Actual source code: ex46.c

      1: static char help[] = "Time dependent Navier-Stokes problem in 2d and 3d with finite elements.\n\
      2: We solve the Navier-Stokes in a rectangular\n\
      3: domain, using a parallel unstructured mesh (DMPLEX) to discretize it.\n\
      4: This example supports discretized auxiliary fields (Re) as well as\n\
      5: multilevel nonlinear solvers.\n\
      6: Contributed by: Julian Andrej <juan@tf.uni-kiel.de>\n\n\n";

      8: #include <petscdmplex.h>
      9: #include <petscsnes.h>
     10: #include <petscts.h>
     11: #include <petscds.h>

     13: /*
     14:   Navier-Stokes equation:

     16:   du/dt + u . grad u - \Delta u - grad p = f
     17:   div u  = 0
     18: */

     20: typedef struct {
     21:   PetscInt mms;
     22: } AppCtx;

     24: #define REYN 400.0

     26: /* MMS1

     28:   u = t + x^2 + y^2;
     29:   v = t + 2*x^2 - 2*x*y;
     30:   p = x + y - 1;

     32:   f_x = -2*t*(x + y) + 2*x*y^2 - 4*x^2*y - 2*x^3 + 4.0/Re - 1.0
     33:   f_y = -2*t*x       + 2*y^3 - 4*x*y^2 - 2*x^2*y + 4.0/Re - 1.0

     35:   so that

     37:     u_t + u \cdot \nabla u - 1/Re \Delta u + \nabla p + f = <1, 1> + <t (2x + 2y) + 2x^3 + 4x^2y - 2xy^2, t 2x + 2x^2y + 4xy^2 - 2y^3> - 1/Re <4, 4> + <1, 1>
     38:                                                     + <-t (2x + 2y) + 2xy^2 - 4x^2y - 2x^3 + 4/Re - 1, -2xt + 2y^3 - 4xy^2 - 2x^2y + 4/Re - 1> = 0
     39:     \nabla \cdot u                                  = 2x - 2x = 0

     41:   where

     43:     <u, v> . <<u_x, v_x>, <u_y, v_y>> = <u u_x + v u_y, u v_x + v v_y>
     44: */
     45: PetscErrorCode mms1_u_2d(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nf, PetscScalar *u, void *ctx)
     46: {
     47:   u[0] = time + x[0] * x[0] + x[1] * x[1];
     48:   u[1] = time + 2.0 * x[0] * x[0] - 2.0 * x[0] * x[1];
     49:   return PETSC_SUCCESS;
     50: }

     52: PetscErrorCode mms1_p_2d(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nf, PetscScalar *p, void *ctx)
     53: {
     54:   *p = x[0] + x[1] - 1.0;
     55:   return PETSC_SUCCESS;
     56: }

     58: /* MMS 2*/

     60: static PetscErrorCode mms2_u_2d(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nf, PetscScalar *u, void *ctx)
     61: {
     62:   u[0] = PetscSinReal(time + x[0]) * PetscSinReal(time + x[1]);
     63:   u[1] = PetscCosReal(time + x[0]) * PetscCosReal(time + x[1]);
     64:   return PETSC_SUCCESS;
     65: }

     67: static PetscErrorCode mms2_p_2d(PetscInt dim, PetscReal time, const PetscReal x[], PetscInt Nf, PetscScalar *p, void *ctx)
     68: {
     69:   *p = PetscSinReal(time + x[0] - x[1]);
     70:   return PETSC_SUCCESS;
     71: }

     73: static void f0_mms1_u(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[])
     74: {
     75:   const PetscReal Re    = REYN;
     76:   const PetscInt  Ncomp = dim;
     77:   PetscInt        c, d;

     79:   for (c = 0; c < Ncomp; ++c) {
     80:     for (d = 0; d < dim; ++d) f0[c] += u[d] * u_x[c * dim + d];
     81:   }
     82:   f0[0] += u_t[0];
     83:   f0[1] += u_t[1];

     85:   f0[0] += -2.0 * t * (x[0] + x[1]) + 2.0 * x[0] * x[1] * x[1] - 4.0 * x[0] * x[0] * x[1] - 2.0 * x[0] * x[0] * x[0] + 4.0 / Re - 1.0;
     86:   f0[1] += -2.0 * t * x[0] + 2.0 * x[1] * x[1] * x[1] - 4.0 * x[0] * x[1] * x[1] - 2.0 * x[0] * x[0] * x[1] + 4.0 / Re - 1.0;
     87: }

     89: static void f0_mms2_u(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[])
     90: {
     91:   const PetscReal Re    = REYN;
     92:   const PetscInt  Ncomp = dim;
     93:   PetscInt        c, d;

     95:   for (c = 0; c < Ncomp; ++c) {
     96:     for (d = 0; d < dim; ++d) f0[c] += u[d] * u_x[c * dim + d];
     97:   }
     98:   f0[0] += u_t[0];
     99:   f0[1] += u_t[1];

    101:   f0[0] -= (Re * ((1.0L / 2.0L) * PetscSinReal(2 * t + 2 * x[0]) + PetscSinReal(2 * t + x[0] + x[1]) + PetscCosReal(t + x[0] - x[1])) + 2.0 * PetscSinReal(t + x[0]) * PetscSinReal(t + x[1])) / Re;
    102:   f0[1] -= (-Re * ((1.0L / 2.0L) * PetscSinReal(2 * t + 2 * x[1]) + PetscSinReal(2 * t + x[0] + x[1]) + PetscCosReal(t + x[0] - x[1])) + 2.0 * PetscCosReal(t + x[0]) * PetscCosReal(t + x[1])) / Re;
    103: }

    105: static void f1_u(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[])
    106: {
    107:   const PetscReal Re    = REYN;
    108:   const PetscInt  Ncomp = dim;
    109:   PetscInt        comp, d;

    111:   for (comp = 0; comp < Ncomp; ++comp) {
    112:     for (d = 0; d < dim; ++d) f1[comp * dim + d] = 1.0 / Re * u_x[comp * dim + d];
    113:     f1[comp * dim + comp] -= u[Ncomp];
    114:   }
    115: }

    117: static void f0_p(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[])
    118: {
    119:   PetscInt d;
    120:   for (d = 0, f0[0] = 0.0; d < dim; ++d) f0[0] += u_x[d * dim + d];
    121: }

    123: static void f1_p(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[])
    124: {
    125:   PetscInt d;
    126:   for (d = 0; d < dim; ++d) f1[d] = 0.0;
    127: }

    129: /*
    130:   (psi_i, u_j grad_j u_i) ==> (\psi_i, \phi_j grad_j u_i)
    131: */
    132: static void g0_uu(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[])
    133: {
    134:   PetscInt NcI = dim, NcJ = dim;
    135:   PetscInt fc, gc;
    136:   PetscInt d;

    138:   for (d = 0; d < dim; ++d) g0[d * dim + d] = u_tShift;

    140:   for (fc = 0; fc < NcI; ++fc) {
    141:     for (gc = 0; gc < NcJ; ++gc) g0[fc * NcJ + gc] += u_x[fc * NcJ + gc];
    142:   }
    143: }

    145: /*
    146:   (psi_i, u_j grad_j u_i) ==> (\psi_i, \u_j grad_j \phi_i)
    147: */
    148: static void g1_uu(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[])
    149: {
    150:   PetscInt NcI = dim;
    151:   PetscInt NcJ = dim;
    152:   PetscInt fc, gc, dg;
    153:   for (fc = 0; fc < NcI; ++fc) {
    154:     for (gc = 0; gc < NcJ; ++gc) {
    155:       for (dg = 0; dg < dim; ++dg) {
    156:         /* kronecker delta */
    157:         if (fc == gc) g1[(fc * NcJ + gc) * dim + dg] += u[dg];
    158:       }
    159:     }
    160:   }
    161: }

    163: /* < q, \nabla\cdot u >
    164:    NcompI = 1, NcompJ = dim */
    165: static void g1_pu(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[])
    166: {
    167:   PetscInt d;
    168:   for (d = 0; d < dim; ++d) g1[d * dim + d] = 1.0; /* \frac{\partial\phi^{u_d}}{\partial x_d} */
    169: }

    171: /* -< \nabla\cdot v, p >
    172:     NcompI = dim, NcompJ = 1 */
    173: static void g2_up(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 g2[])
    174: {
    175:   PetscInt d;
    176:   for (d = 0; d < dim; ++d) g2[d * dim + d] = -1.0; /* \frac{\partial\psi^{u_d}}{\partial x_d} */
    177: }

    179: /* < \nabla v, \nabla u + {\nabla u}^T >
    180:    This just gives \nabla u, give the perdiagonal for the transpose */
    181: static void g3_uu(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[])
    182: {
    183:   const PetscReal Re    = REYN;
    184:   const PetscInt  Ncomp = dim;
    185:   PetscInt        compI, d;

    187:   for (compI = 0; compI < Ncomp; ++compI) {
    188:     for (d = 0; d < dim; ++d) g3[((compI * Ncomp + compI) * dim + d) * dim + d] = 1.0 / Re;
    189:   }
    190: }

    192: static PetscErrorCode ProcessOptions(MPI_Comm comm, AppCtx *options)
    193: {
    194:   PetscFunctionBeginUser;
    195:   options->mms = 1;

    197:   PetscOptionsBegin(comm, "", "Navier-Stokes Equation Options", "DMPLEX");
    198:   PetscCall(PetscOptionsInt("-mms", "The manufactured solution to use", "ex46.c", options->mms, &options->mms, NULL));
    199:   PetscOptionsEnd();
    200:   PetscFunctionReturn(PETSC_SUCCESS);
    201: }

    203: static PetscErrorCode CreateMesh(MPI_Comm comm, DM *dm, AppCtx *ctx)
    204: {
    205:   PetscFunctionBeginUser;
    206:   PetscCall(DMCreate(comm, dm));
    207:   PetscCall(DMSetType(*dm, DMPLEX));
    208:   PetscCall(DMSetFromOptions(*dm));
    209:   PetscCall(DMViewFromOptions(*dm, NULL, "-dm_view"));
    210:   PetscFunctionReturn(PETSC_SUCCESS);
    211: }

    213: static PetscErrorCode SetupProblem(DM dm, AppCtx *ctx)
    214: {
    215:   PetscDS        ds;
    216:   DMLabel        label;
    217:   const PetscInt id = 1;
    218:   PetscInt       dim;

    220:   PetscFunctionBeginUser;
    221:   PetscCall(DMGetDimension(dm, &dim));
    222:   PetscCall(DMGetDS(dm, &ds));
    223:   PetscCall(DMGetLabel(dm, "marker", &label));
    224:   switch (dim) {
    225:   case 2:
    226:     switch (ctx->mms) {
    227:     case 1:
    228:       PetscCall(PetscDSSetResidual(ds, 0, f0_mms1_u, f1_u));
    229:       PetscCall(PetscDSSetResidual(ds, 1, f0_p, f1_p));
    230:       PetscCall(PetscDSSetJacobian(ds, 0, 0, g0_uu, g1_uu, NULL, g3_uu));
    231:       PetscCall(PetscDSSetJacobian(ds, 0, 1, NULL, NULL, g2_up, NULL));
    232:       PetscCall(PetscDSSetJacobian(ds, 1, 0, NULL, g1_pu, NULL, NULL));
    233:       PetscCall(PetscDSSetExactSolution(ds, 0, mms1_u_2d, ctx));
    234:       PetscCall(PetscDSSetExactSolution(ds, 1, mms1_p_2d, ctx));
    235:       PetscCall(DMAddBoundary(dm, DM_BC_ESSENTIAL, "wall", label, 1, &id, 0, 0, NULL, (void (*)(void))mms1_u_2d, NULL, ctx, NULL));
    236:       break;
    237:     case 2:
    238:       PetscCall(PetscDSSetResidual(ds, 0, f0_mms2_u, f1_u));
    239:       PetscCall(PetscDSSetResidual(ds, 1, f0_p, f1_p));
    240:       PetscCall(PetscDSSetJacobian(ds, 0, 0, g0_uu, g1_uu, NULL, g3_uu));
    241:       PetscCall(PetscDSSetJacobian(ds, 0, 1, NULL, NULL, g2_up, NULL));
    242:       PetscCall(PetscDSSetJacobian(ds, 1, 0, NULL, g1_pu, NULL, NULL));
    243:       PetscCall(PetscDSSetExactSolution(ds, 0, mms2_u_2d, ctx));
    244:       PetscCall(PetscDSSetExactSolution(ds, 1, mms2_p_2d, ctx));
    245:       PetscCall(DMAddBoundary(dm, DM_BC_ESSENTIAL, "wall", label, 1, &id, 0, 0, NULL, (void (*)(void))mms2_u_2d, NULL, ctx, NULL));
    246:       break;
    247:     default:
    248:       SETERRQ(PETSC_COMM_WORLD, PETSC_ERR_ARG_OUTOFRANGE, "Invalid MMS %" PetscInt_FMT, ctx->mms);
    249:     }
    250:     break;
    251:   default:
    252:     SETERRQ(PETSC_COMM_WORLD, PETSC_ERR_ARG_OUTOFRANGE, "Invalid dimension %" PetscInt_FMT, dim);
    253:   }
    254:   PetscFunctionReturn(PETSC_SUCCESS);
    255: }

    257: static PetscErrorCode SetupDiscretization(DM dm, AppCtx *ctx)
    258: {
    259:   MPI_Comm  comm;
    260:   DM        cdm = dm;
    261:   PetscFE   fe[2];
    262:   PetscInt  dim;
    263:   PetscBool simplex;

    265:   PetscFunctionBeginUser;
    266:   PetscCall(PetscObjectGetComm((PetscObject)dm, &comm));
    267:   PetscCall(DMGetDimension(dm, &dim));
    268:   PetscCall(DMPlexIsSimplex(dm, &simplex));
    269:   PetscCall(PetscFECreateDefault(comm, dim, dim, simplex, "vel_", PETSC_DEFAULT, &fe[0]));
    270:   PetscCall(PetscObjectSetName((PetscObject)fe[0], "velocity"));
    271:   PetscCall(PetscFECreateDefault(comm, dim, 1, simplex, "pres_", PETSC_DEFAULT, &fe[1]));
    272:   PetscCall(PetscFECopyQuadrature(fe[0], fe[1]));
    273:   PetscCall(PetscObjectSetName((PetscObject)fe[1], "pressure"));
    274:   /* Set discretization and boundary conditions for each mesh */
    275:   PetscCall(DMSetField(dm, 0, NULL, (PetscObject)fe[0]));
    276:   PetscCall(DMSetField(dm, 1, NULL, (PetscObject)fe[1]));
    277:   PetscCall(DMCreateDS(dm));
    278:   PetscCall(SetupProblem(dm, ctx));
    279:   while (cdm) {
    280:     PetscObject  pressure;
    281:     MatNullSpace nsp;

    283:     PetscCall(DMGetField(cdm, 1, NULL, &pressure));
    284:     PetscCall(MatNullSpaceCreate(PetscObjectComm(pressure), PETSC_TRUE, 0, NULL, &nsp));
    285:     PetscCall(PetscObjectCompose(pressure, "nullspace", (PetscObject)nsp));
    286:     PetscCall(MatNullSpaceDestroy(&nsp));

    288:     PetscCall(DMCopyDisc(dm, cdm));
    289:     PetscCall(DMGetCoarseDM(cdm, &cdm));
    290:   }
    291:   PetscCall(PetscFEDestroy(&fe[0]));
    292:   PetscCall(PetscFEDestroy(&fe[1]));
    293:   PetscFunctionReturn(PETSC_SUCCESS);
    294: }

    296: static PetscErrorCode MonitorError(TS ts, PetscInt step, PetscReal crtime, Vec u, void *ctx)
    297: {
    298:   PetscSimplePointFn *funcs[2];
    299:   void               *ctxs[2];
    300:   DM                  dm;
    301:   PetscDS             ds;
    302:   PetscReal           ferrors[2];

    304:   PetscFunctionBeginUser;
    305:   PetscCall(TSGetDM(ts, &dm));
    306:   PetscCall(DMGetDS(dm, &ds));
    307:   PetscCall(PetscDSGetExactSolution(ds, 0, &funcs[0], &ctxs[0]));
    308:   PetscCall(PetscDSGetExactSolution(ds, 1, &funcs[1], &ctxs[1]));
    309:   PetscCall(DMComputeL2FieldDiff(dm, crtime, funcs, ctxs, u, ferrors));
    310:   PetscCall(PetscPrintf(PETSC_COMM_WORLD, "Timestep: %04d time = %-8.4g \t L_2 Error: [%2.3g, %2.3g]\n", (int)step, (double)crtime, (double)ferrors[0], (double)ferrors[1]));
    311:   PetscFunctionReturn(PETSC_SUCCESS);
    312: }

    314: int main(int argc, char **argv)
    315: {
    316:   AppCtx ctx;
    317:   DM     dm;
    318:   TS     ts;
    319:   Vec    u, r;

    321:   PetscFunctionBeginUser;
    322:   PetscCall(PetscInitialize(&argc, &argv, NULL, help));
    323:   PetscCall(ProcessOptions(PETSC_COMM_WORLD, &ctx));
    324:   PetscCall(CreateMesh(PETSC_COMM_WORLD, &dm, &ctx));
    325:   PetscCall(DMSetApplicationContext(dm, &ctx));
    326:   PetscCall(SetupDiscretization(dm, &ctx));
    327:   PetscCall(DMPlexCreateClosureIndex(dm, NULL));

    329:   PetscCall(DMCreateGlobalVector(dm, &u));
    330:   PetscCall(VecDuplicate(u, &r));

    332:   PetscCall(TSCreate(PETSC_COMM_WORLD, &ts));
    333:   PetscCall(TSMonitorSet(ts, MonitorError, &ctx, NULL));
    334:   PetscCall(TSSetDM(ts, dm));
    335:   PetscCall(DMTSSetBoundaryLocal(dm, DMPlexTSComputeBoundary, &ctx));
    336:   PetscCall(DMTSSetIFunctionLocal(dm, DMPlexTSComputeIFunctionFEM, &ctx));
    337:   PetscCall(DMTSSetIJacobianLocal(dm, DMPlexTSComputeIJacobianFEM, &ctx));
    338:   PetscCall(TSSetExactFinalTime(ts, TS_EXACTFINALTIME_STEPOVER));
    339:   PetscCall(TSSetFromOptions(ts));
    340:   PetscCall(DMTSCheckFromOptions(ts, u));

    342:   {
    343:     PetscSimplePointFn *funcs[2];
    344:     void               *ctxs[2];
    345:     PetscDS             ds;

    347:     PetscCall(DMGetDS(dm, &ds));
    348:     PetscCall(PetscDSGetExactSolution(ds, 0, &funcs[0], &ctxs[0]));
    349:     PetscCall(PetscDSGetExactSolution(ds, 1, &funcs[1], &ctxs[1]));
    350:     PetscCall(DMProjectFunction(dm, 0.0, funcs, ctxs, INSERT_ALL_VALUES, u));
    351:   }
    352:   PetscCall(TSSolve(ts, u));
    353:   PetscCall(VecViewFromOptions(u, NULL, "-sol_vec_view"));

    355:   PetscCall(VecDestroy(&u));
    356:   PetscCall(VecDestroy(&r));
    357:   PetscCall(TSDestroy(&ts));
    358:   PetscCall(DMDestroy(&dm));
    359:   PetscCall(PetscFinalize());
    360:   return 0;
    361: }

    363: /*TEST

    365:   # Full solves
    366:   test:
    367:     suffix: 2d_p2p1_r1
    368:     requires: !single triangle
    369:     filter: sed -e "s~ATOL~RTOL~g" -e "s~ABS~RELATIVE~g"
    370:     args: -dm_refine 1 -vel_petscspace_degree 2 -pres_petscspace_degree 1 \
    371:           -ts_type beuler -ts_max_steps 10 -ts_dt 0.1 -ts_monitor -dmts_check \
    372:           -snes_monitor_short -snes_converged_reason \
    373:           -ksp_monitor_short -ksp_converged_reason \
    374:           -pc_type fieldsplit -pc_fieldsplit_type schur -pc_fieldsplit_schur_fact_type full \
    375:             -fieldsplit_velocity_pc_type lu \
    376:             -fieldsplit_pressure_ksp_rtol 1.0e-10 -fieldsplit_pressure_pc_type jacobi

    378:   test:
    379:     suffix: 2d_q2q1_r1
    380:     requires: !single
    381:     filter: sed -e "s~ATOL~RTOL~g" -e "s~ABS~RELATIVE~g" -e "s~ 0\]~ 0.0\]~g"
    382:     args: -dm_plex_simplex 0 -dm_refine 1 -vel_petscspace_degree 2 -pres_petscspace_degree 1 \
    383:           -ts_type beuler -ts_max_steps 10 -ts_dt 0.1 -ts_monitor -dmts_check \
    384:           -snes_monitor_short -snes_converged_reason \
    385:           -ksp_monitor_short -ksp_converged_reason \
    386:           -pc_type fieldsplit -pc_fieldsplit_type schur -pc_fieldsplit_schur_fact_type full \
    387:             -fieldsplit_velocity_pc_type lu \
    388:             -fieldsplit_pressure_ksp_rtol 1.0e-10 -fieldsplit_pressure_pc_type jacobi

    390: TEST*/