1: static const char help[] = "Minimum surface problem in 2D.\n\
  2: Uses 2-dimensional distributed arrays.\n\
  3: \n\
  4:   Solves the linear systems via multilevel methods \n\
  5: \n\n";

  7: /*

  9:     This example models the partial differential equation

 11:          - Div((1 + ||GRAD T||^2)^(1/2) (GRAD T)) = 0.

 13:     in the unit square, which is uniformly discretized in each of x and
 14:     y in this simple encoding.  The degrees of freedom are vertex centered

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

 20: */

 22: #include <petscsnes.h>
 23: #include <petscdm.h>
 24: #include <petscdmda.h>

 26: extern PetscErrorCode FormFunctionLocal(DMDALocalInfo *, PetscScalar **, PetscScalar **, void *);

 28: int main(int argc, char **argv)
 29: {
 30:   SNES      snes;
 31:   PetscInt  its, lits;
 32:   PetscReal litspit;
 33:   DM        da;

 35:   PetscFunctionBeginUser;
 36:   PetscCall(PetscInitialize(&argc, &argv, NULL, help));
 37:   /*
 38:       Set the DMDA (grid structure) for the grids.
 39:   */
 40:   PetscCall(DMDACreate2d(PETSC_COMM_WORLD, DM_BOUNDARY_NONE, DM_BOUNDARY_NONE, DMDA_STENCIL_STAR, 5, 5, PETSC_DECIDE, PETSC_DECIDE, 1, 1, 0, 0, &da));
 41:   PetscCall(DMSetFromOptions(da));
 42:   PetscCall(DMSetUp(da));
 43:   PetscCall(DMDASNESSetFunctionLocal(da, INSERT_VALUES, (PetscErrorCode (*)(DMDALocalInfo *, void *, void *, void *))FormFunctionLocal, NULL));
 44:   PetscCall(SNESCreate(PETSC_COMM_WORLD, &snes));
 45:   PetscCall(SNESSetDM(snes, da));
 46:   PetscCall(DMDestroy(&da));

 48:   PetscCall(SNESSetFromOptions(snes));

 50:   PetscCall(SNESSolve(snes, 0, 0));
 51:   PetscCall(SNESGetIterationNumber(snes, &its));
 52:   PetscCall(SNESGetLinearSolveIterations(snes, &lits));
 53:   litspit = ((PetscReal)lits) / ((PetscReal)its);
 54:   PetscCall(PetscPrintf(PETSC_COMM_WORLD, "Number of SNES iterations = %" PetscInt_FMT "\n", its));
 55:   PetscCall(PetscPrintf(PETSC_COMM_WORLD, "Number of Linear iterations = %" PetscInt_FMT "\n", lits));
 56:   PetscCall(PetscPrintf(PETSC_COMM_WORLD, "Average Linear its / SNES = %e\n", (double)litspit));

 58:   PetscCall(SNESDestroy(&snes));
 59:   PetscCall(PetscFinalize());
 60:   return 0;
 61: }

 63: PetscErrorCode FormFunctionLocal(DMDALocalInfo *info, PetscScalar **t, PetscScalar **f, void *ptr)
 64: {
 65:   PetscInt    i, j;
 66:   PetscScalar hx, hy;
 67:   PetscScalar gradup, graddown, gradleft, gradright, gradx, grady;
 68:   PetscScalar coeffup, coeffdown, coeffleft, coeffright;

 70:   PetscFunctionBeginUser;
 71:   hx = 1.0 / (PetscReal)(info->mx - 1);
 72:   hy = 1.0 / (PetscReal)(info->my - 1);

 74:   /* Evaluate function */
 75:   for (j = info->ys; j < info->ys + info->ym; j++) {
 76:     for (i = info->xs; i < info->xs + info->xm; i++) {
 77:       if (i == 0 || i == info->mx - 1 || j == 0 || j == info->my - 1) {
 78:         f[j][i] = t[j][i] - (1.0 - (2.0 * hx * (PetscReal)i - 1.0) * (2.0 * hx * (PetscReal)i - 1.0));
 79:       } else {
 80:         gradup    = (t[j + 1][i] - t[j][i]) / hy;
 81:         graddown  = (t[j][i] - t[j - 1][i]) / hy;
 82:         gradright = (t[j][i + 1] - t[j][i]) / hx;
 83:         gradleft  = (t[j][i] - t[j][i - 1]) / hx;

 85:         gradx = .5 * (t[j][i + 1] - t[j][i - 1]) / hx;
 86:         grady = .5 * (t[j + 1][i] - t[j - 1][i]) / hy;

 88:         coeffup   = 1.0 / PetscSqrtScalar(1.0 + gradup * gradup + gradx * gradx);
 89:         coeffdown = 1.0 / PetscSqrtScalar(1.0 + graddown * graddown + gradx * gradx);

 91:         coeffleft  = 1.0 / PetscSqrtScalar(1.0 + gradleft * gradleft + grady * grady);
 92:         coeffright = 1.0 / PetscSqrtScalar(1.0 + gradright * gradright + grady * grady);

 94:         f[j][i] = (coeffup * gradup - coeffdown * graddown) * hx + (coeffright * gradright - coeffleft * gradleft) * hy;
 95:       }
 96:     }
 97:   }
 98:   PetscFunctionReturn(PETSC_SUCCESS);
 99: }

101: /*TEST

103:    test:
104:       args: -pc_type mg -da_refine 1 -ksp_type fgmres

106:    test:
107:       suffix: 2
108:       nsize: 2
109:       args: -pc_type mg -da_refine 1 -ksp_type fgmres

111:    test:
112:       suffix: 3
113:       nsize: 2
114:       args: -pc_type mg -da_refine 1 -ksp_type fgmres -snes_type newtontrdc -snes_trdc_use_cauchy false

116:    test:
117:       suffix: 4
118:       nsize: 2
119:       args: -pc_type mg -da_refine 1 -ksp_type fgmres -snes_type newtontrdc
120:       filter: sed -e "s/SNES iterations = 1[0-3]/SNES iterations = 13/g" |sed -e "s/Linear iterations = 2[7-9]/Linear iterations = 29/g" |sed -e "s/Linear iterations = 3[0-1]/Linear iterations = 29/g"

122: TEST*/