xref: /petsc/src/snes/tutorials/ex70.c (revision 3ba1676111f5c958fe6c2729b46ca4d523958bb3)

static char help[] = "Poiseuille flow problem. Viscous, laminar flow in a 2D channel with parabolic velocity\n\
                      profile and linear pressure drop, exact solution of the 2D Stokes\n";

/*
     M A R I T I M E  R E S E A R C H  I N S T I T U T E  N E T H E R L A N D S
   author : Christiaan M. Klaij

   Poiseuille flow problem.

   Viscous, laminar flow in a 2D channel with parabolic velocity
   profile and linear pressure drop, exact solution of the 2D Stokes
   equations.

   Discretized with the cell-centered finite-volume method on a
   Cartesian grid with co-located variables. Variables ordered as
   [u1...uN v1...vN p1...pN]^T. Matrix [A00 A01; A10, A11] solved with
   PCFIELDSPLIT. Lower factorization is used to mimic the Semi-Implicit
   Method for Pressure Linked Equations (SIMPLE) used as preconditioner
   instead of solver.

   Disclaimer: does not contain the pressure-weighed interpolation
   method needed to suppress spurious pressure modes in real-life
   problems.

   Usage:
     mpiexec -n 2 ./ex70 -nx 32 -ny 48 -ksp_type fgmres -pc_type fieldsplit -pc_fieldsplit_type schur -pc_fieldsplit_schur_fact_type lower -fieldsplit_1_pc_type none

     Runs with PCFIELDSPLIT on 32x48 grid, no PC for the Schur
     complement because A11 is zero. FGMRES is needed because
     PCFIELDSPLIT is a variable preconditioner.

     mpiexec -n 2 ./ex70 -nx 32 -ny 48 -ksp_type fgmres -pc_type fieldsplit -pc_fieldsplit_type schur -pc_fieldsplit_schur_fact_type lower -user_pc

     Same as above but with user defined PC for the true Schur
     complement. PC based on the SIMPLE-type approximation (inverse of
     A00 approximated by inverse of its diagonal).

     mpiexec -n 2 ./ex70 -nx 32 -ny 48 -ksp_type fgmres -pc_type fieldsplit -pc_fieldsplit_type schur -pc_fieldsplit_schur_fact_type lower -user_ksp

     Replace the true Schur complement with a user defined Schur
     complement based on the SIMPLE-type approximation. Same matrix is
     used as PC.

     mpiexec -n 2 ./ex70 -nx 32 -ny 48 -ksp_type fgmres -pc_type fieldsplit -pc_fieldsplit_type schur -pc_fieldsplit_schur_fact_type lower -fieldsplit_0_ksp_type gmres -fieldsplit_0_pc_type bjacobi -fieldsplit_1_pc_type jacobi -fieldsplit_1_inner_ksp_type preonly -fieldsplit_1_inner_pc_type jacobi -fieldsplit_1_upper_ksp_type preonly -fieldsplit_1_upper_pc_type jacobi

     Out-of-the-box SIMPLE-type preconditioning. The major advantage
     is that the user neither needs to provide the approximation of
     the Schur complement, nor the corresponding preconditioner.
*/

#include <petscksp.h>

typedef struct {
  PetscBool userPC, userKSP, matsymmetric; /* user defined preconditioner and matrix for the Schur complement */
  PetscInt  nx, ny;                        /* nb of cells in x- and y-direction */
  PetscReal hx, hy;                        /* mesh size in x- and y-direction */
  Mat       A;                             /* block matrix */
  Mat       subA[4];                       /* the four blocks */
  Mat       myS;                           /* the approximation of the Schur complement */
  Vec       x, b, y;                       /* solution, rhs and temporary vector */
  IS        isg[2];                        /* index sets of split "0" and "1" */
} Stokes;

PetscErrorCode StokesSetupMatBlock00(Stokes *); /* setup the block Q */
PetscErrorCode StokesSetupMatBlock01(Stokes *); /* setup the block G */
PetscErrorCode StokesSetupMatBlock10(Stokes *); /* setup the block D (equal to the transpose of G) */
PetscErrorCode StokesSetupMatBlock11(Stokes *); /* setup the block C (equal to zero) */

PetscErrorCode StokesGetPosition(Stokes *, PetscInt, PetscInt *, PetscInt *); /* row number j*nx+i corresponds to position (i,j) in grid */

PetscErrorCode StokesStencilLaplacian(Stokes *, PetscInt, PetscInt, PetscInt *, PetscInt *, PetscScalar *); /* stencil of the Laplacian operator */
PetscErrorCode StokesStencilGradientX(Stokes *, PetscInt, PetscInt, PetscInt *, PetscInt *, PetscScalar *); /* stencil of the Gradient operator (x-component) */
PetscErrorCode StokesStencilGradientY(Stokes *, PetscInt, PetscInt, PetscInt *, PetscInt *, PetscScalar *); /* stencil of the Gradient operator (y-component) */

PetscErrorCode StokesRhs(Stokes *);                                        /* rhs vector */
PetscErrorCode StokesRhsMomX(Stokes *, PetscInt, PetscInt, PetscScalar *); /* right hand side of velocity (x-component) */
PetscErrorCode StokesRhsMomY(Stokes *, PetscInt, PetscInt, PetscScalar *); /* right hand side of velocity (y-component) */
PetscErrorCode StokesRhsMass(Stokes *, PetscInt, PetscInt, PetscScalar *); /* right hand side of pressure */

PetscErrorCode StokesSetupApproxSchur(Stokes *); /* approximation of the Schur complement */

PetscErrorCode StokesExactSolution(Stokes *); /* exact solution vector */
PetscErrorCode StokesWriteSolution(Stokes *); /* write solution to file */

/* exact solution for the velocity (x-component, y-component is zero) */
PetscScalar StokesExactVelocityX(const PetscScalar y)
{
  return 4.0 * y * (1.0 - y);
}

/* exact solution for the pressure */
PetscScalar StokesExactPressure(const PetscScalar x)
{
  return 8.0 * (2.0 - x);
}

PetscErrorCode StokesSetupPC(Stokes *s, KSP ksp)
{
  KSP     *subksp;
  PC       pc;
  PetscInt n = 1;

  PetscFunctionBeginUser;
  PetscCall(KSPGetPC(ksp, &pc));
  PetscCall(PCFieldSplitSetIS(pc, "0", s->isg[0]));
  PetscCall(PCFieldSplitSetIS(pc, "1", s->isg[1]));
  if (s->userPC) PetscCall(PCFieldSplitSetSchurPre(pc, PC_FIELDSPLIT_SCHUR_PRE_USER, s->myS));
  if (s->userKSP) {
    PetscCall(PCSetUp(pc));
    PetscCall(PCFieldSplitGetSubKSP(pc, &n, &subksp));
    PetscCall(KSPSetOperators(subksp[1], s->myS, s->myS));
    PetscCall(PetscFree(subksp));
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode StokesWriteSolution(Stokes *s)
{
  PetscMPIInt        size;
  PetscInt           n, i, j;
  const PetscScalar *array;

  PetscFunctionBeginUser;
  /* write data (*warning* only works sequential) */
  PetscCallMPI(MPI_Comm_size(MPI_COMM_WORLD, &size));
  if (size == 1) {
    PetscViewer viewer;
    PetscCall(VecGetArrayRead(s->x, &array));
    PetscCall(PetscViewerASCIIOpen(PETSC_COMM_WORLD, "solution.dat", &viewer));
    PetscCall(PetscViewerASCIIPrintf(viewer, "# x, y, u, v, p\n"));
    for (j = 0; j < s->ny; j++) {
      for (i = 0; i < s->nx; i++) {
        n = j * s->nx + i;
        PetscCall(PetscViewerASCIIPrintf(viewer, "%.12g %.12g %.12g %.12g %.12g\n", (double)(i * s->hx + s->hx / 2), (double)(j * s->hy + s->hy / 2), (double)PetscRealPart(array[n]), (double)PetscRealPart(array[n + s->nx * s->ny]), (double)PetscRealPart(array[n + 2 * s->nx * s->ny])));
      }
    }
    PetscCall(VecRestoreArrayRead(s->x, &array));
    PetscCall(PetscViewerDestroy(&viewer));
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode StokesSetupIndexSets(Stokes *s)
{
  PetscFunctionBeginUser;
  /* the two index sets */
  PetscCall(MatNestGetISs(s->A, s->isg, NULL));
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode StokesSetupVectors(Stokes *s)
{
  PetscFunctionBeginUser;
  /* solution vector x */
  PetscCall(VecCreate(PETSC_COMM_WORLD, &s->x));
  PetscCall(VecSetSizes(s->x, PETSC_DECIDE, 3 * s->nx * s->ny));
  PetscCall(VecSetType(s->x, VECMPI));

  /* exact solution y */
  PetscCall(VecDuplicate(s->x, &s->y));
  PetscCall(StokesExactSolution(s));

  /* rhs vector b */
  PetscCall(VecDuplicate(s->x, &s->b));
  PetscCall(StokesRhs(s));
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode StokesGetPosition(Stokes *s, PetscInt row, PetscInt *i, PetscInt *j)
{
  PetscInt n;

  PetscFunctionBeginUser;
  /* cell number n=j*nx+i has position (i,j) in grid */
  n  = row % (s->nx * s->ny);
  *i = n % s->nx;
  *j = (n - (*i)) / s->nx;
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode StokesExactSolution(Stokes *s)
{
  PetscInt    row, start, end, i, j;
  PetscScalar val;
  Vec         y0, y1;

  PetscFunctionBeginUser;
  /* velocity part */
  PetscCall(VecGetSubVector(s->y, s->isg[0], &y0));
  PetscCall(VecGetOwnershipRange(y0, &start, &end));
  for (row = start; row < end; row++) {
    PetscCall(StokesGetPosition(s, row, &i, &j));
    if (row < s->nx * s->ny) {
      val = StokesExactVelocityX(j * s->hy + s->hy / 2);
    } else {
      val = 0;
    }
    PetscCall(VecSetValue(y0, row, val, INSERT_VALUES));
  }
  PetscCall(VecRestoreSubVector(s->y, s->isg[0], &y0));

  /* pressure part */
  PetscCall(VecGetSubVector(s->y, s->isg[1], &y1));
  PetscCall(VecGetOwnershipRange(y1, &start, &end));
  for (row = start; row < end; row++) {
    PetscCall(StokesGetPosition(s, row, &i, &j));
    val = StokesExactPressure(i * s->hx + s->hx / 2);
    PetscCall(VecSetValue(y1, row, val, INSERT_VALUES));
  }
  PetscCall(VecRestoreSubVector(s->y, s->isg[1], &y1));
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode StokesRhs(Stokes *s)
{
  PetscInt    row, start, end, i, j;
  PetscScalar val;
  Vec         b0, b1;

  PetscFunctionBeginUser;
  /* velocity part */
  PetscCall(VecGetSubVector(s->b, s->isg[0], &b0));
  PetscCall(VecGetOwnershipRange(b0, &start, &end));
  for (row = start; row < end; row++) {
    PetscCall(StokesGetPosition(s, row, &i, &j));
    if (row < s->nx * s->ny) {
      PetscCall(StokesRhsMomX(s, i, j, &val));
    } else {
      PetscCall(StokesRhsMomY(s, i, j, &val));
    }
    PetscCall(VecSetValue(b0, row, val, INSERT_VALUES));
  }
  PetscCall(VecRestoreSubVector(s->b, s->isg[0], &b0));

  /* pressure part */
  PetscCall(VecGetSubVector(s->b, s->isg[1], &b1));
  PetscCall(VecGetOwnershipRange(b1, &start, &end));
  for (row = start; row < end; row++) {
    PetscCall(StokesGetPosition(s, row, &i, &j));
    PetscCall(StokesRhsMass(s, i, j, &val));
    if (s->matsymmetric) val = -1.0 * val;
    PetscCall(VecSetValue(b1, row, val, INSERT_VALUES));
  }
  PetscCall(VecRestoreSubVector(s->b, s->isg[1], &b1));
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode StokesSetupMatBlock00(Stokes *s)
{
  PetscInt    row, start, end, sz, i, j;
  PetscInt    cols[5];
  PetscScalar vals[5];

  PetscFunctionBeginUser;
  /* A[0] is 2N-by-2N */
  PetscCall(MatCreate(PETSC_COMM_WORLD, &s->subA[0]));
  PetscCall(MatSetOptionsPrefix(s->subA[0], "a00_"));
  PetscCall(MatSetSizes(s->subA[0], PETSC_DECIDE, PETSC_DECIDE, 2 * s->nx * s->ny, 2 * s->nx * s->ny));
  PetscCall(MatSetType(s->subA[0], MATMPIAIJ));
  PetscCall(MatMPIAIJSetPreallocation(s->subA[0], 5, NULL, 5, NULL));
  PetscCall(MatGetOwnershipRange(s->subA[0], &start, &end));

  for (row = start; row < end; row++) {
    PetscCall(StokesGetPosition(s, row, &i, &j));
    /* first part: rows 0 to (nx*ny-1) */
    PetscCall(StokesStencilLaplacian(s, i, j, &sz, cols, vals));
    /* second part: rows (nx*ny) to (2*nx*ny-1) */
    if (row >= s->nx * s->ny) {
      for (i = 0; i < sz; i++) cols[i] += s->nx * s->ny;
    }
    for (i = 0; i < sz; i++) vals[i] = -1.0 * vals[i]; /* dynamic viscosity coef mu=-1 */
    PetscCall(MatSetValues(s->subA[0], 1, &row, sz, cols, vals, INSERT_VALUES));
  }
  PetscCall(MatAssemblyBegin(s->subA[0], MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(s->subA[0], MAT_FINAL_ASSEMBLY));
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode StokesSetupMatBlock01(Stokes *s)
{
  PetscInt    row, start, end, sz, i, j;
  PetscInt    cols[5];
  PetscScalar vals[5];

  PetscFunctionBeginUser;
  /* A[1] is 2N-by-N */
  PetscCall(MatCreate(PETSC_COMM_WORLD, &s->subA[1]));
  PetscCall(MatSetOptionsPrefix(s->subA[1], "a01_"));
  PetscCall(MatSetSizes(s->subA[1], PETSC_DECIDE, PETSC_DECIDE, 2 * s->nx * s->ny, s->nx * s->ny));
  PetscCall(MatSetType(s->subA[1], MATMPIAIJ));
  PetscCall(MatMPIAIJSetPreallocation(s->subA[1], 5, NULL, 5, NULL));
  PetscCall(MatGetOwnershipRange(s->subA[1], &start, &end));

  PetscCall(MatSetOption(s->subA[1], MAT_IGNORE_ZERO_ENTRIES, PETSC_TRUE));

  for (row = start; row < end; row++) {
    PetscCall(StokesGetPosition(s, row, &i, &j));
    /* first part: rows 0 to (nx*ny-1) */
    if (row < s->nx * s->ny) {
      PetscCall(StokesStencilGradientX(s, i, j, &sz, cols, vals));
    } else { /* second part: rows (nx*ny) to (2*nx*ny-1) */
      PetscCall(StokesStencilGradientY(s, i, j, &sz, cols, vals));
    }
    PetscCall(MatSetValues(s->subA[1], 1, &row, sz, cols, vals, INSERT_VALUES));
  }
  PetscCall(MatAssemblyBegin(s->subA[1], MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(s->subA[1], MAT_FINAL_ASSEMBLY));
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode StokesSetupMatBlock10(Stokes *s)
{
  PetscFunctionBeginUser;
  /* A[2] is minus transpose of A[1] */
  PetscCall(MatTranspose(s->subA[1], MAT_INITIAL_MATRIX, &s->subA[2]));
  if (!s->matsymmetric) PetscCall(MatScale(s->subA[2], -1.0));
  PetscCall(MatSetOptionsPrefix(s->subA[2], "a10_"));
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode StokesSetupMatBlock11(Stokes *s)
{
  PetscFunctionBeginUser;
  /* A[3] is N-by-N null matrix */
  PetscCall(MatCreate(PETSC_COMM_WORLD, &s->subA[3]));
  PetscCall(MatSetOptionsPrefix(s->subA[3], "a11_"));
  PetscCall(MatSetSizes(s->subA[3], PETSC_DECIDE, PETSC_DECIDE, s->nx * s->ny, s->nx * s->ny));
  PetscCall(MatSetType(s->subA[3], MATMPIAIJ));
  PetscCall(MatMPIAIJSetPreallocation(s->subA[3], 0, NULL, 0, NULL));
  PetscCall(MatAssemblyBegin(s->subA[3], MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(s->subA[3], MAT_FINAL_ASSEMBLY));
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode StokesSetupApproxSchur(Stokes *s)
{
  Vec diag;

  PetscFunctionBeginUser;
  /* Schur complement approximation: myS = A11 - A10 inv(DIAGFORM(A00)) A01 */
  /* note: A11 is zero */
  /* note: in real life this matrix would be build directly, */
  /* i.e. without MatMatMult */

  /* inverse of diagonal of A00 */
  PetscCall(VecCreate(PETSC_COMM_WORLD, &diag));
  PetscCall(VecSetSizes(diag, PETSC_DECIDE, 2 * s->nx * s->ny));
  PetscCall(VecSetType(diag, VECMPI));
  PetscCall(MatGetDiagonal(s->subA[0], diag));
  PetscCall(VecReciprocal(diag));

  /* compute: - A10 inv(DIAGFORM(A00)) A01 */
  PetscCall(MatDiagonalScale(s->subA[1], diag, NULL)); /* (*warning* overwrites subA[1]) */
  PetscCall(MatMatMult(s->subA[2], s->subA[1], MAT_INITIAL_MATRIX, PETSC_DEFAULT, &s->myS));
  PetscCall(MatScale(s->myS, -1.0));

  /* restore A10 */
  PetscCall(MatGetDiagonal(s->subA[0], diag));
  PetscCall(MatDiagonalScale(s->subA[1], diag, NULL));
  PetscCall(VecDestroy(&diag));
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode StokesSetupMatrix(Stokes *s)
{
  PetscFunctionBeginUser;
  PetscCall(StokesSetupMatBlock00(s));
  PetscCall(StokesSetupMatBlock01(s));
  PetscCall(StokesSetupMatBlock10(s));
  PetscCall(StokesSetupMatBlock11(s));
  PetscCall(MatCreateNest(PETSC_COMM_WORLD, 2, NULL, 2, NULL, s->subA, &s->A));
  PetscCall(StokesSetupApproxSchur(s));
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode StokesStencilLaplacian(Stokes *s, PetscInt i, PetscInt j, PetscInt *sz, PetscInt *cols, PetscScalar *vals)
{
  PetscInt    p = j * s->nx + i, w = p - 1, e = p + 1, s2 = p - s->nx, n = p + s->nx;
  PetscScalar ae = s->hy / s->hx, aeb = 0;
  PetscScalar aw = s->hy / s->hx, awb = s->hy / (s->hx / 2);
  PetscScalar as = s->hx / s->hy, asb = s->hx / (s->hy / 2);
  PetscScalar an = s->hx / s->hy, anb = s->hx / (s->hy / 2);

  PetscFunctionBeginUser;
  if (i == 0 && j == 0) { /* south-west corner */
    *sz     = 3;
    cols[0] = p;
    vals[0] = -(ae + awb + asb + an);
    cols[1] = e;
    vals[1] = ae;
    cols[2] = n;
    vals[2] = an;
  } else if (i == 0 && j == s->ny - 1) { /* north-west corner */
    *sz     = 3;
    cols[0] = s2;
    vals[0] = as;
    cols[1] = p;
    vals[1] = -(ae + awb + as + anb);
    cols[2] = e;
    vals[2] = ae;
  } else if (i == s->nx - 1 && j == 0) { /* south-east corner */
    *sz     = 3;
    cols[0] = w;
    vals[0] = aw;
    cols[1] = p;
    vals[1] = -(aeb + aw + asb + an);
    cols[2] = n;
    vals[2] = an;
  } else if (i == s->nx - 1 && j == s->ny - 1) { /* north-east corner */
    *sz     = 3;
    cols[0] = s2;
    vals[0] = as;
    cols[1] = w;
    vals[1] = aw;
    cols[2] = p;
    vals[2] = -(aeb + aw + as + anb);
  } else if (i == 0) { /* west boundary */
    *sz     = 4;
    cols[0] = s2;
    vals[0] = as;
    cols[1] = p;
    vals[1] = -(ae + awb + as + an);
    cols[2] = e;
    vals[2] = ae;
    cols[3] = n;
    vals[3] = an;
  } else if (i == s->nx - 1) { /* east boundary */
    *sz     = 4;
    cols[0] = s2;
    vals[0] = as;
    cols[1] = w;
    vals[1] = aw;
    cols[2] = p;
    vals[2] = -(aeb + aw + as + an);
    cols[3] = n;
    vals[3] = an;
  } else if (j == 0) { /* south boundary */
    *sz     = 4;
    cols[0] = w;
    vals[0] = aw;
    cols[1] = p;
    vals[1] = -(ae + aw + asb + an);
    cols[2] = e;
    vals[2] = ae;
    cols[3] = n;
    vals[3] = an;
  } else if (j == s->ny - 1) { /* north boundary */
    *sz     = 4;
    cols[0] = s2;
    vals[0] = as;
    cols[1] = w;
    vals[1] = aw;
    cols[2] = p;
    vals[2] = -(ae + aw + as + anb);
    cols[3] = e;
    vals[3] = ae;
  } else { /* interior */
    *sz     = 5;
    cols[0] = s2;
    vals[0] = as;
    cols[1] = w;
    vals[1] = aw;
    cols[2] = p;
    vals[2] = -(ae + aw + as + an);
    cols[3] = e;
    vals[3] = ae;
    cols[4] = n;
    vals[4] = an;
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode StokesStencilGradientX(Stokes *s, PetscInt i, PetscInt j, PetscInt *sz, PetscInt *cols, PetscScalar *vals)
{
  PetscInt    p = j * s->nx + i, w = p - 1, e = p + 1;
  PetscScalar ae = s->hy / 2, aeb = s->hy;
  PetscScalar aw = -s->hy / 2, awb = 0;

  PetscFunctionBeginUser;
  if (i == 0 && j == 0) { /* south-west corner */
    *sz     = 2;
    cols[0] = p;
    vals[0] = -(ae + awb);
    cols[1] = e;
    vals[1] = ae;
  } else if (i == 0 && j == s->ny - 1) { /* north-west corner */
    *sz     = 2;
    cols[0] = p;
    vals[0] = -(ae + awb);
    cols[1] = e;
    vals[1] = ae;
  } else if (i == s->nx - 1 && j == 0) { /* south-east corner */
    *sz     = 2;
    cols[0] = w;
    vals[0] = aw;
    cols[1] = p;
    vals[1] = -(aeb + aw);
  } else if (i == s->nx - 1 && j == s->ny - 1) { /* north-east corner */
    *sz     = 2;
    cols[0] = w;
    vals[0] = aw;
    cols[1] = p;
    vals[1] = -(aeb + aw);
  } else if (i == 0) { /* west boundary */
    *sz     = 2;
    cols[0] = p;
    vals[0] = -(ae + awb);
    cols[1] = e;
    vals[1] = ae;
  } else if (i == s->nx - 1) { /* east boundary */
    *sz     = 2;
    cols[0] = w;
    vals[0] = aw;
    cols[1] = p;
    vals[1] = -(aeb + aw);
  } else if (j == 0) { /* south boundary */
    *sz     = 3;
    cols[0] = w;
    vals[0] = aw;
    cols[1] = p;
    vals[1] = -(ae + aw);
    cols[2] = e;
    vals[2] = ae;
  } else if (j == s->ny - 1) { /* north boundary */
    *sz     = 3;
    cols[0] = w;
    vals[0] = aw;
    cols[1] = p;
    vals[1] = -(ae + aw);
    cols[2] = e;
    vals[2] = ae;
  } else { /* interior */
    *sz     = 3;
    cols[0] = w;
    vals[0] = aw;
    cols[1] = p;
    vals[1] = -(ae + aw);
    cols[2] = e;
    vals[2] = ae;
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode StokesStencilGradientY(Stokes *s, PetscInt i, PetscInt j, PetscInt *sz, PetscInt *cols, PetscScalar *vals)
{
  PetscInt    p = j * s->nx + i, s2 = p - s->nx, n = p + s->nx;
  PetscScalar as = -s->hx / 2, asb = 0;
  PetscScalar an = s->hx / 2, anb = 0;

  PetscFunctionBeginUser;
  if (i == 0 && j == 0) { /* south-west corner */
    *sz     = 2;
    cols[0] = p;
    vals[0] = -(asb + an);
    cols[1] = n;
    vals[1] = an;
  } else if (i == 0 && j == s->ny - 1) { /* north-west corner */
    *sz     = 2;
    cols[0] = s2;
    vals[0] = as;
    cols[1] = p;
    vals[1] = -(as + anb);
  } else if (i == s->nx - 1 && j == 0) { /* south-east corner */
    *sz     = 2;
    cols[0] = p;
    vals[0] = -(asb + an);
    cols[1] = n;
    vals[1] = an;
  } else if (i == s->nx - 1 && j == s->ny - 1) { /* north-east corner */
    *sz     = 2;
    cols[0] = s2;
    vals[0] = as;
    cols[1] = p;
    vals[1] = -(as + anb);
  } else if (i == 0) { /* west boundary */
    *sz     = 3;
    cols[0] = s2;
    vals[0] = as;
    cols[1] = p;
    vals[1] = -(as + an);
    cols[2] = n;
    vals[2] = an;
  } else if (i == s->nx - 1) { /* east boundary */
    *sz     = 3;
    cols[0] = s2;
    vals[0] = as;
    cols[1] = p;
    vals[1] = -(as + an);
    cols[2] = n;
    vals[2] = an;
  } else if (j == 0) { /* south boundary */
    *sz     = 2;
    cols[0] = p;
    vals[0] = -(asb + an);
    cols[1] = n;
    vals[1] = an;
  } else if (j == s->ny - 1) { /* north boundary */
    *sz     = 2;
    cols[0] = s2;
    vals[0] = as;
    cols[1] = p;
    vals[1] = -(as + anb);
  } else { /* interior */
    *sz     = 3;
    cols[0] = s2;
    vals[0] = as;
    cols[1] = p;
    vals[1] = -(as + an);
    cols[2] = n;
    vals[2] = an;
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode StokesRhsMomX(Stokes *s, PetscInt i, PetscInt j, PetscScalar *val)
{
  PetscScalar y   = j * s->hy + s->hy / 2;
  PetscScalar awb = s->hy / (s->hx / 2);

  PetscFunctionBeginUser;
  if (i == 0) { /* west boundary */
    *val = awb * StokesExactVelocityX(y);
  } else {
    *val = 0.0;
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode StokesRhsMomY(Stokes *s, PetscInt i, PetscInt j, PetscScalar *val)
{
  PetscFunctionBeginUser;
  *val = 0.0;
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode StokesRhsMass(Stokes *s, PetscInt i, PetscInt j, PetscScalar *val)
{
  PetscScalar y   = j * s->hy + s->hy / 2;
  PetscScalar aeb = s->hy;

  PetscFunctionBeginUser;
  if (i == 0) { /* west boundary */
    *val = aeb * StokesExactVelocityX(y);
  } else {
    *val = 0.0;
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode StokesCalcResidual(Stokes *s)
{
  PetscReal val;
  Vec       b0, b1;

  PetscFunctionBeginUser;
  /* residual Ax-b (*warning* overwrites b) */
  PetscCall(VecScale(s->b, -1.0));
  PetscCall(MatMultAdd(s->A, s->x, s->b, s->b));

  /* residual velocity */
  PetscCall(VecGetSubVector(s->b, s->isg[0], &b0));
  PetscCall(VecNorm(b0, NORM_2, &val));
  PetscCall(PetscPrintf(PETSC_COMM_WORLD, " residual u = %g\n", (double)val));
  PetscCall(VecRestoreSubVector(s->b, s->isg[0], &b0));

  /* residual pressure */
  PetscCall(VecGetSubVector(s->b, s->isg[1], &b1));
  PetscCall(VecNorm(b1, NORM_2, &val));
  PetscCall(PetscPrintf(PETSC_COMM_WORLD, " residual p = %g\n", (double)val));
  PetscCall(VecRestoreSubVector(s->b, s->isg[1], &b1));

  /* total residual */
  PetscCall(VecNorm(s->b, NORM_2, &val));
  PetscCall(PetscPrintf(PETSC_COMM_WORLD, " residual [u,p] = %g\n", (double)val));
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode StokesCalcError(Stokes *s)
{
  PetscScalar scale = PetscSqrtReal((double)s->nx * s->ny);
  PetscReal   val;
  Vec         y0, y1;

  PetscFunctionBeginUser;
  /* error y-x */
  PetscCall(VecAXPY(s->y, -1.0, s->x));

  /* error in velocity */
  PetscCall(VecGetSubVector(s->y, s->isg[0], &y0));
  PetscCall(VecNorm(y0, NORM_2, &val));
  PetscCall(PetscPrintf(PETSC_COMM_WORLD, " discretization error u = %g\n", (double)(PetscRealPart(val / scale))));
  PetscCall(VecRestoreSubVector(s->y, s->isg[0], &y0));

  /* error in pressure */
  PetscCall(VecGetSubVector(s->y, s->isg[1], &y1));
  PetscCall(VecNorm(y1, NORM_2, &val));
  PetscCall(PetscPrintf(PETSC_COMM_WORLD, " discretization error p = %g\n", (double)(PetscRealPart(val / scale))));
  PetscCall(VecRestoreSubVector(s->y, s->isg[1], &y1));

  /* total error */
  PetscCall(VecNorm(s->y, NORM_2, &val));
  PetscCall(PetscPrintf(PETSC_COMM_WORLD, " discretization error [u,p] = %g\n", (double)PetscRealPart((val / scale))));
  PetscFunctionReturn(PETSC_SUCCESS);
}

int main(int argc, char **argv)
{
  Stokes s;
  KSP    ksp;

  PetscFunctionBeginUser;
  PetscCall(PetscInitialize(&argc, &argv, NULL, help));
  s.nx = 4;
  s.ny = 6;
  PetscCall(PetscOptionsGetInt(NULL, NULL, "-nx", &s.nx, NULL));
  PetscCall(PetscOptionsGetInt(NULL, NULL, "-ny", &s.ny, NULL));
  s.hx           = 2.0 / s.nx;
  s.hy           = 1.0 / s.ny;
  s.matsymmetric = PETSC_FALSE;
  PetscCall(PetscOptionsGetBool(NULL, NULL, "-mat_set_symmetric", &s.matsymmetric, NULL));
  s.userPC = s.userKSP = PETSC_FALSE;
  PetscCall(PetscOptionsHasName(NULL, NULL, "-user_pc", &s.userPC));
  PetscCall(PetscOptionsHasName(NULL, NULL, "-user_ksp", &s.userKSP));

  PetscCall(StokesSetupMatrix(&s));
  PetscCall(StokesSetupIndexSets(&s));
  PetscCall(StokesSetupVectors(&s));

  PetscCall(KSPCreate(PETSC_COMM_WORLD, &ksp));
  PetscCall(KSPSetOperators(ksp, s.A, s.A));
  PetscCall(KSPSetFromOptions(ksp));
  PetscCall(StokesSetupPC(&s, ksp));
  PetscCall(KSPSolve(ksp, s.b, s.x));

  /* don't trust, verify! */
  PetscCall(StokesCalcResidual(&s));
  PetscCall(StokesCalcError(&s));
  PetscCall(StokesWriteSolution(&s));

  PetscCall(KSPDestroy(&ksp));
  PetscCall(MatDestroy(&s.subA[0]));
  PetscCall(MatDestroy(&s.subA[1]));
  PetscCall(MatDestroy(&s.subA[2]));
  PetscCall(MatDestroy(&s.subA[3]));
  PetscCall(MatDestroy(&s.A));
  PetscCall(VecDestroy(&s.x));
  PetscCall(VecDestroy(&s.b));
  PetscCall(VecDestroy(&s.y));
  PetscCall(MatDestroy(&s.myS));
  PetscCall(PetscFinalize());
  return 0;
}

/*TEST

   test:
      nsize: 2
      args: -nx 16 -ny 24 -ksp_type fgmres -pc_type fieldsplit -pc_fieldsplit_type schur -pc_fieldsplit_schur_fact_type lower -fieldsplit_1_pc_type none

   test:
      suffix: 2
      nsize: 2
      args: -nx 16 -ny 24 -ksp_type fgmres -pc_type fieldsplit -pc_fieldsplit_type schur -pc_fieldsplit_schur_fact_type lower -user_pc

   test:
      suffix: 3
      nsize: 2
      args: -nx 16 -ny 24 -ksp_type fgmres -pc_type fieldsplit -pc_fieldsplit_type schur -pc_fieldsplit_schur_fact_type lower -user_pc

   test:
      suffix: 4
      nsize: 2
      args: -nx 16 -ny 24 -ksp_type fgmres -pc_type fieldsplit -pc_fieldsplit_type schur -pc_fieldsplit_schur_fact_type lower -fieldsplit_0_ksp_type gmres -fieldsplit_0_pc_type bjacobi -fieldsplit_1_pc_type jacobi -fieldsplit_1_inner_ksp_type preonly -fieldsplit_1_inner_pc_type jacobi -fieldsplit_1_upper_ksp_type preonly -fieldsplit_1_upper_pc_type jacobi

   test:
      suffix: 4_pcksp
      nsize: 2
      args: -nx 16 -ny 24 -ksp_type fgmres -pc_type fieldsplit -pc_fieldsplit_type schur -pc_fieldsplit_schur_fact_type lower -fieldsplit_0_ksp_type gmres -fieldsplit_0_pc_type bjacobi -fieldsplit_1_pc_type jacobi -fieldsplit_1_inner_ksp_type preonly -fieldsplit_1_upper_ksp_type preonly -fieldsplit_1_upper_pc_type jacobi

   test:
      suffix: 5
      nsize: 2
      args: -nx 4 -ny 8 -mat_set_symmetric -ksp_type fgmres -pc_type fieldsplit -pc_fieldsplit_type gkb -fieldsplit_0_ksp_type cg -fieldsplit_0_pc_type jacobi -fieldsplit_0_ksp_rtol 1e-10

   test:
      suffix: 6
      nsize: 2
      args: -nx 4 -ny 8 -mat_set_symmetric -ksp_type preonly -pc_type fieldsplit -pc_fieldsplit_type gkb -fieldsplit_0_ksp_type cg -fieldsplit_0_pc_type jacobi -fieldsplit_0_ksp_rtol 1e-10

   test:
      suffix: 7
      nsize: 2
      args: -nx 4 -ny 8 -mat_set_symmetric -ksp_type preonly -pc_type fieldsplit -pc_fieldsplit_type gkb -pc_fieldsplit_gkb_tol 1e-4 -pc_fieldsplit_gkb_nu 5 -fieldsplit_0_ksp_type cg -fieldsplit_0_pc_type jacobi -fieldsplit_0_ksp_rtol 1e-6

TEST*/