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: /*T
  8:    Concepts: SNES^solving a system of nonlinear equations
  9:    Concepts: DMDA^using distributed arrays
 10:    Concepts: multigrid;
 11:    Processors: n
 12: T*/

 14: /*

 16:     This example models the partial differential equation

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


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

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

 28: */

 30: #include <petscsnes.h>
 31: #include <petscdmda.h>

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

 37: int main(int argc,char **argv)
 38: {
 39:   SNES           snes;
 41:   PetscInt       its,lits;
 42:   PetscReal      litspit;
 43:   DM             da;

 45:   PetscInitialize(&argc,&argv,NULL,help);

 47:   /*
 48:       Set the DMDA (grid structure) for the grids.
 49:   */
 50:   DMDACreate2d(PETSC_COMM_WORLD, DMDA_BOUNDARY_NONE, DMDA_BOUNDARY_NONE,DMDA_STENCIL_STAR,-5,-5,PETSC_DECIDE,PETSC_DECIDE,1,1,0,0,&da);
 51:   DMDASNESSetFunctionLocal(da,INSERT_VALUES,(PetscErrorCode (*)(DMDALocalInfo*,void*,void*,void*))FormFunctionLocal,NULL);
 52:   SNESCreate(PETSC_COMM_WORLD,&snes);
 53:   SNESSetDM(snes,da);
 54:   DMDestroy(&da);

 56:   SNESSetFromOptions(snes);

 58:   SNESSolve(snes,0,0);
 59:   SNESGetIterationNumber(snes,&its);
 60:   SNESGetLinearSolveIterations(snes,&lits);
 61:   litspit = ((PetscReal)lits)/((PetscReal)its);
 62:   PetscPrintf(PETSC_COMM_WORLD,"Number of SNES iterations = %D\n",its);
 63:   PetscPrintf(PETSC_COMM_WORLD,"Number of Linear iterations = %D\n",lits);
 64:   PetscPrintf(PETSC_COMM_WORLD,"Average Linear its / SNES = %e\n",litspit);

 66:   SNESDestroy(&snes);
 67:   PetscFinalize();

 69:   return 0;
 70: }

 74: PetscErrorCode FormFunctionLocal(DMDALocalInfo *info,PetscScalar **t,PetscScalar **f,void *ptr)
 75: {
 76:   PetscInt    i,j;
 77:   PetscScalar hx,hy;
 78:   PetscScalar gradup,graddown,gradleft,gradright,gradx,grady;
 79:   PetscScalar coeffup,coeffdown,coeffleft,coeffright;

 82:   hx = 1.0/(PetscReal)(info->mx-1);  hy    = 1.0/(PetscReal)(info->my-1);

 84:   /* Evaluate function */
 85:   for (j=info->ys; j<info->ys+info->ym; j++) {
 86:     for (i=info->xs; i<info->xs+info->xm; i++) {

 88:       if (i == 0 || i == info->mx-1 || j == 0 || j == info->my-1) {
 89:         f[j][i] = t[j][i] - (1.0 - (2.0*hx*(PetscReal)i - 1.0)*(2.0*hx*(PetscReal)i - 1.0));
 90:       } else {

 92:         gradup    = (t[j+1][i] - t[j][i])/hy;
 93:         graddown  = (t[j][i] - t[j-1][i])/hy;
 94:         gradright = (t[j][i+1] - t[j][i])/hx;
 95:         gradleft  = (t[j][i] - t[j][i-1])/hx;

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

100:         coeffup   = 1.0/PetscSqrtScalar(1.0 + gradup*gradup + gradx*gradx);
101:         coeffdown = 1.0/PetscSqrtScalar(1.0 + graddown*graddown + gradx*gradx);

103:         coeffleft  = 1.0/PetscSqrtScalar(1.0 + gradleft*gradleft + grady*grady);
104:         coeffright = 1.0/PetscSqrtScalar(1.0 + gradright*gradright + grady*grady);

106:         f[j][i] = (coeffup*gradup - coeffdown*graddown)*hx + (coeffright*gradright - coeffleft*gradleft)*hy;
107:       }

109:     }
110:   }
111:   return(0);
112: }