impls/plex/plexfem.c

#include <petsc-private/dmpleximpl.h>   /*I      "petscdmplex.h"   I*/
#include <petscsf.h>

#include <petsc-private/petscfeimpl.h>
#include <petsc-private/petscfvimpl.h>

#undef __FUNCT__
#define __FUNCT__ "DMPlexGetScale"
PetscErrorCode DMPlexGetScale(DM dm, PetscUnit unit, PetscReal *scale)
{
  DM_Plex *mesh = (DM_Plex*) dm->data;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(dm, DM_CLASSID, 1);
  PetscValidPointer(scale, 3);
  *scale = mesh->scale[unit];
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "DMPlexSetScale"
PetscErrorCode DMPlexSetScale(DM dm, PetscUnit unit, PetscReal scale)
{
  DM_Plex *mesh = (DM_Plex*) dm->data;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(dm, DM_CLASSID, 1);
  mesh->scale[unit] = scale;
  PetscFunctionReturn(0);
}

PETSC_STATIC_INLINE PetscInt epsilon(PetscInt i, PetscInt j, PetscInt k)
{
  switch (i) {
  case 0:
    switch (j) {
    case 0: return 0;
    case 1:
      switch (k) {
      case 0: return 0;
      case 1: return 0;
      case 2: return 1;
      }
    case 2:
      switch (k) {
      case 0: return 0;
      case 1: return -1;
      case 2: return 0;
      }
    }
  case 1:
    switch (j) {
    case 0:
      switch (k) {
      case 0: return 0;
      case 1: return 0;
      case 2: return -1;
      }
    case 1: return 0;
    case 2:
      switch (k) {
      case 0: return 1;
      case 1: return 0;
      case 2: return 0;
      }
    }
  case 2:
    switch (j) {
    case 0:
      switch (k) {
      case 0: return 0;
      case 1: return 1;
      case 2: return 0;
      }
    case 1:
      switch (k) {
      case 0: return -1;
      case 1: return 0;
      case 2: return 0;
      }
    case 2: return 0;
    }
  }
  return 0;
}

#undef __FUNCT__
#define __FUNCT__ "DMPlexCreateRigidBody"
/*@C
  DMPlexCreateRigidBody - create rigid body modes from coordinates

  Collective on DM

  Input Arguments:
+ dm - the DM
. section - the local section associated with the rigid field, or NULL for the default section
- globalSection - the global section associated with the rigid field, or NULL for the default section

  Output Argument:
. sp - the null space

  Note: This is necessary to take account of Dirichlet conditions on the displacements

  Level: advanced

.seealso: MatNullSpaceCreate()
@*/
PetscErrorCode DMPlexCreateRigidBody(DM dm, PetscSection section, PetscSection globalSection, MatNullSpace *sp)
{
  MPI_Comm       comm;
  Vec            coordinates, localMode, mode[6];
  PetscSection   coordSection;
  PetscScalar   *coords;
  PetscInt       dim, vStart, vEnd, v, n, m, d, i, j;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  ierr = PetscObjectGetComm((PetscObject)dm,&comm);CHKERRQ(ierr);
  ierr = DMGetDimension(dm, &dim);CHKERRQ(ierr);
  if (dim == 1) {
    ierr = MatNullSpaceCreate(comm, PETSC_TRUE, 0, NULL, sp);CHKERRQ(ierr);
    PetscFunctionReturn(0);
  }
  if (!section)       {ierr = DMGetDefaultSection(dm, &section);CHKERRQ(ierr);}
  if (!globalSection) {ierr = DMGetDefaultGlobalSection(dm, &globalSection);CHKERRQ(ierr);}
  ierr = PetscSectionGetConstrainedStorageSize(globalSection, &n);CHKERRQ(ierr);
  ierr = DMPlexGetDepthStratum(dm, 0, &vStart, &vEnd);CHKERRQ(ierr);
  ierr = DMGetCoordinateSection(dm, &coordSection);CHKERRQ(ierr);
  ierr = DMGetCoordinatesLocal(dm, &coordinates);CHKERRQ(ierr);
  m    = (dim*(dim+1))/2;
  ierr = VecCreate(comm, &mode[0]);CHKERRQ(ierr);
  ierr = VecSetSizes(mode[0], n, PETSC_DETERMINE);CHKERRQ(ierr);
  ierr = VecSetUp(mode[0]);CHKERRQ(ierr);
  for (i = 1; i < m; ++i) {ierr = VecDuplicate(mode[0], &mode[i]);CHKERRQ(ierr);}
  /* Assume P1 */
  ierr = DMGetLocalVector(dm, &localMode);CHKERRQ(ierr);
  for (d = 0; d < dim; ++d) {
    PetscScalar values[3] = {0.0, 0.0, 0.0};

    values[d] = 1.0;
    ierr      = VecSet(localMode, 0.0);CHKERRQ(ierr);
    for (v = vStart; v < vEnd; ++v) {
      ierr = DMPlexVecSetClosure(dm, section, localMode, v, values, INSERT_VALUES);CHKERRQ(ierr);
    }
    ierr = DMLocalToGlobalBegin(dm, localMode, INSERT_VALUES, mode[d]);CHKERRQ(ierr);
    ierr = DMLocalToGlobalEnd(dm, localMode, INSERT_VALUES, mode[d]);CHKERRQ(ierr);
  }
  ierr = VecGetArray(coordinates, &coords);CHKERRQ(ierr);
  for (d = dim; d < dim*(dim+1)/2; ++d) {
    PetscInt i, j, k = dim > 2 ? d - dim : d;

    ierr = VecSet(localMode, 0.0);CHKERRQ(ierr);
    for (v = vStart; v < vEnd; ++v) {
      PetscScalar values[3] = {0.0, 0.0, 0.0};
      PetscInt    off;

      ierr = PetscSectionGetOffset(coordSection, v, &off);CHKERRQ(ierr);
      for (i = 0; i < dim; ++i) {
        for (j = 0; j < dim; ++j) {
          values[j] += epsilon(i, j, k)*PetscRealPart(coords[off+i]);
        }
      }
      ierr = DMPlexVecSetClosure(dm, section, localMode, v, values, INSERT_VALUES);CHKERRQ(ierr);
    }
    ierr = DMLocalToGlobalBegin(dm, localMode, INSERT_VALUES, mode[d]);CHKERRQ(ierr);
    ierr = DMLocalToGlobalEnd(dm, localMode, INSERT_VALUES, mode[d]);CHKERRQ(ierr);
  }
  ierr = VecRestoreArray(coordinates, &coords);CHKERRQ(ierr);
  ierr = DMRestoreLocalVector(dm, &localMode);CHKERRQ(ierr);
  for (i = 0; i < dim; ++i) {ierr = VecNormalize(mode[i], NULL);CHKERRQ(ierr);}
  /* Orthonormalize system */
  for (i = dim; i < m; ++i) {
    PetscScalar dots[6];

    ierr = VecMDot(mode[i], i, mode, dots);CHKERRQ(ierr);
    for (j = 0; j < i; ++j) dots[j] *= -1.0;
    ierr = VecMAXPY(mode[i], i, dots, mode);CHKERRQ(ierr);
    ierr = VecNormalize(mode[i], NULL);CHKERRQ(ierr);
  }
  ierr = MatNullSpaceCreate(comm, PETSC_FALSE, m, mode, sp);CHKERRQ(ierr);
  for (i = 0; i< m; ++i) {ierr = VecDestroy(&mode[i]);CHKERRQ(ierr);}
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "DMPlexSetMaxProjectionHeight"
/*@
  DMPlexSetMaxProjectionHeight - In DMPlexProjectXXXLocal() functions, the projected values of a basis function's dofs
  are computed by associating the basis function with one of the mesh points in its transitively-closed support, and
  evaluating the dual space basis of that point.  A basis function is associated with the point in its
  transitively-closed support whose mesh height is highest (w.r.t. DAG height), but not greater than the maximum
  projection height, which is set with this function.  By default, the maximum projection height is zero, which means
  that only mesh cells are used to project basis functions.  A height of one, for example, evaluates a cell-interior
  basis functions using its cells dual space basis, but all other basis functions with the dual space basis of a face.

  Input Parameters:
+ dm - the DMPlex object
- height - the maximum projection height >= 0

  Level: advanced

.seealso: DMPlexGetMaxProjectionHeight(), DMPlexProjectFunctionLocal(), DMPlexProjectFunctionLabelLocal()
@*/
PetscErrorCode DMPlexSetMaxProjectionHeight(DM dm, PetscInt height)
{
  DM_Plex *plex = (DM_Plex *) dm->data;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(dm, DM_CLASSID, 1);
  plex->maxProjectionHeight = height;
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "DMPlexGetMaxProjectionHeight"
/*@
  DMPlexGetMaxProjectionHeight - Get the maximum height (w.r.t. DAG) of mesh points used to evaluate dual bases in
  DMPlexProjectXXXLocal() functions.

  Input Parameters:
. dm - the DMPlex object

  Output Parameters:
. height - the maximum projection height

  Level: intermediate

.seealso: DMPlexSetMaxProjectionHeight(), DMPlexProjectFunctionLocal(), DMPlexProjectFunctionLabelLocal()
@*/
PetscErrorCode DMPlexGetMaxProjectionHeight(DM dm, PetscInt *height)
{
  DM_Plex *plex = (DM_Plex *) dm->data;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(dm, DM_CLASSID, 1);
  *height = plex->maxProjectionHeight;
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "DMPlexProjectFunctionLabelLocal"
PetscErrorCode DMPlexProjectFunctionLabelLocal(DM dm, DMLabel label, PetscInt numIds, const PetscInt ids[], void (**funcs)(const PetscReal [], PetscScalar *, void *), void **ctxs, InsertMode mode, Vec localX)
{
  PetscDualSpace *sp, *cellsp;
  PetscInt       *numComp;
  PetscSection    section;
  PetscScalar    *values;
  PetscBool      *fieldActive;
  PetscInt        numFields, dim, dimEmbed, spDim, totDim = 0, numValues, pStart, pEnd, cStart, cEnd, cEndInterior, f, d, v, i, comp, maxHeight, h;
  PetscErrorCode  ierr;

  PetscFunctionBegin;
  ierr = DMPlexGetHeightStratum(dm, 0, &cStart, &cEnd);CHKERRQ(ierr);
  ierr = DMPlexGetHybridBounds(dm, &cEndInterior, NULL, NULL, NULL);CHKERRQ(ierr);
  cEnd = cEndInterior < 0 ? cEnd : cEndInterior;
  if (cEnd <= cStart) PetscFunctionReturn(0);
  ierr = DMGetDimension(dm, &dim);CHKERRQ(ierr);
  ierr = DMGetCoordinateDim(dm, &dimEmbed);CHKERRQ(ierr);
  ierr = DMGetDefaultSection(dm, &section);CHKERRQ(ierr);
  ierr = PetscSectionGetNumFields(section, &numFields);CHKERRQ(ierr);
  ierr = PetscMalloc2(numFields,&sp,numFields,&numComp);CHKERRQ(ierr);
  ierr = DMPlexGetMaxProjectionHeight(dm,&maxHeight);CHKERRQ(ierr);
  if (maxHeight < 0 || maxHeight > dim) {SETERRQ2(PETSC_COMM_SELF,PETSC_ERR_USER,"maximum projection height %d not in [0, %d)\n", maxHeight,dim);}
  if (maxHeight > 0) {ierr = PetscMalloc1(numFields,&cellsp);CHKERRQ(ierr);}
  else               {cellsp = sp;}
  for (h = 0; h <= maxHeight; h++) {
    ierr = DMPlexGetHeightStratum(dm, h, &pStart, &pEnd);CHKERRQ(ierr);
    if (!h) {pStart = cStart; pEnd = cEnd;}
    if (pEnd <= pStart) continue;
    totDim = 0;
    for (f = 0; f < numFields; ++f) {
      PetscObject  obj;
      PetscClassId id;

      ierr = DMGetField(dm, f, &obj);CHKERRQ(ierr);
      ierr = PetscObjectGetClassId(obj, &id);CHKERRQ(ierr);
      if (id == PETSCFE_CLASSID) {
        PetscFE fe = (PetscFE) obj;

        ierr = PetscFEGetNumComponents(fe, &numComp[f]);CHKERRQ(ierr);
        if (!h) {
          ierr = PetscFEGetDualSpace(fe, &cellsp[f]);CHKERRQ(ierr);
          sp[f] = cellsp[f];
          ierr = PetscObjectReference((PetscObject) sp[f]);CHKERRQ(ierr);
        } else {
          ierr = PetscDualSpaceGetHeightSubspace(cellsp[f], h, &sp[f]);CHKERRQ(ierr);
          if (!sp[f]) continue;
        }
      } else if (id == PETSCFV_CLASSID) {
        PetscFV         fv = (PetscFV) obj;
        PetscQuadrature q;

        ierr = PetscFVGetNumComponents(fv, &numComp[f]);CHKERRQ(ierr);
        ierr = PetscFVGetQuadrature(fv, &q);CHKERRQ(ierr);
        ierr = PetscDualSpaceCreate(PetscObjectComm((PetscObject) fv), &sp[f]);CHKERRQ(ierr);
        ierr = PetscDualSpaceSetDM(sp[f], dm);CHKERRQ(ierr);
        ierr = PetscDualSpaceSetType(sp[f], PETSCDUALSPACESIMPLE);CHKERRQ(ierr);
        ierr = PetscDualSpaceSimpleSetDimension(sp[f], 1);CHKERRQ(ierr);
        ierr = PetscDualSpaceSimpleSetFunctional(sp[f], 0, q);CHKERRQ(ierr);
      } else SETERRQ1(PetscObjectComm((PetscObject)dm), PETSC_ERR_ARG_WRONG, "Unknown discretization type for field %d", f);
      ierr = PetscDualSpaceGetDimension(sp[f], &spDim);CHKERRQ(ierr);
      totDim += spDim*numComp[f];
    }
    ierr = DMPlexVecGetClosure(dm, section, localX, pStart, &numValues, NULL);CHKERRQ(ierr);
    if (numValues != totDim) SETERRQ2(PETSC_COMM_SELF, PETSC_ERR_ARG_WRONG, "The section point closure size %d != dual space dimension %d", numValues, totDim);
    if (!totDim) continue;
    ierr = DMGetWorkArray(dm, numValues, PETSC_SCALAR, &values);CHKERRQ(ierr);
    ierr = DMGetWorkArray(dm, numFields, PETSC_BOOL, &fieldActive);CHKERRQ(ierr);
    for (f = 0; f < numFields; ++f) fieldActive[f] = (funcs[f] && sp[f]) ? PETSC_TRUE : PETSC_FALSE;
    for (i = 0; i < numIds; ++i) {
      IS              pointIS;
      const PetscInt *points;
      PetscInt        n, p;

      ierr = DMLabelGetStratumIS(label, ids[i], &pointIS);CHKERRQ(ierr);
      ierr = ISGetLocalSize(pointIS, &n);CHKERRQ(ierr);
      ierr = ISGetIndices(pointIS, &points);CHKERRQ(ierr);
      for (p = 0; p < n; ++p) {
        const PetscInt    point = points[p];
        PetscFECellGeom   geom;

        if ((point < pStart) || (point >= pEnd)) continue;
        ierr          = DMPlexComputeCellGeometryFEM(dm, point, NULL, geom.v0, geom.J, NULL, &geom.detJ);CHKERRQ(ierr);
        geom.dim      = dim - h;
        geom.dimEmbed = dimEmbed;
        for (f = 0, v = 0; f < numFields; ++f) {
          void * const ctx = ctxs ? ctxs[f] : NULL;

          if (!sp[f]) continue;
          ierr = PetscDualSpaceGetDimension(sp[f], &spDim);CHKERRQ(ierr);
          for (d = 0; d < spDim; ++d) {
            if (funcs[f]) {
              ierr = PetscDualSpaceApply(sp[f], d, &geom, numComp[f], funcs[f], ctx, &values[v]);CHKERRQ(ierr);
            } else {
              for (comp = 0; comp < numComp[f]; ++comp) values[v+comp] = 0.0;
            }
            v += numComp[f];
          }
        }
        ierr = DMPlexVecSetFieldClosure_Internal(dm, section, localX, fieldActive, point, values, mode);CHKERRQ(ierr);
      }
      ierr = ISRestoreIndices(pointIS, &points);CHKERRQ(ierr);
      ierr = ISDestroy(&pointIS);CHKERRQ(ierr);
    }
    if (h) {
      for (f = 0; f < numFields; ++f) {ierr = PetscDualSpaceDestroy(&sp[f]);CHKERRQ(ierr);}
    }
  }
  ierr = DMRestoreWorkArray(dm, numValues, PETSC_SCALAR, &values);CHKERRQ(ierr);
  ierr = DMRestoreWorkArray(dm, numFields, PETSC_BOOL, &fieldActive);CHKERRQ(ierr);
  for (f = 0; f < numFields; ++f) {ierr = PetscDualSpaceDestroy(&sp[f]);CHKERRQ(ierr);}
  ierr = PetscFree2(sp, numComp);CHKERRQ(ierr);
  if (maxHeight > 0) {
    ierr = PetscFree(cellsp);CHKERRQ(ierr);
  }
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "DMPlexProjectFunctionLocal"
PetscErrorCode DMPlexProjectFunctionLocal(DM dm, void (**funcs)(const PetscReal [], PetscScalar *, void *), void **ctxs, InsertMode mode, Vec localX)
{
  PetscDualSpace *sp, *cellsp;
  PetscInt       *numComp;
  PetscSection    section;
  PetscScalar    *values;
  PetscInt        numFields, dim, dimEmbed, spDim, totDim = 0, numValues, pStart, pEnd, p, cStart, cEnd, cEndInterior, f, d, v, comp, h, maxHeight;
  PetscErrorCode  ierr;

  PetscFunctionBegin;
  ierr = DMPlexGetHeightStratum(dm, 0, &cStart, &cEnd);CHKERRQ(ierr);
  ierr = DMPlexGetHybridBounds(dm, &cEndInterior, NULL, NULL, NULL);CHKERRQ(ierr);
  cEnd = cEndInterior < 0 ? cEnd : cEndInterior;
  if (cEnd <= cStart) PetscFunctionReturn(0);
  ierr = DMGetDefaultSection(dm, &section);CHKERRQ(ierr);
  ierr = PetscSectionGetNumFields(section, &numFields);CHKERRQ(ierr);
  ierr = PetscMalloc2(numFields, &sp, numFields, &numComp);CHKERRQ(ierr);
  ierr = DMGetDimension(dm, &dim);CHKERRQ(ierr);
  ierr = DMGetCoordinateDim(dm, &dimEmbed);CHKERRQ(ierr);
  ierr = DMPlexGetMaxProjectionHeight(dm,&maxHeight);CHKERRQ(ierr);
  if (maxHeight < 0 || maxHeight > dim) {SETERRQ2(PETSC_COMM_SELF,PETSC_ERR_USER,"maximum projection height %d not in [0, %d)\n", maxHeight,dim);}
  if (maxHeight > 0) {
    ierr = PetscMalloc1(numFields,&cellsp);CHKERRQ(ierr);
  }
  else {
    cellsp = sp;
  }
  for (h = 0; h <= maxHeight; h++) {
    ierr = DMPlexGetHeightStratum(dm, h, &pStart, &pEnd);CHKERRQ(ierr);
    if (!h) {pStart = cStart; pEnd = cEnd;}
    if (pEnd <= pStart) continue;
    totDim = 0;
    for (f = 0; f < numFields; ++f) {
      PetscObject  obj;
      PetscClassId id;

      ierr = DMGetField(dm, f, &obj);CHKERRQ(ierr);
      ierr = PetscObjectGetClassId(obj, &id);CHKERRQ(ierr);
      if (id == PETSCFE_CLASSID) {
        PetscFE fe = (PetscFE) obj;

        ierr = PetscFEGetNumComponents(fe, &numComp[f]);CHKERRQ(ierr);
        if (!h) {
          ierr  = PetscFEGetDualSpace(fe, &cellsp[f]);CHKERRQ(ierr);
          sp[f] = cellsp[f];
          ierr  = PetscObjectReference((PetscObject) sp[f]);CHKERRQ(ierr);
        }
        else {
          ierr = PetscDualSpaceGetHeightSubspace(cellsp[f], h, &sp[f]);CHKERRQ(ierr);
          if (!sp[f]) {
            continue;
          }
        }
      } else if (id == PETSCFV_CLASSID) {
        PetscFV         fv = (PetscFV) obj;
        PetscQuadrature q;

        if (h) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_ARG_INCOMP, "Projection height > 0 not supported for finite volume");
        ierr = PetscFVGetNumComponents(fv, &numComp[f]);CHKERRQ(ierr);
        ierr = PetscFVGetQuadrature(fv, &q);CHKERRQ(ierr);
        ierr = PetscDualSpaceCreate(PetscObjectComm((PetscObject) fv), &sp[f]);CHKERRQ(ierr);
        ierr = PetscDualSpaceSetDM(sp[f], dm);CHKERRQ(ierr);
        ierr = PetscDualSpaceSetType(sp[f], PETSCDUALSPACESIMPLE);CHKERRQ(ierr);
        ierr = PetscDualSpaceSimpleSetDimension(sp[f], 1);CHKERRQ(ierr);
        ierr = PetscDualSpaceSimpleSetFunctional(sp[f], 0, q);CHKERRQ(ierr);
      } else SETERRQ1(PetscObjectComm((PetscObject)dm), PETSC_ERR_ARG_WRONG, "Unknown discretization type for field %d", f);
      ierr = PetscDualSpaceGetDimension(sp[f], &spDim);CHKERRQ(ierr);
      totDim += spDim*numComp[f];
    }
    ierr = DMPlexVecGetClosure(dm, section, localX, pStart, &numValues, NULL);CHKERRQ(ierr);
    if (numValues != totDim) SETERRQ2(PETSC_COMM_SELF, PETSC_ERR_ARG_WRONG, "The section point closure size %d != dual space dimension %d", numValues, totDim);
    if (!totDim) continue;
    ierr = DMGetWorkArray(dm, numValues, PETSC_SCALAR, &values);CHKERRQ(ierr);
    for (p = pStart; p < pEnd; ++p) {
      PetscFECellGeom geom;

      ierr          = DMPlexComputeCellGeometryFEM(dm, p, NULL, geom.v0, geom.J, NULL, &geom.detJ);CHKERRQ(ierr);
      geom.dim      = dim - h;
      geom.dimEmbed = dimEmbed;
      for (f = 0, v = 0; f < numFields; ++f) {
        void * const ctx = ctxs ? ctxs[f] : NULL;

        if (!sp[f]) continue;
        ierr = PetscDualSpaceGetDimension(sp[f], &spDim);CHKERRQ(ierr);
        for (d = 0; d < spDim; ++d) {
          if (funcs[f]) {
            ierr = PetscDualSpaceApply(sp[f], d, &geom, numComp[f], funcs[f], ctx, &values[v]);CHKERRQ(ierr);
          } else {
            for (comp = 0; comp < numComp[f]; ++comp) values[v+comp] = 0.0;
          }
          v += numComp[f];
        }
      }
      ierr = DMPlexVecSetClosure(dm, section, localX, p, values, mode);CHKERRQ(ierr);
    }
    ierr = DMRestoreWorkArray(dm, numValues, PETSC_SCALAR, &values);CHKERRQ(ierr);
    if (h || !maxHeight) {
      for (f = 0; f < numFields; f++) {ierr = PetscDualSpaceDestroy(&sp[f]);CHKERRQ(ierr);}
    }
  }
  ierr = PetscFree2(sp, numComp);CHKERRQ(ierr);
  if (maxHeight > 0) {
    for (f = 0; f < numFields; f++) {ierr = PetscDualSpaceDestroy(&cellsp[f]);CHKERRQ(ierr);}
    ierr = PetscFree(cellsp);CHKERRQ(ierr);
  }
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "DMPlexProjectFunction"
/*@C
  DMPlexProjectFunction - This projects the given function into the function space provided.

  Input Parameters:
+ dm      - The DM
. funcs   - The coordinate functions to evaluate, one per field
. ctxs    - Optional array of contexts to pass to each coordinate function.  ctxs itself may be null.
- mode    - The insertion mode for values

  Output Parameter:
. X - vector

  Level: developer

.seealso: DMPlexComputeL2Diff()
@*/
PetscErrorCode DMPlexProjectFunction(DM dm, void (**funcs)(const PetscReal [], PetscScalar *, void *), void **ctxs, InsertMode mode, Vec X)
{
  Vec            localX;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(dm, DM_CLASSID, 1);
  ierr = DMGetLocalVector(dm, &localX);CHKERRQ(ierr);
  ierr = DMPlexProjectFunctionLocal(dm, funcs, ctxs, mode, localX);CHKERRQ(ierr);
  ierr = DMLocalToGlobalBegin(dm, localX, mode, X);CHKERRQ(ierr);
  ierr = DMLocalToGlobalEnd(dm, localX, mode, X);CHKERRQ(ierr);
  ierr = DMRestoreLocalVector(dm, &localX);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "DMPlexProjectFieldLocal"
PetscErrorCode DMPlexProjectFieldLocal(DM dm, Vec localU, void (**funcs)(const PetscScalar[], const PetscScalar[], const PetscScalar[], const PetscScalar[], const PetscScalar[], const PetscScalar[], const PetscReal [], PetscScalar []), InsertMode mode, Vec localX)
{
  DM                dmAux;
  PetscDS           prob, probAux = NULL;
  Vec               A;
  PetscSection      section, sectionAux = NULL;
  PetscScalar      *values, *u, *u_x, *a, *a_x;
  PetscReal        *x, *v0, *J, *invJ, detJ;
  PetscInt          Nf, dim, spDim, totDim, numValues, cStart, cEnd, cEndInterior, c, f, d, v, comp, maxHeight;
  PetscErrorCode    ierr;

  PetscFunctionBegin;
  /* TODO: Move out dual space extraction so that we do not create one for every cell (see ProjectFunctionLocal) */
  ierr = DMPlexGetMaxProjectionHeight(dm,&maxHeight);CHKERRQ(ierr);
  if (maxHeight > 0) {SETERRQ(PETSC_COMM_SELF,PETSC_ERR_SUP,"Field projection for height > 0 not supported yet");}
  ierr = DMGetDS(dm, &prob);CHKERRQ(ierr);
  ierr = DMGetDimension(dm, &dim);CHKERRQ(ierr);
  ierr = DMGetDefaultSection(dm, &section);CHKERRQ(ierr);
  ierr = PetscSectionGetNumFields(section, &Nf);CHKERRQ(ierr);
  ierr = DMPlexGetHeightStratum(dm, 0, &cStart, &cEnd);CHKERRQ(ierr);
  ierr = PetscDSGetTotalDimension(prob, &totDim);CHKERRQ(ierr);
  ierr = PetscDSGetEvaluationArrays(prob, &u, NULL, &u_x);CHKERRQ(ierr);
  ierr = PetscDSGetRefCoordArrays(prob, &x, NULL);CHKERRQ(ierr);
  ierr = PetscObjectQuery((PetscObject) dm, "dmAux", (PetscObject *) &dmAux);CHKERRQ(ierr);
  ierr = PetscObjectQuery((PetscObject) dm, "A", (PetscObject *) &A);CHKERRQ(ierr);
  if (dmAux) {
    ierr = DMGetDS(dmAux, &probAux);CHKERRQ(ierr);
    ierr = DMGetDefaultSection(dmAux, &sectionAux);CHKERRQ(ierr);
    ierr = PetscDSGetEvaluationArrays(probAux, &a, NULL, &a_x);CHKERRQ(ierr);
  }
  ierr = DMPlexInsertBoundaryValues(dm, localU, 0.0, NULL, NULL, NULL);CHKERRQ(ierr);
  ierr = DMPlexVecGetClosure(dm, section, localX, cStart, &numValues, NULL);CHKERRQ(ierr);
  if (numValues != totDim) SETERRQ2(PETSC_COMM_SELF, PETSC_ERR_ARG_WRONG, "The section cell closure size %d != dual space dimension %d", numValues, totDim);
  ierr = DMGetWorkArray(dm, numValues, PETSC_SCALAR, &values);CHKERRQ(ierr);
  ierr = PetscMalloc3(dim,&v0,dim*dim,&J,dim*dim,&invJ);CHKERRQ(ierr);
  ierr = DMPlexGetHybridBounds(dm, &cEndInterior, NULL, NULL, NULL);CHKERRQ(ierr);
  cEnd = cEndInterior < 0 ? cEnd : cEndInterior;
  for (c = cStart; c < cEnd; ++c) {
    PetscScalar *coefficients = NULL, *coefficientsAux = NULL;

    ierr = DMPlexComputeCellGeometryFEM(dm, c, NULL, v0, J, invJ, &detJ);CHKERRQ(ierr);
    ierr = DMPlexVecGetClosure(dm, section, localU, c, NULL, &coefficients);CHKERRQ(ierr);
    if (dmAux) {ierr = DMPlexVecGetClosure(dmAux, sectionAux, A, c, NULL, &coefficientsAux);CHKERRQ(ierr);}
    for (f = 0, v = 0; f < Nf; ++f) {
      PetscObject    obj;
      PetscClassId   id;
      PetscDualSpace sp;
      PetscInt       Ncf;

      ierr = PetscDSGetDiscretization(prob, f, &obj);CHKERRQ(ierr);
      ierr = PetscObjectGetClassId(obj, &id);CHKERRQ(ierr);
      if (id == PETSCFE_CLASSID) {
        PetscFE fe = (PetscFE) obj;

        ierr = PetscFEGetDualSpace(fe, &sp);CHKERRQ(ierr);
        ierr = PetscFEGetNumComponents(fe, &Ncf);CHKERRQ(ierr);
      } else if (id == PETSCFV_CLASSID) {
        PetscFV         fv = (PetscFV) obj;
        PetscQuadrature q;

        ierr = PetscFVGetQuadrature(fv, &q);CHKERRQ(ierr);
        ierr = PetscFVGetNumComponents(fv, &Ncf);CHKERRQ(ierr);
        ierr = PetscDualSpaceCreate(PetscObjectComm((PetscObject) fv), &sp);CHKERRQ(ierr);
        ierr = PetscDualSpaceSetDM(sp, dm);CHKERRQ(ierr);
        ierr = PetscDualSpaceSetType(sp, PETSCDUALSPACESIMPLE);CHKERRQ(ierr);
        ierr = PetscDualSpaceSimpleSetDimension(sp, 1);CHKERRQ(ierr);
        ierr = PetscDualSpaceSimpleSetFunctional(sp, 0, q);CHKERRQ(ierr);
      }
      ierr = PetscDualSpaceGetDimension(sp, &spDim);CHKERRQ(ierr);
      for (d = 0; d < spDim; ++d) {
        PetscQuadrature  quad;
        const PetscReal *points, *weights;
        PetscInt         numPoints, q;

        if (funcs[f]) {
          ierr = PetscDualSpaceGetFunctional(sp, d, &quad);CHKERRQ(ierr);
          ierr = PetscQuadratureGetData(quad, NULL, &numPoints, &points, &weights);CHKERRQ(ierr);
          for (q = 0; q < numPoints; ++q) {
            CoordinatesRefToReal(dim, dim, v0, J, &points[q*dim], x);
            ierr = EvaluateFieldJets(prob,    PETSC_FALSE, q, invJ, coefficients,    NULL, u, u_x, NULL);CHKERRQ(ierr);
            ierr = EvaluateFieldJets(probAux, PETSC_FALSE, q, invJ, coefficientsAux, NULL, a, a_x, NULL);CHKERRQ(ierr);
            (*funcs[f])(u, NULL, u_x, a, NULL, a_x, x, &values[v]);
          }
        } else {
          for (comp = 0; comp < Ncf; ++comp) values[v+comp] = 0.0;
        }
        v += Ncf;
      }
      if (id == PETSCFV_CLASSID) {ierr = PetscDualSpaceDestroy(&sp);CHKERRQ(ierr);}
    }
    ierr = DMPlexVecRestoreClosure(dm, section, localU, c, NULL, &coefficients);CHKERRQ(ierr);
    if (dmAux) {ierr = DMPlexVecRestoreClosure(dmAux, sectionAux, A, c, NULL, &coefficientsAux);CHKERRQ(ierr);}
    ierr = DMPlexVecSetClosure(dm, section, localX, c, values, mode);CHKERRQ(ierr);
  }
  ierr = DMRestoreWorkArray(dm, numValues, PETSC_SCALAR, &values);CHKERRQ(ierr);
  ierr = PetscFree3(v0,J,invJ);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "DMPlexInsertBoundaryValues_FEM_Internal"
static PetscErrorCode DMPlexInsertBoundaryValues_FEM_Internal(DM dm, PetscInt field, DMLabel label, PetscInt numids, const PetscInt ids[], void (*func)(const PetscReal[], PetscScalar *, void *), void *ctx, Vec locX)
{
  void        (**funcs)(const PetscReal x[], PetscScalar *u, void *ctx);
  void         **ctxs;
  PetscInt       numFields;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  ierr = DMGetNumFields(dm, &numFields);CHKERRQ(ierr);
  ierr = PetscCalloc2(numFields,&funcs,numFields,&ctxs);CHKERRQ(ierr);
  funcs[field] = func;
  ctxs[field]  = ctx;
  ierr = DMPlexProjectFunctionLabelLocal(dm, label, numids, ids, funcs, ctxs, INSERT_BC_VALUES, locX);CHKERRQ(ierr);
  ierr = PetscFree2(funcs,ctxs);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "DMPlexInsertBoundaryValues_FVM_Internal"
static PetscErrorCode DMPlexInsertBoundaryValues_FVM_Internal(DM dm, PetscReal time, Vec faceGeometry, Vec cellGeometry, Vec Grad,
                                                              PetscInt field, DMLabel label, PetscInt numids, const PetscInt ids[], PetscErrorCode (*func)(PetscReal,const PetscReal*,const PetscReal*,const PetscScalar*,PetscScalar*,void*), void *ctx, Vec locX)
{
  PetscDS            prob;
  PetscSF            sf;
  DM                 dmFace, dmCell, dmGrad;
  const PetscScalar *facegeom, *cellgeom, *grad;
  const PetscInt    *leaves;
  PetscScalar       *x, *fx;
  PetscInt           dim, nleaves, loc, fStart, fEnd, pdim, i;
  PetscErrorCode     ierr;

  PetscFunctionBegin;
  ierr = DMGetPointSF(dm, &sf);CHKERRQ(ierr);
  ierr = PetscSFGetGraph(sf, NULL, &nleaves, &leaves, NULL);CHKERRQ(ierr);
  nleaves = PetscMax(0, nleaves);
  ierr = DMGetDimension(dm, &dim);CHKERRQ(ierr);
  ierr = DMPlexGetHeightStratum(dm, 1, &fStart, &fEnd);CHKERRQ(ierr);
  ierr = DMGetDS(dm, &prob);CHKERRQ(ierr);
  ierr = VecGetDM(faceGeometry, &dmFace);CHKERRQ(ierr);
  ierr = VecGetArrayRead(faceGeometry, &facegeom);CHKERRQ(ierr);
  ierr = VecGetDM(cellGeometry, &dmCell);CHKERRQ(ierr);
  ierr = VecGetArrayRead(cellGeometry, &cellgeom);CHKERRQ(ierr);
  if (Grad) {
    PetscFV fv;

    ierr = PetscDSGetDiscretization(prob, field, (PetscObject *) &fv);CHKERRQ(ierr);
    ierr = VecGetDM(Grad, &dmGrad);CHKERRQ(ierr);
    ierr = VecGetArrayRead(Grad, &grad);CHKERRQ(ierr);
    ierr = PetscFVGetNumComponents(fv, &pdim);CHKERRQ(ierr);
    ierr = DMGetWorkArray(dm, pdim, PETSC_SCALAR, &fx);CHKERRQ(ierr);
  }
  ierr = VecGetArray(locX, &x);CHKERRQ(ierr);
  for (i = 0; i < numids; ++i) {
    IS              faceIS;
    const PetscInt *faces;
    PetscInt        numFaces, f;

    ierr = DMLabelGetStratumIS(label, ids[i], &faceIS);CHKERRQ(ierr);
    if (!faceIS) continue; /* No points with that id on this process */
    ierr = ISGetLocalSize(faceIS, &numFaces);CHKERRQ(ierr);
    ierr = ISGetIndices(faceIS, &faces);CHKERRQ(ierr);
    for (f = 0; f < numFaces; ++f) {
      const PetscInt         face = faces[f], *cells;
      const PetscFVFaceGeom *fg;

      if ((face < fStart) || (face >= fEnd)) continue; /* Refinement adds non-faces to labels */
      ierr = PetscFindInt(face, nleaves, (PetscInt *) leaves, &loc);CHKERRQ(ierr);
      if (loc >= 0) continue;
      ierr = DMPlexPointLocalRead(dmFace, face, facegeom, &fg);CHKERRQ(ierr);
      ierr = DMPlexGetSupport(dm, face, &cells);CHKERRQ(ierr);
      if (Grad) {
        const PetscFVCellGeom *cg;
        const PetscScalar     *cx, *cgrad;
        PetscScalar           *xG;
        PetscReal              dx[3];
        PetscInt               d;

        ierr = DMPlexPointLocalRead(dmCell, cells[0], cellgeom, &cg);CHKERRQ(ierr);
        ierr = DMPlexPointLocalRead(dm, cells[0], x, &cx);CHKERRQ(ierr);
        ierr = DMPlexPointLocalRead(dmGrad, cells[0], grad, &cgrad);CHKERRQ(ierr);
        ierr = DMPlexPointLocalFieldRef(dm, cells[1], field, x, &xG);CHKERRQ(ierr);
        DMPlex_WaxpyD_Internal(dim, -1, cg->centroid, fg->centroid, dx);
        for (d = 0; d < pdim; ++d) fx[d] = cx[d] + DMPlex_DotD_Internal(dim, &cgrad[d*dim], dx);
        ierr = (*func)(time, fg->centroid, fg->normal, fx, xG, ctx);CHKERRQ(ierr);
      } else {
        const PetscScalar *xI;
        PetscScalar       *xG;

        ierr = DMPlexPointLocalRead(dm, cells[0], x, &xI);CHKERRQ(ierr);
        ierr = DMPlexPointLocalFieldRef(dm, cells[1], field, x, &xG);CHKERRQ(ierr);
        ierr = (*func)(time, fg->centroid, fg->normal, xI, xG, ctx);CHKERRQ(ierr);
      }
    }
    ierr = ISRestoreIndices(faceIS, &faces);CHKERRQ(ierr);
    ierr = ISDestroy(&faceIS);CHKERRQ(ierr);
  }
  ierr = VecRestoreArray(locX, &x);CHKERRQ(ierr);
  if (Grad) {
    ierr = DMRestoreWorkArray(dm, pdim, PETSC_SCALAR, &fx);CHKERRQ(ierr);
    ierr = VecRestoreArrayRead(Grad, &grad);CHKERRQ(ierr);
  }
  ierr = VecRestoreArrayRead(cellGeometry, &cellgeom);CHKERRQ(ierr);
  ierr = VecRestoreArrayRead(faceGeometry, &facegeom);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "DMPlexInsertBoundaryValues"
PetscErrorCode DMPlexInsertBoundaryValues(DM dm, Vec locX, PetscReal time, Vec faceGeomFVM, Vec cellGeomFVM, Vec gradFVM)
{
  PetscInt       numBd, b;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(dm, DM_CLASSID, 1);
  PetscValidHeaderSpecific(locX, VEC_CLASSID, 2);
  if (faceGeomFVM) {PetscValidHeaderSpecific(faceGeomFVM, VEC_CLASSID, 4);}
  if (cellGeomFVM) {PetscValidHeaderSpecific(cellGeomFVM, VEC_CLASSID, 5);}
  if (gradFVM)     {PetscValidHeaderSpecific(gradFVM, VEC_CLASSID, 6);}
  ierr = DMPlexGetNumBoundary(dm, &numBd);CHKERRQ(ierr);
  for (b = 0; b < numBd; ++b) {
    PetscBool       isEssential;
    const char     *labelname;
    DMLabel         label;
    PetscInt        field;
    PetscObject     obj;
    PetscClassId    id;
    void          (*func)();
    PetscInt        numids;
    const PetscInt *ids;
    void           *ctx;

    ierr = DMPlexGetBoundary(dm, b, &isEssential, NULL, &labelname, &field, &func, &numids, &ids, &ctx);CHKERRQ(ierr);
    if (!isEssential) continue;
    ierr = DMPlexGetLabel(dm, labelname, &label);CHKERRQ(ierr);
    ierr = DMGetField(dm, field, &obj);CHKERRQ(ierr);
    ierr = PetscObjectGetClassId(obj, &id);CHKERRQ(ierr);
    if (id == PETSCFE_CLASSID) {
      ierr = DMPlexInsertBoundaryValues_FEM_Internal(dm, field, label, numids, ids, (void (*)(const PetscReal[], PetscScalar *, void *)) func, ctx, locX);CHKERRQ(ierr);
    } else if (id == PETSCFV_CLASSID) {
      if (!faceGeomFVM) continue;
      ierr = DMPlexInsertBoundaryValues_FVM_Internal(dm, time, faceGeomFVM, cellGeomFVM, gradFVM,
                                                     field, label, numids, ids, (PetscErrorCode (*)(PetscReal,const PetscReal*,const PetscReal*,const PetscScalar*,PetscScalar*,void*)) func, ctx, locX);CHKERRQ(ierr);
    } else SETERRQ1(PetscObjectComm((PetscObject)dm), PETSC_ERR_ARG_WRONG, "Unknown discretization type for field %d", field);
  }
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "DMPlexComputeL2Diff"
/*@C
  DMPlexComputeL2Diff - This function computes the L_2 difference between a function u and an FEM interpolant solution u_h.

  Input Parameters:
+ dm    - The DM
. funcs - The functions to evaluate for each field component
. ctxs  - Optional array of contexts to pass to each function, or NULL.
- X     - The coefficient vector u_h

  Output Parameter:
. diff - The diff ||u - u_h||_2

  Level: developer

.seealso: DMPlexProjectFunction(), DMPlexComputeL2FieldDiff(), DMPlexComputeL2GradientDiff()
@*/
PetscErrorCode DMPlexComputeL2Diff(DM dm, void (**funcs)(const PetscReal [], PetscScalar *, void *), void **ctxs, Vec X, PetscReal *diff)
{
  const PetscInt   debug = 0;
  PetscSection     section;
  PetscQuadrature  quad;
  Vec              localX;
  PetscScalar     *funcVal, *interpolant;
  PetscReal       *coords, *v0, *J, *invJ, detJ;
  PetscReal        localDiff = 0.0;
  const PetscReal *quadPoints, *quadWeights;
  PetscInt         dim, numFields, numComponents = 0, numQuadPoints, cStart, cEnd, cEndInterior, c, field, fieldOffset;
  PetscErrorCode   ierr;

  PetscFunctionBegin;
  ierr = DMGetDimension(dm, &dim);CHKERRQ(ierr);
  ierr = DMGetDefaultSection(dm, &section);CHKERRQ(ierr);
  ierr = PetscSectionGetNumFields(section, &numFields);CHKERRQ(ierr);
  ierr = DMGetLocalVector(dm, &localX);CHKERRQ(ierr);
  ierr = DMGlobalToLocalBegin(dm, X, INSERT_VALUES, localX);CHKERRQ(ierr);
  ierr = DMGlobalToLocalEnd(dm, X, INSERT_VALUES, localX);CHKERRQ(ierr);
  for (field = 0; field < numFields; ++field) {
    PetscObject  obj;
    PetscClassId id;
    PetscInt     Nc;

    ierr = DMGetField(dm, field, &obj);CHKERRQ(ierr);
    ierr = PetscObjectGetClassId(obj, &id);CHKERRQ(ierr);
    if (id == PETSCFE_CLASSID) {
      PetscFE fe = (PetscFE) obj;

      ierr = PetscFEGetQuadrature(fe, &quad);CHKERRQ(ierr);
      ierr = PetscFEGetNumComponents(fe, &Nc);CHKERRQ(ierr);
    } else if (id == PETSCFV_CLASSID) {
      PetscFV fv = (PetscFV) obj;

      ierr = PetscFVGetQuadrature(fv, &quad);CHKERRQ(ierr);
      ierr = PetscFVGetNumComponents(fv, &Nc);CHKERRQ(ierr);
    } else SETERRQ1(PetscObjectComm((PetscObject)dm), PETSC_ERR_ARG_WRONG, "Unknown discretization type for field %d", field);
    numComponents += Nc;
  }
  ierr = PetscQuadratureGetData(quad, NULL, &numQuadPoints, &quadPoints, &quadWeights);CHKERRQ(ierr);
  ierr = DMPlexProjectFunctionLocal(dm, funcs, ctxs, INSERT_BC_VALUES, localX);CHKERRQ(ierr);
  ierr = PetscMalloc6(numComponents,&funcVal,numComponents,&interpolant,dim,&coords,dim,&v0,dim*dim,&J,dim*dim,&invJ);CHKERRQ(ierr);
  ierr = DMPlexGetHeightStratum(dm, 0, &cStart, &cEnd);CHKERRQ(ierr);
  ierr = DMPlexGetHybridBounds(dm, &cEndInterior, NULL, NULL, NULL);CHKERRQ(ierr);
  cEnd = cEndInterior < 0 ? cEnd : cEndInterior;
  for (c = cStart; c < cEnd; ++c) {
    PetscScalar *x = NULL;
    PetscReal    elemDiff = 0.0;

    ierr = DMPlexComputeCellGeometryFEM(dm, c, NULL, v0, J, invJ, &detJ);CHKERRQ(ierr);
    if (detJ <= 0.0) SETERRQ2(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Invalid determinant %g for element %d", detJ, c);
    ierr = DMPlexVecGetClosure(dm, NULL, localX, c, NULL, &x);CHKERRQ(ierr);

    for (field = 0, fieldOffset = 0; field < numFields; ++field) {
      PetscObject  obj;
      PetscClassId id;
      void * const ctx = ctxs ? ctxs[field] : NULL;
      PetscInt     Nb, Nc, q, fc;

      ierr = DMGetField(dm, field, &obj);CHKERRQ(ierr);
      ierr = PetscObjectGetClassId(obj, &id);CHKERRQ(ierr);
      if (id == PETSCFE_CLASSID)      {ierr = PetscFEGetNumComponents((PetscFE) obj, &Nc);CHKERRQ(ierr);ierr = PetscFEGetDimension((PetscFE) obj, &Nb);CHKERRQ(ierr);}
      else if (id == PETSCFV_CLASSID) {ierr = PetscFVGetNumComponents((PetscFV) obj, &Nc);CHKERRQ(ierr);Nb = 1;}
      else SETERRQ1(PetscObjectComm((PetscObject)dm), PETSC_ERR_ARG_WRONG, "Unknown discretization type for field %d", field);
      if (debug) {
        char title[1024];
        ierr = PetscSNPrintf(title, 1023, "Solution for Field %d", field);CHKERRQ(ierr);
        ierr = DMPrintCellVector(c, title, Nb*Nc, &x[fieldOffset]);CHKERRQ(ierr);
      }
      for (q = 0; q < numQuadPoints; ++q) {
        CoordinatesRefToReal(dim, dim, v0, J, &quadPoints[q*dim], coords);
        (*funcs[field])(coords, funcVal, ctx);
        if (id == PETSCFE_CLASSID)      {ierr = PetscFEInterpolate_Static((PetscFE) obj, &x[fieldOffset], q, interpolant);CHKERRQ(ierr);}
        else if (id == PETSCFV_CLASSID) {ierr = PetscFVInterpolate_Static((PetscFV) obj, &x[fieldOffset], q, interpolant);CHKERRQ(ierr);}
        else SETERRQ1(PetscObjectComm((PetscObject)dm), PETSC_ERR_ARG_WRONG, "Unknown discretization type for field %d", field);
        for (fc = 0; fc < Nc; ++fc) {
          if (debug) {ierr = PetscPrintf(PETSC_COMM_SELF, "    elem %d field %d diff %g\n", c, field, PetscSqr(PetscRealPart(interpolant[fc] - funcVal[fc]))*quadWeights[q]*detJ);CHKERRQ(ierr);}
          elemDiff += PetscSqr(PetscRealPart(interpolant[fc] - funcVal[fc]))*quadWeights[q]*detJ;
        }
      }
      fieldOffset += Nb*Nc;
    }
    ierr = DMPlexVecRestoreClosure(dm, NULL, localX, c, NULL, &x);CHKERRQ(ierr);
    if (debug) {ierr = PetscPrintf(PETSC_COMM_SELF, "  elem %d diff %g\n", c, elemDiff);CHKERRQ(ierr);}
    localDiff += elemDiff;
  }
  ierr  = PetscFree6(funcVal,interpolant,coords,v0,J,invJ);CHKERRQ(ierr);
  ierr  = DMRestoreLocalVector(dm, &localX);CHKERRQ(ierr);
  ierr  = MPI_Allreduce(&localDiff, diff, 1, MPIU_REAL, MPI_SUM, PetscObjectComm((PetscObject)dm));CHKERRQ(ierr);
  *diff = PetscSqrtReal(*diff);
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "DMPlexComputeL2GradientDiff"
/*@C
  DMPlexComputeL2GradientDiff - This function computes the L_2 difference between the gradient of a function u and an FEM interpolant solution grad u_h.

  Input Parameters:
+ dm    - The DM
. funcs - The gradient functions to evaluate for each field component
. ctxs  - Optional array of contexts to pass to each function, or NULL.
. X     - The coefficient vector u_h
- n     - The vector to project along

  Output Parameter:
. diff - The diff ||(grad u - grad u_h) . n||_2

  Level: developer

.seealso: DMPlexProjectFunction(), DMPlexComputeL2Diff()
@*/
PetscErrorCode DMPlexComputeL2GradientDiff(DM dm, void (**funcs)(const PetscReal [], const PetscReal [], PetscScalar *, void *), void **ctxs, Vec X, const PetscReal n[], PetscReal *diff)
{
  const PetscInt  debug = 0;
  PetscSection    section;
  PetscQuadrature quad;
  Vec             localX;
  PetscScalar    *funcVal, *interpolantVec;
  PetscReal      *coords, *realSpaceDer, *v0, *J, *invJ, detJ;
  PetscReal       localDiff = 0.0;
  PetscInt        dim, numFields, numComponents = 0, cStart, cEnd, cEndInterior, c, field, fieldOffset, comp;
  PetscErrorCode  ierr;

  PetscFunctionBegin;
  ierr = DMGetDimension(dm, &dim);CHKERRQ(ierr);
  ierr = DMGetDefaultSection(dm, &section);CHKERRQ(ierr);
  ierr = PetscSectionGetNumFields(section, &numFields);CHKERRQ(ierr);
  ierr = DMGetLocalVector(dm, &localX);CHKERRQ(ierr);
  ierr = DMGlobalToLocalBegin(dm, X, INSERT_VALUES, localX);CHKERRQ(ierr);
  ierr = DMGlobalToLocalEnd(dm, X, INSERT_VALUES, localX);CHKERRQ(ierr);
  for (field = 0; field < numFields; ++field) {
    PetscFE  fe;
    PetscInt Nc;

    ierr = DMGetField(dm, field, (PetscObject *) &fe);CHKERRQ(ierr);
    ierr = PetscFEGetQuadrature(fe, &quad);CHKERRQ(ierr);
    ierr = PetscFEGetNumComponents(fe, &Nc);CHKERRQ(ierr);
    numComponents += Nc;
  }
  /* ierr = DMPlexProjectFunctionLocal(dm, fe, funcs, INSERT_BC_VALUES, localX);CHKERRQ(ierr); */
  ierr = PetscMalloc7(numComponents,&funcVal,dim,&coords,dim,&realSpaceDer,dim,&v0,dim*dim,&J,dim*dim,&invJ,dim,&interpolantVec);CHKERRQ(ierr);
  ierr = DMPlexGetHeightStratum(dm, 0, &cStart, &cEnd);CHKERRQ(ierr);
  ierr = DMPlexGetHybridBounds(dm, &cEndInterior, NULL, NULL, NULL);CHKERRQ(ierr);
  cEnd = cEndInterior < 0 ? cEnd : cEndInterior;
  for (c = cStart; c < cEnd; ++c) {
    PetscScalar *x = NULL;
    PetscReal    elemDiff = 0.0;

    ierr = DMPlexComputeCellGeometryFEM(dm, c, NULL, v0, J, invJ, &detJ);CHKERRQ(ierr);
    if (detJ <= 0.0) SETERRQ2(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Invalid determinant %g for element %d", detJ, c);
    ierr = DMPlexVecGetClosure(dm, NULL, localX, c, NULL, &x);CHKERRQ(ierr);

    for (field = 0, comp = 0, fieldOffset = 0; field < numFields; ++field) {
      PetscFE          fe;
      void * const     ctx = ctxs ? ctxs[field] : NULL;
      const PetscReal *quadPoints, *quadWeights;
      PetscReal       *basisDer;
      PetscInt         numQuadPoints, Nb, Ncomp, q, d, e, fc, f, g;

      ierr = DMGetField(dm, field, (PetscObject *) &fe);CHKERRQ(ierr);
      ierr = PetscQuadratureGetData(quad, NULL, &numQuadPoints, &quadPoints, &quadWeights);CHKERRQ(ierr);
      ierr = PetscFEGetDimension(fe, &Nb);CHKERRQ(ierr);
      ierr = PetscFEGetNumComponents(fe, &Ncomp);CHKERRQ(ierr);
      ierr = PetscFEGetDefaultTabulation(fe, NULL, &basisDer, NULL);CHKERRQ(ierr);
      if (debug) {
        char title[1024];
        ierr = PetscSNPrintf(title, 1023, "Solution for Field %d", field);CHKERRQ(ierr);
        ierr = DMPrintCellVector(c, title, Nb*Ncomp, &x[fieldOffset]);CHKERRQ(ierr);
      }
      for (q = 0; q < numQuadPoints; ++q) {
        for (d = 0; d < dim; d++) {
          coords[d] = v0[d];
          for (e = 0; e < dim; e++) {
            coords[d] += J[d*dim+e]*(quadPoints[q*dim+e] + 1.0);
          }
        }
        (*funcs[field])(coords, n, funcVal, ctx);
        for (fc = 0; fc < Ncomp; ++fc) {
          PetscScalar interpolant = 0.0;

          for (d = 0; d < dim; ++d) interpolantVec[d] = 0.0;
          for (f = 0; f < Nb; ++f) {
            const PetscInt fidx = f*Ncomp+fc;

            for (d = 0; d < dim; ++d) {
              realSpaceDer[d] = 0.0;
              for (g = 0; g < dim; ++g) {
                realSpaceDer[d] += invJ[g*dim+d]*basisDer[(q*Nb*Ncomp+fidx)*dim+g];
              }
              interpolantVec[d] += x[fieldOffset+fidx]*realSpaceDer[d];
            }
          }
          for (d = 0; d < dim; ++d) interpolant += interpolantVec[d]*n[d];
          if (debug) {ierr = PetscPrintf(PETSC_COMM_SELF, "    elem %d fieldDer %d diff %g\n", c, field, PetscSqr(PetscRealPart(interpolant - funcVal[fc]))*quadWeights[q]*detJ);CHKERRQ(ierr);}
          elemDiff += PetscSqr(PetscRealPart(interpolant - funcVal[fc]))*quadWeights[q]*detJ;
        }
      }
      comp        += Ncomp;
      fieldOffset += Nb*Ncomp;
    }
    ierr = DMPlexVecRestoreClosure(dm, NULL, localX, c, NULL, &x);CHKERRQ(ierr);
    if (debug) {ierr = PetscPrintf(PETSC_COMM_SELF, "  elem %d diff %g\n", c, elemDiff);CHKERRQ(ierr);}
    localDiff += elemDiff;
  }
  ierr  = PetscFree7(funcVal,coords,realSpaceDer,v0,J,invJ,interpolantVec);CHKERRQ(ierr);
  ierr  = DMRestoreLocalVector(dm, &localX);CHKERRQ(ierr);
  ierr  = MPI_Allreduce(&localDiff, diff, 1, MPIU_REAL, MPI_SUM, PetscObjectComm((PetscObject)dm));CHKERRQ(ierr);
  *diff = PetscSqrtReal(*diff);
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "DMPlexComputeL2FieldDiff"
/*@C
  DMPlexComputeL2FieldDiff - This function computes the L_2 difference between a function u and an FEM interpolant solution u_h, separated into field components.

  Input Parameters:
+ dm    - The DM
. funcs - The functions to evaluate for each field component
. ctxs  - Optional array of contexts to pass to each function, or NULL.
- X     - The coefficient vector u_h

  Output Parameter:
. diff - The array of differences, ||u^f - u^f_h||_2

  Level: developer

.seealso: DMPlexProjectFunction(), DMPlexComputeL2Diff(), DMPlexComputeL2GradientDiff()
@*/
PetscErrorCode DMPlexComputeL2FieldDiff(DM dm, void (**funcs)(const PetscReal [], PetscScalar *, void *), void **ctxs, Vec X, PetscReal diff[])
{
  const PetscInt   debug = 0;
  PetscSection     section;
  PetscQuadrature  quad;
  Vec              localX;
  PetscScalar     *funcVal, *interpolant;
  PetscReal       *coords, *v0, *J, *invJ, detJ;
  PetscReal       *localDiff;
  const PetscReal *quadPoints, *quadWeights;
  PetscInt         dim, numFields, numComponents = 0, numQuadPoints, cStart, cEnd, cEndInterior, c, field, fieldOffset;
  PetscErrorCode   ierr;

  PetscFunctionBegin;
  ierr = DMGetDimension(dm, &dim);CHKERRQ(ierr);
  ierr = DMGetDefaultSection(dm, &section);CHKERRQ(ierr);
  ierr = PetscSectionGetNumFields(section, &numFields);CHKERRQ(ierr);
  ierr = DMGetLocalVector(dm, &localX);CHKERRQ(ierr);
  ierr = DMGlobalToLocalBegin(dm, X, INSERT_VALUES, localX);CHKERRQ(ierr);
  ierr = DMGlobalToLocalEnd(dm, X, INSERT_VALUES, localX);CHKERRQ(ierr);
  for (field = 0; field < numFields; ++field) {
    PetscObject  obj;
    PetscClassId id;
    PetscInt     Nc;

    ierr = DMGetField(dm, field, &obj);CHKERRQ(ierr);
    ierr = PetscObjectGetClassId(obj, &id);CHKERRQ(ierr);
    if (id == PETSCFE_CLASSID) {
      PetscFE fe = (PetscFE) obj;

      ierr = PetscFEGetQuadrature(fe, &quad);CHKERRQ(ierr);
      ierr = PetscFEGetNumComponents(fe, &Nc);CHKERRQ(ierr);
    } else if (id == PETSCFV_CLASSID) {
      PetscFV fv = (PetscFV) obj;

      ierr = PetscFVGetQuadrature(fv, &quad);CHKERRQ(ierr);
      ierr = PetscFVGetNumComponents(fv, &Nc);CHKERRQ(ierr);
    } else SETERRQ1(PetscObjectComm((PetscObject)dm), PETSC_ERR_ARG_WRONG, "Unknown discretization type for field %d", field);
    numComponents += Nc;
  }
  ierr = PetscQuadratureGetData(quad, NULL, &numQuadPoints, &quadPoints, &quadWeights);CHKERRQ(ierr);
  ierr = DMPlexProjectFunctionLocal(dm, funcs, ctxs, INSERT_BC_VALUES, localX);CHKERRQ(ierr);
  ierr = PetscCalloc7(numFields,&localDiff,numComponents,&funcVal,numComponents,&interpolant,dim,&coords,dim,&v0,dim*dim,&J,dim*dim,&invJ);CHKERRQ(ierr);
  ierr = DMPlexGetHeightStratum(dm, 0, &cStart, &cEnd);CHKERRQ(ierr);
  ierr = DMPlexGetHybridBounds(dm, &cEndInterior, NULL, NULL, NULL);CHKERRQ(ierr);
  cEnd = cEndInterior < 0 ? cEnd : cEndInterior;
  for (c = cStart; c < cEnd; ++c) {
    PetscScalar *x = NULL;

    ierr = DMPlexComputeCellGeometryFEM(dm, c, NULL, v0, J, invJ, &detJ);CHKERRQ(ierr);
    if (detJ <= 0.0) SETERRQ2(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Invalid determinant %g for element %d", detJ, c);
    ierr = DMPlexVecGetClosure(dm, NULL, localX, c, NULL, &x);CHKERRQ(ierr);

    for (field = 0, fieldOffset = 0; field < numFields; ++field) {
      PetscObject  obj;
      PetscClassId id;
      void * const ctx = ctxs ? ctxs[field] : NULL;
      PetscInt     Nb, Nc, q, fc;

      PetscReal       elemDiff = 0.0;

      ierr = DMGetField(dm, field, &obj);CHKERRQ(ierr);
      ierr = PetscObjectGetClassId(obj, &id);CHKERRQ(ierr);
      if (id == PETSCFE_CLASSID)      {ierr = PetscFEGetNumComponents((PetscFE) obj, &Nc);CHKERRQ(ierr);ierr = PetscFEGetDimension((PetscFE) obj, &Nb);CHKERRQ(ierr);}
      else if (id == PETSCFV_CLASSID) {ierr = PetscFVGetNumComponents((PetscFV) obj, &Nc);CHKERRQ(ierr);Nb = 1;}
      else SETERRQ1(PetscObjectComm((PetscObject)dm), PETSC_ERR_ARG_WRONG, "Unknown discretization type for field %d", field);
      if (debug) {
        char title[1024];
        ierr = PetscSNPrintf(title, 1023, "Solution for Field %d", field);CHKERRQ(ierr);
        ierr = DMPrintCellVector(c, title, Nb*Nc, &x[fieldOffset]);CHKERRQ(ierr);
      }
      for (q = 0; q < numQuadPoints; ++q) {
        CoordinatesRefToReal(dim, dim, v0, J, &quadPoints[q*dim], coords);
        (*funcs[field])(coords, funcVal, ctx);
        if (id == PETSCFE_CLASSID)      {ierr = PetscFEInterpolate_Static((PetscFE) obj, &x[fieldOffset], q, interpolant);CHKERRQ(ierr);}
        else if (id == PETSCFV_CLASSID) {ierr = PetscFVInterpolate_Static((PetscFV) obj, &x[fieldOffset], q, interpolant);CHKERRQ(ierr);}
        else SETERRQ1(PetscObjectComm((PetscObject)dm), PETSC_ERR_ARG_WRONG, "Unknown discretization type for field %d", field);
        for (fc = 0; fc < Nc; ++fc) {
          if (debug) {ierr = PetscPrintf(PETSC_COMM_SELF, "    elem %d field %d diff %g\n", c, field, PetscSqr(PetscRealPart(interpolant[fc] - funcVal[fc]))*quadWeights[q]*detJ);CHKERRQ(ierr);}
          elemDiff += PetscSqr(PetscRealPart(interpolant[fc] - funcVal[fc]))*quadWeights[q]*detJ;
        }
      }
      fieldOffset += Nb*Nc;
      localDiff[field] += elemDiff;
    }
    ierr = DMPlexVecRestoreClosure(dm, NULL, localX, c, NULL, &x);CHKERRQ(ierr);
  }
  ierr = DMRestoreLocalVector(dm, &localX);CHKERRQ(ierr);
  ierr = MPI_Allreduce(localDiff, diff, numFields, MPIU_REAL, MPI_SUM, PetscObjectComm((PetscObject)dm));CHKERRQ(ierr);
  for (field = 0; field < numFields; ++field) diff[field] = PetscSqrtReal(diff[field]);
  ierr = PetscFree7(localDiff,funcVal,interpolant,coords,v0,J,invJ);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "DMPlexComputeIntegralFEM"
/*@
  DMPlexComputeIntegralFEM - Form the local integral F from the local input X using pointwise functions specified by the user

  Input Parameters:
+ dm - The mesh
. X  - Local input vector
- user - The user context

  Output Parameter:
. integral - Local integral for each field

  Level: developer

.seealso: DMPlexComputeResidualFEM()
@*/
PetscErrorCode DMPlexComputeIntegralFEM(DM dm, Vec X, PetscReal *integral, void *user)
{
  DM_Plex          *mesh  = (DM_Plex *) dm->data;
  DM                dmAux;
  Vec               localX, A;
  PetscDS           prob, probAux = NULL;
  PetscQuadrature   q;
  PetscSection      section, sectionAux;
  PetscFECellGeom  *cgeom;
  PetscScalar      *u, *a = NULL;
  PetscReal        *lintegral, *vol;
  PetscInt          dim, Nf, f, numCells, cStart, cEnd, cEndInterior, c;
  PetscInt          totDim, totDimAux;
  PetscErrorCode    ierr;

  PetscFunctionBegin;
  ierr = PetscLogEventBegin(DMPLEX_IntegralFEM,dm,0,0,0);CHKERRQ(ierr);
  ierr = DMGetLocalVector(dm, &localX);CHKERRQ(ierr);
  ierr = DMGlobalToLocalBegin(dm, X, INSERT_VALUES, localX);CHKERRQ(ierr);
  ierr = DMGlobalToLocalEnd(dm, X, INSERT_VALUES, localX);CHKERRQ(ierr);
  ierr = DMGetDimension(dm, &dim);CHKERRQ(ierr);
  ierr = DMGetDefaultSection(dm, &section);CHKERRQ(ierr);
  ierr = DMGetDS(dm, &prob);CHKERRQ(ierr);
  ierr = PetscDSGetTotalDimension(prob, &totDim);CHKERRQ(ierr);
  ierr = PetscSectionGetNumFields(section, &Nf);CHKERRQ(ierr);
  ierr = DMPlexGetHeightStratum(dm, 0, &cStart, &cEnd);CHKERRQ(ierr);
  ierr = DMPlexGetHybridBounds(dm, &cEndInterior, NULL, NULL, NULL);CHKERRQ(ierr);
  cEnd = cEndInterior < 0 ? cEnd : cEndInterior;
  numCells = cEnd - cStart;
  ierr = PetscObjectQuery((PetscObject) dm, "dmAux", (PetscObject *) &dmAux);CHKERRQ(ierr);
  ierr = PetscObjectQuery((PetscObject) dm, "A", (PetscObject *) &A);CHKERRQ(ierr);
  if (dmAux) {
    ierr = DMGetDefaultSection(dmAux, &sectionAux);CHKERRQ(ierr);
    ierr = DMGetDS(dmAux, &probAux);CHKERRQ(ierr);
    ierr = PetscDSGetTotalDimension(probAux, &totDimAux);CHKERRQ(ierr);
  }
  ierr = DMPlexInsertBoundaryValues(dm, localX, 0.0, NULL, NULL, NULL);CHKERRQ(ierr);
  ierr = PetscMalloc4(Nf,&lintegral,numCells*totDim,&u,numCells,&cgeom,numCells,&vol);CHKERRQ(ierr);
  if (dmAux) {ierr = PetscMalloc1(numCells*totDimAux, &a);CHKERRQ(ierr);}
  for (f = 0; f < Nf; ++f) {lintegral[f] = 0.0;}
  for (c = cStart; c < cEnd; ++c) {
    PetscScalar *x = NULL;
    PetscInt     i;

    ierr = DMPlexComputeCellGeometryFEM(dm, c, NULL, cgeom[c].v0, cgeom[c].J, cgeom[c].invJ, &cgeom[c].detJ);CHKERRQ(ierr);
    ierr = DMPlexComputeCellGeometryFVM(dm, c, &vol[c], NULL, NULL);CHKERRQ(ierr);
    if (cgeom[c].detJ <= 0.0) SETERRQ2(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Invalid determinant %g for element %d", cgeom[c].detJ, c);
    ierr = DMPlexVecGetClosure(dm, section, localX, c, NULL, &x);CHKERRQ(ierr);
    for (i = 0; i < totDim; ++i) u[c*totDim+i] = x[i];
    ierr = DMPlexVecRestoreClosure(dm, section, localX, c, NULL, &x);CHKERRQ(ierr);
    if (dmAux) {
      ierr = DMPlexVecGetClosure(dmAux, sectionAux, A, c, NULL, &x);CHKERRQ(ierr);
      for (i = 0; i < totDimAux; ++i) a[c*totDimAux+i] = x[i];
      ierr = DMPlexVecRestoreClosure(dmAux, sectionAux, A, c, NULL, &x);CHKERRQ(ierr);
    }
  }
  for (f = 0; f < Nf; ++f) {
    PetscObject  obj;
    PetscClassId id;
    PetscInt     numChunks, numBatches, batchSize, numBlocks, blockSize, Ne, Nr, offset;

    ierr = PetscDSGetDiscretization(prob, f, &obj);CHKERRQ(ierr);
    ierr = PetscObjectGetClassId(obj, &id);CHKERRQ(ierr);
    if (id == PETSCFE_CLASSID) {
      PetscFE         fe = (PetscFE) obj;
      PetscQuadrature q;
      PetscInt        Nq, Nb;

      ierr = PetscFEGetTileSizes(fe, NULL, &numBlocks, NULL, &numBatches);CHKERRQ(ierr);
      ierr = PetscFEGetQuadrature(fe, &q);CHKERRQ(ierr);
      ierr = PetscQuadratureGetData(q, NULL, &Nq, NULL, NULL);CHKERRQ(ierr);
      ierr = PetscFEGetDimension(fe, &Nb);CHKERRQ(ierr);
      blockSize = Nb*Nq;
      batchSize = numBlocks * blockSize;
      ierr = PetscFESetTileSizes(fe, blockSize, numBlocks, batchSize, numBatches);CHKERRQ(ierr);
      numChunks = numCells / (numBatches*batchSize);
      Ne        = numChunks*numBatches*batchSize;
      Nr        = numCells % (numBatches*batchSize);
      offset    = numCells - Nr;
      ierr = PetscFEIntegrate(fe, prob, f, Ne, cgeom, u, probAux, a, lintegral);CHKERRQ(ierr);
      ierr = PetscFEIntegrate(fe, prob, f, Nr, &cgeom[offset], &u[offset*totDim], probAux, &a[offset*totDimAux], lintegral);CHKERRQ(ierr);
    } else if (id == PETSCFV_CLASSID) {
      PetscFV  fv = (PetscFV) obj;
      PetscInt foff;
      void   (*obj_func)(const PetscScalar u[], const PetscScalar u_x[], const PetscScalar u_t[], const PetscScalar a[], const PetscScalar a_x[], const PetscScalar a_t[], const PetscReal x[], PetscScalar obj[]);

      ierr = PetscDSGetObjective(prob, f, &obj_func);CHKERRQ(ierr);
      ierr = PetscDSGetFieldOffset(prob, f, &foff);CHKERRQ(ierr);
      if (obj_func) {
        for (c = 0; c < numCells; ++c) {
          /* TODO: Need full pointwise interpolation and get centroid for x argument */
          obj_func(&u[totDim*c+foff], NULL, NULL, NULL, NULL, NULL, NULL, &lintegral[f]);
          lintegral[f] *= vol[c];
        }
      }
    } else SETERRQ1(PetscObjectComm((PetscObject) dm), PETSC_ERR_ARG_WRONG, "Unknown discretization type for field %d", f);
  }
  if (dmAux) {ierr = PetscFree(a);CHKERRQ(ierr);}
  if (mesh->printFEM) {
    ierr = PetscPrintf(PetscObjectComm((PetscObject) dm), "Local integral:");CHKERRQ(ierr);
    for (f = 0; f < Nf; ++f) {ierr = PetscPrintf(PetscObjectComm((PetscObject) dm), " %g", lintegral[f]);CHKERRQ(ierr);}
    ierr = PetscPrintf(PetscObjectComm((PetscObject) dm), "\n");CHKERRQ(ierr);
  }
  ierr = DMRestoreLocalVector(dm, &localX);CHKERRQ(ierr);
  ierr = MPI_Allreduce(lintegral, integral, Nf, MPIU_REAL, MPI_SUM, PetscObjectComm((PetscObject) dm));CHKERRQ(ierr);
  ierr = PetscFree4(lintegral,u,cgeom,vol);CHKERRQ(ierr);
  ierr = PetscLogEventEnd(DMPLEX_IntegralFEM,dm,0,0,0);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "DMPlexComputeInterpolatorFEM"
/*@
  DMPlexComputeInterpolatorFEM - Form the local portion of the interpolation matrix I from the coarse DM to the uniformly refined DM.

  Input Parameters:
+ dmf  - The fine mesh
. dmc  - The coarse mesh
- user - The user context

  Output Parameter:
. In  - The interpolation matrix

  Note:
  The first member of the user context must be an FEMContext.

  We form the residual one batch of elements at a time. This allows us to offload work onto an accelerator,
  like a GPU, or vectorize on a multicore machine.

  Level: developer

.seealso: DMPlexComputeJacobianFEM()
@*/
PetscErrorCode DMPlexComputeInterpolatorFEM(DM dmc, DM dmf, Mat In, void *user)
{
  DM_Plex          *mesh  = (DM_Plex *) dmc->data;
  const char       *name  = "Interpolator";
  PetscDS           prob;
  PetscFE          *feRef;
  PetscSection      fsection, fglobalSection;
  PetscSection      csection, cglobalSection;
  PetscScalar      *elemMat;
  PetscInt          dim, Nf, f, fieldI, fieldJ, offsetI, offsetJ, cStart, cEnd, cEndInterior, c;
  PetscInt          cTotDim, rTotDim = 0;
  PetscErrorCode    ierr;

  PetscFunctionBegin;
  ierr = PetscLogEventBegin(DMPLEX_InterpolatorFEM,dmc,dmf,0,0);CHKERRQ(ierr);
  ierr = DMGetDimension(dmf, &dim);CHKERRQ(ierr);
  ierr = DMGetDefaultSection(dmf, &fsection);CHKERRQ(ierr);
  ierr = DMGetDefaultGlobalSection(dmf, &fglobalSection);CHKERRQ(ierr);
  ierr = DMGetDefaultSection(dmc, &csection);CHKERRQ(ierr);
  ierr = DMGetDefaultGlobalSection(dmc, &cglobalSection);CHKERRQ(ierr);
  ierr = PetscSectionGetNumFields(fsection, &Nf);CHKERRQ(ierr);
  ierr = DMPlexGetHeightStratum(dmc, 0, &cStart, &cEnd);CHKERRQ(ierr);
  ierr = DMPlexGetHybridBounds(dmc, &cEndInterior, NULL, NULL, NULL);CHKERRQ(ierr);
  cEnd = cEndInterior < 0 ? cEnd : cEndInterior;
  ierr = DMGetDS(dmf, &prob);CHKERRQ(ierr);
  ierr = PetscMalloc1(Nf,&feRef);CHKERRQ(ierr);
  for (f = 0; f < Nf; ++f) {
    PetscFE  fe;
    PetscInt rNb, Nc;

    ierr = PetscDSGetDiscretization(prob, f, (PetscObject *) &fe);CHKERRQ(ierr);
    ierr = PetscFERefine(fe, &feRef[f]);CHKERRQ(ierr);
    ierr = PetscFEGetDimension(feRef[f], &rNb);CHKERRQ(ierr);
    ierr = PetscFEGetNumComponents(fe, &Nc);CHKERRQ(ierr);
    rTotDim += rNb*Nc;
  }
  ierr = PetscDSGetTotalDimension(prob, &cTotDim);CHKERRQ(ierr);
  ierr = PetscMalloc1(rTotDim*cTotDim,&elemMat);CHKERRQ(ierr);
  ierr = PetscMemzero(elemMat, rTotDim*cTotDim * sizeof(PetscScalar));CHKERRQ(ierr);
  for (fieldI = 0, offsetI = 0; fieldI < Nf; ++fieldI) {
    PetscDualSpace   Qref;
    PetscQuadrature  f;
    const PetscReal *qpoints, *qweights;
    PetscReal       *points;
    PetscInt         npoints = 0, Nc, Np, fpdim, i, k, p, d;

    /* Compose points from all dual basis functionals */
    ierr = PetscFEGetDualSpace(feRef[fieldI], &Qref);CHKERRQ(ierr);
    ierr = PetscFEGetNumComponents(feRef[fieldI], &Nc);CHKERRQ(ierr);
    ierr = PetscDualSpaceGetDimension(Qref, &fpdim);CHKERRQ(ierr);
    for (i = 0; i < fpdim; ++i) {
      ierr = PetscDualSpaceGetFunctional(Qref, i, &f);CHKERRQ(ierr);
      ierr = PetscQuadratureGetData(f, NULL, &Np, NULL, NULL);CHKERRQ(ierr);
      npoints += Np;
    }
    ierr = PetscMalloc1(npoints*dim,&points);CHKERRQ(ierr);
    for (i = 0, k = 0; i < fpdim; ++i) {
      ierr = PetscDualSpaceGetFunctional(Qref, i, &f);CHKERRQ(ierr);
      ierr = PetscQuadratureGetData(f, NULL, &Np, &qpoints, NULL);CHKERRQ(ierr);
      for (p = 0; p < Np; ++p, ++k) for (d = 0; d < dim; ++d) points[k*dim+d] = qpoints[p*dim+d];
    }

    for (fieldJ = 0, offsetJ = 0; fieldJ < Nf; ++fieldJ) {
      PetscFE    fe;
      PetscReal *B;
      PetscInt   NcJ, cpdim, j;

      /* Evaluate basis at points */
      ierr = PetscDSGetDiscretization(prob, fieldJ, (PetscObject *) &fe);CHKERRQ(ierr);
      ierr = PetscFEGetNumComponents(fe, &NcJ);CHKERRQ(ierr);
      ierr = PetscFEGetDimension(fe, &cpdim);CHKERRQ(ierr);
      /* For now, fields only interpolate themselves */
      if (fieldI == fieldJ) {
        if (Nc != NcJ) SETERRQ2(PETSC_COMM_SELF, PETSC_ERR_ARG_WRONG, "Number of components in fine space field %d does not match coarse field %d", Nc, NcJ);
        ierr = PetscFEGetTabulation(fe, npoints, points, &B, NULL, NULL);CHKERRQ(ierr);
        for (i = 0, k = 0; i < fpdim; ++i) {
          ierr = PetscDualSpaceGetFunctional(Qref, i, &f);CHKERRQ(ierr);
          ierr = PetscQuadratureGetData(f, NULL, &Np, NULL, &qweights);CHKERRQ(ierr);
          for (p = 0; p < Np; ++p, ++k) {
            for (j = 0; j < cpdim; ++j) {
              for (c = 0; c < Nc; ++c) elemMat[(offsetI + i*Nc + c)*cTotDim + offsetJ + j*NcJ + c] += B[k*cpdim*NcJ+j*Nc+c]*qweights[p];
            }
          }
        }
        ierr = PetscFERestoreTabulation(fe, npoints, points, &B, NULL, NULL);CHKERRQ(ierr);CHKERRQ(ierr);
      }
      offsetJ += cpdim*NcJ;
    }
    offsetI += fpdim*Nc;
    ierr = PetscFree(points);CHKERRQ(ierr);
  }
  if (mesh->printFEM > 1) {ierr = DMPrintCellMatrix(0, name, rTotDim, cTotDim, elemMat);CHKERRQ(ierr);}
  /* Preallocate matrix */
  {
    PetscHashJK ht;
    PetscLayout rLayout;
    PetscInt   *dnz, *onz, *cellCIndices, *cellFIndices;
    PetscInt    locRows, rStart, rEnd, cell, r;

    ierr = MatGetLocalSize(In, &locRows, NULL);CHKERRQ(ierr);
    ierr = PetscLayoutCreate(PetscObjectComm((PetscObject) In), &rLayout);CHKERRQ(ierr);
    ierr = PetscLayoutSetLocalSize(rLayout, locRows);CHKERRQ(ierr);
    ierr = PetscLayoutSetBlockSize(rLayout, 1);CHKERRQ(ierr);
    ierr = PetscLayoutSetUp(rLayout);CHKERRQ(ierr);
    ierr = PetscLayoutGetRange(rLayout, &rStart, &rEnd);CHKERRQ(ierr);
    ierr = PetscLayoutDestroy(&rLayout);CHKERRQ(ierr);
    ierr = PetscCalloc4(locRows,&dnz,locRows,&onz,cTotDim,&cellCIndices,rTotDim,&cellFIndices);CHKERRQ(ierr);
    ierr = PetscHashJKCreate(&ht);CHKERRQ(ierr);
    for (cell = cStart; cell < cEnd; ++cell) {
      ierr = DMPlexMatGetClosureIndicesRefined(dmf, fsection, fglobalSection, dmc, csection, cglobalSection, cell, cellCIndices, cellFIndices);CHKERRQ(ierr);
      for (r = 0; r < rTotDim; ++r) {
        PetscHashJKKey  key;
        PetscHashJKIter missing, iter;

        key.j = cellFIndices[r];
        if (key.j < 0) continue;
        for (c = 0; c < cTotDim; ++c) {
          key.k = cellCIndices[c];
          if (key.k < 0) continue;
          ierr = PetscHashJKPut(ht, key, &missing, &iter);CHKERRQ(ierr);
          if (missing) {
            ierr = PetscHashJKSet(ht, iter, 1);CHKERRQ(ierr);
            if ((key.k >= rStart) && (key.k < rEnd)) ++dnz[key.j-rStart];
            else                                     ++onz[key.j-rStart];
          }
        }
      }
    }
    ierr = PetscHashJKDestroy(&ht);CHKERRQ(ierr);
    ierr = MatXAIJSetPreallocation(In, 1, dnz, onz, NULL, NULL);CHKERRQ(ierr);
    ierr = MatSetOption(In, MAT_NEW_NONZERO_ALLOCATION_ERR,PETSC_TRUE);CHKERRQ(ierr);
    ierr = PetscFree4(dnz,onz,cellCIndices,cellFIndices);CHKERRQ(ierr);
  }
  /* Fill matrix */
  ierr = MatZeroEntries(In);CHKERRQ(ierr);
  for (c = cStart; c < cEnd; ++c) {
    ierr = DMPlexMatSetClosureRefined(dmf, fsection, fglobalSection, dmc, csection, cglobalSection, In, c, elemMat, INSERT_VALUES);CHKERRQ(ierr);
  }
  for (f = 0; f < Nf; ++f) {ierr = PetscFEDestroy(&feRef[f]);CHKERRQ(ierr);}
  ierr = PetscFree(feRef);CHKERRQ(ierr);
  ierr = PetscFree(elemMat);CHKERRQ(ierr);
  ierr = MatAssemblyBegin(In, MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  ierr = MatAssemblyEnd(In, MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  if (mesh->printFEM) {
    ierr = PetscPrintf(PETSC_COMM_WORLD, "%s:\n", name);CHKERRQ(ierr);
    ierr = MatChop(In, 1.0e-10);CHKERRQ(ierr);
    ierr = MatView(In, PETSC_VIEWER_STDOUT_WORLD);CHKERRQ(ierr);
  }
  ierr = PetscLogEventEnd(DMPLEX_InterpolatorFEM,dmc,dmf,0,0);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

#undef __FUNCT__
#define __FUNCT__ "DMPlexComputeInjectorFEM"
PetscErrorCode DMPlexComputeInjectorFEM(DM dmc, DM dmf, VecScatter *sc, void *user)
{
  PetscDS        prob;
  PetscFE       *feRef;
  Vec            fv, cv;
  IS             fis, cis;
  PetscSection   fsection, fglobalSection, csection, cglobalSection;
  PetscInt      *cmap, *cellCIndices, *cellFIndices, *cindices, *findices;
  PetscInt       cTotDim, fTotDim = 0, Nf, f, field, cStart, cEnd, cEndInterior, c, dim, d, startC, offsetC, offsetF, m;
  PetscErrorCode ierr;

  PetscFunctionBegin;
  ierr = PetscLogEventBegin(DMPLEX_InjectorFEM,dmc,dmf,0,0);CHKERRQ(ierr);
  ierr = DMGetDimension(dmf, &dim);CHKERRQ(ierr);
  ierr = DMGetDefaultSection(dmf, &fsection);CHKERRQ(ierr);
  ierr = DMGetDefaultGlobalSection(dmf, &fglobalSection);CHKERRQ(ierr);
  ierr = DMGetDefaultSection(dmc, &csection);CHKERRQ(ierr);
  ierr = DMGetDefaultGlobalSection(dmc, &cglobalSection);CHKERRQ(ierr);
  ierr = PetscSectionGetNumFields(fsection, &Nf);CHKERRQ(ierr);
  ierr = DMPlexGetHeightStratum(dmc, 0, &cStart, &cEnd);CHKERRQ(ierr);
  ierr = DMPlexGetHybridBounds(dmc, &cEndInterior, NULL, NULL, NULL);CHKERRQ(ierr);
  cEnd = cEndInterior < 0 ? cEnd : cEndInterior;
  ierr = DMGetDS(dmc, &prob);CHKERRQ(ierr);
  ierr = PetscMalloc1(Nf,&feRef);CHKERRQ(ierr);
  for (f = 0; f < Nf; ++f) {
    PetscFE  fe;
    PetscInt fNb, Nc;

    ierr = PetscDSGetDiscretization(prob, f, (PetscObject *) &fe);CHKERRQ(ierr);
    ierr = PetscFERefine(fe, &feRef[f]);CHKERRQ(ierr);
    ierr = PetscFEGetDimension(feRef[f], &fNb);CHKERRQ(ierr);
    ierr = PetscFEGetNumComponents(fe, &Nc);CHKERRQ(ierr);
    fTotDim += fNb*Nc;
  }
  ierr = PetscDSGetTotalDimension(prob, &cTotDim);CHKERRQ(ierr);
  ierr = PetscMalloc1(cTotDim,&cmap);CHKERRQ(ierr);
  for (field = 0, offsetC = 0, offsetF = 0; field < Nf; ++field) {
    PetscFE        feC;
    PetscDualSpace QF, QC;
    PetscInt       NcF, NcC, fpdim, cpdim;

    ierr = PetscDSGetDiscretization(prob, field, (PetscObject *) &feC);CHKERRQ(ierr);
    ierr = PetscFEGetNumComponents(feC, &NcC);CHKERRQ(ierr);
    ierr = PetscFEGetNumComponents(feRef[field], &NcF);CHKERRQ(ierr);
    if (NcF != NcC) SETERRQ2(PETSC_COMM_SELF, PETSC_ERR_ARG_WRONG, "Number of components in fine space field %d does not match coarse field %d", NcF, NcC);
    ierr = PetscFEGetDualSpace(feRef[field], &QF);CHKERRQ(ierr);
    ierr = PetscDualSpaceGetDimension(QF, &fpdim);CHKERRQ(ierr);
    ierr = PetscFEGetDualSpace(feC, &QC);CHKERRQ(ierr);
    ierr = PetscDualSpaceGetDimension(QC, &cpdim);CHKERRQ(ierr);
    for (c = 0; c < cpdim; ++c) {
      PetscQuadrature  cfunc;
      const PetscReal *cqpoints;
      PetscInt         NpC;

      ierr = PetscDualSpaceGetFunctional(QC, c, &cfunc);CHKERRQ(ierr);
      ierr = PetscQuadratureGetData(cfunc, NULL, &NpC, &cqpoints, NULL);CHKERRQ(ierr);
      if (NpC != 1) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_WRONG, "Do not know how to do injection for moments");
      for (f = 0; f < fpdim; ++f) {
        PetscQuadrature  ffunc;
        const PetscReal *fqpoints;
        PetscReal        sum = 0.0;
        PetscInt         NpF, comp;

        ierr = PetscDualSpaceGetFunctional(QF, f, &ffunc);CHKERRQ(ierr);
        ierr = PetscQuadratureGetData(ffunc, NULL, &NpF, &fqpoints, NULL);CHKERRQ(ierr);
        if (NpC != NpF) continue;
        for (d = 0; d < dim; ++d) sum += PetscAbsReal(cqpoints[d] - fqpoints[d]);
        if (sum > 1.0e-9) continue;
        for (comp = 0; comp < NcC; ++comp) {
          cmap[(offsetC+c)*NcC+comp] = (offsetF+f)*NcF+comp;
        }
        break;
      }
    }
    offsetC += cpdim*NcC;
    offsetF += fpdim*NcF;
  }
  for (f = 0; f < Nf; ++f) {ierr = PetscFEDestroy(&feRef[f]);CHKERRQ(ierr);}
  ierr = PetscFree(feRef);CHKERRQ(ierr);

  ierr = DMGetGlobalVector(dmf, &fv);CHKERRQ(ierr);
  ierr = DMGetGlobalVector(dmc, &cv);CHKERRQ(ierr);
  ierr = VecGetOwnershipRange(cv, &startC, NULL);CHKERRQ(ierr);
  ierr = PetscSectionGetConstrainedStorageSize(cglobalSection, &m);CHKERRQ(ierr);
  ierr = PetscMalloc2(cTotDim,&cellCIndices,fTotDim,&cellFIndices);CHKERRQ(ierr);
  ierr = PetscMalloc1(m,&cindices);CHKERRQ(ierr);
  ierr = PetscMalloc1(m,&findices);CHKERRQ(ierr);
  for (d = 0; d < m; ++d) cindices[d] = findices[d] = -1;
  for (c = cStart; c < cEnd; ++c) {
    ierr = DMPlexMatGetClosureIndicesRefined(dmf, fsection, fglobalSection, dmc, csection, cglobalSection, c, cellCIndices, cellFIndices);CHKERRQ(ierr);
    for (d = 0; d < cTotDim; ++d) {
      if (cellCIndices[d] < 0) continue;
      if ((findices[cellCIndices[d]-startC] >= 0) && (findices[cellCIndices[d]-startC] != cellFIndices[cmap[d]])) SETERRQ3(PETSC_COMM_SELF, PETSC_ERR_PLIB, "Coarse dof %d maps to both %d and %d", cindices[cellCIndices[d]-startC], findices[cellCIndices[d]-startC], cellFIndices[cmap[d]]);
      cindices[cellCIndices[d]-startC] = cellCIndices[d];
      findices[cellCIndices[d]-startC] = cellFIndices[cmap[d]];
    }
  }
  ierr = PetscFree(cmap);CHKERRQ(ierr);
  ierr = PetscFree2(cellCIndices,cellFIndices);CHKERRQ(ierr);

  ierr = ISCreateGeneral(PETSC_COMM_SELF, m, cindices, PETSC_OWN_POINTER, &cis);CHKERRQ(ierr);
  ierr = ISCreateGeneral(PETSC_COMM_SELF, m, findices, PETSC_OWN_POINTER, &fis);CHKERRQ(ierr);
  ierr = VecScatterCreate(cv, cis, fv, fis, sc);CHKERRQ(ierr);
  ierr = ISDestroy(&cis);CHKERRQ(ierr);
  ierr = ISDestroy(&fis);CHKERRQ(ierr);
  ierr = DMRestoreGlobalVector(dmf, &fv);CHKERRQ(ierr);
  ierr = DMRestoreGlobalVector(dmc, &cv);CHKERRQ(ierr);
  ierr = PetscLogEventEnd(DMPLEX_InjectorFEM,dmc,dmf,0,0);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}