xref: /petsc/src/ts/tutorials/ex45.c (revision d0609ced746bc51b019815ca91d747429db24893)

static char help[] = "Heat Equation in 2d and 3d with finite elements.\n\
We solve the heat equation in a rectangular\n\
domain, using a parallel unstructured mesh (DMPLEX) to discretize it.\n\
Contributed by: Julian Andrej <juan@tf.uni-kiel.de>\n\n\n";

#include <petscdmplex.h>
#include <petscds.h>
#include <petscts.h>

/*
  Heat equation:

    du/dt - \Delta u + f = 0
*/

typedef enum {SOL_QUADRATIC_LINEAR, SOL_QUADRATIC_TRIG, SOL_TRIG_LINEAR, SOL_TRIG_TRIG, NUM_SOLUTION_TYPES} SolutionType;
const char *solutionTypes[NUM_SOLUTION_TYPES+1] = {"quadratic_linear", "quadratic_trig", "trig_linear", "trig_trig", "unknown"};

typedef struct {
  SolutionType solType; /* Type of exact solution */
  /* Solver setup */
  PetscBool expTS;      /* Flag for explicit timestepping */
  PetscBool lumped;     /* Lump the mass matrix */
} AppCtx;

/*
Exact 2D solution:
  u    = 2t + x^2 + y^2
  u_t  = 2
  \Delta u = 2 + 2 = 4
  f    = 2
  F(u) = 2 - (2 + 2) + 2 = 0

Exact 3D solution:
  u = 3t + x^2 + y^2 + z^2
  F(u) = 3 - (2 + 2 + 2) + 3 = 0
*/
static PetscErrorCode mms_quad_lin(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nc, PetscScalar *u, void *ctx)
{
  PetscInt d;

  *u = dim*time;
  for (d = 0; d < dim; ++d) *u += x[d]*x[d];
  return 0;
}

static PetscErrorCode mms_quad_lin_t(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nc, PetscScalar *u, void *ctx)
{
  *u = dim;
  return 0;
}

static void f0_quad_lin_exp(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[])
{
  f0[0] = -(PetscScalar) dim;
}
static void f0_quad_lin(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[])
{
  PetscScalar exp[1] = {0.};
  f0_quad_lin_exp(dim, Nf, NfAux, uOff, uOff_x, u, u_t, u_x, aOff, aOff_x, a, a_t, a_x, t, x, numConstants, constants, exp);
  f0[0] = u_t[0] - exp[0];
}

/*
Exact 2D solution:
  u = 2*cos(t) + x^2 + y^2
  F(u) = -2*sint(t) - (2 + 2) + 2*sin(t) + 4 = 0

Exact 3D solution:
  u = 3*cos(t) + x^2 + y^2 + z^2
  F(u) = -3*sin(t) - (2 + 2 + 2) + 3*sin(t) + 6 = 0
*/
static PetscErrorCode mms_quad_trig(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nc, PetscScalar *u, void *ctx)
{
  PetscInt d;

  *u = dim*PetscCosReal(time);
  for (d = 0; d < dim; ++d) *u += x[d]*x[d];
  return 0;
}

static PetscErrorCode mms_quad_trig_t(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nc, PetscScalar *u, void *ctx)
{
  *u = -dim*PetscSinReal(time);
  return 0;
}

static void f0_quad_trig_exp(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[])
{
  f0[0] = -dim*(PetscSinReal(t) + 2.0);
}
static void f0_quad_trig(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[])
{
  PetscScalar exp[1] = {0.};
  f0_quad_trig_exp(dim, Nf, NfAux, uOff, uOff_x, u, u_t, u_x, aOff, aOff_x, a, a_t, a_x, t, x, numConstants, constants, exp);
  f0[0] = u_t[0] - exp[0];
}

/*
Exact 2D solution:
  u = 2\pi^2 t + cos(\pi x) + cos(\pi y)
  F(u) = 2\pi^2 - \pi^2 (cos(\pi x) + cos(\pi y)) + \pi^2 (cos(\pi x) + cos(\pi y)) - 2\pi^2 = 0

Exact 3D solution:
  u = 3\pi^2 t + cos(\pi x) + cos(\pi y) + cos(\pi z)
  F(u) = 3\pi^2 - \pi^2 (cos(\pi x) + cos(\pi y) + cos(\pi z)) + \pi^2 (cos(\pi x) + cos(\pi y) + cos(\pi z)) - 3\pi^2 = 0
*/
static PetscErrorCode mms_trig_lin(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nc, PetscScalar *u, void *ctx)
{
  PetscInt d;

  *u = dim*PetscSqr(PETSC_PI)*time;
  for (d = 0; d < dim; ++d) *u += PetscCosReal(PETSC_PI*x[d]);
  return 0;
}

static PetscErrorCode mms_trig_lin_t(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nc, PetscScalar *u, void *ctx)
{
  *u = dim*PetscSqr(PETSC_PI);
  return 0;
}

static void f0_trig_lin(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 d;
  f0[0] = u_t[0];
  for (d = 0; d < dim; ++d) f0[0] += PetscSqr(PETSC_PI)*(PetscCosReal(PETSC_PI*x[d]) - 1.0);
}

/*
Exact 2D solution:
  u    = pi^2 cos(t) + cos(\pi x) + cos(\pi y)
  u_t  = -pi^2 sin(t)
  \Delta u = -\pi^2 (cos(\pi x) + cos(\pi y))
  f    = pi^2 sin(t) - \pi^2 (cos(\pi x) + cos(\pi y))
  F(u) = -\pi^2 sin(t) + \pi^2 (cos(\pi x) + cos(\pi y)) - \pi^2 (cos(\pi x) + cos(\pi y)) + \pi^2 sin(t) = 0

Exact 3D solution:
  u    = pi^2 cos(t) + cos(\pi x) + cos(\pi y) + cos(\pi z)
  u_t  = -pi^2 sin(t)
  \Delta u = -\pi^2 (cos(\pi x) + cos(\pi y) + cos(\pi z))
  f    = pi^2 sin(t) - \pi^2 (cos(\pi x) + cos(\pi y) + cos(\pi z))
  F(u) = -\pi^2 sin(t) + \pi^2 (cos(\pi x) + cos(\pi y) + cos(\pi z)) - \pi^2 (cos(\pi x) + cos(\pi y) + cos(\pi z)) + \pi^2 sin(t) = 0
*/
static PetscErrorCode mms_trig_trig(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nc, PetscScalar *u, void *ctx)
{
  PetscInt d;

  *u = PetscSqr(PETSC_PI)*PetscCosReal(time);
  for (d = 0; d < dim; ++d) *u += PetscCosReal(PETSC_PI*x[d]);
  return 0;
}

static PetscErrorCode mms_trig_trig_t(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nc, PetscScalar *u, void *ctx)
{
  *u = -PetscSqr(PETSC_PI)*PetscSinReal(time);
  return 0;
}

static void f0_trig_trig_exp(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 d;
  f0[0] -= PetscSqr(PETSC_PI)*PetscSinReal(t);
  for (d = 0; d < dim; ++d) f0[0] += PetscSqr(PETSC_PI)*PetscCosReal(PETSC_PI*x[d]);
}
static void f0_trig_trig(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[])
{
  PetscScalar exp[1] = {0.};
  f0_trig_trig_exp(dim, Nf, NfAux, uOff, uOff_x, u, u_t, u_x, aOff, aOff_x, a, a_t, a_x, t, x, numConstants, constants, exp);
  f0[0] = u_t[0] - exp[0];
}

static void f1_temp_exp(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 f1[])
{
  PetscInt d;
  for (d = 0; d < dim; ++d) f1[d] = -u_x[d];
}
static void f1_temp(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 f1[])
{
  PetscInt d;
  for (d = 0; d < dim; ++d) f1[d] = u_x[d];
}

static void g3_temp(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 g3[])
{
  PetscInt d;
  for (d = 0; d < dim; ++d) g3[d*dim+d] = 1.0;
}

static void g0_temp(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[])
{
  g0[0] = u_tShift*1.0;
}

static PetscErrorCode ProcessOptions(MPI_Comm comm, AppCtx *options)
{
  PetscInt       sol;

  PetscFunctionBeginUser;
  options->solType  = SOL_QUADRATIC_LINEAR;
  options->expTS    = PETSC_FALSE;
  options->lumped   = PETSC_TRUE;

  PetscOptionsBegin(comm, "", "Heat Equation Options", "DMPLEX");
  PetscCall(PetscOptionsEList("-sol_type", "Type of exact solution", "ex45.c", solutionTypes, NUM_SOLUTION_TYPES, solutionTypes[options->solType], &sol, NULL));
  options->solType = (SolutionType) sol;
  PetscCall(PetscOptionsBool("-explicit", "Use explicit timestepping", "ex45.c", options->expTS, &options->expTS, NULL));
  PetscCall(PetscOptionsBool("-lumped", "Lump the mass matrix", "ex45.c", options->lumped, &options->lumped, NULL));
  PetscOptionsEnd();
  PetscFunctionReturn(0);
}

static PetscErrorCode CreateMesh(MPI_Comm comm, DM *dm, AppCtx *ctx)
{
  PetscFunctionBeginUser;
  PetscCall(DMCreate(comm, dm));
  PetscCall(DMSetType(*dm, DMPLEX));
  PetscCall(DMSetFromOptions(*dm));
  PetscCall(DMViewFromOptions(*dm, NULL, "-dm_view"));
  PetscFunctionReturn(0);
}

static PetscErrorCode SetupProblem(DM dm, AppCtx *ctx)
{
  PetscDS        ds;
  DMLabel        label;
  const PetscInt id = 1;

  PetscFunctionBeginUser;
  PetscCall(DMGetLabel(dm, "marker", &label));
  PetscCall(DMGetDS(dm, &ds));
  PetscCall(PetscDSSetJacobian(ds, 0, 0, g0_temp, NULL, NULL, g3_temp));
  switch (ctx->solType) {
    case SOL_QUADRATIC_LINEAR:
      PetscCall(PetscDSSetResidual(ds, 0, f0_quad_lin,  f1_temp));
      PetscCall(PetscDSSetRHSResidual(ds, 0, f0_quad_lin_exp, f1_temp_exp));
      PetscCall(PetscDSSetExactSolution(ds, 0, mms_quad_lin, ctx));
      PetscCall(PetscDSSetExactSolutionTimeDerivative(ds, 0, mms_quad_lin_t, ctx));
      PetscCall(DMAddBoundary(dm, DM_BC_ESSENTIAL, "wall", label, 1, &id, 0, 0, NULL, (void (*)(void)) mms_quad_lin, (void (*)(void)) mms_quad_lin_t, ctx, NULL));
      break;
    case SOL_QUADRATIC_TRIG:
      PetscCall(PetscDSSetResidual(ds, 0, f0_quad_trig, f1_temp));
      PetscCall(PetscDSSetRHSResidual(ds, 0, f0_quad_trig_exp, f1_temp_exp));
      PetscCall(PetscDSSetExactSolution(ds, 0, mms_quad_trig, ctx));
      PetscCall(PetscDSSetExactSolutionTimeDerivative(ds, 0, mms_quad_trig_t, ctx));
      PetscCall(DMAddBoundary(dm, DM_BC_ESSENTIAL, "wall", label, 1, &id, 0, 0, NULL, (void (*)(void)) mms_quad_trig, (void (*)(void)) mms_quad_trig_t, ctx, NULL));
      break;
    case SOL_TRIG_LINEAR:
      PetscCall(PetscDSSetResidual(ds, 0, f0_trig_lin,  f1_temp));
      PetscCall(PetscDSSetExactSolution(ds, 0, mms_trig_lin, ctx));
      PetscCall(PetscDSSetExactSolutionTimeDerivative(ds, 0, mms_trig_lin_t, ctx));
      PetscCall(DMAddBoundary(dm, DM_BC_ESSENTIAL, "wall", label, 1, &id, 0, 0, NULL, (void (*)(void)) mms_trig_lin, (void (*)(void)) mms_trig_lin_t, ctx, NULL));
      break;
    case SOL_TRIG_TRIG:
      PetscCall(PetscDSSetResidual(ds, 0, f0_trig_trig, f1_temp));
      PetscCall(PetscDSSetRHSResidual(ds, 0, f0_trig_trig_exp, f1_temp_exp));
      PetscCall(PetscDSSetExactSolution(ds, 0, mms_trig_trig, ctx));
      PetscCall(PetscDSSetExactSolutionTimeDerivative(ds, 0, mms_trig_trig_t, ctx));
      break;
    default: SETERRQ(PetscObjectComm((PetscObject) dm), PETSC_ERR_ARG_WRONG, "Invalid solution type: %s (%D)", solutionTypes[PetscMin(ctx->solType, NUM_SOLUTION_TYPES)], ctx->solType);
  }
  PetscFunctionReturn(0);
}

static PetscErrorCode SetupDiscretization(DM dm, AppCtx* ctx)
{
  DM             cdm = dm;
  PetscFE        fe;
  DMPolytopeType ct;
  PetscBool      simplex;
  PetscInt       dim, cStart;

  PetscFunctionBeginUser;
  PetscCall(DMGetDimension(dm, &dim));
  PetscCall(DMPlexGetHeightStratum(dm, 0, &cStart, NULL));
  PetscCall(DMPlexGetCellType(dm, cStart, &ct));
  simplex = DMPolytopeTypeGetNumVertices(ct) == DMPolytopeTypeGetDim(ct)+1 ? PETSC_TRUE : PETSC_FALSE;
  /* Create finite element */
  PetscCall(PetscFECreateDefault(PETSC_COMM_SELF, dim, 1, simplex, "temp_", -1, &fe));
  PetscCall(PetscObjectSetName((PetscObject) fe, "temperature"));
  /* Set discretization and boundary conditions for each mesh */
  PetscCall(DMSetField(dm, 0, NULL, (PetscObject) fe));
  PetscCall(DMCreateDS(dm));
  if (ctx->expTS) {
    PetscDS ds;

    PetscCall(DMGetDS(dm, &ds));
    PetscCall(PetscDSSetImplicit(ds, 0, PETSC_FALSE));
  }
  PetscCall(SetupProblem(dm, ctx));
  while (cdm) {
    PetscCall(DMCopyDisc(dm, cdm));
    PetscCall(DMGetCoarseDM(cdm, &cdm));
  }
  PetscCall(PetscFEDestroy(&fe));
  PetscFunctionReturn(0);
}

static PetscErrorCode SetInitialConditions(TS ts, Vec u)
{
  DM             dm;
  PetscReal      t;

  PetscFunctionBegin;
  PetscCall(TSGetDM(ts, &dm));
  PetscCall(TSGetTime(ts, &t));
  PetscCall(DMComputeExactSolution(dm, t, u, NULL));
  PetscFunctionReturn(0);
}

int main(int argc, char **argv)
{
  DM             dm;
  TS             ts;
  Vec            u;
  AppCtx         ctx;

  PetscCall(PetscInitialize(&argc, &argv, NULL, help));
  PetscCall(ProcessOptions(PETSC_COMM_WORLD, &ctx));
  PetscCall(CreateMesh(PETSC_COMM_WORLD, &dm, &ctx));
  PetscCall(DMSetApplicationContext(dm, &ctx));
  PetscCall(SetupDiscretization(dm, &ctx));

  PetscCall(TSCreate(PETSC_COMM_WORLD, &ts));
  PetscCall(TSSetDM(ts, dm));
  PetscCall(DMTSSetBoundaryLocal(dm, DMPlexTSComputeBoundary, &ctx));
  if (ctx.expTS) {
    PetscCall(DMTSSetRHSFunctionLocal(dm, DMPlexTSComputeRHSFunctionFEM, &ctx));
    if (ctx.lumped) PetscCall(DMTSCreateRHSMassMatrixLumped(dm));
    else            PetscCall(DMTSCreateRHSMassMatrix(dm));
  } else {
    PetscCall(DMTSSetIFunctionLocal(dm, DMPlexTSComputeIFunctionFEM, &ctx));
    PetscCall(DMTSSetIJacobianLocal(dm, DMPlexTSComputeIJacobianFEM, &ctx));
  }
  PetscCall(TSSetExactFinalTime(ts, TS_EXACTFINALTIME_MATCHSTEP));
  PetscCall(TSSetFromOptions(ts));
  PetscCall(TSSetComputeInitialCondition(ts, SetInitialConditions));

  PetscCall(DMCreateGlobalVector(dm, &u));
  PetscCall(DMTSCheckFromOptions(ts, u));
  PetscCall(SetInitialConditions(ts, u));
  PetscCall(PetscObjectSetName((PetscObject) u, "temperature"));
  PetscCall(TSSolve(ts, u));
  PetscCall(DMTSCheckFromOptions(ts, u));
  if (ctx.expTS) PetscCall(DMTSDestroyRHSMassMatrix(dm));

  PetscCall(VecDestroy(&u));
  PetscCall(TSDestroy(&ts));
  PetscCall(DMDestroy(&dm));
  PetscCall(PetscFinalize());
  return 0;
}

/*TEST

  test:
    suffix: 2d_p1
    requires: triangle
    args: -sol_type quadratic_linear -dm_refine 1 -temp_petscspace_degree 1 -dmts_check .0001 \
          -ts_type beuler -ts_max_steps 5 -ts_dt 0.1 -snes_error_if_not_converged -pc_type lu
  test:
    suffix: 2d_p1_exp
    requires: triangle
    args: -sol_type quadratic_linear -dm_refine 1 -temp_petscspace_degree 1 -explicit \
          -ts_type euler -ts_max_steps 4 -ts_dt 1e-3 -ts_monitor_error
  test:
    # -dm_refine 2 -convest_num_refine 3 get L_2 convergence rate: [1.9]
    suffix: 2d_p1_sconv
    requires: triangle
    args: -sol_type quadratic_linear -temp_petscspace_degree 1 -ts_convergence_estimate -ts_convergence_temporal 0 -convest_num_refine 1 \
          -ts_type beuler -ts_max_steps 1 -ts_dt 0.00001 -snes_error_if_not_converged -pc_type lu
  test:
    # -dm_refine 2 -convest_num_refine 3 get L_2 convergence rate: [2.1]
    suffix: 2d_p1_sconv_2
    requires: triangle
    args: -sol_type trig_trig -temp_petscspace_degree 1 -ts_convergence_estimate -ts_convergence_temporal 0 -convest_num_refine 1 \
          -ts_type beuler -ts_max_steps 1 -ts_dt 1e-6 -snes_error_if_not_converged -pc_type lu
  test:
    # -dm_refine 4 -convest_num_refine 3 get L_2 convergence rate: [1.2]
    suffix: 2d_p1_tconv
    requires: triangle
    args: -sol_type quadratic_trig -temp_petscspace_degree 1 -ts_convergence_estimate -convest_num_refine 1 \
          -ts_type beuler -ts_max_steps 4 -ts_dt 0.1 -snes_error_if_not_converged -pc_type lu
  test:
    # -dm_refine 6 -convest_num_refine 3 get L_2 convergence rate: [1.0]
    suffix: 2d_p1_tconv_2
    requires: triangle
    args: -sol_type trig_trig -temp_petscspace_degree 1 -ts_convergence_estimate -convest_num_refine 1 \
          -ts_type beuler -ts_max_steps 4 -ts_dt 0.1 -snes_error_if_not_converged -pc_type lu
  test:
    # The L_2 convergence rate cannot be seen since stability of the explicit integrator requires that is be more accurate than the grid
    suffix: 2d_p1_exp_tconv_2
    requires: triangle
    args: -sol_type trig_trig -temp_petscspace_degree 1 -explicit -ts_convergence_estimate -convest_num_refine 1 \
          -ts_type euler -ts_max_steps 4 -ts_dt 1e-4 -lumped 0 -mass_pc_type lu
  test:
    # The L_2 convergence rate cannot be seen since stability of the explicit integrator requires that is be more accurate than the grid
    suffix: 2d_p1_exp_tconv_2_lump
    requires: triangle
    args: -sol_type trig_trig -temp_petscspace_degree 1 -explicit -ts_convergence_estimate -convest_num_refine 1 \
          -ts_type euler -ts_max_steps 4 -ts_dt 1e-4
  test:
    suffix: 2d_p2
    requires: triangle
    args: -sol_type quadratic_linear -dm_refine 0 -temp_petscspace_degree 2 -dmts_check .0001 \
          -ts_type beuler -ts_max_steps 5 -ts_dt 0.1 -snes_error_if_not_converged -pc_type lu
  test:
    # -dm_refine 2 -convest_num_refine 3 get L_2 convergence rate: [2.9]
    suffix: 2d_p2_sconv
    requires: triangle
    args: -sol_type trig_linear -temp_petscspace_degree 2 -ts_convergence_estimate -ts_convergence_temporal 0 -convest_num_refine 1 \
          -ts_type beuler -ts_max_steps 1 -ts_dt 0.00000001 -snes_error_if_not_converged -pc_type lu
  test:
    # -dm_refine 2 -convest_num_refine 3 get L_2 convergence rate: [3.1]
    suffix: 2d_p2_sconv_2
    requires: triangle
    args: -sol_type trig_trig -temp_petscspace_degree 2 -ts_convergence_estimate -ts_convergence_temporal 0 -convest_num_refine 1 \
          -ts_type beuler -ts_max_steps 1 -ts_dt 0.00000001 -snes_error_if_not_converged -pc_type lu
  test:
    # -dm_refine 3 -convest_num_refine 3 get L_2 convergence rate: [1.0]
    suffix: 2d_p2_tconv
    requires: triangle
    args: -sol_type quadratic_trig -temp_petscspace_degree 2 -ts_convergence_estimate -convest_num_refine 1 \
          -ts_type beuler -ts_max_steps 4 -ts_dt 0.1 -snes_error_if_not_converged -pc_type lu
  test:
    # -dm_refine 3 -convest_num_refine 3 get L_2 convergence rate: [1.0]
    suffix: 2d_p2_tconv_2
    requires: triangle
    args: -sol_type trig_trig -temp_petscspace_degree 2 -ts_convergence_estimate -convest_num_refine 1 \
          -ts_type beuler -ts_max_steps 4 -ts_dt 0.1 -snes_error_if_not_converged -pc_type lu
  test:
    suffix: 2d_q1
    args: -sol_type quadratic_linear -dm_plex_simplex 0 -dm_refine 1 -temp_petscspace_degree 1 -dmts_check .0001 \
          -ts_type beuler -ts_max_steps 5 -ts_dt 0.1 -snes_error_if_not_converged -pc_type lu
  test:
    # -dm_refine 2 -convest_num_refine 3 get L_2 convergence rate: [1.9]
    suffix: 2d_q1_sconv
    args: -sol_type quadratic_linear -dm_plex_simplex 0 -temp_petscspace_degree 1 -ts_convergence_estimate -ts_convergence_temporal 0 -convest_num_refine 1 \
          -ts_type beuler -ts_max_steps 1 -ts_dt 0.00001 -snes_error_if_not_converged -pc_type lu
  test:
    # -dm_refine 4 -convest_num_refine 3 get L_2 convergence rate: [1.2]
    suffix: 2d_q1_tconv
    args: -sol_type quadratic_trig -dm_plex_simplex 0 -temp_petscspace_degree 1 -ts_convergence_estimate -convest_num_refine 1 \
          -ts_type beuler -ts_max_steps 4 -ts_dt 0.1 -snes_error_if_not_converged -pc_type lu
  test:
    suffix: 2d_q2
    args: -sol_type quadratic_linear -dm_plex_simplex 0 -dm_refine 0 -temp_petscspace_degree 2 -dmts_check .0001 \
          -ts_type beuler -ts_max_steps 5 -ts_dt 0.1 -snes_error_if_not_converged -pc_type lu
  test:
    # -dm_refine 2 -convest_num_refine 3 get L_2 convergence rate: [2.9]
    suffix: 2d_q2_sconv
    args: -sol_type trig_linear -dm_plex_simplex 0 -temp_petscspace_degree 2 -ts_convergence_estimate -ts_convergence_temporal 0 -convest_num_refine 1 \
          -ts_type beuler -ts_max_steps 1 -ts_dt 0.00000001 -snes_error_if_not_converged -pc_type lu
  test:
    # -dm_refine 3 -convest_num_refine 3 get L_2 convergence rate: [1.0]
    suffix: 2d_q2_tconv
    args: -sol_type quadratic_trig -dm_plex_simplex 0 -temp_petscspace_degree 2 -ts_convergence_estimate -convest_num_refine 1 \
          -ts_type beuler -ts_max_steps 4 -ts_dt 0.1 -snes_error_if_not_converged -pc_type lu

  test:
    suffix: 3d_p1
    requires: ctetgen
    args: -sol_type quadratic_linear -dm_refine 1 -temp_petscspace_degree 1 -dmts_check .0001 \
          -ts_type beuler -ts_max_steps 5 -ts_dt 0.1 -snes_error_if_not_converged -pc_type lu
  test:
    # -dm_refine 2 -convest_num_refine 3 get L_2 convergence rate: [1.9]
    suffix: 3d_p1_sconv
    requires: ctetgen
    args: -sol_type quadratic_linear -temp_petscspace_degree 1 -ts_convergence_estimate -ts_convergence_temporal 0 -convest_num_refine 1 \
          -ts_type beuler -ts_max_steps 1 -ts_dt 0.00001 -snes_error_if_not_converged -pc_type lu
  test:
    # -dm_refine 4 -convest_num_refine 3 get L_2 convergence rate: [1.2]
    suffix: 3d_p1_tconv
    requires: ctetgen
    args: -sol_type quadratic_trig -temp_petscspace_degree 1 -ts_convergence_estimate -convest_num_refine 1 \
          -ts_type beuler -ts_max_steps 4 -ts_dt 0.1 -snes_error_if_not_converged -pc_type lu
  test:
    suffix: 3d_p2
    requires: ctetgen
    args: -sol_type quadratic_linear -dm_refine 0 -temp_petscspace_degree 2 -dmts_check .0001 \
          -ts_type beuler -ts_max_steps 5 -ts_dt 0.1 -snes_error_if_not_converged -pc_type lu
  test:
    # -dm_refine 2 -convest_num_refine 3 get L_2 convergence rate: [2.9]
    suffix: 3d_p2_sconv
    requires: ctetgen
    args: -sol_type trig_linear -temp_petscspace_degree 2 -ts_convergence_estimate -ts_convergence_temporal 0 -convest_num_refine 1 \
          -ts_type beuler -ts_max_steps 1 -ts_dt 0.00000001 -snes_error_if_not_converged -pc_type lu
  test:
    # -dm_refine 3 -convest_num_refine 3 get L_2 convergence rate: [1.0]
    suffix: 3d_p2_tconv
    requires: ctetgen
    args: -sol_type quadratic_trig -temp_petscspace_degree 2 -ts_convergence_estimate -convest_num_refine 1 \
          -ts_type beuler -ts_max_steps 4 -ts_dt 0.1 -snes_error_if_not_converged -pc_type lu
  test:
    suffix: 3d_q1
    args: -sol_type quadratic_linear -dm_plex_simplex 0 -dm_refine 1 -temp_petscspace_degree 1 -dmts_check .0001 \
          -ts_type beuler -ts_max_steps 5 -ts_dt 0.1 -snes_error_if_not_converged -pc_type lu
  test:
    # -dm_refine 2 -convest_num_refine 3 get L_2 convergence rate: [1.9]
    suffix: 3d_q1_sconv
    args: -sol_type quadratic_linear -dm_plex_simplex 0 -temp_petscspace_degree 1 -ts_convergence_estimate -ts_convergence_temporal 0 -convest_num_refine 1 \
          -ts_type beuler -ts_max_steps 1 -ts_dt 0.00001 -snes_error_if_not_converged -pc_type lu
  test:
    # -dm_refine 4 -convest_num_refine 3 get L_2 convergence rate: [1.2]
    suffix: 3d_q1_tconv
    args: -sol_type quadratic_trig -dm_plex_simplex 0 -temp_petscspace_degree 1 -ts_convergence_estimate -convest_num_refine 1 \
          -ts_type beuler -ts_max_steps 4 -ts_dt 0.1 -snes_error_if_not_converged -pc_type lu
  test:
    suffix: 3d_q2
    args: -sol_type quadratic_linear -dm_plex_simplex 0 -dm_refine 0 -temp_petscspace_degree 2 -dmts_check .0001 \
          -ts_type beuler -ts_max_steps 5 -ts_dt 0.1 -snes_error_if_not_converged -pc_type lu
  test:
    # -dm_refine 2 -convest_num_refine 3 get L_2 convergence rate: [2.9]
    suffix: 3d_q2_sconv
    args: -sol_type trig_linear -dm_plex_simplex 0 -temp_petscspace_degree 2 -ts_convergence_estimate -ts_convergence_temporal 0 -convest_num_refine 1 \
          -ts_type beuler -ts_max_steps 1 -ts_dt 0.00000001 -snes_error_if_not_converged -pc_type lu
  test:
    # -dm_refine 3 -convest_num_refine 3 get L_2 convergence rate: [1.0]
    suffix: 3d_q2_tconv
    args: -sol_type quadratic_trig -dm_plex_simplex 0 -temp_petscspace_degree 2 -ts_convergence_estimate -convest_num_refine 1 \
          -ts_type beuler -ts_max_steps 4 -ts_dt 0.1 -snes_error_if_not_converged -pc_type lu

  test:
    # For a nice picture, -bd_dm_refine 2 -dm_refine 1 -dm_view hdf5:${PETSC_DIR}/sol.h5 -ts_monitor_solution hdf5:${PETSC_DIR}/sol.h5::append
    suffix: egads_sphere
    requires: egads ctetgen
    args: -sol_type quadratic_linear \
          -dm_plex_boundary_filename ${wPETSC_DIR}/share/petsc/datafiles/meshes/unit_sphere.egadslite -dm_plex_boundary_label marker -bd_dm_plex_scale 40 \
          -temp_petscspace_degree 2 -dmts_check .0001 \
          -ts_type beuler -ts_max_steps 5 -ts_dt 0.1 -snes_error_if_not_converged -pc_type lu

TEST*/