Merge master in lor-gpu

Merge pull request #3199 from mfem/mfem-4.5-dev
Final changes for mfem-4.5
2022-10-23 09:54:23 -07:00 · 2022-10-22 15:33:35 -07:00 · 2022-10-22 15:10:18 -07:00 · 2022-10-21 16:58:01 -07:00 · 2022-10-21 16:57:53 -07:00 · 2022-10-21 16:50:22 -07:00
253 changed files with 17432 additions and 3661 deletions
@@ -23,9 +23,8 @@ install:
 - set MSMPI_LIB64=C:\Program Files (x86)\Microsoft SDKs\MPI\Lib\x64
 - set MSMPI_INC=C:\Program Files (x86)\Microsoft SDKs\MPI\Include

-# Install METIS, use a mirror because the original source server is not always
-# up. Original url:
-#    http://glaros.dtc.umn.edu/gkhome/fetch/sw/metis/metis-5.1.0.tar.gz
+# Install METIS, use MFEM's mirror because the original source server is often
+# down and we don't support yet the new repo https://github.com/KarypisLab/METIS
 - ps: Start-FileDownload 'https://mfem.github.io/tpls/metis-5.1.0.tar.gz'
 - 7z x metis-5.1.0.tar.gz -so | 7z x -si -ttar > nul
 - cd metis-5.1.0
@@ -1,4 +1,4 @@
-name: Build Deploy Container
+name: "Docker"

 on:

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

 # In this CI section, we build different variants of mfem and run test on them.
-name: builds-and-tests
+name: "Tests"

 # Github actions can use the default "GITHUB_TOKEN". By default, this token
 # is set to have permissive access. However, this is not a good practice
@@ -47,17 +47,17 @@ jobs:
  builds-and-tests:
    strategy:
      matrix:
-        os: [ubuntu-20.04, macos-10.15, windows-2022]
+        os: [ubuntu-latest, macos-latest, windows-latest]
        target: [dbg, opt]
        mpi: [seq, par]
        build-system: [make, cmake]
        hypre-target: [int32]
        exclude:
-          - os: ubuntu-20.04
+          - os: ubuntu-latest
            build-system: cmake
-          - os: macos-10.15
+          - os: macos-latest
            build-system: cmake
-          - os: windows-2022
+          - os: windows-latest
            build-system: make
        # 'include' allows us to:
        # - Add a variable to all jobs without creating a new matrix dimension.
@@ -72,15 +72,15 @@ jobs:
            codecov: NO
          - target: opt
            codecov: YES
-          - os: windows-2022
+          - os: windows-latest
            codecov: NO
-          - os: ubuntu-20.04
+          - os: ubuntu-latest
            target: opt
            codecov: NO
            mpi: par
            build-system: cmake
            hypre-target: int32
-          - os: ubuntu-20.04
+          - os: ubuntu-latest
            target: opt
            codecov: NO
            mpi: par
@@ -112,35 +112,35 @@ jobs:
    # TODO: It would be nice to have only one step, e.g. with a dedicated
    # action, but I (@adrienbernede) don't see how at the moment.
    - name: get MPI (Linux)
-      if: matrix.mpi == 'par' && matrix.os == 'ubuntu-20.04'
+      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-20.04'
+      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-10.15'
+      if: ( matrix.mpi == 'par' || matrix.codecov == 'YES' ) && matrix.os == 'macos-latest'
      uses: Homebrew/actions/setup-homebrew@c4aafe8c4620bf08883dd4679c374f11e73329d3

    - name: get MPI (MacOS)
-      if: matrix.mpi == 'par' && matrix.os == 'macos-10.15'
+      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)
-      if: matrix.codecov == 'YES' && matrix.os == 'macos-10.15'
+      if: matrix.codecov == 'YES' && matrix.os == 'macos-latest'
      run: |
        export HOMEBREW_NO_INSTALL_CLEANUP=1
        brew install lcov

    - name: get MPI (Windows)
-      if: matrix.mpi == 'par' && matrix.os == 'windows-2022'
+      if: matrix.mpi == 'par' && matrix.os == 'windows-latest'
      uses: mpi4py/setup-mpi@v1.0.3

    # Get Hypre through cache, or build it.
@@ -154,7 +154,7 @@ jobs:
        key: ${{ runner.os }}-build-${{ env.HYPRE_TOP_DIR }}-${{ matrix.hypre-target }}-v2.2

    - name: get hypre
-      if: matrix.mpi == 'par' && steps.hypre-cache.outputs.cache-hit != 'true' && matrix.os != 'windows-2022'
+      if: matrix.mpi == 'par' && steps.hypre-cache.outputs.cache-hit != 'true' && matrix.os != 'windows-latest'
      uses: mfem/github-actions/build-hypre@v2.2
      with:
        archive: ${{ env.HYPRE_ARCHIVE }}
@@ -163,7 +163,7 @@ jobs:
        build-system: make

    - name: get hypre (Windows)
-      if: matrix.mpi == 'par' && steps.hypre-cache.outputs.cache-hit != 'true' && matrix.os == 'windows-2022'
+      if: matrix.mpi == 'par' && steps.hypre-cache.outputs.cache-hit != 'true' && matrix.os == 'windows-latest'
      uses: mfem/github-actions/build-hypre@v2.2
      with:
        archive: ${{ env.HYPRE_ARCHIVE }}
@@ -175,14 +175,14 @@ jobs:
    # Install will only run on cache miss.
    - name: cache metis
      id: metis-cache
-      if: matrix.mpi == 'par' && matrix.os != 'windows-2022'
+      if: matrix.mpi == 'par' && matrix.os != 'windows-latest'
      uses: actions/cache@v2
      with:
        path: ${{ env.METIS_TOP_DIR }}
        key: ${{ runner.os }}-build-${{ env.METIS_TOP_DIR }}-v2.2

    - name: install metis
-      if: matrix.mpi == 'par' && matrix.os != 'windows-2022' && steps.metis-cache.outputs.cache-hit != 'true'
+      if: matrix.mpi == 'par' && matrix.os != 'windows-latest' && steps.metis-cache.outputs.cache-hit != 'true'
      uses: mfem/github-actions/build-metis@v2.2
      with:
        archive: ${{ env.METIS_ARCHIVE }}
@@ -196,20 +196,16 @@ jobs:
        key: ${{ runner.os }}-${{ matrix.mpi }}-vcpkg-v1

    - name: prepare binary cache location
-      if: matrix.os == 'windows-2022' && steps.vcpkg-cache.outputs.cache-hit != 'true'
+      if: matrix.os == 'windows-latest' && steps.vcpkg-cache.outputs.cache-hit != 'true'
      run: |
        mkdir -p vcpkg_cache

    - name: install metis (Windows)
-      if: matrix.mpi == 'par' && matrix.os == 'windows-2022'
+      if: matrix.mpi == 'par' && matrix.os == 'windows-latest'
      env:
        VCPKG_DEFAULT_BINARY_CACHE: ${{ github.workspace }}/vcpkg_cache
      run: |
-        $PortFile = 'C:\vcpkg\ports\metis\portfile.cmake'
-        $OriginalURL = 'http://glaros.dtc.umn.edu/gkhome/fetch/sw/metis/metis-${METIS_VERSION}.tar.gz'
-        $NewURL = 'https://github.com/mfem/tpls/raw/gh-pages/metis-5.1.0.tar.gz'
-        (Get-Content $PortFile).replace($OriginalURL, $NewURL) | Set-Content $PortFile
-        vcpkg install metis --triplet=x64-windows-static
+        vcpkg install metis-mfem --triplet=x64-windows-static --overlay-ports=${{ env.MFEM_TOP_DIR }}/config/vcpkg/ports

    # MFEM build and test
    - name: build
@@ -252,7 +248,7 @@ jobs:
      shell: bash

    - name: cmake unit tests (Ubuntu 20.04)
-      if: matrix.build-system == 'cmake' && matrix.target == 'opt' && matrix.os == 'ubuntu-20.04'
+      if: matrix.build-system == 'cmake' && matrix.target == 'opt' && matrix.os == 'ubuntu-latest'
      run: |
        CTEST_CONFIG="Release"
        [[ ${{ matrix.target }} == 'dbg' ]] && CTEST_CONFIG="Debug"
@@ -260,7 +256,7 @@ jobs:
      shell: bash

    - name: cmake tests
-      if: matrix.build-system == 'cmake' && matrix.target == 'opt' && matrix.os != 'ubuntu-20.04'
+      if: matrix.build-system == 'cmake' && matrix.target == 'opt' && matrix.os != 'ubuntu-latest'
      run: |
        CTEST_CONFIG="Release"
        cd ${{ env.MFEM_TOP_DIR }}/build && ctest --output-on-failure -C ${CTEST_CONFIG}
@@ -0,0 +1,70 @@
+# For most projects, this workflow file will not need changing; you simply need
+# to commit it to your repository.
+#
+# You may wish to alter this file to override the set of languages analyzed,
+# or to provide custom queries or build logic.
+#
+# ******** NOTE ********
+# We have attempted to detect the languages in your repository. Please check
+# the `language` matrix defined below to confirm you have the correct set of
+# supported CodeQL languages.
+#
+name: "Static Analysis"
+
+on:
+  push:
+    branches: [ "master", "next"]
+  pull_request:
+    # The branches below must be a subset of the branches above
+    branches: [ "master" ]
+
+jobs:
+  analyze:
+    name: Analyze
+    runs-on: ubuntu-latest
+    permissions:
+      actions: read
+      contents: read
+      security-events: write
+
+    strategy:
+      fail-fast: false
+      matrix:
+        language: [ 'cpp' ]
+        # CodeQL supports [ 'cpp', 'csharp', 'go', 'java', 'javascript', 'python', 'ruby' ]
+        # Learn more about CodeQL language support at https://aka.ms/codeql-docs/language-support
+
+    steps:
+    - name: Checkout repository
+      uses: actions/checkout@v3
+
+    # Initializes the CodeQL tools for scanning.
+    - name: Initialize CodeQL
+      uses: github/codeql-action/init@v2
+      with:
+        languages: ${{ matrix.language }}
+        # If you wish to specify custom queries, you can do so here or in a config file.
+        # By default, queries listed here will override any specified in a config file.
+        # Prefix the list here with "+" to use these queries and those in the config file.
+
+        # Details on CodeQL's query packs refer to : https://docs.github.com/en/code-security/code-scanning/automatically-scanning-your-code-for-vulnerabilities-and-errors/configuring-code-scanning#using-queries-in-ql-packs
+        # queries: security-extended,security-and-quality
+
+
+    # Autobuild attempts to build any compiled languages  (C/C++, C#, or Java).
+    # If this step fails, then you should remove it and run the build manually (see below)
+    - name: Autobuild
+      uses: github/codeql-action/autobuild@v2
+
+    # ℹ️ Command-line programs to run using the OS shell.
+    # 📚 See https://docs.github.com/en/actions/using-workflows/workflow-syntax-for-github-actions#jobsjob_idstepsrun
+
+    #   If the Autobuild fails above, remove it and uncomment the following three lines.
+    #   modify them (or add more) to build your code if your project, please refer to the EXAMPLE below for guidance.
+
+    # - run: |
+    #   echo "Run, Build Application using script"
+    #   ./location_of_script_within_repo/buildscript.sh
+
+    - name: Perform CodeQL Analysis
+      uses: github/codeql-action/analyze@v2
@@ -9,7 +9,7 @@
 # terms of the BSD-3 license. We welcome feedback and contributions, see file
 # CONTRIBUTING.md for details.

-name: build-analysis
+name: "Build Analysis"

 permissions:
  actions: write
@@ -31,7 +31,7 @@ env:

 jobs:
  gitignore:
-    runs-on: ubuntu-18.04
+    runs-on: ubuntu-latest

    steps:
    - name: Cancel Previous Runs
@@ -9,7 +9,7 @@
 # terms of the BSD-3 license. We welcome feedback and contributions, see file
 # CONTRIBUTING.md for details.

-name: repo-check
+name: "Checks"

 permissions:
  actions: write
@@ -28,7 +28,7 @@ on:

 jobs:
  file-headers-check:
-    runs-on: ubuntu-18.04
+    runs-on: ubuntu-latest
    if: |
      (github.event_name == 'push' ||
       github.event.pull_request.head.repo.full_name != github.repository)
@@ -78,7 +78,7 @@ jobs:
        exit 1

  code-style:
-    runs-on: ubuntu-18.04
+    runs-on: ubuntu-latest
    if: |
      (github.event_name == 'push' ||
       github.event.pull_request.head.repo.full_name != github.repository)
@@ -88,14 +88,14 @@ jobs:

    - name: get astyle
      run: |
-        sudo apt-get install astyle=3.1-1ubuntu2
+        sudo apt-get install astyle

    - name: style check
      run: |
        ./config/githooks/pre-push --style

  documentation:
-    runs-on: ubuntu-18.04
+    runs-on: ubuntu-latest
    if: |
      (github.event_name == 'push' ||
       github.event.pull_request.head.repo.full_name != github.repository)
@@ -106,6 +106,8 @@ jobs:
    - name: get doxygen and graphviz
      run: |
        sudo apt-get install doxygen graphviz
+        cd doc
+        doxygen -u CodeDocumentation.conf.in

    - name: build documentation
      run: |
@@ -118,7 +120,7 @@ jobs:
      github.ref != 'refs/heads/master' &&
      (github.event_name == 'push' ||
       github.event.pull_request.head.repo.full_name != github.repository)
-    runs-on: ubuntu-18.04
+    runs-on: ubuntu-latest
    steps:
    - name: checkout mfem
      uses: actions/checkout@v2
@@ -18,6 +18,11 @@ CMakeFiles/
 # Backup files
 *~

+*.sqlite
+*.nsys-rep
+*.qdstrm
+*.csv
+
 # Default install location
 /mfem/

@@ -275,6 +280,7 @@ miniapps/tools/load-dc
 miniapps/tools/convert-dc
 miniapps/tools/lor-transfer
 miniapps/tools/get-values
+miniapps/tools/check-tmop-metric

 miniapps/toys/automata
 miniapps/toys/life
@@ -307,6 +313,7 @@ miniapps/solvers/sol.*
 miniapps/parelag/MultilevelHcurlHdivSolver
 miniapps/parelag/*.mesh

+miniapps/multidomain/multidomain
 miniapps/hooke/hooke

 # Unit test binary and outputs
@@ -327,6 +334,7 @@ tests/benchmarks/bench_ceed
 tests/benchmarks/bench_tmop
 tests/benchmarks/bench_vector
 tests/benchmarks/bench_virtuals
+tests/benchmarks/bench_lor

 # Test script output
 tests/scripts/*.err
@@ -8,31 +8,48 @@
                               https://mfem.org


-Version 4.4.1 (development)
-===========================
+Version 4.5, released on October 22, 2022
+=========================================

 Meshing improvements
 --------------------
+- Added new SubMesh and ParSubMesh classes that can be used to extract a subset
+  of a given Mesh. These classes have the same functionality as Mesh and ParMesh
+  and work with all existing MFEM interfaces like finite element spaces etc.
+
+- Added a method, ParMesh::GetSerialMesh(), that reconstructs a partitioned
+  parallel mesh on a given single rank. Also, added ParMesh::PrintAsSerial(),
+  which saves the reconstructed serial mesh to a C++ stream on rank 0.
+
+- Added more 3D TMOP metrics, as well as specialized metrics for mesh
+  untangling and worst-case quality improvement.
+
+- Added a new method, Mesh::NodesUpdated, which should be called after the mesh
+  node coordinates have changed, e.g. after the mesh has moved. This is
+  necessary, for example, with device assembly of linear and bilinear forms.
+
 - Added support for mixed meshes and pyramids in GSLIB-FindPoints.

 Discretization improvements
 ---------------------------
- Added support for assembling low-order-refined matrices using a GPU-enabled
-  "batched" algorithm. The lor_solvers and plor_solvers now fully support GPU
-  acceleration.
-
- Added support for partial assembly and fully matrix-free operators on mixed
-  meshes (different element types and p-adaptivity) through libCEED, including
-  device acceleration, e.g. with NVIDIA and AMD GPUs. The p-adaptivity is
-  currently limited by MFEM capabilities, i.e. 2D serial meshes. All mixed
-  element topologies are supported in serial and parallel: segment, triangle,
-  square, tetrahedron, cube, prism, and pyramid.
-
 - Added full assembly and device support for several LinearForm integrators:
  * DomainLF: (f, v)
  * VectorDomainLF: ((f1,...,fn), (v1,...,vn))
  * DomainLFGrad: (f, grad(v))
  * VectorDomainLFGrad: ((f1x,f1y,f1z,...,fnx,fny,fnz), grad(v1,...,vn))
+  The device assembly of linear forms has to be explicitly enabled by calling
+  LinearForm::UseFastAssembly(true), otherwise the legacy linear form assembly
+  is used by default.
+
+- Added support for assembling low-order-refined matrices using a GPU-enabled
+  "batched" algorithm. The lor_solvers and plor_solvers now fully support GPU
+  acceleration with arbitrary user-supplied coefficients.
+
+- Added a new class FaceQuadratureSpace that allows for the construction of
+  QuadratureFunctions on the interior or boundary faces of a mesh.
+
+- Added a class CoefficientVector for efficient access of variable coefficient
+  values at quadrature points (in particular for GPU/device kernels).

 - Added WhiteGaussianNoiseDomainLFIntegrator: a LinearFormIntegrator class for
  spatial Gaussian white noise.
@@ -40,8 +57,25 @@ Discretization improvements
 - Added a new Zienkiewicz-Zhu patch recovery-based a posteriori error estimator.
  See fem/estimators.hpp.

+- Various fixes and improvements in LinearFormExtension.
+
 Linear and nonlinear solvers
 ----------------------------
+- Added a new class DGMassInverse that performs a local element-wise CG
+  iteration to solve systems involving the discontinuous Galerkin mass matrix,
+  including support for device/GPU acceleration.
+
+- Added more flexibility to the constrained solver classes:
+  * PenaltyConstrainedSolver now allows for a vector of penalty parameters
+    (necessary for penalty contact)
+  * PenaltyConstrainedSolver and EliminationSolver can use GMRES or PCG
+  * All constraint solver classes can take a user-defined preconditioner
+
+ - Added functions to toggle additional options for the SuperLU_Dist and Hypre
+   preconditioners (ParaSails, Euclid, ILU).
+
+- Added boundary elimination with device support for `SparseMatrix` and
+  `HypreParMatrix`.

 New and updated examples and miniapps
 -------------------------------------
@@ -51,12 +85,28 @@ New and updated examples and miniapps
  automatic differentiation tools like a native dual number implementation or a
  third party library such as Enzyme. See miniapps/elasticity for more details.

+- Added example for body-fitted volumetric and shape integration using the
+  Algoim library in miniapps/shifted.
+
 - Add a new example code, Example 33/33p, to demonstrate the solution of
  spectral fractional PDEs with MFEM.

 Integrations, testing and documentation
 ---------------------------------------
 - Added a Dockerfile for a simple MFEM container, see config/docker/README.md.
+  More sophisticated developer containers are available in the new repo
+  https://github.com/mfem/containers.
+
+- Added support for the LLVM-based automatic differentiation tool Enzyme, see
+  https://github.com/EnzymeAD/Enzyme. Build system flags and a convenience
+  header are provided. The functionality and interaction are demonstrated in a
+  new miniapp in miniapps/elasticity.
+
+- Added support for partial assembly and fully matrix-free operators on mixed
+  meshes (different element types and p-adaptivity) through libCEED, including
+  device acceleration, e.g. with NVIDIA and AMD GPUs. The p-adaptivity is
+  currently limited to 2D serial meshes. All mixed element topologies are
+  supported in both serial and parallel.

 - Added support for ParMoonolith, https://bitbucket.org/zulianp/par_moonolith,
  which provides parallel non-conforming, non-matching, variational, volumetric
@@ -64,29 +114,40 @@ Integrations, testing and documentation
  between arbitrarily distributed and unrelated finite element meshes in a
  variationally consistent way.

- Added support for the LLVM-based automatic differentiation tool Enzyme, see
-  https://github.com/EnzymeAD/Enzyme. Build system flags and a convenience
-  header are provided. The functionality and interaction are demonstrated in a
-  new miniapp in miniapps/elasticity.
+- Fully encapsulated SUNDIALS `N_Vector` object within the `SundialsNVector`
+  class by removing deprecated (e.g. `HypreParVector::ToNVector`) and
+  non-deprecated (e.g. `Vector::ToNVector`) functions in other classes.

- Added example for body-fitted volumetric and shape integration using the
-  Algoim library.
+- New benchmark for the different assembly levels inspired by the CEED
+  Bake-Off Problems, see tests/benchmarks/bench_assembly_levels.cpp.

 - Added Windows 2022 CI testing with GitHub actions.

 Miscellaneous
 -------------
- Various other simplifications, extensions, and bugfixes in the code.
+- The method SparseMatrix::EnsureMultTranspose() is now automatically called
+  by the methods AddMultTranspose(), MultTranspose(), and AbsMultTranspose().
+  Added a method with the same name to class HypreParMatrix which is also called
+  automatically by the HypreParMatrix::MultTranspose() methods.

+- Updated various MemoryUsage methods to return 'std::size_t' instead of 'long'
+  since the latter is 32-bit in Win64 builds.

- Added boundary elimination with device support for `SparseMatrix` and
-  `HypreParMatrix`.
-  
 - When using `AssemblyLevel::FULL`, `FABilinearFormExtension::FormSystemMatrix`
  outputs an `OperatorHandle` containing a `SparseMatrix` in serial, and an
  `HypreParMatrix` in parallel (instead of a `ConstrainedOperator`).

- Added TMOP metrics for mesh untangling and worst-case quality improvement.
+- In various places in the library, replace the use of 'long' with 'long long'
+  to better support Win64 builds where 'long' is 32-bit and 'long long' is
+  64-bit. On Linux and MacOS, both types are typically 64-bit.
+
+- The behavior of GridFunction::GetTrueVector() has been changed to not return
+  an empty true vector.
+
+- Added support for ordering search points byVDIM in FindPointsGSLIB.
+
+- Various other simplifications, extensions, and bugfixes in the code.
+

 Version 4.4, released on March 21, 2022
 =======================================
@@ -119,11 +180,6 @@ Meshing improvements
 - Added a simpler interface to access mesh face information, see FaceInformation
  and GetFaceInformation in the Mesh class.

- Added the method ParMesh::GetSerialMesh() that reconstructs a partitioned
-  parallel mesh on a given single rank. Also, added the method
-  ParMesh::PrintAsSerial() that saves the reconstructed serial mesh to a C++
-  stream on rank 0.
-
 - Gmsh meshes where all elements have zero physical tag (the default Gmsh output
  format if no physical groups are defined) are now successfully loaded, and
  elements are reassigned attribute number 1.
@@ -213,9 +269,6 @@ Integrations, testing and documentation
 - Switched from Artistic Style (astyle) version 2.05.1 to version 3.1 for code
  formatting. See the "make style" target.

- New benchmark for the different assembly levels inspired by the CEED
-  Bake-Off Problems, see tests/benchmarks/bench_assembly_levels.cpp.
-
 Miscellaneous
 -------------
 - Added a simple singleton class, Mpi, as a replacement for MPI_Session. New
@@ -226,13 +279,6 @@ Miscellaneous

 - Fixed several MinGW build issues on Windows.

- In various places in the library, replace the use of 'long' with 'long long'
-  to better support Win64 builds where 'long' is 32-bit and 'long long' is
-  64-bit. On Linux and MacOS, both types are typically 64-bit.
-
- Update various "MemoryUsage" methods to return 'std::size_t' instead of 'long'
-  since the latter is 32-bit in Win64 builds.
-
 - Added 'double' atomicAdd implementation for previous versions of CUDA.

 - HypreParVector and Vector now support C++ move semantics, and the copy
@@ -51,7 +51,7 @@ project(mfem NONE)
 # Current version of MFEM, see also `makefile`.
 #   mfem_VERSION = (string)
 #   MFEM_VERSION = (int)   [automatically derived from mfem_VERSION]
-set(${PROJECT_NAME}_VERSION 4.4.1)
+set(${PROJECT_NAME}_VERSION 4.5.0)

 # Prohibit in-source build
 if (${PROJECT_SOURCE_DIR} STREQUAL ${PROJECT_BINARY_DIR})
@@ -81,6 +81,10 @@ if (MFEM_USE_STRUMPACK)
  # Just needed to find the MPI_Fortran libraries to link with
  set(XSDK_ENABLE_Fortran ON)
 endif()
+# SUNDIALS >= 6.4.0 requires C++14:
+if (MFEM_USE_SUNDIALS AND ("${CMAKE_CXX_STANDARD}" LESS "14"))
+  set(CMAKE_CXX_STANDARD 14)
+endif()
 if (MFEM_USE_GINKGO AND ("${CMAKE_CXX_STANDARD}" LESS "14"))
  set(CMAKE_CXX_STANDARD 14)
 endif()
@@ -137,7 +141,7 @@ if (MFEM_USE_CUDA)
   set(CUSPARSE_FOUND TRUE)
   set(CUSPARSE_LIBRARIES "cusparse")
   set(CUBLAS_FOUND TRUE)
-   set(CUSBLAS_LIBRARIES "cublas")
+   set(CUBLAS_LIBRARIES "cublas")
 endif()

 if (XSDK_ENABLE_C)
@@ -200,10 +204,10 @@ if (MFEM_USE_MPI)
  find_package(MPI REQUIRED)
  set(MPI_CXX_INCLUDE_DIRS ${MPI_CXX_INCLUDE_PATH})
  if (MFEM_MPIEXEC)
-    set(MPIEXEC ${MFEM_MPIEXEC})
+    string(REPLACE " " ";" MPIEXEC ${MFEM_MPIEXEC})
  endif()
  if (MFEM_MPIEXEC_NP)
-    set(MPIEXEC_NUMPROC_FLAG ${MFEM_MPIEXEC_NP})
+    string(REPLACE " " ";" MPIEXEC_NUMPROC_FLAG ${MFEM_MPIEXEC_NP})
  endif()
  # Parallel MFEM depends on hypre
  find_package(HYPRE REQUIRED)
@@ -477,12 +481,21 @@ if (NOT DEFINED MFEM_TIMER_TYPE)
  endif()
 endif()

+# Without this, CMake 3.21.1 (and 3.20.2) run into CMake Errors like the following:
+# CMake Error at config/cmake/modules/MfemCmakeUtilities.cmake:60 (add_library):
+#   Target "mfem" links to target "Threads::Threads" but the target was not
+#   found.  Perhaps a find_package() call is missing for an IMPORTED target, or
+#   an ALIAS target is missing?
+# Call Stack (most recent call first):
+#   CMakeLists.txt:474 (mfem_add_library)
+find_package(Threads REQUIRED)
+
 # List all possible libraries in order of dependencies.
 # [METIS < SuiteSparse]:
 #    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 BLAS LAPACK SuperLUDist METIS SuiteSparse SUNDIALS
+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
    ADIOS2 CUBLAS CUSPARSE MKL_CPARDISO AMGX CALIPER CODIPACK BENCHMARK PARELAG
@@ -102,7 +102,9 @@ The MFEM source code has the following structure:
  .
  ├── config
  │   ├── cmake
-  │   └── githooks
+  │   ├── docker
+  │   ├── githooks
+  │   └── vcpkg
  ├── data
  ├── doc
  ├── examples
@@ -111,6 +113,7 @@ The MFEM source code has the following structure:
  │   ├── ginkgo
  │   ├── hiop
  │   ├── jupyter
+  │   ├── moonolith
  │   ├── petsc
  │   ├── pumi
  │   ├── sundials
@@ -118,13 +121,15 @@ The MFEM source code has the following structure:
  ├── fem
  │   ├── ceed
  │   ├── fe
-  │   ├── qinterp
+  │   ├── lor
  │   ├── moonolith
+  │   ├── qinterp
  │   └── tmop
  ├── general
  ├── linalg
  │   └── simd
  ├── mesh
+  │   └── submesh
  ├── miniapps
  │   ├── adjoint
  │   ├── autodiff
@@ -134,6 +139,7 @@ The MFEM source code has the following structure:
  │   ├── hooke
  │   ├── meshing
  │   ├── mtop
+  │   ├── multidomain
  │   ├── navier
  │   ├── nurbs
  │   ├── parelag
@@ -7,6 +7,10 @@

                               https://mfem.org

+This file provides a detailed description of how to build and install the MFEM
+library. For a simple build, see the step-by-step instructions on the website
+at https://mfem.org/building.
+
 The MFEM library has a serial and an MPI-based parallel version, which largely
 share the same code base. The only prerequisite for building the serial version
 of MFEM is a (modern) C++ compiler, such as g++. The parallel version of MFEM
@@ -16,7 +20,11 @@ requires an MPI C++ compiler, as well as the following external libraries:
  https://github.com/hypre-space/hypre

 - METIS (a family of multilevel partitioning algorithms)
-  http://glaros.dtc.umn.edu/gkhome/metis/metis/overview
+  https://github.com/mfem/tpls
+
+  Note: We recommend our mirror of metis-4.0.3/5.1.0 above because the METIS
+  webpage, http://glaros.dtc.umn.edu/gkhome/metis/metis/overview, is often down
+  and we don't support yet the new repo https://github.com/KarypisLab/METIS.

 The hypre dependency can be downloaded as a tarball from GitHub or from the
 project webpage https://www.llnl.gov/casc/hypre. For example, the 2.24.0 release
@@ -472,10 +480,10 @@ MFEM_USE_CODIPACK = YES/NO
  www.scicomp.uni-kl.de/codi/

 MFEM_USE_ALGOIM = YES/NO
-  Enable the usage of Algoim - a collection of high-order accurate numerical 
-  methods and C++ algorithms for working with implicitly-defined geometry and 
-  level set methods. The Algoim library requires the Blitz++ library. The MFEM 
-  provides interface to Algoim v1. Thus, to check out the specific state use: 
+  Enable the usage of Algoim - a collection of high-order accurate numerical
+  methods and C++ algorithms for working with implicitly-defined geometry and
+  level set methods. The Algoim library requires the Blitz++ library. The MFEM
+  provides interface to Algoim v1. Thus, to check out the specific state use:
  git checkout 9c9ca0ef094d8ab0390ed36367a1151b459bbe0a
  https://algoim.github.io

@@ -550,7 +558,7 @@ MFEM_USE_FMS = YES/NO
   Enables support for the FMS library which consists of the DataCollection
   sub-class mfem::FMSDataCollection for I/O in FMS formats, see the header file
   fem/fmsdatacollection.hpp. In addition, this option enables in-memory
-   convetion routines between FMS's FmsDataCollection structure and MFEM's
+   conversion routines between FMS's FmsDataCollection structure and MFEM's
   DataCollection class, see the header file fem/fmsconvert.hpp.

 MFEM_USE_PARELAG = YES/NO
@@ -597,7 +605,7 @@ The specific libraries and their options are:

 - METIS, used when MFEM_USE_METIS = YES. If using METIS 5, set
  MFEM_USE_METIS_5 = YES (default is to use METIS 4).
-  URL: http://glaros.dtc.umn.edu/gkhome/metis/metis/overview
+  URL: https://github.com/mfem/tpls (MFEM mirror, see above)
  Options: METIS_OPT, METIS_LIB.
  Versions: METIS 4.0.3 or 5.1.0.

@@ -754,12 +762,12 @@ The specific libraries and their options are:
  Options: GSLIB_OPT, GSLIB_LIB.
  Versions: GSLIB >= 1.0.7.

- ALGOIM (optional), used when MFE_USE_ALGOIM=YES. The library provides only
+- ALGOIM (optional), used when MFEM_USE_ALGOIM=YES. The library provides only
  headers so it just needs to be downloaded at the same level as MFEM. Download
  the specific version we use as:
  "git clone https://github.com/algoim/algoim.git;
   git checkout 9c9ca0ef094d8ab0390ed36367a1151b459bbe0a"
-  ALGOIM depends on BLITZ and rhe library must be built prior to the MFEM build.
+  ALGOIM depends on BLITZ and the library must be built prior to the MFEM build.
  Download v1.0.2, untar it at the same level as MFEM and create a symbolic link:
  "ln -s blitz-1.0.2 blitz".
  Build Blitz using CMake as:
@@ -19,7 +19,7 @@ if(EXISTS "${ENZYME_DIR}/ClangEnzyme-${ENZYME_VERSION}.so")
  # Set ENZYME_FOUND
  set(ENZYME_FOUND TRUE CACHE BOOL "ENZYME was found." FORCE)

-  # Set CXX flags to accomodate the Enzyme Clang plugin
+  # Set CXX flags to accommodate the Enzyme Clang plugin
  set(CMAKE_CXX_FLAGS "${CMAKE_CXX_FLAGS} -Xclang -load -Xclang ${ENZYME_DIR}/ClangEnzyme-${ENZYME_VERSION}.so -mllvm -enzyme-loose-types=1")
  set(MFEM_USE_ENZYME YES)
 else()
@@ -0,0 +1,57 @@
+# 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.
+
+# Defines the following variables:
+#   - HDF5_FOUND - If HDF5 was found
+#   - HDF5_LIBRARIES - The HDF5 libraries
+#   - HDF5_INCLUDE_DIRS - The HDF5 include directories
+
+# First Check for HDF5_DIR
+if(NOT HDF5_DIR)
+    MESSAGE(FATAL_ERROR "Could not find HDF5. HDF5 support needs explicit HDF5_DIR")
+endif()
+
+# Find includes
+find_path( HDF5_INCLUDE_DIRS hdf5.h
+           PATHS  ${HDF5_DIR}/include/
+           NO_DEFAULT_PATH
+           NO_CMAKE_ENVIRONMENT_PATH
+           NO_CMAKE_PATH
+           NO_SYSTEM_ENVIRONMENT_PATH
+           NO_CMAKE_SYSTEM_PATH)
+
+find_library( __HDF5_LIBRARY NAMES hdf5 libhdf5 libhdf5_D libhdf5_debug
+              PATHS ${HDF5_DIR}/lib
+              NO_DEFAULT_PATH
+              NO_CMAKE_ENVIRONMENT_PATH
+              NO_CMAKE_PATH
+              NO_SYSTEM_ENVIRONMENT_PATH
+              NO_CMAKE_SYSTEM_PATH)
+
+find_library( __HDF5_HL_LIBRARY NAMES hdf5_hl libhdf5_hl libhdf5_hl_D libhdf5_hl_debug
+              PATHS ${HDF5_DIR}/lib
+              NO_DEFAULT_PATH
+              NO_CMAKE_ENVIRONMENT_PATH
+              NO_CMAKE_PATH
+              NO_SYSTEM_ENVIRONMENT_PATH
+              NO_CMAKE_SYSTEM_PATH)
+
+set(HDF5_LIBRARIES ${__HDF5_HL_LIBRARY} ${__HDF5_LIBRARY})
+
+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
+                                  HDF5_INCLUDE_DIRS
+                                  __HDF5_LIBRARY
+                                  __HDF5_HL_LIBRARY
+                                  HDF5_LIBRARIES )
@@ -14,6 +14,6 @@
 #   - UMPIRE_LIBRARIES
 #   - UMPIRE_INCLUDE_DIRS

-include(MfemCmakeUtilities)
-mfem_find_package(UMPIRE UMPIRE UMPIRE_DIR "include" "umpire/Umpire.hpp" "lib" "umpire"
-  "Paths to headers required by UMPIRE." "Libraries required by UMPIRE.")
+find_package(umpire REQUIRED CONFIG)
+set(UMPIRE_FOUND ${umpire_FOUND})
+set(UMPIRE_LIBRARIES "umpire")
@@ -43,22 +43,14 @@ function(convert_filenames_to_full_paths NAMES)
  set(${NAMES} ${tmp_names} PARENT_SCOPE)
 endfunction()

-# Wrapper for add_executable that calls the HIP wrapper if applicable
+# Wrapper for add_executable
 macro(mfem_add_executable NAME)
-  if (MFEM_USE_HIP)
-    add_executable(${NAME} ${ARGN})
-  else()
-    add_executable(${NAME} ${ARGN})
-  endif()
+  add_executable(${NAME} ${ARGN})
 endmacro()

-# Wrapper for add_library that calls the HIP wrapper if applicable
+# Wrapper for add_library
 macro(mfem_add_library NAME)
-  if (MFEM_USE_HIP)
-    add_library(${NAME} ${ARGN})
-  else()
-    add_library(${NAME} ${ARGN})
-  endif()
+  add_library(${NAME} ${ARGN})
 endmacro()

 # Simple shortcut to add_custom_target() with option to add the target to the
@@ -31,9 +31,11 @@

 // Windows specific options
 #ifdef _WIN32
+#ifndef _USE_MATH_DEFINES
 // Macro needed to get defines like M_PI from <cmath>. (Visual Studio C++ only?)
 #define _USE_MATH_DEFINES
 #endif
+#endif
 // On Cygwin the option -std=c++11 prevents the definition of M_PI. Defining
 // the following macro allows us to get M_PI and some needed functions, e.g.
 // posix_memalign(), strdup(), strerror_r().
@@ -179,7 +179,7 @@ ifeq ($(MFEM_USE_MPI)$(MFEM_USE_HIP),YESYES)
 endif

 # ROCM/HIP directory such that ROCM/HIP libraries like rocsparse and rocrand are
-# found in $(HIP_DIR)/lib, usually as links. Typically, this directoory is of
+# found in $(HIP_DIR)/lib, usually as links. Typically, this directory is of
 # the form /opt/rocm-X.Y.Z which is called ROCM_PATH by hipconfig.
 ifeq ($(MFEM_USE_HIP),YES)
   HIP_DIR := $(patsubst %/,%,$(dir $(shell which $(HIP_CXX))))
@@ -251,12 +251,16 @@ POSIX_CLOCKS_LIB = -lrt
 # SUNDIALS library configuration
 # For sundials_nvecmpiplusx and nvecparallel remember to build with MPI_ENABLE=ON
 # and modify cmake variables for hypre for sundials
-SUNDIALS_DIR    = @MFEM_DIR@/../sundials-5.0.0/instdir
-SUNDIALS_OPT    = -I$(SUNDIALS_DIR)/include
-SUNDIALS_LIBDIR = $(wildcard $(SUNDIALS_DIR)/lib*)
-SUNDIALS_LIB    = $(XLINKER)-rpath,$(SUNDIALS_LIBDIR) -L$(SUNDIALS_LIBDIR)\
+SUNDIALS_DIR = @MFEM_DIR@/../sundials-5.0.0/instdir
+# SUNDIALS >= 6.4.0 requires C++14:
+ifeq ($(MFEM_USE_SUNDIALS),YES)
+   BASE_FLAGS = -std=c++14
+endif
+SUNDIALS_OPT = -I$(SUNDIALS_DIR)/include
+SUNDIALS_LIB = $(XLINKER)-rpath,$(SUNDIALS_DIR)/lib64\
+ $(XLINKER)-rpath,$(SUNDIALS_DIR)/lib\
+ -L$(SUNDIALS_DIR)/lib64 -L$(SUNDIALS_DIR)/lib\
 -lsundials_arkode -lsundials_cvodes -lsundials_nvecserial -lsundials_kinsol
-
 ifeq ($(MFEM_USE_MPI),YES)
   SUNDIALS_LIB += -lsundials_nvecparallel -lsundials_nvecmpiplusx
 endif
@@ -309,7 +313,7 @@ SCALAPACK_LIB = -L$(SCALAPACK_DIR)/lib -lscalapack $(LAPACK_LIB)
 MPI_FORTRAN_LIB = -lmpifort
 # OpenMPI:
 # MPI_FORTRAN_LIB = -lmpi_mpifh
-# Additional Fortan library:
+# Additional Fortran library:
 # MPI_FORTRAN_LIB += -lgfortran

 # MUMPS library configuration
@@ -554,15 +554,14 @@ function go()
   local cmd_line="${1##+( )}"
   cmd_line="${cmd_line%%+( )}"
   shopt -u extglob
-   eval local cmd=(${cmd_line})
   local res=""
   echo $sep
   echo "<${group}>" "${cmd_line}"
   echo $sep
   if [ "${timing}" == "yes" ]; then
-      timed_run "${cmd[@]}"
+      timed_run eval "${cmd_line}"
   else
-      "${cmd[@]}"
+      eval "${cmd_line}"
   fi
   if [ "$?" -eq 0 ]; then
      res="${green}  OK  ${none}"
@@ -0,0 +1,8 @@
+--- a/CMakeLists.txt	Wed Dec 21 18:24:22 2016
+++ b/CMakeLists.txt	Wed Dec 21 18:24:26 2016
+@@ -20,4 +20,4 @@
+ # Recursively look for CMakeLists.txt in subdirs.
+ add_subdirectory("include")
+ add_subdirectory("libmetis")
+-add_subdirectory("programs")
+# add_subdirectory("programs")
@@ -0,0 +1,15 @@
+--- a/CMakeLists.txt	Sat Mar 30 17:24:45 2013
+++ b/CMakeLists.txt	Wed Dec 21 18:23:43 2016
+@@ -4,11 +4,7 @@
+ set(GKLIB_PATH "GKlib" CACHE PATH "path to GKlib")
+ set(SHARED FALSE CACHE BOOL "build a shared library")
+ 
+-if(MSVC)
+-  set(METIS_INSTALL FALSE)
+-else()
+-  set(METIS_INSTALL TRUE)
+-endif()
+set(METIS_INSTALL TRUE)
+ 
+ # Configure libmetis library.
+ if(SHARED)
@@ -0,0 +1,34 @@
+diff --git a/include/metis.h b/include/metis.h
+index dc5406a..7732437 100644
+--- a/include/metis.h
+++ b/include/metis.h
+@@ -72,10 +72,14 @@ typedef __int64 int64_t;
+ #define PRId64       "I64d"
+ #define SCNd32       "ld"
+ #define SCNd64       "I64d"
+#ifdef _WIN32
+#include <stdint.h>
+#else
+ #define INT32_MIN    ((int32_t)_I32_MIN)
+ #define INT32_MAX    _I32_MAX
+ #define INT64_MIN    ((int64_t)_I64_MIN)
+ #define INT64_MAX    _I64_MAX
+#endif
+ #else
+ #include <inttypes.h>
+ #endif
+diff --git a/GKlib/gk_arch.h b/GKlib/gk_arch.h
+index 78b1431..7258763 100644
+--- a/GKlib/gk_arch.h
+++ b/GKlib/gk_arch.h
+@@ -32,8 +32,8 @@
+ 
+ 
+ #ifdef __MSC__ 
+-  #include "ms_stdint.h"
+-  #include "ms_inttypes.h"
+  #include <stdint.h>
+  #include <inttypes.h>
+   #include "ms_stat.h"
+ #else
+ #ifndef SUNOS
@@ -0,0 +1,11 @@
+--- a/GKlib/gk_arch.h	Wed Dec 21 18:34:18 2016
+++ b/GKlib/gk_arch.h	Wed Dec 21 18:30:49 2016
+@@ -58,7 +58,7 @@
+ #define PTRDIFF_MAX  INT64_MAX
+ #endif
+ 
+-#ifdef __MSC__
+#if defined(__MSC__) && (_MSC_VER < 1900)
+ /* MSC does not have rint() function */
+ #define rint(x) ((int)((x)+0.5))  
+ 
@@ -0,0 +1,14 @@
+diff --git a/CMakeLists.txt b/CMakeLists.txt
+index e94f050..b9613a7 100644
+--- a/CMakeLists.txt
+++ b/CMakeLists.txt
+@@ -1,7 +1,8 @@
+ cmake_minimum_required(VERSION 2.8)
+ project(METIS)
+ 
+-set(GKLIB_PATH "GKlib" CACHE PATH "path to GKlib")
+set(GKLIB_PATH "${CMAKE_SOURCE_DIR}/GKlib" CACHE PATH "path to GKlib")
+
+ set(SHARED FALSE CACHE BOOL "build a shared library")
+ 
+ set(METIS_INSTALL TRUE)
@@ -0,0 +1,11 @@
+--- a/libmetis/metislib.h	Sat Mar 30 17:24:45 2013
+++ b/libmetis/metislib.h	Wed Dec 21 18:30:59 2016
+@@ -31,7 +31,7 @@
+ #include <proto.h>
+ 
+ 
+-#if defined(COMPILER_MSC)
+#if defined(COMPILER_MSC) && (_MSC_VER < 1900)
+ #if defined(rint)
+   #undef rint
+ #endif
@@ -0,0 +1,10 @@
+--- a/libmetis/CMakeLists.txt	Sat Mar 30 17:24:45 2013
+++ b/libmetis/CMakeLists.txt	Wed Dec 21 17:41:37 2016
+@@ -11,6 +11,6 @@
+ if(METIS_INSTALL)
+   install(TARGETS metis
+     LIBRARY DESTINATION lib
+-    RUNTIME DESTINATION lib
+    RUNTIME DESTINATION bin
+     ARCHIVE DESTINATION lib)
+ endif()
@@ -0,0 +1,44 @@
+diff --git a/CMakeLists.txt b/CMakeLists.txt
+index b9613a7..e43ffee 100644
+--- a/CMakeLists.txt
+++ b/CMakeLists.txt
+@@ -22,3 +22,23 @@ include_directories(include)
+ add_subdirectory("include")
+ add_subdirectory("libmetis")
+ # add_subdirectory("programs")
+
+if(METIS_INSTALL)
+  set(PRJ_NAME metis)
+  set(PRJ_VER 5.1.0)
+  install(EXPORT metisTargets 
+    FILE ${PRJ_NAME}Targets.cmake
+    DESTINATION lib/cmake/${PRJ_NAME})
+  include(CMakePackageConfigHelpers)
+  write_basic_package_version_file(
+    ${CMAKE_CURRENT_BINARY_DIR}/${PRJ_NAME}ConfigVersion.cmake
+    VERSION ${PRJ_VER}
+    COMPATIBILITY SameMajorVersion)
+  file(WRITE ${CMAKE_CURRENT_BINARY_DIR}/${PRJ_NAME}Config.cmake
+    "include(\${CMAKE_CURRENT_LIST_DIR}/${PRJ_NAME}Targets.cmake)")
+  install(FILES 
+    ${CMAKE_CURRENT_BINARY_DIR}/${PRJ_NAME}ConfigVersion.cmake
+    ${CMAKE_CURRENT_BINARY_DIR}/${PRJ_NAME}Config.cmake
+    DESTINATION lib/cmake/${PRJ_NAME})
+endif()
+
+diff --git a/libmetis/CMakeLists.txt b/libmetis/CMakeLists.txt
+index 7a5fc74..5a68cf0 100644
+--- a/libmetis/CMakeLists.txt
+++ b/libmetis/CMakeLists.txt
+@@ -9,8 +9,9 @@ if(UNIX)
+ endif()
+ 
+ if(METIS_INSTALL)
+-  install(TARGETS metis
+  install(TARGETS metis EXPORT metisTargets
+     LIBRARY DESTINATION lib
+     RUNTIME DESTINATION bin
+-    ARCHIVE DESTINATION lib)
+    ARCHIVE DESTINATION lib
+    INCLUDES DESTINATION include)
+ endif()
@@ -0,0 +1,41 @@
+vcpkg_check_linkage(ONLY_STATIC_LIBRARY)
+set(OPTIONS -DSHARED=OFF)
+
+set(METIS_VERSION 5.1.0)
+
+vcpkg_download_distfile(ARCHIVE
+    URLS "https://github.com/mfem/tpls/raw/gh-pages/metis-${METIS_VERSION}.tar.gz"
+    FILENAME "metis-${METIS_VERSION}.tar.gz"
+    SHA512 deea47749d13bd06fbeaf98a53c6c0b61603ddc17a43dae81d72c8015576f6495fd83c11b0ef68d024879ed5415c14ebdbd87ce49c181bdac680573bea8bdb25
+)
+
+vcpkg_extract_source_archive_ex(
+    OUT_SOURCE_PATH SOURCE_PATH
+    ARCHIVE ${ARCHIVE}
+    REF ${METIS_VERSION}
+    PATCHES
+        enable-install.patch
+        disable-programs.patch
+        fix-runtime-install-destination.patch
+        fix-metis-vs14-math.patch
+        fix-gklib-vs14-math.patch
+        fix-linux-build-error.patch
+        install-metisConfig.patch
+        fix-INT_MIN_define.patch
+)
+
+vcpkg_configure_cmake(
+    SOURCE_PATH ${SOURCE_PATH}
+    PREFER_NINJA
+    OPTIONS ${OPTIONS}
+)
+
+vcpkg_install_cmake()
+vcpkg_copy_pdbs()
+vcpkg_fixup_cmake_targets(CONFIG_PATH lib/cmake/metis)
+
+file(REMOVE_RECURSE ${CURRENT_PACKAGES_DIR}/debug/include)
+
+# Handle copyright
+file(COPY ${SOURCE_PATH}/LICENSE.txt DESTINATION ${CURRENT_PACKAGES_DIR}/share/metis)
+file(INSTALL ${SOURCE_PATH}/LICENSE.txt DESTINATION ${CURRENT_PACKAGES_DIR}/share/${PORT} RENAME copyright)
@@ -0,0 +1,7 @@
+{
+  "name": "metis-mfem",
+  "version-string": "5.1.0",
+  "port-version": 0,
+  "description": "Serial Graph Partitioning and Fill-reducing Matrix Ordering",
+  "homepage": "http://glaros.dtc.umn.edu/gkhome/metis/metis/overview"
+}
@@ -1,7 +1,7 @@
 MFEM mesh v1.0

 #
-# MFEM Geomety Types (see mesh/geom.hpp):
+# MFEM Geometry Types (see mesh/geom.hpp):
 #
 # POINT       = 0
 # SEGMENT     = 1
@@ -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         = v4.4.1
+PROJECT_NUMBER         = v4.5.0

 # 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
@@ -2349,7 +2349,7 @@ PLANTUML_INCLUDE_PATH  =
 # Minimum value: 0, maximum value: 10000, default value: 50.
 # This tag requires that the tag HAVE_DOT is set to YES.

-DOT_GRAPH_MAX_NODES    = 50
+DOT_GRAPH_MAX_NODES    = 100

 # The MAX_DOT_GRAPH_DEPTH tag can be used to set the maximum depth of the graphs
 # generated by dot. A depth value of 3 means that only nodes reachable from the
@@ -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
@@ -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
@@ -195,7 +195,7 @@ int main(int argc, char *argv[])
   Array<int> ess_tdof_list(0);
   if (h1 && pmesh.bdr_attributes.Size())
   {
-      // For a continuous basis the linear system must be modifed to enforce an
+      // For a continuous basis the linear system must be modified to enforce an
      // essential (Dirichlet) boundary condition. In the DG case this is not
      // necessary as the boundary condition will only be enforced weakly.
      fespace.GetEssentialTrueDofs(dbc_bdr, ess_tdof_list);
@@ -197,7 +197,6 @@ int main(int argc, char *argv[])
      SparseMatrix &M(mVarf->SpMat());
      SparseMatrix &B(bVarf->SpMat());
      B *= -1.;
-      B.EnsureMultTranspose();
      Bt = new TransposeOperator(&B);

      darcyOp.SetBlock(0,0, &M);
@@ -187,7 +187,6 @@ int main(int argc, char *argv[])
 {
   // 1. Initialize MPI and HYPRE.
   Mpi::Init(argc, argv);
-   int num_procs = Mpi::WorldSize();
   int myid = Mpi::WorldRank();
   Hypre::Init();

@@ -248,7 +248,10 @@ int main(int argc, char *argv[])
   //     constraints for non-conforming AMR, static condensation, etc.
   if (myid == 0) { cout << "matrix ... " << flush; }
   if (static_cond) { a->EnableStaticCondensation(); }
-   a->Assemble();
+   // Here we want to try out block-size aware AMG solver in PETSc.
+   // For that to work properly, we need a fully-compliant block-size
+   // structure and we do not skip zeros when assembling.
+   a->Assemble(use_petsc ? 0 : 1);

   Vector B, X;
   if (!use_petsc)
@@ -294,13 +297,14 @@ int main(int argc, char *argv[])
         cout << "done." << endl;
         cout << "Size of linear system: " << A.M() << endl;
      }
-      PetscPCGSolver *pcg = new PetscPCGSolver(A);
+      // Tell PETSc the matrix has a block structure
+      A.SetBlockSize(dim);

-      // The preconditioner for the PCG solver defined below is specified in the
-      // PETSc config file, rc_ex2p, since a Krylov solver in PETSc can also
-      // customize its preconditioner.
+      // The preconditioner for the PCG solver can be specified in the
+      // PETSc config file
+      PetscPCGSolver *pcg = new PetscPCGSolver(A);
      PetscPreconditioner *prec = NULL;
-      if (use_nonoverlapping)
+      if (use_nonoverlapping) // Specialized BDDC construction
      {
         // Compute dofs belonging to the natural boundary
         Array<int> nat_tdof_list, nat_bdr(pmesh->bdr_attributes.Max());
@@ -450,7 +450,7 @@ int main(int argc, char *argv[])
      for (int ti = 0; !done; )
      {
         // We cannot match exactly the time history of the Run method
-         // since we are explictly telling PETSc to use a time step
+         // since we are explicitly telling PETSc to use a time step
         double dt_real = min(dt, t_final - t);
         ode_solver->Step(*U, t, dt_real);
         ti++;
@@ -78,6 +78,7 @@ EX1_ARGS_CUDA         := -m ../../data/star.mesh --usepetsc --partial-assembly -
 EX1_ARGS_CUDAAMG      := -m ../../data/star.mesh --usepetsc --device cuda --petscopts rc_ex1p_cudaamg
 EX2_ARGS              := -m ../../data/beam-quad.mesh --usepetsc --petscopts rc_ex2p
 EX2_ARGS_BDDC         := -m ../../data/beam-tri.mesh --usepetsc --nonoverlapping --petscopts rc_ex2p_bddc
+EX2_ARGS_ASM          := -m ../../data/beam-quad.mesh --usepetsc --petscopts rc_ex2p_asm
 EX3_ARGS              := -m ../../data/klein-bottle.mesh -o 2 -f 0.1 --usepetsc --petscopts rc_ex3p_bddc --nonoverlapping
 EX4_ARGS              := -m ../../data/klein-bottle.mesh -o 2 --usepetsc --petscopts rc_ex4p_bddc --nonoverlapping
 EX4_HYB_ARGS          := -m ../../data/klein-bottle.mesh -o 2 --usepetsc --petscopts rc_ex4p_bddc --nonoverlapping --hybridization
@@ -109,6 +110,7 @@ endif
 ex2p-test-par: ex2p
 	@$(call mfem-test,$<, $(RUN_MPI), $(TESTNAME),$(EX2_ARGS))
 	@$(call mfem-test,$<, $(RUN_MPI), $(TESTNAME),$(EX2_ARGS_BDDC))
+	@$(call mfem-test,$<, $(RUN_MPI), $(TESTNAME),$(EX2_ARGS_ASM))
 ex3p-test-par: ex3p
 	@$(call mfem-test,$<, $(RUN_MPI), $(TESTNAME),$(EX3_ARGS))
 ex4p-test-par: ex4p
@@ -1,8 +1,7 @@
 -ksp_converged_reason

 # GAMG is still not used at its best,
-# since we are not exploiting the
-# block size (Ordering::byVDIM) and the RBMs
+# since we are not exploiting the RBMs

 -ksp_view
 -pc_type gamg
@@ -0,0 +1,10 @@
+# Additive Schwarz with Overlap
+# This is not a good solver for elasticity
+# These options are here only to describe
+# the setup of the solver
+-ksp_converged_reason
+-ksp_view
+-ksp_max_it 10
+-pc_type asm
+-pc_asm_overlap 1
+-sub_pc_type icc
@@ -210,6 +210,9 @@ void visualize(ostream &os, Mesh *mesh, GridFunction *deformed_nodes,

 int main(int argc, char *argv[])
 {
+   // 0. Initialize SUNDIALS.
+   Sundials::Init();
+
   // 1. Parse command-line options.
   const char *mesh_file = "../../data/beam-quad.mesh";
   int ref_levels = 2;
@@ -215,10 +215,11 @@ void visualize(ostream &os, ParMesh *mesh, ParGridFunction *deformed_nodes,

 int main(int argc, char *argv[])
 {
-   // 1. Initialize MPI and HYPRE.
+   // 1. Initialize MPI, HYPRE, and SUNDIALS.
   Mpi::Init(argc, argv);
   int myid = Mpi::WorldRank();
   Hypre::Init();
+   Sundials::Init();

   // 2. Parse command-line options.
   const char *mesh_file = "../../data/beam-quad.mesh";
@@ -109,6 +109,9 @@ double InitialTemperature(const Vector &x);

 int main(int argc, char *argv[])
 {
+   // 0. Initialize SUNDIALS.
+   Sundials::Init();
+
   // 1. Parse command-line options.
   const char *mesh_file = "../../data/star.mesh";
   int ref_levels = 2;
@@ -290,7 +293,10 @@ int main(int argc, char *argv[])
         arkode->Init(oper);
         arkode->SetSStolerances(reltol, abstol);
         arkode->SetMaxStep(dt);
-         if (ode_solver_type == 11) { arkode->SetERKTableNum(FEHLBERG_13_7_8); }
+         if (ode_solver_type == 11)
+         {
+            arkode->SetERKTableNum(ARKODE_FEHLBERG_13_7_8);
+         }
         ode_solver = arkode; break;
      case 12:
         arkode = new ARKStepSolver(ARKStepSolver::IMPLICIT);
@@ -101,11 +101,12 @@ double InitialTemperature(const Vector &x);

 int main(int argc, char *argv[])
 {
-   // 1. Initialize MPI and HYPRE.
+   // 1. Initialize MPI, HYPRE, and SUNDIALS.
   Mpi::Init(argc, argv);
   int num_procs = Mpi::WorldSize();
   int myid = Mpi::WorldRank();
   Hypre::Init();
+   Sundials::Init();

   // 2. Parse command-line options.
   const char *mesh_file = "../../data/star.mesh";
@@ -327,7 +328,10 @@ int main(int argc, char *argv[])
         arkode->Init(oper);
         arkode->SetSStolerances(reltol, abstol);
         arkode->SetMaxStep(dt);
-         if (ode_solver_type == 11) { arkode->SetERKTableNum(FEHLBERG_13_7_8); }
+         if (ode_solver_type == 11)
+         {
+            arkode->SetERKTableNum(ARKODE_FEHLBERG_13_7_8);
+         }
         ode_solver = arkode; break;
      case 12:
         arkode = new ARKStepSolver(MPI_COMM_WORLD, ARKStepSolver::IMPLICIT);
@@ -140,6 +140,9 @@ public:

 int main(int argc, char *argv[])
 {
+   // 0. Initialize SUNDIALS.
+   Sundials::Init();
+
   // 1. Parse command-line options.
   problem = 0;
   const char *mesh_file = "../../data/periodic-hexagon.mesh";
@@ -408,7 +411,7 @@ int main(int argc, char *argv[])
         arkode->Init(adv);
         arkode->SetSStolerances(reltol, abstol);
         arkode->SetMaxStep(dt);
-         arkode->SetERKTableNum(FEHLBERG_13_7_8);
+         arkode->SetERKTableNum(ARKODE_FEHLBERG_13_7_8);
         ode_solver = arkode; break;
   }

@@ -152,11 +152,12 @@ public:

 int main(int argc, char *argv[])
 {
-   // 1. Initialize MPI and HYPRE.
+   // 1. Initialize MPI, HYPRE, and SUNDIALS.
   Mpi::Init(argc, argv);
   int num_procs = Mpi::WorldSize();
   int myid = Mpi::WorldRank();
   Hypre::Init();
+   Sundials::Init();

   // 2. Parse command-line options.
   problem = 0;
@@ -487,7 +488,10 @@ int main(int argc, char *argv[])
         arkode->Init(adv);
         arkode->SetSStolerances(reltol, abstol);
         arkode->SetMaxStep(dt);
-         if (ode_solver_type == 9) { arkode->SetERKTableNum(FEHLBERG_13_7_8); }
+         if (ode_solver_type == 9)
+         {
+            arkode->SetERKTableNum(ARKODE_FEHLBERG_13_7_8);
+         }
         ode_solver = arkode; break;
   }

@@ -35,7 +35,7 @@ add_mfem_examples(SUPERLU_EXAMPLES_SRCS ${PFX} "" test_superlu)
 if (MFEM_ENABLE_TESTING)
  # Command line options for the tests.
  # Example 1: Test SuperLU on the simple Poisson problem
-  set(EX1_COMMON_OPTS -m ../../data/star.mesh -p 2)
+  set(EX1_COMMON_OPTS -m ../../data/star.mesh)
  set(EX1P_TEST_OPTS ${EX1_COMMON_OPTS})

  # Add the tests: one test per source file.
@@ -39,6 +39,7 @@ set(SRCS
  complex_fem.cpp
  convergence.cpp
  datacollection.cpp
+  dgmassinv.cpp
  doftrans.cpp
  eltrans.cpp
  estimators.cpp
@@ -72,6 +73,7 @@ set(SRCS
  linearform.cpp
  linearform_ext.cpp
  lininteg.cpp
+  lininteg_boundary.cpp
  lininteg_domain.cpp
  lininteg_domain_grad.cpp
  lor/lor.cpp
@@ -88,6 +90,7 @@ set(SRCS
  fespacehierarchy.cpp
  nonlininteg_vectorconvection.cpp
  nonlininteg_vectorconvection_mf.cpp
+  qfunction.cpp
  qinterp/det.cpp
  qinterp/eval_by_nodes.cpp
  qinterp/eval_by_vdim.cpp
@@ -95,6 +98,7 @@ set(SRCS
  qinterp/grad_by_vdim.cpp
  qinterp/grad_phys_by_nodes.cpp
  qinterp/grad_phys_by_vdim.cpp
+  qspace.cpp
  quadinterpolator.cpp
  quadinterpolator_face.cpp
  restriction.cpp
@@ -136,10 +140,13 @@ set(HDRS
  bilinearform.hpp
  bilinearform_ext.hpp
  bilininteg.hpp
+  bilininteg_mass_pa.hpp
  coefficient.hpp
  complex_fem.hpp
  convergence.hpp
  datacollection.hpp
+  dgmassinv.hpp
+  dgmassinv_kernels.hpp
  doftrans.hpp
  eltrans.hpp
  estimators.hpp
@@ -189,9 +196,11 @@ set(HDRS
  nonlinearform.hpp
  nonlinearform_ext.hpp
  nonlininteg.hpp
+  qfunction.hpp
  qinterp/dispatch.hpp
  qinterp/eval.hpp
  qinterp/grad.hpp
+  qspace.hpp
  quadinterpolator.hpp
  quadinterpolator_face.hpp
  restriction.hpp
@@ -136,7 +136,7 @@ void BilinearForm::SetAssemblyLevel(AssemblyLevel assembly_level)
         ext = new MFBilinearFormExtension(this);
         break;
      default:
-         mfem_error("Unknown assembly level");
+         MFEM_ABORT("BilinearForm: unknown assembly level");
   }
 }

@@ -992,6 +992,7 @@ void BilinearForm::EliminateVDofs(const Array<int> &vdofs_,
      mat_e = new SparseMatrix(height);
   }

+   vdofs_.HostRead();
   for (int i = 0; i < vdofs_.Size(); i++)
   {
      int vdof = vdofs_[i];
@@ -26,7 +26,8 @@ namespace mfem
 {

 /** @brief Enumeration defining the assembly level for bilinear and nonlinear
-    form classes derived from Operator. */
+    form classes derived from Operator. For more details, see
+    https://mfem.org/howto/assembly_levels */
 enum class AssemblyLevel
 {
   /// In the case of a BilinearForm LEGACY corresponds to a fully assembled
@@ -177,7 +178,7 @@ public:
       - AssemblyLevel::ELEMENT
       - AssemblyLevel::NONE

-       This method must be called before assembly. */
+       If used, this method must be called before assembly. */
   void SetAssemblyLevel(AssemblyLevel assembly_level);

   /// Returns the assembly level
@@ -333,7 +334,7 @@ public:


   /**  @brief Nullifies the internal matrix \f$ M \f$ and returns a pointer
-        to it.  Used for transfering ownership. */
+        to it.  Used for transferring ownership. */
   SparseMatrix *LoseMat() { SparseMatrix *tmp = mat; mat = NULL; return tmp; }

   /** @brief Returns a const reference to the sparse matrix of eliminated b.c.:
@@ -774,7 +775,7 @@ public:
   SparseMatrix &SpMat() { return *mat; }

   /**  @brief Nullifies the internal matrix \f$ M \f$ and returns a pointer
-        to it.  Used for transfering ownership. */
+        to it.  Used for transferring ownership. */
   SparseMatrix *LoseMat() { SparseMatrix *tmp = mat; mat = NULL; return tmp; }

   /// Adds a domain integrator. Assumes ownership of @a bfi.
@@ -18,6 +18,8 @@
 #include "pgridfunc.hpp"
 #include "ceed/interface/util.hpp"

+#include "../general/nvtx.hpp"
+
 namespace mfem
 {

@@ -160,7 +162,7 @@ void MFBilinearFormExtension::Mult(const Vector &x, Vector &y) const
         {
            intFaceIntegrators[i]->AddMultMF(int_face_X, int_face_Y);
         }
-         int_face_restrict_lex->AddMultTranspose(int_face_Y, y);
+         int_face_restrict_lex->AddMultTransposeInPlace(int_face_Y, y);
      }
   }

@@ -176,7 +178,7 @@ void MFBilinearFormExtension::Mult(const Vector &x, Vector &y) const
         {
            bdrFaceIntegrators[i]->AddMultMF(bdr_face_X, bdr_face_Y);
         }
-         bdr_face_restrict_lex->AddMultTranspose(bdr_face_Y, y);
+         bdr_face_restrict_lex->AddMultTransposeInPlace(bdr_face_Y, y);
      }
   }
 }
@@ -217,7 +219,7 @@ void MFBilinearFormExtension::MultTranspose(const Vector &x, Vector &y) const
         {
            intFaceIntegrators[i]->AddMultTransposeMF(int_face_X, int_face_Y);
         }
-         int_face_restrict_lex->AddMultTranspose(int_face_Y, y);
+         int_face_restrict_lex->AddMultTransposeInPlace(int_face_Y, y);
      }
   }

@@ -233,7 +235,7 @@ void MFBilinearFormExtension::MultTranspose(const Vector &x, Vector &y) const
         {
            bdrFaceIntegrators[i]->AddMultTransposeMF(bdr_face_X, bdr_face_Y);
         }
-         bdr_face_restrict_lex->AddMultTranspose(bdr_face_Y, y);
+         bdr_face_restrict_lex->AddMultTransposeInPlace(bdr_face_Y, y);
      }
   }
 }
@@ -289,6 +291,10 @@ void PABilinearFormExtension::SetupRestrictionOperators(const L2FaceValues m)

 void PABilinearFormExtension::Assemble()
 {
+#undef MFEM_NVTX_COLOR
+#define MFEM_NVTX_COLOR NavyBlue
+   NVTX("HO Assemble");
+
   SetupRestrictionOperators(L2FaceValues::DoubleValued);

   Array<BilinearFormIntegrator*> &integrators = *a->GetDBFI();
@@ -383,6 +389,10 @@ void PABilinearFormExtension::FormLinearSystem(const Array<int> &ess_tdof_list,

 void PABilinearFormExtension::Mult(const Vector &x, Vector &y) const
 {
+#undef MFEM_NVTX_COLOR
+#define MFEM_NVTX_COLOR MediumSpringGreen
+   NVTX("HO Apply");
+
   Array<BilinearFormIntegrator*> &integrators = *a->GetDBFI();

   const int iSz = integrators.Size();
@@ -418,7 +428,7 @@ void PABilinearFormExtension::Mult(const Vector &x, Vector &y) const
         {
            intFaceIntegrators[i]->AddMultPA(int_face_X, int_face_Y);
         }
-         int_face_restrict_lex->AddMultTranspose(int_face_Y, y);
+         int_face_restrict_lex->AddMultTransposeInPlace(int_face_Y, y);
      }
   }

@@ -434,7 +444,7 @@ void PABilinearFormExtension::Mult(const Vector &x, Vector &y) const
         {
            bdrFaceIntegrators[i]->AddMultPA(bdr_face_X, bdr_face_Y);
         }
-         bdr_face_restrict_lex->AddMultTranspose(bdr_face_Y, y);
+         bdr_face_restrict_lex->AddMultTransposeInPlace(bdr_face_Y, y);
      }
   }
 }
@@ -475,7 +485,7 @@ void PABilinearFormExtension::MultTranspose(const Vector &x, Vector &y) const
         {
            intFaceIntegrators[i]->AddMultTransposePA(int_face_X, int_face_Y);
         }
-         int_face_restrict_lex->AddMultTranspose(int_face_Y, y);
+         int_face_restrict_lex->AddMultTransposeInPlace(int_face_Y, y);
      }
   }

@@ -491,7 +501,7 @@ void PABilinearFormExtension::MultTranspose(const Vector &x, Vector &y) const
         {
            bdrFaceIntegrators[i]->AddMultTransposePA(bdr_face_X, bdr_face_Y);
         }
-         bdr_face_restrict_lex->AddMultTranspose(bdr_face_Y, y);
+         bdr_face_restrict_lex->AddMultTransposeInPlace(bdr_face_Y, y);
      }
   }
 }
@@ -668,7 +678,7 @@ void EABilinearFormExtension::Mult(const Vector &x, Vector &y) const
            Y(j, 0, f) += res;
         });
         // Apply the Interior Face Restriction transposed
-         int_face_restrict_lex->AddMultTranspose(int_face_Y, y);
+         int_face_restrict_lex->AddMultTransposeInPlace(int_face_Y, y);
      }
   }

@@ -699,7 +709,7 @@ void EABilinearFormExtension::Mult(const Vector &x, Vector &y) const
            Y(j, f) += res;
         });
         // Apply the Boundary Face Restriction transposed
-         bdr_face_restrict_lex->AddMultTranspose(bdr_face_Y, y);
+         bdr_face_restrict_lex->AddMultTransposeInPlace(bdr_face_Y, y);
      }
   }
 }
@@ -796,7 +806,7 @@ void EABilinearFormExtension::MultTranspose(const Vector &x, Vector &y) const
            Y(j, 0, f) += res;
         });
         // Apply the Interior Face Restriction transposed
-         int_face_restrict_lex->AddMultTranspose(int_face_Y, y);
+         int_face_restrict_lex->AddMultTransposeInPlace(int_face_Y, y);
      }
   }

@@ -827,7 +837,7 @@ void EABilinearFormExtension::MultTranspose(const Vector &x, Vector &y) const
            Y(j, f) += res;
         });
         // Apply the Boundary Face Restriction transposed
-         bdr_face_restrict_lex->AddMultTranspose(bdr_face_Y, y);
+         bdr_face_restrict_lex->AddMultTransposeInPlace(bdr_face_Y, y);
      }
   }
 }
@@ -975,6 +985,9 @@ void FABilinearFormExtension::RAP(OperatorHandle &A)
 void FABilinearFormExtension::EliminateBC(const Array<int> &ess_dofs,
                                          OperatorHandle &A)
 {
+   MFEM_VERIFY(a->diag_policy == DiagonalPolicy::DIAG_ONE,
+               "Only DiagonalPolicy::DIAG_ONE supported with"
+               " FABilinearFormExtension.");
 #ifdef MFEM_USE_MPI
   if ( dynamic_cast<ParBilinearForm*>(a) )
   {
@@ -2003,6 +2003,83 @@ void CurlCurlIntegrator::AssembleElementMatrix
   }
 }

+void CurlCurlIntegrator::AssembleElementMatrix2(const FiniteElement &trial_fe,
+                                                const FiniteElement &test_fe,
+                                                ElementTransformation &Trans,
+                                                DenseMatrix &elmat)
+{
+   int tr_nd = trial_fe.GetDof();
+   int te_nd = test_fe.GetDof();
+   dim = trial_fe.GetDim();
+   int dimc = trial_fe.GetCurlDim();
+   double w;
+
+#ifdef MFEM_THREAD_SAFE
+   Vector D;
+   DenseMatrix curlshape(tr_nd,dimc), curlshape_dFt(tr_nd,dimc), M;
+   DenseMatrix te_curlshape(te_nd,dimc), te_curlshape_dFt(te_nd,dimc);
+#else
+   curlshape.SetSize(tr_nd,dimc);
+   curlshape_dFt.SetSize(tr_nd,dimc);
+   te_curlshape.SetSize(te_nd,dimc);
+   te_curlshape_dFt.SetSize(te_nd,dimc);
+#endif
+   elmat.SetSize(te_nd, tr_nd);
+
+   if (MQ) { M.SetSize(dimc); }
+   if (DQ) { D.SetSize(dimc); }
+
+   const IntegrationRule *ir = IntRule;
+   if (ir == NULL)
+   {
+      int order;
+      if (trial_fe.Space() == FunctionSpace::Pk)
+      {
+         order = test_fe.GetOrder() + trial_fe.GetOrder() - 2;
+      }
+      else
+      {
+         order = test_fe.GetOrder() + trial_fe.GetOrder() + trial_fe.GetDim() - 1;
+      }
+      ir = &IntRules.Get(trial_fe.GetGeomType(), order);
+   }
+
+   elmat = 0.0;
+   for (int i = 0; i < ir->GetNPoints(); i++)
+   {
+      const IntegrationPoint &ip = ir->IntPoint(i);
+
+      Trans.SetIntPoint(&ip);
+
+      w = ip.weight * Trans.Weight();
+      trial_fe.CalcPhysCurlShape(Trans, curlshape_dFt);
+      test_fe.CalcPhysCurlShape(Trans, te_curlshape_dFt);
+
+      if (MQ)
+      {
+         MQ->Eval(M, Trans, ip);
+         M *= w;
+         Mult(te_curlshape_dFt, M, te_curlshape);
+         AddMultABt(te_curlshape, curlshape_dFt, elmat);
+      }
+      else if (DQ)
+      {
+         DQ->Eval(D, Trans, ip);
+         D *= w;
+         AddMultADBt(te_curlshape_dFt,D,curlshape_dFt,elmat);
+      }
+      else
+      {
+         if (Q)
+         {
+            w *= Q->Eval(Trans, ip);
+         }
+         curlshape_dFt *= w;
+         AddMultABt(te_curlshape_dFt, curlshape_dFt, elmat);
+      }
+   }
+}
+
 void CurlCurlIntegrator
 ::ComputeElementFlux(const FiniteElement &el, ElementTransformation &Trans,
                     Vector &u, const FiniteElement &fluxelem, Vector &flux,
@@ -2240,6 +2317,84 @@ double VectorCurlCurlIntegrator::GetElementEnergy(
   return 0.5 * energy;
 }

+void MixedCurlIntegrator::AssembleElementMatrix2(
+   const FiniteElement &trial_fe, const FiniteElement &test_fe,
+   ElementTransformation &Trans, DenseMatrix &elmat)
+{
+   int dim = trial_fe.GetDim();
+   int trial_dof = trial_fe.GetDof();
+   int test_dof = test_fe.GetDof();
+   int dimc = (dim == 3) ? 3 : 1;
+
+   MFEM_VERIFY(trial_fe.GetMapType() == mfem::FiniteElement::H_CURL ||
+               (dim == 2 && trial_fe.GetMapType() == mfem::FiniteElement::VALUE),
+               "Trial finite element must be either 2D/3D H(Curl) or 2D H1");
+   MFEM_VERIFY(test_fe.GetMapType() == mfem::FiniteElement::VALUE ||
+               test_fe.GetMapType() == mfem::FiniteElement::INTEGRAL,
+               "Test finite element must be in H1/L2");
+
+   bool spaceH1 = (trial_fe.GetMapType() == mfem::FiniteElement::VALUE);
+
+   if (spaceH1)
+   {
+      dshape.SetSize(trial_dof,dim);
+      curlshape.SetSize(dim*trial_dof,1);
+      dimc = dim;
+   }
+   else
+   {
+      curlshape.SetSize(trial_dof,dimc);
+      elmat_comp.SetSize(test_dof, trial_dof);
+   }
+   elmat.SetSize(dimc * test_dof, trial_dof);
+   shape.SetSize(test_dof);
+   elmat = 0.0;
+
+   double c;
+   Vector d_col;
+   const IntegrationRule *ir = IntRule;
+
+   if (ir == NULL)
+   {
+      int order = trial_fe.GetOrder() + test_fe.GetOrder() + Trans.OrderJ();
+      ir = &IntRules.Get(trial_fe.GetGeomType(), order);
+   }
+
+   for (int i = 0; i < ir->GetNPoints(); i++)
+   {
+      const IntegrationPoint &ip = ir->IntPoint(i);
+      Trans.SetIntPoint(&ip);
+      if (spaceH1)
+      {
+         trial_fe.CalcPhysDShape(Trans, dshape);
+         dshape.GradToCurl(curlshape);
+      }
+      else
+      {
+         trial_fe.CalcPhysCurlShape(Trans, curlshape);
+      }
+      test_fe.CalcPhysShape(Trans, shape);
+      c = ip.weight*Trans.Weight();
+      if (Q)
+      {
+         c *= Q->Eval(Trans, ip);
+      }
+      shape *= c;
+
+      for (int d = 0; d < dimc; ++d)
+      {
+         double * curldata = &(curlshape.GetData())[d*trial_dof];
+         for (int jj = 0; jj < trial_dof; ++jj)
+         {
+            for (int ii = 0; ii < test_dof; ++ii)
+            {
+               elmat(d * test_dof + ii, jj) += shape(ii) * curldata[jj];
+            }
+         }
+      }
+   }
+}
+

 void VectorFEMassIntegrator::AssembleElementMatrix(
   const FiniteElement &el,
@@ -2586,6 +2741,54 @@ void DivDivIntegrator::AssembleElementMatrix(
   }
 }

+void DivDivIntegrator::AssembleElementMatrix2(
+   const FiniteElement &trial_fe,
+   const FiniteElement &test_fe,
+   ElementTransformation &Trans,
+   DenseMatrix &elmat)
+{
+   int tr_nd = trial_fe.GetDof();
+   int te_nd = test_fe.GetDof();
+   double c;
+
+#ifdef MFEM_THREAD_SAFE
+   Vector divshape(tr_nd);
+   Vector te_divshape(te_nd);
+#else
+   divshape.SetSize(tr_nd);
+   te_divshape.SetSize(te_nd);
+#endif
+   elmat.SetSize(te_nd,tr_nd);
+
+   const IntegrationRule *ir = IntRule;
+   if (ir == NULL)
+   {
+      int order = 2 * max(test_fe.GetOrder(),
+                          trial_fe.GetOrder()) - 2; // <--- OK for RTk
+      ir = &IntRules.Get(test_fe.GetGeomType(), order);
+   }
+
+   elmat = 0.0;
+
+   for (int i = 0; i < ir -> GetNPoints(); i++)
+   {
+      const IntegrationPoint &ip = ir->IntPoint(i);
+
+      trial_fe.CalcDivShape(ip,divshape);
+      test_fe.CalcDivShape(ip,te_divshape);
+
+      Trans.SetIntPoint (&ip);
+      c = ip.weight / Trans.Weight();
+
+      if (Q)
+      {
+         c *= Q -> Eval (Trans, ip);
+      }
+
+      te_divshape *= c;
+      AddMultVWt(te_divshape, divshape, elmat);
+   }
+}

 void VectorDiffusionIntegrator::AssembleElementMatrix(
   const FiniteElement &el,
@@ -3780,7 +3983,7 @@ void NormalTraceJumpIntegrator::AssembleFaceMatrix(
      for (i = 0; i < ndof1; i++)
         for (j = 0; j < face_ndof; j++)
         {
-            elmat(i, j) -= shape1_n(i) * face_shape(j);
+            elmat(i, j) += shape1_n(i) * face_shape(j);
         }
      if (ndof2)
      {
@@ -3788,7 +3991,7 @@ void NormalTraceJumpIntegrator::AssembleFaceMatrix(
         for (i = 0; i < ndof2; i++)
            for (j = 0; j < face_ndof; j++)
            {
-               elmat(ndof1+i, j) += shape2_n(i) * face_shape(j);
+               elmat(ndof1+i, j) -= shape2_n(i) * face_shape(j);
            }
      }
   }
@@ -215,10 +215,10 @@ public:
                             function by any coefficients describing the
                             integrator.
       @param[in] ir  If passed (the default value is NULL), the implementation
-                        of the method will ignore the integration rule provided
-                        by the @a fluxelem parameter and, instead, compute the
-                        discrete flux at the points specified by the integration
-                        rule @a ir.
+                      of the method will ignore the integration rule provided
+                      by the @a fluxelem parameter and, instead, compute the
+                      discrete flux at the points specified by the integration
+                      rule @a ir.
    */
   virtual void ComputeElementFlux(const FiniteElement &el,
                                   ElementTransformation &Trans,
@@ -2174,6 +2174,7 @@ public:
 /** Class for local mass matrix assembling a(u,v) := (Q u, v) */
 class MassIntegrator: public BilinearFormIntegrator
 {
+   friend class DGMassInverse;
 protected:
 #ifndef MFEM_THREAD_SAFE
   Vector shape, te_shape;
@@ -2524,6 +2525,7 @@ private:
 #ifndef MFEM_THREAD_SAFE
   Vector D;
   DenseMatrix curlshape, curlshape_dFt, M;
+   DenseMatrix te_curlshape, te_curlshape_dFt;
   DenseMatrix vshape, projcurl;
 #endif

@@ -2557,6 +2559,11 @@ public:
                                      ElementTransformation &Trans,
                                      DenseMatrix &elmat);

+   virtual void AssembleElementMatrix2(const FiniteElement &trial_fe,
+                                       const FiniteElement &test_fe,
+                                       ElementTransformation &Trans,
+                                       DenseMatrix &elmat);
+
   virtual void ComputeElementFlux(const FiniteElement &el,
                                   ElementTransformation &Trans,
                                   Vector &u, const FiniteElement &fluxelem,
@@ -2602,6 +2609,35 @@ public:
                                   const Vector &elfun);
 };

+/** Class for integrating the bilinear form a(u,v) := (Q curl u, v) where Q is
+    an optional scalar coefficient, and v is a vector with components v_i in
+    the L2 or H1 space. This integrator handles 3 cases:
+    (a) u ∈ H(curl) in 3D, v is a 3D vector with components v_i in L^2 or H^1
+    (b) u ∈ H(curl) in 2D, v is a scalar field in L^2 or H^1
+    (c) u is a scalar field in H^1, i.e, curl u := [0 1;-1 0]grad u and v is a
+        2D vector field with components v_i in L^2 or H^1 space.
+    Note: Case (b) can also be handled by MixedScalarCurlIntegrator  */
+class MixedCurlIntegrator : public BilinearFormIntegrator
+{
+protected:
+   Coefficient *Q;
+
+private:
+   Vector shape;
+   DenseMatrix dshape;
+   DenseMatrix curlshape;
+   DenseMatrix elmat_comp;
+public:
+   MixedCurlIntegrator() : Q{NULL} { }
+   MixedCurlIntegrator(Coefficient *q_) :  Q{q_} { }
+   MixedCurlIntegrator(Coefficient &q) :  Q{&q} { }
+
+   virtual void AssembleElementMatrix2(const FiniteElement &trial_fe,
+                                       const FiniteElement &test_fe,
+                                       ElementTransformation &Trans,
+                                       DenseMatrix &elmat);
+};
+
 /** Integrator for (Q u, v), where Q is an optional coefficient (of type scalar,
    vector (diagonal matrix), or matrix), trial function u is in H(Curl) or
    H(Div), and test function v is in H(Curl), H(Div), or v=(v1,...,vn), where
@@ -2725,7 +2761,7 @@ protected:

 private:
 #ifndef MFEM_THREAD_SAFE
-   Vector divshape;
+   Vector divshape, te_divshape;
 #endif

   // PA extension
@@ -2743,6 +2779,12 @@ public:
   virtual void AssembleElementMatrix(const FiniteElement &el,
                                      ElementTransformation &Trans,
                                      DenseMatrix &elmat);
+
+   virtual void AssembleElementMatrix2(const FiniteElement &trial_fe,
+                                       const FiniteElement &test_fe,
+                                       ElementTransformation &Trans,
+                                       DenseMatrix &elmat);
+
   const Coefficient *GetCoefficient() const { return Q; }
 };

@@ -12,6 +12,7 @@
 #include "../general/forall.hpp"
 #include "bilininteg.hpp"
 #include "gridfunc.hpp"
+#include "qfunction.hpp"
 #include "ceed/integrators/convection/convection.hpp"
 #include "quadinterpolator.hpp"

@@ -1408,66 +1409,10 @@ void ConvectionIntegrator::AssemblePA(const FiniteElementSpace &fes)
   dofs1D = maps->ndof;
   quad1D = maps->nqpt;
   pa_data.SetSize(symmDims * nq * ne, mt);
-   Vector vel;
-   if (VectorConstantCoefficient *cQ =
-          dynamic_cast<VectorConstantCoefficient*>(Q))
-   {
-      vel = cQ->GetVec();
-   }
-   else if (VectorGridFunctionCoefficient *vgfQ =
-               dynamic_cast<VectorGridFunctionCoefficient*>(Q))
-   {
-      vel.SetSize(dim * nq * ne, mt);

-      const GridFunction *gf = vgfQ->GetGridFunction();
-      const FiniteElementSpace &gf_fes = *gf->FESpace();
-      const QuadratureInterpolator *qi(gf_fes.GetQuadratureInterpolator(*ir));
-      const bool use_tensor_products = UsesTensorBasis(gf_fes);
-      const ElementDofOrdering ordering = use_tensor_products ?
-                                          ElementDofOrdering::LEXICOGRAPHIC :
-                                          ElementDofOrdering::NATIVE;
-      const Operator *R = gf_fes.GetElementRestriction(ordering);
+   QuadratureSpace qs(*mesh, *ir);
+   CoefficientVector vel(*Q, qs, CoefficientStorage::COMPRESSED);

-      Vector xe(R->Height(), mt);
-      xe.UseDevice(true);
-
-      R->Mult(*gf, xe);
-      qi->SetOutputLayout(QVectorLayout::byVDIM);
-      qi->DisableTensorProducts(!use_tensor_products);
-      qi->Values(xe,vel);
-   }
-   else if (VectorQuadratureFunctionCoefficient* vqfQ =
-               dynamic_cast<VectorQuadratureFunctionCoefficient*>(Q))
-   {
-      const QuadratureFunction &qFun = vqfQ->GetQuadFunction();
-      MFEM_VERIFY(qFun.Size() == dim * nq * ne,
-                  "Incompatible QuadratureFunction dimension \n");
-
-      MFEM_VERIFY(ir == &qFun.GetSpace()->GetElementIntRule(0),
-                  "IntegrationRule used within integrator and in"
-                  " QuadratureFunction appear to be different");
-
-      qFun.Read();
-      vel.MakeRef(const_cast<QuadratureFunction &>(qFun),0);
-   }
-   else
-   {
-      vel.SetSize(dim * nq * ne);
-      auto C = Reshape(vel.HostWrite(), dim, nq, ne);
-      DenseMatrix MQ_ir;
-      for (int e = 0; e < ne; ++e)
-      {
-         ElementTransformation& T = *fes.GetElementTransformation(e);
-         Q->Eval(MQ_ir, T, *ir);
-         for (int q = 0; q < nq; ++q)
-         {
-            for (int i = 0; i < dim; ++i)
-            {
-               C(i,q,e) = MQ_ir(i,q);
-            }
-         }
-      }
-   }
   PAConvectionSetup(dim, nq, ne, ir->GetWeights(), geom->J,
                     vel, alpha, pa_data);
 }
@@ -12,6 +12,7 @@
 #include "../general/forall.hpp"
 #include "bilininteg.hpp"
 #include "gridfunc.hpp"
+#include "qfunction.hpp"
 #include "restriction.hpp"

 using namespace std;
@@ -161,88 +162,24 @@ void DGTraceIntegrator::SetupPA(const FiniteElementSpace &fes, FaceType type)
   dofs1D = maps->ndof;
   quad1D = maps->nqpt;
   pa_data.SetSize(symmDims * nq * nf, Device::GetMemoryType());
-   Vector vel;
-   if (VectorConstantCoefficient *c_u = dynamic_cast<VectorConstantCoefficient*>
-                                        (u))
-   {
-      vel = c_u->GetVec();
-   }
-   else if (VectorQuadratureFunctionCoefficient* qf_u =
-               dynamic_cast<VectorQuadratureFunctionCoefficient*>(u))
-   {
-      // Assumed to be in lexicographical ordering
-      const QuadratureFunction &qFun = qf_u->GetQuadFunction();
-      MFEM_VERIFY(qFun.Size() == dim * nq * nf,
-                  "Incompatible QuadratureFunction dimension \n");

-      MFEM_VERIFY(ir == &qFun.GetSpace()->GetElementIntRule(0),
-                  "IntegrationRule used within integrator and in"
-                  " QuadratureFunction appear to be different");
-      qFun.Read();
-      vel.MakeRef(const_cast<QuadratureFunction &>(qFun),0);
-   }
-   else
+   FaceQuadratureSpace qs(*mesh, *ir, type);
+   CoefficientVector vel(*u, qs, CoefficientStorage::COMPRESSED);
+
+   CoefficientVector r(qs, CoefficientStorage::COMPRESSED);
+   if (rho == nullptr)
   {
-      vel.SetSize(dim * nq * nf);
-      auto C = Reshape(vel.HostWrite(), dim, nq, nf);
-      Vector Vq(dim);
-      int f_ind = 0;
-      for (int f = 0; f < mesh->GetNumFacesWithGhost(); ++f)
-      {
-         Mesh::FaceInformation face = mesh->GetFaceInformation(f);
-         if (face.IsNonconformingCoarse())
-         {
-            // We skip nonconforming coarse faces as they are treated
-            // by the corresponding nonconforming fine faces.
-            continue;
-         }
-         else if ( face.IsOfFaceType(type) )
-         {
-            const int mask = FaceElementTransformations::HAVE_ELEM1 |
-                             FaceElementTransformations::HAVE_LOC1;
-            FaceElementTransformations &T =
-               *fes.GetMesh()->GetFaceElementTransformations(f, mask);
-            for (int q = 0; q < nq; ++q)
-            {
-               // Convert to lexicographic ordering
-               int iq = ToLexOrdering(dim, face.element[0].local_face_id,
-                                      quad1D, q);
-               T.SetAllIntPoints(&ir->IntPoint(q));
-               const IntegrationPoint &eip1 = T.GetElement1IntPoint();
-               u->Eval(Vq, *T.Elem1, eip1);
-               for (int i = 0; i < dim; ++i)
-               {
-                  C(i,iq,f_ind) = Vq(i);
-               }
-            }
-            f_ind++;
-         }
-      }
-      MFEM_VERIFY(f_ind==nf, "Incorrect number of faces.");
+      r.SetConstant(1.0);
   }
-   Vector r;
-   if (rho==nullptr)
+   else if (ConstantCoefficient *const_rho = dynamic_cast<ConstantCoefficient*>
+                                             (rho))
   {
-      r.SetSize(1);
-      r(0) = 1.0;
-   }
-   else if (ConstantCoefficient *c_rho = dynamic_cast<ConstantCoefficient*>(rho))
-   {
-      r.SetSize(1);
-      r(0) = c_rho->constant;
+      r.SetConstant(const_rho->constant);
   }
   else if (QuadratureFunctionCoefficient* qf_rho =
               dynamic_cast<QuadratureFunctionCoefficient*>(rho))
   {
-      const QuadratureFunction &qFun = qf_rho->GetQuadFunction();
-      MFEM_VERIFY(qFun.Size() == nq * nf,
-                  "Incompatible QuadratureFunction dimension \n");
-
-      MFEM_VERIFY(ir == &qFun.GetSpace()->GetElementIntRule(0),
-                  "IntegrationRule used within integrator and in"
-                  " QuadratureFunction appear to be different");
-      qFun.Read();
-      r.MakeRef(const_cast<QuadratureFunction &>(qFun),0);
+      r.MakeRef(qf_rho->GetQuadFunction());
   }
   else
   {
@@ -254,45 +191,42 @@ void DGTraceIntegrator::SetupPA(const FiniteElementSpace &fes, FaceType type)
      for (int f = 0; f < mesh->GetNumFacesWithGhost(); ++f)
      {
         Mesh::FaceInformation face = mesh->GetFaceInformation(f);
-         if (face.IsNonconformingCoarse())
+         if (face.IsNonconformingCoarse() || !face.IsOfFaceType(type))
         {
            // We skip nonconforming coarse faces as they are treated
            // by the corresponding nonconforming fine faces.
            continue;
         }
-         else if ( face.IsOfFaceType(type) )
+         FaceElementTransformations &T =
+            *fes.GetMesh()->GetFaceElementTransformations(f);
+         for (int q = 0; q < nq; ++q)
         {
-            FaceElementTransformations &T =
-               *fes.GetMesh()->GetFaceElementTransformations(f);
-            for (int q = 0; q < nq; ++q)
+            // Convert to lexicographic ordering
+            int iq = ToLexOrdering(dim, face.element[0].local_face_id,
+                                   quad1D, q);
+
+            T.SetAllIntPoints(&ir->IntPoint(q));
+            const IntegrationPoint &eip1 = T.GetElement1IntPoint();
+            const IntegrationPoint &eip2 = T.GetElement2IntPoint();
+            double rq;
+
+            if (face.IsBoundary())
            {
-               // Convert to lexicographic ordering
-               int iq = ToLexOrdering(dim, face.element[0].local_face_id,
-                                      quad1D, q);
-
-               T.SetAllIntPoints(&ir->IntPoint(q));
-               const IntegrationPoint &eip1 = T.GetElement1IntPoint();
-               const IntegrationPoint &eip2 = T.GetElement2IntPoint();
-               double rq;
-
-               if ( face.IsBoundary() )
-               {
-                  rq = rho->Eval(*T.Elem1, eip1);
-               }
-               else
-               {
-                  double udotn = 0.0;
-                  for (int d=0; d<dim; ++d)
-                  {
-                     udotn += C_vel(d,iq,f_ind)*n(iq,d,f_ind);
-                  }
-                  if (udotn >= 0.0) { rq = rho->Eval(*T.Elem2, eip2); }
-                  else { rq = rho->Eval(*T.Elem1, eip1); }
-               }
-               C(iq,f_ind) = rq;
+               rq = rho->Eval(*T.Elem1, eip1);
            }
-            f_ind++;
+            else
+            {
+               double udotn = 0.0;
+               for (int d=0; d<dim; ++d)
+               {
+                  udotn += C_vel(d,iq,f_ind)*n(iq,d,f_ind);
+               }
+               if (udotn >= 0.0) { rq = rho->Eval(*T.Elem2, eip2); }
+               else { rq = rho->Eval(*T.Elem1, eip1); }
+            }
+            C(iq,f_ind) = rq;
         }
+         f_ind++;
      }
      MFEM_VERIFY(f_ind==nf, "Incorrect number of faces.");
   }
@@ -12,6 +12,7 @@
 #include "../general/forall.hpp"
 #include "bilininteg.hpp"
 #include "gridfunc.hpp"
+#include "qfunction.hpp"
 #include "ceed/integrators/diffusion/diffusion.hpp"

 using namespace std;
@@ -390,120 +391,21 @@ void DiffusionIntegrator::AssemblePA(const FiniteElementSpace &fes)
   maps = &el.GetDofToQuad(*ir, DofToQuad::TENSOR);
   dofs1D = maps->ndof;
   quad1D = maps->nqpt;
-   int coeffDim = 1;
-   Vector coeff;
-   const int MQfullDim = MQ ? MQ->GetHeight() * MQ->GetWidth() : 0;
-   if (auto *SMQ = dynamic_cast<SymmetricMatrixCoefficient *>(MQ))
-   {
-      MFEM_VERIFY(SMQ->GetSize() == dim, "");
-      coeffDim = symmDims;
-      coeff.SetSize(symmDims * nq * ne);

-      DenseSymmetricMatrix sym_mat;
-      sym_mat.SetSize(dim);
+   QuadratureSpace qs(*mesh, *ir);
+   CoefficientVector coeff(qs, CoefficientStorage::COMPRESSED);

-      auto C = Reshape(coeff.HostWrite(), symmDims, nq, ne);
+   if (MQ) { coeff.ProjectTranspose(*MQ); }
+   else if (VQ) { coeff.Project(*VQ); }
+   else if (Q) { coeff.Project(*Q); }
+   else { coeff.SetConstant(1.0); }

-      for (int e=0; e<ne; ++e)
-      {
-         ElementTransformation *tr = mesh->GetElementTransformation(e);
-         for (int p=0; p<nq; ++p)
-         {
-            SMQ->Eval(sym_mat, *tr, ir->IntPoint(p));
-            int cnt = 0;
-            for (int i=0; i<dim; ++i)
-               for (int j=i; j<dim; ++j, ++cnt)
-               {
-                  C(cnt, p, e) = sym_mat(i,j);
-               }
-         }
-      }
-   }
-   else if (MQ)
-   {
-      symmetric = false;
-      MFEM_VERIFY(MQ->GetHeight() == dim && MQ->GetWidth() == dim, "");
+   const int coeff_dim = coeff.GetVDim();
+   symmetric = (coeff_dim != dims*dims);
+   const int pa_size = symmetric ? symmDims : dims*dims;

-      coeffDim = MQfullDim;
-
-      coeff.SetSize(MQfullDim * nq * ne);
-
-      DenseMatrix mat;
-      mat.SetSize(dim);
-
-      auto C = Reshape(coeff.HostWrite(), MQfullDim, nq, ne);
-      for (int e=0; e<ne; ++e)
-      {
-         ElementTransformation *tr = mesh->GetElementTransformation(e);
-         for (int p=0; p<nq; ++p)
-         {
-            MQ->Eval(mat, *tr, ir->IntPoint(p));
-            for (int i=0; i<dim; ++i)
-               for (int j=0; j<dim; ++j)
-               {
-                  C(j+(i*dim), p, e) = mat(i,j);
-               }
-         }
-      }
-   }
-   else if (VQ)
-   {
-      MFEM_VERIFY(VQ->GetVDim() == dim, "");
-      coeffDim = VQ->GetVDim();
-      coeff.SetSize(coeffDim * nq * ne);
-      auto C = Reshape(coeff.HostWrite(), coeffDim, nq, ne);
-      Vector DM(coeffDim);
-      for (int e=0; e<ne; ++e)
-      {
-         ElementTransformation *tr = mesh->GetElementTransformation(e);
-         for (int p=0; p<nq; ++p)
-         {
-            VQ->Eval(DM, *tr, ir->IntPoint(p));
-            for (int i=0; i<coeffDim; ++i)
-            {
-               C(i, p, e) = DM[i];
-            }
-         }
-      }
-   }
-   else if (Q == nullptr)
-   {
-      coeff.SetSize(1);
-      coeff(0) = 1.0;
-   }
-   else if (ConstantCoefficient* cQ = dynamic_cast<ConstantCoefficient*>(Q))
-   {
-      coeff.SetSize(1);
-      coeff(0) = cQ->constant;
-   }
-   else if (QuadratureFunctionCoefficient* qfQ =
-               dynamic_cast<QuadratureFunctionCoefficient*>(Q))
-   {
-      const QuadratureFunction &qFun = qfQ->GetQuadFunction();
-      MFEM_VERIFY(qFun.Size() == ne*nq,
-                  "Incompatible QuadratureFunction dimension \n");
-
-      MFEM_VERIFY(ir == &qFun.GetSpace()->GetElementIntRule(0),
-                  "IntegrationRule used within integrator and in"
-                  " QuadratureFunction appear to be different");
-      qFun.Read();
-      coeff.MakeRef(const_cast<QuadratureFunction &>(qFun),0);
-   }
-   else
-   {
-      coeff.SetSize(nq * ne);
-      auto C = Reshape(coeff.HostWrite(), nq, ne);
-      for (int e = 0; e < ne; ++e)
-      {
-         ElementTransformation& T = *fes.GetElementTransformation(e);
-         for (int q = 0; q < nq; ++q)
-         {
-            C(q,e) = Q->Eval(T, ir->IntPoint(q));
-         }
-      }
-   }
-   pa_data.SetSize((symmetric ? symmDims : MQfullDim) * nq * ne, mt);
-   PADiffusionSetup(dim, sdim, dofs1D, quad1D, coeffDim, ne, ir->GetWeights(),
+   pa_data.SetSize(pa_size * nq * ne, mt);
+   PADiffusionSetup(dim, sdim, dofs1D, quad1D, coeff_dim, ne, ir->GetWeights(),
                    geom->J, coeff, pa_data);
 }

@@ -1784,7 +1686,7 @@ static void PADiffusionApply(const int dim,
         case 0x77: return SmemPADiffusionApply2D<7,7,4>(NE,symm,B,G,D,X,Y);
         case 0x88: return SmemPADiffusionApply2D<8,8,2>(NE,symm,B,G,D,X,Y);
         case 0x99: return SmemPADiffusionApply2D<9,9,2>(NE,symm,B,G,D,X,Y);
-         default:   return PADiffusionApply2D(NE,symm,B,G,Bt,Gt,D,X,Y,D1D,Q1D);
+            // default:   return PADiffusionApply2D(NE,symm,B,G,Bt,Gt,D,X,Y,D1D,Q1D);
      }
   }

@@ -1802,7 +1704,14 @@ static void PADiffusionApply(const int dim,
         case 0x67: return SmemPADiffusionApply3D<6,7>(NE,symm,B,G,D,X,Y);
         case 0x78: return SmemPADiffusionApply3D<7,8>(NE,symm,B,G,D,X,Y);
         case 0x89: return SmemPADiffusionApply3D<8,9>(NE,symm,B,G,D,X,Y);
-         default:   return PADiffusionApply3D(NE,symm,B,G,Bt,Gt,D,X,Y,D1D,Q1D);
+         case 0x33: return SmemPADiffusionApply3D<3,3>(NE,symm,B,G,D,X,Y);
+         case 0x44: return SmemPADiffusionApply3D<4,4>(NE,symm,B,G,D,X,Y);
+         case 0x55: return SmemPADiffusionApply3D<5,5>(NE,symm,B,G,D,X,Y);
+         case 0x66: return SmemPADiffusionApply3D<6,6>(NE,symm,B,G,D,X,Y);
+         case 0x77: return SmemPADiffusionApply3D<7,7>(NE,symm,B,G,D,X,Y);
+         case 0x88: return SmemPADiffusionApply3D<8,8>(NE,symm,B,G,D,X,Y);
+         case 0x99: return SmemPADiffusionApply3D<9,9>(NE,symm,B,G,D,X,Y);
+            // default:   return PADiffusionApply3D(NE,symm,B,G,Bt,Gt,D,X,Y,D1D,Q1D);
      }
   }
   MFEM_ABORT("Unknown kernel: 0x"<<std::hex << id << std::dec);
@@ -12,6 +12,7 @@
 #include "../general/forall.hpp"
 #include "bilininteg.hpp"
 #include "gridfunc.hpp"
+#include "qfunction.hpp"

 using namespace std;

@@ -209,44 +210,8 @@ void GradientIntegrator::AssemblePA(const FiniteElementSpace &trial_fes,
               "PA requires test and trial space to have same number of quadrature points!");
   pa_data.SetSize(nq * dimsToStore * ne, Device::GetMemoryType());

-   Vector coeff;
-
-   if (Q == nullptr)
-   {
-      coeff.SetSize(1);
-      coeff(0) = 1.0;
-   }
-   else if (ConstantCoefficient* cQ = dynamic_cast<ConstantCoefficient*>(Q))
-   {
-      coeff.SetSize(1);
-      coeff(0) = cQ->constant;
-   }
-   else if (QuadratureFunctionCoefficient* qfQ =
-               dynamic_cast<QuadratureFunctionCoefficient*>(Q))
-   {
-      const QuadratureFunction &qFun = qfQ->GetQuadFunction();
-      MFEM_VERIFY(qFun.Size() == ne*nq,
-                  "Incompatible QuadratureFunction dimension \n");
-
-      MFEM_VERIFY(ir == &qFun.GetSpace()->GetElementIntRule(0),
-                  "IntegrationRule used within integrator and in"
-                  " QuadratureFunction appear to be different");
-      qFun.Read();
-      coeff.MakeRef(const_cast<QuadratureFunction &>(qFun),0);
-   }
-   else
-   {
-      coeff.SetSize(nq * ne);
-      auto C = Reshape(coeff.HostWrite(), nq, ne);
-      for (int e = 0; e < ne; ++e)
-      {
-         ElementTransformation& T = *trial_fes.GetElementTransformation(e);
-         for (int q = 0; q < nq; ++q)
-         {
-            C(q,e) = Q->Eval(T, ir->IntPoint(q));
-         }
-      }
-   }
+   QuadratureSpace qs(*mesh, *ir);
+   CoefficientVector coeff(Q, qs, CoefficientStorage::COMPRESSED);

   PAGradientSetup(dim, trial_dofs1D, test_dofs1D, quad1D,
                   ne, ir->GetWeights(), geom->J, coeff, pa_data);
@@ -865,4 +830,3 @@ void GradientIntegrator::AddMultTransposePA(const Vector &x, Vector &y) const
 }

 } // namespace mfem
-
@@ -12,6 +12,7 @@
 #include "../general/forall.hpp"
 #include "bilininteg.hpp"
 #include "gridfunc.hpp"
+#include "qspace.hpp"

 using namespace std;

@@ -967,8 +968,6 @@ void CurlCurlIntegrator::AssemblePA(const FiniteElementSpace &fes)
   dim = mesh->Dimension();
   MFEM_VERIFY(dim == 2 || dim == 3, "");

-   const int dimc = (dim == 3) ? 3 : 1;
-
   ne = fes.GetNE();
   geom = mesh->GetGeometricFactors(*ir, GeometricFactors::JACOBIANS);
   mapsC = &el->GetDofToQuad(*ir, DofToQuad::TENSOR);
@@ -978,88 +977,19 @@ void CurlCurlIntegrator::AssemblePA(const FiniteElementSpace &fes)

   MFEM_VERIFY(dofs1D == mapsO->ndof + 1 && quad1D == mapsO->nqpt, "");

-   auto SMQ = dynamic_cast<SymmetricMatrixCoefficient *>(MQ);
+   QuadratureSpace qs(*mesh, *ir);
+   CoefficientVector coeff(qs, CoefficientStorage::SYMMETRIC);
+   if (Q) { coeff.Project(*Q); }
+   else if (MQ) { coeff.ProjectTranspose(*MQ); }
+   else if (DQ) { coeff.Project(*DQ); }
+   else { coeff.SetConstant(1.0); }

-   const int MQsymmDim = SMQ ? (SMQ->GetSize() * (SMQ->GetSize() + 1)) / 2 : 0;
-   const int MQfullDim = MQ ? (MQ->GetHeight() * MQ->GetWidth()) : 0;
-   const int MQdim = SMQ ? MQsymmDim : MQfullDim;
-   const int coeffDim = MQ ? MQdim : (DQ ? DQ->GetVDim() : 1);
-
-   symmetric = (SMQ || MQ == NULL);
-
-   const int symmDims = (dims * (dims + 1)) / 2; // 1x1: 1, 2x2: 3, 3x3: 6
-   const int ndata = (dim == 2) ? 1 : (symmetric ? symmDims : MQfullDim);
+   const int coeff_dim = coeff.GetVDim();
+   symmetric = (coeff_dim != dim*dim);
+   const int sym_dims = (dims * (dims + 1)) / 2; // 1x1: 1, 2x2: 3, 3x3: 6
+   const int ndata = (dim == 2) ? 1 : (symmetric ? sym_dims : dim*dim);
   pa_data.SetSize(ndata * nq * ne, Device::GetMemoryType());

-   Vector coeff(coeffDim * ne * nq);
-   coeff = 1.0;
-   auto coeffh = Reshape(coeff.HostWrite(), coeffDim, nq, ne);
-   if (Q || DQ || MQ)
-   {
-      Vector DM(DQ ? coeffDim : 0);
-      DenseMatrix GM;
-      DenseSymmetricMatrix SM;
-
-      if (DQ)
-      {
-         MFEM_VERIFY(coeffDim == dimc, "");
-      }
-      if (SMQ)
-      {
-         SM.SetSize(dimc);
-         MFEM_VERIFY(SMQ->GetSize() == dimc, "");
-      }
-      else if (MQ)
-      {
-         GM.SetSize(dimc);
-         MFEM_VERIFY(coeffDim == MQdim, "");
-         MFEM_VERIFY(MQ->GetHeight() == dimc && MQ->GetWidth() == dimc, "");
-      }
-
-      for (int e=0; e<ne; ++e)
-      {
-         ElementTransformation *tr = mesh->GetElementTransformation(e);
-         for (int p=0; p<nq; ++p)
-         {
-            if (SMQ)
-            {
-               SMQ->Eval(SM, *tr, ir->IntPoint(p));
-
-               int cnt = 0;
-               for (int i=0; i<dimc; ++i)
-                  for (int j=i; j<dimc; ++j, ++cnt)
-                  {
-                     coeffh(cnt, p, e) = SM(i,j);
-                  }
-
-            }
-            else if (MQ)
-            {
-               MQ->Eval(GM, *tr, ir->IntPoint(p));
-
-               for (int i=0; i<dimc; ++i)
-                  for (int j=0; j<dimc; ++j)
-                  {
-                     coeffh(j+(i*dimc), p, e) = GM(i,j);
-                  }
-
-            }
-            else if (DQ)
-            {
-               DQ->Eval(DM, *tr, ir->IntPoint(p));
-               for (int i=0; i<coeffDim; ++i)
-               {
-                  coeffh(i, p, e) = DM[i];
-               }
-            }
-            else
-            {
-               coeffh(0, p, e) = Q->Eval(*tr, ir->IntPoint(p));
-            }
-         }
-      }
-   }
-
   if (el->GetDerivType() != mfem::FiniteElement::CURL)
   {
      MFEM_ABORT("Unknown kernel.");
@@ -1067,7 +997,7 @@ void CurlCurlIntegrator::AssemblePA(const FiniteElementSpace &fes)

   if (dim == 3)
   {
-      PACurlCurlSetup3D(quad1D, coeffDim, ne, ir->GetWeights(), geom->J, coeff,
+      PACurlCurlSetup3D(quad1D, coeff_dim, ne, ir->GetWeights(), geom->J, coeff,
                        pa_data);
   }
   else
@@ -2780,7 +2710,7 @@ void CurlCurlIntegrator::AssembleDiagonalPA(Vector& diag)
   }
 }

-// Apply to x corresponding to DOF's in H^1 (trial), whose gradients are
+// Apply to x corresponding to DOFs in H^1 (trial), whose gradients are
 // integrated against H(curl) test functions corresponding to y.
 void PAHcurlH1Apply3D(const int D1D,
                      const int Q1D,
@@ -2970,7 +2900,7 @@ void PAHcurlH1Apply3D(const int D1D,
   }); // end of element loop
 }

-// Apply to x corresponding to DOF's in H(curl), integrated
+// Apply to x corresponding to DOFs in H(curl), integrated
 // against gradients of H^1 functions corresponding to y.
 void PAHcurlH1ApplyTranspose3D(const int D1D,
                               const int Q1D,
@@ -3169,7 +3099,7 @@ void PAHcurlH1ApplyTranspose3D(const int D1D,
   }); // end of element loop
 }

-// Apply to x corresponding to DOF's in H^1 (trial), whose gradients are
+// Apply to x corresponding to DOFs in H^1 (trial), whose gradients are
 // integrated against H(curl) test functions corresponding to y.
 void PAHcurlH1Apply2D(const int D1D,
                      const int Q1D,
@@ -3293,7 +3223,7 @@ void PAHcurlH1Apply2D(const int D1D,
   }); // end of element loop
 }

-// Apply to x corresponding to DOF's in H(curl), integrated
+// Apply to x corresponding to DOFs in H(curl), integrated
 // against gradients of H^1 functions corresponding to y.
 void PAHcurlH1ApplyTranspose2D(const int D1D,
                               const int Q1D,
@@ -3489,20 +3419,8 @@ void MixedScalarCurlIntegrator::AssemblePA(const FiniteElementSpace &trial_fes,

   pa_data.SetSize(nq * ne, Device::GetMemoryType());

-   Vector coeff(ne * nq);
-   coeff = 1.0;
-   auto coeffh = Reshape(coeff.HostWrite(), nq, ne);
-   if (Q)
-   {
-      for (int e=0; e<ne; ++e)
-      {
-         ElementTransformation *tr = mesh->GetElementTransformation(e);
-         for (int p=0; p<nq; ++p)
-         {
-            coeffh(p, e) = Q->Eval(*tr, ir->IntPoint(p));
-         }
-      }
-   }
+   QuadratureSpace qs(*mesh, *ir);
+   CoefficientVector coeff(Q, qs, CoefficientStorage::FULL);

   if (dim == 2)
   {
@@ -3593,38 +3511,11 @@ void MixedVectorCurlIntegrator::AssemblePA(const FiniteElementSpace &trial_fes,
   const int ndata = curlSpaces ? (coeffDim == 1 ? 1 : 9) : symmDims;
   pa_data.SetSize(ndata * nq * ne, Device::GetMemoryType());

-   Vector coeff(coeffDim * nq * ne);
-   coeff = 1.0;
-   auto coeffh = Reshape(coeff.HostWrite(), coeffDim, nq, ne);
-   if (Q || DQ)
-   {
-      Vector V(coeffDim);
-      if (DQ)
-      {
-         MFEM_VERIFY(DQ->GetVDim() == coeffDim, "");
-      }
-
-      for (int e=0; e<ne; ++e)
-      {
-         ElementTransformation *tr = mesh->GetElementTransformation(e);
-
-         for (int p=0; p<nq; ++p)
-         {
-            if (DQ)
-            {
-               DQ->Eval(V, *tr, ir->IntPoint(p));
-               for (int i=0; i<coeffDim; ++i)
-               {
-                  coeffh(i, p, e) = V[i];
-               }
-            }
-            else
-            {
-               coeffh(0, p, e) = Q->Eval(*tr, ir->IntPoint(p));
-            }
-         }
-      }
-   }
+   QuadratureSpace qs(*mesh, *ir);
+   CoefficientVector coeff(qs, CoefficientStorage::FULL);
+   if (Q) { coeff.Project(*Q); }
+   else if (DQ) { coeff.Project(*DQ); }
+   else { coeff.SetConstant(1.0); }

   if (testType == mfem::FiniteElement::CURL &&
       trialType == mfem::FiniteElement::CURL && dim == 3)
@@ -3652,7 +3543,7 @@ void MixedVectorCurlIntegrator::AssemblePA(const FiniteElementSpace &trial_fes,
   }
 }

-// Apply to x corresponding to DOF's in H(curl) (trial), whose curl is
+// Apply to x corresponding to DOFs in H(curl) (trial), whose curl is
 // integrated against H(curl) test functions corresponding to y.
 template<int MAX_D1D = HCURL_MAX_D1D, int MAX_Q1D = HCURL_MAX_Q1D>
 static void PAHcurlL2Apply3D(const int D1D,
@@ -4015,7 +3906,7 @@ static void PAHcurlL2Apply3D(const int D1D,
   }); // end of element loop
 }

-// Apply to x corresponding to DOF's in H(curl) (trial), whose curl is
+// Apply to x corresponding to DOFs in H(curl) (trial), whose curl is
 // integrated against H(curl) test functions corresponding to y.
 template<int MAX_D1D = HCURL_MAX_D1D, int MAX_Q1D = HCURL_MAX_Q1D>
 static void SmemPAHcurlL2Apply3D(const int D1D,
@@ -4325,7 +4216,7 @@ static void SmemPAHcurlL2Apply3D(const int D1D,
   ForallWrap<3>(true, NE, device_kernel, host_kernel, Q1D, Q1D, Q1D);
 }

-// Apply to x corresponding to DOF's in H(curl) (trial), whose curl is
+// Apply to x corresponding to DOFs in H(curl) (trial), whose curl is
 // integrated against H(div) test functions corresponding to y.
 template<int MAX_D1D = HCURL_MAX_D1D, int MAX_Q1D = HCURL_MAX_Q1D>
 static void PAHcurlHdivApply3D(const int D1D,
@@ -4681,7 +4572,7 @@ static void PAHcurlHdivApply3D(const int D1D,
   }); // end of element loop
 }

-// Apply to x corresponding to DOF's in H(div) (test), integrated against the
+// Apply to x corresponding to DOFs in H(div) (test), integrated against the
 // curl of H(curl) trial functions corresponding to y.
 template<int MAX_D1D = HCURL_MAX_D1D, int MAX_Q1D = HCURL_MAX_Q1D>
 static void PAHcurlHdivApply3DTranspose(const int D1D,
@@ -5146,38 +5037,11 @@ void MixedVectorWeakCurlIntegrator::AssemblePA(const FiniteElementSpace

   pa_data.SetSize(ndata * nq * ne, Device::GetMemoryType());

-   Vector coeff(coeffDim * nq * ne);
-   coeff = 1.0;
-   auto coeffh = Reshape(coeff.HostWrite(), coeffDim, nq, ne);
-   if (Q || DQ)
-   {
-      Vector V(coeffDim);
-      if (DQ)
-      {
-         MFEM_VERIFY(DQ->GetVDim() == coeffDim, "");
-      }
-
-      for (int e=0; e<ne; ++e)
-      {
-         ElementTransformation *tr = mesh->GetElementTransformation(e);
-
-         for (int p=0; p<nq; ++p)
-         {
-            if (DQ)
-            {
-               DQ->Eval(V, *tr, ir->IntPoint(p));
-               for (int i=0; i<coeffDim; ++i)
-               {
-                  coeffh(i, p, e) = V[i];
-               }
-            }
-            else
-            {
-               coeffh(0, p, e) = Q->Eval(*tr, ir->IntPoint(p));
-            }
-         }
-      }
-   }
+   QuadratureSpace qs(*mesh, *ir);
+   CoefficientVector coeff(qs, CoefficientStorage::FULL);
+   if (Q) { coeff.Project(*Q); }
+   else if (DQ) { coeff.Project(*DQ); }
+   else { coeff.SetConstant(1.0); }

   if (trialType == mfem::FiniteElement::CURL && dim == 3)
   {
@@ -5203,7 +5067,7 @@ void MixedVectorWeakCurlIntegrator::AssemblePA(const FiniteElementSpace
   }
 }

-// Apply to x corresponding to DOF's in H(curl) (trial), integrated against curl
+// Apply to x corresponding to DOFs in H(curl) (trial), integrated against curl
 // of H(curl) test functions corresponding to y.
 template<int MAX_D1D = HCURL_MAX_D1D, int MAX_Q1D = HCURL_MAX_Q1D>
 static void PAHcurlL2Apply3DTranspose(const int D1D,
@@ -12,6 +12,7 @@
 #include "../general/forall.hpp"
 #include "bilininteg.hpp"
 #include "gridfunc.hpp"
+#include "qspace.hpp"

 using namespace std;

@@ -1513,19 +1514,8 @@ void DivDivIntegrator::AssemblePA(const FiniteElementSpace &fes)

   pa_data.SetSize(nq * ne, Device::GetMemoryType());

-   Vector coeff(ne * nq);
-   coeff = 1.0;
-   if (Q)
-   {
-      for (int e=0; e<ne; ++e)
-      {
-         ElementTransformation *tr = mesh->GetElementTransformation(e);
-         for (int p=0; p<nq; ++p)
-         {
-            coeff[p + (e * nq)] = Q->Eval(*tr, ir->IntPoint(p));
-         }
-      }
-   }
+   QuadratureSpace qs(*mesh, *ir);
+   CoefficientVector coeff(Q, qs, CoefficientStorage::FULL);

   if (el->GetDerivType() == mfem::FiniteElement::DIV && dim == 3)
   {
@@ -1783,19 +1773,8 @@ VectorFEDivergenceIntegrator::AssemblePA(const FiniteElementSpace &trial_fes,

   pa_data.SetSize(nq * ne, Device::GetMemoryType());

-   Vector coeff(ne * nq);
-   coeff = 1.0;
-   if (Q)
-   {
-      for (int e=0; e<ne; ++e)
-      {
-         ElementTransformation *tr = mesh->GetElementTransformation(e);
-         for (int p=0; p<nq; ++p)
-         {
-            coeff[p + (e * nq)] = Q->Eval(*tr, ir->IntPoint(p));
-         }
-      }
-   }
+   QuadratureSpace qs(*mesh, *ir);
+   CoefficientVector coeff(Q, qs, CoefficientStorage::FULL);

   if (test_el->GetMapType() == FiniteElement::INTEGRAL)
   {
@@ -1818,7 +1797,7 @@ VectorFEDivergenceIntegrator::AssemblePA(const FiniteElementSpace &trial_fes,
   }
 }

-// Apply to x corresponding to DOF's in H(div) (trial), whose divergence is
+// Apply to x corresponding to DOFs in H(div) (trial), whose divergence is
 // integrated against L_2 test functions corresponding to y.
 static void PAHdivL2Apply3D(const int D1D,
                            const int Q1D,
@@ -1981,7 +1960,7 @@ static void PAHdivL2Apply3D(const int D1D,
   }); // end of element loop
 }

-// Apply to x corresponding to DOF's in H(div) (trial), whose divergence is
+// Apply to x corresponding to DOFs in H(div) (trial), whose divergence is
 // integrated against L_2 test functions corresponding to y.
 static void PAHdivL2Apply2D(const int D1D,
                            const int Q1D,
@@ -12,7 +12,9 @@
 #include "../general/forall.hpp"
 #include "bilininteg.hpp"
 #include "gridfunc.hpp"
+#include "qfunction.hpp"
 #include "ceed/integrators/mass/mass.hpp"
+#include "bilininteg_mass_pa.hpp"

 using namespace std;

@@ -60,43 +62,10 @@ void MassIntegrator::AssemblePA(const FiniteElementSpace &fes)
   dofs1D = maps->ndof;
   quad1D = maps->nqpt;
   pa_data.SetSize(ne*nq, mt);
-   Vector coeff;
-   if (Q == nullptr)
-   {
-      coeff.SetSize(1);
-      coeff(0) = 1.0;
-   }
-   else if (ConstantCoefficient* cQ = dynamic_cast<ConstantCoefficient*>(Q))
-   {
-      coeff.SetSize(1);
-      coeff(0) = cQ->constant;
-   }
-   else if (QuadratureFunctionCoefficient* qfQ =
-               dynamic_cast<QuadratureFunctionCoefficient*>(Q))
-   {
-      const QuadratureFunction &qFun = qfQ->GetQuadFunction();
-      MFEM_VERIFY(qFun.Size() == nq * ne,
-                  "Incompatible QuadratureFunction dimension \n");

-      MFEM_VERIFY(ir == &qFun.GetSpace()->GetElementIntRule(0),
-                  "IntegrationRule used within integrator and in"
-                  " QuadratureFunction appear to be different");
-      qFun.Read();
-      coeff.MakeRef(const_cast<QuadratureFunction &>(qFun),0);
-   }
-   else
-   {
-      coeff.SetSize(nq * ne);
-      auto C = Reshape(coeff.HostWrite(), nq, ne);
-      for (int e = 0; e < ne; ++e)
-      {
-         ElementTransformation& T = *fes.GetElementTransformation(e);
-         for (int q = 0; q < nq; ++q)
-         {
-            C(q,e) = Q->Eval(T, ir->IntPoint(q));
-         }
-      }
-   }
+   QuadratureSpace qs(*mesh, *ir);
+   CoefficientVector coeff(Q, qs, CoefficientStorage::COMPRESSED);
+
   if (dim==1) { MFEM_ABORT("Not supported yet... stay tuned!"); }
   if (dim==2)
   {
@@ -590,85 +559,18 @@ static void PAMassApply2D(const int NE,
                          const int d1d = 0,
                          const int q1d = 0)
 {
-   const int D1D = T_D1D ? T_D1D : d1d;
-   const int Q1D = T_Q1D ? T_Q1D : q1d;
-   MFEM_VERIFY(D1D <= MAX_D1D, "");
-   MFEM_VERIFY(Q1D <= MAX_Q1D, "");
-   auto B = Reshape(b_.Read(), Q1D, D1D);
-   auto Bt = Reshape(bt_.Read(), D1D, Q1D);
-   auto D = Reshape(d_.Read(), Q1D, Q1D, NE);
-   auto X = Reshape(x_.Read(), D1D, D1D, NE);
-   auto Y = Reshape(y_.ReadWrite(), D1D, D1D, NE);
+   MFEM_VERIFY(T_D1D ? T_D1D : d1d <= MAX_D1D, "");
+   MFEM_VERIFY(T_Q1D ? T_Q1D : q1d <= MAX_Q1D, "");
+
+   const auto B = b_.Read();
+   const auto Bt = bt_.Read();
+   const auto D = d_.Read();
+   const auto X = x_.Read();
+   auto Y = y_.ReadWrite();
+
   MFEM_FORALL(e, NE,
   {
-      const int D1D = T_D1D ? T_D1D : d1d; // nvcc workaround
-      const int Q1D = T_Q1D ? T_Q1D : q1d;
-      // the following variables are evaluated at compile time
-      constexpr int max_D1D = T_D1D ? T_D1D : MAX_D1D;
-      constexpr int max_Q1D = T_Q1D ? T_Q1D : MAX_Q1D;
-      double sol_xy[max_Q1D][max_Q1D];
-      for (int qy = 0; qy < Q1D; ++qy)
-      {
-         for (int qx = 0; qx < Q1D; ++qx)
-         {
-            sol_xy[qy][qx] = 0.0;
-         }
-      }
-      for (int dy = 0; dy < D1D; ++dy)
-      {
-         double sol_x[max_Q1D];
-         for (int qy = 0; qy < Q1D; ++qy)
-         {
-            sol_x[qy] = 0.0;
-         }
-         for (int dx = 0; dx < D1D; ++dx)
-         {
-            const double s = X(dx,dy,e);
-            for (int qx = 0; qx < Q1D; ++qx)
-            {
-               sol_x[qx] += B(qx,dx)* s;
-            }
-         }
-         for (int qy = 0; qy < Q1D; ++qy)
-         {
-            const double d2q = B(qy,dy);
-            for (int qx = 0; qx < Q1D; ++qx)
-            {
-               sol_xy[qy][qx] += d2q * sol_x[qx];
-            }
-         }
-      }
-      for (int qy = 0; qy < Q1D; ++qy)
-      {
-         for (int qx = 0; qx < Q1D; ++qx)
-         {
-            sol_xy[qy][qx] *= D(qx,qy,e);
-         }
-      }
-      for (int qy = 0; qy < Q1D; ++qy)
-      {
-         double sol_x[max_D1D];
-         for (int dx = 0; dx < D1D; ++dx)
-         {
-            sol_x[dx] = 0.0;
-         }
-         for (int qx = 0; qx < Q1D; ++qx)
-         {
-            const double s = sol_xy[qy][qx];
-            for (int dx = 0; dx < D1D; ++dx)
-            {
-               sol_x[dx] += Bt(dx,qx) * s;
-            }
-         }
-         for (int dy = 0; dy < D1D; ++dy)
-         {
-            const double q2d = Bt(dy,qy);
-            for (int dx = 0; dx < D1D; ++dx)
-            {
-               Y(dx,dy,e) += q2d * sol_x[dx];
-            }
-         }
-      }
+      internal::PAMassApply2D_Element(e, NE, B, Bt, D, X, Y, d1d, q1d);
   });
 }

@@ -690,108 +592,13 @@ static void SmemPAMassApply2D(const int NE,
   constexpr int MD1 = T_D1D ? T_D1D : MAX_D1D;
   MFEM_VERIFY(D1D <= MD1, "");
   MFEM_VERIFY(Q1D <= MQ1, "");
-   auto b = Reshape(b_.Read(), Q1D, D1D);
-   auto D = Reshape(d_.Read(), Q1D, Q1D, NE);
-   auto x = Reshape(x_.Read(), D1D, D1D, NE);
-   auto Y = Reshape(y_.ReadWrite(), D1D, D1D, NE);
+   const auto b = b_.Read();
+   const auto D = d_.Read();
+   const auto x = x_.Read();
+   auto Y = y_.ReadWrite();
   MFEM_FORALL_2D(e, NE, Q1D, Q1D, NBZ,
   {
-      const int tidz = MFEM_THREAD_ID(z);
-      const int D1D = T_D1D ? T_D1D : d1d;
-      const int Q1D = T_Q1D ? T_Q1D : q1d;
-      constexpr int NBZ = T_NBZ ? T_NBZ : 1;
-      constexpr int MQ1 = T_Q1D ? T_Q1D : MAX_Q1D;
-      constexpr int MD1 = T_D1D ? T_D1D : MAX_D1D;
-      constexpr int MDQ = (MQ1 > MD1) ? MQ1 : MD1;
-      MFEM_SHARED double BBt[MQ1*MD1];
-      double (*B)[MD1] = (double (*)[MD1]) BBt;
-      double (*Bt)[MQ1] = (double (*)[MQ1]) BBt;
-      MFEM_SHARED double sm0[NBZ][MDQ*MDQ];
-      MFEM_SHARED double sm1[NBZ][MDQ*MDQ];
-      double (*X)[MD1] = (double (*)[MD1]) (sm0 + tidz);
-      double (*DQ)[MQ1] = (double (*)[MQ1]) (sm1 + tidz);
-      double (*QQ)[MQ1] = (double (*)[MQ1]) (sm0 + tidz);
-      double (*QD)[MD1] = (double (*)[MD1]) (sm1 + tidz);
-      MFEM_FOREACH_THREAD(dy,y,D1D)
-      {
-         MFEM_FOREACH_THREAD(dx,x,D1D)
-         {
-            X[dy][dx] = x(dx,dy,e);
-         }
-      }
-      if (tidz == 0)
-      {
-         MFEM_FOREACH_THREAD(dy,y,D1D)
-         {
-            MFEM_FOREACH_THREAD(q,x,Q1D)
-            {
-               B[q][dy] = b(q,dy);
-            }
-         }
-      }
-      MFEM_SYNC_THREAD;
-      MFEM_FOREACH_THREAD(dy,y,D1D)
-      {
-         MFEM_FOREACH_THREAD(qx,x,Q1D)
-         {
-            double dq = 0.0;
-            for (int dx = 0; dx < D1D; ++dx)
-            {
-               dq += X[dy][dx] * B[qx][dx];
-            }
-            DQ[dy][qx] = dq;
-         }
-      }
-      MFEM_SYNC_THREAD;
-      MFEM_FOREACH_THREAD(qy,y,Q1D)
-      {
-         MFEM_FOREACH_THREAD(qx,x,Q1D)
-         {
-            double qq = 0.0;
-            for (int dy = 0; dy < D1D; ++dy)
-            {
-               qq += DQ[dy][qx] * B[qy][dy];
-            }
-            QQ[qy][qx] = qq * D(qx, qy, e);
-         }
-      }
-      MFEM_SYNC_THREAD;
-      if (tidz == 0)
-      {
-         MFEM_FOREACH_THREAD(dy,y,D1D)
-         {
-            MFEM_FOREACH_THREAD(q,x,Q1D)
-            {
-               Bt[dy][q] = b(q,dy);
-            }
-         }
-      }
-      MFEM_SYNC_THREAD;
-      MFEM_FOREACH_THREAD(qy,y,Q1D)
-      {
-         MFEM_FOREACH_THREAD(dx,x,D1D)
-         {
-            double dq = 0.0;
-            for (int qx = 0; qx < Q1D; ++qx)
-            {
-               dq += QQ[qy][qx] * Bt[dx][qx];
-            }
-            QD[qy][dx] = dq;
-         }
-      }
-      MFEM_SYNC_THREAD;
-      MFEM_FOREACH_THREAD(dy,y,D1D)
-      {
-         MFEM_FOREACH_THREAD(dx,x,D1D)
-         {
-            double dd = 0.0;
-            for (int qy = 0; qy < Q1D; ++qy)
-            {
-               dd += (QD[qy][dx] * Bt[dy][qy]);
-            }
-            Y(dx, dy, e) += dd;
-         }
-      }
+      internal::SmemPAMassApply2D_Element<T_D1D,T_Q1D,T_NBZ>(e, NE, b, D, x, Y, d1d, q1d);
   });
 }

@@ -805,134 +612,18 @@ static void PAMassApply3D(const int NE,
                          const int d1d = 0,
                          const int q1d = 0)
 {
-   const int D1D = T_D1D ? T_D1D : d1d;
-   const int Q1D = T_Q1D ? T_Q1D : q1d;
-   MFEM_VERIFY(D1D <= MAX_D1D, "");
-   MFEM_VERIFY(Q1D <= MAX_Q1D, "");
-   auto B = Reshape(b_.Read(), Q1D, D1D);
-   auto Bt = Reshape(bt_.Read(), D1D, Q1D);
-   auto D = Reshape(d_.Read(), Q1D, Q1D, Q1D, NE);
-   auto X = Reshape(x_.Read(), D1D, D1D, D1D, NE);
-   auto Y = Reshape(y_.ReadWrite(), D1D, D1D, D1D, NE);
+   MFEM_VERIFY(T_D1D ? T_D1D : d1d <= MAX_D1D, "");
+   MFEM_VERIFY(T_Q1D ? T_Q1D : q1d <= MAX_Q1D, "");
+
+   const auto B = b_.Read();
+   const auto Bt = bt_.Read();
+   const auto D = d_.Read();
+   const auto X = x_.Read();
+   auto Y = y_.ReadWrite();
+
   MFEM_FORALL(e, NE,
   {
-      const int D1D = T_D1D ? T_D1D : d1d;
-      const int Q1D = T_Q1D ? T_Q1D : q1d;
-      constexpr int max_D1D = T_D1D ? T_D1D : MAX_D1D;
-      constexpr int max_Q1D = T_Q1D ? T_Q1D : MAX_Q1D;
-      double sol_xyz[max_Q1D][max_Q1D][max_Q1D];
-      for (int qz = 0; qz < Q1D; ++qz)
-      {
-         for (int qy = 0; qy < Q1D; ++qy)
-         {
-            for (int qx = 0; qx < Q1D; ++qx)
-            {
-               sol_xyz[qz][qy][qx] = 0.0;
-            }
-         }
-      }
-      for (int dz = 0; dz < D1D; ++dz)
-      {
-         double sol_xy[max_Q1D][max_Q1D];
-         for (int qy = 0; qy < Q1D; ++qy)
-         {
-            for (int qx = 0; qx < Q1D; ++qx)
-            {
-               sol_xy[qy][qx] = 0.0;
-            }
-         }
-         for (int dy = 0; dy < D1D; ++dy)
-         {
-            double sol_x[max_Q1D];
-            for (int qx = 0; qx < Q1D; ++qx)
-            {
-               sol_x[qx] = 0;
-            }
-            for (int dx = 0; dx < D1D; ++dx)
-            {
-               const double s = X(dx,dy,dz,e);
-               for (int qx = 0; qx < Q1D; ++qx)
-               {
-                  sol_x[qx] += B(qx,dx) * s;
-               }
-            }
-            for (int qy = 0; qy < Q1D; ++qy)
-            {
-               const double wy = B(qy,dy);
-               for (int qx = 0; qx < Q1D; ++qx)
-               {
-                  sol_xy[qy][qx] += wy * sol_x[qx];
-               }
-            }
-         }
-         for (int qz = 0; qz < Q1D; ++qz)
-         {
-            const double wz = B(qz,dz);
-            for (int qy = 0; qy < Q1D; ++qy)
-            {
-               for (int qx = 0; qx < Q1D; ++qx)
-               {
-                  sol_xyz[qz][qy][qx] += wz * sol_xy[qy][qx];
-               }
-            }
-         }
-      }
-      for (int qz = 0; qz < Q1D; ++qz)
-      {
-         for (int qy = 0; qy < Q1D; ++qy)
-         {
-            for (int qx = 0; qx < Q1D; ++qx)
-            {
-               sol_xyz[qz][qy][qx] *= D(qx,qy,qz,e);
-            }
-         }
-      }
-      for (int qz = 0; qz < Q1D; ++qz)
-      {
-         double sol_xy[max_D1D][max_D1D];
-         for (int dy = 0; dy < D1D; ++dy)
-         {
-            for (int dx = 0; dx < D1D; ++dx)
-            {
-               sol_xy[dy][dx] = 0;
-            }
-         }
-         for (int qy = 0; qy < Q1D; ++qy)
-         {
-            double sol_x[max_D1D];
-            for (int dx = 0; dx < D1D; ++dx)
-            {
-               sol_x[dx] = 0;
-            }
-            for (int qx = 0; qx < Q1D; ++qx)
-            {
-               const double s = sol_xyz[qz][qy][qx];
-               for (int dx = 0; dx < D1D; ++dx)
-               {
-                  sol_x[dx] += Bt(dx,qx) * s;
-               }
-            }
-            for (int dy = 0; dy < D1D; ++dy)
-            {
-               const double wy = Bt(dy,qy);
-               for (int dx = 0; dx < D1D; ++dx)
-               {
-                  sol_xy[dy][dx] += wy * sol_x[dx];
-               }
-            }
-         }
-         for (int dz = 0; dz < D1D; ++dz)
-         {
-            const double wz = Bt(dz,qz);
-            for (int dy = 0; dy < D1D; ++dy)
-            {
-               for (int dx = 0; dx < D1D; ++dx)
-               {
-                  Y(dx,dy,dz,e) += wz * sol_xy[dy][dx];
-               }
-            }
-         }
-      }
+      internal::PAMassApply3D_Element(e, NE, B, Bt, D, X, Y, d1d, q1d);
   });
 }

@@ -953,213 +644,13 @@ static void SmemPAMassApply3D(const int NE,
   constexpr int M1D = T_D1D ? T_D1D : MAX_D1D;
   MFEM_VERIFY(D1D <= M1D, "");
   MFEM_VERIFY(Q1D <= M1Q, "");
-   auto b = Reshape(b_.Read(), Q1D, D1D);
-   auto d = Reshape(d_.Read(), Q1D, Q1D, Q1D, NE);
-   auto x = Reshape(x_.Read(), D1D, D1D, D1D, NE);
-   auto y = Reshape(y_.ReadWrite(), D1D, D1D, D1D, NE);
+   auto b = b_.Read();
+   auto d = d_.Read();
+   auto x = x_.Read();
+   auto y = y_.ReadWrite();
   MFEM_FORALL_3D(e, NE, Q1D, Q1D, 1,
   {
-      const int D1D = T_D1D ? T_D1D : d1d;
-      const int Q1D = T_Q1D ? T_Q1D : q1d;
-      constexpr int MQ1 = T_Q1D ? T_Q1D : MAX_Q1D;
-      constexpr int MD1 = T_D1D ? T_D1D : MAX_D1D;
-      constexpr int MDQ = (MQ1 > MD1) ? MQ1 : MD1;
-      MFEM_SHARED double sDQ[MQ1*MD1];
-      double (*B)[MD1] = (double (*)[MD1]) sDQ;
-      double (*Bt)[MQ1] = (double (*)[MQ1]) sDQ;
-      MFEM_SHARED double sm0[MDQ*MDQ*MDQ];
-      MFEM_SHARED double sm1[MDQ*MDQ*MDQ];
-      double (*X)[MD1][MD1]   = (double (*)[MD1][MD1]) sm0;
-      double (*DDQ)[MD1][MQ1] = (double (*)[MD1][MQ1]) sm1;
-      double (*DQQ)[MQ1][MQ1] = (double (*)[MQ1][MQ1]) sm0;
-      double (*QQQ)[MQ1][MQ1] = (double (*)[MQ1][MQ1]) sm1;
-      double (*QQD)[MQ1][MD1] = (double (*)[MQ1][MD1]) sm0;
-      double (*QDD)[MD1][MD1] = (double (*)[MD1][MD1]) sm1;
-      MFEM_FOREACH_THREAD(dy,y,D1D)
-      {
-         MFEM_FOREACH_THREAD(dx,x,D1D)
-         {
-            MFEM_UNROLL(MD1)
-            for (int dz = 0; dz < D1D; ++dz)
-            {
-               X[dz][dy][dx] = x(dx,dy,dz,e);
-            }
-         }
-         MFEM_FOREACH_THREAD(dx,x,Q1D)
-         {
-            B[dx][dy] = b(dx,dy);
-         }
-      }
-      MFEM_SYNC_THREAD;
-      MFEM_FOREACH_THREAD(dy,y,D1D)
-      {
-         MFEM_FOREACH_THREAD(qx,x,Q1D)
-         {
-            double u[D1D];
-            MFEM_UNROLL(MD1)
-            for (int dz = 0; dz < D1D; dz++)
-            {
-               u[dz] = 0;
-            }
-            MFEM_UNROLL(MD1)
-            for (int dx = 0; dx < D1D; ++dx)
-            {
-               MFEM_UNROLL(MD1)
-               for (int dz = 0; dz < D1D; ++dz)
-               {
-                  u[dz] += X[dz][dy][dx] * B[qx][dx];
-               }
-            }
-            MFEM_UNROLL(MD1)
-            for (int dz = 0; dz < D1D; ++dz)
-            {
-               DDQ[dz][dy][qx] = u[dz];
-            }
-         }
-      }
-      MFEM_SYNC_THREAD;
-      MFEM_FOREACH_THREAD(qy,y,Q1D)
-      {
-         MFEM_FOREACH_THREAD(qx,x,Q1D)
-         {
-            double u[D1D];
-            MFEM_UNROLL(MD1)
-            for (int dz = 0; dz < D1D; dz++)
-            {
-               u[dz] = 0;
-            }
-            MFEM_UNROLL(MD1)
-            for (int dy = 0; dy < D1D; ++dy)
-            {
-               MFEM_UNROLL(MD1)
-               for (int dz = 0; dz < D1D; dz++)
-               {
-                  u[dz] += DDQ[dz][dy][qx] * B[qy][dy];
-               }
-            }
-            MFEM_UNROLL(MD1)
-            for (int dz = 0; dz < D1D; dz++)
-            {
-               DQQ[dz][qy][qx] = u[dz];
-            }
-         }
-      }
-      MFEM_SYNC_THREAD;
-      MFEM_FOREACH_THREAD(qy,y,Q1D)
-      {
-         MFEM_FOREACH_THREAD(qx,x,Q1D)
-         {
-            double u[Q1D];
-            MFEM_UNROLL(MQ1)
-            for (int qz = 0; qz < Q1D; qz++)
-            {
-               u[qz] = 0;
-            }
-            MFEM_UNROLL(MD1)
-            for (int dz = 0; dz < D1D; ++dz)
-            {
-               MFEM_UNROLL(MQ1)
-               for (int qz = 0; qz < Q1D; qz++)
-               {
-                  u[qz] += DQQ[dz][qy][qx] * B[qz][dz];
-               }
-            }
-            MFEM_UNROLL(MQ1)
-            for (int qz = 0; qz < Q1D; qz++)
-            {
-               QQQ[qz][qy][qx] = u[qz] * d(qx,qy,qz,e);
-            }
-         }
-      }
-      MFEM_SYNC_THREAD;
-      MFEM_FOREACH_THREAD(d,y,D1D)
-      {
-         MFEM_FOREACH_THREAD(q,x,Q1D)
-         {
-            Bt[d][q] = b(q,d);
-         }
-      }
-      MFEM_SYNC_THREAD;
-      MFEM_FOREACH_THREAD(qy,y,Q1D)
-      {
-         MFEM_FOREACH_THREAD(dx,x,D1D)
-         {
-            double u[Q1D];
-            MFEM_UNROLL(MQ1)
-            for (int qz = 0; qz < Q1D; ++qz)
-            {
-               u[qz] = 0;
-            }
-            MFEM_UNROLL(MQ1)
-            for (int qx = 0; qx < Q1D; ++qx)
-            {
-               MFEM_UNROLL(MQ1)
-               for (int qz = 0; qz < Q1D; ++qz)
-               {
-                  u[qz] += QQQ[qz][qy][qx] * Bt[dx][qx];
-               }
-            }
-            MFEM_UNROLL(MQ1)
-            for (int qz = 0; qz < Q1D; ++qz)
-            {
-               QQD[qz][qy][dx] = u[qz];
-            }
-         }
-      }
-      MFEM_SYNC_THREAD;
-      MFEM_FOREACH_THREAD(dy,y,D1D)
-      {
-         MFEM_FOREACH_THREAD(dx,x,D1D)
-         {
-            double u[Q1D];
-            MFEM_UNROLL(MQ1)
-            for (int qz = 0; qz < Q1D; ++qz)
-            {
-               u[qz] = 0;
-            }
-            MFEM_UNROLL(MQ1)
-            for (int qy = 0; qy < Q1D; ++qy)
-            {
-               MFEM_UNROLL(MQ1)
-               for (int qz = 0; qz < Q1D; ++qz)
-               {
-                  u[qz] += QQD[qz][qy][dx] * Bt[dy][qy];
-               }
-            }
-            MFEM_UNROLL(MQ1)
-            for (int qz = 0; qz < Q1D; ++qz)
-            {
-               QDD[qz][dy][dx] = u[qz];
-            }
-         }
-      }
-      MFEM_SYNC_THREAD;
-      MFEM_FOREACH_THREAD(dy,y,D1D)
-      {
-         MFEM_FOREACH_THREAD(dx,x,D1D)
-         {
-            double u[D1D];
-            MFEM_UNROLL(MD1)
-            for (int dz = 0; dz < D1D; ++dz)
-            {
-               u[dz] = 0;
-            }
-            MFEM_UNROLL(MQ1)
-            for (int qz = 0; qz < Q1D; ++qz)
-            {
-               MFEM_UNROLL(MD1)
-               for (int dz = 0; dz < D1D; ++dz)
-               {
-                  u[dz] += QDD[qz][dy][dx] * Bt[dz][qz];
-               }
-            }
-            MFEM_UNROLL(MD1)
-            for (int dz = 0; dz < D1D; ++dz)
-            {
-               y(dx,dy,dz,e) += u[dz];
-            }
-         }
-      }
+      internal::SmemPAMassApply3D_Element<T_D1D,T_Q1D>(e, NE, b, d, x, y, d1d, q1d);
   });
 }

@@ -0,0 +1,632 @@
+// 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.
+
+#ifndef MFEM_BILININTEG_MASS_PA_HPP
+#define MFEM_BILININTEG_MASS_PA_HPP
+
+#include "../config/config.hpp"
+#include "../general/forall.hpp"
+#include "../linalg/dtensor.hpp"
+
+namespace mfem
+{
+
+namespace internal
+{
+
+template <bool ACCUMULATE = true>
+MFEM_HOST_DEVICE inline
+void PAMassApply2D_Element(const int e,
+                           const int NE,
+                           const double *b_,
+                           const double *bt_,
+                           const double *d_,
+                           const double *x_,
+                           double *y_,
+                           const int d1d = 0,
+                           const int q1d = 0)
+{
+   const int D1D = d1d;
+   const int Q1D = q1d;
+   auto B = ConstDeviceMatrix(b_, Q1D, D1D);
+   auto Bt = ConstDeviceMatrix(bt_, D1D, Q1D);
+   auto D = ConstDeviceCube(d_, Q1D, Q1D, NE);
+   auto X = ConstDeviceCube(x_, D1D, D1D, NE);
+   auto Y = DeviceCube(y_, D1D, D1D, NE);
+
+   if (!ACCUMULATE)
+   {
+      for (int dy = 0; dy < D1D; ++dy)
+      {
+         for (int dx = 0; dx < D1D; ++dx)
+         {
+            Y(dx, dy, e) = 0.0;
+         }
+      }
+   }
+
+   constexpr int max_D1D = MAX_D1D;
+   constexpr int max_Q1D = MAX_Q1D;
+   double sol_xy[max_Q1D][max_Q1D];
+   for (int qy = 0; qy < Q1D; ++qy)
+   {
+      for (int qx = 0; qx < Q1D; ++qx)
+      {
+         sol_xy[qy][qx] = 0.0;
+      }
+   }
+   for (int dy = 0; dy < D1D; ++dy)
+   {
+      double sol_x[max_Q1D];
+      for (int qy = 0; qy < Q1D; ++qy)
+      {
+         sol_x[qy] = 0.0;
+      }
+      for (int dx = 0; dx < D1D; ++dx)
+      {
+         const double s = X(dx,dy,e);
+         for (int qx = 0; qx < Q1D; ++qx)
+         {
+            sol_x[qx] += B(qx,dx)* s;
+         }
+      }
+      for (int qy = 0; qy < Q1D; ++qy)
+      {
+         const double d2q = B(qy,dy);
+         for (int qx = 0; qx < Q1D; ++qx)
+         {
+            sol_xy[qy][qx] += d2q * sol_x[qx];
+         }
+      }
+   }
+   for (int qy = 0; qy < Q1D; ++qy)
+   {
+      for (int qx = 0; qx < Q1D; ++qx)
+      {
+         sol_xy[qy][qx] *= D(qx,qy,e);
+      }
+   }
+   for (int qy = 0; qy < Q1D; ++qy)
+   {
+      double sol_x[max_D1D];
+      for (int dx = 0; dx < D1D; ++dx)
+      {
+         sol_x[dx] = 0.0;
+      }
+      for (int qx = 0; qx < Q1D; ++qx)
+      {
+         const double s = sol_xy[qy][qx];
+         for (int dx = 0; dx < D1D; ++dx)
+         {
+            sol_x[dx] += Bt(dx,qx) * s;
+         }
+      }
+      for (int dy = 0; dy < D1D; ++dy)
+      {
+         const double q2d = Bt(dy,qy);
+         for (int dx = 0; dx < D1D; ++dx)
+         {
+            Y(dx,dy,e) += q2d * sol_x[dx];
+         }
+      }
+   }
+}
+
+template<int T_D1D, int T_Q1D, int T_NBZ, bool ACCUMULATE = true>
+MFEM_HOST_DEVICE inline
+void SmemPAMassApply2D_Element(const int e,
+                               const int NE,
+                               const double *b_,
+                               const double *d_,
+                               const double *x_,
+                               double *y_,
+                               int d1d = 0,
+                               int q1d = 0)
+{
+   const int D1D = T_D1D ? T_D1D : d1d;
+   const int Q1D = T_Q1D ? T_Q1D : q1d;
+   constexpr int NBZ = T_NBZ ? T_NBZ : 1;
+
+   constexpr int MQ1 = T_Q1D ? T_Q1D : MAX_Q1D;
+   constexpr int MD1 = T_D1D ? T_D1D : MAX_D1D;
+   constexpr int MDQ = (MQ1 > MD1) ? MQ1 : MD1;
+
+   auto b = ConstDeviceMatrix(b_, Q1D, D1D);
+   auto D = ConstDeviceCube(d_, Q1D, Q1D, NE);
+   auto x = ConstDeviceCube(x_, D1D, D1D, NE);
+   auto Y = DeviceCube(y_, D1D, D1D, NE);
+
+   const int tidz = MFEM_THREAD_ID(z);
+
+   MFEM_SHARED double BBt[MQ1*MD1];
+   double (*B)[MD1] = (double (*)[MD1]) BBt;
+   double (*Bt)[MQ1] = (double (*)[MQ1]) BBt;
+   MFEM_SHARED double sm0[NBZ][MDQ*MDQ];
+   MFEM_SHARED double sm1[NBZ][MDQ*MDQ];
+   double (*X)[MD1] = (double (*)[MD1]) (sm0 + tidz);
+   double (*DQ)[MQ1] = (double (*)[MQ1]) (sm1 + tidz);
+   double (*QQ)[MQ1] = (double (*)[MQ1]) (sm0 + tidz);
+   double (*QD)[MD1] = (double (*)[MD1]) (sm1 + tidz);
+
+
+   MFEM_FOREACH_THREAD(dy,y,D1D)
+   {
+      MFEM_FOREACH_THREAD(dx,x,D1D)
+      {
+         X[dy][dx] = x(dx,dy,e);
+      }
+   }
+   if (tidz == 0)
+   {
+      MFEM_FOREACH_THREAD(dy,y,D1D)
+      {
+         MFEM_FOREACH_THREAD(q,x,Q1D)
+         {
+            B[q][dy] = b(q,dy);
+         }
+      }
+   }
+   MFEM_SYNC_THREAD;
+   MFEM_FOREACH_THREAD(dy,y,D1D)
+   {
+      MFEM_FOREACH_THREAD(qx,x,Q1D)
+      {
+         double dq = 0.0;
+         for (int dx = 0; dx < D1D; ++dx)
+         {
+            dq += X[dy][dx] * B[qx][dx];
+         }
+         DQ[dy][qx] = dq;
+      }
+   }
+   MFEM_SYNC_THREAD;
+   MFEM_FOREACH_THREAD(qy,y,Q1D)
+   {
+      MFEM_FOREACH_THREAD(qx,x,Q1D)
+      {
+         double qq = 0.0;
+         for (int dy = 0; dy < D1D; ++dy)
+         {
+            qq += DQ[dy][qx] * B[qy][dy];
+         }
+         QQ[qy][qx] = qq * D(qx, qy, e);
+      }
+   }
+   MFEM_SYNC_THREAD;
+   if (tidz == 0)
+   {
+      MFEM_FOREACH_THREAD(dy,y,D1D)
+      {
+         MFEM_FOREACH_THREAD(q,x,Q1D)
+         {
+            Bt[dy][q] = b(q,dy);
+         }
+      }
+   }
+   MFEM_SYNC_THREAD;
+   MFEM_FOREACH_THREAD(qy,y,Q1D)
+   {
+      MFEM_FOREACH_THREAD(dx,x,D1D)
+      {
+         double dq = 0.0;
+         for (int qx = 0; qx < Q1D; ++qx)
+         {
+            dq += QQ[qy][qx] * Bt[dx][qx];
+         }
+         QD[qy][dx] = dq;
+      }
+   }
+   MFEM_SYNC_THREAD;
+   MFEM_FOREACH_THREAD(dy,y,D1D)
+   {
+      MFEM_FOREACH_THREAD(dx,x,D1D)
+      {
+         double dd = 0.0;
+         for (int qy = 0; qy < Q1D; ++qy)
+         {
+            dd += (QD[qy][dx] * Bt[dy][qy]);
+         }
+         if (ACCUMULATE)
+         {
+            Y(dx, dy, e) += dd;
+         }
+         else
+         {
+            Y(dx, dy, e) = dd;
+         }
+      }
+   }
+}
+
+template <bool ACCUMULATE = true>
+MFEM_HOST_DEVICE inline
+void PAMassApply3D_Element(const int e,
+                           const int NE,
+                           const double *b_,
+                           const double *bt_,
+                           const double *d_,
+                           const double *x_,
+                           double *y_,
+                           const int d1d,
+                           const int q1d)
+{
+   const int D1D = d1d;
+   const int Q1D = q1d;
+   auto B = ConstDeviceMatrix(b_, Q1D, D1D);
+   auto Bt = ConstDeviceMatrix(bt_, D1D, Q1D);
+   auto D = DeviceTensor<4,const double>(d_, Q1D, Q1D, Q1D, NE);
+   auto X = DeviceTensor<4,const double>(x_, D1D, D1D, D1D, NE);
+   auto Y = DeviceTensor<4,double>(y_, D1D, D1D, D1D, NE);
+
+   if (!ACCUMULATE)
+   {
+      for (int dz = 0; dz < D1D; ++dz)
+      {
+         for (int dy = 0; dy < D1D; ++dy)
+         {
+            for (int dx = 0; dx < D1D; ++dx)
+            {
+               Y(dx, dy, dz, e) = 0.0;
+            }
+         }
+      }
+   }
+
+   constexpr int max_D1D = MAX_D1D;
+   constexpr int max_Q1D = MAX_Q1D;
+   double sol_xyz[max_Q1D][max_Q1D][max_Q1D];
+   for (int qz = 0; qz < Q1D; ++qz)
+   {
+      for (int qy = 0; qy < Q1D; ++qy)
+      {
+         for (int qx = 0; qx < Q1D; ++qx)
+         {
+            sol_xyz[qz][qy][qx] = 0.0;
+         }
+      }
+   }
+   for (int dz = 0; dz < D1D; ++dz)
+   {
+      double sol_xy[max_Q1D][max_Q1D];
+      for (int qy = 0; qy < Q1D; ++qy)
+      {
+         for (int qx = 0; qx < Q1D; ++qx)
+         {
+            sol_xy[qy][qx] = 0.0;
+         }
+      }
+      for (int dy = 0; dy < D1D; ++dy)
+      {
+         double sol_x[max_Q1D];
+         for (int qx = 0; qx < Q1D; ++qx)
+         {
+            sol_x[qx] = 0;
+         }
+         for (int dx = 0; dx < D1D; ++dx)
+         {
+            const double s = X(dx,dy,dz,e);
+            for (int qx = 0; qx < Q1D; ++qx)
+            {
+               sol_x[qx] += B(qx,dx) * s;
+            }
+         }
+         for (int qy = 0; qy < Q1D; ++qy)
+         {
+            const double wy = B(qy,dy);
+            for (int qx = 0; qx < Q1D; ++qx)
+            {
+               sol_xy[qy][qx] += wy * sol_x[qx];
+            }
+         }
+      }
+      for (int qz = 0; qz < Q1D; ++qz)
+      {
+         const double wz = B(qz,dz);
+         for (int qy = 0; qy < Q1D; ++qy)
+         {
+            for (int qx = 0; qx < Q1D; ++qx)
+            {
+               sol_xyz[qz][qy][qx] += wz * sol_xy[qy][qx];
+            }
+         }
+      }
+   }
+   for (int qz = 0; qz < Q1D; ++qz)
+   {
+      for (int qy = 0; qy < Q1D; ++qy)
+      {
+         for (int qx = 0; qx < Q1D; ++qx)
+         {
+            sol_xyz[qz][qy][qx] *= D(qx,qy,qz,e);
+         }
+      }
+   }
+   for (int qz = 0; qz < Q1D; ++qz)
+   {
+      double sol_xy[max_D1D][max_D1D];
+      for (int dy = 0; dy < D1D; ++dy)
+      {
+         for (int dx = 0; dx < D1D; ++dx)
+         {
+            sol_xy[dy][dx] = 0;
+         }
+      }
+      for (int qy = 0; qy < Q1D; ++qy)
+      {
+         double sol_x[max_D1D];
+         for (int dx = 0; dx < D1D; ++dx)
+         {
+            sol_x[dx] = 0;
+         }
+         for (int qx = 0; qx < Q1D; ++qx)
+         {
+            const double s = sol_xyz[qz][qy][qx];
+            for (int dx = 0; dx < D1D; ++dx)
+            {
+               sol_x[dx] += Bt(dx,qx) * s;
+            }
+         }
+         for (int dy = 0; dy < D1D; ++dy)
+         {
+            const double wy = Bt(dy,qy);
+            for (int dx = 0; dx < D1D; ++dx)
+            {
+               sol_xy[dy][dx] += wy * sol_x[dx];
+            }
+         }
+      }
+      for (int dz = 0; dz < D1D; ++dz)
+      {
+         const double wz = Bt(dz,qz);
+         for (int dy = 0; dy < D1D; ++dy)
+         {
+            for (int dx = 0; dx < D1D; ++dx)
+            {
+               Y(dx,dy,dz,e) += wz * sol_xy[dy][dx];
+            }
+         }
+      }
+   }
+}
+
+template<int T_D1D, int T_Q1D, bool ACCUMULATE = true>
+MFEM_HOST_DEVICE inline
+void SmemPAMassApply3D_Element(const int e,
+                               const int NE,
+                               const double *b_,
+                               const double *d_,
+                               const double *x_,
+                               double *y_,
+                               const int d1d = 0,
+                               const int q1d = 0)
+{
+   constexpr int D1D = T_D1D ? T_D1D : d1d;
+   constexpr int Q1D = T_Q1D ? T_Q1D : q1d;
+   constexpr int MQ1 = T_Q1D ? T_Q1D : MAX_Q1D;
+   constexpr int MD1 = T_D1D ? T_D1D : MAX_D1D;
+   constexpr int MDQ = (MQ1 > MD1) ? MQ1 : MD1;
+
+   auto b = ConstDeviceMatrix(b_, Q1D, D1D);
+   auto d = DeviceTensor<4,const double>(d_, Q1D, Q1D, Q1D, NE);
+   auto x = DeviceTensor<4,const double>(x_, D1D, D1D, D1D, NE);
+   auto y = DeviceTensor<4,double>(y_, D1D, D1D, D1D, NE);
+
+   MFEM_SHARED double sDQ[MQ1*MD1];
+   double (*B)[MD1] = (double (*)[MD1]) sDQ;
+   double (*Bt)[MQ1] = (double (*)[MQ1]) sDQ;
+   MFEM_SHARED double sm0[MDQ*MDQ*MDQ];
+   MFEM_SHARED double sm1[MDQ*MDQ*MDQ];
+   double (*X)[MD1][MD1]   = (double (*)[MD1][MD1]) sm0;
+   double (*DDQ)[MD1][MQ1] = (double (*)[MD1][MQ1]) sm1;
+   double (*DQQ)[MQ1][MQ1] = (double (*)[MQ1][MQ1]) sm0;
+   double (*QQQ)[MQ1][MQ1] = (double (*)[MQ1][MQ1]) sm1;
+   double (*QQD)[MQ1][MD1] = (double (*)[MQ1][MD1]) sm0;
+   double (*QDD)[MD1][MD1] = (double (*)[MD1][MD1]) sm1;
+   MFEM_FOREACH_THREAD(dy,y,D1D)
+   {
+      MFEM_FOREACH_THREAD(dx,x,D1D)
+      {
+         MFEM_UNROLL(MD1)
+         for (int dz = 0; dz < D1D; ++dz)
+         {
+            X[dz][dy][dx] = x(dx,dy,dz,e);
+         }
+      }
+      MFEM_FOREACH_THREAD(dx,x,Q1D)
+      {
+         B[dx][dy] = b(dx,dy);
+      }
+   }
+   MFEM_SYNC_THREAD;
+   MFEM_FOREACH_THREAD(dy,y,D1D)
+   {
+      MFEM_FOREACH_THREAD(qx,x,Q1D)
+      {
+         double u[D1D];
+         MFEM_UNROLL(MD1)
+         for (int dz = 0; dz < D1D; dz++)
+         {
+            u[dz] = 0;
+         }
+         MFEM_UNROLL(MD1)
+         for (int dx = 0; dx < D1D; ++dx)
+         {
+            MFEM_UNROLL(MD1)
+            for (int dz = 0; dz < D1D; ++dz)
+            {
+               u[dz] += X[dz][dy][dx] * B[qx][dx];
+            }
+         }
+         MFEM_UNROLL(MD1)
+         for (int dz = 0; dz < D1D; ++dz)
+         {
+            DDQ[dz][dy][qx] = u[dz];
+         }
+      }
+   }
+   MFEM_SYNC_THREAD;
+   MFEM_FOREACH_THREAD(qy,y,Q1D)
+   {
+      MFEM_FOREACH_THREAD(qx,x,Q1D)
+      {
+         double u[D1D];
+         MFEM_UNROLL(MD1)
+         for (int dz = 0; dz < D1D; dz++)
+         {
+            u[dz] = 0;
+         }
+         MFEM_UNROLL(MD1)
+         for (int dy = 0; dy < D1D; ++dy)
+         {
+            MFEM_UNROLL(MD1)
+            for (int dz = 0; dz < D1D; dz++)
+            {
+               u[dz] += DDQ[dz][dy][qx] * B[qy][dy];
+            }
+         }
+         MFEM_UNROLL(MD1)
+         for (int dz = 0; dz < D1D; dz++)
+         {
+            DQQ[dz][qy][qx] = u[dz];
+         }
+      }
+   }
+   MFEM_SYNC_THREAD;
+   MFEM_FOREACH_THREAD(qy,y,Q1D)
+   {
+      MFEM_FOREACH_THREAD(qx,x,Q1D)
+      {
+         double u[Q1D];
+         MFEM_UNROLL(MQ1)
+         for (int qz = 0; qz < Q1D; qz++)
+         {
+            u[qz] = 0;
+         }
+         MFEM_UNROLL(MD1)
+         for (int dz = 0; dz < D1D; ++dz)
+         {
+            MFEM_UNROLL(MQ1)
+            for (int qz = 0; qz < Q1D; qz++)
+            {
+               u[qz] += DQQ[dz][qy][qx] * B[qz][dz];
+            }
+         }
+         MFEM_UNROLL(MQ1)
+         for (int qz = 0; qz < Q1D; qz++)
+         {
+            QQQ[qz][qy][qx] = u[qz] * d(qx,qy,qz,e);
+         }
+      }
+   }
+   MFEM_SYNC_THREAD;
+   MFEM_FOREACH_THREAD(di,y,D1D)
+   {
+      MFEM_FOREACH_THREAD(q,x,Q1D)
+      {
+         Bt[di][q] = b(q,di);
+      }
+   }
+   MFEM_SYNC_THREAD;
+   MFEM_FOREACH_THREAD(qy,y,Q1D)
+   {
+      MFEM_FOREACH_THREAD(dx,x,D1D)
+      {
+         double u[Q1D];
+         MFEM_UNROLL(MQ1)
+         for (int qz = 0; qz < Q1D; ++qz)
+         {
+            u[qz] = 0;
+         }
+         MFEM_UNROLL(MQ1)
+         for (int qx = 0; qx < Q1D; ++qx)
+         {
+            MFEM_UNROLL(MQ1)
+            for (int qz = 0; qz < Q1D; ++qz)
+            {
+               u[qz] += QQQ[qz][qy][qx] * Bt[dx][qx];
+            }
+         }
+         MFEM_UNROLL(MQ1)
+         for (int qz = 0; qz < Q1D; ++qz)
+         {
+            QQD[qz][qy][dx] = u[qz];
+         }
+      }
+   }
+   MFEM_SYNC_THREAD;
+   MFEM_FOREACH_THREAD(dy,y,D1D)
+   {
+      MFEM_FOREACH_THREAD(dx,x,D1D)
+      {
+         double u[Q1D];
+         MFEM_UNROLL(MQ1)
+         for (int qz = 0; qz < Q1D; ++qz)
+         {
+            u[qz] = 0;
+         }
+         MFEM_UNROLL(MQ1)
+         for (int qy = 0; qy < Q1D; ++qy)
+         {
+            MFEM_UNROLL(MQ1)
+            for (int qz = 0; qz < Q1D; ++qz)
+            {
+               u[qz] += QQD[qz][qy][dx] * Bt[dy][qy];
+            }
+         }
+         MFEM_UNROLL(MQ1)
+         for (int qz = 0; qz < Q1D; ++qz)
+         {
+            QDD[qz][dy][dx] = u[qz];
+         }
+      }
+   }
+   MFEM_SYNC_THREAD;
+   MFEM_FOREACH_THREAD(dy,y,D1D)
+   {
+      MFEM_FOREACH_THREAD(dx,x,D1D)
+      {
+         double u[D1D];
+         MFEM_UNROLL(MD1)
+         for (int dz = 0; dz < D1D; ++dz)
+         {
+            u[dz] = 0;
+         }
+         MFEM_UNROLL(MQ1)
+         for (int qz = 0; qz < Q1D; ++qz)
+         {
+            MFEM_UNROLL(MD1)
+            for (int dz = 0; dz < D1D; ++dz)
+            {
+               u[dz] += QDD[qz][dy][dx] * Bt[dz][qz];
+            }
+         }
+         MFEM_UNROLL(MD1)
+         for (int dz = 0; dz < D1D; ++dz)
+         {
+            if (ACCUMULATE)
+            {
+               y(dx,dy,dz,e) += u[dz];
+            }
+            else
+            {
+               y(dx,dy,dz,e) = u[dz];
+            }
+         }
+      }
+   }
+   MFEM_SYNC_THREAD;
+}
+
+} // namespace internal
+
+} // namespace mfem
+
+#endif
@@ -12,6 +12,7 @@
 #include "../general/forall.hpp"
 #include "bilininteg.hpp"
 #include "gridfunc.hpp"
+#include "qfunction.hpp"
 #include "ceed/integrators/diffusion/diffusion.hpp"

 using namespace std;
@@ -175,43 +176,9 @@ void VectorDiffusionIntegrator::AssemblePA(const FiniteElementSpace &fes)

   MFEM_VERIFY(!VQ && !MQ,
               "Only scalar coefficient supported for partial assembly for VectorDiffusionIntegrator");
-   Vector coeff;
-   if (Q == nullptr)
-   {
-      coeff.SetSize(1);
-      coeff(0) = 1.0;
-   }
-   else if (ConstantCoefficient* cQ = dynamic_cast<ConstantCoefficient*>(Q))
-   {
-      coeff.SetSize(1);
-      coeff(0) = cQ->constant;
-   }
-   else if (QuadratureFunctionCoefficient* qfQ =
-               dynamic_cast<QuadratureFunctionCoefficient*>(Q))
-   {
-      const QuadratureFunction &qFun = qfQ->GetQuadFunction();
-      MFEM_VERIFY(qFun.Size() == ne*nq,
-                  "Incompatible QuadratureFunction dimension \n");

-      MFEM_VERIFY(ir == &qFun.GetSpace()->GetElementIntRule(0),
-                  "IntegrationRule used within integrator and in"
-                  " QuadratureFunction appear to be different");
-      qFun.Read();
-      coeff.MakeRef(const_cast<QuadratureFunction &>(qFun),0);
-   }
-   else
-   {
-      coeff.SetSize(nq * ne);
-      auto Co = Reshape(coeff.HostWrite(), nq, ne);
-      for (int e = 0; e < ne; ++e)
-      {
-         ElementTransformation& T = *fes.GetElementTransformation(e);
-         for (int q = 0; q < nq; ++q)
-         {
-            Co(q,e) = Q->Eval(T, ir->IntPoint(q));
-         }
-      }
-   }
+   QuadratureSpace qs(*mesh, *ir);
+   CoefficientVector coeff(Q, qs, CoefficientStorage::COMPRESSED);

   const Array<double> &w = ir->GetWeights();
   const Vector &j = geom->J;
@@ -11,6 +11,7 @@

 #include "../general/forall.hpp"
 #include "bilininteg.hpp"
+#include "qspace.hpp"
 #include "gridfunc.hpp"

 namespace mfem
@@ -793,140 +794,63 @@ void VectorFEMassIntegrator::AssemblePA(const FiniteElementSpace &trial_fes,
   trial_fetype = trial_el->GetDerivType();
   test_fetype = test_el->GetDerivType();

-   auto SMQ = dynamic_cast<SymmetricMatrixCoefficient *>(MQ);
-
-   const int MQsymmDim = SMQ ? (SMQ->GetSize() * (SMQ->GetSize() + 1)) / 2 : 0;
-   const int MQfullDim = MQ ? (MQ->GetHeight() * MQ->GetWidth()) : 0;
-   const int MQdim = SMQ ? MQsymmDim : MQfullDim;
-   const int coeffDim = MQ ? MQdim : (DQ ? DQ->GetVDim() : 1);
-
-   symmetric = (SMQ || MQ == NULL);
-
   const bool trial_curl = (trial_fetype == mfem::FiniteElement::CURL);
   const bool trial_div = (trial_fetype == mfem::FiniteElement::DIV);
   const bool test_curl = (test_fetype == mfem::FiniteElement::CURL);
   const bool test_div = (test_fetype == mfem::FiniteElement::DIV);

+   QuadratureSpace qs(*mesh, *ir);
+   CoefficientVector coeff(qs, CoefficientStorage::SYMMETRIC);
+   if (Q) { coeff.Project(*Q); }
+   else if (MQ) { coeff.ProjectTranspose(*MQ); }
+   else if (DQ) { coeff.Project(*DQ); }
+   else { coeff.SetConstant(1.0); }
+
+   const int coeff_dim = coeff.GetVDim();
+   symmetric = (coeff_dim != dim*dim);
+
   if ((trial_curl && test_div) || (trial_div && test_curl))
-      pa_data.SetSize((coeffDim == 1 ? 1 : dim*dim) * nq * ne,
+      pa_data.SetSize((coeff_dim == 1 ? 1 : dim*dim) * nq * ne,
                      Device::GetMemoryType());
   else
-      pa_data.SetSize((symmetric ? symmDims : MQfullDim) * nq * ne,
+      pa_data.SetSize((symmetric ? symmDims : dims*dims) * nq * ne,
                      Device::GetMemoryType());

-   Vector coeff;
-
-   auto *qf_c = dynamic_cast<QuadratureFunctionCoefficient*>(Q);
-   if (qf_c)
-   {
-      const QuadratureFunction &qf = qf_c->GetQuadFunction();
-      qf.Read();
-      coeff.MakeRef(const_cast<QuadratureFunction&>(qf), 0);
-   }
-   else
-   {
-      coeff.SetSize(coeffDim * ne * nq);
-      coeff = 1.0;
-      auto coeffh = Reshape(coeff.HostWrite(), coeffDim, nq, ne);
-      if (Q || DQ || MQ)
-      {
-         Vector DM(DQ ? coeffDim : 0);
-         DenseMatrix M;
-         DenseSymmetricMatrix SM;
-
-         if (DQ)
-         {
-            MFEM_VERIFY(coeffDim == dim, "");
-         }
-         if (SMQ)
-         {
-            MFEM_VERIFY(SMQ->GetSize() == dim, "");
-            SM.SetSize(dim);
-         }
-         else if (MQ)
-         {
-            MFEM_VERIFY(coeffDim == MQdim, "");
-            MFEM_VERIFY(MQ->GetHeight() == dim && MQ->GetWidth() == dim, "");
-            M.SetSize(dim);
-         }
-
-         for (int e=0; e<ne; ++e)
-         {
-            ElementTransformation *tr = mesh->GetElementTransformation(e);
-            for (int p=0; p<nq; ++p)
-            {
-               if (SMQ)
-               {
-                  SMQ->Eval(SM, *tr, ir->IntPoint(p));
-                  int cnt = 0;
-                  for (int i=0; i<dim; ++i)
-                     for (int j=i; j<dim; ++j, ++cnt)
-                     {
-                        coeffh(cnt, p, e) = SM(i,j);
-                     }
-               }
-               else if (MQ)
-               {
-                  MQ->Eval(M, *tr, ir->IntPoint(p));
-
-                  for (int i=0; i<dim; ++i)
-                     for (int j=0; j<dim; ++j)
-                     {
-                        coeffh(j+(i*dim), p, e) = M(i,j);
-                     }
-               }
-               else if (DQ)
-               {
-                  DQ->Eval(DM, *tr, ir->IntPoint(p));
-                  for (int i=0; i<coeffDim; ++i)
-                  {
-                     coeffh(i, p, e) = DM[i];
-                  }
-               }
-               else
-               {
-                  coeffh(0, p, e) = Q->Eval(*tr, ir->IntPoint(p));
-               }
-            }
-         }
-      }
-   }
-
   if (trial_curl && test_curl && dim == 3)
   {
-      PADiffusionSetup3D(quad1D, coeffDim, ne, ir->GetWeights(), geom->J,
+      PADiffusionSetup3D(quad1D, coeff_dim, ne, ir->GetWeights(), geom->J,
                         coeff, pa_data);
   }
   else if (trial_curl && test_curl && dim == 2)
   {
-      PADiffusionSetup2D<2>(quad1D, coeffDim, ne, ir->GetWeights(), geom->J,
+      PADiffusionSetup2D<2>(quad1D, coeff_dim, ne, ir->GetWeights(), geom->J,
                            coeff, pa_data);
   }
   else if (trial_div && test_div && dim == 3)
   {
-      PAHdivSetup3D(quad1D, coeffDim, ne, ir->GetWeights(), geom->J,
+      PAHdivSetup3D(quad1D, coeff_dim, ne, ir->GetWeights(), geom->J,
                    coeff, pa_data);
   }
   else if (trial_div && test_div && dim == 2)
   {
-      PAHdivSetup2D(quad1D, coeffDim, ne, ir->GetWeights(), geom->J,
+      PAHdivSetup2D(quad1D, coeff_dim, ne, ir->GetWeights(), geom->J,
                    coeff, pa_data);
   }
   else if (((trial_curl && test_div) || (trial_div && test_curl)) &&
            test_fel->GetOrder() == trial_fel->GetOrder())
   {
-      if (coeffDim == 1)
+      if (coeff_dim == 1)
      {
-         PAHcurlL2Setup(nq, coeffDim, ne, ir->GetWeights(), coeff, pa_data);
+         PAHcurlL2Setup(nq, coeff_dim, ne, ir->GetWeights(), coeff, pa_data);
      }
      else
      {
         const bool tr = (trial_div && test_curl);
         if (dim == 3)
-            PAHcurlHdivSetup3D(quad1D, coeffDim, ne, tr, ir->GetWeights(),
+            PAHcurlHdivSetup3D(quad1D, coeff_dim, ne, tr, ir->GetWeights(),
                               geom->J, coeff, pa_data);
         else
-            PAHcurlHdivSetup2D(quad1D, coeffDim, ne, tr, ir->GetWeights(),
+            PAHcurlHdivSetup2D(quad1D, coeff_dim, ne, tr, ir->GetWeights(),
                               geom->J, coeff, pa_data);
      }
   }
@@ -1168,19 +1092,8 @@ void MixedVectorGradientIntegrator::AssemblePA(const FiniteElementSpace

   pa_data.SetSize(symmDims * nq * ne, Device::GetMemoryType());

-   Vector coeff(ne * nq);
-   coeff = 1.0;
-   if (Q)
-   {
-      for (int e=0; e<ne; ++e)
-      {
-         ElementTransformation *tr = mesh->GetElementTransformation(e);
-         for (int p=0; p<nq; ++p)
-         {
-            coeff[p + (e * nq)] = Q->Eval(*tr, ir->IntPoint(p));
-         }
-      }
-   }
+   QuadratureSpace qs(*mesh, *ir);
+   CoefficientVector coeff(Q, qs, CoefficientStorage::FULL);

   // Use the same setup functions as VectorFEMassIntegrator.
   if (test_el->GetDerivType() == mfem::FiniteElement::CURL && dim == 3)
@@ -112,7 +112,7 @@ static void InitBasisImpl(const FiniteElementSpace &fes,
   const bool tensor = dynamic_cast<const mfem::TensorBasisElement *>
                       (&fe) != nullptr;

-   // Init or retreive key values
+   // Init or retrieve key values
   if (basis_itr == mfem::internal::ceed_basis_map.end())
   {
      if ( tensor )
@@ -20,6 +20,7 @@
 #include "../../../linalg/dtensor.hpp"
 #include "../../../mesh/mesh.hpp"
 #include "../../gridfunc.hpp"
+#include "../../qfunction.hpp"
 #include "util.hpp"
 #include "ceed.hpp"

@@ -121,7 +122,7 @@ void InitCoefficient(mfem::Coefficient *Q, mfem::Mesh &mesh,
      MFEM_VERIFY(qFun.Size() == nq * ne,
                  "Incompatible QuadratureFunction dimension \n");

-      MFEM_VERIFY(&ir == &qFun.GetSpace()->GetElementIntRule(0),
+      MFEM_VERIFY(&ir == &qFun.GetSpace()->GetIntRule(0),
                  "IntegrationRule used within integrator and in"
                  " QuadratureFunction appear to be different");
      qFun.Read();
@@ -195,7 +196,7 @@ void InitCoefficient(mfem::VectorCoefficient *VQ, mfem::Mesh &mesh,
      MFEM_VERIFY(qFun.Size() == dim * nq * ne,
                  "Incompatible QuadratureFunction dimension \n");

-      MFEM_VERIFY(&ir == &qFun.GetSpace()->GetElementIntRule(0),
+      MFEM_VERIFY(&ir == &qFun.GetSpace()->GetIntRule(0),
                  "IntegrationRule used within integrator and in"
                  " QuadratureFunction appear to be different");
      qFun.Read();
@@ -279,7 +280,7 @@ void InitCoefficientWithIndices(mfem::Coefficient *Q, mfem::Mesh &mesh,
      MFEM_VERIFY(qFun.Size() == nq * ne,
                  "Incompatible QuadratureFunction dimension \n");

-      MFEM_VERIFY(&ir == &qFun.GetSpace()->GetElementIntRule(0),
+      MFEM_VERIFY(&ir == &qFun.GetSpace()->GetIntRule(0),
                  "IntegrationRule used within integrator and in"
                  " QuadratureFunction appear to be different");
      ceedCoeff->coeff.SetSize(nq * nelem);
@@ -369,7 +370,7 @@ void InitCoefficientWithIndices(mfem::VectorCoefficient *VQ, mfem::Mesh &mesh,
      MFEM_VERIFY(qFun.Size() == dim * nq * ne,
                  "Incompatible QuadratureFunction dimension \n");

-      MFEM_VERIFY(&ir == &qFun.GetSpace()->GetElementIntRule(0),
+      MFEM_VERIFY(&ir == &qFun.GetSpace()->GetIntRule(0),
                  "IntegrationRule used within integrator and in"
                  " QuadratureFunction appear to be different");
      ceedCoeff->coeff.SetSize(dim * nq * nelem);
@@ -232,7 +232,7 @@ void InitRestriction(const FiniteElementSpace &fes,
   RestrKey restr_key(&fes, nelem, P, ncomp, restr_type::Standard);
   auto restr_itr = mfem::internal::ceed_restr_map.find(restr_key);

-   // Init or retreive key values
+   // Init or retrieve key values
   if (restr_itr == mfem::internal::ceed_restr_map.end())
   {
      InitRestrictionImpl(fes, ceed, restr);
@@ -257,7 +257,7 @@ void InitRestrictionWithIndices(const FiniteElementSpace &fes,
   RestrKey restr_key(&fes, nelem, P, ncomp, restr_type::Standard);
   auto restr_itr = mfem::internal::ceed_restr_map.find(restr_key);

-   // Init or retreive key values
+   // Init or retrieve key values
   if (restr_itr == mfem::internal::ceed_restr_map.end())
   {
      InitRestrictionWithIndicesImpl(fes, nelem, indices, ceed, restr);
@@ -281,7 +281,7 @@ void InitCoeffRestrictionWithIndices(const FiniteElementSpace &fes,
   RestrKey restr_key(&fes, nelem, nquads, ncomp, restr_type::Coeff);
   auto restr_itr = mfem::internal::ceed_restr_map.find(restr_key);

-   // Init or retreive key values
+   // Init or retrieve key values
   if (restr_itr == mfem::internal::ceed_restr_map.end())
   {
      InitCoeffRestrictionWithIndicesImpl(fes, nelem, indices, nquads, ncomp,
@@ -676,7 +676,6 @@ AlgebraicSpaceHierarchy::AlgebraicSpaceHierarchy(FiniteElementSpace &fes)
      const SparseMatrix *R = fespaces[ilevel+1]->GetRestrictionMatrix();
      if (R)
      {
-         R->EnsureMultTranspose();
         R_tr[ilevel] = new TransposeOperator(*R);
      }
      else
@@ -745,7 +744,7 @@ ParAlgebraicCoarseSpace::ParAlgebraicCoarseSpace(
   ldof_group.SetSize(lsize);
   ldof_group = 0;

-   GroupTopology &group_topo = gc_fine->GetGroupTopology();
+   const GroupTopology &group_topo = gc_fine->GetGroupTopology();
   gc = new GroupCommunicator(group_topo);
   Table &group_ldof = gc->GroupLDofTable();
   group_ldof.MakeI(group_ldof_fine.Size());
@@ -822,11 +821,11 @@ HypreParMatrix *ParAlgebraicCoarseSpace::GetProlongationHypreParMatrix()

   ParMesh *pmesh = dynamic_cast<ParMesh*>(mesh);
   MFEM_VERIFY(pmesh != NULL, "");
-   Array<HYPRE_Int> dof_offsets, tdof_offsets, tdof_nb_offsets;
-   Array<HYPRE_Int> *offsets[2] = {&dof_offsets, &tdof_offsets};
+   Array<HYPRE_BigInt> dof_offsets, tdof_offsets, tdof_nb_offsets;
+   Array<HYPRE_BigInt> *offsets[2] = {&dof_offsets, &tdof_offsets};
   int lsize = P->Height();
   int ltsize = P->Width();
-   HYPRE_Int loc_sizes[2] = {lsize, ltsize};
+   HYPRE_BigInt loc_sizes[2] = {lsize, ltsize};
   pmesh->GenerateOffsets(2, loc_sizes, offsets);

   MPI_Comm comm = pmesh->GetComm();
@@ -870,12 +869,12 @@ HypreParMatrix *ParAlgebraicCoarseSpace::GetProlongationHypreParMatrix()
   HYPRE_Int *j_offd = Memory<HYPRE_Int>(lsize-ltsize);
   int offd_counter;

-   HYPRE_Int *cmap   = Memory<HYPRE_Int>(lsize-ltsize);
+   HYPRE_BigInt *cmap   = Memory<HYPRE_BigInt>(lsize-ltsize);

-   HYPRE_Int *col_starts = tdof_offsets;
-   HYPRE_Int *row_starts = dof_offsets;
+   HYPRE_BigInt *col_starts = tdof_offsets;
+   HYPRE_BigInt *row_starts = dof_offsets;

-   Array<Pair<HYPRE_Int, int> > cmap_j_offd(lsize-ltsize);
+   Array<Pair<HYPRE_BigInt, int> > cmap_j_offd(lsize-ltsize);

   i_diag[0] = i_offd[0] = 0;
   diag_counter = offd_counter = 0;
@@ -909,7 +908,7 @@ HypreParMatrix *ParAlgebraicCoarseSpace::GetProlongationHypreParMatrix()
      i_offd[i_ldof+1] = offd_counter;
   }

-   SortPairs<HYPRE_Int, int>(cmap_j_offd, offd_counter);
+   SortPairs<HYPRE_BigInt, int>(cmap_j_offd, offd_counter);

   for (int i = 0; i < offd_counter; i++)
   {
@@ -48,6 +48,31 @@ ElementTransformation *RefinedToCoarse(
   return coarse_T;
 }

+void Coefficient::Project(QuadratureFunction &qf)
+{
+   QuadratureSpaceBase &qspace = *qf.GetSpace();
+   const int ne = qspace.GetNE();
+   Vector values;
+   for (int iel = 0; iel < ne; ++iel)
+   {
+      qf.GetValues(iel, values);
+      const IntegrationRule &ir = qspace.GetIntRule(iel);
+      ElementTransformation& T = *qspace.GetTransformation(iel);
+      for (int iq = 0; iq < ir.Size(); ++iq)
+      {
+         const IntegrationPoint &ip = ir[iq];
+         T.SetIntPoint(&ip);
+         const int iq_p = qspace.GetPermutedIndex(iel, iq);
+         values[iq_p] = Eval(T, ip);
+      }
+   }
+}
+
+void ConstantCoefficient::Project(QuadratureFunction &qf)
+{
+   qf = constant;
+}
+
 double PWConstCoefficient::Eval(ElementTransformation & T,
                                const IntegrationPoint & ip)
 {
@@ -135,6 +160,11 @@ double GridFunctionCoefficient::Eval (ElementTransformation &T,
   }
 }

+void GridFunctionCoefficient::Project(QuadratureFunction &qf)
+{
+   qf.ProjectGridFunction(*GridF);
+}
+
 void TransformedCoefficient::SetTime(double t)
 {
   if (Q1) { Q1->SetTime(t); }
@@ -203,6 +233,29 @@ void VectorCoefficient::Eval(DenseMatrix &M, ElementTransformation &T,
   }
 }

+void VectorCoefficient::Project(QuadratureFunction &qf)
+{
+   MFEM_VERIFY(vdim == qf.GetVDim(), "Wrong sizes.");
+   QuadratureSpaceBase &qspace = *qf.GetSpace();
+   const int ne = qspace.GetNE();
+   DenseMatrix values;
+   Vector col;
+   for (int iel = 0; iel < ne; ++iel)
+   {
+      qf.GetValues(iel, values);
+      const IntegrationRule &ir = qspace.GetIntRule(iel);
+      ElementTransformation& T = *qspace.GetTransformation(iel);
+      for (int iq = 0; iq < ir.Size(); ++iq)
+      {
+         const IntegrationPoint &ip = ir[iq];
+         T.SetIntPoint(&ip);
+         const int iq_p = qspace.GetPermutedIndex(iel, iq);
+         values.GetColumnReference(iq_p, col);
+         Eval(col, T, ip);
+      }
+   }
+}
+
 void PWVectorCoefficient::InitMap(const Array<int> & attr,
                                  const Array<VectorCoefficient*> & coefs)
 {
@@ -368,6 +421,11 @@ void VectorGridFunctionCoefficient::Eval(
   }
 }

+void VectorGridFunctionCoefficient::Project(QuadratureFunction &qf)
+{
+   qf.ProjectGridFunction(*GridFunc);
+}
+
 GradientGridFunctionCoefficient::GradientGridFunctionCoefficient (
   const GridFunction *gf)
   : VectorCoefficient((gf) ?
@@ -517,6 +575,29 @@ void VectorRestrictedCoefficient::Eval(
   }
 }

+void MatrixCoefficient::Project(QuadratureFunction &qf, bool transpose)
+{
+   MFEM_VERIFY(qf.GetVDim() == height*width, "Wrong sizes.");
+   QuadratureSpaceBase &qspace = *qf.GetSpace();
+   const int ne = qspace.GetNE();
+   DenseMatrix values, matrix;
+   for (int iel = 0; iel < ne; ++iel)
+   {
+      qf.GetValues(iel, values);
+      const IntegrationRule &ir = qspace.GetIntRule(iel);
+      ElementTransformation& T = *qspace.GetTransformation(iel);
+      for (int iq = 0; iq < ir.Size(); ++iq)
+      {
+         const IntegrationPoint &ip = ir[iq];
+         T.SetIntPoint(&ip);
+         const int iq_p = qspace.GetPermutedIndex(iel, iq);
+         matrix.UseExternalData(&values(0, iq_p), height, width);
+         Eval(matrix, T, ip);
+         if (transpose) { matrix.Transpose(); }
+      }
+   }
+}
+
 void PWMatrixCoefficient::InitMap(const Array<int> & attr,
                                  const Array<MatrixCoefficient*> & coefs)
 {
@@ -669,6 +750,31 @@ void MatrixFunctionCoefficient::EvalSymmetric(Vector &K,
   }
 }

+void SymmetricMatrixCoefficient::ProjectSymmetric(QuadratureFunction &qf)
+{
+   const int vdim = qf.GetVDim();
+   MFEM_VERIFY(vdim == height*(height+1)/2, "Wrong sizes.");
+
+   QuadratureSpaceBase &qspace = *qf.GetSpace();
+   const int ne = qspace.GetNE();
+   DenseMatrix values;
+   DenseSymmetricMatrix matrix;
+   for (int iel = 0; iel < ne; ++iel)
+   {
+      qf.GetValues(iel, values);
+      const IntegrationRule &ir = qspace.GetIntRule(iel);
+      ElementTransformation& T = *qspace.GetTransformation(iel);
+      for (int iq = 0; iq < ir.Size(); ++iq)
+      {
+         const IntegrationPoint &ip = ir[iq];
+         T.SetIntPoint(&ip);
+         matrix.UseExternalData(&values(0, iq), vdim);
+         Eval(matrix, T, ip);
+      }
+   }
+}
+
+
 void SymmetricMatrixCoefficient::Eval(DenseMatrix &K, ElementTransformation &T,
                                      const IntegrationPoint &ip)
 {
@@ -1437,12 +1543,12 @@ void VectorQuadratureFunctionCoefficient::Eval(Vector &V,

   if (index == 0 && vdim == QuadF.GetVDim())
   {
-      QuadF.GetElementValues(T.ElementNo, ip.index, V);
+      QuadF.GetValues(T.ElementNo, ip.index, V);
   }
   else
   {
      Vector temp;
-      QuadF.GetElementValues(T.ElementNo, ip.index, temp);
+      QuadF.GetValues(T.ElementNo, ip.index, temp);
      V.SetSize(vdim);
      for (int i = 0; i < vdim; i++)
      {
@@ -1453,6 +1559,11 @@ void VectorQuadratureFunctionCoefficient::Eval(Vector &V,
   return;
 }

+void VectorQuadratureFunctionCoefficient::Project(QuadratureFunction &qf)
+{
+   qf = QuadF;
+}
+
 QuadratureFunctionCoefficient::QuadratureFunctionCoefficient(
   QuadratureFunction &qf) : QuadF(qf)
 {
@@ -1464,8 +1575,210 @@ double QuadratureFunctionCoefficient::Eval(ElementTransformation &T,
 {
   QuadF.HostRead();
   Vector temp(1);
-   QuadF.GetElementValues(T.ElementNo, ip.index, temp);
+   QuadF.GetValues(T.ElementNo, ip.index, temp);
   return temp[0];
 }

+void QuadratureFunctionCoefficient::Project(QuadratureFunction &qf)
+{
+   qf = QuadF;
+}
+
+
+CoefficientVector::CoefficientVector(
+   QuadratureSpaceBase &qs_, CoefficientStorage storage_)
+   : Vector(), storage(storage_), vdim(0), qs(qs_), qf(NULL)
+{
+   UseDevice(true);
+}
+
+CoefficientVector::CoefficientVector(Coefficient *coeff,
+                                     QuadratureSpaceBase &qs_,
+                                     CoefficientStorage storage_)
+   : CoefficientVector(qs_, storage_)
+{
+   if (coeff == NULL)
+   {
+      SetConstant(1.0);
+   }
+   else
+   {
+      Project(*coeff);
+   }
+}
+
+CoefficientVector::CoefficientVector(Coefficient &coeff,
+                                     QuadratureSpaceBase &qs_,
+                                     CoefficientStorage storage_)
+   : CoefficientVector(qs_, storage_)
+{
+   Project(coeff);
+}
+
+CoefficientVector::CoefficientVector(VectorCoefficient &coeff,
+                                     QuadratureSpaceBase &qs_,
+                                     CoefficientStorage storage_)
+   : CoefficientVector(qs_, storage_)
+{
+   Project(coeff);
+}
+
+CoefficientVector::CoefficientVector(MatrixCoefficient &coeff,
+                                     QuadratureSpaceBase &qs_,
+                                     CoefficientStorage storage_)
+   : CoefficientVector(qs_, storage_)
+{
+   Project(coeff);
+}
+
+void CoefficientVector::Project(Coefficient &coeff)
+{
+   vdim = 1;
+   if (auto *const_coeff = dynamic_cast<ConstantCoefficient*>(&coeff))
+   {
+      SetConstant(const_coeff->constant);
+   }
+   else if (auto *qf_coeff = dynamic_cast<QuadratureFunctionCoefficient*>(&coeff))
+   {
+      MakeRef(qf_coeff->GetQuadFunction());
+   }
+   else
+   {
+      if (qf == nullptr) { qf = new QuadratureFunction(qs); }
+      qf->SetVDim(1);
+      coeff.Project(*qf);
+      Vector::MakeRef(*qf, 0, qf->Size());
+   }
+}
+
+void CoefficientVector::Project(VectorCoefficient &coeff)
+{
+   vdim = coeff.GetVDim();
+   if (auto *const_coeff = dynamic_cast<VectorConstantCoefficient*>(&coeff))
+   {
+      SetConstant(const_coeff->GetVec());
+   }
+   else if (auto *qf_coeff =
+               dynamic_cast<VectorQuadratureFunctionCoefficient*>(&coeff))
+   {
+      MakeRef(qf_coeff->GetQuadFunction());
+   }
+   else
+   {
+      if (qf == nullptr) { qf = new QuadratureFunction(qs, vdim); }
+      qf->SetVDim(vdim);
+      coeff.Project(*qf);
+      Vector::MakeRef(*qf, 0, qf->Size());
+   }
+}
+
+void CoefficientVector::Project(MatrixCoefficient &coeff, bool transpose)
+{
+   if (auto *const_coeff = dynamic_cast<MatrixConstantCoefficient*>(&coeff))
+   {
+      SetConstant(const_coeff->GetMatrix());
+   }
+   else if (auto *const_sym_coeff =
+               dynamic_cast<SymmetricMatrixConstantCoefficient*>(&coeff))
+   {
+      SetConstant(const_sym_coeff->GetMatrix());
+   }
+   else
+   {
+      auto *sym_coeff = dynamic_cast<SymmetricMatrixCoefficient*>(&coeff);
+      const bool sym = sym_coeff && (storage & CoefficientStorage::SYMMETRIC);
+      const int height = coeff.GetHeight();
+      const int width = coeff.GetWidth();
+      vdim = sym ? height*(height + 1)/2 : width*height;
+
+      if (qf == nullptr) { qf = new QuadratureFunction(qs, vdim); }
+      qf->SetVDim(vdim);
+      if (sym) { sym_coeff->ProjectSymmetric(*qf); }
+      else { coeff.Project(*qf, transpose); }
+      Vector::MakeRef(*qf, 0, qf->Size());
+   }
+}
+
+void CoefficientVector::ProjectTranspose(MatrixCoefficient &coeff)
+{
+   Project(coeff, true);
+}
+
+void CoefficientVector::MakeRef(const QuadratureFunction &qf_)
+{
+   vdim = qf_.GetVDim();
+   const QuadratureSpaceBase *qs2 = qf_.GetSpace();
+   MFEM_CONTRACT_VAR(qs2); // qs2 used only for asserts
+   MFEM_VERIFY(qs2 != NULL, "Invalid QuadratureSpace.")
+   MFEM_VERIFY(qs2->GetMesh() == qs.GetMesh(), "Meshes differ.");
+   MFEM_VERIFY(qs2->GetOrder() == qs.GetOrder(), "Orders differ.");
+   Vector::MakeRef(const_cast<QuadratureFunction&>(qf_), 0, qf_.Size());
+}
+
+void CoefficientVector::SetConstant(double constant)
+{
+   const int nq = (storage & CoefficientStorage::CONSTANTS) ? 1 : qs.GetSize();
+   vdim = 1;
+   SetSize(nq);
+   Vector::operator=(constant);
+}
+
+void CoefficientVector::SetConstant(const Vector &constant)
+{
+   const int nq = (storage & CoefficientStorage::CONSTANTS) ? 1 : qs.GetSize();
+   vdim = constant.Size();
+   SetSize(nq*vdim);
+   for (int iq = 0; iq < nq; ++iq)
+   {
+      for (int vd = 0; vd<vdim; ++vd)
+      {
+         (*this)[vd + iq*vdim] = constant[vd];
+      }
+   }
+}
+
+void CoefficientVector::SetConstant(const DenseMatrix &constant)
+{
+   const int nq = (storage & CoefficientStorage::CONSTANTS) ? 1 : qs.GetSize();
+   const int width = constant.Width();
+   const int height = constant.Height();
+   vdim = width*height;
+   SetSize(nq*vdim);
+   for (int iq = 0; iq < nq; ++iq)
+   {
+      for (int j = 0; j < width; ++j)
+      {
+         for (int i = 0; i < height; ++i)
+         {
+            (*this)[i + j*height + iq*vdim] = constant(i, j);
+         }
+      }
+   }
+}
+
+void CoefficientVector::SetConstant(const DenseSymmetricMatrix &constant)
+{
+   const int nq = (storage & CoefficientStorage::CONSTANTS) ? 1 : qs.GetSize();
+   const int height = constant.Height();
+   const bool sym = storage & CoefficientStorage::SYMMETRIC;
+   vdim = sym ? height*(height + 1)/2 : height*height;
+   SetSize(nq*vdim);
+   for (int iq = 0; iq < nq; ++iq)
+   {
+      for (int vd = 0; vd < vdim; ++vd)
+      {
+         const double value = sym ? constant.GetData()[vd] : constant(vd % height,
+                                                                      vd / height);
+         (*this)[vd + iq*vdim] = value;
+      }
+   }
+}
+
+int CoefficientVector::GetVDim() const { return vdim; }
+
+CoefficientVector::~CoefficientVector()
+{
+   delete qf;
+}
+
 }
@@ -23,6 +23,8 @@ namespace mfem
 {

 class Mesh;
+class QuadratureSpaceBase;
+class QuadratureFunction;

 #ifdef MFEM_USE_MPI
 class ParMesh;
@@ -70,6 +72,10 @@ public:
      return Eval(T, ip);
   }

+   /// @brief Fill the QuadratureFunction @a qf by evaluating the coefficient at
+   /// the quadrature points.
+   virtual void Project(QuadratureFunction &qf);
+
   virtual ~Coefficient() { }
 };

@@ -87,6 +93,9 @@ public:
   virtual double Eval(ElementTransformation &T,
                       const IntegrationPoint &ip)
   { return (constant); }
+
+   /// Fill the QuadratureFunction @a qf with the constant value.
+   void Project(QuadratureFunction &qf);
 };

 /** @brief A piecewise constant coefficient with the constants keyed
@@ -274,6 +283,13 @@ public:
   /// Evaluate the coefficient at @a ip.
   virtual double Eval(ElementTransformation &T,
                       const IntegrationPoint &ip);
+
+   /// @brief Fill the QuadratureFunction @a qf by evaluating the coefficient at
+   /// the quadrature points.
+   ///
+   /// This function uses the efficient QuadratureFunction::ProjectGridFunction
+   /// to fill the QuadratureFunction.
+   virtual void Project(QuadratureFunction &qf);
 };


@@ -471,6 +487,13 @@ public:
   virtual void Eval(DenseMatrix &M, ElementTransformation &T,
                     const IntegrationRule &ir);

+   /// @brief Fill the QuadratureFunction @a qf by evaluating the coefficient at
+   /// the quadrature points.
+   ///
+   /// The @a vdim of the VectorCoefficient should be equal to the @a vdim of
+   /// the QuadratureFunction.
+   virtual void Project(QuadratureFunction &qf);
+
   virtual ~VectorCoefficient() { }
 };

@@ -491,7 +514,7 @@ public:
                     const IntegrationPoint &ip) { V = vec; }

   /// Return a reference to the constant vector in this class.
-   const Vector& GetVec() { return vec; }
+   const Vector& GetVec() const { return vec; }
 };

 /** @brief A piecewise vector-valued coefficient with the pieces keyed off the
@@ -688,6 +711,13 @@ public:
   virtual void Eval(DenseMatrix &M, ElementTransformation &T,
                     const IntegrationRule &ir);

+   /// @brief Fill the QuadratureFunction @a qf by evaluating the coefficient at
+   /// the quadrature points.
+   ///
+   /// This function uses the efficient QuadratureFunction::ProjectGridFunction
+   /// to fill the QuadratureFunction.
+   virtual void Project(QuadratureFunction &qf);
+
   virtual ~VectorGridFunctionCoefficient() { }
 };

@@ -915,6 +945,14 @@ public:
   virtual void Eval(DenseMatrix &K, ElementTransformation &T,
                     const IntegrationPoint &ip) = 0;

+   /// @brief Fill the QuadratureFunction @a qf by evaluating the coefficient at
+   /// the quadrature points. The matrix will be transposed or not according to
+   /// the boolean argument @a transpose.
+   ///
+   /// The @a vdim of the QuadratureFunction should be equal to the height times
+   /// the width of the matrix.
+   virtual void Project(QuadratureFunction &qf, bool transpose=false);
+
   /// (DEPRECATED) Evaluate a symmetric matrix coefficient.
   /** @brief Evaluate the upper triangular entries of the matrix coefficient
       in the symmetric case, similarly to Eval. Matrix entry (i,j) is stored
@@ -943,6 +981,8 @@ public:
   /// Evaluate the matrix coefficient at @a ip.
   virtual void Eval(DenseMatrix &M, ElementTransformation &T,
                     const IntegrationPoint &ip) { M = mat; }
+   /// Return a reference to the constant matrix.
+   const DenseMatrix& GetMatrix() { return mat; }
 };


@@ -1146,6 +1186,8 @@ public:
       can be overridden with the @a own parameter. */
   void Set(int i, int j, Coefficient * c, bool own=true);

+   using MatrixCoefficient::Eval;
+
   /// Evaluate coefficient located at (i,j) in the matrix using integration
   /// point @a ip.
   double Eval(int i, int j, ElementTransformation &T, const IntegrationPoint &ip)
@@ -1260,6 +1302,15 @@ public:
   /// Get the size of the matrix.
   int GetSize() const { return height; }

+   /// @brief Fill the QuadratureFunction @a qf by evaluating the coefficient at
+   /// the quadrature points.
+   ///
+   /// @note As opposed to MatrixCoefficient::Project, this function stores only
+   /// the @a symmetric part of the matrix at each quadrature point.
+   ///
+   /// The @a vdim of the coefficient should be equal to height*(height+1)/2.
+   virtual void ProjectSymmetric(QuadratureFunction &qf);
+
   /** @brief Evaluate the matrix coefficient in the element described by @a T
       at the point @a ip, storing the result as a symmetric matrix @a K. */
   /** @note When this method is called, the caller must make sure that the
@@ -1280,6 +1331,9 @@ public:
   virtual void Eval(DenseMatrix &K, ElementTransformation &T,
                     const IntegrationPoint &ip);

+   /// Return a reference to the constant matrix.
+   const DenseSymmetricMatrix& GetMatrix() { return mat; }
+
   virtual ~SymmetricMatrixCoefficient() { }
 };

@@ -2049,8 +2103,6 @@ public:
 };
 ///@}

-class QuadratureFunction;
-
 /** @brief Vector quadrature function coefficient which requires that the
    quadrature rules used for this vector coefficient be the same as those that
    live within the supplied QuadratureFunction. */
@@ -2075,6 +2127,8 @@ public:
   virtual void Eval(Vector &V, ElementTransformation &T,
                     const IntegrationPoint &ip);

+   virtual void Project(QuadratureFunction &qf);
+
   virtual ~VectorQuadratureFunctionCoefficient() { }
 };

@@ -2094,9 +2148,123 @@ public:

   virtual double Eval(ElementTransformation &T, const IntegrationPoint &ip);

+   virtual void Project(QuadratureFunction &qf);
+
   virtual ~QuadratureFunctionCoefficient() { }
 };

+/// Flags that determine what storage optimizations to use in CoefficientVector
+enum class CoefficientStorage : int
+{
+   FULL = 0, ///< Store the coefficient as a full QuadratureFunction.
+   CONSTANTS = 1 << 0, ///< Store constants using only @a vdim entries.
+   SYMMETRIC = 1 << 1, ///< Store the triangular part of symmetric matrices.
+   COMPRESSED = CONSTANTS | SYMMETRIC ///< Enable all above compressions.
+};
+
+inline CoefficientStorage operator|(CoefficientStorage a, CoefficientStorage b)
+{
+   return CoefficientStorage(int(a) | int(b));
+}
+
+inline int operator&(CoefficientStorage a, CoefficientStorage b)
+{
+   return int(a) & int(b);
+}
+
+
+/// @brief Class to represent a coefficient evaluated at quadrature points.
+///
+/// In the general case, a CoefficientVector is the same as a QuadratureFunction
+/// with a coefficient projected onto it.
+///
+/// This class allows for some "compression" of the coefficient data, according
+/// to the storage flags given by CoefficientStorage. For example, constant
+/// coefficients can be stored using only @a vdim values, and symmetric matrices
+/// can be stored using e.g. the upper triangular part of the matrix.
+class CoefficientVector : public Vector
+{
+protected:
+   CoefficientStorage storage; ///< Storage optimizations (see CoefficientStorage).
+   int vdim; ///< Number of values per quadrature point.
+   QuadratureSpaceBase &qs; ///< Associated QuadratureSpaceBase.
+   QuadratureFunction *qf; ///< Internal QuadratureFunction (owned, may be NULL).
+public:
+   /// Create an empty CoefficientVector.
+   CoefficientVector(QuadratureSpaceBase &qs_,
+                     CoefficientStorage storage_ = CoefficientStorage::FULL);
+
+   /// @brief Create a CoefficientVector from the given Coefficient and
+   /// QuadratureSpaceBase.
+   ///
+   /// If @a coeff is NULL, it will be interpreted as a constant with value one.
+   /// @sa CoefficientStorage for a description of @a storage_.
+   CoefficientVector(Coefficient *coeff, QuadratureSpaceBase &qs,
+                     CoefficientStorage storage_ = CoefficientStorage::FULL);
+
+   /// @brief Create a CoefficientVector from the given Coefficient and
+   /// QuadratureSpaceBase.
+   ///
+   /// @sa CoefficientStorage for a description of @a storage_.
+   CoefficientVector(Coefficient &coeff, QuadratureSpaceBase &qs,
+                     CoefficientStorage storage_ = CoefficientStorage::FULL);
+
+   /// @brief Create a CoefficientVector from the given VectorCoefficient and
+   /// QuadratureSpaceBase.
+   ///
+   /// @sa CoefficientStorage for a description of @a storage_.
+   CoefficientVector(VectorCoefficient &coeff, QuadratureSpaceBase &qs,
+                     CoefficientStorage storage_ = CoefficientStorage::FULL);
+
+   /// @brief Create a CoefficientVector from the given MatrixCoefficient and
+   /// QuadratureSpaceBase.
+   ///
+   /// @sa CoefficientStorage for a description of @a storage_.
+   CoefficientVector(MatrixCoefficient &coeff, QuadratureSpaceBase &qs,
+                     CoefficientStorage storage_ = CoefficientStorage::FULL);
+
+   /// @brief Evaluate the given Coefficient at the quadrature points defined by
+   /// @ref qs.
+   void Project(Coefficient &coeff);
+
+   /// @brief Evaluate the given VectorCoefficient at the quadrature points
+   /// defined by @ref qs.
+   ///
+   /// @sa CoefficientVector for a description of the @a compress argument.
+   void Project(VectorCoefficient &coeff);
+
+   /// @brief Evaluate the given MatrixCoefficient at the quadrature points
+   /// defined by @ref qs.
+   ///
+   /// @sa CoefficientVector for a description of the @a compress argument.
+   void Project(MatrixCoefficient &coeff, bool transpose=false);
+
+   /// @brief Project the transpose of @a coeff.
+   ///
+   /// @sa Project(MatrixCoefficient&, QuadratureSpace&, bool, bool)
+   void ProjectTranspose(MatrixCoefficient &coeff);
+
+   /// Make this vector a reference to the given QuadratureFunction.
+   void MakeRef(const QuadratureFunction &qf_);
+
+   /// Set this vector to the given constant.
+   void SetConstant(double constant);
+
+   /// Set this vector to the given constant vector.
+   void SetConstant(const Vector &constant);
+
+   /// Set this vector to the given constant matrix.
+   void SetConstant(const DenseMatrix &constant);
+
+   /// Set this vector to the given constant symmetric matrix.
+   void SetConstant(const DenseSymmetricMatrix &constant);
+
+   /// Return the number of values per quadrature point.
+   int GetVDim() const;
+
+   ~CoefficientVector();
+};
+
 /** @brief Compute the Lp norm of a function f.
    \f$ \| f \|_{Lp} = ( \int_\Omega | f |^p d\Omega)^{1/p} \f$ */
 double ComputeLpNorm(double p, Coefficient &coeff, Mesh &mesh,
@@ -442,7 +442,7 @@ void VisItDataCollection::RegisterQField(const std::string& name,
   {
      int locLOD = GlobGeometryRefiner.GetRefinementLevelFromElems(
                      mesh->GetElementBaseGeometry(e),
-                      qf->GetElementIntRule(e).GetNPoints());
+                      qf->GetIntRule(e).GetNPoints());

      LOD = std::max(LOD,locLOD);
   }
@@ -1078,7 +1078,7 @@ void ParaViewDataCollection::SaveGFieldVTU(std::ostream &os, int ref_,
         it->second->GetValues(i, RefG->RefPts, val, pmat);
         for (int j = 0; j < val.Size(); j++)
         {
-            WriteBinaryOrASCII(out, buf, val(j), "\n", pv_data_format);
+            WriteBinaryOrASCII(os, buf, val(j), "\n", pv_data_format);
         }
      }
   }
@@ -1094,7 +1094,7 @@ void ParaViewDataCollection::SaveGFieldVTU(std::ostream &os, int ref_,
         {
            for (int ii = 0; ii < vval.Height(); ii++)
            {
-               WriteBinaryOrASCII(out, buf, vval(ii,jj), " ", pv_data_format);
+               WriteBinaryOrASCII(os, buf, vval(ii,jj), " ", pv_data_format);
            }
            if (pv_data_format == VTKFormat::ASCII) { os << '\n'; }
         }
@@ -14,6 +14,7 @@

 #include "../config/config.hpp"
 #include "gridfunc.hpp"
+#include "qfunction.hpp"
 #ifdef MFEM_USE_MPI
 #include "pgridfunc.hpp"
 #endif
@@ -0,0 +1,315 @@
+// 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.
+
+#include "dgmassinv.hpp"
+#include "bilinearform.hpp"
+#include "dgmassinv_kernels.hpp"
+#include "../general/forall.hpp"
+
+namespace mfem
+{
+
+DGMassInverse::DGMassInverse(FiniteElementSpace &fes_orig, Coefficient *coeff,
+                             const IntegrationRule *ir,
+                             int btype)
+   : Solver(fes_orig.GetTrueVSize()),
+     fec(fes_orig.GetMaxElementOrder(),
+         fes_orig.GetMesh()->Dimension(),
+         btype,
+         fes_orig.GetFE(0)->GetMapType()),
+     fes(fes_orig.GetMesh(), &fec)
+{
+   MFEM_VERIFY(fes.IsDGSpace(), "Space must be DG.");
+   MFEM_VERIFY(!fes.IsVariableOrder(), "Variable orders not supported.");
+
+   const int btype_orig =
+      static_cast<const L2_FECollection*>(fes_orig.FEColl())->GetBasisType();
+
+   if (btype_orig == btype)
+   {
+      // No change of basis required
+      d2q = nullptr;
+   }
+   else
+   {
+      // original basis to solver basis
+      const auto mode = DofToQuad::TENSOR;
+      d2q = &fes_orig.GetFE(0)->GetDofToQuad(fes.GetFE(0)->GetNodes(), mode);
+
+      int n = d2q->ndof;
+      Array<double> B_inv = d2q->B; // deep copy
+      Array<int> ipiv(n);
+      // solver basis to original
+      LUFactors lu(B_inv.HostReadWrite(), ipiv.HostWrite());
+      lu.Factor(n);
+      B_.SetSize(n*n);
+      lu.GetInverseMatrix(n, B_.HostWrite());
+      Bt_.SetSize(n*n);
+      DenseMatrix B_matrix(B_.HostReadWrite(), n, n);
+      DenseMatrix Bt_matrix(Bt_.HostWrite(), n, n);
+      Bt_matrix.Transpose(B_matrix);
+   }
+
+   if (coeff) { m = new MassIntegrator(*coeff, ir); }
+   else { m = new MassIntegrator(ir); }
+
+   diag_inv.SetSize(height);
+   // Workspace vectors used for CG
+   r_.SetSize(height);
+   d_.SetSize(height);
+   z_.SetSize(height);
+   // Only need transformed RHS if basis is different
+   if (btype_orig != btype) { b2_.SetSize(height); }
+
+   M = new BilinearForm(&fes);
+   M->AddDomainIntegrator(m); // M assumes ownership of m
+   M->SetAssemblyLevel(AssemblyLevel::PARTIAL);
+
+   // Assemble the bilinear form and its diagonal (for preconditioning).
+   Update();
+}
+
+DGMassInverse::DGMassInverse(FiniteElementSpace &fes_, Coefficient &coeff,
+                             int btype)
+   : DGMassInverse(fes_, &coeff, nullptr, btype) { }
+
+DGMassInverse::DGMassInverse(FiniteElementSpace &fes_, Coefficient &coeff,
+                             const IntegrationRule &ir, int btype)
+   : DGMassInverse(fes_, &coeff, &ir, btype) { }
+
+DGMassInverse::DGMassInverse(FiniteElementSpace &fes_,
+                             const IntegrationRule &ir, int btype)
+   : DGMassInverse(fes_, nullptr, &ir, btype) { }
+
+DGMassInverse::DGMassInverse(FiniteElementSpace &fes_, int btype)
+   : DGMassInverse(fes_, nullptr, nullptr, btype) { }
+
+void DGMassInverse::SetOperator(const Operator &op)
+{
+   MFEM_ABORT("SetOperator not supported with DGMassInverse.")
+}
+
+void DGMassInverse::SetRelTol(const double rel_tol_) { rel_tol = rel_tol_; }
+
+void DGMassInverse::SetAbsTol(const double abs_tol_) { abs_tol = abs_tol_; }
+
+void DGMassInverse::SetMaxIter(const double max_iter_) { max_iter = max_iter_; }
+
+void DGMassInverse::Update()
+{
+   M->Assemble();
+   M->AssembleDiagonal(diag_inv);
+   internal::MakeReciprocal(diag_inv.Size(), diag_inv.ReadWrite());
+}
+
+DGMassInverse::~DGMassInverse()
+{
+   delete M;
+}
+
+template<int DIM, int D1D, int Q1D>
+void DGMassInverse::DGMassCGIteration(const Vector &b_, Vector &u_) const
+{
+   using namespace internal; // host/device kernel functions
+
+   const int NE = fes.GetNE();
+   const int d1d = m->dofs1D;
+   const int q1d = m->quad1D;
+
+   const int ND = static_cast<int>(pow(d1d, DIM));
+
+   const auto B = m->maps->B.Read();
+   const auto Bt = m->maps->Bt.Read();
+   const auto pa_data = m->pa_data.Read();
+   const auto dinv = diag_inv.Read();
+   auto r = r_.Write();
+   auto d = d_.Write();
+   auto z = z_.Write();
+   auto u = u_.ReadWrite();
+
+   const double RELTOL = rel_tol;
+   const double ABSTOL = abs_tol;
+   const double MAXIT = max_iter;
+   const bool IT_MODE = iterative_mode;
+   const bool CHANGE_BASIS = (d2q != nullptr);
+
+   // b is the right-hand side (if no change of basis, this just points to the
+   // incoming RHS vector, if we have to change basis, this points to the
+   // internal b2 vector where we put the transformed RHS)
+   const double *b;
+   // the following are non-null if we have to change basis
+   double *b2 = nullptr; // non-const access to b2
+   const double *b_orig = nullptr; // RHS vector in "original" basis
+   const double *d2q_B = nullptr; // matrix to transform initial guess
+   const double *q2d_B = nullptr; // matrix to transform solution
+   const double *q2d_Bt = nullptr; // matrix to transform RHS
+   if (CHANGE_BASIS)
+   {
+      d2q_B = d2q->B.Read();
+      q2d_B = B_.Read();
+      q2d_Bt = Bt_.Read();
+
+      b2 = b2_.Write();
+      b_orig = b_.Read();
+      b = b2;
+   }
+   else
+   {
+      b = b_.Read();
+   }
+
+   constexpr int NB = Q1D ? Q1D : 1; // block size
+
+   MFEM_FORALL_2D(e, NE, NB, NB, 1,
+   {
+      constexpr int NB = Q1D ? Q1D : 1; // redefine here for some compilers
+
+      // Perform change of basis if needed
+      if (CHANGE_BASIS)
+      {
+         // Transform RHS
+         DGMassBasis<DIM,D1D,MAX_D1D>(e, NE, q2d_Bt, b_orig, b2, d1d);
+         if (IT_MODE)
+         {
+            // Transform initial guess
+            DGMassBasis<DIM,D1D,MAX_D1D>(e, NE, d2q_B, u, u, d1d);
+         }
+      }
+
+      const int tid = MFEM_THREAD_ID(x) + NB*MFEM_THREAD_ID(y);
+
+      // Compute first residual
+      if (IT_MODE)
+      {
+         DGMassApply<DIM,D1D,Q1D>(e, NE, B, Bt, pa_data, u, r, d1d, q1d);
+         DGMassAxpy(e, NE, ND, 1.0, b, -1.0, r, r); // r = b - r
+      }
+      else
+      {
+         // if not in iterative mode, use zero initial guess
+         const int BX = MFEM_THREAD_SIZE(x);
+         const int BY = MFEM_THREAD_SIZE(y);
+         const int bxy = BX*BY;
+         const auto B = ConstDeviceMatrix(b, ND, NE);
+         auto U = DeviceMatrix(u, ND, NE);
+         auto R = DeviceMatrix(r, ND, NE);
+         for (int i = tid; i < ND; i += bxy)
+         {
+            U(i, e) = 0.0;
+            R(i, e) = B(i, e);
+         }
+         MFEM_SYNC_THREAD;
+      }
+
+      DGMassPreconditioner(e, NE, ND, dinv, r, z);
+      DGMassAxpy(e, NE, ND, 1.0, z, 0.0, z, d); // d = z
+
+      double nom = DGMassDot<NB>(e, NE, ND, d, r);
+      if (nom < 0.0) { return; /* Not positive definite */ }
+      double r0 = fmax(nom*RELTOL*RELTOL, ABSTOL*ABSTOL);
+      if (nom <= r0) { return; /* Converged */ }
+
+      DGMassApply<DIM,D1D,Q1D>(e, NE, B, Bt, pa_data, d, z, d1d, q1d);
+      double den = DGMassDot<NB>(e, NE, ND, z, d);
+      if (den <= 0.0)
+      {
+         DGMassDot<NB>(e, NE, ND, d, d);
+         // d2 > 0 => not positive definite
+         if (den == 0.0) { return; }
+      }
+
+      // start iteration
+      int i = 1;
+      while (true)
+      {
+         const double alpha = nom/den;
+         DGMassAxpy(e, NE, ND, 1.0, u, alpha, d, u); // u = u + alpha*d
+         DGMassAxpy(e, NE, ND, 1.0, r, -alpha, z, r); // r = r - alpha*A*d
+
+         DGMassPreconditioner(e, NE, ND, dinv, r, z);
+
+         double betanom = DGMassDot<NB>(e, NE, ND, r, z);
+         if (betanom < 0.0) { return; /* Not positive definite */ }
+         if (betanom <= r0) { break; /* Converged */ }
+
+         if (++i > MAXIT) { break; }
+
+         const double beta = betanom/nom;
+         DGMassAxpy(e, NE, ND, 1.0, z, beta, d, d); // d = z + beta*d
+         DGMassApply<DIM,D1D,Q1D>(e, NE, B, Bt, pa_data, d, z, d1d, q1d); // z = A d
+         den = DGMassDot<NB>(e, NE, ND, d, z);
+         if (den <= 0.0)
+         {
+            DGMassDot<NB>(e, NE, ND, d, d);
+            // d2 > 0 => not positive definite
+            if (den == 0.0) { break; }
+         }
+         nom = betanom;
+      }
+
+      if (CHANGE_BASIS)
+      {
+         DGMassBasis<DIM,D1D,MAX_D1D>(e, NE, q2d_B, u, u, d1d);
+      }
+   });
+}
+
+void DGMassInverse::Mult(const Vector &Mu, Vector &u) const
+{
+   // Dispatch to templated version based on dim, d1d, and q1d.
+   const int dim = fes.GetMesh()->Dimension();
+   const int d1d = m->dofs1D;
+   const int q1d = m->quad1D;
+
+   const int id = (d1d << 4) | q1d;
+
+   if (dim == 2)
+   {
+      switch (id)
+      {
+         case 0x11: return DGMassCGIteration<2,1,1>(Mu, u);
+         case 0x22: return DGMassCGIteration<2,2,2>(Mu, u);
+         case 0x33: return DGMassCGIteration<2,3,3>(Mu, u);
+         case 0x35: return DGMassCGIteration<2,3,5>(Mu, u);
+         case 0x44: return DGMassCGIteration<2,4,4>(Mu, u);
+         case 0x46: return DGMassCGIteration<2,4,6>(Mu, u);
+         case 0x55: return DGMassCGIteration<2,5,5>(Mu, u);
+         case 0x57: return DGMassCGIteration<2,5,7>(Mu, u);
+         case 0x66: return DGMassCGIteration<2,6,6>(Mu, u);
+         case 0x68: return DGMassCGIteration<2,6,8>(Mu, u);
+         default: return DGMassCGIteration<2>(Mu, u); // Fallback
+      }
+   }
+   else if (dim == 3)
+   {
+      switch (id)
+      {
+         case 0x22: return DGMassCGIteration<3,2,2>(Mu, u);
+         case 0x23: return DGMassCGIteration<3,2,3>(Mu, u);
+         case 0x33: return DGMassCGIteration<3,3,3>(Mu, u);
+         case 0x34: return DGMassCGIteration<3,3,4>(Mu, u);
+         case 0x35: return DGMassCGIteration<3,3,5>(Mu, u);
+         case 0x44: return DGMassCGIteration<3,4,4>(Mu, u);
+         case 0x45: return DGMassCGIteration<3,4,5>(Mu, u);
+         case 0x46: return DGMassCGIteration<3,4,6>(Mu, u);
+         case 0x48: return DGMassCGIteration<3,4,8>(Mu, u);
+         case 0x55: return DGMassCGIteration<3,5,5>(Mu, u);
+         case 0x56: return DGMassCGIteration<3,5,6>(Mu, u);
+         case 0x57: return DGMassCGIteration<3,5,7>(Mu, u);
+         case 0x58: return DGMassCGIteration<3,5,8>(Mu, u);
+         case 0x66: return DGMassCGIteration<3,6,6>(Mu, u);
+         case 0x67: return DGMassCGIteration<3,6,7>(Mu, u);
+         default: return DGMassCGIteration<3>(Mu, u); // Fallback
+      }
+   }
+}
+
+} // namespace mfem
@@ -0,0 +1,112 @@
+// 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.
+
+#ifndef MFEM_DGMASSINV_HPP
+#define MFEM_DGMASSINV_HPP
+
+#include "../linalg/operator.hpp"
+#include "fespace.hpp"
+
+namespace mfem
+{
+
+/// @brief Solver for the discontinuous Galerkin mass matrix.
+///
+/// This class performs a @a local (diagonally preconditioned) conjugate
+/// gradient iteration for each element. Optionally, a change of basis is
+/// performed to iterate on a better-conditioned system. This class fully
+/// supports execution on device (GPU).
+class DGMassInverse : public Solver
+{
+protected:
+   DG_FECollection fec; ///< FE collection in requested basis.
+   FiniteElementSpace fes; ///< FE space in requested basis.
+   const DofToQuad *d2q; ///< Change of basis. Not owned.
+   Array<double> B_; ///< Inverse of change of basis.
+   Array<double> Bt_; ///< Inverse of change of basis, transposed.
+   class BilinearForm *M; ///< Mass bilinear form, owned.
+   class MassIntegrator *m; ///< Mass integrator, owned by the form @ref M.
+   Vector diag_inv; ///< Jacobi preconditioner.
+   double rel_tol = 1e-12; ///< Relative CG tolerance.
+   double abs_tol = 1e-12; ///< Absolute CG tolerance.
+   int max_iter = 100; ///< Maximum number of CG iterations;
+
+   /// @name Intermediate vectors needed for CG three-term recurrence.
+   ///@{
+   mutable Vector r_, d_, z_, b2_;
+   ///@}
+
+   /// @brief Protected constructor, used internally.
+   ///
+   /// Custom coefficient and integration rule are used if @a coeff and @a ir
+   /// are non-NULL.
+   DGMassInverse(FiniteElementSpace &fes_, Coefficient *coeff,
+                 const IntegrationRule *ir, int btype);
+public:
+   /// @brief Construct the DG inverse mass operator for @a fes_.
+   ///
+   /// The basis type @a btype determines which basis should be used internally
+   /// in the solver. This <b>does not</b> have to be the same basis as @a fes_.
+   /// The best choice is typically BasisType::GaussLegendre because it is
+   /// well-preconditioned by its diagonal.
+   ///
+   /// The solution and right-hand side used for the solver are not affected by
+   /// this basis (they correspond to the basis of @a fes_). @a btype is only
+   /// used internally, and only has an effect on the convergence rate.
+   DGMassInverse(FiniteElementSpace &fes_, int btype=BasisType::GaussLegendre);
+   /// @brief Construct the DG inverse mass operator for @a fes_ with
+   /// Coefficient @a coeff.
+   ///
+   /// @sa DGMassInverse(FiniteElementSpace&, int) for information about @a
+   /// btype.
+   DGMassInverse(FiniteElementSpace &fes_, Coefficient &coeff,
+                 int btype=BasisType::GaussLegendre);
+   /// @brief Construct the DG inverse mass operator for @a fes_ with
+   /// Coefficient @a coeff and IntegrationRule @a ir.
+   ///
+   /// @sa DGMassInverse(FiniteElementSpace&, int) for information about @a
+   /// btype.
+   DGMassInverse(FiniteElementSpace &fes_, Coefficient &coeff,
+                 const IntegrationRule &ir, int btype=BasisType::GaussLegendre);
+   /// @brief Construct the DG inverse mass operator for @a fes_ with
+   /// IntegrationRule @a ir.
+   ///
+   /// @sa DGMassInverse(FiniteElementSpace&, int) for information about @a
+   /// btype.
+   DGMassInverse(FiniteElementSpace &fes_, const IntegrationRule &ir,
+                 int btype=BasisType::GaussLegendre);
+   /// @brief Solve the system M b = u.
+   ///
+   /// If @ref iterative_mode is @a true, @a u is used as an initial guess.
+   void Mult(const Vector &b, Vector &u) const;
+   /// Not implemented. Aborts.
+   void SetOperator(const Operator &op);
+   /// Set the relative tolerance.
+   void SetRelTol(const double rel_tol_);
+   /// Set the absolute tolerance.
+   void SetAbsTol(const double abs_tol_);
+   /// Set the maximum number of iterations.
+   void SetMaxIter(const double max_iter_);
+   /// Recompute operator and preconditioner (when coefficient or mesh changes).
+   void Update();
+
+   ~DGMassInverse();
+
+   /// @brief Solve the system M b = u. <b>Not part of the public interface.</b>
+   /// @note This member function must be public because it contains an
+   /// MFEM_FORALL kernel (nvcc limitation)
+   template<int DIM, int D1D = 0, int Q1D = 0>
+   void DGMassCGIteration(const Vector &b_, Vector &u_) const;
+};
+
+} // namespace mfem
+
+#endif
@@ -0,0 +1,295 @@
+// 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.
+
+#ifndef MFEM_DGMASSINV_KERNELS_HPP
+#define MFEM_DGMASSINV_KERNELS_HPP
+
+#include "bilininteg_mass_pa.hpp"
+#include "../linalg/kernels.hpp"
+#include "kernels.hpp"
+
+namespace mfem
+{
+
+namespace internal
+{
+
+void MakeReciprocal(int n, double *x)
+{
+   MFEM_FORALL(i, n, x[i] = 1.0/x[i]; );
+}
+
+template <int DIM, int D1D, int Q1D>
+MFEM_HOST_DEVICE inline
+void DGMassApply(const int e,
+                 const int NE,
+                 const double *B,
+                 const double *Bt,
+                 const double *pa_data,
+                 const double *x,
+                 double *y,
+                 const int d1d = 0,
+                 const int q1d = 0)
+{
+   constexpr bool use_smem = (D1D > 0 && Q1D > 0);
+   constexpr bool ACCUM = false;
+   constexpr int NBZ = 1;
+   if (use_smem)
+   {
+      // cannot specialize functions below with D1D or Q1D equal to zero
+      // (this branch only runs with D1D and Q1D are both positive)
+      constexpr int TD1D = D1D ? D1D : 1;
+      constexpr int TQ1D = Q1D ? Q1D : 1;
+      if (DIM == 2)
+      {
+         SmemPAMassApply2D_Element<TD1D,TQ1D,NBZ,ACCUM>(e, NE, B, pa_data, x, y);
+      }
+      else if (DIM == 3)
+      {
+         SmemPAMassApply3D_Element<TD1D,TQ1D,ACCUM>(e, NE, B, pa_data, x, y);
+      }
+      else
+      {
+         MFEM_ABORT_KERNEL("Unsupported dimension.");
+      }
+   }
+   else
+   {
+      if (DIM == 2)
+      {
+         PAMassApply2D_Element<ACCUM>(e, NE, B, Bt, pa_data, x, y, d1d, q1d);
+      }
+      else if (DIM == 3)
+      {
+         PAMassApply3D_Element<ACCUM>(e, NE, B, Bt, pa_data, x, y, d1d, q1d);
+      }
+      else
+      {
+         MFEM_ABORT_KERNEL("Unsupported dimension.");
+      }
+   }
+}
+
+MFEM_HOST_DEVICE inline
+void DGMassPreconditioner(const int e,
+                          const int NE,
+                          const int ND,
+                          const double *dinv,
+                          const double *x,
+                          double *y)
+{
+   const auto X = ConstDeviceMatrix(x, ND, NE);
+   const auto D = ConstDeviceMatrix(dinv, ND, NE);
+   auto Y = DeviceMatrix(y, ND, NE);
+
+   const int tid = MFEM_THREAD_ID(x) + MFEM_THREAD_SIZE(x)*MFEM_THREAD_ID(y);
+   const int bxy = MFEM_THREAD_SIZE(x)*MFEM_THREAD_SIZE(y);
+
+   for (int i = tid; i < ND; i += bxy)
+   {
+      Y(i, e) = D(i, e)*X(i, e);
+   }
+   MFEM_SYNC_THREAD;
+}
+
+MFEM_HOST_DEVICE inline
+void DGMassAxpy(const int e,
+                const int NE,
+                const int ND,
+                const double a,
+                const double *x,
+                const double b,
+                const double *y,
+                double *z)
+{
+   const auto X = ConstDeviceMatrix(x, ND, NE);
+   const auto Y = ConstDeviceMatrix(y, ND, NE);
+   auto Z = DeviceMatrix(z, ND, NE);
+
+   const int tid = MFEM_THREAD_ID(x) + MFEM_THREAD_SIZE(x)*MFEM_THREAD_ID(y);
+   const int bxy = MFEM_THREAD_SIZE(x)*MFEM_THREAD_SIZE(y);
+
+   for (int i = tid; i < ND; i += bxy)
+   {
+      Z(i, e) = a*X(i, e) + b*Y(i, e);
+   }
+   MFEM_SYNC_THREAD;
+}
+
+template <int NB>
+MFEM_HOST_DEVICE inline
+double DGMassDot(const int e,
+                 const int NE,
+                 const int ND,
+                 const double *x,
+                 const double *y)
+{
+   const auto X = ConstDeviceMatrix(x, ND, NE);
+   const auto Y = ConstDeviceMatrix(y, ND, NE);
+
+   const int tid = MFEM_THREAD_ID(x) + MFEM_THREAD_SIZE(x)*MFEM_THREAD_ID(y);
+   const int bxy = MFEM_THREAD_SIZE(x)*MFEM_THREAD_SIZE(y);
+
+   MFEM_SHARED double s_dot[NB*NB];
+   s_dot[tid] = 0.0;
+
+   for (int i = tid; i < ND; i += bxy) { s_dot[tid] += X(i,e)*Y(i,e); }
+   MFEM_SYNC_THREAD;
+
+   if (bxy > 512 && tid + 512 < bxy) { s_dot[tid] += s_dot[tid + 512]; }
+   MFEM_SYNC_THREAD;
+
+   if (bxy > 256 && tid < 256 && tid + 256 < bxy) { s_dot[tid] += s_dot[tid + 256]; }
+   MFEM_SYNC_THREAD;
+
+   if (bxy > 128 && tid < 128 && tid + 128 < bxy) { s_dot[tid] += s_dot[tid + 128]; }
+   MFEM_SYNC_THREAD;
+
+   if (bxy > 64 && tid < 64 && tid + 64 < bxy) { s_dot[tid] += s_dot[tid + 64]; }
+   MFEM_SYNC_THREAD;
+
+   if (bxy > 32 && tid < 32 && tid + 32 < bxy) { s_dot[tid] += s_dot[tid + 32]; }
+   MFEM_SYNC_THREAD;
+
+   if (bxy > 16 && tid < 16 && tid + 16 < bxy) { s_dot[tid] += s_dot[tid + 16]; }
+   MFEM_SYNC_THREAD;
+
+   if (bxy > 8 && tid < 8 && tid + 8 < bxy) { s_dot[tid] += s_dot[tid + 8]; }
+   MFEM_SYNC_THREAD;
+
+   if (bxy > 4 && tid < 4 && tid + 4 < bxy) { s_dot[tid] += s_dot[tid + 4]; }
+   MFEM_SYNC_THREAD;
+
+   if (bxy > 2 && tid < 2 && tid + 2 < bxy) { s_dot[tid] += s_dot[tid + 2]; }
+   MFEM_SYNC_THREAD;
+
+   if (bxy > 1 && tid < 1 && tid + 1 < bxy) { s_dot[tid] += s_dot[tid + 1]; }
+   MFEM_SYNC_THREAD;
+
+   return s_dot[0];
+}
+
+template<int T_D1D = 0, int MAX_D1D = 0>
+MFEM_HOST_DEVICE inline
+void DGMassBasis2D(const int e,
+                   const int NE,
+                   const double *b_,
+                   const double *x_,
+                   double *y_,
+                   const int d1d = 0)
+{
+   constexpr int MD1 = T_D1D ? T_D1D : MAX_D1D;
+   const int D1D = T_D1D ? T_D1D : d1d;
+
+   const auto b = Reshape(b_, D1D, D1D);
+   const auto x = Reshape(x_, D1D, D1D, NE);
+   auto y = Reshape(y_, D1D, D1D, NE);
+
+   MFEM_SHARED double sB[MD1*MD1];
+   MFEM_SHARED double sm0[MD1*MD1];
+   MFEM_SHARED double sm1[MD1*MD1];
+
+   kernels::internal::LoadB<MD1,MD1>(D1D,D1D,b,sB);
+
+   ConstDeviceMatrix B(sB, D1D,D1D);
+   DeviceMatrix DD(sm0, MD1, MD1);
+   DeviceMatrix DQ(sm1, MD1, MD1);
+   DeviceMatrix QQ(sm0, MD1, MD1);
+
+   kernels::internal::LoadX(e,D1D,x,DD);
+   kernels::internal::EvalX(D1D,D1D,B,DD,DQ);
+   kernels::internal::EvalY(D1D,D1D,B,DQ,QQ);
+   MFEM_SYNC_THREAD; // sync here to allow in-place evaluations
+   MFEM_FOREACH_THREAD(qy,y,D1D)
+   {
+      MFEM_FOREACH_THREAD(qx,x,D1D)
+      {
+         y(qx,qy,e) = QQ(qx,qy);
+      }
+   }
+   MFEM_SYNC_THREAD;
+}
+
+template<int T_D1D = 0, int MAX_D1D = 0>
+MFEM_HOST_DEVICE inline
+void DGMassBasis3D(const int e,
+                   const int NE,
+                   const double *b_,
+                   const double *x_,
+                   double *y_,
+                   const int d1d = 0)
+{
+   const int D1D = T_D1D ? T_D1D : d1d;
+
+   const auto b = Reshape(b_, D1D, D1D);
+   const auto x = Reshape(x_, D1D, D1D, D1D, NE);
+   auto y = Reshape(y_, D1D, D1D, D1D, NE);
+
+   constexpr int MD1 = T_D1D ? T_D1D : MAX_D1D;
+
+   MFEM_SHARED double sB[MD1*MD1];
+   MFEM_SHARED double sm0[MD1*MD1*MD1];
+   MFEM_SHARED double sm1[MD1*MD1*MD1];
+
+   kernels::internal::LoadB<MD1,MD1>(D1D,D1D,b,sB);
+
+   ConstDeviceMatrix B(sB, D1D,D1D);
+   DeviceCube DDD(sm0, MD1,MD1,MD1);
+   DeviceCube DDQ(sm1, MD1,MD1,MD1);
+   DeviceCube DQQ(sm0, MD1,MD1,MD1);
+   DeviceCube QQQ(sm1, MD1,MD1,MD1);
+
+   kernels::internal::LoadX(e,D1D,x,DDD);
+   kernels::internal::EvalX(D1D,D1D,B,DDD,DDQ);
+   kernels::internal::EvalY(D1D,D1D,B,DDQ,DQQ);
+   kernels::internal::EvalZ(D1D,D1D,B,DQQ,QQQ);
+   MFEM_SYNC_THREAD; // sync here to allow in-place evaluation
+   MFEM_FOREACH_THREAD(qz,z,D1D)
+   {
+      MFEM_FOREACH_THREAD(qy,y,D1D)
+      {
+         for (int qx = 0; qx < D1D; ++qx)
+         {
+            y(qx,qy,qz,e) = QQQ(qz,qy,qx);
+         }
+      }
+   }
+   MFEM_SYNC_THREAD;
+}
+
+template<int DIM, int T_D1D = 0, int MAX_D1D = 0>
+MFEM_HOST_DEVICE inline
+void DGMassBasis(const int e,
+                 const int NE,
+                 const double *b_,
+                 const double *x_,
+                 double *y_,
+                 const int d1d = 0)
+{
+   if (DIM == 2)
+   {
+      DGMassBasis2D<T_D1D, MAX_D1D>(e, NE, b_, x_, y_, d1d);
+   }
+   else if (DIM == 3)
+   {
+      DGMassBasis3D<T_D1D, MAX_D1D>(e, NE, b_, x_, y_, d1d);
+   }
+   else
+   {
+      MFEM_ABORT_KERNEL("Dimension not supported.");
+   }
+}
+
+} // namespace internal
+
+} // namespace mfem
+
+#endif
@@ -159,7 +159,7 @@ void KellyErrorEstimator::ComputeEstimates()
   // the FaceInfo class [1]. Also, the FaceElementTransformations
   // documentation [2] may be helpful to grasp what is going on. Note
   // that the FaceElementTransformations also works in the non-
-   // conforming case to transfer the gauss points from the slave to
+   // conforming case to transfer the Gauss points from the slave to
   // the master element.
   // [1]
   // https://github.com/mfem/mfem/blob/02d0bfe9c18ce049c3c93a6a4208080fcfc96991/mesh/mesh.hpp#L94
@@ -417,7 +417,7 @@ void KellyErrorEstimator::ComputeEstimates()
         Vector val(flux_space->GetVDim());
         flux->GetVectorValue(FT->Elem2No, ip, val);

-         // Evaluate gauss point
+         // Evaluate Gauss point
         Vector normal(mesh->SpaceDimension());
         FT->Face->SetIntPoint(&fip);
         if (mesh->Dimension() == mesh->SpaceDimension())
@@ -314,7 +314,7 @@ 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, dshape_cy, dshape_cz;
+   Vector dshape_cx(p + 1), dshape_cy(p + 1), dshape_cz(p + 1);
 #endif

   if (obasis1d.IsIntegratedType())
@@ -656,7 +656,7 @@ 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, dshape_cy;
+   Vector dshape_cx(p + 1), dshape_cy(p + 1);
 #endif

   if (obasis1d.IsIntegratedType())
@@ -145,7 +145,7 @@ 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, dshape_cy;
+   Vector dshape_cx(pp1 + 1), dshape_cy(pp1 + 1);
 #endif

   if (obasis1d.IsIntegratedType())
@@ -473,7 +473,7 @@ 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, dshape_cy, dshape_cz;
+   Vector dshape_cx(pp1 + 1), dshape_cy(pp1 + 1), dshape_cz(pp1 + 1);
 #endif

   if (obasis1d.IsIntegratedType())
@@ -53,8 +53,12 @@ public:

   virtual int DofForGeometry(Geometry::Type GeomType) const = 0;

-   /** @brief Returns an array, say p, that maps a local permuted index i to
-       a local base index: base_i = p[i]. */
+   /** @brief Returns an array, say p, that maps a local permuted index i to a
+       local base index: base_i = p[i].
+
+       @note Only provides information about interior dofs. See
+       FiniteElementCollection::SubDofOrder if interior \a and boundary dof
+       order is needed. */
   virtual const int *DofOrderForOrientation(Geometry::Type GeomType,
                                             int Or) const = 0;

@@ -95,10 +99,10 @@ public:
   | RT_ValTrace_[DIM]_[ORDER] | H^{1/2} | * | 1 / 0 | VALUE | H^{1/2}-conforming trace elements for H(div) defined on the interface between mesh elements (faces) |
   | RT_Trace@[BTYPE]_[DIM]_[ORDER] | H^{1/2} | * | 1 / 0 | INTEGRAL | H^{1/2}-conforming trace elements for H(div) defined on the interface between mesh elements (faces) |
   | RT_ValTrace@[BTYPE]_[DIM]_[ORDER] |  H^{1/2} | * | 1 / 0 | VALUE | H^{1/2}-conforming trace elements for H(div) defined on the interface between mesh elements (faces) |
-   | L2_[DIM]_[ORDER] | L2 | * | 0 | VALUE | Discontinous L2 elements |
-   | L2_T[BTYPE]_[DIM]_[ORDER] | L2 | * | 0 | VALUE | Discontinous L2 elements |
-   | L2Int_[DIM]_[ORDER] | L2 | * | 0 | INTEGRAL | Discontinous L2 elements |
-   | L2Int_T[BTYPE]_[DIM]_[ORDER] | L2 | * | 0 | INTEGRAL | Discontinous L2 elements |
+   | L2_[DIM]_[ORDER] | L2 | * | 0 | VALUE | Discontinuous L2 elements |
+   | L2_T[BTYPE]_[DIM]_[ORDER] | L2 | * | 0 | VALUE | Discontinuous L2 elements |
+   | L2Int_[DIM]_[ORDER] | L2 | * | 0 | INTEGRAL | Discontinuous L2 elements |
+   | L2Int_T[BTYPE]_[DIM]_[ORDER] | L2 | * | 0 | INTEGRAL | Discontinuous L2 elements |
   | DG_Iface_[DIM]_[ORDER] | - | * | 0 | VALUE | Discontinuous elements on the interface between mesh elements (faces) |
   | DG_Iface@[BTYPE]_[DIM]_[ORDER] | - | * | 0 | VALUE | Discontinuous elements on the interface between mesh elements (faces) |
   | DG_IntIface_[DIM]_[ORDER] | - | * | 0 | INTEGRAL | Discontinuous elements on the interface between mesh elements (faces) |
@@ -45,6 +45,7 @@
 #include "multigrid.hpp"
 #include "ceed/solvers/algebraic.hpp"
 #include "lor/lor.hpp"
+#include "dgmassinv.hpp"

 #ifdef MFEM_USE_MPI
 #include "pfespace.hpp"
@@ -1199,8 +1199,6 @@ void FiniteElementSpace::BuildConformingInterpolation() const
      MakeVDimMatrix(*cR);
      if (cR_hp) { MakeVDimMatrix(*cR_hp); }
   }
-
-   cP->EnsureMultTranspose();
 }

 void FiniteElementSpace::MakeVDimMatrix(SparseMatrix &mat) const
@@ -1258,7 +1256,7 @@ int FiniteElementSpace::GetNConformingDofs() const
   return P ? (P->Width() / vdim) : ndofs;
 }

-const Operator *FiniteElementSpace::GetElementRestriction(
+const ElementRestrictionOperator *FiniteElementSpace::GetElementRestriction(
   ElementDofOrdering e_ordering) const
 {
   // Check if we have a discontinuous space using the FE collection:
@@ -1273,7 +1271,7 @@ const Operator *FiniteElementSpace::GetElementRestriction(
         // The output E-vector layout is: ND x VDIM x NE.
         L2E_nat.Reset(new L2ElementRestriction(*this));
      }
-      return L2E_nat.Ptr();
+      return L2E_nat.Is<ElementRestrictionOperator>();
   }
   if (e_ordering == ElementDofOrdering::LEXICOGRAPHIC)
   {
@@ -1281,14 +1279,14 @@ const Operator *FiniteElementSpace::GetElementRestriction(
      {
         L2E_lex.Reset(new ElementRestriction(*this, e_ordering));
      }
-      return L2E_lex.Ptr();
+      return L2E_lex.Is<ElementRestrictionOperator>();
   }
   // e_ordering == ElementDofOrdering::NATIVE
   if (L2E_nat.Ptr() == NULL)
   {
      L2E_nat.Reset(new ElementRestriction(*this, e_ordering));
   }
-   return L2E_nat.Ptr();
+   return L2E_nat.Is<ElementRestrictionOperator>();
 }

 const FaceRestriction *FiniteElementSpace::GetFaceRestriction(
@@ -3613,58 +3611,4 @@ FiniteElementCollection *FiniteElementSpace::Load(Mesh *m, std::istream &input)
   return r_fec;
 }

-
-void QuadratureSpace::Construct()
-{
-   // protected method
-   int offset = 0;
-   const int num_elem = mesh->GetNE();
-   element_offsets = new int[num_elem + 1];
-   for (int g = 0; g < Geometry::NumGeom; g++)
-   {
-      int_rule[g] = NULL;
-   }
-   for (int i = 0; i < num_elem; i++)
-   {
-      element_offsets[i] = offset;
-      int geom = mesh->GetElementBaseGeometry(i);
-      if (int_rule[geom] == NULL)
-      {
-         int_rule[geom] = &IntRules.Get(geom, order);
-      }
-      offset += int_rule[geom]->GetNPoints();
-   }
-   element_offsets[num_elem] = size = offset;
-}
-
-QuadratureSpace::QuadratureSpace(Mesh *mesh_, std::istream &in)
-   : mesh(mesh_)
-{
-   const char *msg = "invalid input stream";
-   string ident;
-
-   in >> ident; MFEM_VERIFY(ident == "QuadratureSpace", msg);
-   in >> ident; MFEM_VERIFY(ident == "Type:", msg);
-   in >> ident;
-   if (ident == "default_quadrature")
-   {
-      in >> ident; MFEM_VERIFY(ident == "Order:", msg);
-      in >> order;
-   }
-   else
-   {
-      MFEM_ABORT("unknown QuadratureSpace type: " << ident);
-      return;
-   }
-
-   Construct();
-}
-
-void QuadratureSpace::Save(std::ostream &os) const
-{
-   os << "QuadratureSpace\n"
-      << "Type: default_quadrature\n"
-      << "Order: " << order << '\n';
-}
-
 } // namespace mfem
@@ -47,6 +47,14 @@ public:
   static void DofsToVDofs(int ndofs, int vdim, Array<int> &dofs);
 };

+/// @brief Type describing possible layouts for Q-vectors.
+/// @sa QuadratureInterpolator and FaceQuadratureInterpolator.
+enum class QVectorLayout
+{
+   byNODES,  ///< NQPT x VDIM x NE (values) / NQPT x VDIM x DIM x NE (grads)
+   byVDIM    ///< VDIM x NQPT x NE (values) / VDIM x DIM x NQPT x NE (grads)
+};
+
 template <> inline int
 Ordering::Map<Ordering::byNODES>(int ndofs, int vdim, int dof, int vd)
 {
@@ -396,7 +404,7 @@ public:
   FiniteElementSpace();

   /** @brief Copy constructor: deep copy all data from @a orig except the Mesh,
-       the FiniteElementCollection, ans some derived data. */
+       the FiniteElementCollection, and some derived data. */
   /** If the @a mesh or @a fec pointers are NULL (default), then the new
       FiniteElementSpace will reuse the respective pointers from @a orig. If
       any of these pointers is not NULL, the given pointer will be used instead
@@ -508,7 +516,8 @@ public:
       L2ElementRestriction class.

       The returned Operator is owned by the FiniteElementSpace. */
-   const Operator *GetElementRestriction(ElementDofOrdering e_ordering) const;
+   const ElementRestrictionOperator *GetElementRestriction(
+      ElementDofOrdering e_ordering) const;

   /// Return an Operator that converts L-vectors to E-vectors on each face.
   virtual const FaceRestriction *GetFaceRestriction(
@@ -522,7 +531,12 @@ public:
       Operator returned by GetElementRestriction().

       All elements will use the same IntegrationRule, @a ir as the target
-       quadrature points. */
+       quadrature points.
+
+       @note The returned pointer is shared. A good practice, before using it,
+       is to set all its properties to their expected values, as other parts of
+       the code may also change them. That is, it's good to call
+       SetOutputLayout() and DisableTensorProducts() before interpolating. */
   const QuadratureInterpolator *GetQuadratureInterpolator(
      const IntegrationRule &ir) const;

@@ -533,12 +547,22 @@ public:
       Operator returned by GetElementRestriction().

       The target quadrature points in the elements are described by the given
-       QuadratureSpace, @a qs. */
+       QuadratureSpace, @a qs.
+
+       @note The returned pointer is shared. A good practice, before using it,
+       is to set all its properties to their expected values, as other parts of
+       the code may also change them. That is, it's good to call
+       SetOutputLayout() and DisableTensorProducts() before interpolating. */
   const QuadratureInterpolator *GetQuadratureInterpolator(
      const QuadratureSpace &qs) const;

   /** @brief Return a FaceQuadratureInterpolator that interpolates E-vectors to
-       quadrature point values and/or derivatives (Q-vectors). */
+       quadrature point values and/or derivatives (Q-vectors).
+
+       @note The returned pointer is shared. A good practice, before using it,
+       is to set all its properties to their expected values, as other parts of
+       the code may also change them. That is, it's good to call
+       SetOutputLayout() and DisableTensorProducts() before interpolating. */
   const FaceQuadratureInterpolator *GetFaceQuadratureInterpolator(
      const IntegrationRule &ir, FaceType type) const;

@@ -929,57 +953,6 @@ public:
   virtual ~FiniteElementSpace();
 };

-
-/// Class representing the storage layout of a QuadratureFunction.
-/** Multiple QuadratureFunction%s can share the same QuadratureSpace. */
-class QuadratureSpace
-{
-protected:
-   friend class QuadratureFunction; // Uses the element_offsets.
-
-   Mesh *mesh;
-   int order;
-   int size;
-
-   const IntegrationRule *int_rule[Geometry::NumGeom];
-   int *element_offsets; // scalar offsets; size = number of elements + 1
-
-   // protected functions
-
-   // Assuming mesh and order are set, construct the members: int_rule,
-   // element_offsets, and size.
-   void Construct();
-
-public:
-   /// Create a QuadratureSpace based on the global rules from #IntRules.
-   QuadratureSpace(Mesh *mesh_, int order_)
-      : mesh(mesh_), order(order_) { Construct(); }
-
-   /// Read a QuadratureSpace from the stream @a in.
-   QuadratureSpace(Mesh *mesh_, std::istream &in);
-
-   virtual ~QuadratureSpace() { delete [] element_offsets; }
-
-   /// Return the total number of quadrature points.
-   int GetSize() const { return size; }
-
-   /// Return the order of the quadrature rule(s) used by all elements.
-   int GetOrder() const { return order; }
-
-   /// Returns the mesh
-   inline Mesh *GetMesh() const { return mesh; }
-
-   /// Returns number of elements in the mesh.
-   inline int GetNE() const { return mesh->GetNE(); }
-
-   /// Get the IntegrationRule associated with mesh element @a idx.
-   const IntegrationRule &GetElementIntRule(int idx) const
-   { return *int_rule[mesh->GetElementBaseGeometry(idx)]; }
-
-   /// Write the QuadratureSpace to the stream @a out.
-   void Save(std::ostream &out) const;
-};
-
 /// @brief Return true if the mesh contains only one topology and the elements are tensor elements.
 inline bool UsesTensorBasis(const FiniteElementSpace& fes)
 {
@@ -12,6 +12,7 @@
 // Implementation of GridFunction

 #include "gridfunc.hpp"
+#include "quadinterpolator.hpp"
 #include "../mesh/nurbs.hpp"
 #include "../general/text.hpp"

@@ -188,6 +189,8 @@ void GridFunction::Update()
   {
      SetSize(fes->GetVSize());
   }
+
+   if (t_vec.Size() > 0) { SetTrueVector(); }
 }

 void GridFunction::SetSpace(FiniteElementSpace *f)
@@ -2761,7 +2764,8 @@ void GridFunction::ProjectBdrCoefficientTangent(
 }

 double GridFunction::ComputeL2Error(
-   Coefficient *exsol[], const IntegrationRule *irs[]) const
+   Coefficient *exsol[], const IntegrationRule *irs[],
+   const Array<int> *elems) const
 {
   double error = 0.0, a;
   const FiniteElement *fe;
@@ -2772,6 +2776,7 @@ double GridFunction::ComputeL2Error(

   for (i = 0; i < fes->GetNE(); i++)
   {
+      if (elems != NULL && (*elems)[i] == 0) { continue; }
      fe = fes->GetFE(i);
      fdof = fe->GetDof();
      transf = fes->GetElementTransformation(i);
@@ -2815,7 +2820,7 @@ double GridFunction::ComputeL2Error(

 double GridFunction::ComputeL2Error(
   VectorCoefficient &exsol, const IntegrationRule *irs[],
-   Array<int> *elems) const
+   const Array<int> *elems) const
 {
   double error = 0.0;
   const FiniteElement *fe;
@@ -3234,7 +3239,7 @@ double GridFunction::ComputeMaxError(

 double GridFunction::ComputeW11Error(
   Coefficient *exsol, VectorCoefficient *exgrad, int norm_type,
-   Array<int> *elems, const IntegrationRule *irs[]) const
+   const Array<int> *elems, const IntegrationRule *irs[]) const
 {
   // assuming vdim is 1
   int i, fdof, dim, intorder, j, k;
@@ -3340,7 +3345,8 @@ double GridFunction::ComputeW11Error(

 double GridFunction::ComputeLpError(const double p, Coefficient &exsol,
                                    Coefficient *weight,
-                                    const IntegrationRule *irs[]) const
+                                    const IntegrationRule *irs[],
+                                    const Array<int> *elems) const
 {
   double error = 0.0;
   const FiniteElement *fe;
@@ -3349,6 +3355,7 @@ double GridFunction::ComputeLpError(const double p, Coefficient &exsol,

   for (int i = 0; i < fes->GetNE(); i++)
   {
+      if (elems != NULL && (*elems)[i] == 0) { continue; }
      fe = fes->GetFE(i);
      const IntegrationRule *ir;
      if (irs)
@@ -3968,178 +3975,6 @@ void GridFunction::LegacyNCReorder()
   Vector::Swap(tmp);
 }

-
-QuadratureFunction::QuadratureFunction(Mesh *mesh, std::istream &in)
-{
-   const char *msg = "invalid input stream";
-   string ident;
-
-   qspace = new QuadratureSpace(mesh, in);
-   own_qspace = true;
-
-   in >> ident; MFEM_VERIFY(ident == "VDim:", msg);
-   in >> vdim;
-
-   Load(in, vdim*qspace->GetSize());
-}
-
-QuadratureFunction & QuadratureFunction::operator=(double value)
-{
-   Vector::operator=(value);
-   return *this;
-}
-
-QuadratureFunction & QuadratureFunction::operator=(const Vector &v)
-{
-   MFEM_ASSERT(qspace && v.Size() == this->Size(), "");
-   Vector::operator=(v);
-   return *this;
-}
-
-QuadratureFunction & QuadratureFunction::operator=(const QuadratureFunction &v)
-{
-   return this->operator=((const Vector &)v);
-}
-
-void QuadratureFunction::Save(std::ostream &os) const
-{
-   qspace->Save(os);
-   os << "VDim: " << vdim << '\n'
-      << '\n';
-   Vector::Print(os, vdim);
-   os.flush();
-}
-
-std::ostream &operator<<(std::ostream &os, const QuadratureFunction &qf)
-{
-   qf.Save(os);
-   return os;
-}
-
-void QuadratureFunction::SaveVTU(std::ostream &os, VTKFormat format,
-                                 int compression_level) const
-{
-   os << R"(<VTKFile type="UnstructuredGrid" version="0.1")";
-   if (compression_level != 0)
-   {
-      os << R"( compressor="vtkZLibDataCompressor")";
-   }
-   os << " byte_order=\"" << VTKByteOrder() << "\">\n";
-   os << "<UnstructuredGrid>\n";
-
-   const char *fmt_str = (format == VTKFormat::ASCII) ? "ascii" : "binary";
-   const char *type_str = (format != VTKFormat::BINARY32) ? "Float64" : "Float32";
-   std::vector<char> buf;
-
-   int np = qspace->GetSize();
-   int ne = qspace->GetNE();
-   int sdim = qspace->GetMesh()->SpaceDimension();
-
-   // For quadrature functions, each point is a vertex cell, so number of cells
-   // is equal to number of points
-   os << "<Piece NumberOfPoints=\"" << np
-      << "\" NumberOfCells=\"" << np << "\">\n";
-
-   // print out the points
-   os << "<Points>\n";
-   os << "<DataArray type=\"" << type_str
-      << "\" NumberOfComponents=\"3\" format=\"" << fmt_str << "\">\n";
-
-   Vector pt(sdim);
-   for (int i = 0; i < ne; i++)
-   {
-      ElementTransformation &T = *qspace->GetMesh()->GetElementTransformation(i);
-      const IntegrationRule &ir = GetElementIntRule(i);
-      for (int j = 0; j < ir.Size(); j++)
-      {
-         T.Transform(ir[j], pt);
-         WriteBinaryOrASCII(os, buf, pt[0], " ", format);
-         if (sdim > 1) { WriteBinaryOrASCII(os, buf, pt[1], " ", format); }
-         else { WriteBinaryOrASCII(os, buf, 0.0, " ", format); }
-         if (sdim > 2) { WriteBinaryOrASCII(os, buf, pt[2], "", format); }
-         else { WriteBinaryOrASCII(os, buf, 0.0, "", format); }
-         if (format == VTKFormat::ASCII) { os << '\n'; }
-      }
-   }
-   if (format != VTKFormat::ASCII)
-   {
-      WriteBase64WithSizeAndClear(os, buf, compression_level);
-   }
-   os << "</DataArray>\n";
-   os << "</Points>\n";
-
-   // Write cells (each cell is just a vertex)
-   os << "<Cells>\n";
-   // Connectivity
-   os << R"(<DataArray type="Int32" Name="connectivity" format=")"
-      << fmt_str << "\">\n";
-
-   for (int i=0; i<np; ++i) { WriteBinaryOrASCII(os, buf, i, "\n", format); }
-   if (format != VTKFormat::ASCII)
-   {
-      WriteBase64WithSizeAndClear(os, buf, compression_level);
-   }
-   os << "</DataArray>\n";
-   // Offsets
-   os << R"(<DataArray type="Int32" Name="offsets" format=")"
-      << fmt_str << "\">\n";
-   for (int i=0; i<np; ++i) { WriteBinaryOrASCII(os, buf, i, "\n", format); }
-   if (format != VTKFormat::ASCII)
-   {
-      WriteBase64WithSizeAndClear(os, buf, compression_level);
-   }
-   os << "</DataArray>\n";
-   // Types
-   os << R"(<DataArray type="UInt8" Name="types" format=")"
-      << fmt_str << "\">\n";
-   for (int i = 0; i < np; i++)
-   {
-      uint8_t vtk_cell_type = VTKGeometry::POINT;
-      WriteBinaryOrASCII(os, buf, vtk_cell_type, "\n", format);
-   }
-   if (format != VTKFormat::ASCII)
-   {
-      WriteBase64WithSizeAndClear(os, buf, compression_level);
-   }
-   os << "</DataArray>\n";
-   os << "</Cells>\n";
-
-   os << "<PointData>\n";
-   os << "<DataArray type=\"" << type_str << "\" Name=\"u\" format=\""
-      << fmt_str << "\" NumberOfComponents=\"" << vdim << "\">\n";
-   for (int i = 0; i < ne; i++)
-   {
-      DenseMatrix vals;
-      GetElementValues(i, vals);
-      for (int j = 0; j < vals.Size(); ++j)
-      {
-         for (int vd = 0; vd < vdim; ++vd)
-         {
-            WriteBinaryOrASCII(os, buf, vals(vd, j), " ", format);
-         }
-         if (format == VTKFormat::ASCII) { os << '\n'; }
-      }
-   }
-   if (format != VTKFormat::ASCII)
-   {
-      WriteBase64WithSizeAndClear(os, buf, compression_level);
-   }
-   os << "</DataArray>\n";
-   os << "</PointData>\n";
-
-   os << "</Piece>\n";
-   os << "</UnstructuredGrid>\n";
-   os << "</VTKFile>" << std::endl;
-}
-
-void QuadratureFunction::SaveVTU(const std::string &filename, VTKFormat format,
-                                 int compression_level) const
-{
-   std::ofstream f(filename + ".vtu");
-   SaveVTU(f, format, compression_level);
-}
-
-
 double ZZErrorEstimator(BilinearFormIntegrator &blfi,
                        GridFunction &u,
                        GridFunction &flux, Vector &error_estimates,
@@ -4257,7 +4092,7 @@ void TensorProductLegendre(int dim,                // input
   }
   else
   {
-      // Bounding box is not reorientated no need to change orientation
+      // Bounding box is not reoriented no need to change orientation
      x = x_in;
   }

@@ -4281,44 +4116,44 @@ void TensorProductLegendre(int dim,                // input
   switch (dim)
   {
      case 1:
+      {
+         for (int i = 0; i <= order; i++)
+         {
+            poly(i) = poly_x(i);
+         }
+      }
+      break;
+      case 2:
+      {
+         for (int j = 0; j <= order; j++)
         {
            for (int i = 0; i <= order; i++)
            {
-               poly(i) = poly_x(i);
+               int cnt = i + (order+1) * j;
+               poly(cnt) = poly_x(i) * poly_y(j);
            }
         }
-         break;
-      case 2:
+      }
+      break;
+      case 3:
+      {
+         for (int k = 0; k <= order; k++)
         {
            for (int j = 0; j <= order; j++)
            {
               for (int i = 0; i <= order; i++)
               {
-                  int cnt = i + (order+1) * j;
-                  poly(cnt) = poly_x(i) * poly_y(j);
+                  int cnt = i + (order+1) * j + (order+1) * (order+1) * k;
+                  poly(cnt) = poly_x(i) * poly_y(j) * poly_z(k);
               }
            }
         }
-         break;
-      case 3:
-         {
-            for (int k = 0; k <= order; k++)
-            {
-               for (int j = 0; j <= order; j++)
-               {
-                  for (int i = 0; i <= order; i++)
-                  {
-                     int cnt = i + (order+1) * j + (order+1) * (order+1) * k;
-                     poly(cnt) = poly_x(i) * poly_y(j) * poly_z(k);
-                  }
-               }
-            }
-         }
-         break;
+      }
+      break;
      default:
-         {
-            MFEM_ABORT("TensorProductLegendre: invalid value of dim");
-         }
+      {
+         MFEM_ABORT("TensorProductLegendre: invalid value of dim");
+      }
   }
 }

@@ -129,19 +129,21 @@ public:
   int CurlDim() const;

   /// Read only access to the (optional) internal true-dof Vector.
-   /** Note that the returned Vector may be empty, if not previously allocated
-       or set. */
-   const Vector &GetTrueVector() const { return t_vec; }
+   const Vector &GetTrueVector() const
+   {
+      MFEM_VERIFY(t_vec.Size() > 0, "SetTrueVector() before GetTrueVector()");
+      return t_vec;
+   }
   /// Read and write access to the (optional) internal true-dof Vector.
-   /** Note that the returned Vector may be empty, if not previously allocated
-       or set. */
-   Vector &GetTrueVector() { return t_vec; }
+   /** Note that @a t_vec is set if it is not allocated or set already.*/
+   Vector &GetTrueVector()
+   { if (t_vec.Size() == 0) { SetTrueVector(); } return t_vec; }

   /// Extract the true-dofs from the GridFunction.
   void GetTrueDofs(Vector &tv) const;

   /// Shortcut for calling GetTrueDofs() with GetTrueVector() as argument.
-   void SetTrueVector() { GetTrueDofs(GetTrueVector()); }
+   void SetTrueVector() { GetTrueDofs(t_vec); }

   /// Set the GridFunction from the given true-dof vector.
   virtual void SetFromTrueDofs(const Vector &tv);
@@ -476,21 +478,27 @@ public:
                                             Array<int> &bdr_attr);


-   virtual double ComputeL2Error(Coefficient &exsol,
-                                 const IntegrationRule *irs[] = NULL) const
-   { return ComputeLpError(2.0, exsol, NULL, irs); }
-
   virtual double ComputeL2Error(Coefficient *exsol[],
-                                 const IntegrationRule *irs[] = NULL) const;
-
-   virtual double ComputeL2Error(VectorCoefficient &exsol,
                                 const IntegrationRule *irs[] = NULL,
-                                 Array<int> *elems = NULL) const;
+                                 const Array<int> *elems = NULL) const;

   /// Returns ||grad u_ex - grad u_h||_L2 in element ielem for H1 or L2 elements
   virtual double ComputeElementGradError(int ielem, VectorCoefficient *exgrad,
                                          const IntegrationRule *irs[] = NULL) const;

+   /// Returns ||u_ex - u_h||_L2 for H1 or L2 elements
+   /* The @a elems input variable expects a list of markers:
+      an elem marker equal to 1 will compute the L2 error on that element
+      an elem marker equal to 0 will not compute the L2 error on that element */
+   virtual double ComputeL2Error(Coefficient &exsol,
+                                 const IntegrationRule *irs[] = NULL,
+                                 const Array<int> *elems = NULL) const
+   { return GridFunction::ComputeLpError(2.0, exsol, NULL, irs, elems); }
+
+   virtual double ComputeL2Error(VectorCoefficient &exsol,
+                                 const IntegrationRule *irs[] = NULL,
+                                 const Array<int> *elems = NULL) const;
+
   /// Returns ||grad u_ex - grad u_h||_L2 for H1 or L2 elements
   virtual double ComputeGradError(VectorCoefficient *exgrad,
                                   const IntegrationRule *irs[] = NULL) const;
@@ -564,16 +572,20 @@ public:
   { return ComputeLpError(1.0, exsol, NULL, irs); }

   virtual double ComputeW11Error(Coefficient *exsol, VectorCoefficient *exgrad,
-                                  int norm_type, Array<int> *elems = NULL,
+                                  int norm_type, const Array<int> *elems = NULL,
                                  const IntegrationRule *irs[] = NULL) const;

   virtual double ComputeL1Error(VectorCoefficient &exsol,
                                 const IntegrationRule *irs[] = NULL) const
   { return ComputeLpError(1.0, exsol, NULL, NULL, irs); }

+   /* The @a elems input variable expects a list of markers:
+    an elem marker equal to 1 will compute the L2 error on that element
+    an elem marker equal to 0 will not compute the L2 error on that element */
   virtual double ComputeLpError(const double p, Coefficient &exsol,
                                 Coefficient *weight = NULL,
-                                 const IntegrationRule *irs[] = NULL) const;
+                                 const IntegrationRule *irs[] = NULL,
+                                 const Array<int> *elems = NULL) const;

   /** Compute the Lp error in each element of the mesh and store the results in
       the Vector @a error. The result should be of length number of elements,
@@ -755,176 +767,6 @@ public:
   }
 };

-
-/** @brief Class representing a function through its values (scalar or vector)
-    at quadrature points. */
-class QuadratureFunction : public Vector
-{
-protected:
-   QuadratureSpace *qspace; ///< Associated QuadratureSpace
-   int vdim;                ///< Vector dimension
-   bool own_qspace;         ///< QuadratureSpace ownership flag
-
-public:
-   /// Create an empty QuadratureFunction.
-   /** The object can be initialized later using the SetSpace() methods. */
-   QuadratureFunction()
-      : qspace(NULL), vdim(0), own_qspace(false) { }
-
-   /** @brief Copy constructor. The QuadratureSpace ownership flag, #own_qspace,
-       in the new object is set to false. */
-   QuadratureFunction(const QuadratureFunction &orig)
-      : Vector(orig),
-        qspace(orig.qspace), vdim(orig.vdim), own_qspace(false) { }
-
-   /// Create a QuadratureFunction based on the given QuadratureSpace.
-   /** The QuadratureFunction does not assume ownership of the QuadratureSpace.
-       @note The Vector data is not initialized. */
-   QuadratureFunction(QuadratureSpace *qspace_, int vdim_ = 1)
-      : Vector(vdim_*qspace_->GetSize()),
-        qspace(qspace_), vdim(vdim_), own_qspace(false) { }
-
-   /** @brief Create a QuadratureFunction based on the given QuadratureSpace,
-       using the external data, @a qf_data. */
-   /** The QuadratureFunction does not assume ownership of neither the
-       QuadratureSpace nor the external data. */
-   QuadratureFunction(QuadratureSpace *qspace_, double *qf_data, int vdim_ = 1)
-      : Vector(qf_data, vdim_*qspace_->GetSize()),
-        qspace(qspace_), vdim(vdim_), own_qspace(false) { }
-
-   /// Read a QuadratureFunction from the stream @a in.
-   /** The QuadratureFunction assumes ownership of the read QuadratureSpace. */
-   QuadratureFunction(Mesh *mesh, std::istream &in);
-
-   virtual ~QuadratureFunction() { if (own_qspace) { delete qspace; } }
-
-   /// Get the associated QuadratureSpace.
-   QuadratureSpace *GetSpace() const { return qspace; }
-
-   /// Change the QuadratureSpace and optionally the vector dimension.
-   /** If the new QuadratureSpace is different from the current one, the
-       QuadratureFunction will not assume ownership of the new space; otherwise,
-       the ownership flag remains the same.
-
-       If the new vector dimension @a vdim_ < 0, the vector dimension remains
-       the same.
-
-       The data size is updated by calling Vector::SetSize(). */
-   inline void SetSpace(QuadratureSpace *qspace_, int vdim_ = -1);
-
-   /** @brief Change the QuadratureSpace, the data array, and optionally the
-       vector dimension. */
-   /** If the new QuadratureSpace is different from the current one, the
-       QuadratureFunction will not assume ownership of the new space; otherwise,
-       the ownership flag remains the same.
-
-       If the new vector dimension @a vdim_ < 0, the vector dimension remains
-       the same.
-
-       The data array is replaced by calling Vector::NewDataAndSize(). */
-   inline void SetSpace(QuadratureSpace *qspace_, double *qf_data,
-                        int vdim_ = -1);
-
-   /// Get the vector dimension.
-   int GetVDim() const { return vdim; }
-
-   /// Set the vector dimension, updating the size by calling Vector::SetSize().
-   void SetVDim(int vdim_)
-   { vdim = vdim_; SetSize(vdim*qspace->GetSize()); }
-
-   /// Get the QuadratureSpace ownership flag.
-   bool OwnsSpace() { return own_qspace; }
-
-   /// Set the QuadratureSpace ownership flag.
-   void SetOwnsSpace(bool own) { own_qspace = own; }
-
-   /// Redefine '=' for QuadratureFunction = constant.
-   QuadratureFunction &operator=(double value);
-
-   /// Copy the data from @a v.
-   /** The size of @a v must be equal to the size of the associated
-       QuadratureSpace #qspace times the QuadratureFunction dimension
-       i.e. QuadratureFunction::Size(). */
-   QuadratureFunction &operator=(const Vector &v);
-
-   /// Copy assignment. Only the data of the base class Vector is copied.
-   /** The QuadratureFunctions @a v and @a *this must have QuadratureSpaces with
-       the same size.
-
-       @note Defining this method overwrites the implicitly defined copy
-       assignment operator. */
-   QuadratureFunction &operator=(const QuadratureFunction &v);
-
-   /// Get the IntegrationRule associated with mesh element @a idx.
-   const IntegrationRule &GetElementIntRule(int idx) const
-   { return qspace->GetElementIntRule(idx); }
-
-   /// Return all values associated with mesh element @a idx in a Vector.
-   /** The result is stored in the Vector @a values as a reference to the
-       global values.
-
-       Inside the Vector @a values, the index `i+vdim*j` corresponds to the
-       `i`-th vector component at the `j`-th quadrature point.
-    */
-   inline void GetElementValues(int idx, Vector &values);
-
-   /// Return all values associated with mesh element @a idx in a Vector.
-   /** The result is stored in the Vector @a values as a copy of the
-       global values.
-
-       Inside the Vector @a values, the index `i+vdim*j` corresponds to the
-       `i`-th vector component at the `j`-th quadrature point.
-    */
-   inline void GetElementValues(int idx, Vector &values) const;
-
-   /// Return the quadrature function values at an integration point.
-   /** The result is stored in the Vector @a values as a reference to the
-       global values. */
-   inline void GetElementValues(int idx, const int ip_num, Vector &values);
-
-   /// Return the quadrature function values at an integration point.
-   /** The result is stored in the Vector @a values as a copy to the
-       global values. */
-   inline void GetElementValues(int idx, const int ip_num, Vector &values) const;
-
-   /// Return all values associated with mesh element @a idx in a DenseMatrix.
-   /** The result is stored in the DenseMatrix @a values as a reference to the
-       global values.
-
-       Inside the DenseMatrix @a values, the `(i,j)` entry corresponds to the
-       `i`-th vector component at the `j`-th quadrature point.
-    */
-   inline void GetElementValues(int idx, DenseMatrix &values);
-
-   /// Return all values associated with mesh element @a idx in a const DenseMatrix.
-   /** The result is stored in the DenseMatrix @a values as a copy of the
-       global values.
-
-       Inside the DenseMatrix @a values, the `(i,j)` entry corresponds to the
-       `i`-th vector component at the `j`-th quadrature point.
-    */
-   inline void GetElementValues(int idx, DenseMatrix &values) const;
-
-   /// Write the QuadratureFunction to the stream @a out.
-   void Save(std::ostream &out) const;
-
-   /// @brief Write the QuadratureFunction to @a out in VTU (ParaView) format.
-   ///
-   /// The data will be uncompressed if @a compression_level is zero, or if the
-   /// format is VTKFormat::ASCII. Otherwise, zlib compression will be used for
-   /// binary data.
-   void SaveVTU(std::ostream &out, VTKFormat format=VTKFormat::ASCII,
-                int compression_level=0) const;
-
-   /// @brief Save the QuadratureFunction to a VTU (ParaView) file.
-   ///
-   /// The extension ".vtu" will be appended to @a filename.
-   /// @sa SaveVTU(std::ostream &out, VTKFormat format=VTKFormat::ASCII,
-   ///             int compression_level=0)
-   void SaveVTU(const std::string &filename, VTKFormat format=VTKFormat::ASCII,
-                int compression_level=0) const;
-};
-
 /// Overload operator<< for std::ostream and QuadratureFunction.
 std::ostream &operator<<(std::ostream &out, const QuadratureFunction &qf);

@@ -1012,95 +854,6 @@ public:
 GridFunction *Extrude1DGridFunction(Mesh *mesh, Mesh *mesh2d,
                                    GridFunction *sol, const int ny);

-
-// Inline methods
-
-inline void QuadratureFunction::SetSpace(QuadratureSpace *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(QuadratureSpace *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());
-}
-
-inline void QuadratureFunction::GetElementValues(int idx, Vector &values)
-{
-   const int s_offset = qspace->element_offsets[idx];
-   const int sl_size = qspace->element_offsets[idx+1] - s_offset;
-   values.NewDataAndSize(data + vdim*s_offset, vdim*sl_size);
-}
-
-inline void QuadratureFunction::GetElementValues(int idx, Vector &values) const
-{
-   const int s_offset = qspace->element_offsets[idx];
-   const int sl_size = qspace->element_offsets[idx+1] - s_offset;
-   values.SetSize(vdim*sl_size);
-   const double *q = data + vdim*s_offset;
-   for (int i = 0; i<values.Size(); i++)
-   {
-      values(i) = *(q++);
-   }
-}
-
-inline void QuadratureFunction::GetElementValues(int idx, const int ip_num,
-                                                 Vector &values)
-{
-   const int s_offset = qspace->element_offsets[idx] * vdim + ip_num * vdim;
-   values.NewDataAndSize(data + s_offset, vdim);
-}
-
-inline void QuadratureFunction::GetElementValues(int idx, const int ip_num,
-                                                 Vector &values) const
-{
-   const int s_offset = qspace->element_offsets[idx] * vdim + ip_num * vdim;
-   values.SetSize(vdim);
-   const double *q = data + s_offset;
-   for (int i = 0; i < values.Size(); i++)
-   {
-      values(i) = *(q++);
-   }
-}
-
-inline void QuadratureFunction::GetElementValues(int idx, DenseMatrix &values)
-{
-   const int s_offset = qspace->element_offsets[idx];
-   const int sl_size = qspace->element_offsets[idx+1] - s_offset;
-   values.Reset(data + vdim*s_offset, vdim, sl_size);
-}
-
-inline void QuadratureFunction::GetElementValues(int idx,
-                                                 DenseMatrix &values) const
-{
-   const int s_offset = qspace->element_offsets[idx];
-   const int sl_size = qspace->element_offsets[idx+1] - s_offset;
-   values.SetSize(vdim, sl_size);
-   const double *q = data + vdim*s_offset;
-   for (int j = 0; j<sl_size; j++)
-   {
-      for (int i = 0; i<vdim; i++)
-      {
-         values(i,j) = *(q++);
-      }
-   }
-}
-
 } // namespace mfem

 #endif
@@ -168,7 +168,8 @@ void FindPointsGSLIB::Setup(Mesh &m, const double bb_t, const double newt_tol,
   setupflag = true;
 }

-void FindPointsGSLIB::FindPoints(const Vector &point_pos)
+void FindPointsGSLIB::FindPoints(const Vector &point_pos,
+                                 int point_pos_ordering)
 {
   MFEM_VERIFY(setupflag, "Use FindPointsGSLIB::Setup before finding points.");
   points_cnt = point_pos.Size() / dim;
@@ -178,14 +179,24 @@ 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++)
+   {
+      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);
+      }
+   }
+
   if (dim == 2)
   {
-      const double *xv_base[2];
-      xv_base[0] = point_pos.GetData();
-      xv_base[1] = point_pos.GetData() + points_cnt;
-      unsigned xv_stride[2];
-      xv_stride[0] = sizeof(double);
-      xv_stride[1] = sizeof(double);
      findpts_2(gsl_code.GetData(), sizeof(unsigned int),
                gsl_proc.GetData(), sizeof(unsigned int),
                gsl_elem.GetData(), sizeof(unsigned int),
@@ -195,14 +206,6 @@ void FindPointsGSLIB::FindPoints(const Vector &point_pos)
   }
   else
   {
-      const double *xv_base[3];
-      xv_base[0] = point_pos.GetData();
-      xv_base[1] = point_pos.GetData() + points_cnt;
-      xv_base[2] = point_pos.GetData() + 2*points_cnt;
-      unsigned xv_stride[3];
-      xv_stride[0] = sizeof(double);
-      xv_stride[1] = sizeof(double);
-      xv_stride[2] = sizeof(double);
      findpts_3(gsl_code.GetData(), sizeof(unsigned int),
                gsl_proc.GetData(), sizeof(unsigned int),
                gsl_elem.GetData(), sizeof(unsigned int),
@@ -227,27 +230,29 @@ void FindPointsGSLIB::FindPoints(const Vector &point_pos)
 }

 void FindPointsGSLIB::FindPoints(Mesh &m, const Vector &point_pos,
-                                 const double bb_t, const double newt_tol,
-                                 const int npt_max)
+                                 int point_pos_ordering, const double bb_t,
+                                 const double newt_tol, const int npt_max)
 {
   if (!setupflag || (mesh != &m) )
   {
      Setup(m, bb_t, newt_tol, npt_max);
   }
-   FindPoints(point_pos);
+   FindPoints(point_pos, point_pos_ordering);
 }

 void FindPointsGSLIB::Interpolate(const Vector &point_pos,
-                                  const GridFunction &field_in, Vector &field_out)
+                                  const GridFunction &field_in, Vector &field_out,
+                                  int point_pos_ordering)
 {
-   FindPoints(point_pos);
+   FindPoints(point_pos, point_pos_ordering);
   Interpolate(field_in, field_out);
 }

 void FindPointsGSLIB::Interpolate(Mesh &m, const Vector &point_pos,
-                                  const GridFunction &field_in, Vector &field_out)
+                                  const GridFunction &field_in, Vector &field_out,
+                                  int point_pos_ordering)
 {
-   FindPoints(m, point_pos);
+   FindPoints(m, point_pos, point_pos_ordering);
   Interpolate(field_in, field_out);
 }

@@ -860,7 +865,9 @@ void FindPointsGSLIB::Interpolate(const GridFunction &field_in,
      {
         for (int i = 0; i < indl2.Size(); i++)
         {
-            int idx = indl2[i] + j*points_cnt;
+            int idx = field_in.FESpace()->GetOrdering() == Ordering::byNODES ?
+                      indl2[i] + j*points_cnt:
+                      indl2[i]*ncomp + j;
            field_out(idx) = field_out_l2(idx);
         }
      }
@@ -885,7 +892,17 @@ void FindPointsGSLIB::InterpolateH1(const GridFunction &field_in,
   {
      const int dataptrin  = i*points_fld,
                dataptrout = i*points_cnt;
-      field_in_scalar.NewDataAndSize(field_in.GetData()+dataptrin, points_fld);
+      if (field_in.FESpace()->GetOrdering() == Ordering::byNODES)
+      {
+         field_in_scalar.NewDataAndSize(field_in.GetData()+dataptrin, points_fld);
+      }
+      else
+      {
+         for (int j = 0; j < points_fld; j++)
+         {
+            field_in_scalar(j) = field_in(i + j*ncomp);
+         }
+      }
      GetNodalValues(&field_in_scalar, node_vals);

      if (dim==2)
@@ -907,6 +924,17 @@ void FindPointsGSLIB::InterpolateH1(const GridFunction &field_in,
                        points_cnt, node_vals.GetData(), fdata3D);
      }
   }
+   if (field_in.FESpace()->GetOrdering() == Ordering::byVDIM)
+   {
+      Vector field_out_temp = field_out;
+      for (int i = 0; i < ncomp; i++)
+      {
+         for (int j = 0; j < points_cnt; j++)
+         {
+            field_out(i + j*ncomp) = field_out_temp(j + i*points_cnt);
+         }
+      }
+   }
 }

 void FindPointsGSLIB::InterpolateGeneral(const GridFunction &field_in,
@@ -929,9 +957,19 @@ void FindPointsGSLIB::InterpolateGeneral(const GridFunction &field_in,
         if (dim == 3) { ip.z = gsl_mfem_ref(index*dim + 2); }
         Vector localval(ncomp);
         field_in.GetVectorValue(gsl_mfem_elem[index], ip, localval);
-         for (int i = 0; i < ncomp; i++)
+         if (field_in.FESpace()->GetOrdering() == Ordering::byNODES)
         {
-            field_out(index + i*npt) = localval(i);
+            for (int i = 0; i < ncomp; i++)
+            {
+               field_out(index + i*npt) = localval(i);
+            }
+         }
+         else //byVDIM
+         {
+            for (int i = 0; i < ncomp; i++)
+            {
+               field_out(index*ncomp + i) = localval(i);
+            }
         }
      }
   }
@@ -1044,7 +1082,9 @@ void FindPointsGSLIB::InterpolateGeneral(const GridFunction &field_in,
            sdpt = (struct send_pt *)sendpt->ptr;
            for (int index = 0; index < sendpt->n; index++)
            {
-               int idx = sdpt->index + j*nptorig;
+               int idx = field_in.FESpace()->GetOrdering() == Ordering::byNODES ?
+                         sdpt->index + j*nptorig :
+                         sdpt->index*ncomp + j;
               field_out(idx) = sdpt->ival;
               ++sdpt;
            }
@@ -1139,7 +1179,8 @@ void OversetFindPointsGSLIB::Setup(Mesh &m, const int meshid,
 }

 void OversetFindPointsGSLIB::FindPoints(const Vector &point_pos,
-                                        Array<unsigned int> &point_id)
+                                        Array<unsigned int> &point_id,
+                                        int point_pos_ordering)
 {
   MFEM_VERIFY(setupflag, "Use OversetFindPointsGSLIB::Setup before "
               "finding points.");
@@ -1153,14 +1194,24 @@ 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++)
+   {
+      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);
+      }
+   }
+
   if (dim == 2)
   {
-      const double *xv_base[2];
-      xv_base[0] = point_pos.GetData();
-      xv_base[1] = point_pos.GetData() + points_cnt;
-      unsigned xv_stride[2];
-      xv_stride[0] = sizeof(double);
-      xv_stride[1] = sizeof(double);
      findptsms_2(gsl_code.GetData(), sizeof(unsigned int),
                  gsl_proc.GetData(), sizeof(unsigned int),
                  gsl_elem.GetData(), sizeof(unsigned int),
@@ -1172,14 +1223,6 @@ void OversetFindPointsGSLIB::FindPoints(const Vector &point_pos,
   }
   else
   {
-      const double *xv_base[3];
-      xv_base[0] = point_pos.GetData();
-      xv_base[1] = point_pos.GetData() + points_cnt;
-      xv_base[2] = point_pos.GetData() + 2*points_cnt;
-      unsigned xv_stride[3];
-      xv_stride[0] = sizeof(double);
-      xv_stride[1] = sizeof(double);
-      xv_stride[2] = sizeof(double);
      findptsms_3(gsl_code.GetData(), sizeof(unsigned int),
                  gsl_proc.GetData(), sizeof(unsigned int),
                  gsl_elem.GetData(), sizeof(unsigned int),
@@ -1208,9 +1251,10 @@ void OversetFindPointsGSLIB::FindPoints(const Vector &point_pos,
 void OversetFindPointsGSLIB::Interpolate(const Vector &point_pos,
                                         Array<unsigned int> &point_id,
                                         const GridFunction &field_in,
-                                         Vector &field_out)
+                                         Vector &field_out,
+                                         int point_pos_ordering)
 {
-   FindPoints(point_pos, point_id);
+   FindPoints(point_pos, point_id, point_pos_ordering);
   Interpolate(field_in, field_out);
 }

@@ -121,8 +121,9 @@ public:
   void Setup(Mesh &m, const double bb_t = 0.1,
              const double newt_tol = 1.0e-12,
              const int npt_max = 256);
-   /** Searches positions given in physical space by @a point_pos. These positions
-       must by ordered by nodes: (XXX...,YYY...,ZZZ).
+   /** Searches positions given in physical space by @a point_pos.
+       These positions can be ordered byNodes: (XXX...,YYY...,ZZZ) or
+       byVDim: (XYZ,XYZ,....XYZ) specified by @a point_pos_ordering.
       This function populates the following member variables:
       #gsl_code        Return codes for each point: inside element (0),
                        element boundary (1), not found (2).
@@ -140,9 +141,11 @@ public:
                        Defaults to 0 for points that were not found.
       #gsl_dist        Distance between the sought and the found point
                        in physical space. */
-   void FindPoints(const Vector &point_pos);
+   void FindPoints(const Vector &point_pos,
+                   int point_pos_ordering = Ordering::byNODES);
   /// Setup FindPoints and search positions
   void FindPoints(Mesh &m, const Vector &point_pos,
+                   int point_pos_ordering = Ordering::byNODES,
                   const double bb_t = 0.1,
                   const double newt_tol = 1.0e-12,  const int npt_max = 256);

@@ -154,12 +157,18 @@ public:
       @param[out] field_out  Interpolated values. For points that are not found
                              the value is set to #default_interp_value. */
   virtual void Interpolate(const GridFunction &field_in, Vector &field_out);
-   /** Search positions and interpolate */
+   /** Search positions and interpolate. The ordering (byNODES or byVDIM) of
+       the output values in @a field_out corresponds to the ordering used
+       in the input GridFunction @a field_in. */
   void Interpolate(const Vector &point_pos, const GridFunction &field_in,
-                    Vector &field_out);
-   /** Setup FindPoints, search positions and interpolate */
+                    Vector &field_out,
+                    int point_pos_ordering = Ordering::byNODES);
+   /** Setup FindPoints, search positions and interpolate. The ordering (byNODES
+       or byVDIM) of the output values in @a field_out corresponds to the
+       ordering used in the input GridFunction @a field_in. */
   void Interpolate(Mesh &m, const Vector &point_pos,
-                    const GridFunction &field_in, Vector &field_out);
+                    const GridFunction &field_in, Vector &field_out,
+                    int point_pos_ordering = Ordering::byNODES);

   /// Average type to be used for L2 functions in-case a point is located at
   /// an element boundary where the function might be multi-valued.
@@ -247,15 +256,20 @@ public:
   /** Searches positions given in physical space by @a point_pos. All output
       Arrays and Vectors are expected to have the correct size.

-       @param[in]  point_pos  Positions to be found. Must by ordered by nodes
-                              (XXX...,YYY...,ZZZ).
-       @param[in]  point_id   Index of the mesh that the point belongs to
-                              (corresponding to @a meshid in Setup). */
-   void FindPoints(const Vector &point_pos, Array<unsigned int> &point_id);
+       @param[in]  point_pos           Positions to be found.
+       @param[in]  point_id            Index of the mesh that the point belongs
+                                       to (corresponding to @a meshid in Setup).
+       @param[in]  point_pos_ordering  Ordering of the points:
+                                       byNodes: (XXX...,YYY...,ZZZ) or
+                                       byVDim: (XYZ,XYZ,....XYZ) */
+   void FindPoints(const Vector &point_pos,
+                   Array<unsigned int> &point_id,
+                   int point_pos_ordering = Ordering::byNODES);

   /** Search positions and interpolate */
   void Interpolate(const Vector &point_pos, Array<unsigned int> &point_id,
-                    const GridFunction &field_in, Vector &field_out);
+                    const GridFunction &field_in, Vector &field_out,
+                    int point_pos_ordering = Ordering::byNODES);
   using FindPointsGSLIB::Interpolate;
 };

@@ -827,7 +827,6 @@ void Hybridization::ReduceRHS(const Vector &b, Vector &b_r) const
      }
      else
      {
-         Ct->EnsureMultTranspose();
         Ct->MultTranspose(bf, bl);
      }
      b_r.SetSize(pH.Ptr()->Height());
@@ -120,6 +120,22 @@ MFEM_HOST_DEVICE inline void LoadBGt(const int D1D, const int Q1D,
   MFEM_SYNC_THREAD;
 }

+/// Load 2D input scalar into given DeviceMatrix
+MFEM_HOST_DEVICE inline void LoadX(const int e, const int D1D,
+                                   const DeviceTensor<3, const double> &x,
+                                   DeviceMatrix &DD)
+{
+   MFEM_FOREACH_THREAD(dy,y,D1D)
+   {
+      MFEM_FOREACH_THREAD(dx,x,D1D)
+      {
+         DD(dx,dy) = x(dx,dy,e);
+      }
+   }
+   MFEM_SYNC_THREAD;
+}
+
+
 /// Load 2D input scalar into shared memory
 template<int MD1, int NBZ>
 MFEM_HOST_DEVICE inline void LoadX(const int e, const int D1D,
@@ -128,15 +144,7 @@ MFEM_HOST_DEVICE inline void LoadX(const int e, const int D1D,
 {
   const int tidz = MFEM_THREAD_ID(z);
   DeviceMatrix X(sX[tidz], D1D, D1D);
-
-   MFEM_FOREACH_THREAD(dy,y,D1D)
-   {
-      MFEM_FOREACH_THREAD(dx,x,D1D)
-      {
-         X(dx,dy) = x(dx,dy,e);
-      }
-   }
-   MFEM_SYNC_THREAD;
+   LoadX(e, D1D, x, X);
 }

 /// Load 2D input scalar into shared memory, with comp
@@ -19,12 +19,13 @@ namespace mfem
 LinearForm::LinearForm(FiniteElementSpace *f, LinearForm *lf)
   : Vector(f->GetVSize())
 {
-   // Linear forms are stored on the device
-   UseDevice(true);
-
-   fes = f;
   ext = nullptr;
   extern_lfs = 1;
+   fast_assembly = false;
+   fes = f;
+
+   // Linear forms are stored on the device
+   UseDevice(true);

   // Copy the pointers to the integrators
   domain_integs = lf->domain_integs;
@@ -102,25 +103,45 @@ void LinearForm::AddInteriorFaceIntegrator(LinearFormIntegrator *lfi)

 bool LinearForm::SupportsDevice()
 {
-   // return false for NURBS meshs, so we don’t convert it to non-NURBS
+   // return false for NURBS meshes, so we don’t convert it to non-NURBS
   // through Assemble, AssembleDevice, GetGeometricFactors and EnsureNodes
-   if (fes->GetMesh()->NURBSext != nullptr) { return false; }
+   const Mesh &mesh = *fes->GetMesh();
+   if (mesh.NURBSext != nullptr) { return false; }

-   // scan domain integrator to verify that all can use device assembly
-   if (domain_integs.Size() > 0)
+   // scan integrators to verify that all can use device assembly
+   auto IntegratorsSupportDevice = [](const Array<LinearFormIntegrator*> &integ)
   {
-      for (int k = 0; k < domain_integs.Size(); k++)
+      for (int k = 0; k < integ.Size(); k++)
      {
-         if (!domain_integs[k]->SupportsDevice()) { return false; }
+         if (!integ[k]->SupportsDevice()) { return false; }
+      }
+      return true;
+   };
+
+   if (!IntegratorsSupportDevice(domain_integs)) { return false; }
+   if (!IntegratorsSupportDevice(boundary_integs)) { return false; }
+   if (boundary_face_integs.Size() > 0 || interior_face_integs.Size() > 0 ||
+       domain_delta_integs.Size() > 0) { return false; }
+
+   if (boundary_integs.Size() > 0)
+   {
+      // Make sure there are no boundary faces that are not boundary elements
+      if (fes->GetNFbyType(FaceType::Boundary) != fes->GetNBE())
+      {
+         return false;
+      }
+      // Make sure every boundary element corresponds to a boundary face
+      for (int be = 0; be < fes->GetNBE(); ++be)
+      {
+         const int f = mesh.GetBdrElementEdgeIndex(be);
+         const auto face_info = mesh.GetFaceInformation(f);
+         if (!face_info.IsBoundary())
+         {
+            return false;
+         }
      }
   }

-   // boundary, delta and face integrators are not supported yet
-   if (GetBLFI()->Size() > 0 || GetFLFI()->Size() > 0 ||
-       GetDLFI_Delta()->Size() > 0 || GetIFLFI()->Size() > 0) { return false; }
-
-   const Mesh &mesh = *fes->GetMesh();
-
   // no support for elements with varying polynomial orders
   if (fes->IsVariableOrder()) { return false; }

@@ -134,25 +155,30 @@ bool LinearForm::SupportsDevice()
   return true;
 }

-void LinearForm::Assemble(bool use_device)
+void LinearForm::UseFastAssembly(bool use_fa)
+{
+   fast_assembly = use_fa;
+
+   if (fast_assembly && SupportsDevice() && !ext)
+   {
+      ext = new LinearFormExtension(this);
+   }
+}
+
+void LinearForm::Assemble()
 {
   Array<int> vdofs;
   ElementTransformation *eltrans;
   DofTransformation *doftrans;
   Vector elemvect;

-   if (!ext && use_device && SupportsDevice())
-   {
-      ext = new LinearFormExtension(this);
-   }
-
   Vector::operator=(0.0);

   // The above operation is executed on device because of UseDevice().
   // The first use of AddElementVector() below will move it back to host
   // because both 'vdofs' and 'elemvect' are on host.

-   if (ext) { return ext->Assemble(); }
+   if (fast_assembly && ext) { return ext->Assemble(); }

   if (domain_integs.Size())
   {
@@ -173,8 +199,8 @@ void LinearForm::Assemble(bool use_device)
         int elem_attr = fes->GetMesh()->GetAttribute(i);
         for (int k = 0; k < domain_integs.Size(); k++)
         {
-            if ( domain_integs_marker[k] == NULL ||
-                 (*(domain_integs_marker[k]))[elem_attr-1] == 1 )
+            const Array<int> * const markers = domain_integs_marker[k];
+            if ( markers == NULL || (*markers)[elem_attr-1] == 1 )
            {
               doftrans = fes -> GetElementVDofs (i, vdofs);
               eltrans = fes -> GetElementTransformation (i);
@@ -30,12 +30,16 @@ protected:
   FiniteElementSpace *fes;

   /** @brief Extension for supporting different assembly levels. */
-   LinearFormExtension *ext;
+   LinearFormExtension *ext = nullptr;
+
+   /// @brief Should we use the device-compatible fast assembly algorithm (false
+   /// by default)
+   bool fast_assembly = false;

   /** @brief Indicates the LinearFormIntegrator%s stored in #domain_integs,
       #domain_delta_integs, #boundary_integs, and #boundary_face_integs are
       owned by another LinearForm. */
-   int extern_lfs;
+   int extern_lfs = 0;

   /// Set of Domain Integrators to be applied.
   Array<LinearFormIntegrator*> domain_integs;
@@ -82,7 +86,7 @@ public:
   /// Creates linear form associated with FE space @a *f.
   /** The pointer @a f is not owned by the newly constructed object. */
   LinearForm(FiniteElementSpace *f) : Vector(f->GetVSize())
-   { fes = f; ext = nullptr; extern_lfs = 0; UseDevice(true); }
+   { fes = f; UseDevice(true); }

   /** @brief Create a LinearForm on the FiniteElementSpace @a f, using the
       same integrators as the LinearForm @a lf.
@@ -97,7 +101,8 @@ public:
   /** The associated FiniteElementSpace can be set later using one of the
       methods: Update(FiniteElementSpace *) or
       Update(FiniteElementSpace *, Vector &, int). */
-   LinearForm() { fes = NULL; ext = nullptr; extern_lfs = 0; UseDevice(true); }
+   LinearForm()
+   { fes = NULL; UseDevice(true); }

   /// Construct a LinearForm using previously allocated array @a data.
   /** The LinearForm does not assume ownership of @a data which is assumed to
@@ -105,7 +110,7 @@ public:
       for externally allocated array, the pointer @a data can be NULL. The data
       array can be replaced later using the method SetData(). */
   LinearForm(FiniteElementSpace *f, double *data) : Vector(data, f->GetVSize())
-   { fes = f; ext = nullptr; extern_lfs = 0; }
+   { fes = f; }

   /// Copy assignment. Only the data of the base class Vector is copied.
   /** It is assumed that this object and @a rhs use FiniteElementSpace%s that
@@ -185,13 +190,20 @@ public:
       corresponding pointer (to Array<int>) will be NULL. */
   Array<Array<int>*> *GetFLFI_Marker() { return &boundary_face_integs_marker; }

+   /// @brief Which assembly algorithm to use: the new device-compatible fast
+   /// assembly (true), or the legacy CPU-only algorithm (false).
+   /** If not set, the default value is false.  If used, this method must be
+       called before assembly. */
+   void UseFastAssembly(bool use_fa);
+
   /// Assembles the linear form i.e. sums over all domain/bdr integrators.
-   /// When @a use_device is set to true and the linearform assembly is
-   /// compatible with device execution, it will be executed on the device.
-   void Assemble(bool use_device = true);
+   /** When @ref UseFastAssembly "UseFastAssembly(true)" has been called and the
+       linearform assembly is compatible with device execution, it will be
+       executed on the device. */
+   void Assemble();

   /// Return true if assembly on device is supported, false otherwise.
-   bool SupportsDevice();
+   virtual bool SupportsDevice();

   /// Assembles delta functions of the linear form
   void AssembleDelta();
@@ -25,7 +25,8 @@ void LinearFormExtension::Assemble()
               "match the number of vector dofs!");

   const Array<Array<int>*> &domain_integs_marker = *lf->GetDLFI_Marker();
-   const int mesh_attributes_size = fes.GetMesh()->attributes.Size();
+   const int mesh_attributes_max = fes.GetMesh()->attributes.Size() ?
+                                   fes.GetMesh()->attributes.Max() : 0;
   const Array<LinearFormIntegrator*> &domain_integs = *lf->GetDLFI();

   for (int k = 0; k < domain_integs.Size(); ++k)
@@ -39,7 +40,7 @@ void LinearFormExtension::Assemble()
      if (has_markers_k)
      {
         // Element attribute marker should be of length mesh->attributes
-         MFEM_VERIFY(mesh_attributes_size == domain_integs_marker_k->Size(),
+         MFEM_VERIFY(mesh_attributes_max == domain_integs_marker_k->Size(),
                     "invalid element marker for domain linear form "
                     "integrator #" << k << ", counting from zero");
      }
@@ -59,7 +60,47 @@ void LinearFormExtension::Assemble()
      // Assemble the linear form
      b = 0.0;
      domain_integs[k]->AssembleDevice(fes, markers, b);
-      elem_restrict_lex->MultTranspose(b, *lf);
+      if (k == 0) { elem_restrict_lex->MultTranspose(b, *lf); }
+      else { elem_restrict_lex->AddMultTranspose(b, *lf); }
+   }
+
+   const Array<Array<int>*> &boundary_integs_marker = lf->boundary_integs_marker;
+   const int bdr_attributes_max = fes.GetMesh()->bdr_attributes.Size() ?
+                                  fes.GetMesh()->bdr_attributes.Max() : 0;
+   const Array<LinearFormIntegrator*> &boundary_integs = lf->boundary_integs;
+
+   for (int k = 0; k < boundary_integs.Size(); ++k)
+   {
+      // Get the markers for this integrator
+      const Array<int> *boundary_integs_marker_k = boundary_integs_marker[k];
+
+      // check if there are markers for this integrator
+      const bool has_markers_k = boundary_integs_marker_k != nullptr;
+
+      if (has_markers_k)
+      {
+         // Element attribute marker should be of length mesh->attributes
+         MFEM_VERIFY(bdr_attributes_max == boundary_integs_marker_k->Size(),
+                     "invalid boundary marker for boundary linear form "
+                     "integrator #" << k << ", counting from zero");
+      }
+
+      // if there are no markers, just use the whole linear form (1)
+      if (!has_markers_k) { bdr_markers.HostReadWrite(); bdr_markers = 1; }
+      else
+      {
+         // scan the attributes to set the markers to 0 or 1
+         const int NBE = bdr_attributes.Size();
+         const auto attr = bdr_attributes.Read();
+         const auto attr_markers = boundary_integs_marker_k->Read();
+         auto markers_w = bdr_markers.Write();
+         MFEM_FORALL(e, NBE, markers_w[e] = attr_markers[attr[e]-1] == 1;);
+      }
+
+      // Assemble the linear form
+      bdr_b = 0.0;
+      boundary_integs[k]->AssembleDevice(fes, bdr_markers, bdr_b);
+      bdr_restrict_lex->AddMultTranspose(bdr_b, *lf);
   }
 }

@@ -67,22 +108,64 @@ void LinearFormExtension::Update()
 {
   const FiniteElementSpace &fes = *lf->FESpace();
   const Mesh &mesh = *fes.GetMesh();
-   const int NE = fes.GetNE();
+   constexpr ElementDofOrdering ordering = ElementDofOrdering::LEXICOGRAPHIC;

   MFEM_VERIFY(lf->Size() == fes.GetVSize(), "");

-   markers.SetSize(NE);
-   //markers.UseDevice(true);
+   if (lf->domain_integs.Size() > 0)
+   {
+      const int NE = fes.GetNE();
+      markers.SetSize(NE);
+      //markers.UseDevice(true);

-   // Gather the attributes on the host from all the elements
-   attributes.SetSize(NE);
-   for (int i = 0; i < NE; ++i) { attributes[i] = mesh.GetAttribute(i); }
+      // Gather the attributes on the host from all the elements
+      attributes.SetSize(NE);
+      for (int i = 0; i < NE; ++i) { attributes[i] = mesh.GetAttribute(i); }

-   constexpr ElementDofOrdering ordering = ElementDofOrdering::LEXICOGRAPHIC;
-   elem_restrict_lex = fes.GetElementRestriction(ordering);
-   MFEM_VERIFY(elem_restrict_lex, "Element restriction not available");
-   b.SetSize(elem_restrict_lex->Height(), Device::GetMemoryType());
-   b.UseDevice(true);
+      elem_restrict_lex = fes.GetElementRestriction(ordering);
+      MFEM_VERIFY(elem_restrict_lex, "Element restriction not available");
+      b.SetSize(elem_restrict_lex->Height(), Device::GetMemoryType());
+      b.UseDevice(true);
+   }
+
+   if (lf->boundary_integs.Size() > 0)
+   {
+      const int nf_bdr = fes.GetNFbyType(FaceType::Boundary);
+      bdr_markers.SetSize(nf_bdr);
+      // bdr_markers.UseDevice(true);
+
+      // The face restriction will give us "face E-vectors" on the boundary that
+      // are numbered in the order of the faces of mesh. This numbering will be
+      // different than the numbering of the boundary elements. We compute
+      // mappings so that the array `bdr_attributes[i]` gives the boundary
+      // attribute of the `i`th boundary face in the mesh face order.
+      std::unordered_map<int,int> f_to_be;
+      for (int i = 0; i < mesh.GetNBE(); ++i)
+      {
+         const int f = mesh.GetBdrElementEdgeIndex(i);
+         f_to_be[f] = i;
+      }
+      MFEM_VERIFY(size_t(nf_bdr) == f_to_be.size(), "Incompatible sizes");
+      bdr_attributes.SetSize(nf_bdr);
+      int f_ind = 0;
+      for (int f = 0; f < mesh.GetNumFaces(); ++f)
+      {
+         if (f_to_be.find(f) != f_to_be.end())
+         {
+            const int be = f_to_be[f];
+            bdr_attributes[f_ind] = mesh.GetBdrAttribute(be);
+            ++f_ind;
+         }
+      }
+
+      bdr_restrict_lex =
+         dynamic_cast<const FaceRestriction*>(
+            fes.GetFaceRestriction(ordering, FaceType::Boundary,
+                                   L2FaceValues::SingleValued));
+      MFEM_VERIFY(bdr_restrict_lex, "Face restriction not available");
+      bdr_b.SetSize(bdr_restrict_lex->Height(), Device::GetMemoryType());
+      bdr_b.UseDevice(true);
+   }
 }

 } // namespace mfem
@@ -25,19 +25,22 @@ class LinearForm;
 class LinearFormExtension
 {
   /// Attributes of all mesh elements.
-   Array<int> attributes;
+   Array<int> attributes, bdr_attributes;

   /// Temporary markers for device kernels.
-   Array<int> markers;
+   Array<int> markers, bdr_markers;

   /// Linear form from which this extension depends. Not owned.
   LinearForm *lf;

   /// Operator that converts FiniteElementSpace L-vectors to E-vectors.
-   const Operator *elem_restrict_lex; // Not owned
+   const ElementRestrictionOperator *elem_restrict_lex; // Not owned
+
+   /// Operator that converts L-vectors to boundary E-vectors.
+   const FaceRestriction *bdr_restrict_lex; // Not owned

   /// Internal E-vectors.
-   mutable Vector b;
+   mutable Vector b, bdr_b;

 public:

@@ -1046,7 +1046,7 @@ void VectorQuadratureLFIntegrator::AssembleRHSElementVect(
   const FiniteElement &fe, ElementTransformation &Tr, Vector &elvect)
 {
   const IntegrationRule *ir =
-      &vqfc.GetQuadFunction().GetSpace()->GetElementIntRule(Tr.ElementNo);
+      &vqfc.GetQuadFunction().GetSpace()->GetIntRule(Tr.ElementNo);

   const int nqp = ir->GetNPoints();
   const int vdim = vqfc.GetVDim();
@@ -1078,7 +1078,7 @@ void QuadratureLFIntegrator::AssembleRHSElementVect(const FiniteElement &fe,
                                                    Vector &elvect)
 {
   const IntegrationRule *ir =
-      &qfc.GetQuadFunction().GetSpace()->GetElementIntRule(Tr.ElementNo);
+      &qfc.GetQuadFunction().GetSpace()->GetIntRule(Tr.ElementNo);

   const int nqp = ir->GetNPoints();
   const int ndofs = fe.GetDof();
@@ -187,6 +187,13 @@ public:
   BoundaryLFIntegrator(Coefficient &QG, int a = 1, int b = 1)
      : Q(QG), oa(a), ob(b) { }

+   virtual bool SupportsDevice() { return true; }
+
+   /// Method defining assembly on device
+   virtual void AssembleDevice(const FiniteElementSpace &fes,
+                               const Array<int> &markers,
+                               Vector &b);
+
   /** Given a particular boundary Finite Element and a transformation (Tr)
       computes the element boundary vector, elvect. */
   virtual void AssembleRHSElementVect(const FiniteElement &el,
@@ -210,6 +217,13 @@ public:
   BoundaryNormalLFIntegrator(VectorCoefficient &QG, int a = 1, int b = 1)
      : Q(QG), oa(a), ob(b) { }

+   virtual bool SupportsDevice() { return true; }
+
+   /// Method defining assembly on device
+   virtual void AssembleDevice(const FiniteElementSpace &fes,
+                               const Array<int> &markers,
+                               Vector &b);
+
   virtual void AssembleRHSElementVect(const FiniteElement &el,
                                       ElementTransformation &Tr,
                                       Vector &elvect);
@@ -0,0 +1,241 @@
+// 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.
+
+#include "fem.hpp"
+#include "../fem/kernels.hpp"
+#include "../general/forall.hpp"
+
+namespace mfem
+{
+
+template<int T_D1D = 0, int T_Q1D = 0> static
+void BLFEvalAssemble2D(const int vdim, const int nbe, const int d, const int q,
+                       const bool normals, const int *markers, const double *b,
+                       const double *detj, const double *n, const double *weights,
+                       const Vector &coeff, double *y)
+{
+   const auto F = coeff.Read();
+   const auto M = Reshape(markers, nbe);
+   const auto B = Reshape(b, q, d);
+   const auto detJ = Reshape(detj, q, nbe);
+   const auto N = Reshape(n, q, 2, nbe);
+   const auto W = Reshape(weights, q);
+   const int cvdim = normals ? 2 : 1;
+   const bool cst = coeff.Size() == cvdim;
+   const auto C = cst ? Reshape(F,cvdim,1,1) : Reshape(F,cvdim,q,nbe);
+   auto Y = Reshape(y, d, vdim, nbe);
+
+   MFEM_FORALL(e, nbe,
+   {
+      if (M(e) == 0) { return; } // ignore
+
+      constexpr int Q = T_Q1D ? T_Q1D : MAX_Q1D;
+      double QQ[Q];
+
+      for (int c = 0; c < vdim; ++c)
+      {
+         for (int qx = 0; qx < q; ++qx)
+         {
+            double coeff_val = 0.0;
+            if (normals)
+            {
+               for (int cd = 0; cd < 2; ++cd)
+               {
+                  const double cval = cst ? C(cd,0,0) : C(cd,qx,e);
+                  coeff_val += cval * N(qx, cd, e);
+               }
+            }
+            else
+            {
+               coeff_val = cst ? C(0,0,0) : C(0,qx,e);
+            }
+            QQ[qx] = W(qx) * coeff_val * detJ(qx,e);
+         }
+         for (int dx = 0; dx < d; ++dx)
+         {
+            double u = 0;
+            for (int qx = 0; qx < q; ++qx) { u += QQ[qx] * B(qx,dx); }
+            Y(dx,c,e) += u;
+         }
+      }
+   });
+}
+
+template<int T_D1D = 0, int T_Q1D = 0> static
+void BLFEvalAssemble3D(const int vdim, const int nbe, const int d, const int q,
+                       const bool normals, const int *markers, const double *b,
+                       const double *detj, const double *n, const double *weights,
+                       const Vector &coeff, double *y)
+{
+   const auto F = coeff.Read();
+   const auto M = Reshape(markers, nbe);
+   const auto B = Reshape(b, q, d);
+   const auto detJ = Reshape(detj, q, q, nbe);
+   const auto N = Reshape(n, q, q, 3, nbe);
+   const auto W = Reshape(weights, q, q);
+   const int cvdim = normals ? 3 : 1;
+   const bool cst = coeff.Size() == cvdim;
+   const auto C = cst ? Reshape(F,cvdim,1,1,1) : Reshape(F,cvdim,q,q,nbe);
+   auto Y = Reshape(y, d, d, vdim, nbe);
+
+   MFEM_FORALL_2D(e, nbe, q, q, 1,
+   {
+      if (M(e) == 0) { return; } // ignore
+
+      constexpr int Q = T_Q1D ? T_Q1D : MAX_Q1D;
+      constexpr int D = T_D1D ? T_D1D : MAX_D1D;
+
+      MFEM_SHARED double sBt[Q*D];
+      MFEM_SHARED double sQQ[Q*Q];
+      MFEM_SHARED double sQD[Q*D];
+
+      const DeviceMatrix Bt(sBt, d, q);
+      kernels::internal::LoadB<D,Q>(d, q, B, sBt);
+
+      const DeviceMatrix QQ(sQQ, q, q);
+      const DeviceMatrix QD(sQD, q, d);
+
+      for (int c = 0; c < vdim; ++c)
+      {
+         MFEM_FOREACH_THREAD(x,x,q)
+         {
+            MFEM_FOREACH_THREAD(y,y,q)
+            {
+               double coeff_val = 0.0;
+               if (normals)
+               {
+                  for (int cd = 0; cd < 3; ++cd)
+                  {
+                     double cval = cst ? C(cd,0,0,0) : C(cd,x,y,e);
+                     coeff_val += cval * N(x,y,cd,e);
+                  }
+               }
+               else
+               {
+                  coeff_val = cst ? C(0,0,0,0) : C(0,x,y,e);
+               }
+               QQ(y,x) = W(x,y) * coeff_val * detJ(x,y,e);
+            }
+         }
+         MFEM_SYNC_THREAD;
+         MFEM_FOREACH_THREAD(qy,y,q)
+         {
+            MFEM_FOREACH_THREAD(dx,x,d)
+            {
+               double u = 0.0;
+               for (int qx = 0; qx < q; ++qx) { u += QQ(qy,qx) * Bt(dx,qx); }
+               QD(qy,dx) = u;
+            }
+         }
+         MFEM_SYNC_THREAD;
+         MFEM_FOREACH_THREAD(dy,y,d)
+         {
+            MFEM_FOREACH_THREAD(dx,x,d)
+            {
+               double u = 0.0;
+               for (int qy = 0; qy < q; ++qy) { u += QD(qy,dx) * Bt(dy,qy); }
+               Y(dx,dy,c,e) += u;
+            }
+         }
+         MFEM_SYNC_THREAD;
+      }
+   });
+}
+
+static void BLFEvalAssemble(const FiniteElementSpace &fes,
+                            const IntegrationRule &ir,
+                            const Array<int> &markers,
+                            const Vector &coeff,
+                            const bool normals,
+                            Vector &y)
+{
+   Mesh &mesh = *fes.GetMesh();
+   const int dim = mesh.Dimension();
+   const FiniteElement &el = *fes.GetBE(0);
+   const MemoryType mt = Device::GetDeviceMemoryType();
+   const DofToQuad &maps = el.GetDofToQuad(ir, DofToQuad::TENSOR);
+   const int d = maps.ndof, q = maps.nqpt;
+   int flags = FaceGeometricFactors::DETERMINANTS;
+   if (normals) { flags |= FaceGeometricFactors::NORMALS; }
+   const FaceGeometricFactors *geom = mesh.GetFaceGeometricFactors(
+                                         ir, flags, FaceType::Boundary, mt);
+   auto ker = (dim == 2) ? BLFEvalAssemble2D<> : BLFEvalAssemble3D<>;
+
+   if (dim==2)
+   {
+      if (d==1 && q==1) { ker=BLFEvalAssemble2D<1,1>; }
+      if (d==2 && q==2) { ker=BLFEvalAssemble2D<2,2>; }
+      if (d==3 && q==3) { ker=BLFEvalAssemble2D<3,3>; }
+      if (d==4 && q==4) { ker=BLFEvalAssemble2D<4,4>; }
+      if (d==5 && q==5) { ker=BLFEvalAssemble2D<5,5>; }
+      if (d==2 && q==3) { ker=BLFEvalAssemble2D<2,3>; }
+      if (d==3 && q==4) { ker=BLFEvalAssemble2D<3,4>; }
+      if (d==4 && q==5) { ker=BLFEvalAssemble2D<4,5>; }
+      if (d==5 && q==6) { ker=BLFEvalAssemble2D<5,6>; }
+   }
+
+   if (dim==3)
+   {
+      if (d==1 && q==1) { ker=BLFEvalAssemble3D<1,1>; }
+      if (d==2 && q==2) { ker=BLFEvalAssemble3D<2,2>; }
+      if (d==3 && q==3) { ker=BLFEvalAssemble3D<3,3>; }
+      if (d==4 && q==4) { ker=BLFEvalAssemble3D<4,4>; }
+      if (d==5 && q==5) { ker=BLFEvalAssemble3D<5,5>; }
+      if (d==2 && q==3) { ker=BLFEvalAssemble3D<2,3>; }
+      if (d==3 && q==4) { ker=BLFEvalAssemble3D<3,4>; }
+      if (d==4 && q==5) { ker=BLFEvalAssemble3D<4,5>; }
+      if (d==5 && q==6) { ker=BLFEvalAssemble3D<5,6>; }
+   }
+
+   MFEM_VERIFY(ker, "No kernel ndof " << d << " nqpt " << q);
+
+   const int vdim = fes.GetVDim();
+   const int nbe = fes.GetMesh()->GetNFbyType(FaceType::Boundary);
+   const int *M = markers.Read();
+   const double *B = maps.B.Read();
+   const double *detJ = geom->detJ.Read();
+   const double *n = geom->normal.Read();
+   const double *W = ir.GetWeights().Read();
+   double *Y = y.ReadWrite();
+   ker(vdim, nbe, d, q, normals, M, B, detJ, n, W, coeff, Y);
+}
+
+void BoundaryLFIntegrator::AssembleDevice(const FiniteElementSpace &fes,
+                                          const Array<int> &markers,
+                                          Vector &b)
+{
+   const FiniteElement &fe = *fes.GetBE(0);
+   const int qorder = oa * fe.GetOrder() + ob;
+   const Geometry::Type gtype = fe.GetGeomType();
+   const IntegrationRule &ir = IntRule ? *IntRule : IntRules.Get(gtype, qorder);
+   Mesh &mesh = *fes.GetMesh();
+
+   FaceQuadratureSpace qs(mesh, ir, FaceType::Boundary);
+   CoefficientVector coeff(Q, qs, CoefficientStorage::COMPRESSED);
+   BLFEvalAssemble(fes, ir, markers, coeff, false, b);
+}
+
+void BoundaryNormalLFIntegrator::AssembleDevice(const FiniteElementSpace &fes,
+                                                const Array<int> &markers,
+                                                Vector &b)
+{
+   const FiniteElement &fe = *fes.GetBE(0);
+   const int qorder = oa * fe.GetOrder() + ob;
+   const Geometry::Type gtype = fe.GetGeomType();
+   const IntegrationRule &ir = IntRule ? *IntRule : IntRules.Get(gtype, qorder);
+   Mesh &mesh = *fes.GetMesh();
+
+   FaceQuadratureSpace qs(mesh, ir, FaceType::Boundary);
+   CoefficientVector coeff(Q, qs, CoefficientStorage::COMPRESSED);
+   BLFEvalAssemble(fes, ir, markers, coeff, true, b);
+}
+
+} // namespace mfem
@@ -19,13 +19,13 @@ namespace mfem
 template<int T_D1D = 0, int T_Q1D = 0> static
 void DLFEvalAssemble2D(const int vdim, const int ne, const int d, const int q,
                       const int map_type, const int *markers, const double *b,
-                       const double *j, const double *weights,
+                       const double *detj, const double *weights,
                       const Vector &coeff, double *y)
 {
   const auto F = coeff.Read();
   const auto M = Reshape(markers, ne);
   const auto B = Reshape(b, q, d);
-   const auto J = Reshape(j, q, q, 2,2, ne);
+   const auto DETJ = Reshape(detj, q, q, ne);
   const auto W = Reshape(weights, q, q);
   const bool cst = coeff.Size() == vdim;
   const auto C = cst ? Reshape(F,vdim,1,1,1) : Reshape(F,vdim,q,q,ne);
@@ -55,19 +55,7 @@ void DLFEvalAssemble2D(const int vdim, const int ne, const int d, const int q,
         {
            MFEM_FOREACH_THREAD(y,y,q)
            {
-               double detJ;
-               if (map_type == FiniteElement::VALUE)
-               {
-                  const double J11 = J(x,y,0,0,e);
-                  const double J21 = J(x,y,1,0,e);
-                  const double J12 = J(x,y,0,1,e);
-                  const double J22 = J(x,y,1,1,e);
-                  detJ = J11 * J22 - J21 * J12;
-               }
-               else
-               {
-                  detJ = 1.0;
-               }
+               const double detJ = (map_type == FiniteElement::VALUE) ? DETJ(x,y,e) : 1.0;
               const double coeff_val = cst ? cst_val : C(c,x,y,e);
               QQ(y,x) = W(x,y) * coeff_val * detJ;
            }
@@ -100,13 +88,13 @@ void DLFEvalAssemble2D(const int vdim, const int ne, const int d, const int q,
 template<int T_D1D = 0, int T_Q1D = 0> static
 void DLFEvalAssemble3D(const int vdim, const int ne, const int d, const int q,
                       const int map_type, const int *markers, const double *b,
-                       const double *j, const double *weights,
+                       const double *detj, const double *weights,
                       const Vector &coeff, double *y)
 {
   const auto F = coeff.Read();
   const auto M = Reshape(markers, ne);
   const auto B = Reshape(b, q,d);
-   const auto J = Reshape(j, q,q,q, 3,3, ne);
+   const auto DETJ = Reshape(detj, q, q, q, ne);
   const auto W = Reshape(weights, q,q,q);
   const bool cst_coeff = coeff.Size() == vdim;
   const auto C = cst_coeff ? Reshape(F,vdim,1,1,1,1):Reshape(F,vdim,q,q,q,ne);
@@ -138,26 +126,7 @@ void DLFEvalAssemble3D(const int vdim, const int ne, const int d, const int q,
            {
               for (int z = 0; z < q; ++z)
               {
-                  double detJ;
-                  if (map_type == FiniteElement::VALUE)
-                  {
-                     const double J11 = J(x,y,z,0,0,e);
-                     const double J21 = J(x,y,z,1,0,e);
-                     const double J31 = J(x,y,z,2,0,e);
-                     const double J12 = J(x,y,z,0,1,e);
-                     const double J22 = J(x,y,z,1,1,e);
-                     const double J32 = J(x,y,z,2,1,e);
-                     const double J13 = J(x,y,z,0,2,e);
-                     const double J23 = J(x,y,z,1,2,e);
-                     const double J33 = J(x,y,z,2,2,e);
-                     detJ = J11 * (J22 * J33 - J32 * J23) -
-                     /* */  J21 * (J12 * J33 - J32 * J13) +
-                     /* */  J31 * (J12 * J23 - J22 * J13);
-                  }
-                  else
-                  {
-                     detJ = 1.0;
-                  }
+                  const double detJ = (map_type == FiniteElement::VALUE) ? DETJ(x,y,z,e) : 1.0;
                  const double coeff_val = cst_coeff ? cst_val : C(c,x,y,z,e);
                  QQQ(z,y,x) = W(x,y,z) * coeff_val * detJ;
               }
@@ -222,7 +191,7 @@ static void DLFEvalAssemble(const FiniteElementSpace &fes,
   const MemoryType mt = Device::GetDeviceMemoryType();
   const DofToQuad &maps = el.GetDofToQuad(*ir, DofToQuad::TENSOR);
   const int d = maps.ndof, q = maps.nqpt;
-   constexpr int flags = GeometricFactors::JACOBIANS;
+   constexpr int flags = GeometricFactors::DETERMINANTS;
   const GeometricFactors *geom = mesh->GetGeometricFactors(*ir, flags, mt);
   const int map_type = fes.GetFE(0)->GetMapType();
   decltype(&DLFEvalAssemble2D<>) ker =
@@ -260,10 +229,10 @@ static void DLFEvalAssemble(const FiniteElementSpace &fes,
   const int ne = fes.GetMesh()->GetNE();
   const int *M = markers.Read();
   const double *B = maps.B.Read();
-   const double *J = geom->J.Read();
+   const double *detJ = geom->detJ.Read();
   const double *W = ir->GetWeights().Read();
   double *Y = y.ReadWrite();
-   ker(vdim, ne, d, q, map_type, M, B, J, W, coeff, Y);
+   ker(vdim, ne, d, q, map_type, M, B, detJ, W, coeff, Y);
 }

 void DomainLFIntegrator::AssembleDevice(const FiniteElementSpace &fes,
@@ -274,42 +243,9 @@ void DomainLFIntegrator::AssembleDevice(const FiniteElementSpace &fes,
   const int qorder = oa * fe.GetOrder() + ob;
   const Geometry::Type gtype = fe.GetGeomType();
   const IntegrationRule *ir = IntRule ? IntRule : &IntRules.Get(gtype, qorder);
-   const int nq = ir->GetNPoints(), ne = fes.GetMesh()->GetNE();

-   Vector coeff;
-   if (ConstantCoefficient *cQ =
-          dynamic_cast<ConstantCoefficient*>(&Q))
-   {
-      coeff.SetSize(1);
-      coeff(0) = cQ->constant;
-   }
-   else if (QuadratureFunctionCoefficient *qfQ =
-               dynamic_cast<QuadratureFunctionCoefficient*>(&Q))
-   {
-      const QuadratureFunction &qfun = qfQ->GetQuadFunction();
-      MFEM_VERIFY(qfun.Size() == fes.GetVDim()*ne*nq,
-                  "Incompatible QuadratureFunction dimension \n");
-      MFEM_VERIFY(ir == &qfun.GetSpace()->GetElementIntRule(0),
-                  "IntegrationRule used within integrator and in"
-                  " QuadratureFunction appear to be different.\n");
-      qfun.Read();
-      coeff.MakeRef(const_cast<QuadratureFunction&>(qfun),0);
-   }
-   else
-   {
-      coeff.SetSize(nq * ne);
-      auto C = Reshape(coeff.HostWrite(), nq, ne);
-      for (int e = 0; e < ne; ++e)
-      {
-         ElementTransformation& Tr = *fes.GetElementTransformation(e);
-         for (int q = 0; q < nq; ++q)
-         {
-            const IntegrationPoint &ip = ir->IntPoint(q);
-            Tr.SetIntPoint(&ip);
-            C(q,e) = Q.Eval(Tr, ip);
-         }
-      }
-   }
+   QuadratureSpace qs(*fes.GetMesh(), *ir);
+   CoefficientVector coeff(Q, qs, CoefficientStorage::COMPRESSED);
   DLFEvalAssemble(fes, ir, markers, coeff, b);
 }

@@ -317,48 +253,14 @@ void VectorDomainLFIntegrator::AssembleDevice(const FiniteElementSpace &fes,
                                              const Array<int> &markers,
                                              Vector &b)
 {
-   const int vdim = fes.GetVDim();
   const FiniteElement &fe = *fes.GetFE(0);
   const int qorder = 2 * fe.GetOrder();
   const Geometry::Type gtype = fe.GetGeomType();
   const IntegrationRule *ir = IntRule ? IntRule : &IntRules.Get(gtype, qorder);
-   const int nq = ir->GetNPoints(), ne = fes.GetMesh()->GetNE();

-   if (VectorConstantCoefficient *vcQ =
-          dynamic_cast<VectorConstantCoefficient*>(&Q))
-   {
-      Qvec = vcQ->GetVec();
-   }
-   else if (VectorQuadratureFunctionCoefficient *vQ =
-               dynamic_cast<VectorQuadratureFunctionCoefficient*>(&Q))
-   {
-      const QuadratureFunction &qfun = vQ->GetQuadFunction();
-      MFEM_VERIFY(qfun.Size() == vdim*ne*nq,
-                  "Incompatible QuadratureFunction dimension \n");
-      MFEM_VERIFY(ir == &qfun.GetSpace()->GetElementIntRule(0),
-                  "IntegrationRule used within integrator and in"
-                  " QuadratureFunction appear to be different.\n");
-      qfun.Read();
-      Qvec.MakeRef(const_cast<QuadratureFunction&>(qfun),0);
-   }
-   else
-   {
-      Vector qv(vdim);
-      Qvec.SetSize(vdim * nq * ne);
-      auto C = Reshape(Qvec.HostWrite(), vdim, nq, ne);
-      for (int e = 0; e < ne; ++e)
-      {
-         ElementTransformation& Tr = *fes.GetElementTransformation(e);
-         for (int q = 0; q < nq; ++q)
-         {
-            const IntegrationPoint &ip = ir->IntPoint(q);
-            Tr.SetIntPoint(&ip);
-            Q.Eval(qv, Tr, ip);
-            for (int c=0; c<vdim; ++c) { C(c,q,e) = qv[c]; }
-         }
-      }
-   }
-   DLFEvalAssemble(fes, ir, markers, Qvec, b);
+   QuadratureSpace qs(*fes.GetMesh(), *ir);
+   CoefficientVector coeff(Q, qs, CoefficientStorage::COMPRESSED);
+   DLFEvalAssemble(fes, ir, markers, coeff, b);
 }

 } // namespace mfem
@@ -324,108 +324,24 @@ void DomainLFGradIntegrator::AssembleDevice(const FiniteElementSpace &fes,
   const int qorder = 2 * fe.GetOrder();
   const Geometry::Type gtype = fe.GetGeomType();
   const IntegrationRule *ir = IntRule ? IntRule : &IntRules.Get(gtype, qorder);
-   const int nq = ir->GetNPoints(), ne = fes.GetMesh()->GetNE();

-   if (VectorConstantCoefficient *vcQ =
-          dynamic_cast<VectorConstantCoefficient*>(&Q))
-   {
-      Qvec = vcQ->GetVec();
-   }
-   else if (VectorQuadratureFunctionCoefficient *vqfQ =
-               dynamic_cast<VectorQuadratureFunctionCoefficient*>(&Q))
-   {
-      const QuadratureFunction &qfun = vqfQ->GetQuadFunction();
-      MFEM_VERIFY(qfun.Size() == ne*nq,
-                  "Incompatible QuadratureFunction dimension \n");
-      MFEM_VERIFY(ir == &qfun.GetSpace()->GetElementIntRule(0),
-                  "IntegrationRule used within integrator and in"
-                  " QuadratureFunction appear to be different.\n");
-      qfun.Read();
-      Qvec.MakeRef(const_cast<QuadratureFunction&>(qfun),0);
-   }
-   else
-   {
-      const int qvdim = Q.GetVDim();
-      Vector qvec(qvdim);
-      Qvec.SetSize(qvdim * nq * ne);
-      auto C = Reshape(Qvec.HostWrite(), qvdim, nq, ne);
-      for (int e = 0; e < ne; ++e)
-      {
-         ElementTransformation& Tr = *fes.GetElementTransformation(e);
-         for (int q = 0; q < nq; ++q)
-         {
-            const IntegrationPoint &ip = ir->IntPoint(q);
-            Tr.SetIntPoint(&ip);
-            Q.Eval(qvec, Tr, ip);
-            for (int c=0; c < qvdim; ++c)
-            {
-               C(c,q,e) = qvec[c];
-            }
-         }
-      }
-   }
-   DLFGradAssemble(fes, ir, markers, Qvec, b);
+   QuadratureSpace qs(*fes.GetMesh(), *ir);
+   CoefficientVector coeff(Q, qs, CoefficientStorage::COMPRESSED);
+   DLFGradAssemble(fes, ir, markers, coeff, b);
 }

 void VectorDomainLFGradIntegrator::AssembleDevice(const FiniteElementSpace &fes,
                                                  const Array<int> &markers,
                                                  Vector &b)
 {
-   const int vdim = fes.GetVDim();
   const FiniteElement &fe = *fes.GetFE(0);
   const int qorder = 2 * fe.GetOrder();
   const Geometry::Type gtype = fe.GetGeomType();
   const IntegrationRule *ir = IntRule ? IntRule : &IntRules.Get(gtype, qorder);
-   const int nq = ir->GetNPoints(), ne = fes.GetMesh()->GetNE(),
-             ns = fes.GetMesh()->SpaceDimension();

-   if (VectorConstantCoefficient *vcQ =
-          dynamic_cast<VectorConstantCoefficient*>(&Q))
-   {
-      Qvec = vcQ->GetVec();
-   }
-   else if (QuadratureFunctionCoefficient *qfQ =
-               dynamic_cast<QuadratureFunctionCoefficient*>(&Q))
-   {
-      const QuadratureFunction &qfun = qfQ->GetQuadFunction();
-      MFEM_VERIFY(qfun.Size() == ne*nq,
-                  "Incompatible QuadratureFunction dimension \n");
-      MFEM_VERIFY(ir == &qfun.GetSpace()->GetElementIntRule(0),
-                  "IntegrationRule used within integrator and in"
-                  " QuadratureFunction appear to be different.\n");
-      qfun.Read();
-      Qvec.MakeRef(const_cast<QuadratureFunction&>(qfun),0);
-   }
-   else if (VectorQuadratureFunctionCoefficient* vqfQ =
-               dynamic_cast<VectorQuadratureFunctionCoefficient*>(&Q))
-   {
-      const QuadratureFunction &qFun = vqfQ->GetQuadFunction();
-      MFEM_VERIFY(qFun.Size() == vdim * ns * nq * ne,
-                  "Incompatible QuadratureFunction dimension \n");
-      MFEM_VERIFY(ir == &qFun.GetSpace()->GetElementIntRule(0),
-                  "IntegrationRule used within integrator and in"
-                  " QuadratureFunction appear to be different");
-      qFun.Read();
-      Qvec.MakeRef(const_cast<QuadratureFunction &>(qFun),0);
-   }
-   else
-   {
-      Vector qvec(vdim);
-      Qvec.SetSize(vdim * nq * ne);
-      auto C = Reshape(Qvec.HostWrite(), vdim, nq, ne);
-      for (int e = 0; e < ne; ++e)
-      {
-         ElementTransformation &Tr = *fes.GetElementTransformation(e);
-         for (int q = 0; q < nq; ++q)
-         {
-            const IntegrationPoint &ip = ir->IntPoint(q);
-            Tr.SetIntPoint(&ip);
-            Q.Eval(qvec, Tr, ip);
-            for (int c = 0; c<vdim; ++c) { C(c,q,e) = qvec[c]; }
-         }
-      }
-   }
-   DLFGradAssemble(fes, ir, markers, Qvec, b);
+   QuadratureSpace qs(*fes.GetMesh(), *ir);
+   CoefficientVector coeff(Q, qs, CoefficientStorage::COMPRESSED);
+   DLFGradAssemble(fes, ir, markers, coeff, b);
 }

 } // namespace mfem
@@ -365,8 +365,11 @@ FiniteElementSpace &LORBase::GetFESpace() const

 void LORBase::AssembleSystem(BilinearForm &a_ho, const Array<int> &ess_dofs)
 {
-   A.Clear();
-   delete a;
+   if (a)
+   {
+      A.Clear();
+      delete a;
+   }
   if (BatchedLORAssembly::FormIsSupported(a_ho))
   {
      // Skip forming the space
@@ -467,7 +470,9 @@ void LORDiscretization::FormLORSpace()
   mesh = new Mesh(Mesh::MakeRefined(mesh_ho, refinements, ref_type));

   fec = fes_ho.FEColl()->Clone(GetLOROrder());
-   fes = new FiniteElementSpace(mesh, fec);
+   const int vdim = fes_ho.GetVDim();
+   const Ordering::Type ordering = fes_ho.GetOrdering();
+   fes = new FiniteElementSpace(mesh, fec, vdim, ordering);
   SetupProlongationAndRestriction();
 }

@@ -511,8 +516,9 @@ void ParLORDiscretization::FormLORSpace()
   mesh = pmesh;

   fec = pfes_ho.FEColl()->Clone(GetLOROrder());
-   ParFiniteElementSpace *pfes = new ParFiniteElementSpace(pmesh, fec);
-   fes = pfes;
+   const int vdim = fes_ho.GetVDim();
+   const Ordering::Type ordering = fes_ho.GetOrdering();
+   fes = new ParFiniteElementSpace(pmesh, fec, vdim, ordering);
   SetupProlongationAndRestriction();
 }

@@ -95,7 +95,7 @@ protected:
   /// Returns the order of the LOR space. 1 for H1 or ND, 0 for L2 or RT.
   int GetLOROrder() const;

-   /// Construct the LOR space (overriden for serial and parallel versions).
+   /// Construct the LOR space (overridden for serial and parallel versions).
   virtual void FormLORSpace() = 0;

   /// Construct the LORBase object for the given FE space and refinement type.
@@ -13,6 +13,9 @@
 #include "../../general/forall.hpp"
 #include "../../fem/pbilinearform.hpp"

+#define MFEM_NVTX_COLOR DeepSkyBlue
+#include "../../general/nvtx.hpp"
+
 namespace mfem
 {

@@ -78,8 +81,10 @@ void BatchedLOR_ADS::Form3DFaceToEdge(Array<int> &face2edge)
   }
 }

-void BatchedLOR_ADS::FormCurlMatrix()
+void BatchedLOR_ADS::FormCurlMatrixLocal()
 {
+   NVTX("Discrete Curl");
+
   // The curl matrix maps from LOR edges to LOR faces. Given a quadrilateral
   // face (defined by its four edges) f_i = (e_j1, e_j2, e_j3, e_j4), the
   // matrix has nonzeros A(i, jk), so there are always exactly four nonzeros
@@ -87,10 +92,9 @@ void BatchedLOR_ADS::FormCurlMatrix()
   const int nface_dof = face_fes.GetNDofs();
   const int nedge_dof = edge_fes.GetNDofs();

-   SparseMatrix C_local;
   C_local.OverrideSize(nface_dof, nedge_dof);
-
-   C_local.GetMemoryI().New(nedge_dof+1, Device::GetDeviceMemoryType());
+   EnsureCapacity(C_local.GetMemoryI(), nedge_dof+1,
+                  Device::GetDeviceMemoryType());
   // Each row always has four nonzeros
   const int nnz = 4*nedge_dof;
   auto I = C_local.WriteI();
@@ -120,8 +124,8 @@ void BatchedLOR_ADS::FormCurlMatrix()
   const auto f2e = Reshape(face2edge.Read(), 4, nface_per_el);

   // Fill J and data
-   C_local.GetMemoryJ().New(nnz, Device::GetDeviceMemoryType());
-   C_local.GetMemoryData().New(nnz, Device::GetDeviceMemoryType());
+   EnsureCapacity(C_local.GetMemoryJ(), nnz, Device::GetDeviceMemoryType());
+   EnsureCapacity(C_local.GetMemoryData(), nnz, Device::GetDeviceMemoryType());

   auto J = C_local.WriteJ();
   auto V = C_local.WriteData();
@@ -146,6 +150,11 @@ void BatchedLOR_ADS::FormCurlMatrix()
         V[i*4 + k] = sgn*sgn_f*sgn_e;
      }
   });
+}
+
+void BatchedLOR_ADS::FormCurlMatrix()
+{
+   FormCurlMatrixLocal();

   // Create a block diagonal parallel matrix
   OperatorHandle C_diag(Operator::Hypre_ParCSR);
@@ -179,6 +188,8 @@ void BatchedLOR_ADS::FormCurlMatrix()
   }
   C->CopyRowStarts();
   C->CopyColStarts();
+
+   C_local.Clear();
 }

 HypreParMatrix *BatchedLOR_ADS::StealCurlMatrix()
@@ -36,7 +36,9 @@ protected:
   ND_FECollection edge_fec; ///< The associated Nedelec collection.
   ParFiniteElementSpace edge_fes; ///< The associated Nedelec space.
   BatchedLOR_AMS ams; ///< The associated AMS object.
-   HypreParMatrix *C; ///< The discrete curl matrix.
+   HypreParMatrix *C = nullptr; ///< The discrete curl matrix.
+
+   SparseMatrix C_local;

   /// Form the local elementwise discrete curl matrix.
   void Form3DFaceToEdge(Array<int> &face2edge);
@@ -64,6 +66,9 @@ public:

   /// Form the discrete curl matrix (not part of the public API).
   void FormCurlMatrix();
+
+   void FormCurlMatrixLocal();
+
   ~BatchedLOR_ADS();
 };

--- a/Show More
+++ b/Show More