Date: 2026-03-31T05:46:14+00:00

Actual source code: ex13.c

      1: static char help[] = "Solves a variable Poisson problem with KSP.\n\n";

      3: /*
      4:   Include "petscksp.h" so that we can use KSP solvers.  Note that this file
      5:   automatically includes:
      6:      petscsys.h       - base PETSc routines   petscvec.h - vectors
      7:      petscmat.h - matrices
      8:      petscis.h     - index sets            petscksp.h - Krylov subspace methods
      9:      petscviewer.h - viewers               petscpc.h  - preconditioners
     10: */
     11: #include <petscksp.h>

     13: /*
     14:     User-defined context that contains all the data structures used
     15:     in the linear solution process.
     16: */
     17: typedef struct {
     18:   Vec         x, b;     /* solution vector, right-hand-side vector */
     19:   Mat         A;        /* sparse matrix */
     20:   KSP         ksp;      /* linear solver context */
     21:   PetscInt    m, n;     /* grid dimensions */
     22:   PetscScalar hx2, hy2; /* 1/(m+1)*(m+1) and 1/(n+1)*(n+1) */
     23: } UserCtx;

     25: extern PetscErrorCode UserInitializeLinearSolver(PetscInt, PetscInt, UserCtx *);
     26: extern PetscErrorCode UserFinalizeLinearSolver(UserCtx *);
     27: extern PetscErrorCode UserDoLinearSolver(PetscScalar *, UserCtx *userctx, PetscScalar *b, PetscScalar *x);

     29: int main(int argc, char **args)
     30: {
     31:   UserCtx      userctx;
     32:   PetscInt     m = 6, n = 7, t, tmax = 2, i, Ii, j, N;
     33:   PetscScalar *userx, *rho, *solution, *userb, hx, hy, x, y;
     34:   PetscReal    enorm;

     36:   /*
     37:      Initialize the PETSc libraries
     38:   */
     39:   PetscFunctionBeginUser;
     40:   PetscCall(PetscInitialize(&argc, &args, NULL, help));
     41:   /*
     42:      The next two lines are for testing only; these allow the user to
     43:      decide the grid size at runtime.
     44:   */
     45:   PetscCall(PetscOptionsGetInt(NULL, NULL, "-m", &m, NULL));
     46:   PetscCall(PetscOptionsGetInt(NULL, NULL, "-n", &n, NULL));

     48:   /*
     49:      Create the empty sparse matrix and linear solver data structures
     50:   */
     51:   PetscCall(UserInitializeLinearSolver(m, n, &userctx));
     52:   N = m * n;

     54:   /*
     55:      Allocate arrays to hold the solution to the linear system.
     56:      This is not normally done in PETSc programs, but in this case,
     57:      since we are calling these routines from a non-PETSc program, we
     58:      would like to reuse the data structures from another code. So in
     59:      the context of a larger application these would be provided by
     60:      other (non-PETSc) parts of the application code.
     61:   */
     62:   PetscCall(PetscMalloc1(N, &userx));
     63:   PetscCall(PetscMalloc1(N, &userb));
     64:   PetscCall(PetscMalloc1(N, &solution));

     66:   /*
     67:       Allocate an array to hold the coefficients in the elliptic operator
     68:   */
     69:   PetscCall(PetscMalloc1(N, &rho));

     71:   /*
     72:      Fill up the array rho[] with the function rho(x,y) = x; fill the
     73:      right-hand side b and the solution with a known problem for testing.
     74:   */
     75:   hx = 1.0 / (m + 1);
     76:   hy = 1.0 / (n + 1);
     77:   y  = hy;
     78:   Ii = 0;
     79:   for (j = 0; j < n; j++) {
     80:     x = hx;
     81:     for (i = 0; i < m; i++) {
     82:       rho[Ii]      = x;
     83:       solution[Ii] = PetscSinScalar(2. * PETSC_PI * x) * PetscSinScalar(2. * PETSC_PI * y);
     84:       userb[Ii]    = -2 * PETSC_PI * PetscCosScalar(2 * PETSC_PI * x) * PetscSinScalar(2 * PETSC_PI * y) + 8 * PETSC_PI * PETSC_PI * x * PetscSinScalar(2 * PETSC_PI * x) * PetscSinScalar(2 * PETSC_PI * y);
     85:       x += hx;
     86:       Ii++;
     87:     }
     88:     y += hy;
     89:   }

     91:   /*
     92:      Loop over a bunch of timesteps, setting up and solver the linear
     93:      system for each time-step.

     95:      Note this is somewhat artificial. It is intended to demonstrate how
     96:      one may reuse the linear solver stuff in each time-step.
     97:   */
     98:   for (t = 0; t < tmax; t++) {
     99:     PetscCall(UserDoLinearSolver(rho, &userctx, userb, userx));

    101:     /*
    102:         Compute error: Note that this could (and usually should) all be done
    103:         using the PETSc vector operations. Here we demonstrate using more
    104:         standard programming practices to show how they may be mixed with
    105:         PETSc.
    106:     */
    107:     enorm = 0.0;
    108:     for (i = 0; i < N; i++) enorm += PetscRealPart(PetscConj(solution[i] - userx[i]) * (solution[i] - userx[i]));
    109:     enorm *= PetscRealPart(hx * hy);
    110:     PetscCall(PetscPrintf(PETSC_COMM_WORLD, "m %" PetscInt_FMT " n %" PetscInt_FMT " error norm %g\n", m, n, (double)enorm));
    111:   }

    113:   /*
    114:      We are all finished solving linear systems, so we clean up the
    115:      data structures.
    116:   */
    117:   PetscCall(PetscFree(rho));
    118:   PetscCall(PetscFree(solution));
    119:   PetscCall(PetscFree(userx));
    120:   PetscCall(PetscFree(userb));
    121:   PetscCall(UserFinalizeLinearSolver(&userctx));
    122:   PetscCall(PetscFinalize());
    123:   return 0;
    124: }

    126: /* ------------------------------------------------------------------------*/
    127: PetscErrorCode UserInitializeLinearSolver(PetscInt m, PetscInt n, UserCtx *userctx)
    128: {
    129:   PetscInt N;

    131:   PetscFunctionBeginUser;
    132:   /*
    133:      Here we assume use of a grid of size m x n, with all points on the
    134:      interior of the domain, i.e., we do not include the points corresponding
    135:      to homogeneous Dirichlet boundary conditions.  We assume that the domain
    136:      is [0,1]x[0,1].
    137:   */
    138:   userctx->m   = m;
    139:   userctx->n   = n;
    140:   userctx->hx2 = (m + 1) * (m + 1);
    141:   userctx->hy2 = (n + 1) * (n + 1);
    142:   N            = m * n;

    144:   /*
    145:      Create the sparse matrix. Preallocate 5 nonzeros per row.
    146:   */
    147:   PetscCall(MatCreateSeqAIJ(PETSC_COMM_SELF, N, N, 5, 0, &userctx->A));

    149:   /*
    150:      Create vectors. Here we create vectors with no memory allocated.
    151:      This way, we can use the data structures already in the program
    152:      by using VecPlaceArray() subroutine at a later stage.
    153:   */
    154:   PetscCall(VecCreateSeqWithArray(PETSC_COMM_SELF, 1, N, NULL, &userctx->b));
    155:   PetscCall(VecDuplicate(userctx->b, &userctx->x));

    157:   /*
    158:      Create linear solver context. This will be used repeatedly for all
    159:      the linear solves needed.
    160:   */
    161:   PetscCall(KSPCreate(PETSC_COMM_SELF, &userctx->ksp));
    162:   PetscFunctionReturn(PETSC_SUCCESS);
    163: }

    165: /*
    166:    Solves -div (rho grad psi) = F using finite differences.
    167:    rho is a 2-dimensional array of size m by n, stored in Fortran
    168:    style by columns. userb is a standard one-dimensional array.
    169: */
    170: /* ------------------------------------------------------------------------*/
    171: PetscErrorCode UserDoLinearSolver(PetscScalar *rho, UserCtx *userctx, PetscScalar *userb, PetscScalar *userx)
    172: {
    173:   PetscInt    i, j, Ii, J, m = userctx->m, n = userctx->n;
    174:   Mat         A = userctx->A;
    175:   PC          pc;
    176:   PetscScalar v, hx2 = userctx->hx2, hy2 = userctx->hy2;

    178:   PetscFunctionBeginUser;
    179:   /*
    180:      This is not the most efficient way of generating the matrix
    181:      but let's not worry about it. We should have separate code for
    182:      the four corners, each edge and then the interior. Then we won't
    183:      have the slow if-tests inside the loop.

    185:      Computes the operator
    186:              -div rho grad
    187:      on an m by n grid with zero Dirichlet boundary conditions. The rho
    188:      is assumed to be given on the same grid as the finite difference
    189:      stencil is applied.  For a staggered grid, one would have to change
    190:      things slightly.
    191:   */
    192:   Ii = 0;
    193:   for (j = 0; j < n; j++) {
    194:     for (i = 0; i < m; i++) {
    195:       if (j > 0) {
    196:         J = Ii - m;
    197:         v = -.5 * (rho[Ii] + rho[J]) * hy2;
    198:         PetscCall(MatSetValues(A, 1, &Ii, 1, &J, &v, INSERT_VALUES));
    199:       }
    200:       if (j < n - 1) {
    201:         J = Ii + m;
    202:         v = -.5 * (rho[Ii] + rho[J]) * hy2;
    203:         PetscCall(MatSetValues(A, 1, &Ii, 1, &J, &v, INSERT_VALUES));
    204:       }
    205:       if (i > 0) {
    206:         J = Ii - 1;
    207:         v = -.5 * (rho[Ii] + rho[J]) * hx2;
    208:         PetscCall(MatSetValues(A, 1, &Ii, 1, &J, &v, INSERT_VALUES));
    209:       }
    210:       if (i < m - 1) {
    211:         J = Ii + 1;
    212:         v = -.5 * (rho[Ii] + rho[J]) * hx2;
    213:         PetscCall(MatSetValues(A, 1, &Ii, 1, &J, &v, INSERT_VALUES));
    214:       }
    215:       v = 2.0 * rho[Ii] * (hx2 + hy2);
    216:       PetscCall(MatSetValues(A, 1, &Ii, 1, &Ii, &v, INSERT_VALUES));
    217:       Ii++;
    218:     }
    219:   }

    221:   /*
    222:      Assemble matrix
    223:   */
    224:   PetscCall(MatAssemblyBegin(A, MAT_FINAL_ASSEMBLY));
    225:   PetscCall(MatAssemblyEnd(A, MAT_FINAL_ASSEMBLY));

    227:   /*
    228:      Set operators. Here the matrix that defines the linear system
    229:      also serves as the matrix from which the preconditioner is constructed. Since all the matrices
    230:      will have the same nonzero pattern here, we indicate this so the
    231:      linear solvers can take advantage of this.
    232:   */
    233:   PetscCall(KSPSetOperators(userctx->ksp, A, A));

    235:   /*
    236:      Set linear solver defaults for this problem (optional).
    237:      - Here we set it to use direct LU factorization for the solution
    238:   */
    239:   PetscCall(KSPGetPC(userctx->ksp, &pc));
    240:   PetscCall(PCSetType(pc, PCLU));

    242:   /*
    243:      Set runtime options, e.g.,
    244:         -ksp_type <type> -pc_type <type> -ksp_monitor -ksp_rtol <rtol>
    245:      These options will override those specified above as long as
    246:      KSPSetFromOptions() is called _after_ any other customization
    247:      routines.

    249:      Run the program with the option -help to see all the possible
    250:      linear solver options.
    251:   */
    252:   PetscCall(KSPSetFromOptions(userctx->ksp));

    254:   /*
    255:      This allows the PETSc linear solvers to compute the solution
    256:      directly in the user's array rather than in the PETSc vector.

    258:      This is essentially a hack and not highly recommend unless you
    259:      are quite comfortable with using PETSc. In general, users should
    260:      write their entire application using PETSc vectors rather than
    261:      arrays.
    262:   */
    263:   PetscCall(VecPlaceArray(userctx->x, userx));
    264:   PetscCall(VecPlaceArray(userctx->b, userb));

    266:   /* - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
    267:                       Solve the linear system
    268:      - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - */

    270:   PetscCall(KSPSolve(userctx->ksp, userctx->b, userctx->x));

    272:   /*
    273:     Put back the PETSc array that belongs in the vector xuserctx->x
    274:   */
    275:   PetscCall(VecResetArray(userctx->x));
    276:   PetscCall(VecResetArray(userctx->b));
    277:   PetscFunctionReturn(PETSC_SUCCESS);
    278: }

    280: /* ------------------------------------------------------------------------*/
    281: PetscErrorCode UserFinalizeLinearSolver(UserCtx *userctx)
    282: {
    283:   /*
    284:      We are all done and don't need to solve any more linear systems, so
    285:      we free the work space.  All PETSc objects should be destroyed when
    286:      they are no longer needed.
    287:   */
    288:   PetscFunctionBeginUser;
    289:   PetscCall(KSPDestroy(&userctx->ksp));
    290:   PetscCall(VecDestroy(&userctx->x));
    291:   PetscCall(VecDestroy(&userctx->b));
    292:   PetscCall(MatDestroy(&userctx->A));
    293:   PetscFunctionReturn(PETSC_SUCCESS);
    294: }

    296: /*TEST

    298:    test:
    299:       args: -m 19 -n 20 -ksp_gmres_cgs_refinement_type refine_always

    301: TEST*/