xref: /petsc/src/snes/utils/dmplexsnes.c (revision ab72731f676526bea82774ba2406315ead495b83)

#include <petsc/private/dmpleximpl.h> /*I "petscdmplex.h" I*/
#include <petsc/private/snesimpl.h>   /*I "petscsnes.h"   I*/
#include <petscds.h>
#include <petsc/private/petscimpl.h>
#include <petsc/private/petscfeimpl.h>

#ifdef PETSC_HAVE_LIBCEED
  #include <petscdmceed.h>
  #include <petscdmplexceed.h>
#endif

static void pressure_Private(PetscInt dim, PetscInt Nf, PetscInt NfAux, const PetscInt uOff[], const PetscInt uOff_x[], const PetscScalar u[], const PetscScalar u_t[], const PetscScalar u_x[], const PetscInt aOff[], const PetscInt aOff_x[], const PetscScalar a[], const PetscScalar a_t[], const PetscScalar a_x[], PetscReal t, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar p[])
{
  p[0] = u[uOff[1]];
}

/*
  SNESCorrectDiscretePressure_Private - Add a vector in the nullspace to make the continuum integral of the pressure field equal to zero.
  This is normally used only to evaluate convergence rates for the pressure accurately.

  Collective

  Input Parameters:
+ snes      - The SNES
. pfield    - The field number for pressure
. nullspace - The pressure nullspace
. u         - The solution vector
- ctx       - An optional user context

  Output Parameter:
. u         - The solution with a continuum pressure integral of zero

  Level: developer

  Notes:
  If int(u) = a and int(n) = b, then int(u - a/b n) = a - a/b b = 0. We assume that the nullspace is a single vector given explicitly.

.seealso: `SNESConvergedCorrectPressure()`
*/
static PetscErrorCode SNESCorrectDiscretePressure_Private(SNES snes, PetscInt pfield, MatNullSpace nullspace, Vec u, void *ctx)
{
  DM          dm;
  PetscDS     ds;
  const Vec  *nullvecs;
  PetscScalar pintd, *intc, *intn;
  MPI_Comm    comm;
  PetscInt    Nf, Nv;

  PetscFunctionBegin;
  PetscCall(PetscObjectGetComm((PetscObject)snes, &comm));
  PetscCall(SNESGetDM(snes, &dm));
  PetscCheck(dm, comm, PETSC_ERR_ARG_WRONG, "Cannot compute test without a SNES DM");
  PetscCheck(nullspace, comm, PETSC_ERR_ARG_WRONG, "Cannot compute test without a Jacobian nullspace");
  PetscCall(DMGetDS(dm, &ds));
  PetscCall(PetscDSSetObjective(ds, pfield, pressure_Private));
  PetscCall(MatNullSpaceGetVecs(nullspace, NULL, &Nv, &nullvecs));
  PetscCheck(Nv == 1, comm, PETSC_ERR_ARG_OUTOFRANGE, "Can only handle a single null vector for pressure, not %" PetscInt_FMT, Nv);
  PetscCall(VecDot(nullvecs[0], u, &pintd));
  PetscCheck(PetscAbsScalar(pintd) <= PETSC_SMALL, comm, PETSC_ERR_ARG_WRONG, "Discrete integral of pressure: %g", (double)PetscRealPart(pintd));
  PetscCall(PetscDSGetNumFields(ds, &Nf));
  PetscCall(PetscMalloc2(Nf, &intc, Nf, &intn));
  PetscCall(DMPlexComputeIntegralFEM(dm, nullvecs[0], intn, ctx));
  PetscCall(DMPlexComputeIntegralFEM(dm, u, intc, ctx));
  PetscCall(VecAXPY(u, -intc[pfield] / intn[pfield], nullvecs[0]));
#if defined(PETSC_USE_DEBUG)
  PetscCall(DMPlexComputeIntegralFEM(dm, u, intc, ctx));
  PetscCheck(PetscAbsScalar(intc[pfield]) <= PETSC_SMALL, comm, PETSC_ERR_ARG_WRONG, "Continuum integral of pressure after correction: %g", (double)PetscRealPart(intc[pfield]));
#endif
  PetscCall(PetscFree2(intc, intn));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@C
  SNESConvergedCorrectPressure - The regular `SNES` convergence test that, up on convergence, adds a vector in the nullspace
  to make the continuum integral of the pressure field equal to zero.

  Logically Collective

  Input Parameters:
+ snes  - the `SNES` context
. it    - the iteration (0 indicates before any Newton steps)
. xnorm - 2-norm of current iterate
. gnorm - 2-norm of current step
. f     - 2-norm of function at current iterate
- ctx   - Optional user context

  Output Parameter:
. reason - `SNES_CONVERGED_ITERATING`, `SNES_CONVERGED_ITS`, or `SNES_DIVERGED_FNORM_NAN`

  Options Database Key:
. -snes_convergence_test correct_pressure - see `SNESSetFromOptions()`

  Level: advanced

  Notes:
  In order to use this convergence test, you must set up several PETSc structures. First fields must be added to the `DM`, and a `PetscDS`
  must be created with discretizations of those fields. We currently assume that the pressure field has index 1.
  The pressure field must have a nullspace, likely created using the `DMSetNullSpaceConstructor()` interface.
  Last we must be able to integrate the pressure over the domain, so the `DM` attached to the SNES `must` be a `DMPLEX` at this time.

  Developer Note:
  This is a total misuse of the `SNES` convergence test handling system. It should be removed. Perhaps a `SNESSetPostSolve()` could
  be constructed to handle this process.

.seealso: `SNES`, `DM`, `SNESConvergedDefault()`, `SNESSetConvergenceTest()`, `DMSetNullSpaceConstructor()`
@*/
PetscErrorCode SNESConvergedCorrectPressure(SNES snes, PetscInt it, PetscReal xnorm, PetscReal gnorm, PetscReal f, SNESConvergedReason *reason, void *ctx)
{
  PetscBool monitorIntegral = PETSC_FALSE;

  PetscFunctionBegin;
  PetscCall(SNESConvergedDefault(snes, it, xnorm, gnorm, f, reason, ctx));
  if (monitorIntegral) {
    Mat          J;
    Vec          u;
    MatNullSpace nullspace;
    const Vec   *nullvecs;
    PetscScalar  pintd;

    PetscCall(SNESGetSolution(snes, &u));
    PetscCall(SNESGetJacobian(snes, &J, NULL, NULL, NULL));
    PetscCall(MatGetNullSpace(J, &nullspace));
    PetscCall(MatNullSpaceGetVecs(nullspace, NULL, NULL, &nullvecs));
    PetscCall(VecDot(nullvecs[0], u, &pintd));
    PetscCall(PetscInfo(snes, "SNES: Discrete integral of pressure: %g\n", (double)PetscRealPart(pintd)));
  }
  if (*reason > 0) {
    Mat          J;
    Vec          u;
    MatNullSpace nullspace;
    PetscInt     pfield = 1;

    PetscCall(SNESGetSolution(snes, &u));
    PetscCall(SNESGetJacobian(snes, &J, NULL, NULL, NULL));
    PetscCall(MatGetNullSpace(J, &nullspace));
    PetscCall(SNESCorrectDiscretePressure_Private(snes, pfield, nullspace, u, ctx));
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

/************************** Interpolation *******************************/

static PetscErrorCode DMSNESConvertPlex(DM dm, DM *plex, PetscBool copy)
{
  PetscBool isPlex;

  PetscFunctionBegin;
  PetscCall(PetscObjectTypeCompare((PetscObject)dm, DMPLEX, &isPlex));
  if (isPlex) {
    *plex = dm;
    PetscCall(PetscObjectReference((PetscObject)dm));
  } else {
    PetscCall(PetscObjectQuery((PetscObject)dm, "dm_plex", (PetscObject *)plex));
    if (!*plex) {
      PetscCall(DMConvert(dm, DMPLEX, plex));
      PetscCall(PetscObjectCompose((PetscObject)dm, "dm_plex", (PetscObject)*plex));
      if (copy) {
        PetscCall(DMCopyDMSNES(dm, *plex));
        PetscCall(DMCopyAuxiliaryVec(dm, *plex));
      }
    } else {
      PetscCall(PetscObjectReference((PetscObject)*plex));
    }
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@C
  DMInterpolationCreate - Creates a `DMInterpolationInfo` context

  Collective

  Input Parameter:
. comm - the communicator

  Output Parameter:
. ctx - the context

  Level: beginner

.seealso: `DMInterpolationInfo`, `DMInterpolationEvaluate()`, `DMInterpolationAddPoints()`, `DMInterpolationDestroy()`
@*/
PetscErrorCode DMInterpolationCreate(MPI_Comm comm, DMInterpolationInfo *ctx)
{
  PetscFunctionBegin;
  PetscAssertPointer(ctx, 2);
  PetscCall(PetscNew(ctx));

  (*ctx)->comm   = comm;
  (*ctx)->dim    = -1;
  (*ctx)->nInput = 0;
  (*ctx)->points = NULL;
  (*ctx)->cells  = NULL;
  (*ctx)->n      = -1;
  (*ctx)->coords = NULL;
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@C
  DMInterpolationSetDim - Sets the spatial dimension for the interpolation context

  Not Collective

  Input Parameters:
+ ctx - the context
- dim - the spatial dimension

  Level: intermediate

.seealso: `DMInterpolationInfo`, `DMInterpolationGetDim()`, `DMInterpolationEvaluate()`, `DMInterpolationAddPoints()`
@*/
PetscErrorCode DMInterpolationSetDim(DMInterpolationInfo ctx, PetscInt dim)
{
  PetscFunctionBegin;
  PetscCheck(!(dim < 1) && !(dim > 3), ctx->comm, PETSC_ERR_ARG_OUTOFRANGE, "Invalid dimension for points: %" PetscInt_FMT, dim);
  ctx->dim = dim;
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@C
  DMInterpolationGetDim - Gets the spatial dimension for the interpolation context

  Not Collective

  Input Parameter:
. ctx - the context

  Output Parameter:
. dim - the spatial dimension

  Level: intermediate

.seealso: `DMInterpolationInfo`, `DMInterpolationSetDim()`, `DMInterpolationEvaluate()`, `DMInterpolationAddPoints()`
@*/
PetscErrorCode DMInterpolationGetDim(DMInterpolationInfo ctx, PetscInt *dim)
{
  PetscFunctionBegin;
  PetscAssertPointer(dim, 2);
  *dim = ctx->dim;
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@C
  DMInterpolationSetDof - Sets the number of fields interpolated at a point for the interpolation context

  Not Collective

  Input Parameters:
+ ctx - the context
- dof - the number of fields

  Level: intermediate

.seealso: `DMInterpolationInfo`, `DMInterpolationGetDof()`, `DMInterpolationEvaluate()`, `DMInterpolationAddPoints()`
@*/
PetscErrorCode DMInterpolationSetDof(DMInterpolationInfo ctx, PetscInt dof)
{
  PetscFunctionBegin;
  PetscCheck(dof >= 1, ctx->comm, PETSC_ERR_ARG_OUTOFRANGE, "Invalid number of components: %" PetscInt_FMT, dof);
  ctx->dof = dof;
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@C
  DMInterpolationGetDof - Gets the number of fields interpolated at a point for the interpolation context

  Not Collective

  Input Parameter:
. ctx - the context

  Output Parameter:
. dof - the number of fields

  Level: intermediate

.seealso: `DMInterpolationInfo`, `DMInterpolationSetDof()`, `DMInterpolationEvaluate()`, `DMInterpolationAddPoints()`
@*/
PetscErrorCode DMInterpolationGetDof(DMInterpolationInfo ctx, PetscInt *dof)
{
  PetscFunctionBegin;
  PetscAssertPointer(dof, 2);
  *dof = ctx->dof;
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@C
  DMInterpolationAddPoints - Add points at which we will interpolate the fields

  Not Collective

  Input Parameters:
+ ctx    - the context
. n      - the number of points
- points - the coordinates for each point, an array of size n * dim

  Level: intermediate

  Note:
  The coordinate information is copied.

.seealso: `DMInterpolationInfo`, `DMInterpolationSetDim()`, `DMInterpolationEvaluate()`, `DMInterpolationCreate()`
@*/
PetscErrorCode DMInterpolationAddPoints(DMInterpolationInfo ctx, PetscInt n, PetscReal points[])
{
  PetscFunctionBegin;
  PetscCheck(ctx->dim >= 0, ctx->comm, PETSC_ERR_ARG_WRONGSTATE, "The spatial dimension has not been set");
  PetscCheck(!ctx->points, ctx->comm, PETSC_ERR_ARG_WRONGSTATE, "Cannot add points multiple times yet");
  ctx->nInput = n;

  PetscCall(PetscMalloc1(n * ctx->dim, &ctx->points));
  PetscCall(PetscArraycpy(ctx->points, points, n * ctx->dim));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@C
  DMInterpolationSetUp - Compute spatial indices for point location during interpolation

  Collective

  Input Parameters:
+ ctx                 - the context
. dm                  - the `DM` for the function space used for interpolation
. redundantPoints     - If `PETSC_TRUE`, all processes are passing in the same array of points. Otherwise, points need to be communicated among processes.
- ignoreOutsideDomain - If `PETSC_TRUE`, ignore points outside the domain, otherwise return an error

  Level: intermediate

.seealso: `DMInterpolationInfo`, `DMInterpolationEvaluate()`, `DMInterpolationAddPoints()`, `DMInterpolationCreate()`
@*/
PetscErrorCode DMInterpolationSetUp(DMInterpolationInfo ctx, DM dm, PetscBool redundantPoints, PetscBool ignoreOutsideDomain)
{
  MPI_Comm           comm = ctx->comm;
  PetscScalar       *a;
  PetscInt           p, q, i;
  PetscMPIInt        rank, size;
  Vec                pointVec;
  PetscSF            cellSF;
  PetscLayout        layout;
  PetscReal         *globalPoints;
  PetscScalar       *globalPointsScalar;
  const PetscInt    *ranges;
  PetscMPIInt       *counts, *displs;
  const PetscSFNode *foundCells;
  const PetscInt    *foundPoints;
  PetscMPIInt       *foundProcs, *globalProcs;
  PetscInt           n, N, numFound;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(dm, DM_CLASSID, 2);
  PetscCallMPI(MPI_Comm_size(comm, &size));
  PetscCallMPI(MPI_Comm_rank(comm, &rank));
  PetscCheck(ctx->dim >= 0, comm, PETSC_ERR_ARG_WRONGSTATE, "The spatial dimension has not been set");
  /* Locate points */
  n = ctx->nInput;
  if (!redundantPoints) {
    PetscCall(PetscLayoutCreate(comm, &layout));
    PetscCall(PetscLayoutSetBlockSize(layout, 1));
    PetscCall(PetscLayoutSetLocalSize(layout, n));
    PetscCall(PetscLayoutSetUp(layout));
    PetscCall(PetscLayoutGetSize(layout, &N));
    /* Communicate all points to all processes */
    PetscCall(PetscMalloc3(N * ctx->dim, &globalPoints, size, &counts, size, &displs));
    PetscCall(PetscLayoutGetRanges(layout, &ranges));
    for (p = 0; p < size; ++p) {
      counts[p] = (ranges[p + 1] - ranges[p]) * ctx->dim;
      displs[p] = ranges[p] * ctx->dim;
    }
    PetscCallMPI(MPI_Allgatherv(ctx->points, n * ctx->dim, MPIU_REAL, globalPoints, counts, displs, MPIU_REAL, comm));
  } else {
    N            = n;
    globalPoints = ctx->points;
    counts = displs = NULL;
    layout          = NULL;
  }
#if 0
  PetscCall(PetscMalloc3(N,&foundCells,N,&foundProcs,N,&globalProcs));
  /* foundCells[p] = m->locatePoint(&globalPoints[p*ctx->dim]); */
#else
  #if defined(PETSC_USE_COMPLEX)
  PetscCall(PetscMalloc1(N * ctx->dim, &globalPointsScalar));
  for (i = 0; i < N * ctx->dim; i++) globalPointsScalar[i] = globalPoints[i];
  #else
  globalPointsScalar = globalPoints;
  #endif
  PetscCall(VecCreateSeqWithArray(PETSC_COMM_SELF, ctx->dim, N * ctx->dim, globalPointsScalar, &pointVec));
  PetscCall(PetscMalloc2(N, &foundProcs, N, &globalProcs));
  for (p = 0; p < N; ++p) foundProcs[p] = size;
  cellSF = NULL;
  PetscCall(DMLocatePoints(dm, pointVec, DM_POINTLOCATION_REMOVE, &cellSF));
  PetscCall(PetscSFGetGraph(cellSF, NULL, &numFound, &foundPoints, &foundCells));
#endif
  for (p = 0; p < numFound; ++p) {
    if (foundCells[p].index >= 0) foundProcs[foundPoints ? foundPoints[p] : p] = rank;
  }
  /* Let the lowest rank process own each point */
  PetscCall(MPIU_Allreduce(foundProcs, globalProcs, N, MPI_INT, MPI_MIN, comm));
  ctx->n = 0;
  for (p = 0; p < N; ++p) {
    if (globalProcs[p] == size) {
      PetscCheck(ignoreOutsideDomain, comm, PETSC_ERR_PLIB, "Point %" PetscInt_FMT ": %g %g %g not located in mesh", p, (double)globalPoints[p * ctx->dim + 0], (double)(ctx->dim > 1 ? globalPoints[p * ctx->dim + 1] : 0.0),
                 (double)(ctx->dim > 2 ? globalPoints[p * ctx->dim + 2] : 0.0));
      if (rank == 0) ++ctx->n;
    } else if (globalProcs[p] == rank) ++ctx->n;
  }
  /* Create coordinates vector and array of owned cells */
  PetscCall(PetscMalloc1(ctx->n, &ctx->cells));
  PetscCall(VecCreate(comm, &ctx->coords));
  PetscCall(VecSetSizes(ctx->coords, ctx->n * ctx->dim, PETSC_DECIDE));
  PetscCall(VecSetBlockSize(ctx->coords, ctx->dim));
  PetscCall(VecSetType(ctx->coords, VECSTANDARD));
  PetscCall(VecGetArray(ctx->coords, &a));
  for (p = 0, q = 0, i = 0; p < N; ++p) {
    if (globalProcs[p] == rank) {
      PetscInt d;

      for (d = 0; d < ctx->dim; ++d, ++i) a[i] = globalPoints[p * ctx->dim + d];
      ctx->cells[q] = foundCells[q].index;
      ++q;
    }
    if (globalProcs[p] == size && rank == 0) {
      PetscInt d;

      for (d = 0; d < ctx->dim; ++d, ++i) a[i] = 0.;
      ctx->cells[q] = -1;
      ++q;
    }
  }
  PetscCall(VecRestoreArray(ctx->coords, &a));
#if 0
  PetscCall(PetscFree3(foundCells,foundProcs,globalProcs));
#else
  PetscCall(PetscFree2(foundProcs, globalProcs));
  PetscCall(PetscSFDestroy(&cellSF));
  PetscCall(VecDestroy(&pointVec));
#endif
  if ((void *)globalPointsScalar != (void *)globalPoints) PetscCall(PetscFree(globalPointsScalar));
  if (!redundantPoints) PetscCall(PetscFree3(globalPoints, counts, displs));
  PetscCall(PetscLayoutDestroy(&layout));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@C
  DMInterpolationGetCoordinates - Gets a `Vec` with the coordinates of each interpolation point

  Collective

  Input Parameter:
. ctx - the context

  Output Parameter:
. coordinates - the coordinates of interpolation points

  Level: intermediate

  Note:
  The local vector entries correspond to interpolation points lying on this process, according to the associated `DM`.
  This is a borrowed vector that the user should not destroy.

.seealso: `DMInterpolationInfo`, `DMInterpolationEvaluate()`, `DMInterpolationAddPoints()`, `DMInterpolationCreate()`
@*/
PetscErrorCode DMInterpolationGetCoordinates(DMInterpolationInfo ctx, Vec *coordinates)
{
  PetscFunctionBegin;
  PetscAssertPointer(coordinates, 2);
  PetscCheck(ctx->coords, ctx->comm, PETSC_ERR_ARG_WRONGSTATE, "The interpolation context has not been setup.");
  *coordinates = ctx->coords;
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@C
  DMInterpolationGetVector - Gets a `Vec` which can hold all the interpolated field values

  Collective

  Input Parameter:
. ctx - the context

  Output Parameter:
. v - a vector capable of holding the interpolated field values

  Level: intermediate

  Note:
  This vector should be returned using `DMInterpolationRestoreVector()`.

.seealso: `DMInterpolationInfo`, `DMInterpolationRestoreVector()`, `DMInterpolationEvaluate()`, `DMInterpolationAddPoints()`, `DMInterpolationCreate()`
@*/
PetscErrorCode DMInterpolationGetVector(DMInterpolationInfo ctx, Vec *v)
{
  PetscFunctionBegin;
  PetscAssertPointer(v, 2);
  PetscCheck(ctx->coords, ctx->comm, PETSC_ERR_ARG_WRONGSTATE, "The interpolation context has not been setup.");
  PetscCall(VecCreate(ctx->comm, v));
  PetscCall(VecSetSizes(*v, ctx->n * ctx->dof, PETSC_DECIDE));
  PetscCall(VecSetBlockSize(*v, ctx->dof));
  PetscCall(VecSetType(*v, VECSTANDARD));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@C
  DMInterpolationRestoreVector - Returns a `Vec` which can hold all the interpolated field values

  Collective

  Input Parameters:
+ ctx - the context
- v   - a vector capable of holding the interpolated field values

  Level: intermediate

.seealso: `DMInterpolationInfo`, `DMInterpolationGetVector()`, `DMInterpolationEvaluate()`, `DMInterpolationAddPoints()`, `DMInterpolationCreate()`
@*/
PetscErrorCode DMInterpolationRestoreVector(DMInterpolationInfo ctx, Vec *v)
{
  PetscFunctionBegin;
  PetscAssertPointer(v, 2);
  PetscCheck(ctx->coords, ctx->comm, PETSC_ERR_ARG_WRONGSTATE, "The interpolation context has not been setup.");
  PetscCall(VecDestroy(v));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static inline PetscErrorCode DMInterpolate_Segment_Private(DMInterpolationInfo ctx, DM dm, PetscInt p, Vec xLocal, Vec v)
{
  const PetscInt     c   = ctx->cells[p];
  const PetscInt     dof = ctx->dof;
  PetscScalar       *x   = NULL;
  const PetscScalar *coords;
  PetscScalar       *a;
  PetscReal          v0, J, invJ, detJ, xir[1];
  PetscInt           xSize, comp;

  PetscFunctionBegin;
  PetscCall(VecGetArrayRead(ctx->coords, &coords));
  PetscCall(VecGetArray(v, &a));
  PetscCall(DMPlexComputeCellGeometryFEM(dm, c, NULL, &v0, &J, &invJ, &detJ));
  PetscCheck(detJ > 0.0, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Invalid determinant %g for element %" PetscInt_FMT, (double)detJ, c);
  xir[0] = invJ * PetscRealPart(coords[p] - v0);
  PetscCall(DMPlexVecGetClosure(dm, NULL, xLocal, c, &xSize, &x));
  if (2 * dof == xSize) {
    for (comp = 0; comp < dof; ++comp) a[p * dof + comp] = x[0 * dof + comp] * (1 - xir[0]) + x[1 * dof + comp] * xir[0];
  } else if (dof == xSize) {
    for (comp = 0; comp < dof; ++comp) a[p * dof + comp] = x[0 * dof + comp];
  } else SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Input closure size %" PetscInt_FMT " must be either %" PetscInt_FMT " or %" PetscInt_FMT, xSize, 2 * dof, dof);
  PetscCall(DMPlexVecRestoreClosure(dm, NULL, xLocal, c, &xSize, &x));
  PetscCall(VecRestoreArray(v, &a));
  PetscCall(VecRestoreArrayRead(ctx->coords, &coords));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static inline PetscErrorCode DMInterpolate_Triangle_Private(DMInterpolationInfo ctx, DM dm, PetscInt p, Vec xLocal, Vec v)
{
  const PetscInt     c = ctx->cells[p];
  PetscScalar       *x = NULL;
  const PetscScalar *coords;
  PetscScalar       *a;
  PetscReal         *v0, *J, *invJ, detJ;
  PetscReal          xi[4];

  PetscFunctionBegin;
  PetscCall(PetscMalloc3(ctx->dim, &v0, ctx->dim * ctx->dim, &J, ctx->dim * ctx->dim, &invJ));
  PetscCall(VecGetArrayRead(ctx->coords, &coords));
  PetscCall(VecGetArray(v, &a));
  PetscCall(DMPlexComputeCellGeometryFEM(dm, c, NULL, v0, J, invJ, &detJ));
  PetscCheck(detJ > 0.0, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Invalid determinant %g for element %" PetscInt_FMT, (double)detJ, c);
  PetscCall(DMPlexVecGetClosure(dm, NULL, xLocal, c, NULL, &x));
  for (PetscInt comp = 0; comp < ctx->dof; ++comp) a[p * ctx->dof + comp] = x[0 * ctx->dof + comp];
  for (PetscInt d = 0; d < ctx->dim; ++d) {
    xi[d] = 0.0;
    for (PetscInt f = 0; f < ctx->dim; ++f) xi[d] += invJ[d * ctx->dim + f] * 0.5 * PetscRealPart(coords[p * ctx->dim + f] - v0[f]);
    for (PetscInt comp = 0; comp < ctx->dof; ++comp) a[p * ctx->dof + comp] += PetscRealPart(x[(d + 1) * ctx->dof + comp] - x[0 * ctx->dof + comp]) * xi[d];
  }
  PetscCall(DMPlexVecRestoreClosure(dm, NULL, xLocal, c, NULL, &x));
  PetscCall(VecRestoreArray(v, &a));
  PetscCall(VecRestoreArrayRead(ctx->coords, &coords));
  PetscCall(PetscFree3(v0, J, invJ));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static inline PetscErrorCode DMInterpolate_Tetrahedron_Private(DMInterpolationInfo ctx, DM dm, PetscInt p, Vec xLocal, Vec v)
{
  const PetscInt     c        = ctx->cells[p];
  const PetscInt     order[3] = {2, 1, 3};
  PetscScalar       *x        = NULL;
  PetscReal         *v0, *J, *invJ, detJ;
  const PetscScalar *coords;
  PetscScalar       *a;
  PetscReal          xi[4];

  PetscFunctionBegin;
  PetscCall(PetscMalloc3(ctx->dim, &v0, ctx->dim * ctx->dim, &J, ctx->dim * ctx->dim, &invJ));
  PetscCall(VecGetArrayRead(ctx->coords, &coords));
  PetscCall(VecGetArray(v, &a));
  PetscCall(DMPlexComputeCellGeometryFEM(dm, c, NULL, v0, J, invJ, &detJ));
  PetscCheck(detJ > 0.0, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Invalid determinant %g for element %" PetscInt_FMT, (double)detJ, c);
  PetscCall(DMPlexVecGetClosure(dm, NULL, xLocal, c, NULL, &x));
  for (PetscInt comp = 0; comp < ctx->dof; ++comp) a[p * ctx->dof + comp] = x[0 * ctx->dof + comp];
  for (PetscInt d = 0; d < ctx->dim; ++d) {
    xi[d] = 0.0;
    for (PetscInt f = 0; f < ctx->dim; ++f) xi[d] += invJ[d * ctx->dim + f] * 0.5 * PetscRealPart(coords[p * ctx->dim + f] - v0[f]);
    for (PetscInt comp = 0; comp < ctx->dof; ++comp) a[p * ctx->dof + comp] += PetscRealPart(x[order[d] * ctx->dof + comp] - x[0 * ctx->dof + comp]) * xi[d];
  }
  PetscCall(DMPlexVecRestoreClosure(dm, NULL, xLocal, c, NULL, &x));
  PetscCall(VecRestoreArray(v, &a));
  PetscCall(VecRestoreArrayRead(ctx->coords, &coords));
  PetscCall(PetscFree3(v0, J, invJ));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static inline PetscErrorCode QuadMap_Private(SNES snes, Vec Xref, Vec Xreal, void *ctx)
{
  const PetscScalar *vertices = (const PetscScalar *)ctx;
  const PetscScalar  x0       = vertices[0];
  const PetscScalar  y0       = vertices[1];
  const PetscScalar  x1       = vertices[2];
  const PetscScalar  y1       = vertices[3];
  const PetscScalar  x2       = vertices[4];
  const PetscScalar  y2       = vertices[5];
  const PetscScalar  x3       = vertices[6];
  const PetscScalar  y3       = vertices[7];
  const PetscScalar  f_1      = x1 - x0;
  const PetscScalar  g_1      = y1 - y0;
  const PetscScalar  f_3      = x3 - x0;
  const PetscScalar  g_3      = y3 - y0;
  const PetscScalar  f_01     = x2 - x1 - x3 + x0;
  const PetscScalar  g_01     = y2 - y1 - y3 + y0;
  const PetscScalar *ref;
  PetscScalar       *real;

  PetscFunctionBegin;
  PetscCall(VecGetArrayRead(Xref, &ref));
  PetscCall(VecGetArray(Xreal, &real));
  {
    const PetscScalar p0 = ref[0];
    const PetscScalar p1 = ref[1];

    real[0] = x0 + f_1 * p0 + f_3 * p1 + f_01 * p0 * p1;
    real[1] = y0 + g_1 * p0 + g_3 * p1 + g_01 * p0 * p1;
  }
  PetscCall(PetscLogFlops(28));
  PetscCall(VecRestoreArrayRead(Xref, &ref));
  PetscCall(VecRestoreArray(Xreal, &real));
  PetscFunctionReturn(PETSC_SUCCESS);
}

#include <petsc/private/dmimpl.h>
static inline PetscErrorCode QuadJacobian_Private(SNES snes, Vec Xref, Mat J, Mat M, void *ctx)
{
  const PetscScalar *vertices = (const PetscScalar *)ctx;
  const PetscScalar  x0       = vertices[0];
  const PetscScalar  y0       = vertices[1];
  const PetscScalar  x1       = vertices[2];
  const PetscScalar  y1       = vertices[3];
  const PetscScalar  x2       = vertices[4];
  const PetscScalar  y2       = vertices[5];
  const PetscScalar  x3       = vertices[6];
  const PetscScalar  y3       = vertices[7];
  const PetscScalar  f_01     = x2 - x1 - x3 + x0;
  const PetscScalar  g_01     = y2 - y1 - y3 + y0;
  const PetscScalar *ref;

  PetscFunctionBegin;
  PetscCall(VecGetArrayRead(Xref, &ref));
  {
    const PetscScalar x       = ref[0];
    const PetscScalar y       = ref[1];
    const PetscInt    rows[2] = {0, 1};
    PetscScalar       values[4];

    values[0] = (x1 - x0 + f_01 * y) * 0.5;
    values[1] = (x3 - x0 + f_01 * x) * 0.5;
    values[2] = (y1 - y0 + g_01 * y) * 0.5;
    values[3] = (y3 - y0 + g_01 * x) * 0.5;
    PetscCall(MatSetValues(J, 2, rows, 2, rows, values, INSERT_VALUES));
  }
  PetscCall(PetscLogFlops(30));
  PetscCall(VecRestoreArrayRead(Xref, &ref));
  PetscCall(MatAssemblyBegin(J, MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(J, MAT_FINAL_ASSEMBLY));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static inline PetscErrorCode DMInterpolate_Quad_Private(DMInterpolationInfo ctx, DM dm, PetscInt p, Vec xLocal, Vec v)
{
  const PetscInt     c   = ctx->cells[p];
  PetscFE            fem = NULL;
  DM                 dmCoord;
  SNES               snes;
  KSP                ksp;
  PC                 pc;
  Vec                coordsLocal, r, ref, real;
  Mat                J;
  PetscTabulation    T = NULL;
  const PetscScalar *coords;
  PetscScalar       *a;
  PetscReal          xir[2] = {0., 0.};
  PetscInt           Nf;
  const PetscInt     dof = ctx->dof;
  PetscScalar       *x = NULL, *vertices = NULL;
  PetscScalar       *xi;
  PetscInt           coordSize, xSize;

  PetscFunctionBegin;
  PetscCall(DMGetNumFields(dm, &Nf));
  if (Nf) {
    PetscObject  obj;
    PetscClassId id;

    PetscCall(DMGetField(dm, 0, NULL, &obj));
    PetscCall(PetscObjectGetClassId(obj, &id));
    if (id == PETSCFE_CLASSID) {
      fem = (PetscFE)obj;
      PetscCall(PetscFECreateTabulation(fem, 1, 1, xir, 0, &T));
    }
  }
  PetscCall(DMGetCoordinatesLocal(dm, &coordsLocal));
  PetscCall(DMGetCoordinateDM(dm, &dmCoord));
  PetscCall(SNESCreate(PETSC_COMM_SELF, &snes));
  PetscCall(SNESSetOptionsPrefix(snes, "quad_interp_"));
  PetscCall(VecCreate(PETSC_COMM_SELF, &r));
  PetscCall(VecSetSizes(r, 2, 2));
  PetscCall(VecSetType(r, dm->vectype));
  PetscCall(VecDuplicate(r, &ref));
  PetscCall(VecDuplicate(r, &real));
  PetscCall(MatCreate(PETSC_COMM_SELF, &J));
  PetscCall(MatSetSizes(J, 2, 2, 2, 2));
  PetscCall(MatSetType(J, MATSEQDENSE));
  PetscCall(MatSetUp(J));
  PetscCall(SNESSetFunction(snes, r, QuadMap_Private, NULL));
  PetscCall(SNESSetJacobian(snes, J, J, QuadJacobian_Private, NULL));
  PetscCall(SNESGetKSP(snes, &ksp));
  PetscCall(KSPGetPC(ksp, &pc));
  PetscCall(PCSetType(pc, PCLU));
  PetscCall(SNESSetFromOptions(snes));

  PetscCall(VecGetArrayRead(ctx->coords, &coords));
  PetscCall(VecGetArray(v, &a));
  PetscCall(DMPlexVecGetClosure(dmCoord, NULL, coordsLocal, c, &coordSize, &vertices));
  PetscCheck(4 * 2 == coordSize, ctx->comm, PETSC_ERR_ARG_SIZ, "Invalid closure size %" PetscInt_FMT " should be %d", coordSize, 4 * 2);
  PetscCall(DMPlexVecGetClosure(dm, NULL, xLocal, c, &xSize, &x));
  PetscCall(SNESSetFunction(snes, NULL, NULL, vertices));
  PetscCall(SNESSetJacobian(snes, NULL, NULL, NULL, vertices));
  PetscCall(VecGetArray(real, &xi));
  xi[0] = coords[p * ctx->dim + 0];
  xi[1] = coords[p * ctx->dim + 1];
  PetscCall(VecRestoreArray(real, &xi));
  PetscCall(SNESSolve(snes, real, ref));
  PetscCall(VecGetArray(ref, &xi));
  xir[0] = PetscRealPart(xi[0]);
  xir[1] = PetscRealPart(xi[1]);
  if (4 * dof == xSize) {
    for (PetscInt comp = 0; comp < dof; ++comp) a[p * dof + comp] = x[0 * dof + comp] * (1 - xir[0]) * (1 - xir[1]) + x[1 * dof + comp] * xir[0] * (1 - xir[1]) + x[2 * dof + comp] * xir[0] * xir[1] + x[3 * dof + comp] * (1 - xir[0]) * xir[1];
  } else if (dof == xSize) {
    for (PetscInt comp = 0; comp < dof; ++comp) a[p * dof + comp] = x[0 * dof + comp];
  } else {
    PetscCheck(fem, ctx->comm, PETSC_ERR_ARG_WRONG, "Cannot have a higher order interpolant if the discretization is not PetscFE");
    xir[0] = 2.0 * xir[0] - 1.0;
    xir[1] = 2.0 * xir[1] - 1.0;
    PetscCall(PetscFEComputeTabulation(fem, 1, xir, 0, T));
    for (PetscInt comp = 0; comp < dof; ++comp) {
      a[p * dof + comp] = 0.0;
      for (PetscInt d = 0; d < xSize / dof; ++d) a[p * dof + comp] += x[d * dof + comp] * T->T[0][d * dof + comp];
    }
  }
  PetscCall(VecRestoreArray(ref, &xi));
  PetscCall(DMPlexVecRestoreClosure(dmCoord, NULL, coordsLocal, c, &coordSize, &vertices));
  PetscCall(DMPlexVecRestoreClosure(dm, NULL, xLocal, c, &xSize, &x));
  PetscCall(PetscTabulationDestroy(&T));
  PetscCall(VecRestoreArray(v, &a));
  PetscCall(VecRestoreArrayRead(ctx->coords, &coords));

  PetscCall(SNESDestroy(&snes));
  PetscCall(VecDestroy(&r));
  PetscCall(VecDestroy(&ref));
  PetscCall(VecDestroy(&real));
  PetscCall(MatDestroy(&J));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static inline PetscErrorCode HexMap_Private(SNES snes, Vec Xref, Vec Xreal, void *ctx)
{
  const PetscScalar *vertices = (const PetscScalar *)ctx;
  const PetscScalar  x0       = vertices[0];
  const PetscScalar  y0       = vertices[1];
  const PetscScalar  z0       = vertices[2];
  const PetscScalar  x1       = vertices[9];
  const PetscScalar  y1       = vertices[10];
  const PetscScalar  z1       = vertices[11];
  const PetscScalar  x2       = vertices[6];
  const PetscScalar  y2       = vertices[7];
  const PetscScalar  z2       = vertices[8];
  const PetscScalar  x3       = vertices[3];
  const PetscScalar  y3       = vertices[4];
  const PetscScalar  z3       = vertices[5];
  const PetscScalar  x4       = vertices[12];
  const PetscScalar  y4       = vertices[13];
  const PetscScalar  z4       = vertices[14];
  const PetscScalar  x5       = vertices[15];
  const PetscScalar  y5       = vertices[16];
  const PetscScalar  z5       = vertices[17];
  const PetscScalar  x6       = vertices[18];
  const PetscScalar  y6       = vertices[19];
  const PetscScalar  z6       = vertices[20];
  const PetscScalar  x7       = vertices[21];
  const PetscScalar  y7       = vertices[22];
  const PetscScalar  z7       = vertices[23];
  const PetscScalar  f_1      = x1 - x0;
  const PetscScalar  g_1      = y1 - y0;
  const PetscScalar  h_1      = z1 - z0;
  const PetscScalar  f_3      = x3 - x0;
  const PetscScalar  g_3      = y3 - y0;
  const PetscScalar  h_3      = z3 - z0;
  const PetscScalar  f_4      = x4 - x0;
  const PetscScalar  g_4      = y4 - y0;
  const PetscScalar  h_4      = z4 - z0;
  const PetscScalar  f_01     = x2 - x1 - x3 + x0;
  const PetscScalar  g_01     = y2 - y1 - y3 + y0;
  const PetscScalar  h_01     = z2 - z1 - z3 + z0;
  const PetscScalar  f_12     = x7 - x3 - x4 + x0;
  const PetscScalar  g_12     = y7 - y3 - y4 + y0;
  const PetscScalar  h_12     = z7 - z3 - z4 + z0;
  const PetscScalar  f_02     = x5 - x1 - x4 + x0;
  const PetscScalar  g_02     = y5 - y1 - y4 + y0;
  const PetscScalar  h_02     = z5 - z1 - z4 + z0;
  const PetscScalar  f_012    = x6 - x0 + x1 - x2 + x3 + x4 - x5 - x7;
  const PetscScalar  g_012    = y6 - y0 + y1 - y2 + y3 + y4 - y5 - y7;
  const PetscScalar  h_012    = z6 - z0 + z1 - z2 + z3 + z4 - z5 - z7;
  const PetscScalar *ref;
  PetscScalar       *real;

  PetscFunctionBegin;
  PetscCall(VecGetArrayRead(Xref, &ref));
  PetscCall(VecGetArray(Xreal, &real));
  {
    const PetscScalar p0 = ref[0];
    const PetscScalar p1 = ref[1];
    const PetscScalar p2 = ref[2];

    real[0] = x0 + f_1 * p0 + f_3 * p1 + f_4 * p2 + f_01 * p0 * p1 + f_12 * p1 * p2 + f_02 * p0 * p2 + f_012 * p0 * p1 * p2;
    real[1] = y0 + g_1 * p0 + g_3 * p1 + g_4 * p2 + g_01 * p0 * p1 + g_01 * p0 * p1 + g_12 * p1 * p2 + g_02 * p0 * p2 + g_012 * p0 * p1 * p2;
    real[2] = z0 + h_1 * p0 + h_3 * p1 + h_4 * p2 + h_01 * p0 * p1 + h_01 * p0 * p1 + h_12 * p1 * p2 + h_02 * p0 * p2 + h_012 * p0 * p1 * p2;
  }
  PetscCall(PetscLogFlops(114));
  PetscCall(VecRestoreArrayRead(Xref, &ref));
  PetscCall(VecRestoreArray(Xreal, &real));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static inline PetscErrorCode HexJacobian_Private(SNES snes, Vec Xref, Mat J, Mat M, void *ctx)
{
  const PetscScalar *vertices = (const PetscScalar *)ctx;
  const PetscScalar  x0       = vertices[0];
  const PetscScalar  y0       = vertices[1];
  const PetscScalar  z0       = vertices[2];
  const PetscScalar  x1       = vertices[9];
  const PetscScalar  y1       = vertices[10];
  const PetscScalar  z1       = vertices[11];
  const PetscScalar  x2       = vertices[6];
  const PetscScalar  y2       = vertices[7];
  const PetscScalar  z2       = vertices[8];
  const PetscScalar  x3       = vertices[3];
  const PetscScalar  y3       = vertices[4];
  const PetscScalar  z3       = vertices[5];
  const PetscScalar  x4       = vertices[12];
  const PetscScalar  y4       = vertices[13];
  const PetscScalar  z4       = vertices[14];
  const PetscScalar  x5       = vertices[15];
  const PetscScalar  y5       = vertices[16];
  const PetscScalar  z5       = vertices[17];
  const PetscScalar  x6       = vertices[18];
  const PetscScalar  y6       = vertices[19];
  const PetscScalar  z6       = vertices[20];
  const PetscScalar  x7       = vertices[21];
  const PetscScalar  y7       = vertices[22];
  const PetscScalar  z7       = vertices[23];
  const PetscScalar  f_xy     = x2 - x1 - x3 + x0;
  const PetscScalar  g_xy     = y2 - y1 - y3 + y0;
  const PetscScalar  h_xy     = z2 - z1 - z3 + z0;
  const PetscScalar  f_yz     = x7 - x3 - x4 + x0;
  const PetscScalar  g_yz     = y7 - y3 - y4 + y0;
  const PetscScalar  h_yz     = z7 - z3 - z4 + z0;
  const PetscScalar  f_xz     = x5 - x1 - x4 + x0;
  const PetscScalar  g_xz     = y5 - y1 - y4 + y0;
  const PetscScalar  h_xz     = z5 - z1 - z4 + z0;
  const PetscScalar  f_xyz    = x6 - x0 + x1 - x2 + x3 + x4 - x5 - x7;
  const PetscScalar  g_xyz    = y6 - y0 + y1 - y2 + y3 + y4 - y5 - y7;
  const PetscScalar  h_xyz    = z6 - z0 + z1 - z2 + z3 + z4 - z5 - z7;
  const PetscScalar *ref;

  PetscFunctionBegin;
  PetscCall(VecGetArrayRead(Xref, &ref));
  {
    const PetscScalar x       = ref[0];
    const PetscScalar y       = ref[1];
    const PetscScalar z       = ref[2];
    const PetscInt    rows[3] = {0, 1, 2};
    PetscScalar       values[9];

    values[0] = (x1 - x0 + f_xy * y + f_xz * z + f_xyz * y * z) / 2.0;
    values[1] = (x3 - x0 + f_xy * x + f_yz * z + f_xyz * x * z) / 2.0;
    values[2] = (x4 - x0 + f_yz * y + f_xz * x + f_xyz * x * y) / 2.0;
    values[3] = (y1 - y0 + g_xy * y + g_xz * z + g_xyz * y * z) / 2.0;
    values[4] = (y3 - y0 + g_xy * x + g_yz * z + g_xyz * x * z) / 2.0;
    values[5] = (y4 - y0 + g_yz * y + g_xz * x + g_xyz * x * y) / 2.0;
    values[6] = (z1 - z0 + h_xy * y + h_xz * z + h_xyz * y * z) / 2.0;
    values[7] = (z3 - z0 + h_xy * x + h_yz * z + h_xyz * x * z) / 2.0;
    values[8] = (z4 - z0 + h_yz * y + h_xz * x + h_xyz * x * y) / 2.0;

    PetscCall(MatSetValues(J, 3, rows, 3, rows, values, INSERT_VALUES));
  }
  PetscCall(PetscLogFlops(152));
  PetscCall(VecRestoreArrayRead(Xref, &ref));
  PetscCall(MatAssemblyBegin(J, MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(J, MAT_FINAL_ASSEMBLY));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static inline PetscErrorCode DMInterpolate_Hex_Private(DMInterpolationInfo ctx, DM dm, PetscInt p, Vec xLocal, Vec v)
{
  const PetscInt     c = ctx->cells[p];
  DM                 dmCoord;
  SNES               snes;
  KSP                ksp;
  PC                 pc;
  Vec                coordsLocal, r, ref, real;
  Mat                J;
  const PetscScalar *coords;
  PetscScalar       *a;
  PetscScalar       *x = NULL, *vertices = NULL;
  PetscScalar       *xi;
  PetscReal          xir[3];
  PetscInt           coordSize, xSize;

  PetscFunctionBegin;
  PetscCall(DMGetCoordinatesLocal(dm, &coordsLocal));
  PetscCall(DMGetCoordinateDM(dm, &dmCoord));
  PetscCall(SNESCreate(PETSC_COMM_SELF, &snes));
  PetscCall(SNESSetOptionsPrefix(snes, "hex_interp_"));
  PetscCall(VecCreate(PETSC_COMM_SELF, &r));
  PetscCall(VecSetSizes(r, 3, 3));
  PetscCall(VecSetType(r, dm->vectype));
  PetscCall(VecDuplicate(r, &ref));
  PetscCall(VecDuplicate(r, &real));
  PetscCall(MatCreate(PETSC_COMM_SELF, &J));
  PetscCall(MatSetSizes(J, 3, 3, 3, 3));
  PetscCall(MatSetType(J, MATSEQDENSE));
  PetscCall(MatSetUp(J));
  PetscCall(SNESSetFunction(snes, r, HexMap_Private, NULL));
  PetscCall(SNESSetJacobian(snes, J, J, HexJacobian_Private, NULL));
  PetscCall(SNESGetKSP(snes, &ksp));
  PetscCall(KSPGetPC(ksp, &pc));
  PetscCall(PCSetType(pc, PCLU));
  PetscCall(SNESSetFromOptions(snes));

  PetscCall(VecGetArrayRead(ctx->coords, &coords));
  PetscCall(VecGetArray(v, &a));
  PetscCall(DMPlexVecGetClosure(dmCoord, NULL, coordsLocal, c, &coordSize, &vertices));
  PetscCheck(8 * 3 == coordSize, ctx->comm, PETSC_ERR_ARG_SIZ, "Invalid coordinate closure size %" PetscInt_FMT " should be %d", coordSize, 8 * 3);
  PetscCall(DMPlexVecGetClosure(dm, NULL, xLocal, c, &xSize, &x));
  PetscCheck((8 * ctx->dof == xSize) || (ctx->dof == xSize), ctx->comm, PETSC_ERR_ARG_SIZ, "Invalid input closure size %" PetscInt_FMT " should be %" PetscInt_FMT " or %" PetscInt_FMT, xSize, 8 * ctx->dof, ctx->dof);
  PetscCall(SNESSetFunction(snes, NULL, NULL, vertices));
  PetscCall(SNESSetJacobian(snes, NULL, NULL, NULL, vertices));
  PetscCall(VecGetArray(real, &xi));
  xi[0] = coords[p * ctx->dim + 0];
  xi[1] = coords[p * ctx->dim + 1];
  xi[2] = coords[p * ctx->dim + 2];
  PetscCall(VecRestoreArray(real, &xi));
  PetscCall(SNESSolve(snes, real, ref));
  PetscCall(VecGetArray(ref, &xi));
  xir[0] = PetscRealPart(xi[0]);
  xir[1] = PetscRealPart(xi[1]);
  xir[2] = PetscRealPart(xi[2]);
  if (8 * ctx->dof == xSize) {
    for (PetscInt comp = 0; comp < ctx->dof; ++comp) {
      a[p * ctx->dof + comp] = x[0 * ctx->dof + comp] * (1 - xir[0]) * (1 - xir[1]) * (1 - xir[2]) + x[3 * ctx->dof + comp] * xir[0] * (1 - xir[1]) * (1 - xir[2]) + x[2 * ctx->dof + comp] * xir[0] * xir[1] * (1 - xir[2]) + x[1 * ctx->dof + comp] * (1 - xir[0]) * xir[1] * (1 - xir[2]) +
                               x[4 * ctx->dof + comp] * (1 - xir[0]) * (1 - xir[1]) * xir[2] + x[5 * ctx->dof + comp] * xir[0] * (1 - xir[1]) * xir[2] + x[6 * ctx->dof + comp] * xir[0] * xir[1] * xir[2] + x[7 * ctx->dof + comp] * (1 - xir[0]) * xir[1] * xir[2];
    }
  } else {
    for (PetscInt comp = 0; comp < ctx->dof; ++comp) a[p * ctx->dof + comp] = x[0 * ctx->dof + comp];
  }
  PetscCall(VecRestoreArray(ref, &xi));
  PetscCall(DMPlexVecRestoreClosure(dmCoord, NULL, coordsLocal, c, &coordSize, &vertices));
  PetscCall(DMPlexVecRestoreClosure(dm, NULL, xLocal, c, &xSize, &x));
  PetscCall(VecRestoreArray(v, &a));
  PetscCall(VecRestoreArrayRead(ctx->coords, &coords));

  PetscCall(SNESDestroy(&snes));
  PetscCall(VecDestroy(&r));
  PetscCall(VecDestroy(&ref));
  PetscCall(VecDestroy(&real));
  PetscCall(MatDestroy(&J));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@C
  DMInterpolationEvaluate - Using the input from dm and x, calculates interpolated field values at the interpolation points.

  Input Parameters:
+ ctx - The `DMInterpolationInfo` context
. dm  - The `DM`
- x   - The local vector containing the field to be interpolated

  Output Parameter:
. v - The vector containing the interpolated values

  Level: beginner

  Note:
  A suitable `v` can be obtained using `DMInterpolationGetVector()`.

.seealso: `DMInterpolationInfo`, `DMInterpolationGetVector()`, `DMInterpolationAddPoints()`, `DMInterpolationCreate()`
@*/
PetscErrorCode DMInterpolationEvaluate(DMInterpolationInfo ctx, DM dm, Vec x, Vec v)
{
  PetscDS   ds;
  PetscInt  n, p, Nf, field;
  PetscBool useDS = PETSC_FALSE;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(dm, DM_CLASSID, 2);
  PetscValidHeaderSpecific(x, VEC_CLASSID, 3);
  PetscValidHeaderSpecific(v, VEC_CLASSID, 4);
  PetscCall(VecGetLocalSize(v, &n));
  PetscCheck(n == ctx->n * ctx->dof, ctx->comm, PETSC_ERR_ARG_SIZ, "Invalid input vector size %" PetscInt_FMT " should be %" PetscInt_FMT, n, ctx->n * ctx->dof);
  if (!n) PetscFunctionReturn(PETSC_SUCCESS);
  PetscCall(DMGetDS(dm, &ds));
  if (ds) {
    useDS = PETSC_TRUE;
    PetscCall(PetscDSGetNumFields(ds, &Nf));
    for (field = 0; field < Nf; ++field) {
      PetscObject  obj;
      PetscClassId id;

      PetscCall(PetscDSGetDiscretization(ds, field, &obj));
      PetscCall(PetscObjectGetClassId(obj, &id));
      if (id != PETSCFE_CLASSID) {
        useDS = PETSC_FALSE;
        break;
      }
    }
  }
  if (useDS) {
    const PetscScalar *coords;
    PetscScalar       *interpolant;
    PetscInt           cdim, d;

    PetscCall(DMGetCoordinateDim(dm, &cdim));
    PetscCall(VecGetArrayRead(ctx->coords, &coords));
    PetscCall(VecGetArrayWrite(v, &interpolant));
    for (p = 0; p < ctx->n; ++p) {
      PetscReal    pcoords[3], xi[3];
      PetscScalar *xa   = NULL;
      PetscInt     coff = 0, foff = 0, clSize;

      if (ctx->cells[p] < 0) continue;
      for (d = 0; d < cdim; ++d) pcoords[d] = PetscRealPart(coords[p * cdim + d]);
      PetscCall(DMPlexCoordinatesToReference(dm, ctx->cells[p], 1, pcoords, xi));
      PetscCall(DMPlexVecGetClosure(dm, NULL, x, ctx->cells[p], &clSize, &xa));
      for (field = 0; field < Nf; ++field) {
        PetscTabulation T;
        PetscFE         fe;

        PetscCall(PetscDSGetDiscretization(ds, field, (PetscObject *)&fe));
        PetscCall(PetscFECreateTabulation(fe, 1, 1, xi, 0, &T));
        {
          const PetscReal *basis = T->T[0];
          const PetscInt   Nb    = T->Nb;
          const PetscInt   Nc    = T->Nc;
          PetscInt         f, fc;

          for (fc = 0; fc < Nc; ++fc) {
            interpolant[p * ctx->dof + coff + fc] = 0.0;
            for (f = 0; f < Nb; ++f) interpolant[p * ctx->dof + coff + fc] += xa[foff + f] * basis[(0 * Nb + f) * Nc + fc];
          }
          coff += Nc;
          foff += Nb;
        }
        PetscCall(PetscTabulationDestroy(&T));
      }
      PetscCall(DMPlexVecRestoreClosure(dm, NULL, x, ctx->cells[p], &clSize, &xa));
      PetscCheck(coff == ctx->dof, PETSC_COMM_SELF, PETSC_ERR_PLIB, "Total components %" PetscInt_FMT " != %" PetscInt_FMT " dof specified for interpolation", coff, ctx->dof);
      PetscCheck(foff == clSize, PETSC_COMM_SELF, PETSC_ERR_PLIB, "Total FE space size %" PetscInt_FMT " != %" PetscInt_FMT " closure size", foff, clSize);
    }
    PetscCall(VecRestoreArrayRead(ctx->coords, &coords));
    PetscCall(VecRestoreArrayWrite(v, &interpolant));
  } else {
    for (PetscInt p = 0; p < ctx->n; ++p) {
      const PetscInt cell = ctx->cells[p];
      DMPolytopeType ct;

      PetscCall(DMPlexGetCellType(dm, cell, &ct));
      switch (ct) {
      case DM_POLYTOPE_SEGMENT:
        PetscCall(DMInterpolate_Segment_Private(ctx, dm, p, x, v));
        break;
      case DM_POLYTOPE_TRIANGLE:
        PetscCall(DMInterpolate_Triangle_Private(ctx, dm, p, x, v));
        break;
      case DM_POLYTOPE_QUADRILATERAL:
        PetscCall(DMInterpolate_Quad_Private(ctx, dm, p, x, v));
        break;
      case DM_POLYTOPE_TETRAHEDRON:
        PetscCall(DMInterpolate_Tetrahedron_Private(ctx, dm, p, x, v));
        break;
      case DM_POLYTOPE_HEXAHEDRON:
        PetscCall(DMInterpolate_Hex_Private(ctx, dm, cell, x, v));
        break;
      default:
        SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP, "No support for cell type %s", DMPolytopeTypes[PetscMax(0, PetscMin(ct, DM_NUM_POLYTOPES))]);
      }
    }
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@C
  DMInterpolationDestroy - Destroys a `DMInterpolationInfo` context

  Collective

  Input Parameter:
. ctx - the context

  Level: beginner

.seealso: `DMInterpolationInfo`, `DMInterpolationEvaluate()`, `DMInterpolationAddPoints()`, `DMInterpolationCreate()`
@*/
PetscErrorCode DMInterpolationDestroy(DMInterpolationInfo *ctx)
{
  PetscFunctionBegin;
  PetscAssertPointer(ctx, 1);
  PetscCall(VecDestroy(&(*ctx)->coords));
  PetscCall(PetscFree((*ctx)->points));
  PetscCall(PetscFree((*ctx)->cells));
  PetscCall(PetscFree(*ctx));
  *ctx = NULL;
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@C
  SNESMonitorFields - Monitors the residual for each field separately

  Collective

  Input Parameters:
+ snes   - the `SNES` context
. its    - iteration number
. fgnorm - 2-norm of residual
- vf     - `PetscViewerAndFormat` of `PetscViewerType` `PETSCVIEWERASCII`

  Level: intermediate

  Note:
  This routine prints the residual norm at each iteration.

.seealso: `SNES`, `SNESMonitorSet()`, `SNESMonitorDefault()`
@*/
PetscErrorCode SNESMonitorFields(SNES snes, PetscInt its, PetscReal fgnorm, PetscViewerAndFormat *vf)
{
  PetscViewer        viewer = vf->viewer;
  Vec                res;
  DM                 dm;
  PetscSection       s;
  const PetscScalar *r;
  PetscReal         *lnorms, *norms;
  PetscInt           numFields, f, pStart, pEnd, p;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(viewer, PETSC_VIEWER_CLASSID, 4);
  PetscCall(SNESGetFunction(snes, &res, NULL, NULL));
  PetscCall(SNESGetDM(snes, &dm));
  PetscCall(DMGetLocalSection(dm, &s));
  PetscCall(PetscSectionGetNumFields(s, &numFields));
  PetscCall(PetscSectionGetChart(s, &pStart, &pEnd));
  PetscCall(PetscCalloc2(numFields, &lnorms, numFields, &norms));
  PetscCall(VecGetArrayRead(res, &r));
  for (p = pStart; p < pEnd; ++p) {
    for (f = 0; f < numFields; ++f) {
      PetscInt fdof, foff, d;

      PetscCall(PetscSectionGetFieldDof(s, p, f, &fdof));
      PetscCall(PetscSectionGetFieldOffset(s, p, f, &foff));
      for (d = 0; d < fdof; ++d) lnorms[f] += PetscRealPart(PetscSqr(r[foff + d]));
    }
  }
  PetscCall(VecRestoreArrayRead(res, &r));
  PetscCall(MPIU_Allreduce(lnorms, norms, numFields, MPIU_REAL, MPIU_SUM, PetscObjectComm((PetscObject)dm)));
  PetscCall(PetscViewerPushFormat(viewer, vf->format));
  PetscCall(PetscViewerASCIIAddTab(viewer, ((PetscObject)snes)->tablevel));
  PetscCall(PetscViewerASCIIPrintf(viewer, "%3" PetscInt_FMT " SNES Function norm %14.12e [", its, (double)fgnorm));
  for (f = 0; f < numFields; ++f) {
    if (f > 0) PetscCall(PetscViewerASCIIPrintf(viewer, ", "));
    PetscCall(PetscViewerASCIIPrintf(viewer, "%14.12e", (double)PetscSqrtReal(norms[f])));
  }
  PetscCall(PetscViewerASCIIPrintf(viewer, "]\n"));
  PetscCall(PetscViewerASCIISubtractTab(viewer, ((PetscObject)snes)->tablevel));
  PetscCall(PetscViewerPopFormat(viewer));
  PetscCall(PetscFree2(lnorms, norms));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/********************* Residual Computation **************************/

/*@
  DMPlexSNESComputeResidualFEM - Sums the local residual into vector F from the local input X using pointwise functions specified by the user

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

  Output Parameter:
. F - Local output vector

  Level: developer

  Note:
  The residual is summed into F; the caller is responsible for using `VecZeroEntries()` or otherwise ensuring that any data in F is intentional.

.seealso: `DM`, `DMPlexComputeJacobianAction()`
@*/
PetscErrorCode DMPlexSNESComputeResidualFEM(DM dm, Vec X, Vec F, void *user)
{
  DM       plex;
  IS       allcellIS;
  PetscInt Nds, s;

  PetscFunctionBegin;
  PetscCall(DMSNESConvertPlex(dm, &plex, PETSC_TRUE));
  PetscCall(DMPlexGetAllCells_Internal(plex, &allcellIS));
  PetscCall(DMGetNumDS(dm, &Nds));
  for (s = 0; s < Nds; ++s) {
    PetscDS      ds;
    IS           cellIS;
    PetscFormKey key;

    PetscCall(DMGetRegionNumDS(dm, s, &key.label, NULL, &ds, NULL));
    key.value = 0;
    key.field = 0;
    key.part  = 0;
    if (!key.label) {
      PetscCall(PetscObjectReference((PetscObject)allcellIS));
      cellIS = allcellIS;
    } else {
      IS pointIS;

      key.value = 1;
      PetscCall(DMLabelGetStratumIS(key.label, key.value, &pointIS));
      PetscCall(ISIntersect_Caching_Internal(allcellIS, pointIS, &cellIS));
      PetscCall(ISDestroy(&pointIS));
    }
    PetscCall(DMPlexComputeResidual_Internal(plex, key, cellIS, PETSC_MIN_REAL, X, NULL, 0.0, F, user));
    PetscCall(ISDestroy(&cellIS));
  }
  PetscCall(ISDestroy(&allcellIS));
  PetscCall(DMDestroy(&plex));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  DMPlexSNESComputeResidualDS - Sums the local residual into vector F from the local input X using all pointwise functions with unique keys in the DS

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

  Output Parameter:
. F - Local output vector

  Level: developer

  Note:
  The residual is summed into F; the caller is responsible for using `VecZeroEntries()` or otherwise ensuring that any data in F is intentional.

.seealso: `DM`, `DMPlexComputeJacobianAction()`
@*/
PetscErrorCode DMPlexSNESComputeResidualDS(DM dm, Vec X, Vec F, void *user)
{
  DM       plex;
  IS       allcellIS;
  PetscInt Nds, s;

  PetscFunctionBegin;
  PetscCall(DMSNESConvertPlex(dm, &plex, PETSC_TRUE));
  PetscCall(DMPlexGetAllCells_Internal(plex, &allcellIS));
  PetscCall(DMGetNumDS(dm, &Nds));
  for (s = 0; s < Nds; ++s) {
    PetscDS ds;
    DMLabel label;
    IS      cellIS;

    PetscCall(DMGetRegionNumDS(dm, s, &label, NULL, &ds, NULL));
    {
      PetscWeakFormKind resmap[2] = {PETSC_WF_F0, PETSC_WF_F1};
      PetscWeakForm     wf;
      PetscInt          Nm = 2, m, Nk = 0, k, kp, off = 0;
      PetscFormKey     *reskeys;

      /* Get unique residual keys */
      for (m = 0; m < Nm; ++m) {
        PetscInt Nkm;
        PetscCall(PetscHMapFormGetSize(ds->wf->form[resmap[m]], &Nkm));
        Nk += Nkm;
      }
      PetscCall(PetscMalloc1(Nk, &reskeys));
      for (m = 0; m < Nm; ++m) PetscCall(PetscHMapFormGetKeys(ds->wf->form[resmap[m]], &off, reskeys));
      PetscCheck(off == Nk, PETSC_COMM_SELF, PETSC_ERR_ARG_SIZ, "Number of keys %" PetscInt_FMT " should be %" PetscInt_FMT, off, Nk);
      PetscCall(PetscFormKeySort(Nk, reskeys));
      for (k = 0, kp = 1; kp < Nk; ++kp) {
        if ((reskeys[k].label != reskeys[kp].label) || (reskeys[k].value != reskeys[kp].value)) {
          ++k;
          if (kp != k) reskeys[k] = reskeys[kp];
        }
      }
      Nk = k;

      PetscCall(PetscDSGetWeakForm(ds, &wf));
      for (k = 0; k < Nk; ++k) {
        DMLabel  label = reskeys[k].label;
        PetscInt val   = reskeys[k].value;

        if (!label) {
          PetscCall(PetscObjectReference((PetscObject)allcellIS));
          cellIS = allcellIS;
        } else {
          IS pointIS;

          PetscCall(DMLabelGetStratumIS(label, val, &pointIS));
          PetscCall(ISIntersect_Caching_Internal(allcellIS, pointIS, &cellIS));
          PetscCall(ISDestroy(&pointIS));
        }
        PetscCall(DMPlexComputeResidual_Internal(plex, reskeys[k], cellIS, PETSC_MIN_REAL, X, NULL, 0.0, F, user));
        PetscCall(ISDestroy(&cellIS));
      }
      PetscCall(PetscFree(reskeys));
    }
  }
  PetscCall(ISDestroy(&allcellIS));
  PetscCall(DMDestroy(&plex));
  PetscFunctionReturn(PETSC_SUCCESS);
}

#ifdef PETSC_HAVE_LIBCEED
PetscErrorCode DMPlexSNESComputeResidualCEED(DM dm, Vec locX, Vec locF, void *user)
{
  Ceed       ceed;
  DMCeed     sd = dm->dmceed;
  CeedVector clocX, clocF;

  PetscFunctionBegin;
  PetscCall(DMGetCeed(dm, &ceed));
  PetscCheck(sd, PetscObjectComm((PetscObject)dm), PETSC_ERR_ARG_WRONGSTATE, "This DM has no CEED data. Call DMCeedCreate() before computing the residual.");
  PetscCall(DMCeedComputeGeometry(dm, sd));

  PetscCall(VecGetCeedVectorRead(locX, ceed, &clocX));
  PetscCall(VecGetCeedVector(locF, ceed, &clocF));
  PetscCallCEED(CeedOperatorApplyAdd(sd->op, clocX, clocF, CEED_REQUEST_IMMEDIATE));
  PetscCall(VecRestoreCeedVectorRead(locX, &clocX));
  PetscCall(VecRestoreCeedVector(locF, &clocF));

  {
    DM_Plex *mesh = (DM_Plex *)dm->data;

    if (mesh->printFEM) {
      PetscSection section;
      Vec          locFbc;
      PetscInt     pStart, pEnd, p, maxDof;
      PetscScalar *zeroes;

      PetscCall(DMGetLocalSection(dm, &section));
      PetscCall(VecDuplicate(locF, &locFbc));
      PetscCall(VecCopy(locF, locFbc));
      PetscCall(PetscSectionGetChart(section, &pStart, &pEnd));
      PetscCall(PetscSectionGetMaxDof(section, &maxDof));
      PetscCall(PetscCalloc1(maxDof, &zeroes));
      for (p = pStart; p < pEnd; ++p) PetscCall(VecSetValuesSection(locFbc, section, p, zeroes, INSERT_BC_VALUES));
      PetscCall(PetscFree(zeroes));
      PetscCall(DMPrintLocalVec(dm, "Residual", mesh->printTol, locFbc));
      PetscCall(VecDestroy(&locFbc));
    }
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}
#endif

/*@
  DMPlexSNESComputeBoundaryFEM - Form the boundary values for the local input X

  Input Parameters:
+ dm   - The mesh
- user - The user context

  Output Parameter:
. X - Local solution

  Level: developer

.seealso: `DMPLEX`, `DMPlexComputeJacobianAction()`
@*/
PetscErrorCode DMPlexSNESComputeBoundaryFEM(DM dm, Vec X, void *user)
{
  DM plex;

  PetscFunctionBegin;
  PetscCall(DMSNESConvertPlex(dm, &plex, PETSC_TRUE));
  PetscCall(DMPlexInsertBoundaryValues(plex, PETSC_TRUE, X, PETSC_MIN_REAL, NULL, NULL, NULL));
  PetscCall(DMDestroy(&plex));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  DMSNESComputeJacobianAction - Compute the action of the Jacobian J(X) on Y

  Input Parameters:
+ dm   - The `DM`
. X    - Local solution vector
. Y    - Local input vector
- user - The user context

  Output Parameter:
. F - local output vector

  Level: developer

  Notes:
  Users will typically use `DMSNESCreateJacobianMF()` followed by `MatMult()` instead of calling this routine directly.

.seealso: `DM`, ``DMSNESCreateJacobianMF()`, `DMPlexSNESComputeResidualFEM()`
@*/
PetscErrorCode DMSNESComputeJacobianAction(DM dm, Vec X, Vec Y, Vec F, void *user)
{
  DM       plex;
  IS       allcellIS;
  PetscInt Nds, s;

  PetscFunctionBegin;
  PetscCall(DMSNESConvertPlex(dm, &plex, PETSC_TRUE));
  PetscCall(DMPlexGetAllCells_Internal(plex, &allcellIS));
  PetscCall(DMGetNumDS(dm, &Nds));
  for (s = 0; s < Nds; ++s) {
    PetscDS ds;
    DMLabel label;
    IS      cellIS;

    PetscCall(DMGetRegionNumDS(dm, s, &label, NULL, &ds, NULL));
    {
      PetscWeakFormKind jacmap[4] = {PETSC_WF_G0, PETSC_WF_G1, PETSC_WF_G2, PETSC_WF_G3};
      PetscWeakForm     wf;
      PetscInt          Nm = 4, m, Nk = 0, k, kp, off = 0;
      PetscFormKey     *jackeys;

      /* Get unique Jacobian keys */
      for (m = 0; m < Nm; ++m) {
        PetscInt Nkm;
        PetscCall(PetscHMapFormGetSize(ds->wf->form[jacmap[m]], &Nkm));
        Nk += Nkm;
      }
      PetscCall(PetscMalloc1(Nk, &jackeys));
      for (m = 0; m < Nm; ++m) PetscCall(PetscHMapFormGetKeys(ds->wf->form[jacmap[m]], &off, jackeys));
      PetscCheck(off == Nk, PETSC_COMM_SELF, PETSC_ERR_ARG_SIZ, "Number of keys %" PetscInt_FMT " should be %" PetscInt_FMT, off, Nk);
      PetscCall(PetscFormKeySort(Nk, jackeys));
      for (k = 0, kp = 1; kp < Nk; ++kp) {
        if ((jackeys[k].label != jackeys[kp].label) || (jackeys[k].value != jackeys[kp].value)) {
          ++k;
          if (kp != k) jackeys[k] = jackeys[kp];
        }
      }
      Nk = k;

      PetscCall(PetscDSGetWeakForm(ds, &wf));
      for (k = 0; k < Nk; ++k) {
        DMLabel  label = jackeys[k].label;
        PetscInt val   = jackeys[k].value;

        if (!label) {
          PetscCall(PetscObjectReference((PetscObject)allcellIS));
          cellIS = allcellIS;
        } else {
          IS pointIS;

          PetscCall(DMLabelGetStratumIS(label, val, &pointIS));
          PetscCall(ISIntersect_Caching_Internal(allcellIS, pointIS, &cellIS));
          PetscCall(ISDestroy(&pointIS));
        }
        PetscCall(DMPlexComputeJacobian_Action_Internal(plex, jackeys[k], cellIS, 0.0, 0.0, X, NULL, Y, F, user));
        PetscCall(ISDestroy(&cellIS));
      }
      PetscCall(PetscFree(jackeys));
    }
  }
  PetscCall(ISDestroy(&allcellIS));
  PetscCall(DMDestroy(&plex));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  DMPlexSNESComputeJacobianFEM - Form the local portion of the Jacobian matrix `Jac` at the local solution `X` using pointwise functions specified by the user.

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

  Output Parameters:
+ Jac  - Jacobian matrix
- JacP - approximate Jacobian from which the preconditioner will be built, often `Jac`

  Level: developer

  Note:
  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.

.seealso: `DMPLEX`, `Mat`
@*/
PetscErrorCode DMPlexSNESComputeJacobianFEM(DM dm, Vec X, Mat Jac, Mat JacP, void *user)
{
  DM        plex;
  IS        allcellIS;
  PetscBool hasJac, hasPrec;
  PetscInt  Nds, s;

  PetscFunctionBegin;
  PetscCall(DMSNESConvertPlex(dm, &plex, PETSC_TRUE));
  PetscCall(DMPlexGetAllCells_Internal(plex, &allcellIS));
  PetscCall(DMGetNumDS(dm, &Nds));
  for (s = 0; s < Nds; ++s) {
    PetscDS      ds;
    IS           cellIS;
    PetscFormKey key;

    PetscCall(DMGetRegionNumDS(dm, s, &key.label, NULL, &ds, NULL));
    key.value = 0;
    key.field = 0;
    key.part  = 0;
    if (!key.label) {
      PetscCall(PetscObjectReference((PetscObject)allcellIS));
      cellIS = allcellIS;
    } else {
      IS pointIS;

      key.value = 1;
      PetscCall(DMLabelGetStratumIS(key.label, key.value, &pointIS));
      PetscCall(ISIntersect_Caching_Internal(allcellIS, pointIS, &cellIS));
      PetscCall(ISDestroy(&pointIS));
    }
    if (!s) {
      PetscCall(PetscDSHasJacobian(ds, &hasJac));
      PetscCall(PetscDSHasJacobianPreconditioner(ds, &hasPrec));
      if (hasJac && hasPrec) PetscCall(MatZeroEntries(Jac));
      PetscCall(MatZeroEntries(JacP));
    }
    PetscCall(DMPlexComputeJacobian_Internal(plex, key, cellIS, 0.0, 0.0, X, NULL, Jac, JacP, user));
    PetscCall(ISDestroy(&cellIS));
  }
  PetscCall(ISDestroy(&allcellIS));
  PetscCall(DMDestroy(&plex));
  PetscFunctionReturn(PETSC_SUCCESS);
}

struct _DMSNESJacobianMFCtx {
  DM    dm;
  Vec   X;
  void *ctx;
};

static PetscErrorCode DMSNESJacobianMF_Destroy_Private(Mat A)
{
  struct _DMSNESJacobianMFCtx *ctx;

  PetscFunctionBegin;
  PetscCall(MatShellGetContext(A, &ctx));
  PetscCall(MatShellSetContext(A, NULL));
  PetscCall(DMDestroy(&ctx->dm));
  PetscCall(VecDestroy(&ctx->X));
  PetscCall(PetscFree(ctx));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode DMSNESJacobianMF_Mult_Private(Mat A, Vec Y, Vec Z)
{
  struct _DMSNESJacobianMFCtx *ctx;

  PetscFunctionBegin;
  PetscCall(MatShellGetContext(A, &ctx));
  PetscCall(DMSNESComputeJacobianAction(ctx->dm, ctx->X, Y, Z, ctx->ctx));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  DMSNESCreateJacobianMF - Create a `Mat` which computes the action of the Jacobian matrix-free

  Collective

  Input Parameters:
+ dm   - The `DM`
. X    - The evaluation point for the Jacobian
- user - A user context, or `NULL`

  Output Parameter:
. J - The `Mat`

  Level: advanced

  Note:
  Vec `X` is kept in `J`, so updating `X` then updates the evaluation point.

.seealso: `DM`, `DMSNESComputeJacobianAction()`
@*/
PetscErrorCode DMSNESCreateJacobianMF(DM dm, Vec X, void *user, Mat *J)
{
  struct _DMSNESJacobianMFCtx *ctx;
  PetscInt                     n, N;

  PetscFunctionBegin;
  PetscCall(MatCreate(PetscObjectComm((PetscObject)dm), J));
  PetscCall(MatSetType(*J, MATSHELL));
  PetscCall(VecGetLocalSize(X, &n));
  PetscCall(VecGetSize(X, &N));
  PetscCall(MatSetSizes(*J, n, n, N, N));
  PetscCall(PetscObjectReference((PetscObject)dm));
  PetscCall(PetscObjectReference((PetscObject)X));
  PetscCall(PetscMalloc1(1, &ctx));
  ctx->dm  = dm;
  ctx->X   = X;
  ctx->ctx = user;
  PetscCall(MatShellSetContext(*J, ctx));
  PetscCall(MatShellSetOperation(*J, MATOP_DESTROY, (void (*)(void))DMSNESJacobianMF_Destroy_Private));
  PetscCall(MatShellSetOperation(*J, MATOP_MULT, (void (*)(void))DMSNESJacobianMF_Mult_Private));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*
     MatComputeNeumannOverlap - Computes an unassembled (Neumann) local overlapping Mat in nonlinear context.

   Input Parameters:
+     X - `SNES` linearization point
.     ovl - index set of overlapping subdomains

   Output Parameter:
.     J - unassembled (Neumann) local matrix

   Level: intermediate

.seealso: `DMCreateNeumannOverlap()`, `MATIS`, `PCHPDDMSetAuxiliaryMat()`
*/
static PetscErrorCode MatComputeNeumannOverlap_Plex(Mat J, PetscReal t, Vec X, Vec X_t, PetscReal s, IS ovl, void *ctx)
{
  SNES   snes;
  Mat    pJ;
  DM     ovldm, origdm;
  DMSNES sdm;
  PetscErrorCode (*bfun)(DM, Vec, void *);
  PetscErrorCode (*jfun)(DM, Vec, Mat, Mat, void *);
  void *bctx, *jctx;

  PetscFunctionBegin;
  PetscCall(PetscObjectQuery((PetscObject)ovl, "_DM_Overlap_HPDDM_MATIS", (PetscObject *)&pJ));
  PetscCheck(pJ, PETSC_COMM_SELF, PETSC_ERR_PLIB, "Missing overlapping Mat");
  PetscCall(PetscObjectQuery((PetscObject)ovl, "_DM_Original_HPDDM", (PetscObject *)&origdm));
  PetscCheck(origdm, PETSC_COMM_SELF, PETSC_ERR_PLIB, "Missing original DM");
  PetscCall(MatGetDM(pJ, &ovldm));
  PetscCall(DMSNESGetBoundaryLocal(origdm, &bfun, &bctx));
  PetscCall(DMSNESSetBoundaryLocal(ovldm, bfun, bctx));
  PetscCall(DMSNESGetJacobianLocal(origdm, &jfun, &jctx));
  PetscCall(DMSNESSetJacobianLocal(ovldm, jfun, jctx));
  PetscCall(PetscObjectQuery((PetscObject)ovl, "_DM_Overlap_HPDDM_SNES", (PetscObject *)&snes));
  if (!snes) {
    PetscCall(SNESCreate(PetscObjectComm((PetscObject)ovl), &snes));
    PetscCall(SNESSetDM(snes, ovldm));
    PetscCall(PetscObjectCompose((PetscObject)ovl, "_DM_Overlap_HPDDM_SNES", (PetscObject)snes));
    PetscCall(PetscObjectDereference((PetscObject)snes));
  }
  PetscCall(DMGetDMSNES(ovldm, &sdm));
  PetscCall(VecLockReadPush(X));
  {
    void *ctx;
    PetscErrorCode (*J)(SNES, Vec, Mat, Mat, void *);
    PetscCall(DMSNESGetJacobian(ovldm, &J, &ctx));
    PetscCallBack("SNES callback Jacobian", (*J)(snes, X, pJ, pJ, ctx));
  }
  PetscCall(VecLockReadPop(X));
  /* this is a no-hop, just in case we decide to change the placeholder for the local Neumann matrix */
  {
    Mat locpJ;

    PetscCall(MatISGetLocalMat(pJ, &locpJ));
    PetscCall(MatCopy(locpJ, J, SAME_NONZERO_PATTERN));
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  DMPlexSetSNESLocalFEM - Use `DMPLEX`'s internal FEM routines to compute `SNES` boundary values, residual, and Jacobian.

  Input Parameters:
+ dm          - The `DM` object
. boundaryctx - the user context that will be passed to pointwise evaluation of boundary values (see `PetscDSAddBoundary()`)
. residualctx - the user context that will be passed to pointwise evaluation of finite element residual computations (see `PetscDSSetResidual()`)
- jacobianctx - the user context that will be passed to pointwise evaluation of finite element Jacobian construction (see `PetscDSSetJacobian()`)

  Level: developer

.seealso: `DMPLEX`, `SNES`
@*/
PetscErrorCode DMPlexSetSNESLocalFEM(DM dm, void *boundaryctx, void *residualctx, void *jacobianctx)
{
  PetscBool useCeed;

  PetscFunctionBegin;
  PetscCall(DMPlexGetUseCeed(dm, &useCeed));
  PetscCall(DMSNESSetBoundaryLocal(dm, DMPlexSNESComputeBoundaryFEM, boundaryctx));
  if (useCeed) {
#ifdef PETSC_HAVE_LIBCEED
    PetscCall(DMSNESSetFunctionLocal(dm, DMPlexSNESComputeResidualCEED, residualctx));
#else
    SETERRQ(PetscObjectComm((PetscObject)dm), PETSC_ERR_SUP, "Cannot use CEED traversals without LibCEED. Rerun configure with --download-ceed");
#endif
  } else PetscCall(DMSNESSetFunctionLocal(dm, DMPlexSNESComputeResidualFEM, residualctx));
  PetscCall(DMSNESSetJacobianLocal(dm, DMPlexSNESComputeJacobianFEM, jacobianctx));
  PetscCall(PetscObjectComposeFunction((PetscObject)dm, "MatComputeNeumannOverlap_C", MatComputeNeumannOverlap_Plex));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@C
  DMSNESCheckDiscretization - Check the discretization error of the exact solution

  Input Parameters:
+ snes - the `SNES` object
. dm   - the `DM`
. t    - the time
. u    - a `DM` vector
- tol  - A tolerance for the check, or -1 to print the results instead

  Output Parameter:
. error - An array which holds the discretization error in each field, or `NULL`

  Level: developer

  Note:
  The user must call `PetscDSSetExactSolution()` beforehand

.seealso: `PetscDSSetExactSolution()`, `DNSNESCheckFromOptions()`, `DMSNESCheckResidual()`, `DMSNESCheckJacobian()`
@*/
PetscErrorCode DMSNESCheckDiscretization(SNES snes, DM dm, PetscReal t, Vec u, PetscReal tol, PetscReal error[])
{
  PetscErrorCode (**exacts)(PetscInt, PetscReal, const PetscReal x[], PetscInt, PetscScalar *u, void *ctx);
  void     **ectxs;
  PetscReal *err;
  MPI_Comm   comm;
  PetscInt   Nf, f;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(snes, SNES_CLASSID, 1);
  PetscValidHeaderSpecific(dm, DM_CLASSID, 2);
  PetscValidHeaderSpecific(u, VEC_CLASSID, 4);
  if (error) PetscAssertPointer(error, 6);

  PetscCall(DMComputeExactSolution(dm, t, u, NULL));
  PetscCall(VecViewFromOptions(u, NULL, "-vec_view"));

  PetscCall(PetscObjectGetComm((PetscObject)snes, &comm));
  PetscCall(DMGetNumFields(dm, &Nf));
  PetscCall(PetscCalloc3(Nf, &exacts, Nf, &ectxs, PetscMax(1, Nf), &err));
  {
    PetscInt Nds, s;

    PetscCall(DMGetNumDS(dm, &Nds));
    for (s = 0; s < Nds; ++s) {
      PetscDS         ds;
      DMLabel         label;
      IS              fieldIS;
      const PetscInt *fields;
      PetscInt        dsNf, f;

      PetscCall(DMGetRegionNumDS(dm, s, &label, &fieldIS, &ds, NULL));
      PetscCall(PetscDSGetNumFields(ds, &dsNf));
      PetscCall(ISGetIndices(fieldIS, &fields));
      for (f = 0; f < dsNf; ++f) {
        const PetscInt field = fields[f];
        PetscCall(PetscDSGetExactSolution(ds, field, &exacts[field], &ectxs[field]));
      }
      PetscCall(ISRestoreIndices(fieldIS, &fields));
    }
  }
  if (Nf > 1) {
    PetscCall(DMComputeL2FieldDiff(dm, t, exacts, ectxs, u, err));
    if (tol >= 0.0) {
      for (f = 0; f < Nf; ++f) PetscCheck(err[f] <= tol, comm, PETSC_ERR_ARG_WRONG, "L_2 Error %g for field %" PetscInt_FMT " exceeds tolerance %g", (double)err[f], f, (double)tol);
    } else if (error) {
      for (f = 0; f < Nf; ++f) error[f] = err[f];
    } else {
      PetscCall(PetscPrintf(comm, "L_2 Error: ["));
      for (f = 0; f < Nf; ++f) {
        if (f) PetscCall(PetscPrintf(comm, ", "));
        PetscCall(PetscPrintf(comm, "%g", (double)err[f]));
      }
      PetscCall(PetscPrintf(comm, "]\n"));
    }
  } else {
    PetscCall(DMComputeL2Diff(dm, t, exacts, ectxs, u, &err[0]));
    if (tol >= 0.0) {
      PetscCheck(err[0] <= tol, comm, PETSC_ERR_ARG_WRONG, "L_2 Error %g exceeds tolerance %g", (double)err[0], (double)tol);
    } else if (error) {
      error[0] = err[0];
    } else {
      PetscCall(PetscPrintf(comm, "L_2 Error: %g\n", (double)err[0]));
    }
  }
  PetscCall(PetscFree3(exacts, ectxs, err));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@C
  DMSNESCheckResidual - Check the residual of the exact solution

  Input Parameters:
+ snes - the `SNES` object
. dm   - the `DM`
. u    - a `DM` vector
- tol  - A tolerance for the check, or -1 to print the results instead

  Output Parameter:
. residual - The residual norm of the exact solution, or `NULL`

  Level: developer

.seealso: `DNSNESCheckFromOptions()`, `DMSNESCheckDiscretization()`, `DMSNESCheckJacobian()`
@*/
PetscErrorCode DMSNESCheckResidual(SNES snes, DM dm, Vec u, PetscReal tol, PetscReal *residual)
{
  MPI_Comm  comm;
  Vec       r;
  PetscReal res;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(snes, SNES_CLASSID, 1);
  PetscValidHeaderSpecific(dm, DM_CLASSID, 2);
  PetscValidHeaderSpecific(u, VEC_CLASSID, 3);
  if (residual) PetscAssertPointer(residual, 5);
  PetscCall(PetscObjectGetComm((PetscObject)snes, &comm));
  PetscCall(DMComputeExactSolution(dm, 0.0, u, NULL));
  PetscCall(VecDuplicate(u, &r));
  PetscCall(SNESComputeFunction(snes, u, r));
  PetscCall(VecNorm(r, NORM_2, &res));
  if (tol >= 0.0) {
    PetscCheck(res <= tol, comm, PETSC_ERR_ARG_WRONG, "L_2 Residual %g exceeds tolerance %g", (double)res, (double)tol);
  } else if (residual) {
    *residual = res;
  } else {
    PetscCall(PetscPrintf(comm, "L_2 Residual: %g\n", (double)res));
    PetscCall(VecFilter(r, 1.0e-10));
    PetscCall(PetscObjectSetName((PetscObject)r, "Initial Residual"));
    PetscCall(PetscObjectSetOptionsPrefix((PetscObject)r, "res_"));
    PetscCall(VecViewFromOptions(r, NULL, "-vec_view"));
  }
  PetscCall(VecDestroy(&r));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@C
  DMSNESCheckJacobian - Check the Jacobian of the exact solution against the residual using the Taylor Test

  Input Parameters:
+ snes - the `SNES` object
. dm   - the `DM`
. u    - a `DM` vector
- tol  - A tolerance for the check, or -1 to print the results instead

  Output Parameters:
+ isLinear - Flag indicaing that the function looks linear, or `NULL`
- convRate - The rate of convergence of the linear model, or `NULL`

  Level: developer

.seealso: `DNSNESCheckFromOptions()`, `DMSNESCheckDiscretization()`, `DMSNESCheckResidual()`
@*/
PetscErrorCode DMSNESCheckJacobian(SNES snes, DM dm, Vec u, PetscReal tol, PetscBool *isLinear, PetscReal *convRate)
{
  MPI_Comm     comm;
  PetscDS      ds;
  Mat          J, M;
  MatNullSpace nullspace;
  PetscReal    slope, intercept;
  PetscBool    hasJac, hasPrec, isLin = PETSC_FALSE;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(snes, SNES_CLASSID, 1);
  PetscValidHeaderSpecific(dm, DM_CLASSID, 2);
  PetscValidHeaderSpecific(u, VEC_CLASSID, 3);
  if (isLinear) PetscAssertPointer(isLinear, 5);
  if (convRate) PetscAssertPointer(convRate, 6);
  PetscCall(PetscObjectGetComm((PetscObject)snes, &comm));
  PetscCall(DMComputeExactSolution(dm, 0.0, u, NULL));
  /* Create and view matrices */
  PetscCall(DMCreateMatrix(dm, &J));
  PetscCall(DMGetDS(dm, &ds));
  PetscCall(PetscDSHasJacobian(ds, &hasJac));
  PetscCall(PetscDSHasJacobianPreconditioner(ds, &hasPrec));
  if (hasJac && hasPrec) {
    PetscCall(DMCreateMatrix(dm, &M));
    PetscCall(SNESComputeJacobian(snes, u, J, M));
    PetscCall(PetscObjectSetName((PetscObject)M, "Preconditioning Matrix"));
    PetscCall(PetscObjectSetOptionsPrefix((PetscObject)M, "jacpre_"));
    PetscCall(MatViewFromOptions(M, NULL, "-mat_view"));
    PetscCall(MatDestroy(&M));
  } else {
    PetscCall(SNESComputeJacobian(snes, u, J, J));
  }
  PetscCall(PetscObjectSetName((PetscObject)J, "Jacobian"));
  PetscCall(PetscObjectSetOptionsPrefix((PetscObject)J, "jac_"));
  PetscCall(MatViewFromOptions(J, NULL, "-mat_view"));
  /* Check nullspace */
  PetscCall(MatGetNullSpace(J, &nullspace));
  if (nullspace) {
    PetscBool isNull;
    PetscCall(MatNullSpaceTest(nullspace, J, &isNull));
    PetscCheck(isNull, comm, PETSC_ERR_PLIB, "The null space calculated for the system operator is invalid.");
  }
  /* Taylor test */
  {
    PetscRandom rand;
    Vec         du, uhat, r, rhat, df;
    PetscReal   h;
    PetscReal  *es, *hs, *errors;
    PetscReal   hMax = 1.0, hMin = 1e-6, hMult = 0.1;
    PetscInt    Nv, v;

    /* Choose a perturbation direction */
    PetscCall(PetscRandomCreate(comm, &rand));
    PetscCall(VecDuplicate(u, &du));
    PetscCall(VecSetRandom(du, rand));
    PetscCall(PetscRandomDestroy(&rand));
    PetscCall(VecDuplicate(u, &df));
    PetscCall(MatMult(J, du, df));
    /* Evaluate residual at u, F(u), save in vector r */
    PetscCall(VecDuplicate(u, &r));
    PetscCall(SNESComputeFunction(snes, u, r));
    /* Look at the convergence of our Taylor approximation as we approach u */
    for (h = hMax, Nv = 0; h >= hMin; h *= hMult, ++Nv)
      ;
    PetscCall(PetscCalloc3(Nv, &es, Nv, &hs, Nv, &errors));
    PetscCall(VecDuplicate(u, &uhat));
    PetscCall(VecDuplicate(u, &rhat));
    for (h = hMax, Nv = 0; h >= hMin; h *= hMult, ++Nv) {
      PetscCall(VecWAXPY(uhat, h, du, u));
      /* F(\hat u) \approx F(u) + J(u) (uhat - u) = F(u) + h * J(u) du */
      PetscCall(SNESComputeFunction(snes, uhat, rhat));
      PetscCall(VecAXPBYPCZ(rhat, -1.0, -h, 1.0, r, df));
      PetscCall(VecNorm(rhat, NORM_2, &errors[Nv]));

      es[Nv] = errors[Nv] == 0 ? -16.0 : PetscLog10Real(errors[Nv]);
      hs[Nv] = PetscLog10Real(h);
    }
    PetscCall(VecDestroy(&uhat));
    PetscCall(VecDestroy(&rhat));
    PetscCall(VecDestroy(&df));
    PetscCall(VecDestroy(&r));
    PetscCall(VecDestroy(&du));
    for (v = 0; v < Nv; ++v) {
      if ((tol >= 0) && (errors[v] > tol)) break;
      else if (errors[v] > PETSC_SMALL) break;
    }
    if (v == Nv) isLin = PETSC_TRUE;
    PetscCall(PetscLinearRegression(Nv, hs, es, &slope, &intercept));
    PetscCall(PetscFree3(es, hs, errors));
    /* Slope should be about 2 */
    if (tol >= 0) {
      PetscCheck(isLin || PetscAbsReal(2 - slope) <= tol, comm, PETSC_ERR_ARG_WRONG, "Taylor approximation convergence rate should be 2, not %0.2f", (double)slope);
    } else if (isLinear || convRate) {
      if (isLinear) *isLinear = isLin;
      if (convRate) *convRate = slope;
    } else {
      if (!isLin) PetscCall(PetscPrintf(comm, "Taylor approximation converging at order %3.2f\n", (double)slope));
      else PetscCall(PetscPrintf(comm, "Function appears to be linear\n"));
    }
  }
  PetscCall(MatDestroy(&J));
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode DMSNESCheck_Internal(SNES snes, DM dm, Vec u)
{
  PetscFunctionBegin;
  PetscCall(DMSNESCheckDiscretization(snes, dm, 0.0, u, -1.0, NULL));
  PetscCall(DMSNESCheckResidual(snes, dm, u, -1.0, NULL));
  PetscCall(DMSNESCheckJacobian(snes, dm, u, -1.0, NULL, NULL));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@C
  DMSNESCheckFromOptions - Check the residual and Jacobian functions using the exact solution by outputting some diagnostic information

  Input Parameters:
+ snes - the `SNES` object
- u    - representative `SNES` vector

  Level: developer

  Note:
  The user must call `PetscDSSetExactSolution()` beforehand

.seealso: `SNES`, `DM`
@*/
PetscErrorCode DMSNESCheckFromOptions(SNES snes, Vec u)
{
  DM        dm;
  Vec       sol;
  PetscBool check;

  PetscFunctionBegin;
  PetscCall(PetscOptionsHasName(((PetscObject)snes)->options, ((PetscObject)snes)->prefix, "-dmsnes_check", &check));
  if (!check) PetscFunctionReturn(PETSC_SUCCESS);
  PetscCall(SNESGetDM(snes, &dm));
  PetscCall(VecDuplicate(u, &sol));
  PetscCall(SNESSetSolution(snes, sol));
  PetscCall(DMSNESCheck_Internal(snes, dm, sol));
  PetscCall(VecDestroy(&sol));
  PetscFunctionReturn(PETSC_SUCCESS);
}