Date: 2022-03-31T04:48:33+00:00

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: /*T
     11:    Concepts: SNES^parallel Bratu example
     12:    Concepts: DMDA^using distributed arrays;
     13:    Concepts: IS coloirng types;
     14:    Processors: n
     15: T*/

     17: /* ------------------------------------------------------------------------

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

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

     24:     with boundary conditions

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

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

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

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

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

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

     44:   ------------------------------------------------------------------------- */

     46: /*
     47:    Include "petscdmda.h" so that we can use distributed arrays (DMDAs).
     48:    Include "petscsnes.h" so that we can use SNES solvers.  Note that this
     49: */
     50: #include <petscdm.h>
     51: #include <petscdmda.h>
     52: #include <petscsnes.h>
     53: #include <petscmatlab.h>
     54: #include <petsc/private/snesimpl.h>

     56: /*
     57:    User-defined application context - contains data needed by the
     58:    application-provided call-back routines, FormJacobianLocal() and
     59:    FormFunctionLocal().
     60: */
     61: typedef struct AppCtx AppCtx;
     62: struct AppCtx {
     63:   PetscReal param;          /* test problem parameter */
     64:   PetscInt  m,n;            /* MMS3 parameters */
     65:   PetscErrorCode (*mms_solution)(AppCtx*,const DMDACoor2d*,PetscScalar*);
     66:   PetscErrorCode (*mms_forcing)(AppCtx*,const DMDACoor2d*,PetscScalar*);
     67: };

     69: /*
     70:    User-defined routines
     71: */
     72: extern PetscErrorCode FormInitialGuess(DM,AppCtx*,Vec);
     73: extern PetscErrorCode FormFunctionLocal(DMDALocalInfo*,PetscScalar**,PetscScalar**,AppCtx*);
     74: extern PetscErrorCode FormExactSolution(DM,AppCtx*,Vec);
     75: extern PetscErrorCode ZeroBCSolution(AppCtx*,const DMDACoor2d*,PetscScalar*);
     76: extern PetscErrorCode MMSSolution1(AppCtx*,const DMDACoor2d*,PetscScalar*);
     77: extern PetscErrorCode MMSForcing1(AppCtx*,const DMDACoor2d*,PetscScalar*);
     78: extern PetscErrorCode MMSSolution2(AppCtx*,const DMDACoor2d*,PetscScalar*);
     79: extern PetscErrorCode MMSForcing2(AppCtx*,const DMDACoor2d*,PetscScalar*);
     80: extern PetscErrorCode MMSSolution3(AppCtx*,const DMDACoor2d*,PetscScalar*);
     81: extern PetscErrorCode MMSForcing3(AppCtx*,const DMDACoor2d*,PetscScalar*);
     82: extern PetscErrorCode MMSSolution4(AppCtx*,const DMDACoor2d*,PetscScalar*);
     83: extern PetscErrorCode MMSForcing4(AppCtx*,const DMDACoor2d*,PetscScalar*);
     84: extern PetscErrorCode FormJacobianLocal(DMDALocalInfo*,PetscScalar**,Mat,Mat,AppCtx*);
     85: extern PetscErrorCode FormObjectiveLocal(DMDALocalInfo*,PetscScalar**,PetscReal*,AppCtx*);
     86: extern PetscErrorCode FormFunctionMatlab(SNES,Vec,Vec,void*);
     87: extern PetscErrorCode NonlinearGS(SNES,Vec,Vec,void*);

     89: int main(int argc,char **argv)
     90: {
     91:   SNES           snes;                         /* nonlinear solver */
     92:   Vec            x;                            /* solution vector */
     93:   AppCtx         user;                         /* user-defined work context */
     94:   PetscInt       its;                          /* iterations for convergence */
     95:   PetscReal      bratu_lambda_max = 6.81;
     96:   PetscReal      bratu_lambda_min = 0.;
     97:   PetscInt       MMS              = 0;
     98:   PetscBool      flg              = PETSC_FALSE;
     99:   DM             da;
    100:   Vec            r               = NULL;
    101:   KSP            ksp;
    102:   PetscInt       lits,slits;

    104:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    105:      Initialize program
    106:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    108:   PetscInitialize(&argc,&argv,(char*)0,help);

    110:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    111:      Initialize problem parameters
    112:   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    113:   user.param = 6.0;
    114:   PetscOptionsGetReal(NULL,NULL,"-par",&user.param,NULL);
    116:   PetscOptionsGetInt(NULL,NULL,"-mms",&MMS,NULL);
    117:   if (MMS == 3) {
    118:     PetscInt mPar = 2, nPar = 1;
    119:     PetscOptionsGetInt(NULL,NULL,"-m_par",&mPar,NULL);
    120:     PetscOptionsGetInt(NULL,NULL,"-n_par",&nPar,NULL);
    121:     user.m = PetscPowInt(2,mPar);
    122:     user.n = PetscPowInt(2,nPar);
    123:   }

    125:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    126:      Create nonlinear solver context
    127:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    128:   SNESCreate(PETSC_COMM_WORLD,&snes);
    129:   SNESSetCountersReset(snes,PETSC_FALSE);
    130:   SNESSetNGS(snes, NonlinearGS, NULL);

    132:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    133:      Create distributed array (DMDA) to manage parallel grid and vectors
    134:   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    135:   DMDACreate2d(PETSC_COMM_WORLD, DM_BOUNDARY_NONE, DM_BOUNDARY_NONE,DMDA_STENCIL_STAR,4,4,PETSC_DECIDE,PETSC_DECIDE,1,1,NULL,NULL,&da);
    136:   DMSetFromOptions(da);
    137:   DMSetUp(da);
    138:   DMDASetUniformCoordinates(da, 0.0, 1.0, 0.0, 1.0, 0.0, 1.0);
    139:   DMSetApplicationContext(da,&user);
    140:   SNESSetDM(snes,da);
    141:   /*  - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    142:      Extract global vectors from DMDA; then duplicate for remaining
    143:      vectors that are the same types
    144:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    145:   DMCreateGlobalVector(da,&x);

    147:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    148:      Set local function evaluation routine
    149:   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    150:   user.mms_solution = ZeroBCSolution;
    151:   switch (MMS) {
    152:   case 0: user.mms_solution = NULL; user.mms_forcing = NULL;
    153:   case 1: user.mms_solution = MMSSolution1; user.mms_forcing = MMSForcing1; break;
    154:   case 2: user.mms_solution = MMSSolution2; user.mms_forcing = MMSForcing2; break;
    155:   case 3: user.mms_solution = MMSSolution3; user.mms_forcing = MMSForcing3; break;
    156:   case 4: user.mms_solution = MMSSolution4; user.mms_forcing = MMSForcing4; break;
    157:   default: SETERRQ(PETSC_COMM_WORLD,PETSC_ERR_USER,"Unknown MMS type %d",MMS);
    158:   }
    159:   DMDASNESSetFunctionLocal(da,INSERT_VALUES,(DMDASNESFunction)FormFunctionLocal,&user);
    160:   PetscOptionsGetBool(NULL,NULL,"-fd",&flg,NULL);
    161:   if (!flg) {
    162:     DMDASNESSetJacobianLocal(da,(DMDASNESJacobian)FormJacobianLocal,&user);
    163:   }

    165:   PetscOptionsGetBool(NULL,NULL,"-obj",&flg,NULL);
    166:   if (flg) {
    167:     DMDASNESSetObjectiveLocal(da,(DMDASNESObjective)FormObjectiveLocal,&user);
    168:   }

    170:   if (PetscDefined(HAVE_MATLAB_ENGINE)) {
    171:     PetscBool matlab_function = PETSC_FALSE;
    172:     PetscOptionsGetBool(NULL,NULL,"-matlab_function",&matlab_function,0);
    173:     if (matlab_function) {
    174:       VecDuplicate(x,&r);
    175:       SNESSetFunction(snes,r,FormFunctionMatlab,&user);
    176:     }
    177:   }

    179:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    180:      Customize nonlinear solver; set runtime options
    181:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    182:   SNESSetFromOptions(snes);

    184:   FormInitialGuess(da,&user,x);

    186:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    187:      Solve nonlinear system
    188:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    189:   SNESSolve(snes,NULL,x);
    190:   SNESGetIterationNumber(snes,&its);

    192:   SNESGetLinearSolveIterations(snes,&slits);
    193:   SNESGetKSP(snes,&ksp);
    194:   KSPGetTotalIterations(ksp,&lits);
    196:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    197:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    199:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    200:      If using MMS, check the l_2 error
    201:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    202:   if (MMS) {
    203:     Vec       e;
    204:     PetscReal errorl2, errorinf;
    205:     PetscInt  N;

    207:     VecDuplicate(x, &e);
    208:     PetscObjectViewFromOptions((PetscObject) x, NULL, "-sol_view");
    209:     FormExactSolution(da, &user, e);
    210:     PetscObjectViewFromOptions((PetscObject) e, NULL, "-exact_view");
    211:     VecAXPY(e, -1.0, x);
    212:     PetscObjectViewFromOptions((PetscObject) e, NULL, "-error_view");
    213:     VecNorm(e, NORM_2, &errorl2);
    214:     VecNorm(e, NORM_INFINITY, &errorinf);
    215:     VecGetSize(e, &N);
    216:     PetscPrintf(PETSC_COMM_WORLD, "N: %D error L2 %g inf %g\n", N, (double) errorl2/PetscSqrtReal((PetscReal)N), (double) errorinf);
    217:     VecDestroy(&e);
    218:     PetscLogEventSetDof(SNES_Solve, 0, N);
    219:     PetscLogEventSetError(SNES_Solve, 0, errorl2/PetscSqrtReal(N));
    220:   }

    222:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    223:      Free work space.  All PETSc objects should be destroyed when they
    224:      are no longer needed.
    225:    - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */
    226:   VecDestroy(&r);
    227:   VecDestroy(&x);
    228:   SNESDestroy(&snes);
    229:   DMDestroy(&da);
    230:   PetscFinalize();
    231:   return 0;
    232: }
    233: /* ------------------------------------------------------------------- */
    234: /*
    235:    FormInitialGuess - Forms initial approximation.

    237:    Input Parameters:
    238:    da - The DM
    239:    user - user-defined application context

    241:    Output Parameter:
    242:    X - vector
    243:  */
    244: PetscErrorCode FormInitialGuess(DM da,AppCtx *user,Vec X)
    245: {
    246:   PetscInt       i,j,Mx,My,xs,ys,xm,ym;
    247:   PetscReal      lambda,temp1,temp,hx,hy;
    248:   PetscScalar    **x;

    251:   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);

    253:   lambda = user->param;
    254:   hx     = 1.0/(PetscReal)(Mx-1);
    255:   hy     = 1.0/(PetscReal)(My-1);
    256:   temp1  = lambda/(lambda + 1.0);

    258:   /*
    259:      Get a pointer to vector data.
    260:        - For default PETSc vectors, VecGetArray() returns a pointer to
    261:          the data array.  Otherwise, the routine is implementation dependent.
    262:        - You MUST call VecRestoreArray() when you no longer need access to
    263:          the array.
    264:   */
    265:   DMDAVecGetArray(da,X,&x);

    267:   /*
    268:      Get local grid boundaries (for 2-dimensional DMDA):
    269:        xs, ys   - starting grid indices (no ghost points)
    270:        xm, ym   - widths of local grid (no ghost points)

    272:   */
    273:   DMDAGetCorners(da,&xs,&ys,NULL,&xm,&ym,NULL);

    275:   /*
    276:      Compute initial guess over the locally owned part of the grid
    277:   */
    278:   for (j=ys; j<ys+ym; j++) {
    279:     temp = (PetscReal)(PetscMin(j,My-j-1))*hy;
    280:     for (i=xs; i<xs+xm; i++) {
    281:       if (i == 0 || j == 0 || i == Mx-1 || j == My-1) {
    282:         /* boundary conditions are all zero Dirichlet */
    283:         x[j][i] = 0.0;
    284:       } else {
    285:         x[j][i] = temp1*PetscSqrtReal(PetscMin((PetscReal)(PetscMin(i,Mx-i-1))*hx,temp));
    286:       }
    287:     }
    288:   }

    290:   /*
    291:      Restore vector
    292:   */
    293:   DMDAVecRestoreArray(da,X,&x);
    294:   return 0;
    295: }

    297: /*
    298:   FormExactSolution - Forms MMS solution

    300:   Input Parameters:
    301:   da - The DM
    302:   user - user-defined application context

    304:   Output Parameter:
    305:   X - vector
    306:  */
    307: PetscErrorCode FormExactSolution(DM da, AppCtx *user, Vec U)
    308: {
    309:   DM             coordDA;
    310:   Vec            coordinates;
    311:   DMDACoor2d   **coords;
    312:   PetscScalar  **u;
    313:   PetscInt       xs, ys, xm, ym, i, j;

    316:   DMDAGetCorners(da, &xs, &ys, NULL, &xm, &ym, NULL);
    317:   DMGetCoordinateDM(da, &coordDA);
    318:   DMGetCoordinates(da, &coordinates);
    319:   DMDAVecGetArray(coordDA, coordinates, &coords);
    320:   DMDAVecGetArray(da, U, &u);
    321:   for (j = ys; j < ys+ym; ++j) {
    322:     for (i = xs; i < xs+xm; ++i) {
    323:       user->mms_solution(user,&coords[j][i],&u[j][i]);
    324:     }
    325:   }
    326:   DMDAVecRestoreArray(da, U, &u);
    327:   DMDAVecRestoreArray(coordDA, coordinates, &coords);
    328:   return 0;
    329: }

    331: PetscErrorCode ZeroBCSolution(AppCtx *user,const DMDACoor2d *c,PetscScalar *u)
    332: {
    333:   u[0] = 0.;
    334:   return 0;
    335: }

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

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

    341:   such that u(x,y) is an exact solution with f(x,y) as the right hand side forcing term.
    342:  */
    343: PetscErrorCode MMSSolution1(AppCtx *user,const DMDACoor2d *c,PetscScalar *u)
    344: {
    345:   PetscReal x = PetscRealPart(c->x), y = PetscRealPart(c->y);
    346:   u[0] = x*(1 - x)*y*(1 - y);
    347:   PetscLogFlops(5);
    348:   return 0;
    349: }
    350: PetscErrorCode MMSForcing1(AppCtx *user,const DMDACoor2d *c,PetscScalar *f)
    351: {
    352:   PetscReal x = PetscRealPart(c->x), y = PetscRealPart(c->y);
    353:   f[0] = 2*x*(1 - x) + 2*y*(1 - y) - user->param*PetscExpReal(x*(1 - x)*y*(1 - y));
    354:   return 0;
    355: }

    357: PetscErrorCode MMSSolution2(AppCtx *user,const DMDACoor2d *c,PetscScalar *u)
    358: {
    359:   PetscReal x = PetscRealPart(c->x), y = PetscRealPart(c->y);
    360:   u[0] = PetscSinReal(PETSC_PI*x)*PetscSinReal(PETSC_PI*y);
    361:   PetscLogFlops(5);
    362:   return 0;
    363: }
    364: PetscErrorCode MMSForcing2(AppCtx *user,const DMDACoor2d *c,PetscScalar *f)
    365: {
    366:   PetscReal x = PetscRealPart(c->x), y = PetscRealPart(c->y);
    367:   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));
    368:   return 0;
    369: }

    371: PetscErrorCode MMSSolution3(AppCtx *user,const DMDACoor2d *c,PetscScalar *u)
    372: {
    373:   PetscReal x = PetscRealPart(c->x), y = PetscRealPart(c->y);
    374:   u[0] = PetscSinReal(user->m*PETSC_PI*x*(1-y))*PetscSinReal(user->n*PETSC_PI*y*(1-x));
    375:   PetscLogFlops(5);
    376:   return 0;
    377: }
    378: PetscErrorCode MMSForcing3(AppCtx *user,const DMDACoor2d *c,PetscScalar *f)
    379: {
    380:   PetscReal x = PetscRealPart(c->x), y = PetscRealPart(c->y);
    381:   PetscReal m = user->m, n = user->n, lambda = user->param;
    382:   f[0] = (-(PetscExpReal(PetscSinReal(m*PETSC_PI*x*(1 - y))*PetscSinReal(n*PETSC_PI*(1 - x)*y))*lambda)
    383:           + 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)
    384:                                 + (PetscSqr(m)*(PetscSqr(x) + PetscSqr(-1 + y)) + PetscSqr(n)*(PetscSqr(-1 + x) + PetscSqr(y)))
    385:                                 *PetscSinReal(m*PETSC_PI*x*(-1 + y))*PetscSinReal(n*PETSC_PI*(-1 + x)*y)));
    386:   return 0;
    387: }

    389: PetscErrorCode MMSSolution4(AppCtx *user,const DMDACoor2d *c,PetscScalar *u)
    390: {
    391:   const PetscReal Lx = 1.,Ly = 1.;
    392:   PetscReal x = PetscRealPart(c->x), y = PetscRealPart(c->y);
    393:   u[0] = (PetscPowReal(x,4)-PetscSqr(Lx)*PetscSqr(x))*(PetscPowReal(y,4)-PetscSqr(Ly)*PetscSqr(y));
    394:   PetscLogFlops(9);
    395:   return 0;
    396: }
    397: PetscErrorCode MMSForcing4(AppCtx *user,const DMDACoor2d *c,PetscScalar *f)
    398: {
    399:   const PetscReal Lx = 1.,Ly = 1.;
    400:   PetscReal x = PetscRealPart(c->x), y = PetscRealPart(c->y);
    401:   f[0] = (2*PetscSqr(x)*(PetscSqr(x)-PetscSqr(Lx))*(PetscSqr(Ly)-6*PetscSqr(y))
    402:           + 2*PetscSqr(y)*(PetscSqr(Lx)-6*PetscSqr(x))*(PetscSqr(y)-PetscSqr(Ly))
    403:           - user->param*PetscExpReal((PetscPowReal(x,4)-PetscSqr(Lx)*PetscSqr(x))*(PetscPowReal(y,4)-PetscSqr(Ly)*PetscSqr(y))));
    404:   return 0;
    405: }

    407: /* ------------------------------------------------------------------- */
    408: /*
    409:    FormFunctionLocal - Evaluates nonlinear function, F(x) on local process patch

    411:  */
    412: PetscErrorCode FormFunctionLocal(DMDALocalInfo *info,PetscScalar **x,PetscScalar **f,AppCtx *user)
    413: {
    414:   PetscInt       i,j;
    415:   PetscReal      lambda,hx,hy,hxdhy,hydhx;
    416:   PetscScalar    u,ue,uw,un,us,uxx,uyy,mms_solution,mms_forcing;
    417:   DMDACoor2d     c;

    420:   lambda = user->param;
    421:   hx     = 1.0/(PetscReal)(info->mx-1);
    422:   hy     = 1.0/(PetscReal)(info->my-1);
    423:   hxdhy  = hx/hy;
    424:   hydhx  = hy/hx;
    425:   /*
    426:      Compute function over the locally owned part of the grid
    427:   */
    428:   for (j=info->ys; j<info->ys+info->ym; j++) {
    429:     for (i=info->xs; i<info->xs+info->xm; i++) {
    430:       if (i == 0 || j == 0 || i == info->mx-1 || j == info->my-1) {
    431:         c.x = i*hx; c.y = j*hy;
    432:         user->mms_solution(user,&c,&mms_solution);
    433:         f[j][i] = 2.0*(hydhx+hxdhy)*(x[j][i] - mms_solution);
    434:       } else {
    435:         u  = x[j][i];
    436:         uw = x[j][i-1];
    437:         ue = x[j][i+1];
    438:         un = x[j-1][i];
    439:         us = x[j+1][i];

    441:         /* Enforce boundary conditions at neighboring points -- setting these values causes the Jacobian to be symmetric. */
    442:         if (i-1 == 0) {c.x = (i-1)*hx; c.y = j*hy; user->mms_solution(user,&c,&uw);}
    443:         if (i+1 == info->mx-1) {c.x = (i+1)*hx; c.y = j*hy; user->mms_solution(user,&c,&ue);}
    444:         if (j-1 == 0) {c.x = i*hx; c.y = (j-1)*hy; user->mms_solution(user,&c,&un);}
    445:         if (j+1 == info->my-1) {c.x = i*hx; c.y = (j+1)*hy; user->mms_solution(user,&c,&us);}

    447:         uxx     = (2.0*u - uw - ue)*hydhx;
    448:         uyy     = (2.0*u - un - us)*hxdhy;
    449:         mms_forcing = 0;
    450:         c.x = i*hx; c.y = j*hy;
    451:         if (user->mms_forcing) user->mms_forcing(user,&c,&mms_forcing);
    452:         f[j][i] = uxx + uyy - hx*hy*(lambda*PetscExpScalar(u) + mms_forcing);
    453:       }
    454:     }
    455:   }
    456:   PetscLogFlops(11.0*info->ym*info->xm);
    457:   return 0;
    458: }

    460: /* FormObjectiveLocal - Evaluates nonlinear function, F(x) on local process patch */
    461: PetscErrorCode FormObjectiveLocal(DMDALocalInfo *info,PetscScalar **x,PetscReal *obj,AppCtx *user)
    462: {
    463:   PetscInt       i,j;
    464:   PetscReal      lambda,hx,hy,hxdhy,hydhx,sc,lobj=0;
    465:   PetscScalar    u,ue,uw,un,us,uxux,uyuy;
    466:   MPI_Comm       comm;

    469:   *obj   = 0;
    470:   PetscObjectGetComm((PetscObject)info->da,&comm);
    471:   lambda = user->param;
    472:   hx     = 1.0/(PetscReal)(info->mx-1);
    473:   hy     = 1.0/(PetscReal)(info->my-1);
    474:   sc     = hx*hy*lambda;
    475:   hxdhy  = hx/hy;
    476:   hydhx  = hy/hx;
    477:   /*
    478:      Compute function over the locally owned part of the grid
    479:   */
    480:   for (j=info->ys; j<info->ys+info->ym; j++) {
    481:     for (i=info->xs; i<info->xs+info->xm; i++) {
    482:       if (i == 0 || j == 0 || i == info->mx-1 || j == info->my-1) {
    483:         lobj += PetscRealPart((hydhx + hxdhy)*x[j][i]*x[j][i]);
    484:       } else {
    485:         u  = x[j][i];
    486:         uw = x[j][i-1];
    487:         ue = x[j][i+1];
    488:         un = x[j-1][i];
    489:         us = x[j+1][i];

    491:         if (i-1 == 0) uw = 0.;
    492:         if (i+1 == info->mx-1) ue = 0.;
    493:         if (j-1 == 0) un = 0.;
    494:         if (j+1 == info->my-1) us = 0.;

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

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

    501:         lobj += PetscRealPart(0.5*(uxux + uyuy) - sc*PetscExpScalar(u));
    502:       }
    503:     }
    504:   }
    505:   PetscLogFlops(12.0*info->ym*info->xm);
    506:   MPI_Allreduce(&lobj,obj,1,MPIU_REAL,MPIU_SUM,comm);
    507:   return 0;
    508: }

    510: /*
    511:    FormJacobianLocal - Evaluates Jacobian matrix on local process patch
    512: */
    513: PetscErrorCode FormJacobianLocal(DMDALocalInfo *info,PetscScalar **x,Mat jac,Mat jacpre,AppCtx *user)
    514: {
    515:   PetscInt       i,j,k;
    516:   MatStencil     col[5],row;
    517:   PetscScalar    lambda,v[5],hx,hy,hxdhy,hydhx,sc;
    518:   DM             coordDA;
    519:   Vec            coordinates;
    520:   DMDACoor2d   **coords;

    523:   lambda = user->param;
    524:   /* Extract coordinates */
    525:   DMGetCoordinateDM(info->da, &coordDA);
    526:   DMGetCoordinates(info->da, &coordinates);
    527:   DMDAVecGetArray(coordDA, coordinates, &coords);
    528:   hx     = info->xm > 1 ? PetscRealPart(coords[info->ys][info->xs+1].x) - PetscRealPart(coords[info->ys][info->xs].x) : 1.0;
    529:   hy     = info->ym > 1 ? PetscRealPart(coords[info->ys+1][info->xs].y) - PetscRealPart(coords[info->ys][info->xs].y) : 1.0;
    530:   DMDAVecRestoreArray(coordDA, coordinates, &coords);
    531:   hxdhy  = hx/hy;
    532:   hydhx  = hy/hx;
    533:   sc     = hx*hy*lambda;

    535:   /*
    536:      Compute entries for the locally owned part of the Jacobian.
    537:       - Currently, all PETSc parallel matrix formats are partitioned by
    538:         contiguous chunks of rows across the processors.
    539:       - Each processor needs to insert only elements that it owns
    540:         locally (but any non-local elements will be sent to the
    541:         appropriate processor during matrix assembly).
    542:       - Here, we set all entries for a particular row at once.
    543:       - We can set matrix entries either using either
    544:         MatSetValuesLocal() or MatSetValues(), as discussed above.
    545:   */
    546:   for (j=info->ys; j<info->ys+info->ym; j++) {
    547:     for (i=info->xs; i<info->xs+info->xm; i++) {
    548:       row.j = j; row.i = i;
    549:       /* boundary points */
    550:       if (i == 0 || j == 0 || i == info->mx-1 || j == info->my-1) {
    551:         v[0] =  2.0*(hydhx + hxdhy);
    552:         MatSetValuesStencil(jacpre,1,&row,1,&row,v,INSERT_VALUES);
    553:       } else {
    554:         k = 0;
    555:         /* interior grid points */
    556:         if (j-1 != 0) {
    557:           v[k]     = -hxdhy;
    558:           col[k].j = j - 1; col[k].i = i;
    559:           k++;
    560:         }
    561:         if (i-1 != 0) {
    562:           v[k]     = -hydhx;
    563:           col[k].j = j;     col[k].i = i-1;
    564:           k++;
    565:         }

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

    569:         if (i+1 != info->mx-1) {
    570:           v[k]     = -hydhx;
    571:           col[k].j = j;     col[k].i = i+1;
    572:           k++;
    573:         }
    574:         if (j+1 != info->mx-1) {
    575:           v[k]     = -hxdhy;
    576:           col[k].j = j + 1; col[k].i = i;
    577:           k++;
    578:         }
    579:         MatSetValuesStencil(jacpre,1,&row,k,col,v,INSERT_VALUES);
    580:       }
    581:     }
    582:   }

    584:   /*
    585:      Assemble matrix, using the 2-step process:
    586:        MatAssemblyBegin(), MatAssemblyEnd().
    587:   */
    588:   MatAssemblyBegin(jacpre,MAT_FINAL_ASSEMBLY);
    589:   MatAssemblyEnd(jacpre,MAT_FINAL_ASSEMBLY);
    590:   /*
    591:      Tell the matrix we will never add a new nonzero location to the
    592:      matrix. If we do, it will generate an error.
    593:   */
    594:   MatSetOption(jac,MAT_NEW_NONZERO_LOCATION_ERR,PETSC_TRUE);
    595:   return 0;
    596: }

    598: PetscErrorCode FormFunctionMatlab(SNES snes,Vec X,Vec F,void *ptr)
    599: {
    600: #if PetscDefined(HAVE_MATLAB_ENGINE)
    601:   AppCtx         *user = (AppCtx*)ptr;
    602:   PetscInt       Mx,My;
    603:   PetscReal      lambda,hx,hy;
    604:   Vec            localX,localF;
    605:   MPI_Comm       comm;
    606:   DM             da;

    609:   SNESGetDM(snes,&da);
    610:   DMGetLocalVector(da,&localX);
    611:   DMGetLocalVector(da,&localF);
    612:   PetscObjectSetName((PetscObject)localX,"localX");
    613:   PetscObjectSetName((PetscObject)localF,"localF");
    614:   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);

    616:   lambda = user->param;
    617:   hx     = 1.0/(PetscReal)(Mx-1);
    618:   hy     = 1.0/(PetscReal)(My-1);

    620:   PetscObjectGetComm((PetscObject)snes,&comm);
    621:   /*
    622:      Scatter ghost points to local vector,using the 2-step process
    623:         DMGlobalToLocalBegin(),DMGlobalToLocalEnd().
    624:      By placing code between these two statements, computations can be
    625:      done while messages are in transition.
    626:   */
    627:   DMGlobalToLocalBegin(da,X,INSERT_VALUES,localX);
    628:   DMGlobalToLocalEnd(da,X,INSERT_VALUES,localX);
    629:   PetscMatlabEnginePut(PETSC_MATLAB_ENGINE_(comm),(PetscObject)localX);
    630:   PetscMatlabEngineEvaluate(PETSC_MATLAB_ENGINE_(comm),"localF=ex5m(localX,%18.16e,%18.16e,%18.16e)",hx,hy,lambda);
    631:   PetscMatlabEngineGet(PETSC_MATLAB_ENGINE_(comm),(PetscObject)localF);

    633:   /*
    634:      Insert values into global vector
    635:   */
    636:   DMLocalToGlobalBegin(da,localF,INSERT_VALUES,F);
    637:   DMLocalToGlobalEnd(da,localF,INSERT_VALUES,F);
    638:   DMRestoreLocalVector(da,&localX);
    639:   DMRestoreLocalVector(da,&localF);
    640:   return 0;
    641: #else
    642:     return 0;                     /* Never called */
    643: #endif
    644: }

    646: /* ------------------------------------------------------------------- */
    647: /*
    648:       Applies some sweeps on nonlinear Gauss-Seidel on each process

    650:  */
    651: PetscErrorCode NonlinearGS(SNES snes,Vec X, Vec B, void *ctx)
    652: {
    653:   PetscInt       i,j,k,Mx,My,xs,ys,xm,ym,its,tot_its,sweeps,l;
    654:   PetscReal      lambda,hx,hy,hxdhy,hydhx,sc;
    655:   PetscScalar    **x,**b,bij,F,F0=0,J,u,un,us,ue,eu,uw,uxx,uyy,y;
    656:   PetscReal      atol,rtol,stol;
    657:   DM             da;
    658:   AppCtx         *user;
    659:   Vec            localX,localB;

    662:   tot_its = 0;
    663:   SNESNGSGetSweeps(snes,&sweeps);
    664:   SNESNGSGetTolerances(snes,&atol,&rtol,&stol,&its);
    665:   SNESGetDM(snes,&da);
    666:   DMGetApplicationContext(da,&user);

    668:   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);

    670:   lambda = user->param;
    671:   hx     = 1.0/(PetscReal)(Mx-1);
    672:   hy     = 1.0/(PetscReal)(My-1);
    673:   sc     = hx*hy*lambda;
    674:   hxdhy  = hx/hy;
    675:   hydhx  = hy/hx;

    677:   DMGetLocalVector(da,&localX);
    678:   if (B) {
    679:     DMGetLocalVector(da,&localB);
    680:   }
    681:   for (l=0; l<sweeps; l++) {

    683:     DMGlobalToLocalBegin(da,X,INSERT_VALUES,localX);
    684:     DMGlobalToLocalEnd(da,X,INSERT_VALUES,localX);
    685:     if (B) {
    686:       DMGlobalToLocalBegin(da,B,INSERT_VALUES,localB);
    687:       DMGlobalToLocalEnd(da,B,INSERT_VALUES,localB);
    688:     }
    689:     /*
    690:      Get a pointer to vector data.
    691:      - For default PETSc vectors, VecGetArray() returns a pointer to
    692:      the data array.  Otherwise, the routine is implementation dependent.
    693:      - You MUST call VecRestoreArray() when you no longer need access to
    694:      the array.
    695:      */
    696:     DMDAVecGetArray(da,localX,&x);
    697:     if (B) DMDAVecGetArray(da,localB,&b);
    698:     /*
    699:      Get local grid boundaries (for 2-dimensional DMDA):
    700:      xs, ys   - starting grid indices (no ghost points)
    701:      xm, ym   - widths of local grid (no ghost points)
    702:      */
    703:     DMDAGetCorners(da,&xs,&ys,NULL,&xm,&ym,NULL);

    705:     for (j=ys; j<ys+ym; j++) {
    706:       for (i=xs; i<xs+xm; i++) {
    707:         if (i == 0 || j == 0 || i == Mx-1 || j == My-1) {
    708:           /* boundary conditions are all zero Dirichlet */
    709:           x[j][i] = 0.0;
    710:         } else {
    711:           if (B) bij = b[j][i];
    712:           else   bij = 0.;

    714:           u  = x[j][i];
    715:           un = x[j-1][i];
    716:           us = x[j+1][i];
    717:           ue = x[j][i-1];
    718:           uw = x[j][i+1];

    720:           for (k=0; k<its; k++) {
    721:             eu  = PetscExpScalar(u);
    722:             uxx = (2.0*u - ue - uw)*hydhx;
    723:             uyy = (2.0*u - un - us)*hxdhy;
    724:             F   = uxx + uyy - sc*eu - bij;
    725:             if (k == 0) F0 = F;
    726:             J  = 2.0*(hydhx + hxdhy) - sc*eu;
    727:             y  = F/J;
    728:             u -= y;
    729:             tot_its++;

    731:             if (atol > PetscAbsReal(PetscRealPart(F)) ||
    732:                 rtol*PetscAbsReal(PetscRealPart(F0)) > PetscAbsReal(PetscRealPart(F)) ||
    733:                 stol*PetscAbsReal(PetscRealPart(u)) > PetscAbsReal(PetscRealPart(y))) {
    734:               break;
    735:             }
    736:           }
    737:           x[j][i] = u;
    738:         }
    739:       }
    740:     }
    741:     /*
    742:      Restore vector
    743:      */
    744:     DMDAVecRestoreArray(da,localX,&x);
    745:     DMLocalToGlobalBegin(da,localX,INSERT_VALUES,X);
    746:     DMLocalToGlobalEnd(da,localX,INSERT_VALUES,X);
    747:   }
    748:   PetscLogFlops(tot_its*(21.0));
    749:   DMRestoreLocalVector(da,&localX);
    750:   if (B) {
    751:     DMDAVecRestoreArray(da,localB,&b);
    752:     DMRestoreLocalVector(da,&localB);
    753:   }
    754:   return 0;
    755: }

    757: /*TEST

    759:    test:
    760:      suffix: asm_0
    761:      requires: !single
    762:      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

    764:    test:
    765:      suffix: msm_0
    766:      requires: !single
    767:      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

    769:    test:
    770:      suffix: asm_1
    771:      requires: !single
    772:      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

    774:    test:
    775:      suffix: msm_1
    776:      requires: !single
    777:      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

    779:    test:
    780:      suffix: asm_2
    781:      requires: !single
    782:      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

    784:    test:
    785:      suffix: msm_2
    786:      requires: !single
    787:      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

    789:    test:
    790:      suffix: asm_3
    791:      requires: !single
    792:      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

    794:    test:
    795:      suffix: msm_3
    796:      requires: !single
    797:      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

    799:    test:
    800:      suffix: asm_4
    801:      requires: !single
    802:      nsize: 2
    803:      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

    805:    test:
    806:      suffix: msm_4
    807:      requires: !single
    808:      nsize: 2
    809:      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

    811:    test:
    812:      suffix: asm_5
    813:      requires: !single
    814:      nsize: 2
    815:      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

    817:    test:
    818:      suffix: msm_5
    819:      requires: !single
    820:      nsize: 2
    821:      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

    823:    test:
    824:      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
    825:      requires: !single

    827:    test:
    828:      suffix: 2
    829:      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 -1.

    831:    test:
    832:      suffix: 3
    833:      nsize: 2
    834:      args: -snes_grid_sequence 2 -snes_mf_operator -snes_converged_reason -snes_view -pc_type mg -snes_atol -1 -snes_rtol 1.e-2
    835:      filter: grep -v "otal number of function evaluations"

    837:    test:
    838:      suffix: 4
    839:      nsize: 2
    840:      args: -snes_grid_sequence 2 -snes_monitor_short -ksp_converged_reason -snes_converged_reason -snes_view -pc_type mg -snes_atol -1 -ksp_atol -1

    842:    test:
    843:      suffix: 5_anderson
    844:      args: -da_grid_x 81 -da_grid_y 81 -snes_monitor_short -snes_max_it 50 -par 6.0 -snes_type anderson

    846:    test:
    847:      suffix: 5_aspin
    848:      nsize: 4
    849:      args: -snes_monitor_short -ksp_monitor_short -snes_converged_reason -da_refine 4 -da_overlap 3 -snes_type aspin -snes_view

    851:    test:
    852:      suffix: 5_broyden
    853:      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

    855:    test:
    856:      suffix: 5_fas
    857:      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
    858:      requires: !single

    860:    test:
    861:      suffix: 5_fas_additive
    862:      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

    864:    test:
    865:      suffix: 5_fas_monitor
    866:      args: -da_refine 1 -snes_type fas -snes_fas_monitor
    867:      requires: !single

    869:    test:
    870:      suffix: 5_ls
    871:      args: -da_grid_x 81 -da_grid_y 81 -snes_monitor_short -snes_max_it 50 -par 6.0 -snes_type newtonls

    873:    test:
    874:      suffix: 5_ls_sell_sor
    875:      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
    876:      output_file: output/ex5_5_ls.out

    878:    test:
    879:      suffix: 5_nasm
    880:      nsize: 4
    881:      args: -snes_monitor_short -snes_converged_reason -da_refine 4 -da_overlap 3 -snes_type nasm -snes_nasm_type restrict -snes_max_it 10

    883:    test:
    884:      suffix: 5_ncg
    885:      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

    887:    test:
    888:      suffix: 5_newton_asm_dmda
    889:      nsize: 4
    890:      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
    891:      requires: !single

    893:    test:
    894:      suffix: 5_newton_gasm_dmda
    895:      nsize: 4
    896:      args: -snes_monitor_short -ksp_monitor_short -snes_converged_reason -da_refine 4 -da_overlap 3 -snes_type newtonls -pc_type gasm -malloc_dump
    897:      requires: !single

    899:    test:
    900:      suffix: 5_ngmres
    901:      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

    903:    test:
    904:      suffix: 5_ngmres_fas
    905:      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

    907:    test:
    908:      suffix: 5_ngmres_ngs
    909:      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

    911:    test:
    912:      suffix: 5_ngmres_nrichardson
    913:      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
    914:      output_file: output/ex5_5_ngmres_richardson.out

    916:    test:
    917:      suffix: 5_nrichardson
    918:      args: -da_grid_x 81 -da_grid_y 81 -snes_monitor_short -snes_max_it 50 -par 6.0 -snes_type nrichardson

    920:    test:
    921:      suffix: 5_qn
    922:      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

    924:    test:
    925:      suffix: 6
    926:      nsize: 4
    927:      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

    929:    test:
    930:      requires: complex !single
    931:      suffix: complex
    932:      args: -snes_mf_operator -mat_mffd_complex -snes_monitor

    934: TEST*/