plex/tests/ex39.c

const char help[] = "A test of H-div conforming discretizations on different cell types.\n";

#include <petscdmplex.h>
#include <petscds.h>
#include <petscsnes.h>
#include <petscconvest.h>
#include <petscfe.h>
#include <petsc/private/petscfeimpl.h>

/*
  We are using the system

  \vec{u} = \vec{\hat{u}}
  p = \div{\vec{u}} in low degree approximation space
  d = \div{\vec{u}} - p == 0 in higher degree approximation space

  That is, we are using the field d to compute the error between \div{\vec{u}}
  computed in a space 1 degree higher than p and the value of p which is
  \div{u} computed in the low degree space. If H-div
  elements are implemented correctly then this should be identically zero since
  the divergence of a function in H(div) should be exactly representable in L_2
  by definition.
*/
static PetscErrorCode zero_func(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nc, PetscScalar *u, void *ctx)
{
  PetscInt c;
  for (c = 0; c < Nc; ++c) u[c] = 0;
  return PETSC_SUCCESS;
}
/* Linear Exact Functions
   \vec{u} = \vec{x};
   p = dim;
   */
static PetscErrorCode linear_u(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nc, PetscScalar *u, void *ctx)
{
  PetscInt c;
  for (c = 0; c < Nc; ++c) u[c] = x[c];
  return PETSC_SUCCESS;
}
static PetscErrorCode linear_p(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nc, PetscScalar *u, void *ctx)
{
  u[0] = dim;
  return PETSC_SUCCESS;
}

/* Sinusoidal Exact Functions
 * u_i = \sin{2*\pi*x_i}
 * p = \Sum_{i=1}^{dim} 2*\pi*cos{2*\pi*x_i}
 * */

static PetscErrorCode sinusoid_u(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nc, PetscScalar *u, void *ctx)
{
  PetscInt c;
  for (c = 0; c < Nc; ++c) u[c] = PetscSinReal(2 * PETSC_PI * x[c]);
  return PETSC_SUCCESS;
}
static PetscErrorCode sinusoid_p(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nc, PetscScalar *u, void *ctx)
{
  PetscInt d;
  u[0] = 0;
  for (d = 0; d < dim; ++d) u[0] += 2 * PETSC_PI * PetscCosReal(2 * PETSC_PI * x[d]);
  return PETSC_SUCCESS;
}

/* Pointwise residual for u = u*. Need one of these for each possible u* */
static void f0_v_linear(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 f0[])
{
  PetscInt     i;
  PetscScalar *u_rhs;

  PetscCallAbort(PETSC_COMM_SELF, PetscCalloc1(dim, &u_rhs));
  PetscCallAbort(PETSC_COMM_SELF, linear_u(dim, t, x, dim, u_rhs, NULL));
  for (i = 0; i < dim; ++i) f0[i] = u[uOff[0] + i] - u_rhs[i];
  PetscCallAbort(PETSC_COMM_SELF, PetscFree(u_rhs));
}

static void f0_v_sinusoid(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 f0[])
{
  PetscInt     i;
  PetscScalar *u_rhs;

  PetscCallAbort(PETSC_COMM_SELF, PetscCalloc1(dim, &u_rhs));
  PetscCallAbort(PETSC_COMM_SELF, sinusoid_u(dim, t, x, dim, u_rhs, NULL));
  for (i = 0; i < dim; ++i) f0[i] = u[uOff[0] + i] - u_rhs[i];
  PetscCallAbort(PETSC_COMM_SELF, PetscFree(u_rhs));
}

/* Residual function for enforcing p = \div{u}. */
static void f0_q(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 f0[])
{
  PetscInt    i;
  PetscScalar divu;

  divu = 0.;
  for (i = 0; i < dim; ++i) divu += u_x[uOff_x[0] + i * dim + i];
  f0[0] = u[uOff[1]] - divu;
}

/* Residual function for p_err = \div{u} - p. */
static void f0_w(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 f0[])
{
  PetscInt    i;
  PetscScalar divu;

  divu = 0.;
  for (i = 0; i < dim; ++i) divu += u_x[uOff_x[0] + i * dim + i];
  f0[0] = u[uOff[2]] - u[uOff[1]] + divu;
}

/* Boundary residual for the embedding system. Need one for each form of
 * solution. These enforce u = \hat{u} at the boundary. */
static void f0_bd_u_sinusoid(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[], const PetscReal n[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f0[])
{
  PetscInt     d;
  PetscScalar *u_rhs;

  PetscCallAbort(PETSC_COMM_SELF, PetscCalloc1(dim, &u_rhs));
  PetscCallAbort(PETSC_COMM_SELF, sinusoid_u(dim, t, x, dim, u_rhs, NULL));
  for (d = 0; d < dim; ++d) f0[d] = u_rhs[d];
  PetscCallAbort(PETSC_COMM_SELF, PetscFree(u_rhs));
}

static void f0_bd_u_linear(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[], const PetscReal n[], PetscInt numConstants, const PetscScalar constants[], PetscScalar f0[])
{
  PetscInt     d;
  PetscScalar *u_rhs;

  PetscCallAbort(PETSC_COMM_SELF, PetscCalloc1(dim, &u_rhs));
  PetscCallAbort(PETSC_COMM_SELF, linear_u(dim, t, x, dim, u_rhs, NULL));
  for (d = 0; d < dim; ++d) f0[d] = u_rhs[d];
  PetscCallAbort(PETSC_COMM_SELF, PetscFree(u_rhs));
}
/* Jacobian functions. For the following, v is the test function associated with
 * u, q the test function associated with p, and w the test function associated
 * with d. */
/* <v, u> */
static void g0_vu(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, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g0[])
{
  PetscInt c;
  for (c = 0; c < dim; ++c) g0[c * dim + c] = 1.0;
}

/* <q, p> */
static void g0_qp(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, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g0[])
{
  PetscInt d;
  for (d = 0; d < dim; ++d) g0[d * dim + d] = 1.0;
}

/* -<q, \div{u}> For the embedded system. This is different from the method of
 * manufactured solution because instead of computing <q,\div{u}> - <q,f> we
 * need <q,p> - <q,\div{u}.*/
static void g1_qu(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, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g1[])
{
  PetscInt d;
  for (d = 0; d < dim; ++d) g1[d * dim + d] = -1.0;
}

/* <w, p> This is only used by the embedded system. Where we need to compute
 * <w,d> - <w,p> + <w, \div{u}>*/
static void g0_wp(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, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g0[])
{
  PetscInt d;
  for (d = 0; d < dim; ++d) g0[d * dim + d] = -1.0;
}

/* <w, d> */
static void g0_wd(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, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g0[])
{
  PetscInt c;
  for (c = 0; c < dim; ++c) g0[c * dim + c] = 1.0;
}

/* <w, \div{u}> for the embedded system. */
static void g1_wu(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, PetscReal u_tShift, const PetscReal x[], PetscInt numConstants, const PetscScalar constants[], PetscScalar g1[])
{
  PetscInt d;
  for (d = 0; d < dim; ++d) g1[d * dim + d] = 1.0;
}

/* Enum and string array for selecting mesh perturbation options */
typedef enum {
  NONE         = 0,
  PERTURB      = 1,
  SKEW         = 2,
  SKEW_PERTURB = 3
} Transform;
const char *const TransformTypes[] = {"none", "perturb", "skew", "skew_perturb", "Perturbation", "", NULL};

/* Enum and string array for selecting the form of the exact solution*/
typedef enum {
  LINEAR     = 0,
  SINUSOIDAL = 1
} Solution;
const char *const SolutionTypes[] = {"linear", "sinusoidal", "Solution", "", NULL};

typedef struct {
  Transform mesh_transform;
  Solution  sol_form;
} UserCtx;

/* Process command line options and initialize the UserCtx struct */
static PetscErrorCode ProcessOptions(MPI_Comm comm, UserCtx *user)
{
  PetscFunctionBegin;
  /* Default to  2D, unperturbed triangle mesh and Linear solution.*/
  user->mesh_transform = NONE;
  user->sol_form       = LINEAR;

  PetscOptionsBegin(comm, "", "H-div Test Options", "DMPLEX");
  PetscCall(PetscOptionsEnum("-mesh_transform", "Method used to perturb the mesh vertices. Options are skew, perturb, skew_perturb,or none", "ex39.c", TransformTypes, (PetscEnum)user->mesh_transform, (PetscEnum *)&user->mesh_transform, NULL));
  PetscCall(PetscOptionsEnum("-sol_form", "Form of the exact solution. Options are Linear or Sinusoidal", "ex39.c", SolutionTypes, (PetscEnum)user->sol_form, (PetscEnum *)&user->sol_form, NULL));
  PetscOptionsEnd();
  PetscFunctionReturn(PETSC_SUCCESS);
}

/* Perturb the position of each mesh vertex by a small amount.*/
static PetscErrorCode PerturbMesh(DM *mesh, PetscScalar *coordVals, PetscInt npoints, PetscInt dim)
{
  PetscInt    i, j, k;
  PetscReal   minCoords[3], maxCoords[3], maxPert[3], randVal, amp;
  PetscRandom ran;

  PetscFunctionBegin;
  PetscCall(DMGetCoordinateDim(*mesh, &dim));
  PetscCall(DMGetLocalBoundingBox(*mesh, minCoords, maxCoords));
  PetscCall(PetscRandomCreate(PETSC_COMM_WORLD, &ran));

  /* Compute something approximately equal to half an edge length. This is the
   * most we can perturb points and guarantee that there won't be any topology
   * issues. */
  for (k = 0; k < dim; ++k) maxPert[k] = 0.025 * (maxCoords[k] - minCoords[k]) / (PetscPowReal(npoints, 1. / dim) - 1);
  /* For each mesh vertex */
  for (i = 0; i < npoints; ++i) {
    /* For each coordinate of the vertex */
    for (j = 0; j < dim; ++j) {
      /* Generate a random amplitude in [-0.5*maxPert, 0.5*maxPert] */
      PetscCall(PetscRandomGetValueReal(ran, &randVal));
      amp = maxPert[j] * (randVal - 0.5);
      /* Add the perturbation to the vertex*/
      coordVals[dim * i + j] += amp;
    }
  }

  PetscCall(PetscRandomDestroy(&ran));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/* Apply a global skew transformation to the mesh. */
static PetscErrorCode SkewMesh(DM *mesh, PetscScalar *coordVals, PetscInt npoints, PetscInt dim)
{
  PetscInt     i, j, k, l;
  PetscScalar *transMat;
  PetscScalar  tmpcoord[3];
  PetscRandom  ran;
  PetscReal    randVal;

  PetscFunctionBegin;
  PetscCall(PetscCalloc1(dim * dim, &transMat));
  PetscCall(PetscRandomCreate(PETSC_COMM_WORLD, &ran));

  /* Make a matrix representing a skew transformation */
  for (i = 0; i < dim; ++i) {
    for (j = 0; j < dim; ++j) {
      PetscCall(PetscRandomGetValueReal(ran, &randVal));
      if (i == j) transMat[i * dim + j] = 1.;
      else if (j < i) transMat[i * dim + j] = 2 * (j + i) * randVal;
      else transMat[i * dim + j] = 0;
    }
  }

  /* Multiply each coordinate vector by our transformation.*/
  for (i = 0; i < npoints; ++i) {
    for (j = 0; j < dim; ++j) {
      tmpcoord[j] = 0;
      for (k = 0; k < dim; ++k) tmpcoord[j] += coordVals[dim * i + k] * transMat[dim * k + j];
    }
    for (l = 0; l < dim; ++l) coordVals[dim * i + l] = tmpcoord[l];
  }
  PetscCall(PetscFree(transMat));
  PetscCall(PetscRandomDestroy(&ran));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/* Accesses the mesh coordinate array and performs the transformation operations
 * specified by the user options */
static PetscErrorCode TransformMesh(UserCtx *user, DM *mesh)
{
  PetscInt     dim, npoints;
  PetscScalar *coordVals;
  Vec          coords;

  PetscFunctionBegin;
  PetscCall(DMGetCoordinates(*mesh, &coords));
  PetscCall(VecGetArray(coords, &coordVals));
  PetscCall(VecGetLocalSize(coords, &npoints));
  PetscCall(DMGetCoordinateDim(*mesh, &dim));
  npoints = npoints / dim;

  switch (user->mesh_transform) {
  case PERTURB:
    PetscCall(PerturbMesh(mesh, coordVals, npoints, dim));
    break;
  case SKEW:
    PetscCall(SkewMesh(mesh, coordVals, npoints, dim));
    break;
  case SKEW_PERTURB:
    PetscCall(SkewMesh(mesh, coordVals, npoints, dim));
    PetscCall(PerturbMesh(mesh, coordVals, npoints, dim));
    break;
  default:
    SETERRQ(PETSC_COMM_WORLD, PETSC_ERR_ARG_WRONG, "invalid mesh transformation");
  }
  PetscCall(VecRestoreArray(coords, &coordVals));
  PetscCall(DMSetCoordinates(*mesh, coords));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode CreateMesh(MPI_Comm comm, UserCtx *user, DM *mesh)
{
  PetscFunctionBegin;
  PetscCall(DMCreate(comm, mesh));
  PetscCall(DMSetType(*mesh, DMPLEX));
  PetscCall(DMSetFromOptions(*mesh));

  /* Perform any mesh transformations if specified by user */
  if (user->mesh_transform != NONE) PetscCall(TransformMesh(user, mesh));

  /* Get any other mesh options from the command line */
  PetscCall(DMSetApplicationContext(*mesh, user));
  PetscCall(DMViewFromOptions(*mesh, NULL, "-dm_view"));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/* Setup the system of equations that we wish to solve */
static PetscErrorCode SetupProblem(DM dm, UserCtx *user)
{
  PetscDS        prob;
  DMLabel        label;
  const PetscInt id = 1;

  PetscFunctionBegin;
  PetscCall(DMGetDS(dm, &prob));
  /* All of these are independent of the user's choice of solution */
  PetscCall(PetscDSSetResidual(prob, 1, f0_q, NULL));
  PetscCall(PetscDSSetResidual(prob, 2, f0_w, NULL));
  PetscCall(PetscDSSetJacobian(prob, 0, 0, g0_vu, NULL, NULL, NULL));
  PetscCall(PetscDSSetJacobian(prob, 1, 0, NULL, g1_qu, NULL, NULL));
  PetscCall(PetscDSSetJacobian(prob, 1, 1, g0_qp, NULL, NULL, NULL));
  PetscCall(PetscDSSetJacobian(prob, 2, 0, NULL, g1_wu, NULL, NULL));
  PetscCall(PetscDSSetJacobian(prob, 2, 1, g0_wp, NULL, NULL, NULL));
  PetscCall(PetscDSSetJacobian(prob, 2, 2, g0_wd, NULL, NULL, NULL));

  /* Field 2 is the error between \div{u} and pressure in a higher dimensional
   * space. If all is right this should be machine zero. */
  PetscCall(PetscDSSetExactSolution(prob, 2, zero_func, NULL));

  switch (user->sol_form) {
  case LINEAR:
    PetscCall(PetscDSSetResidual(prob, 0, f0_v_linear, NULL));
    PetscCall(PetscDSSetBdResidual(prob, 0, f0_bd_u_linear, NULL));
    PetscCall(PetscDSSetExactSolution(prob, 0, linear_u, NULL));
    PetscCall(PetscDSSetExactSolution(prob, 1, linear_p, NULL));
    break;
  case SINUSOIDAL:
    PetscCall(PetscDSSetResidual(prob, 0, f0_v_sinusoid, NULL));
    PetscCall(PetscDSSetBdResidual(prob, 0, f0_bd_u_sinusoid, NULL));
    PetscCall(PetscDSSetExactSolution(prob, 0, sinusoid_u, NULL));
    PetscCall(PetscDSSetExactSolution(prob, 1, sinusoid_p, NULL));
    break;
  default:
    SETERRQ(PETSC_COMM_WORLD, PETSC_ERR_ARG_WRONG, "invalid solution form");
  }

  PetscCall(DMGetLabel(dm, "marker", &label));
  PetscCall(PetscDSAddBoundary(prob, DM_BC_NATURAL, "Boundary Integral", label, 1, &id, 0, 0, NULL, (void (*)(void))NULL, NULL, user, NULL));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/* Create the finite element spaces we will use for this system */
static PetscErrorCode SetupDiscretization(DM mesh, PetscErrorCode (*setup)(DM, UserCtx *), UserCtx *user)
{
  DM        cdm = mesh;
  PetscFE   fevel, fepres, fedivErr;
  PetscInt  dim;
  PetscBool simplex;

  PetscFunctionBegin;
  PetscCall(DMGetDimension(mesh, &dim));
  PetscCall(DMPlexIsSimplex(mesh, &simplex));
  /* Create FE objects and give them names so that options can be set from
   * command line */
  PetscCall(PetscFECreateDefault(PetscObjectComm((PetscObject)mesh), dim, dim, simplex, "velocity_", -1, &fevel));
  PetscCall(PetscObjectSetName((PetscObject)fevel, "velocity"));

  PetscCall(PetscFECreateDefault(PetscObjectComm((PetscObject)mesh), dim, 1, simplex, "pressure_", -1, &fepres));
  PetscCall(PetscObjectSetName((PetscObject)fepres, "pressure"));

  PetscCall(PetscFECreateDefault(PetscObjectComm((PetscObject)mesh), dim, 1, simplex, "divErr_", -1, &fedivErr));
  PetscCall(PetscObjectSetName((PetscObject)fedivErr, "divErr"));

  PetscCall(PetscFECopyQuadrature(fevel, fepres));
  PetscCall(PetscFECopyQuadrature(fevel, fedivErr));

  /* Associate the FE objects with the mesh and setup the system */
  PetscCall(DMSetField(mesh, 0, NULL, (PetscObject)fevel));
  PetscCall(DMSetField(mesh, 1, NULL, (PetscObject)fepres));
  PetscCall(DMSetField(mesh, 2, NULL, (PetscObject)fedivErr));
  PetscCall(DMCreateDS(mesh));
  PetscCall((*setup)(mesh, user));

  while (cdm) {
    PetscCall(DMCopyDisc(mesh, cdm));
    PetscCall(DMGetCoarseDM(cdm, &cdm));
  }

  /* The Mesh now owns the fields, so we can destroy the FEs created in this
   * function */
  PetscCall(PetscFEDestroy(&fevel));
  PetscCall(PetscFEDestroy(&fepres));
  PetscCall(PetscFEDestroy(&fedivErr));
  PetscCall(DMDestroy(&cdm));
  PetscFunctionReturn(PETSC_SUCCESS);
}

int main(int argc, char **argv)
{
  PetscInt        i;
  UserCtx         user;
  DM              mesh;
  SNES            snes;
  Vec             computed, divErr;
  PetscReal       divErrNorm;
  IS             *fieldIS;
  PetscBool       exampleSuccess = PETSC_FALSE;
  const PetscReal errTol         = 10. * PETSC_SMALL;

  char stdFormat[] = "L2 Norm of the Divergence Error is: %g\n H(div) elements working correctly: %s\n";

  /* Initialize PETSc */
  PetscFunctionBeginUser;
  PetscCall(PetscInitialize(&argc, &argv, NULL, help));
  PetscCall(ProcessOptions(PETSC_COMM_WORLD, &user));

  /* Set up the system, we need to create a solver and a mesh and then assign
   * the correct spaces into the mesh*/
  PetscCall(SNESCreate(PETSC_COMM_WORLD, &snes));
  PetscCall(CreateMesh(PETSC_COMM_WORLD, &user, &mesh));
  PetscCall(SNESSetDM(snes, mesh));
  PetscCall(SetupDiscretization(mesh, SetupProblem, &user));
  PetscCall(DMPlexSetSNESLocalFEM(mesh, PETSC_FALSE, &user));
  PetscCall(SNESSetFromOptions(snes));

  /* Grab field IS so that we can view the solution by field */
  PetscCall(DMCreateFieldIS(mesh, NULL, NULL, &fieldIS));

  /* Create a vector to store the SNES solution, solve the system and grab the
   * solution from SNES */
  PetscCall(DMCreateGlobalVector(mesh, &computed));
  PetscCall(PetscObjectSetName((PetscObject)computed, "computedSolution"));
  PetscCall(VecSet(computed, 0.0));
  PetscCall(SNESSolve(snes, NULL, computed));
  PetscCall(SNESGetSolution(snes, &computed));
  PetscCall(VecViewFromOptions(computed, NULL, "-computedSolution_view"));

  /* Now we pull out the portion of the vector corresponding to the 3rd field
   * which is the error between \div{u} computed in a higher dimensional space
   * and p = \div{u} computed in a low dimension space. We report the L2 norm of
   * this vector which should be zero if the H(div) spaces are implemented
   * correctly. */
  PetscCall(VecGetSubVector(computed, fieldIS[2], &divErr));
  PetscCall(VecNorm(divErr, NORM_2, &divErrNorm));
  PetscCall(VecRestoreSubVector(computed, fieldIS[2], &divErr));
  exampleSuccess = (PetscBool)(divErrNorm <= errTol);

  PetscCall(PetscPrintf(PETSC_COMM_WORLD, stdFormat, divErrNorm, exampleSuccess ? "true" : "false"));

  /* Tear down */
  PetscCall(VecDestroy(&divErr));
  PetscCall(VecDestroy(&computed));
  for (i = 0; i < 3; ++i) PetscCall(ISDestroy(&fieldIS[i]));
  PetscCall(PetscFree(fieldIS));
  PetscCall(SNESDestroy(&snes));
  PetscCall(DMDestroy(&mesh));
  PetscCall(PetscFinalize());
  return 0;
}

/*TEST
  testset:
    suffix: 2d_bdm
    requires: triangle
    args: -velocity_petscfe_default_quadrature_order 1 \
      -velocity_petscspace_degree 1 \
      -velocity_petscdualspace_type bdm \
      -divErr_petscspace_degree 1 \
      -divErr_petscdualspace_lagrange_continuity false \
      -snes_error_if_not_converged \
      -ksp_rtol 1e-10 \
      -ksp_error_if_not_converged \
      -pc_type fieldsplit\
      -pc_fieldsplit_detect_saddle_point\
      -pc_fieldsplit_type schur\
      -pc_fieldsplit_schur_precondition full
    test:
      suffix: linear
      args: -sol_form linear -mesh_transform none
    test:
      suffix: sinusoidal
      args: -sol_form sinusoidal -mesh_transform none
    test:
      suffix: sinusoidal_skew
      args: -sol_form sinusoidal -mesh_transform skew
    test:
      suffix: sinusoidal_perturb
      args: -sol_form sinusoidal -mesh_transform perturb
    test:
      suffix: sinusoidal_skew_perturb
      args: -sol_form sinusoidal -mesh_transform skew_perturb

  testset:
    TODO: broken
    suffix: 2d_bdmq
    output_file: output/empty.out
    args: -dm_plex_simplex false \
      -velocity_petscspace_degree 1 \
      -velocity_petscdualspace_type bdm \
      -velocity_petscdualspace_lagrange_tensor 1 \
      -divErr_petscspace_degree 1 \
      -divErr_petscdualspace_lagrange_continuity false \
      -snes_error_if_not_converged \
      -ksp_rtol 1e-10 \
      -ksp_error_if_not_converged \
      -pc_type fieldsplit\
      -pc_fieldsplit_detect_saddle_point\
      -pc_fieldsplit_type schur\
      -pc_fieldsplit_schur_precondition full
    test:
      suffix: linear
      args: -sol_form linear -mesh_transform none
    test:
      suffix: sinusoidal
      args: -sol_form sinusoidal -mesh_transform none
    test:
      suffix: sinusoidal_skew
      args: -sol_form sinusoidal -mesh_transform skew
    test:
      suffix: sinusoidal_perturb
      args: -sol_form sinusoidal -mesh_transform perturb
    test:
      suffix: sinusoidal_skew_perturb
      args: -sol_form sinusoidal -mesh_transform skew_perturb

  testset:
    suffix: 3d_bdm
    requires: ctetgen
    args: -dm_plex_dim 3 \
      -velocity_petscspace_degree 1 \
      -velocity_petscdualspace_type bdm \
      -divErr_petscspace_degree 1 \
      -divErr_petscdualspace_lagrange_continuity false \
      -snes_error_if_not_converged \
      -ksp_rtol 1e-10 \
      -ksp_error_if_not_converged \
      -pc_type fieldsplit \
      -pc_fieldsplit_detect_saddle_point \
      -pc_fieldsplit_type schur \
      -pc_fieldsplit_schur_precondition full
    test:
      suffix: linear
      args: -sol_form linear -mesh_transform none
    test:
      suffix: sinusoidal
      args: -sol_form sinusoidal -mesh_transform none
    test:
      suffix: sinusoidal_skew
      args: -sol_form sinusoidal -mesh_transform skew
    test:
      suffix: sinusoidal_perturb
      args: -sol_form sinusoidal -mesh_transform perturb
    test:
      suffix: sinusoidal_skew_perturb
      args: -sol_form sinusoidal -mesh_transform skew_perturb

  testset:
    TODO: broken
    suffix: 3d_bdmq
    output_file: output/empty.out
    requires: ctetgen
    args: -dm_plex_dim 3 \
      -dm_plex_simplex false \
      -velocity_petscspace_degree 1 \
      -velocity_petscdualspace_type bdm \
      -velocity_petscdualspace_lagrange_tensor 1 \
      -divErr_petscspace_degree 1 \
      -divErr_petscdualspace_lagrange_continuity false \
      -snes_error_if_not_converged \
      -ksp_rtol 1e-10 \
      -ksp_error_if_not_converged \
      -pc_type fieldsplit \
      -pc_fieldsplit_detect_saddle_point \
      -pc_fieldsplit_type schur \
      -pc_fieldsplit_schur_precondition full
    test:
      suffix: linear
      args: -sol_form linear -mesh_transform none
    test:
      suffix: sinusoidal
      args: -sol_form sinusoidal -mesh_transform none
    test:
      suffix: sinusoidal_skew
      args: -sol_form sinusoidal -mesh_transform skew
    test:
      suffix: sinusoidal_perturb
      args: -sol_form sinusoidal -mesh_transform perturb
    test:
      suffix: sinusoidal_skew_perturb
      args: -sol_form sinusoidal -mesh_transform skew_perturb

  test:
    suffix: quad_rt_0
    args: -dm_plex_simplex false -mesh_transform skew \
          -divErr_petscspace_degree 1 \
          -divErr_petscdualspace_lagrange_continuity false \
          -snes_error_if_not_converged \
          -ksp_rtol 1e-10 \
          -ksp_error_if_not_converged \
          -pc_type fieldsplit\
          -pc_fieldsplit_detect_saddle_point\
          -pc_fieldsplit_type schur\
          -pc_fieldsplit_schur_precondition full \
          -velocity_petscfe_default_quadrature_order 1 \
          -velocity_petscspace_type sum \
          -velocity_petscspace_variables 2 \
          -velocity_petscspace_components 2 \
          -velocity_petscspace_sum_spaces 2 \
          -velocity_petscspace_sum_concatenate true \
          -velocity_sumcomp_0_petscspace_variables 2 \
          -velocity_sumcomp_0_petscspace_type tensor \
          -velocity_sumcomp_0_petscspace_tensor_spaces 2 \
          -velocity_sumcomp_0_petscspace_tensor_uniform false \
          -velocity_sumcomp_0_tensorcomp_0_petscspace_degree 1 \
          -velocity_sumcomp_0_tensorcomp_1_petscspace_degree 0 \
          -velocity_sumcomp_1_petscspace_variables 2 \
          -velocity_sumcomp_1_petscspace_type tensor \
          -velocity_sumcomp_1_petscspace_tensor_spaces 2 \
          -velocity_sumcomp_1_petscspace_tensor_uniform false \
          -velocity_sumcomp_1_tensorcomp_0_petscspace_degree 0 \
          -velocity_sumcomp_1_tensorcomp_1_petscspace_degree 1 \
          -velocity_petscdualspace_form_degree -1 \
          -velocity_petscdualspace_order 1 \
          -velocity_petscdualspace_lagrange_trimmed true
TEST*/