Actual source code: ex11.c

      1: static char help[] = "Poisson problem with finite elements.\n\
      2: We solve the Poisson problem using a parallel unstructured mesh to discretize it.\n\
      3: This example is a simplified version of ex12.c that only solves the linear problem.\n\
      4: It uses discretized auxiliary fields for coefficient and right-hand side, \n\
      5: supports multilevel solvers and non-conforming mesh refinement.\n\n\n";

      7: /*
      8: Here we describe the PETSc approach to solve nonlinear problems arising from Finite Element (FE) discretizations.

     10: The general model requires to solve the residual equations

     12:     residual(u) := \int_\Omega \phi \cdot f_0(u, \nabla u, a, \nabla a) + \nabla \phi : f_1(u, \nabla u, a, \nabla a) = 0

     14: where \phi is a test function, u is the sought FE solution, and a generically describes auxiliary data (for example material properties).

     16: The functions f_0 (scalar) and f_1 (vector) describe the problem, while : is the usual contraction operator for tensors, i.e. A : B = \sum_{ij} A_{ij} B_{ij}.

     18: The discrete residual is (with abuse of notation)

     20:     F(u) := \sum_e E_e^T [ B^T_e W_{0,e} f_0(u_q, \nabla u_q, a_q, \nabla a_q) + D_e W_{1,e} f_1(u_q, \nabla u_q, a_q, \nabla a_q) ]

     22: where E are element restriction matrices (can support non-conforming meshes), W are quadrature weights, B (resp. D) are basis function (resp. derivative of basis function) matrices, and u_q,a_q are vectors sampled at quadrature points.

     24: Having the residual in the above form, it is straightforward to derive its Jacobian (again with abuse of notation)

     26:     J(u) := \sum_e E_e^T [B^T_e D^T_e] W  J_e [ B_e^T, D_e^T ]^T E_e,

     28: where J_e is the 2x2 matrix

     30:    | \partial_u f_0, \partial_{\grad u} f_0 |
     31:    | \partial_u f_1, \partial_{\grad u} f_1 |

     33: Here we use a single-field approach, but the extension to the multi-field case is straightforward.

     35: To keep the presentation simple, here we are interested in solving the Poisson problem in divergence form

     37:    - \nabla \cdot (K * \nabla u) = g

     39: with either u = 0 or K * \partial_n u = 0 on \partial \Omega.
     40: The above problem possesses the weak form

     42:    \int_\Omega \nabla \phi K \nabla u + \phi g = 0,

     44: thus we have:

     46:    f_0 = g, f_1 = K * \nabla u, and the only non-zero term of the Jacobian is J_{11} = K

     48: See https://petsc.org/release/manual/fe the and the paper "Achieving High Performance with Unified Residual Evaluation" (available at https://arxiv.org/abs/1309.1204) for additional information.
     49: */

     51: /* Include the necessary definitions */
     52: #include <petscdmplex.h>
     53: #include <petscsnes.h>
     54: #include <petscds.h>

     56: /* The f_0 function: we read the right-hand side from the first field of the auxiliary data */
     57: static void f_0(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[])
     58: {
     59:   const PetscScalar g = a[0];

     61:   f0[0] = g;
     62: }

     64: /* The f_1 function: we read the conductivity tensor from the second field of the auxiliary data */
     65: static void f_1(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[])
     66: {
     67:   const PetscScalar *K = &a[1];

     69:   for (PetscInt d1 = 0; d1 < dim; ++d1) {
     70:     PetscScalar v = 0;
     71:     for (PetscInt d2 = 0; d2 < dim; ++d2) v += K[d1 * dim + d2] * u_x[d2];
     72:     f1[d1] = v;
     73:   }
     74: }

     76: /* The only non-zero term for the Jacobian is J_11 */
     77: static void J_11(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 J11[])
     78: {
     79:   const PetscScalar *K = &a[1];

     81:   for (PetscInt d1 = 0; d1 < dim; ++d1) {
     82:     for (PetscInt d2 = 0; d2 < dim; ++d2) J11[d1 * dim + d2] = K[d1 * dim + d2];
     83:   }
     84: }

     86: /* The boundary conditions: Dirichlet (essential) or Neumann (natural) */
     87: typedef enum {
     88:   BC_DIRICHLET,
     89:   BC_NEUMANN,
     90: } bcType;

     92: static const char *const bcTypes[] = {"DIRICHLET", "NEUMANN", "bcType", "BC_", 0};

     94: /* The forcing term: constant or analytical */
     95: typedef enum {
     96:   RHS_CONSTANT,
     97:   RHS_ANALYTICAL,
     98: } rhsType;

    100: static const char *const rhsTypes[] = {"CONSTANT", "ANALYTICAL", "rhsType", "RHS_", 0};

    102: /* the constant case */
    103: static PetscErrorCode rhs_constant(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nc, PetscScalar *g, void *ctx)
    104: {
    105:   *g = 1.0;
    106:   return PETSC_SUCCESS;
    107: }

    109: /* the analytical case */
    110: static PetscErrorCode rhs_analytical(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nc, PetscScalar *g, void *ctx)
    111: {
    112:   PetscScalar v = 1.0;
    113:   for (PetscInt d = 0; d < dim; d++) v *= PetscSinReal(2.0 * PETSC_PI * x[d]);
    114:   *g = v;
    115:   return PETSC_SUCCESS;
    116: }

    118: /* For the Neumann BC case we need a functional to be integrated: average -> \int_\Omega u dx */
    119: static void average(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 obj[])
    120: {
    121:   obj[0] = u[0];
    122: }

    124: /* The conductivity coefficient term: constant, checkerboard or analytical */
    125: typedef enum {
    126:   COEFF_CONSTANT,
    127:   COEFF_CHECKERBOARD,
    128:   COEFF_ANALYTICAL,
    129: } coeffType;

    131: static const char *const coeffTypes[] = {"CONSTANT", "CHECKERBOARD", "ANALYTICAL", "coeffType", "COEFF_", 0};

    133: /* the constant coefficient case */
    134: static PetscErrorCode coefficient_constant(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nc, PetscScalar *K, void *ctx)
    135: {
    136:   for (PetscInt d = 0; d < dim; d++) K[d * dim + d] = 1.0;
    137:   return PETSC_SUCCESS;
    138: }

    140: /* the checkerboard coefficient case: 10^2 in odd ranks, 10^-2 in even ranks */
    141: static PetscErrorCode coefficient_checkerboard(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nc, PetscScalar *K, void *ctx)
    142: {
    143:   PetscScalar exponent = PetscGlobalRank % 2 ? 2.0 : -2.0;
    144:   for (PetscInt d = 0; d < dim; d++) K[d * dim + d] = PetscPowScalar(10.0, exponent);
    145:   return PETSC_SUCCESS;
    146: }

    148: /* the analytical case (channels in diagonal with 4 order of magnitude in jumps) */
    149: static PetscErrorCode coefficient_analytical(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nc, PetscScalar *K, void *ctx)
    150: {
    151:   for (PetscInt d = 0; d < dim; d++) {
    152:     const PetscReal v = PetscPowReal(10.0, 4.0 * PetscSinReal(4.0 * PETSC_PI * x[d]) * PetscCosReal(4.0 * PETSC_PI * x[d]));
    153:     K[d * dim + d]    = v;
    154:   }
    155:   return PETSC_SUCCESS;
    156: }

    158: /* The application context that defines our problem */
    159: typedef struct {
    160:   bcType    bc;         /* type of boundary conditions */
    161:   rhsType   rhs;        /* type of right-hand side */
    162:   coeffType coeff;      /* type of conductivity coefficient */
    163:   PetscInt  order;      /* the polynomial order for the solution */
    164:   PetscInt  rhsOrder;   /* the polynomial order for the right-hand side */
    165:   PetscInt  coeffOrder; /* the polynomial order for the coefficient */
    166:   PetscBool p4est;      /* if we want to use non-conforming AMR */
    167: } AppCtx;

    169: /* Process command line options */
    170: static PetscErrorCode ProcessOptions(MPI_Comm comm, AppCtx *options)
    171: {
    172:   PetscFunctionBeginUser;
    173:   options->bc         = BC_DIRICHLET;
    174:   options->rhs        = RHS_CONSTANT;
    175:   options->coeff      = COEFF_CONSTANT;
    176:   options->order      = 1;
    177:   options->rhsOrder   = 1;
    178:   options->coeffOrder = 1;
    179:   options->p4est      = PETSC_FALSE;

    181:   PetscOptionsBegin(comm, "", "Poisson problem options", "DMPLEX");
    182:   PetscCall(PetscOptionsEnum("-bc_type", "Type of boundary condition", __FILE__, bcTypes, (PetscEnum)options->bc, (PetscEnum *)&options->bc, NULL));
    183:   PetscCall(PetscOptionsEnum("-rhs_type", "Type of forcing term", __FILE__, rhsTypes, (PetscEnum)options->rhs, (PetscEnum *)&options->rhs, NULL));
    184:   PetscCall(PetscOptionsEnum("-coefficient_type", "Type of conductivity coefficient", __FILE__, coeffTypes, (PetscEnum)options->coeff, (PetscEnum *)&options->coeff, NULL));
    185:   PetscCall(PetscOptionsInt("-order", "The polynomial order for the approximation of the solution", __FILE__, options->order, &options->order, NULL));
    186:   PetscCall(PetscOptionsInt("-rhs_order", "The polynomial order for the approximation of the right-hand side", __FILE__, options->rhsOrder, &options->rhsOrder, NULL));
    187:   PetscCall(PetscOptionsInt("-coefficient_order", "The polynomial order for the approximation of the coefficient", __FILE__, options->coeffOrder, &options->coeffOrder, NULL));
    188:   PetscCall(PetscOptionsBool("-p4est", "Use p4est to represent the mesh", __FILE__, options->p4est, &options->p4est, NULL));
    189:   PetscOptionsEnd();

    191:   /* View processed options */
    192:   PetscCall(PetscPrintf(comm, "Simulation parameters\n"));
    193:   PetscCall(PetscPrintf(comm, "  polynomial order: %" PetscInt_FMT "\n", options->order));
    194:   PetscCall(PetscPrintf(comm, "  boundary conditions: %s\n", bcTypes[options->bc]));
    195:   PetscCall(PetscPrintf(comm, "  right-hand side: %s, order %" PetscInt_FMT "\n", rhsTypes[options->rhs], options->rhsOrder));
    196:   PetscCall(PetscPrintf(comm, "  coefficient: %s, order %" PetscInt_FMT "\n", coeffTypes[options->coeff], options->coeffOrder));
    197:   PetscCall(PetscPrintf(comm, "  non-conforming AMR: %d\n", options->p4est));
    198:   PetscFunctionReturn(PETSC_SUCCESS);
    199: }

    201: /* Create mesh from command line options */
    202: static PetscErrorCode CreateMesh(MPI_Comm comm, AppCtx *user, DM *dm)
    203: {
    204:   PetscFunctionBeginUser;
    205:   /* Create various types of meshes only with command line options */
    206:   PetscCall(DMCreate(comm, dm));
    207:   PetscCall(DMSetType(*dm, DMPLEX));
    208:   PetscCall(DMSetOptionsPrefix(*dm, "initial_"));
    209:   PetscCall(DMPlexDistributeSetDefault(*dm, PETSC_FALSE));
    210:   PetscCall(DMSetFromOptions(*dm));
    211:   PetscCall(DMLocalizeCoordinates(*dm));
    212:   PetscCall(DMViewFromOptions(*dm, NULL, "-dm_view"));
    213:   PetscCall(DMPlexDistributeSetDefault(*dm, PETSC_TRUE));
    214:   PetscCall(DMSetOptionsPrefix(*dm, "mesh_"));
    215:   PetscCall(DMSetFromOptions(*dm));

    217:   /* If requested convert to a format suitable for non-conforming adaptive mesh refinement */
    218:   if (user->p4est) {
    219:     PetscInt dim;
    220:     DM       dmConv;

    222:     PetscCall(DMGetDimension(*dm, &dim));
    223:     PetscCheck(dim == 2 || dim == 3, comm, PETSC_ERR_SUP, "p4est support not for dimension %" PetscInt_FMT, dim);
    224:     PetscCall(DMConvert(*dm, dim == 3 ? DMP8EST : DMP4EST, &dmConv));
    225:     if (dmConv) {
    226:       PetscCall(DMDestroy(dm));
    227:       PetscCall(DMSetOptionsPrefix(dmConv, "mesh_"));
    228:       PetscCall(DMSetFromOptions(dmConv));
    229:       *dm = dmConv;
    230:     }
    231:   }
    232:   PetscCall(DMSetUp(*dm));

    234:   /* View the mesh.
    235:      With a single call we can dump ASCII information, VTK data for visualization, store the mesh in HDF5 format, etc. */
    236:   PetscCall(PetscObjectSetName((PetscObject)*dm, "Mesh"));
    237:   PetscCall(DMViewFromOptions(*dm, NULL, "-dm_view"));
    238:   PetscFunctionReturn(PETSC_SUCCESS);
    239: }

    241: /* Setup the discrete problem */
    242: static PetscErrorCode SetupProblem(DM dm, DM fdm, AppCtx *user)
    243: {
    244:   DM             plex, dmAux, cdm = NULL, coordDM;
    245:   Vec            auxData, auxDataGlobal;
    246:   PetscDS        ds;
    247:   DMPolytopeType ct;
    248:   PetscInt       dim, cdim, cStart;
    249:   PetscFE        fe, fe_rhs, fe_K;
    250:   PetscErrorCode (*auxFuncs[])(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nc, PetscScalar u[], void *ctx) = {NULL, NULL};
    251:   void *auxCtxs[]                                                                                                          = {NULL, NULL};

    253:   PetscFunctionBeginUser;
    254:   /* Create the Finite Element for the solution and pass it to the problem DM */
    255:   PetscCall(DMGetDimension(dm, &dim));
    256:   PetscCall(DMConvert(dm, DMPLEX, &plex));
    257:   PetscCall(DMPlexGetHeightStratum(plex, 0, &cStart, NULL));
    258:   PetscCall(DMPlexGetCellType(plex, cStart, &ct));
    259:   PetscCall(DMDestroy(&plex));
    260:   PetscCall(PetscFECreateLagrangeByCell(PETSC_COMM_SELF, dim, 1, ct, user->order, PETSC_DETERMINE, &fe));
    261:   PetscCall(PetscObjectSetName((PetscObject)fe, "potential"));
    262:   PetscCall(DMSetField(dm, 0, NULL, (PetscObject)fe));
    263:   PetscCall(DMCreateDS(dm));

    265:   /* Set residual and Jacobian callbacks */
    266:   PetscCall(DMGetDS(dm, &ds));
    267:   PetscCall(PetscDSSetResidual(ds, 0, f_0, f_1));
    268:   PetscCall(PetscDSSetJacobian(ds, 0, 0, NULL, NULL, NULL, J_11));
    269:   /* Tell DMPLEX we are going to use FEM callbacks */
    270:   PetscCall(DMPlexSetSNESLocalFEM(dm, PETSC_FALSE, &user));

    272:   /* Create the Finite Element for the auxiliary data */
    273:   PetscCall(PetscFECreateLagrangeByCell(PETSC_COMM_SELF, dim, 1, ct, user->rhsOrder, PETSC_DETERMINE, &fe_rhs));
    274:   PetscCall(PetscObjectSetName((PetscObject)fe_rhs, "g"));
    275:   PetscCall(PetscFECopyQuadrature(fe, fe_rhs));
    276:   PetscCall(DMGetCoordinateDM(dm, &coordDM));
    277:   PetscCall(DMGetDimension(coordDM, &cdim));
    278:   PetscCall(PetscFECreateLagrangeByCell(PETSC_COMM_SELF, dim, cdim * cdim, ct, user->coeffOrder, PETSC_DETERMINE, &fe_K));
    279:   PetscCall(PetscObjectSetName((PetscObject)fe_K, "K"));
    280:   PetscCall(PetscFECopyQuadrature(fe, fe_K));
    281:   PetscCall(DMClone(dm, &dmAux));
    282:   PetscCall(DMSetField(dmAux, 0, NULL, (PetscObject)fe_rhs));
    283:   PetscCall(DMSetField(dmAux, 1, NULL, (PetscObject)fe_K));
    284:   PetscCall(DMCreateDS(dmAux));

    286:   /* Project the requested rhs and K to the auxiliary DM and pass it to the problem DM */
    287:   PetscCall(DMCreateLocalVector(dmAux, &auxData));
    288:   PetscCall(DMCreateGlobalVector(dmAux, &auxDataGlobal));
    289:   PetscCall(PetscObjectSetName((PetscObject)auxData, ""));
    290:   if (!fdm) {
    291:     switch (user->rhs) {
    292:     case RHS_CONSTANT:
    293:       auxFuncs[0] = rhs_constant;
    294:       auxCtxs[0]  = NULL;
    295:       break;
    296:     case RHS_ANALYTICAL:
    297:       auxFuncs[0] = rhs_analytical;
    298:       auxCtxs[0]  = NULL;
    299:       break;
    300:     default:
    301:       SETERRQ(PetscObjectComm((PetscObject)dm), PETSC_ERR_SUP, "Unsupported rhs type %s", rhsTypes[user->rhs]);
    302:     }
    303:     switch (user->coeff) {
    304:     case COEFF_CONSTANT:
    305:       auxFuncs[1] = coefficient_constant;
    306:       auxCtxs[1]  = NULL;
    307:       break;
    308:     case COEFF_CHECKERBOARD:
    309:       auxFuncs[1] = coefficient_checkerboard;
    310:       auxCtxs[1]  = NULL;
    311:       break;
    312:     case COEFF_ANALYTICAL:
    313:       auxFuncs[1] = coefficient_analytical;
    314:       auxCtxs[1]  = NULL;
    315:       break;
    316:     default:
    317:       SETERRQ(PetscObjectComm((PetscObject)dm), PETSC_ERR_SUP, "Unsupported coefficient type %s", coeffTypes[user->coeff]);
    318:     }
    319:     PetscCall(DMGetDS(dmAux, &ds));
    320:     PetscCall(DMProjectFunction(dmAux, 0.0, auxFuncs, auxCtxs, INSERT_ALL_VALUES, auxDataGlobal));
    321:     if (user->bc == BC_NEUMANN) {
    322:       PetscScalar vals[2];
    323:       PetscInt    rhs_id = 0;
    324:       IS          is;

    326:       PetscCall(PetscDSSetObjective(ds, rhs_id, average));
    327:       PetscCall(DMPlexComputeIntegralFEM(dmAux, auxDataGlobal, vals, NULL));
    328:       PetscCall(DMCreateSubDM(dmAux, 1, &rhs_id, &is, NULL));
    329:       PetscCall(VecISShift(auxDataGlobal, is, -vals[rhs_id]));
    330:       PetscCall(DMPlexComputeIntegralFEM(dmAux, auxDataGlobal, vals, NULL));
    331:       PetscCall(ISDestroy(&is));
    332:     }
    333:   } else {
    334:     Mat J;
    335:     Vec auxDataGlobalf, auxDataf, Jscale;
    336:     DM  dmAuxf;

    338:     PetscCall(DMGetAuxiliaryVec(fdm, NULL, 0, 0, &auxDataf));
    339:     PetscCall(VecGetDM(auxDataf, &dmAuxf));
    340:     PetscCall(DMSetCoarseDM(dmAuxf, dmAux));
    341:     PetscCall(DMCreateGlobalVector(dmAuxf, &auxDataGlobalf));
    342:     PetscCall(DMLocalToGlobal(dmAuxf, auxDataf, INSERT_VALUES, auxDataGlobalf));
    343:     PetscCall(DMCreateInterpolation(dmAux, dmAuxf, &J, &Jscale));
    344:     PetscCall(MatInterpolate(J, auxDataGlobalf, auxDataGlobal));
    345:     PetscCall(VecPointwiseMult(auxDataGlobal, auxDataGlobal, Jscale));
    346:     PetscCall(VecDestroy(&Jscale));
    347:     PetscCall(VecDestroy(&auxDataGlobalf));
    348:     PetscCall(MatDestroy(&J));
    349:     PetscCall(DMSetCoarseDM(dmAuxf, NULL));
    350:   }
    351:   /* auxiliary data must be a local vector */
    352:   PetscCall(DMGlobalToLocal(dmAux, auxDataGlobal, INSERT_VALUES, auxData));
    353:   { /* view auxiliary data */
    354:     PetscInt level;
    355:     char     optionName[PETSC_MAX_PATH_LEN];

    357:     PetscCall(DMGetRefineLevel(dm, &level));
    358:     PetscCall(PetscSNPrintf(optionName, sizeof(optionName), "-aux_%" PetscInt_FMT "_vec_view", level));
    359:     PetscCall(VecViewFromOptions(auxData, NULL, optionName));
    360:   }
    361:   PetscCall(DMSetAuxiliaryVec(dm, NULL, 0, 0, auxData));
    362:   PetscCall(VecDestroy(&auxData));
    363:   PetscCall(VecDestroy(&auxDataGlobal));
    364:   PetscCall(DMDestroy(&dmAux));

    366:   /* Setup boundary conditions
    367:      Since we use homogeneous natural boundary conditions for the Neumann problem we
    368:      only handle the Dirichlet case here */
    369:   if (user->bc == BC_DIRICHLET) {
    370:     DMLabel  label;
    371:     PetscInt id = 1;

    373:     /* Label faces on the mesh boundary */
    374:     PetscCall(DMCreateLabel(dm, "boundary"));
    375:     PetscCall(DMGetLabel(dm, "boundary", &label));
    376:     PetscCall(DMPlexMarkBoundaryFaces(dm, 1, label));
    377:     PetscCall(DMAddBoundary(dm, DM_BC_ESSENTIAL, "dirichlet", label, 1, &id, 0, 0, NULL, NULL, NULL, NULL, NULL));
    378:   }

    380:   /* Iterate on coarser mesh if present */
    381:   PetscCall(DMGetCoarseDM(dm, &cdm));
    382:   if (cdm) PetscCall(SetupProblem(cdm, dm, user));

    384:   PetscCall(PetscFEDestroy(&fe));
    385:   PetscCall(PetscFEDestroy(&fe_rhs));
    386:   PetscCall(PetscFEDestroy(&fe_K));
    387:   PetscFunctionReturn(PETSC_SUCCESS);
    388: }

    390: /* We are now ready to run the simulation */
    391: int main(int argc, char **argv)
    392: {
    393:   DM     dm;   /* problem specification */
    394:   SNES   snes; /* nonlinear solver */
    395:   KSP    ksp;  /* linear solver */
    396:   Vec    u;    /* solution vector */
    397:   AppCtx user; /* user-defined work context */

    399:   PetscFunctionBeginUser;
    400:   PetscCall(PetscInitialize(&argc, &argv, NULL, help));
    401:   PetscCall(ProcessOptions(PETSC_COMM_WORLD, &user));
    402:   PetscCall(CreateMesh(PETSC_COMM_WORLD, &user, &dm));
    403:   PetscCall(SetupProblem(dm, NULL, &user));

    405:   PetscCall(SNESCreate(PETSC_COMM_WORLD, &snes));
    406:   PetscCall(SNESSetDM(snes, dm));
    407:   PetscCall(SNESSetType(snes, SNESKSPONLY));
    408:   PetscCall(SNESGetKSP(snes, &ksp));
    409:   PetscCall(KSPSetType(ksp, KSPCG));
    410:   PetscCall(KSPSetNormType(ksp, KSP_NORM_NATURAL));
    411:   PetscCall(SNESSetFromOptions(snes));

    413:   PetscCall(DMCreateGlobalVector(dm, &u));
    414:   PetscCall(PetscObjectSetName((PetscObject)u, ""));
    415:   PetscCall(VecSetRandom(u, NULL));
    416:   PetscCall(SNESSolve(snes, NULL, u));
    417:   PetscCall(VecDestroy(&u));
    418:   PetscCall(SNESDestroy(&snes));
    419:   PetscCall(DMDestroy(&dm));
    420:   PetscCall(PetscFinalize());
    421:   return 0;
    422: }

    424: /*TEST

    426:   test:
    427:     suffix: 0
    428:     requires: triangle !single
    429:     args: -bc_type {{dirichlet neumann}separate output} -rhs_type {{constant analytical}separate output} -coefficient_type {{constant checkerboard analytical}separate output} -initial_dm_plex_simplex 1 -pc_type svd -snes_type newtonls -snes_error_if_not_converged

    431:   test:
    432:     requires: !single
    433:     suffix: 0_quad
    434:     args: -bc_type {{dirichlet neumann}separate output} -rhs_type {{constant analytical}separate output} -coefficient_type {{constant checkerboard analytical}separate output} -initial_dm_plex_simplex 0 -pc_type svd -snes_type newtonls -snes_error_if_not_converged

    436:   test:
    437:     suffix: 0_p4est
    438:     requires: p4est !single
    439:     args: -bc_type {{dirichlet neumann}separate output} -rhs_type {{constant analytical}separate output} -coefficient_type {{constant checkerboard analytical}separate output} -initial_dm_plex_simplex 0 -pc_type svd -snes_type newtonls -snes_error_if_not_converged

    441:   testset:
    442:     nsize: 2
    443:     requires: hpddm slepc !single defined(PETSC_HAVE_DYNAMIC_LIBRARIES) defined(PETSC_USE_SHARED_LIBRARIES)
    444:     args: -pc_type hpddm -pc_hpddm_coarse_correction balanced -pc_hpddm_coarse_mat_type aij -pc_hpddm_levels_eps_nev 1 -pc_hpddm_levels_sub_pc_type lu -ksp_monitor -initial_dm_plex_simplex 0 -petscpartitioner_type simple
    445:     output_file: output/ex11_hpddm.out
    446:     test:
    447:       suffix: hpddm
    448:       args:
    449:     test:
    450:       suffix: hpddm_harmonic_overlap
    451:       args: -pc_hpddm_harmonic_overlap 1 -pc_hpddm_has_neumann false -pc_hpddm_levels_1_pc_asm_type basic
    452:     test:
    453:       suffix: hpddm_p4est
    454:       requires: p4est
    455:       args: -p4est
    456:       filter: sed -e "s/non-conforming AMR: 1/non-conforming AMR: 0/g"

    458:   test:
    459:     nsize: 4
    460:     suffix: gdsw_corner
    461:     requires: triangle
    462:     args: -pc_type mg -pc_mg_galerkin -pc_mg_adapt_interp_coarse_space gdsw -pc_mg_levels 2 -mg_levels_pc_type asm -initial_dm_plex_shape ball -initial_dm_refine 2 -mesh_dm_mat_type {{aij is}} -mg_levels_sub_pc_type lu -mg_levels_pc_asm_type basic -petscpartitioner_type simple

    464:   test:
    465:     suffix: asm_seq
    466:     args: -pc_type asm -pc_asm_dm_subdomains -sub_pc_type cholesky -snes_type newtonls -snes_error_if_not_converged -initial_dm_plex_simplex 0

    468: TEST*/