,7 @@ ParFiniteElementSpace::ParallelDerefinementMatrix(int old_ndofs,
               int r = DofToVDof(dofs[i], vd);
               int m = (r >= 0) ? r : (-1 - r);

-               if (!mark[m])
+               if (is_dg || !mark[m])
               {
                  lR.GetRow(i, row);
                  diag->SetRow(r, old_vdofs, row);
@@ -3105,7 +3107,7 @@ ParFiniteElementSpace::ParallelDerefinementMatrix(int old_ndofs,
               int r = DofToVDof(dofs[i], vd);
               int m = (r >= 0) ? r : (-1 - r);

-               if (!mark[m])
+               if (is_dg || !mark[m])
               {
                  lR.GetRow(i, row);
                  MFEM_ASSERT(ldof[geom] == row.Size(), "");
@@ -3122,6 +3124,7 @@ ParFiniteElementSpace::ParallelDerefinementMatrix(int old_ndofs,
         }
      }
   }
+
   messages.clear();
   offd->Finalize(0);
   offd->SetWidth(col_map.size());
@@ -151,14 +151,14 @@ void ParGridFunction::ParallelAverage(Vector &tv) const
 {
   MFEM_VERIFY(pfes->Conforming(), "not implemented for NC meshes");
   pfes->GetProlongationMatrix()->MultTranspose(*this, tv);
-   pfes->DivideByGroupSize(tv);
+   pfes->DivideByGroupSize(tv.HostReadWrite());
 }

 void ParGridFunction::ParallelAverage(HypreParVector &tv) const
 {
   MFEM_VERIFY(pfes->Conforming(), "not implemented for NC meshes");
   pfes->GetProlongationMatrix()->MultTranspose(*this, tv);
-   pfes->DivideByGroupSize(tv);
+   pfes->DivideByGroupSize(tv.HostReadWrite());
 }

 HypreParVector *ParGridFunction::ParallelAverage() const
@@ -353,7 +353,7 @@ void ParGridFunction::GetVectorValue(int i, const IntegrationPoint &ip,
         val.SetSize(vdim);
         for (int k = 0; k < vdim; k++)
         {
-            val(k) = shape * ((const double *)loc_data + dof * k);
+            val(k) = shape * (&loc_data[dof * k]);
         }
      }
      else
@@ -468,7 +468,7 @@ void ParGridFunction::GetVectorValue(ElementTransformation &T,
      val.SetSize(vdim);
      for (int k = 0; k < vdim; k++)
      {
-         val(k) = shape * ((const double *)loc_data + dof * k);
+         val(k) = shape * (&loc_data[dof * k]);
      }
   }
   else
@@ -644,9 +644,9 @@ void ParGridFunction::ProjectBdrCoefficient(

   // Count the values globally.
   GroupCommunicator &gcomm = pfes->GroupComm();
-   gcomm.Reduce<int>(values_counter, GroupCommunicator::Sum);
+   gcomm.Reduce<int>(values_counter.HostReadWrite(), GroupCommunicator::Sum);
   // Accumulate the values globally.
-   gcomm.Reduce<double>(values, GroupCommunicator::Sum);
+   gcomm.Reduce<double>(values.HostReadWrite(), GroupCommunicator::Sum);
   // Only the values in the master are guaranteed to be correct!
   for (int i = 0; i < values.Size(); i++)
   {
@@ -682,9 +682,9 @@ void ParGridFunction::ProjectBdrCoefficientTangent(VectorCoefficient &vcoeff,

   // Count the values globally.
   GroupCommunicator &gcomm = pfes->GroupComm();
-   gcomm.Reduce<int>(values_counter, GroupCommunicator::Sum);
+   gcomm.Reduce<int>(values_counter.HostReadWrite(), GroupCommunicator::Sum);
   // Accumulate the values globally.
-   gcomm.Reduce<double>(values, GroupCommunicator::Sum);
+   gcomm.Reduce<double>(values.HostReadWrite(), GroupCommunicator::Sum);
   // Only the values in the master are guaranteed to be correct!
   for (int i = 0; i < values.Size(); i++)
   {
@@ -1109,11 +1109,11 @@ void ParGridFunction::ComputeFlux(
   SumFluxAndCount(blfi, flux, count, wcoef, subdomain);

   // Accumulate flux and counts in parallel
-   ffes->GroupComm().Reduce<double>(flux, GroupCommunicator::Sum);
-   ffes->GroupComm().Bcast<double>(flux);
+   ffes->GroupComm().Reduce<double>(flux.HostReadWrite(), GroupCommunicator::Sum);
+   ffes->GroupComm().Bcast<double>(flux.HostReadWrite());

-   ffes->GroupComm().Reduce<int>(count, GroupCommunicator::Sum);
-   ffes->GroupComm().Bcast<int>(count);
+   ffes->GroupComm().Reduce<int>(count.HostReadWrite(), GroupCommunicator::Sum);
+   ffes->GroupComm().Bcast<int>(count.HostReadWrite());

   // complete averaging
   for (int i = 0; i < count.Size(); i++)
@@ -91,8 +91,10 @@ void ParNCH1FaceRestriction::NonconformingInterpolation(Vector& y) const
   });
 }

-void ParNCH1FaceRestriction::AddMultTranspose(const Vector &x, Vector &y) const
+void ParNCH1FaceRestriction::AddMultTranspose(const Vector &x, Vector &y,
+                                              const double a) const
 {
+   MFEM_VERIFY(a == 1.0, "General coefficient case is not yet supported!");
   if (nf==0) { return; }
   NonconformingTransposeInterpolation(x);
   H1FaceRestriction::AddMultTranspose(x_interp, y);
@@ -774,8 +776,10 @@ void ParNCL2FaceRestriction::Mult(const Vector& x, Vector& y) const
   }
 }

-void ParNCL2FaceRestriction::AddMultTranspose(const Vector &x, Vector &y) const
+void ParNCL2FaceRestriction::AddMultTranspose(const Vector &x, Vector &y,
+                                              const double a) const
 {
+   MFEM_VERIFY(a == 1.0, "General coefficient case is not yet supported!");
   if (nf==0) { return; }
   if (type==FaceType::Interior)
   {
@@ -65,8 +65,10 @@ public:
                     requested by @a type in the constructor.
                     The face_dofs should be ordered according to the given
                     ElementDofOrdering.
-       @param[in,out] y The L-vector degrees of freedom. */
-   void AddMultTranspose(const Vector &x, Vector &y) const override;
+       @param[in,out] y The L-vector degrees of freedom.
+       @param[in]  a Scalar coefficient for addition. */
+   void AddMultTranspose(const Vector &x, Vector &y,
+                         const double a = 1.0) const override;

   /** @brief Gather the degrees of freedom, i.e. goes from face E-Vector to
       L-Vector.
@@ -302,8 +304,10 @@ public:
                     requested by @a type in the constructor.
                     The face_dofs should be ordered according to the given
                     ElementDofOrdering
-       @param[in,out] y The L-vector degrees of freedom. */
-   void AddMultTranspose(const Vector &x, Vector &y) const override;
+       @param[in,out] y The L-vector degrees of freedom.
+       @param[in]  a Scalar coefficient for addition. */
+   void AddMultTranspose(const Vector &x, Vector &y,
+                         const double a = 1.0) const override;

   /** @brief Gather the degrees of freedom, i.e. goes from face E-Vector to
       L-Vector.
@@ -44,31 +44,6 @@ QuadratureFunction::QuadratureFunction(Mesh *mesh, std::istream &in)
   Load(in, vdim*qspace->GetSize());
 }

-void QuadratureFunction::SetSpace(QuadratureSpaceBase *qspace_, int vdim_)
-{
-   if (qspace_ != qspace)
-   {
-      if (own_qspace) { delete qspace; }
-      qspace = qspace_;
-      own_qspace = false;
-   }
-   vdim = (vdim_ < 0) ? vdim : vdim_;
-   SetSize(vdim*qspace->GetSize());
-}
-
-void QuadratureFunction::SetSpace(
-   QuadratureSpaceBase *qspace_, double *qf_data, int vdim_)
-{
-   if (qspace_ != qspace)
-   {
-      if (own_qspace) { delete qspace; }
-      qspace = qspace_;
-      own_qspace = false;
-   }
-   vdim = (vdim_ < 0) ? vdim : vdim_;
-   NewDataAndSize(qf_data, vdim*qspace->GetSize());
-}
-
 void QuadratureFunction::Save(std::ostream &os) const
 {
   GetSpace()->Save(os);
@@ -47,6 +47,17 @@ public:
   QuadratureFunction(QuadratureSpaceBase *qspace_, int vdim_ = 1)
      : QuadratureFunction(*qspace_, vdim_) { }

+   /** @brief Create a QuadratureFunction based on the given QuadratureSpaceBase,
+       using the external (host) data, @a qf_data. */
+   /** The QuadratureFunction does not assume ownership of the
+       QuadratureSpaceBase or the external data.
+       @warning @a qspace_ may not be NULL.
+       @note @a qf_data must be a valid **host** pointer (see the constructor
+       Vector::Vector(double *, int)). */
+   QuadratureFunction(QuadratureSpaceBase *qspace_, double *qf_data, int vdim_ = 1)
+      : Vector(qf_data, vdim_*qspace_->GetSize()),
+        qspace(qspace_), own_qspace(false), vdim(vdim_) { }
+
   /** @brief Copy constructor. The QuadratureSpace ownership flag, #own_qspace,
       in the new object is set to false. */
   QuadratureFunction(const QuadratureFunction &orig)
@@ -262,6 +273,34 @@ inline void QuadratureFunction::GetValues(
   }
 }

+
+inline void QuadratureFunction::SetSpace(QuadratureSpaceBase *qspace_,
+                                         int vdim_)
+{
+   if (qspace_ != qspace)
+   {
+      if (own_qspace) { delete qspace; }
+      qspace = qspace_;
+      own_qspace = false;
+   }
+   vdim = (vdim_ < 0) ? vdim : vdim_;
+   SetSize(vdim*qspace->GetSize());
+}
+
+inline void QuadratureFunction::SetSpace(
+   QuadratureSpaceBase *qspace_, double *qf_data, int vdim_)
+{
+   if (qspace_ != qspace)
+   {
+      if (own_qspace) { delete qspace; }
+      qspace = qspace_;
+      own_qspace = false;
+   }
+   vdim = (vdim_ < 0) ? vdim : vdim_;
+   NewDataAndSize(qf_data, vdim*qspace->GetSize());
+}
+
+
 } // namespace mfem

 #endif
@@ -16,7 +16,7 @@ namespace mfem

 QuadratureSpaceBase::QuadratureSpaceBase(Mesh &mesh_, Geometry::Type geom,
                                         const IntegrationRule &ir)
-   : mesh(mesh_)
+   : mesh(mesh_), order(ir.GetOrder())
 {
   for (int g = 0; g < Geometry::NumGeom; g++)
   {
@@ -588,7 +588,7 @@ void FaceQuadratureInterpolator::Mult(
   const int vdim = fespace->GetVDim();
   const int dim = fespace->GetMesh()->Dimension();
   const FiniteElement *fe =
-      fespace->GetTraceElement(0, fespace->GetMesh()->GetFaceBaseGeometry(0));
+      fespace->GetTraceElement(0, fespace->GetMesh()->GetFaceGeometry(0));
   const IntegrationRule *ir = IntRule;
   const DofToQuad &maps = fe->GetDofToQuad(*ir, DofToQuad::TENSOR);
   const int nd1d = maps.ndof;
@@ -148,7 +148,7 @@ void ElementRestriction::MultUnsigned(const Vector& x, Vector& y) const
 }

 template <bool ADD>
-void ElementRestriction::AddMultTranspose(const Vector& x, Vector& y) const
+void ElementRestriction::TAddMultTranspose(const Vector& x, Vector& y) const
 {
   // Assumes all elements have the same number of dofs
   const int nd = dof;
@@ -180,13 +180,15 @@ void ElementRestriction::AddMultTranspose(const Vector& x, Vector& y) const
 void ElementRestriction::MultTranspose(const Vector& x, Vector& y) const
 {
   constexpr bool ADD = false;
-   AddMultTranspose<ADD>(x, y);
+   TAddMultTranspose<ADD>(x, y);
 }

-void ElementRestriction::AddMultTranspose(const Vector& x, Vector& y) const
+void ElementRestriction::AddMultTranspose(const Vector& x, Vector& y,
+                                          const double a) const
 {
+   MFEM_VERIFY(a == 1.0, "General coefficient case is not yet supported!");
   constexpr bool ADD = true;
-   AddMultTranspose<ADD>(x, y);
+   TAddMultTranspose<ADD>(x, y);
 }

 void ElementRestriction::MultTransposeUnsigned(const Vector& x, Vector& y) const
@@ -521,7 +523,7 @@ void L2ElementRestriction::Mult(const Vector &x, Vector &y) const
 }

 template <bool ADD>
-void L2ElementRestriction::AddMultTranspose(const Vector &x, Vector &y) const
+void L2ElementRestriction::TAddMultTranspose(const Vector &x, Vector &y) const
 {
   const int nd = ndof;
   const int vd = vdim;
@@ -544,13 +546,15 @@ void L2ElementRestriction::AddMultTranspose(const Vector &x, Vector &y) const
 void L2ElementRestriction::MultTranspose(const Vector &x, Vector &y) const
 {
   constexpr bool ADD = false;
-   AddMultTranspose<ADD>(x, y);
+   TAddMultTranspose<ADD>(x, y);
 }

-void L2ElementRestriction::AddMultTranspose(const Vector &x, Vector &y) const
+void L2ElementRestriction::AddMultTranspose(const Vector &x, Vector &y,
+                                            const double a) const
 {
+   MFEM_VERIFY(a == 1.0, "General coefficient case is not yet supported!");
   constexpr bool ADD = true;
-   AddMultTranspose<ADD>(x, y);
+   TAddMultTranspose<ADD>(x, y);
 }

 void L2ElementRestriction::FillI(SparseMatrix &mat) const
@@ -760,8 +764,10 @@ void H1FaceRestriction::Mult(const Vector& x, Vector& y) const
   });
 }

-void H1FaceRestriction::AddMultTranspose(const Vector& x, Vector& y) const
+void H1FaceRestriction::AddMultTranspose(const Vector& x, Vector& y,
+                                         const double a) const
 {
+   MFEM_VERIFY(a == 1.0, "General coefficient case is not yet supported!");
   if (nf==0) { return; }
   // Assumes all elements have the same number of dofs
   const int nface_dofs = face_dofs;
@@ -1105,7 +1111,7 @@ L2FaceRestriction::L2FaceRestriction(const FiniteElementSpace &fes,
     vdim(fes.GetVDim()),
     byvdim(fes.GetOrdering() == Ordering::byVDIM),
     face_dofs(nf > 0 ?
-               fes.GetTraceElement(0, fes.GetMesh()->GetFaceBaseGeometry(0))->GetDof()
+               fes.GetTraceElement(0, fes.GetMesh()->GetFaceGeometry(0))->GetDof()
               : 0),
     elem_dofs(fes.GetFE(0)->GetDof()),
     nfdofs(nf*face_dofs),
@@ -1266,8 +1272,10 @@ void L2FaceRestriction::DoubleValuedConformingAddMultTranspose(
   });
 }

-void L2FaceRestriction::AddMultTranspose(const Vector& x, Vector& y) const
+void L2FaceRestriction::AddMultTranspose(const Vector& x, Vector& y,
+                                         const double a) const
 {
+   MFEM_VERIFY(a == 1.0, "General coefficient case is not yet supported!");
   if (nf==0) { return; }
   if (m == L2FaceValues::DoubleValued)
   {
@@ -1420,7 +1428,7 @@ void L2FaceRestriction::CheckFESpace(const ElementDofOrdering e_ordering)
      for (int f = 0; f < fes.GetNF(); ++f)
      {
         const FiniteElement *fe =
-            fes.GetTraceElement(f, fes.GetMesh()->GetFaceBaseGeometry(f));
+            fes.GetTraceElement(f, fes.GetMesh()->GetFaceGeometry(f));
         const TensorBasisElement* el =
            dynamic_cast<const TensorBasisElement*>(fe);
         if (el) { continue; }
@@ -1780,7 +1788,7 @@ void InterpolationManager::LinearizeInterpolatorMapIntoVector()
 {
   // Assumes all trace elements are the same.
   const FiniteElement *trace_fe =
-      fes.GetTraceElement(0, fes.GetMesh()->GetFaceBaseGeometry(0));
+      fes.GetTraceElement(0, fes.GetMesh()->GetFaceGeometry(0));
   const int face_dofs = trace_fe->GetDof();
   const int nc_size = interp_map.size();
   MFEM_VERIFY(nc_cpt==nc_size, "Unexpected number of interpolators.");
@@ -2050,8 +2058,10 @@ void NCL2FaceRestriction::DoubleValuedNonconformingTransposeInterpolationInPlace
   });
 }

-void NCL2FaceRestriction::AddMultTranspose(const Vector& x, Vector& y) const
+void NCL2FaceRestriction::AddMultTranspose(const Vector& x, Vector& y,
+                                           const double a) const
 {
+   MFEM_VERIFY(a == 1.0, "General coefficient case is not yet supported!");
   if (nf==0) { return; }
   if (type==FaceType::Interior)
   {
@@ -27,7 +27,8 @@ class ElementRestrictionOperator : public Operator
 public:
   /// @brief Add the E-vector degrees of freedom @a x to the L-vector degrees
   /// of freedom @a y.
-   virtual void AddMultTranspose(const Vector &x, Vector &y) const = 0;
+   void AddMultTranspose(const Vector &x, Vector &y,
+                         const double a = 1.0) const override = 0;
 };

 /// Operator that converts FiniteElementSpace L-vectors to E-vectors.
@@ -65,9 +66,10 @@ protected:

 public:
   ElementRestriction(const FiniteElementSpace&, ElementDofOrdering);
-   void Mult(const Vector &x, Vector &y) const;
-   void MultTranspose(const Vector &x, Vector &y) const;
-   void AddMultTranspose(const Vector &x, Vector &y) const;
+   void Mult(const Vector &x, Vector &y) const override;
+   void MultTranspose(const Vector &x, Vector &y) const override;
+   void AddMultTranspose(const Vector &x, Vector &y,
+                         const double a = 1.0) const override;

   /// Compute Mult without applying signs based on DOF orientations.
   void MultUnsigned(const Vector &x, Vector &y) const;
@@ -99,7 +101,7 @@ public:
   ///
   /// Performs either MultTranspose or AddMultTranspose depending on the
   /// boolean template parameter @a ADD.
-   template <bool ADD> void AddMultTranspose(const Vector &x, Vector &y) const;
+   template <bool ADD> void TAddMultTranspose(const Vector &x, Vector &y) const;
 };

 /// Operator that converts L2 FiniteElementSpace L-vectors to E-vectors.
@@ -116,9 +118,10 @@ class L2ElementRestriction : public ElementRestrictionOperator
   const int ndofs;
 public:
   L2ElementRestriction(const FiniteElementSpace&);
-   void Mult(const Vector &x, Vector &y) const;
-   void MultTranspose(const Vector &x, Vector &y) const;
-   void AddMultTranspose(const Vector &x, Vector &y) const;
+   void Mult(const Vector &x, Vector &y) const override;
+   void MultTranspose(const Vector &x, Vector &y) const override;
+   void AddMultTranspose(const Vector &x, Vector &y,
+                         const double a = 1.0) const override;
   /** Fill the I array of SparseMatrix corresponding to the sparsity pattern
       given by this ElementRestriction. */
   void FillI(SparseMatrix &mat) const;
@@ -129,7 +132,7 @@ public:
   ///
   /// Performs either MultTranspose or AddMultTranspose depending on the
   /// boolean template parameter @a ADD.
-   template <bool ADD> void AddMultTranspose(const Vector &x, Vector &y) const;
+   template <bool ADD> void TAddMultTranspose(const Vector &x, Vector &y) const;
 };

 /** An enum type to specify if only e1 value is requested (SingleValued) or both
@@ -180,8 +183,10 @@ public:
       @param[in]     x The face degrees of freedom on the face.
       @param[in,out] y The L-vector of degrees of freedom to which we add the
                        face degrees of freedom.
+       @param[in]     a Scalar coefficient for addition.
   */
-   virtual void AddMultTranspose(const Vector &x, Vector &y) const = 0;
+   virtual void AddMultTranspose(const Vector &x, Vector &y,
+                                 const double a = 1.0) const override = 0;

   /** @brief Add the face degrees of freedom @a x to the element degrees of
       freedom @a y. Perform the same computation as AddMultTranspose, but
@@ -277,8 +282,10 @@ public:
                     requested by @a type in the constructor.
                     The face_dofs should be ordered according to the given
                     ElementDofOrdering
-       @param[in,out] y The L-vector degrees of freedom. */
-   void AddMultTranspose(const Vector &x, Vector &y) const override;
+       @param[in,out] y The L-vector degrees of freedom.
+       @param[in]  a Scalar coefficient for addition. */
+   void AddMultTranspose(const Vector &x, Vector &y,
+                         const double a = 1.0) const override;

 private:
   /** @brief Compute the scatter indices: L-vector to E-vector, and the offsets
@@ -409,8 +416,10 @@ public:
                     requested by @a type in the constructor.
                     The face_dofs should be ordered according to the given
                     ElementDofOrdering
-       @param[in,out] y The L-vector degrees of freedom. */
-   void AddMultTranspose(const Vector &x, Vector &y) const override;
+       @param[in,out] y The L-vector degrees of freedom.
+       @param[in]  a Scalar coefficient for addition. */
+   void AddMultTranspose(const Vector &x, Vector &y,
+                         const double a = 1.0) const override;

   /** @brief Fill the I array of SparseMatrix corresponding to the sparsity
       pattern given by this L2FaceRestriction.
@@ -831,8 +840,10 @@ public:
                     requested by @a type in the constructor.
                     The face_dofs should be ordered according to the given
                     ElementDofOrdering
-       @param[in,out] y The L-vector degrees of freedom. */
-   void AddMultTranspose(const Vector &x, Vector &y) const override;
+       @param[in,out] y The L-vector degrees of freedom.
+       @param[in]  a Scalar coefficient for addition. */
+   void AddMultTranspose(const Vector &x, Vector &y,
+                         const double a = 1.0) const override;

   /** @brief Gather the degrees of freedom, i.e. goes from face E-Vector to
       L-Vector.
@@ -349,7 +349,7 @@ void StaticCondensation::ReduceRHS(const Vector &b, Vector &sc_b) const

      LUFactors lu(const_cast<double*>((const double*)A_data) + A_offsets[i],
                   const_cast<int*>((const int*)A_ipiv) + A_ipiv_offsets[i]);
-      lu.LSolve(npd, 1, b_p);
+      lu.LSolve(npd, 1, b_p.GetData());

      if (symm)
      {
@@ -527,8 +527,9 @@ void StaticCondensation::ComputeSolution(

      LUFactors lu(const_cast<double*>((const double*)A_data) + A_offsets[i],
                   const_cast<int*>((const int*)A_ipiv) + A_ipiv_offsets[i]);
-      lu.LSolve(npd, 1, b_p);
-      lu.BlockBackSolve(npd, ned, 1, lu.data + npd*npd, s_e, b_p);
+      lu.LSolve(npd, 1, b_p.GetData());
+      lu.BlockBackSolve(npd, ned, 1, lu.data + npd*npd, s_e.GetData(),
+                        b_p.GetData());

      for (int j = 0; j < npd; j++)
      {
@@ -615,6 +615,7 @@ public:
         solFES.VectorAssemble(y_dof.layout, y_dof, solVecLayoutLoc, y);
      }
   }
+   using Operator::AddMult;
 };

 } // namespace mfem
--- a/Show More
+++ b/Show More