Actual source code: ex5.c

      1: static char help[] = "Bratu nonlinear PDE in 2d.\n\
      2: We solve the  Bratu (SFI - solid fuel ignition) problem in a 2D rectangular\n\
      3: domain, using distributed arrays (DMDAs) to partition the parallel grid.\n\
      4: The command line options include:\n\
      5:   -par <parameter>, where <parameter> indicates the problem's nonlinearity\n\
      6:      problem SFI:  <parameter> = Bratu parameter (0 <= par <= 6.81)\n\n\
      7:   -m_par/n_par <parameter>, where <parameter> indicates an integer\n \
      8:       that MMS3 will be evaluated with 2^m_par, 2^n_par";

     10: /* ------------------------------------------------------------------------

     12:     Solid Fuel Ignition (SFI) problem.  This problem is modeled by
     13:     the partial differential equation

     15:             -Laplacian u - lambda*exp(u) = 0,  0 < x,y < 1,

     17:     with boundary conditions

     19:              u = 0  for  x = 0, x = 1, y = 0, y = 1.

     21:     A finite difference approximation with the usual 5-point stencil
     22:     is used to discretize the boundary value problem to obtain a nonlinear
     23:     system of equations.

     25:       This example shows how geometric multigrid can be run transparently with a nonlinear solver so long
     26:       as SNESSetDM() is provided. Example usage

     28:       ./ex5 -pc_type mg -ksp_monitor  -snes_view -pc_mg_levels 3 -pc_mg_galerkin pmat -da_grid_x 17 -da_grid_y 17
     29:              -mg_levels_ksp_monitor -snes_monitor -mg_levels_pc_type sor -pc_mg_type full

     31:       or to run with grid sequencing on the nonlinear problem (note that you do not need to provide the number of
     32:          multigrid levels, it will be determined automatically based on the number of refinements done)

     34:       ./ex5 -pc_type mg -ksp_monitor  -snes_view -pc_mg_galerkin pmat -snes_grid_sequence 3
     35:              -mg_levels_ksp_monitor -snes_monitor -mg_levels_pc_type sor -pc_mg_type full

     37:   ------------------------------------------------------------------------- */

     39: /*
     40:    Include "petscdmda.h" so that we can use distributed arrays (DMDAs).
     41:    Include "petscsnes.h" so that we can use SNES solvers.  Note that this
     42: */
     43: #include <petscdm.h>
     44: #include <petscdmda.h>
     45: #include <petscsnes.h>
     46: #include <petscmatlab.h>
     47: #include <petsc/private/snesimpl.h>

     49: /*
     50:    User-defined application context - contains data needed by the
     51:    application-provided call-back routines, FormJacobianLocal() and
     52:    FormFunctionLocal().
     53: */
     54: typedef struct AppCtx AppCtx;
     55: struct AppCtx {
     56:   PetscReal param; /* test problem parameter */
     57:   PetscInt  m, n;  /* MMS3 parameters */
     58:   PetscErrorCode (*mms_solution)(AppCtx *, const DMDACoor2d *, PetscScalar *);
     59:   PetscErrorCode (*mms_forcing)(AppCtx *, const DMDACoor2d *, PetscScalar *);
     60: };

     62: /* ------------------------------------------------------------------- */
     63: /*
     64:    FormInitialGuess - Forms initial approximation.

     66:    Input Parameters:
     67:    da - The DM
     68:    user - user-defined application context

     70:    Output Parameter:
     71:    X - vector
     72:  */
     73: static PetscErrorCode FormInitialGuess(DM da, AppCtx *user, Vec X)
     74: {
     75:   PetscInt      i, j, Mx, My, xs, ys, xm, ym;
     76:   PetscReal     lambda, temp1, temp, hx, hy;
     77:   PetscScalar **x;

     79:   PetscFunctionBeginUser;
     80:   PetscCall(DMDAGetInfo(da, PETSC_IGNORE, &Mx, &My, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE));

     82:   lambda = user->param;
     83:   hx     = 1.0 / (PetscReal)(Mx - 1);
     84:   hy     = 1.0 / (PetscReal)(My - 1);
     85:   temp1  = lambda / (lambda + 1.0);

     87:   /*
     88:      Get a pointer to vector data.
     89:        - For default PETSc vectors, VecGetArray() returns a pointer to
     90:          the data array.  Otherwise, the routine is implementation dependent.
     91:        - You MUST call VecRestoreArray() when you no longer need access to
     92:          the array.
     93:   */
     94:   PetscCall(DMDAVecGetArray(da, X, &x));

     96:   /*
     97:      Get local grid boundaries (for 2-dimensional DMDA):
     98:        xs, ys   - starting grid indices (no ghost points)
     99:        xm, ym   - widths of local grid (no ghost points)

    101:   */
    102:   PetscCall(DMDAGetCorners(da, &xs, &ys, NULL, &xm, &ym, NULL));

    104:   /*
    105:      Compute initial guess over the locally owned part of the grid
    106:   */
    107:   for (j = ys; j < ys + ym; j++) {
    108:     temp = (PetscReal)(PetscMin(j, My - j - 1)) * hy;
    109:     for (i = xs; i < xs + xm; i++) {
    110:       if (i == 0 || j == 0 || i == Mx - 1 || j == My - 1) {
    111:         /* boundary conditions are all zero Dirichlet */
    112:         x[j][i] = 0.0;
    113:       } else {
    114:         x[j][i] = temp1 * PetscSqrtReal(PetscMin((PetscReal)(PetscMin(i, Mx - i - 1)) * hx, temp));
    115:       }
    116:     }
    117:   }

    119:   /*
    120:      Restore vector
    121:   */
    122:   PetscCall(DMDAVecRestoreArray(da, X, &x));
    123:   PetscFunctionReturn(PETSC_SUCCESS);
    124: }

    126: /*
    127:   FormExactSolution - Forms MMS solution

    129:   Input Parameters:
    130:   da - The DM
    131:   user - user-defined application context

    133:   Output Parameter:
    134:   X - vector
    135:  */
    136: static PetscErrorCode FormExactSolution(DM da, AppCtx *user, Vec U)
    137: {
    138:   DM            coordDA;
    139:   Vec           coordinates;
    140:   DMDACoor2d  **coords;
    141:   PetscScalar **u;
    142:   PetscInt      xs, ys, xm, ym, i, j;

    144:   PetscFunctionBeginUser;
    145:   PetscCall(DMDAGetCorners(da, &xs, &ys, NULL, &xm, &ym, NULL));
    146:   PetscCall(DMGetCoordinateDM(da, &coordDA));
    147:   PetscCall(DMGetCoordinates(da, &coordinates));
    148:   PetscCall(DMDAVecGetArray(coordDA, coordinates, &coords));
    149:   PetscCall(DMDAVecGetArray(da, U, &u));
    150:   for (j = ys; j < ys + ym; ++j) {
    151:     for (i = xs; i < xs + xm; ++i) PetscCall(user->mms_solution(user, &coords[j][i], &u[j][i]));
    152:   }
    153:   PetscCall(DMDAVecRestoreArray(da, U, &u));
    154:   PetscCall(DMDAVecRestoreArray(coordDA, coordinates, &coords));
    155:   PetscFunctionReturn(PETSC_SUCCESS);
    156: }

    158: static PetscErrorCode ZeroBCSolution(AppCtx *user, const DMDACoor2d *c, PetscScalar *u)
    159: {
    160:   u[0] = 0.;
    161:   return PETSC_SUCCESS;
    162: }

    164: /* The functions below evaluate the MMS solution u(x,y) and associated forcing

    166:      f(x,y) = -u_xx - u_yy - lambda exp(u)

    168:   such that u(x,y) is an exact solution with f(x,y) as the right-hand side forcing term.
    169:  */
    170: static PetscErrorCode MMSSolution1(AppCtx *user, const DMDACoor2d *c, PetscScalar *u)
    171: {
    172:   PetscReal x = PetscRealPart(c->x), y = PetscRealPart(c->y);

    174:   PetscFunctionBeginUser;
    175:   u[0] = x * (1 - x) * y * (1 - y);
    176:   PetscCall(PetscLogFlops(5));
    177:   PetscFunctionReturn(PETSC_SUCCESS);
    178: }
    179: static PetscErrorCode MMSForcing1(AppCtx *user, const DMDACoor2d *c, PetscScalar *f)
    180: {
    181:   PetscReal x = PetscRealPart(c->x), y = PetscRealPart(c->y);

    183:   PetscFunctionBeginUser;
    184:   f[0] = 2 * x * (1 - x) + 2 * y * (1 - y) - user->param * PetscExpReal(x * (1 - x) * y * (1 - y));
    185:   PetscFunctionReturn(PETSC_SUCCESS);
    186: }

    188: static PetscErrorCode MMSSolution2(AppCtx *user, const DMDACoor2d *c, PetscScalar *u)
    189: {
    190:   PetscReal x = PetscRealPart(c->x), y = PetscRealPart(c->y);

    192:   PetscFunctionBeginUser;
    193:   u[0] = PetscSinReal(PETSC_PI * x) * PetscSinReal(PETSC_PI * y);
    194:   PetscCall(PetscLogFlops(5));
    195:   PetscFunctionReturn(PETSC_SUCCESS);
    196: }
    197: static PetscErrorCode MMSForcing2(AppCtx *user, const DMDACoor2d *c, PetscScalar *f)
    198: {
    199:   PetscReal x = PetscRealPart(c->x), y = PetscRealPart(c->y);

    201:   PetscFunctionBeginUser;
    202:   f[0] = 2 * PetscSqr(PETSC_PI) * PetscSinReal(PETSC_PI * x) * PetscSinReal(PETSC_PI * y) - user->param * PetscExpReal(PetscSinReal(PETSC_PI * x) * PetscSinReal(PETSC_PI * y));
    203:   PetscFunctionReturn(PETSC_SUCCESS);
    204: }

    206: static PetscErrorCode MMSSolution3(AppCtx *user, const DMDACoor2d *c, PetscScalar *u)
    207: {
    208:   PetscReal x = PetscRealPart(c->x), y = PetscRealPart(c->y);

    210:   PetscFunctionBeginUser;
    211:   u[0] = PetscSinReal(user->m * PETSC_PI * x * (1 - y)) * PetscSinReal(user->n * PETSC_PI * y * (1 - x));
    212:   PetscCall(PetscLogFlops(5));
    213:   PetscFunctionReturn(PETSC_SUCCESS);
    214: }
    215: static PetscErrorCode MMSForcing3(AppCtx *user, const DMDACoor2d *c, PetscScalar *f)
    216: {
    217:   PetscReal x = PetscRealPart(c->x), y = PetscRealPart(c->y);
    218:   PetscReal m = user->m, n = user->n, lambda = user->param;

    220:   PetscFunctionBeginUser;
    221:   f[0] = (-(PetscExpReal(PetscSinReal(m * PETSC_PI * x * (1 - y)) * PetscSinReal(n * PETSC_PI * (1 - x) * y)) * lambda) + PetscSqr(PETSC_PI) * (-2 * m * n * ((-1 + x) * x + (-1 + y) * y) * PetscCosReal(m * PETSC_PI * x * (-1 + y)) * PetscCosReal(n * PETSC_PI * (-1 + x) * y) + (PetscSqr(m) * (PetscSqr(x) + PetscSqr(-1 + y)) + PetscSqr(n) * (PetscSqr(-1 + x) + PetscSqr(y))) * PetscSinReal(m * PETSC_PI * x * (-1 + y)) * PetscSinReal(n * PETSC_PI * (-1 + x) * y)));
    222:   PetscFunctionReturn(PETSC_SUCCESS);
    223: }

    225: static PetscErrorCode MMSSolution4(AppCtx *user, const DMDACoor2d *c, PetscScalar *u)
    226: {
    227:   const PetscReal Lx = 1., Ly = 1.;
    228:   PetscReal       x = PetscRealPart(c->x), y = PetscRealPart(c->y);

    230:   PetscFunctionBeginUser;
    231:   u[0] = (PetscPowReal(x, 4) - PetscSqr(Lx) * PetscSqr(x)) * (PetscPowReal(y, 4) - PetscSqr(Ly) * PetscSqr(y));
    232:   PetscCall(PetscLogFlops(9));
    233:   PetscFunctionReturn(PETSC_SUCCESS);
    234: }
    235: static PetscErrorCode MMSForcing4(AppCtx *user, const DMDACoor2d *c, PetscScalar *f)
    236: {
    237:   const PetscReal Lx = 1., Ly = 1.;
    238:   PetscReal       x = PetscRealPart(c->x), y = PetscRealPart(c->y);

    240:   PetscFunctionBeginUser;
    241:   f[0] = (2 * PetscSqr(x) * (PetscSqr(x) - PetscSqr(Lx)) * (PetscSqr(Ly) - 6 * PetscSqr(y)) + 2 * PetscSqr(y) * (PetscSqr(Lx) - 6 * PetscSqr(x)) * (PetscSqr(y) - PetscSqr(Ly)) - user->param * PetscExpReal((PetscPowReal(x, 4) - PetscSqr(Lx) * PetscSqr(x)) * (PetscPowReal(y, 4) - PetscSqr(Ly) * PetscSqr(y))));
    242:   PetscFunctionReturn(PETSC_SUCCESS);
    243: }

    245: /* ------------------------------------------------------------------- */
    246: /*
    247:    FormFunctionLocal - Evaluates nonlinear function, F(x) on local process patch

    249:  */
    250: static PetscErrorCode FormFunctionLocal(DMDALocalInfo *info, PetscScalar **x, PetscScalar **f, AppCtx *user)
    251: {
    252:   PetscInt    i, j;
    253:   PetscReal   lambda, hx, hy, hxdhy, hydhx;
    254:   PetscScalar u, ue, uw, un, us, uxx, uyy, mms_solution, mms_forcing;
    255:   DMDACoor2d  c;

    257:   PetscFunctionBeginUser;
    258:   lambda = user->param;
    259:   hx     = 1.0 / (PetscReal)(info->mx - 1);
    260:   hy     = 1.0 / (PetscReal)(info->my - 1);
    261:   hxdhy  = hx / hy;
    262:   hydhx  = hy / hx;
    263:   /*
    264:      Compute function over the locally owned part of the grid
    265:   */
    266:   for (j = info->ys; j < info->ys + info->ym; j++) {
    267:     for (i = info->xs; i < info->xs + info->xm; i++) {
    268:       if (i == 0 || j == 0 || i == info->mx - 1 || j == info->my - 1) {
    269:         c.x = i * hx;
    270:         c.y = j * hy;
    271:         PetscCall(user->mms_solution(user, &c, &mms_solution));
    272:         f[j][i] = 2.0 * (hydhx + hxdhy) * (x[j][i] - mms_solution);
    273:       } else {
    274:         u  = x[j][i];
    275:         uw = x[j][i - 1];
    276:         ue = x[j][i + 1];
    277:         un = x[j - 1][i];
    278:         us = x[j + 1][i];

    280:         /* Enforce boundary conditions at neighboring points -- setting these values causes the Jacobian to be symmetric. */
    281:         if (i - 1 == 0) {
    282:           c.x = (i - 1) * hx;
    283:           c.y = j * hy;
    284:           PetscCall(user->mms_solution(user, &c, &uw));
    285:         }
    286:         if (i + 1 == info->mx - 1) {
    287:           c.x = (i + 1) * hx;
    288:           c.y = j * hy;
    289:           PetscCall(user->mms_solution(user, &c, &ue));
    290:         }
    291:         if (j - 1 == 0) {
    292:           c.x = i * hx;
    293:           c.y = (j - 1) * hy;
    294:           PetscCall(user->mms_solution(user, &c, &un));
    295:         }
    296:         if (j + 1 == info->my - 1) {
    297:           c.x = i * hx;
    298:           c.y = (j + 1) * hy;
    299:           PetscCall(user->mms_solution(user, &c, &us));
    300:         }

    302:         uxx         = (2.0 * u - uw - ue) * hydhx;
    303:         uyy         = (2.0 * u - un - us) * hxdhy;
    304:         mms_forcing = 0;
    305:         c.x         = i * hx;
    306:         c.y         = j * hy;
    307:         if (user->mms_forcing) PetscCall(user->mms_forcing(user, &c, &mms_forcing));
    308:         f[j][i] = uxx + uyy - hx * hy * (lambda * PetscExpScalar(u) + mms_forcing);
    309:       }
    310:     }
    311:   }
    312:   PetscCall(PetscLogFlops(11.0 * info->ym * info->xm));
    313:   PetscFunctionReturn(PETSC_SUCCESS);
    314: }

    316: /* FormObjectiveLocal - Evaluates nonlinear function, F(x) on local process patch */
    317: static PetscErrorCode FormObjectiveLocal(DMDALocalInfo *info, PetscScalar **x, PetscReal *obj, AppCtx *user)
    318: {
    319:   PetscInt    i, j;
    320:   PetscReal   lambda, hx, hy, hxdhy, hydhx, sc, lobj = 0;
    321:   PetscScalar u, ue, uw, un, us, uxux, uyuy;
    322:   MPI_Comm    comm;

    324:   PetscFunctionBeginUser;
    325:   *obj = 0;
    326:   PetscCall(PetscObjectGetComm((PetscObject)info->da, &comm));
    327:   lambda = user->param;
    328:   hx     = 1.0 / (PetscReal)(info->mx - 1);
    329:   hy     = 1.0 / (PetscReal)(info->my - 1);
    330:   sc     = hx * hy * lambda;
    331:   hxdhy  = hx / hy;
    332:   hydhx  = hy / hx;
    333:   /*
    334:      Compute function over the locally owned part of the grid
    335:   */
    336:   for (j = info->ys; j < info->ys + info->ym; j++) {
    337:     for (i = info->xs; i < info->xs + info->xm; i++) {
    338:       if (i == 0 || j == 0 || i == info->mx - 1 || j == info->my - 1) {
    339:         lobj += PetscRealPart((hydhx + hxdhy) * x[j][i] * x[j][i]);
    340:       } else {
    341:         u  = x[j][i];
    342:         uw = x[j][i - 1];
    343:         ue = x[j][i + 1];
    344:         un = x[j - 1][i];
    345:         us = x[j + 1][i];

    347:         if (i - 1 == 0) uw = 0.;
    348:         if (i + 1 == info->mx - 1) ue = 0.;
    349:         if (j - 1 == 0) un = 0.;
    350:         if (j + 1 == info->my - 1) us = 0.;

    352:         /* F[u] = 1/2\int_{\omega}\nabla^2u(x)*u(x)*dx */

    354:         uxux = u * (2. * u - ue - uw) * hydhx;
    355:         uyuy = u * (2. * u - un - us) * hxdhy;

    357:         lobj += PetscRealPart(0.5 * (uxux + uyuy) - sc * PetscExpScalar(u));
    358:       }
    359:     }
    360:   }
    361:   PetscCall(PetscLogFlops(12.0 * info->ym * info->xm));
    362:   *obj = lobj;
    363:   PetscFunctionReturn(PETSC_SUCCESS);
    364: }

    366: /*
    367:    FormJacobianLocal - Evaluates Jacobian matrix on local process patch
    368: */
    369: static PetscErrorCode FormJacobianLocal(DMDALocalInfo *info, PetscScalar **x, Mat jac, Mat jacpre, AppCtx *user)
    370: {
    371:   PetscInt     i, j, k;
    372:   MatStencil   col[5], row;
    373:   PetscScalar  lambda, v[5], hx, hy, hxdhy, hydhx, sc;
    374:   DM           coordDA;
    375:   Vec          coordinates;
    376:   DMDACoor2d **coords;

    378:   PetscFunctionBeginUser;
    379:   lambda = user->param;
    380:   /* Extract coordinates */
    381:   PetscCall(DMGetCoordinateDM(info->da, &coordDA));
    382:   PetscCall(DMGetCoordinates(info->da, &coordinates));
    383:   PetscCall(DMDAVecGetArray(coordDA, coordinates, &coords));
    384:   hx = info->xm > 1 ? PetscRealPart(coords[info->ys][info->xs + 1].x) - PetscRealPart(coords[info->ys][info->xs].x) : 1.0;
    385:   hy = info->ym > 1 ? PetscRealPart(coords[info->ys + 1][info->xs].y) - PetscRealPart(coords[info->ys][info->xs].y) : 1.0;
    386:   PetscCall(DMDAVecRestoreArray(coordDA, coordinates, &coords));
    387:   hxdhy = hx / hy;
    388:   hydhx = hy / hx;
    389:   sc    = hx * hy * lambda;

    391:   /*
    392:      Compute entries for the locally owned part of the Jacobian.
    393:       - Currently, all PETSc parallel matrix formats are partitioned by
    394:         contiguous chunks of rows across the processors.
    395:       - Each processor needs to insert only elements that it owns
    396:         locally (but any non-local elements will be sent to the
    397:         appropriate processor during matrix assembly).
    398:       - Here, we set all entries for a particular row at once.
    399:       - We can set matrix entries either using either
    400:         MatSetValuesLocal() or MatSetValues(), as discussed above.
    401:   */
    402:   for (j = info->ys; j < info->ys + info->ym; j++) {
    403:     for (i = info->xs; i < info->xs + info->xm; i++) {
    404:       row.j = j;
    405:       row.i = i;
    406:       /* boundary points */
    407:       if (i == 0 || j == 0 || i == info->mx - 1 || j == info->my - 1) {
    408:         v[0] = 2.0 * (hydhx + hxdhy);
    409:         PetscCall(MatSetValuesStencil(jacpre, 1, &row, 1, &row, v, INSERT_VALUES));
    410:       } else {
    411:         k = 0;
    412:         /* interior grid points */
    413:         if (j - 1 != 0) {
    414:           v[k]     = -hxdhy;
    415:           col[k].j = j - 1;
    416:           col[k].i = i;
    417:           k++;
    418:         }
    419:         if (i - 1 != 0) {
    420:           v[k]     = -hydhx;
    421:           col[k].j = j;
    422:           col[k].i = i - 1;
    423:           k++;
    424:         }

    426:         v[k]     = 2.0 * (hydhx + hxdhy) - sc * PetscExpScalar(x[j][i]);
    427:         col[k].j = row.j;
    428:         col[k].i = row.i;
    429:         k++;

    431:         if (i + 1 != info->mx - 1) {
    432:           v[k]     = -hydhx;
    433:           col[k].j = j;
    434:           col[k].i = i + 1;
    435:           k++;
    436:         }
    437:         if (j + 1 != info->mx - 1) {
    438:           v[k]     = -hxdhy;
    439:           col[k].j = j + 1;
    440:           col[k].i = i;
    441:           k++;
    442:         }
    443:         PetscCall(MatSetValuesStencil(jacpre, 1, &row, k, col, v, INSERT_VALUES));
    444:       }
    445:     }
    446:   }

    448:   /*
    449:      Assemble matrix, using the 2-step process:
    450:        MatAssemblyBegin(), MatAssemblyEnd().
    451:   */
    452:   PetscCall(MatAssemblyBegin(jacpre, MAT_FINAL_ASSEMBLY));
    453:   PetscCall(MatAssemblyEnd(jacpre, MAT_FINAL_ASSEMBLY));
    454:   /*
    455:      Tell the matrix we will never add a new nonzero location to the
    456:      matrix. If we do, it will generate an error.
    457:   */
    458:   PetscCall(MatSetOption(jac, MAT_NEW_NONZERO_LOCATION_ERR, PETSC_TRUE));
    459:   PetscFunctionReturn(PETSC_SUCCESS);
    460: }

    462: static PetscErrorCode FormFunctionMatlab(SNES snes, Vec X, Vec F, void *ptr)
    463: {
    464: #if PetscDefined(HAVE_MATLAB)
    465:   AppCtx   *user = (AppCtx *)ptr;
    466:   PetscInt  Mx, My;
    467:   PetscReal lambda, hx, hy;
    468:   Vec       localX, localF;
    469:   MPI_Comm  comm;
    470:   DM        da;

    472:   PetscFunctionBeginUser;
    473:   PetscCall(SNESGetDM(snes, &da));
    474:   PetscCall(DMGetLocalVector(da, &localX));
    475:   PetscCall(DMGetLocalVector(da, &localF));
    476:   PetscCall(PetscObjectSetName((PetscObject)localX, "localX"));
    477:   PetscCall(PetscObjectSetName((PetscObject)localF, "localF"));
    478:   PetscCall(DMDAGetInfo(da, PETSC_IGNORE, &Mx, &My, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE));

    480:   lambda = user->param;
    481:   hx     = 1.0 / (PetscReal)(Mx - 1);
    482:   hy     = 1.0 / (PetscReal)(My - 1);

    484:   PetscCall(PetscObjectGetComm((PetscObject)snes, &comm));
    485:   /*
    486:      Scatter ghost points to local vector,using the 2-step process
    487:         DMGlobalToLocalBegin(),DMGlobalToLocalEnd().
    488:      By placing code between these two statements, computations can be
    489:      done while messages are in transition.
    490:   */
    491:   PetscCall(DMGlobalToLocalBegin(da, X, INSERT_VALUES, localX));
    492:   PetscCall(DMGlobalToLocalEnd(da, X, INSERT_VALUES, localX));
    493:   PetscCall(PetscMatlabEnginePut(PETSC_MATLAB_ENGINE_(comm), (PetscObject)localX));
    494:   PetscCall(PetscMatlabEngineEvaluate(PETSC_MATLAB_ENGINE_(comm), "localF=ex5m(localX,%18.16e,%18.16e,%18.16e)", (double)hx, (double)hy, (double)lambda));
    495:   PetscCall(PetscMatlabEngineGet(PETSC_MATLAB_ENGINE_(comm), (PetscObject)localF));

    497:   /*
    498:      Insert values into global vector
    499:   */
    500:   PetscCall(DMLocalToGlobalBegin(da, localF, INSERT_VALUES, F));
    501:   PetscCall(DMLocalToGlobalEnd(da, localF, INSERT_VALUES, F));
    502:   PetscCall(DMRestoreLocalVector(da, &localX));
    503:   PetscCall(DMRestoreLocalVector(da, &localF));
    504:   PetscFunctionReturn(PETSC_SUCCESS);
    505: #else
    506:   return PETSC_SUCCESS; /* Never called */
    507: #endif
    508: }

    510: /* ------------------------------------------------------------------- */
    511: /*
    512:       Applies some sweeps on nonlinear Gauss-Seidel on each process

    514:  */
    515: static PetscErrorCode NonlinearGS(SNES snes, Vec X, Vec B, PetscCtx ctx)
    516: {
    517:   PetscInt      i, j, k, Mx, My, xs, ys, xm, ym, its, tot_its, sweeps, l;
    518:   PetscReal     lambda, hx, hy, hxdhy, hydhx, sc;
    519:   PetscScalar **x, **b, bij, F, F0 = 0, J, u, un, us, ue, eu, uw, uxx, uyy, y;
    520:   PetscReal     atol, rtol, stol;
    521:   DM            da;
    522:   AppCtx       *user;
    523:   Vec           localX, localB;

    525:   PetscFunctionBeginUser;
    526:   tot_its = 0;
    527:   PetscCall(SNESNGSGetSweeps(snes, &sweeps));
    528:   PetscCall(SNESNGSGetTolerances(snes, &atol, &rtol, &stol, &its));
    529:   PetscCall(SNESGetDM(snes, &da));
    530:   PetscCall(DMGetApplicationContext(da, &user));

    532:   PetscCall(DMDAGetInfo(da, PETSC_IGNORE, &Mx, &My, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE, PETSC_IGNORE));

    534:   lambda = user->param;
    535:   hx     = 1.0 / (PetscReal)(Mx - 1);
    536:   hy     = 1.0 / (PetscReal)(My - 1);
    537:   sc     = hx * hy * lambda;
    538:   hxdhy  = hx / hy;
    539:   hydhx  = hy / hx;

    541:   PetscCall(DMGetLocalVector(da, &localX));
    542:   if (B) PetscCall(DMGetLocalVector(da, &localB));
    543:   for (l = 0; l < sweeps; l++) {
    544:     PetscCall(DMGlobalToLocalBegin(da, X, INSERT_VALUES, localX));
    545:     PetscCall(DMGlobalToLocalEnd(da, X, INSERT_VALUES, localX));
    546:     if (B) {
    547:       PetscCall(DMGlobalToLocalBegin(da, B, INSERT_VALUES, localB));
    548:       PetscCall(DMGlobalToLocalEnd(da, B, INSERT_VALUES, localB));
    549:     }
    550:     /*
    551:      Get a pointer to vector data.
    552:      - For default PETSc vectors, VecGetArray() returns a pointer to
    553:      the data array.  Otherwise, the routine is implementation dependent.
    554:      - You MUST call VecRestoreArray() when you no longer need access to
    555:      the array.
    556:      */
    557:     PetscCall(DMDAVecGetArray(da, localX, &x));
    558:     if (B) PetscCall(DMDAVecGetArray(da, localB, &b));
    559:     /*
    560:      Get local grid boundaries (for 2-dimensional DMDA):
    561:      xs, ys   - starting grid indices (no ghost points)
    562:      xm, ym   - widths of local grid (no ghost points)
    563:      */
    564:     PetscCall(DMDAGetCorners(da, &xs, &ys, NULL, &xm, &ym, NULL));

    566:     for (j = ys; j < ys + ym; j++) {
    567:       for (i = xs; i < xs + xm; i++) {
    568:         if (i == 0 || j == 0 || i == Mx - 1 || j == My - 1) {
    569:           /* boundary conditions are all zero Dirichlet */
    570:           x[j][i] = 0.0;
    571:         } else {
    572:           if (B) bij = b[j][i];
    573:           else bij = 0.;

    575:           u  = x[j][i];
    576:           un = x[j - 1][i];
    577:           us = x[j + 1][i];
    578:           ue = x[j][i - 1];
    579:           uw = x[j][i + 1];

    581:           for (k = 0; k < its; k++) {
    582:             eu  = PetscExpScalar(u);
    583:             uxx = (2.0 * u - ue - uw) * hydhx;
    584:             uyy = (2.0 * u - un - us) * hxdhy;
    585:             F   = uxx + uyy - sc * eu - bij;
    586:             if (k == 0) F0 = F;
    587:             J = 2.0 * (hydhx + hxdhy) - sc * eu;
    588:             y = F / J;
    589:             u -= y;
    590:             tot_its++;

    592:             if (atol > PetscAbsReal(PetscRealPart(F)) || rtol * PetscAbsReal(PetscRealPart(F0)) > PetscAbsReal(PetscRealPart(F)) || stol * PetscAbsReal(PetscRealPart(u)) > PetscAbsReal(PetscRealPart(y))) break;
    593:           }
    594:           x[j][i] = u;
    595:         }
    596:       }
    597:     }
    598:     /*
    599:      Restore vector
    600:      */
    601:     PetscCall(DMDAVecRestoreArray(da, localX, &x));
    602:     PetscCall(DMLocalToGlobalBegin(da, localX, INSERT_VALUES, X));
    603:     PetscCall(DMLocalToGlobalEnd(da, localX, INSERT_VALUES, X));
    604:   }
    605:   PetscCall(PetscLogFlops(tot_its * (21.0)));
    606:   PetscCall(DMRestoreLocalVector(da, &localX));
    607:   if (B) {
    608:     PetscCall(DMDAVecRestoreArray(da, localB, &b));
    609:     PetscCall(DMRestoreLocalVector(da, &localB));
    610:   }
    611:   PetscFunctionReturn(PETSC_SUCCESS);
    612: }

    614: int main(int argc, char **argv)
    615: {
    616:   SNES      snes; /* nonlinear solver */
    617:   Vec       x;    /* solution vector */
    618:   AppCtx    user; /* user-defined work context */
    619:   PetscInt  its;  /* iterations for convergence */
    620:   PetscReal bratu_lambda_max = 6.81;
    621:   PetscReal bratu_lambda_min = 0.;
    622:   PetscInt  MMS              = 1;
    623:   PetscBool flg              = PETSC_FALSE, setMMS;
    624:   DM        da;
    625:   Vec       r = NULL;
    626:   KSP       ksp;
    627:   PetscInt  lits, slits;

    629:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    630:      Initialize program
    631:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    633:   PetscFunctionBeginUser;
    634:   PetscCall(PetscInitialize(&argc, &argv, NULL, help));

    636:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    637:      Initialize problem parameters
    638:   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    639:   user.param = 6.0;
    640:   PetscCall(PetscOptionsGetReal(NULL, NULL, "-par", &user.param, NULL));
    641:   PetscCheck(user.param <= bratu_lambda_max && user.param >= bratu_lambda_min, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Lambda, %g, is out of range, [%g, %g]", (double)user.param, (double)bratu_lambda_min, (double)bratu_lambda_max);
    642:   PetscCall(PetscOptionsGetInt(NULL, NULL, "-mms", &MMS, &setMMS));
    643:   if (MMS == 3) {
    644:     PetscInt mPar = 2, nPar = 1;
    645:     PetscCall(PetscOptionsGetInt(NULL, NULL, "-m_par", &mPar, NULL));
    646:     PetscCall(PetscOptionsGetInt(NULL, NULL, "-n_par", &nPar, NULL));
    647:     user.m = PetscPowInt(2, mPar);
    648:     user.n = PetscPowInt(2, nPar);
    649:   }

    651:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    652:      Create nonlinear solver context
    653:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    654:   PetscCall(SNESCreate(PETSC_COMM_WORLD, &snes));
    655:   PetscCall(SNESSetCountersReset(snes, PETSC_FALSE));
    656:   PetscCall(SNESSetNGS(snes, NonlinearGS, NULL));

    658:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    659:      Create distributed array (DMDA) to manage parallel grid and vectors
    660:   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    661:   PetscCall(DMDACreate2d(PETSC_COMM_WORLD, DM_BOUNDARY_NONE, DM_BOUNDARY_NONE, DMDA_STENCIL_STAR, 4, 4, PETSC_DECIDE, PETSC_DECIDE, 1, 1, NULL, NULL, &da));
    662:   PetscCall(DMSetFromOptions(da));
    663:   PetscCall(DMSetUp(da));
    664:   PetscCall(DMDASetUniformCoordinates(da, 0.0, 1.0, 0.0, 1.0, 0.0, 1.0));
    665:   PetscCall(DMSetApplicationContext(da, &user));
    666:   PetscCall(SNESSetDM(snes, da));
    667:   /*  - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    668:      Extract global vectors from DMDA; then duplicate for remaining
    669:      vectors that are the same types
    670:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    671:   PetscCall(DMCreateGlobalVector(da, &x));

    673:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    674:      Set local function evaluation routine
    675:   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    676:   switch (MMS) {
    677:   case 0:
    678:     user.mms_solution = ZeroBCSolution;
    679:     user.mms_forcing  = NULL;
    680:     break;
    681:   case 1:
    682:     user.mms_solution = MMSSolution1;
    683:     user.mms_forcing  = MMSForcing1;
    684:     break;
    685:   case 2:
    686:     user.mms_solution = MMSSolution2;
    687:     user.mms_forcing  = MMSForcing2;
    688:     break;
    689:   case 3:
    690:     user.mms_solution = MMSSolution3;
    691:     user.mms_forcing  = MMSForcing3;
    692:     break;
    693:   case 4:
    694:     user.mms_solution = MMSSolution4;
    695:     user.mms_forcing  = MMSForcing4;
    696:     break;
    697:   default:
    698:     SETERRQ(PETSC_COMM_WORLD, PETSC_ERR_USER, "Unknown MMS type %" PetscInt_FMT, MMS);
    699:   }
    700:   PetscCall(DMDASNESSetFunctionLocal(da, INSERT_VALUES, (DMDASNESFunctionFn *)FormFunctionLocal, &user));
    701:   PetscCall(PetscOptionsGetBool(NULL, NULL, "-fd", &flg, NULL));
    702:   if (!flg) PetscCall(DMDASNESSetJacobianLocal(da, (DMDASNESJacobianFn *)FormJacobianLocal, &user));

    704:   PetscCall(PetscOptionsGetBool(NULL, NULL, "-obj", &flg, NULL));
    705:   if (flg) PetscCall(DMDASNESSetObjectiveLocal(da, (DMDASNESObjectiveFn *)FormObjectiveLocal, &user));

    707:   if (PetscDefined(HAVE_MATLAB)) {
    708:     PetscBool matlab_function = PETSC_FALSE;
    709:     PetscCall(PetscOptionsGetBool(NULL, NULL, "-matlab_function", &matlab_function, 0));
    710:     if (matlab_function) {
    711:       PetscCall(VecDuplicate(x, &r));
    712:       PetscCall(SNESSetFunction(snes, r, FormFunctionMatlab, &user));
    713:     }
    714:   }

    716:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    717:      Customize nonlinear solver; set runtime options
    718:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    719:   PetscCall(SNESSetFromOptions(snes));

    721:   PetscCall(FormInitialGuess(da, &user, x));

    723:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    724:      Solve nonlinear system
    725:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    726:   PetscCall(SNESSolve(snes, NULL, x));
    727:   PetscCall(SNESGetIterationNumber(snes, &its));

    729:   PetscCall(SNESGetLinearSolveIterations(snes, &slits));
    730:   PetscCall(SNESGetKSP(snes, &ksp));
    731:   PetscCall(KSPGetTotalIterations(ksp, &lits));
    732:   PetscCheck(lits == slits, PETSC_COMM_WORLD, PETSC_ERR_PLIB, "Number of total linear iterations reported by SNES %" PetscInt_FMT " does not match reported by KSP %" PetscInt_FMT, slits, lits);
    733:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    734:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    736:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    737:      If using MMS, check the l_2 error
    738:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    739:   if (setMMS) {
    740:     Vec       e;
    741:     PetscReal errorl2, errorinf;
    742:     PetscInt  N;

    744:     PetscCall(VecDuplicate(x, &e));
    745:     PetscCall(PetscObjectViewFromOptions((PetscObject)x, NULL, "-sol_view"));
    746:     PetscCall(FormExactSolution(da, &user, e));
    747:     PetscCall(PetscObjectViewFromOptions((PetscObject)e, NULL, "-exact_view"));
    748:     PetscCall(VecAXPY(e, -1.0, x));
    749:     PetscCall(PetscObjectViewFromOptions((PetscObject)e, NULL, "-error_view"));
    750:     PetscCall(VecNorm(e, NORM_2, &errorl2));
    751:     PetscCall(VecNorm(e, NORM_INFINITY, &errorinf));
    752:     PetscCall(VecGetSize(e, &N));
    753:     PetscCall(PetscPrintf(PETSC_COMM_WORLD, "N: %" PetscInt_FMT " error L2 %g inf %g\n", N, (double)(errorl2 / PetscSqrtReal((PetscReal)N)), (double)errorinf));
    754:     PetscCall(VecDestroy(&e));
    755:     PetscCall(PetscLogEventSetDof(SNES_Solve, 0, N));
    756:     PetscCall(PetscLogEventSetError(SNES_Solve, 0, errorl2 / PetscSqrtReal(N)));
    757:   }

    759:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    760:      Free work space.  All PETSc objects should be destroyed when they
    761:      are no longer needed.
    762:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    763:   PetscCall(VecDestroy(&r));
    764:   PetscCall(VecDestroy(&x));
    765:   PetscCall(SNESDestroy(&snes));
    766:   PetscCall(DMDestroy(&da));
    767:   PetscCall(PetscFinalize());
    768:   return 0;
    769: }

    771: /*TEST

    773:    test:
    774:      suffix: asm_0
    775:      requires: !single
    776:      args: -mms 1 -par 0.0 -snes_monitor_short -snes_converged_reason -snes_view -ksp_rtol 1.0e-9 -ksp_monitor_short -ksp_type richardson -pc_type asm -pc_asm_blocks 2 -pc_asm_overlap 0 -pc_asm_local_type additive -sub_pc_type lu

    778:    test:
    779:      suffix: msm_0
    780:      requires: !single
    781:      args: -mms 1 -par 0.0 -snes_monitor_short -snes_converged_reason -snes_view -ksp_rtol 1.0e-9 -ksp_monitor_short -ksp_type richardson -pc_type asm -pc_asm_blocks 2 -pc_asm_overlap 0 -pc_asm_local_type multiplicative -sub_pc_type lu

    783:    test:
    784:      suffix: asm_1
    785:      requires: !single
    786:      args: -mms 1 -par 0.0 -snes_monitor_short -snes_converged_reason -snes_view -ksp_rtol 1.0e-9 -ksp_monitor_short -ksp_type richardson -pc_type asm -pc_asm_blocks 2 -pc_asm_overlap 0 -pc_asm_local_type additive -sub_pc_type lu -da_grid_x 8

    788:    test:
    789:      suffix: msm_1
    790:      requires: !single
    791:      args: -mms 1 -par 0.0 -snes_monitor_short -snes_converged_reason -snes_view -ksp_rtol 1.0e-9 -ksp_monitor_short -ksp_type richardson -pc_type asm -pc_asm_blocks 2 -pc_asm_overlap 0 -pc_asm_local_type multiplicative -sub_pc_type lu -da_grid_x 8

    793:    test:
    794:      suffix: asm_2
    795:      requires: !single
    796:      args: -mms 1 -par 0.0 -snes_monitor_short -snes_converged_reason -snes_view -ksp_rtol 1.0e-9 -ksp_monitor_short -ksp_type richardson -pc_type asm -pc_asm_blocks 3 -pc_asm_overlap 0 -pc_asm_local_type additive -sub_pc_type lu -da_grid_x 8

    798:    test:
    799:      suffix: msm_2
    800:      requires: !single
    801:      args: -mms 1 -par 0.0 -snes_monitor_short -snes_converged_reason -snes_view -ksp_rtol 1.0e-9 -ksp_monitor_short -ksp_type richardson -pc_type asm -pc_asm_blocks 3 -pc_asm_overlap 0 -pc_asm_local_type multiplicative -sub_pc_type lu -da_grid_x 8

    803:    test:
    804:      suffix: asm_3
    805:      requires: !single
    806:      args: -mms 1 -par 0.0 -snes_monitor_short -snes_converged_reason -snes_view -ksp_rtol 1.0e-9 -ksp_monitor_short -ksp_type richardson -pc_type asm -pc_asm_blocks 4 -pc_asm_overlap 0 -pc_asm_local_type additive -sub_pc_type lu -da_grid_x 8

    808:    test:
    809:      suffix: msm_3
    810:      requires: !single
    811:      args: -mms 1 -par 0.0 -snes_monitor_short -snes_converged_reason -snes_view -ksp_rtol 1.0e-9 -ksp_monitor_short -ksp_type richardson -pc_type asm -pc_asm_blocks 4 -pc_asm_overlap 0 -pc_asm_local_type multiplicative -sub_pc_type lu -da_grid_x 8

    813:    test:
    814:      suffix: asm_4
    815:      requires: !single
    816:      nsize: 2
    817:      args: -mms 1 -par 0.0 -snes_monitor_short -snes_converged_reason -snes_view -ksp_rtol 1.0e-9 -ksp_monitor_short -ksp_type richardson -pc_type asm -pc_asm_blocks 2 -pc_asm_overlap 0 -pc_asm_local_type additive -sub_pc_type lu -da_grid_x 8

    819:    test:
    820:      suffix: msm_4
    821:      requires: !single
    822:      nsize: 2
    823:      args: -mms 1 -par 0.0 -snes_monitor_short -snes_converged_reason -snes_view -ksp_rtol 1.0e-9 -ksp_monitor_short -ksp_type richardson -pc_type asm -pc_asm_blocks 2 -pc_asm_overlap 0 -pc_asm_local_type multiplicative -sub_pc_type lu -da_grid_x 8

    825:    test:
    826:      suffix: asm_5
    827:      requires: !single
    828:      nsize: 2
    829:      args: -mms 1 -par 0.0 -snes_monitor_short -snes_converged_reason -snes_view -ksp_rtol 1.0e-9 -ksp_monitor_short -ksp_type richardson -pc_type asm -pc_asm_blocks 4 -pc_asm_overlap 0 -pc_asm_local_type additive -sub_pc_type lu -da_grid_x 8

    831:    test:
    832:      suffix: msm_5
    833:      requires: !single
    834:      nsize: 2
    835:      args: -mms 1 -par 0.0 -snes_monitor_short -snes_converged_reason -snes_view -ksp_rtol 1.0e-9 -ksp_monitor_short -ksp_type richardson -pc_type asm -pc_asm_blocks 4 -pc_asm_overlap 0 -pc_asm_local_type multiplicative -sub_pc_type lu -da_grid_x 8

    837:    test:
    838:      args: -snes_rtol 1.e-5 -pc_type mg -ksp_monitor_short -snes_view -pc_mg_levels 3 -pc_mg_galerkin pmat -da_grid_x 17 -da_grid_y 17 -mg_levels_ksp_monitor_short -mg_levels_ksp_norm_type unpreconditioned -snes_monitor_short -mg_levels_ksp_chebyshev_esteig 0.5,1.1 -mg_levels_pc_type sor -pc_mg_type full
    839:      requires: !single

    841:    test:
    842:      suffix: 2
    843:      args: -pc_type mg -ksp_converged_reason -snes_view -pc_mg_galerkin pmat -snes_grid_sequence 3 -mg_levels_ksp_norm_type unpreconditioned -snes_monitor_short -mg_levels_ksp_chebyshev_esteig 0.5,1.1 -mg_levels_pc_type sor -pc_mg_type full -ksp_atol 0

    845:    test:
    846:      suffix: 3
    847:      nsize: 2
    848:      args: -snes_grid_sequence 2 -snes_mf_operator -snes_converged_reason -snes_view -pc_type mg -snes_atol 0 -snes_rtol 1.e-2
    849:      filter: grep -v "otal number of function evaluations"

    851:    test:
    852:      suffix: 4
    853:      nsize: 2
    854:      args: -snes_grid_sequence 2 -snes_monitor_short -ksp_converged_reason -snes_converged_reason -snes_view -pc_type mg -snes_atol 0 -ksp_atol 0

    856:    test:
    857:      suffix: 5_anderson
    858:      args: -da_grid_x 81 -da_grid_y 81 -snes_monitor_short -snes_max_it 50 -par 6.0 -snes_type anderson

    860:    test:
    861:      suffix: 5_aspin
    862:      nsize: 4
    863:      args: -snes_monitor_short -ksp_monitor_short -snes_converged_reason -da_refine 4 -da_overlap 3 -snes_type aspin -snes_view -npc_sub_pc_type lu -npc_sub_ksp_type preonly

    865:    test:
    866:      suffix: 5_broyden
    867:      args: -da_grid_x 81 -da_grid_y 81 -snes_monitor_short -snes_max_it 50 -par 6.0 -snes_type qn -snes_qn_type broyden -snes_qn_m 10

    869:    test:
    870:      suffix: 5_fas
    871:      args: -fas_coarse_snes_max_it 1 -fas_coarse_pc_type lu -fas_coarse_ksp_type preonly -snes_monitor_short -snes_type fas -fas_coarse_ksp_type richardson -da_refine 6
    872:      requires: !single

    874:    test:
    875:      suffix: 5_fas_additive
    876:      args: -fas_coarse_snes_max_it 1 -fas_coarse_pc_type lu -fas_coarse_ksp_type preonly -snes_monitor_short -snes_type fas -fas_coarse_ksp_type richardson -da_refine 6 -snes_fas_type additive -snes_max_it 50 -snes_linesearch_maxlambda 2.0

    878:    test:
    879:      suffix: 5_fas_monitor
    880:      args: -da_refine 1 -snes_type fas -snes_fas_monitor
    881:      requires: !single

    883:    test:
    884:      suffix: 5_ls
    885:      args: -da_grid_x 81 -da_grid_y 81 -snes_monitor_short -snes_max_it 50 -par 6.0 -snes_type newtonls

    887:    test:
    888:      suffix: 5_ls_sell_sor
    889:      args: -da_grid_x 81 -da_grid_y 81 -snes_monitor_short -snes_max_it 50 -par 6.0 -snes_type newtonls -dm_mat_type sell -pc_type sor
    890:      output_file: output/ex5_5_ls.out

    892:    test:
    893:      suffix: 5_nasm
    894:      nsize: 4
    895:      args: -snes_monitor_short -snes_converged_reason -da_refine 4 -da_overlap 3 -snes_type nasm -snes_nasm_type restrict -snes_max_it 10

    897:    test:
    898:      suffix: 5_ncg
    899:      args: -da_grid_x 81 -da_grid_y 81 -snes_monitor_short -snes_max_it 50 -par 6.0 -snes_type ncg -snes_ncg_type fr

    901:    test:
    902:      suffix: 5_newton_asm_dmda
    903:      nsize: 4
    904:      args: -snes_monitor_short -ksp_monitor_short -snes_converged_reason -da_refine 4 -da_overlap 3 -snes_type newtonls -pc_type asm -pc_asm_dm_subdomains -malloc_dump
    905:      requires: !single

    907:    test:
    908:      suffix: 5_newton_gasm_dmda
    909:      nsize: 4
    910:      args: -snes_monitor_short -ksp_monitor_short -snes_converged_reason -da_refine 4 -da_overlap 3 -snes_type newtonls -pc_type gasm -malloc_dump
    911:      requires: !single

    913:    test:
    914:      suffix: 5_ngmres
    915:      args: -da_grid_x 81 -da_grid_y 81 -snes_monitor_short -snes_max_it 50 -par 6.0 -snes_type ngmres -snes_ngmres_m 10

    917:    test:
    918:      suffix: 5_ngmres_fas
    919:      args: -snes_rtol 1.e-4 -snes_type ngmres -npc_fas_coarse_snes_max_it 1 -npc_fas_coarse_snes_type newtonls -npc_fas_coarse_pc_type lu -npc_fas_coarse_ksp_type preonly -snes_ngmres_m 10 -snes_monitor_short -npc_snes_max_it 1 -npc_snes_type fas -npc_fas_coarse_ksp_type richardson -da_refine 6

    921:    test:
    922:      suffix: 5_ngmres_ngs
    923:      args: -da_grid_x 81 -da_grid_y 81 -snes_monitor_short -snes_max_it 50 -par 6.0 -snes_type ngmres -npc_snes_type ngs -npc_snes_max_it 1

    925:    test:
    926:      suffix: 5_ngmres_nrichardson
    927:      args: -da_grid_x 81 -da_grid_y 81 -snes_monitor_short -snes_max_it 50 -par 6.0 -snes_type ngmres -snes_ngmres_m 10 -npc_snes_type nrichardson -npc_snes_max_it 3
    928:      output_file: output/ex5_5_ngmres_richardson.out

    930:    test:
    931:      suffix: 5_nrichardson
    932:      args: -da_grid_x 81 -da_grid_y 81 -snes_monitor_short -snes_max_it 50 -par 6.0 -snes_type nrichardson

    934:    test:
    935:      suffix: 5_qn
    936:      args: -da_grid_x 81 -da_grid_y 81 -snes_monitor_short -snes_max_it 50 -par 6.0 -snes_type qn -snes_linesearch_type cp -snes_qn_m 10

    938:    test:
    939:      suffix: 6
    940:      nsize: 4
    941:      args: -snes_converged_reason -ksp_converged_reason -da_grid_x 129 -da_grid_y 129 -pc_type mg -pc_mg_levels 8 -mg_levels_ksp_type chebyshev -mg_levels_ksp_chebyshev_esteig 0,0.5,0,1.1 -mg_levels_ksp_max_it 2

    943:    test:
    944:      requires: complex !single
    945:      suffix: complex
    946:      args: -snes_mf_operator -mat_mffd_complex -snes_monitor

    948:    test:
    949:      requires: !single
    950:      suffix: 7_ksp_view_pre
    951:      args: -pc_type gamg -ksp_view_pre

    953:    test:
    954:      requires: !single
    955:      suffix: hem_view_detailed
    956:      args: -pc_type gamg -ksp_view ::ascii_info_detail -pc_gamg_mat_coarsen_type hem

    958:    test:
    959:      requires: !single
    960:      suffix: mis_view_detailed
    961:      args: -pc_type gamg -ksp_view ::ascii_info_detail -pc_gamg_mat_coarsen_type mis

    963:    test:
    964:      requires: !single
    965:      suffix: snes_test_jacobian
    966:      nsize: 4
    967:      args: -snes_test_jacobian -snes_test_jacobian_view

    969: TEST*/