xref: /petsc/src/dm/impls/da/fdda.c (revision 2286efddd54511ab18e8e2adb1e023c4bf8f0b92)

#include <petsc/private/dmdaimpl.h> /*I      "petscdmda.h"     I*/
#include <petscmat.h>
#include <petscbt.h>

extern PetscErrorCode DMCreateColoring_DA_1d_MPIAIJ(DM, ISColoringType, ISColoring *);
extern PetscErrorCode DMCreateColoring_DA_2d_MPIAIJ(DM, ISColoringType, ISColoring *);
extern PetscErrorCode DMCreateColoring_DA_2d_5pt_MPIAIJ(DM, ISColoringType, ISColoring *);
extern PetscErrorCode DMCreateColoring_DA_3d_MPIAIJ(DM, ISColoringType, ISColoring *);

/*
   For ghost i that may be negative or greater than the upper bound this
  maps it into the 0:m-1 range using periodicity
*/
#define SetInRange(i, m) ((i < 0) ? m + i : ((i >= m) ? i - m : i))

static PetscErrorCode DMDASetBlockFills_Private(const PetscInt *dfill, PetscInt w, PetscInt **rfill)
{
  PetscInt i, j, nz, *fill;

  PetscFunctionBegin;
  if (!dfill) PetscFunctionReturn(PETSC_SUCCESS);

  /* count number nonzeros */
  nz = 0;
  for (i = 0; i < w; i++) {
    for (j = 0; j < w; j++) {
      if (dfill[w * i + j]) nz++;
    }
  }
  PetscCall(PetscMalloc1(nz + w + 1, &fill));
  /* construct modified CSR storage of nonzero structure */
  /*  fill[0 -- w] marks starts of each row of column indices (and end of last row)
   so fill[1] - fill[0] gives number of nonzeros in first row etc */
  nz = w + 1;
  for (i = 0; i < w; i++) {
    fill[i] = nz;
    for (j = 0; j < w; j++) {
      if (dfill[w * i + j]) {
        fill[nz] = j;
        nz++;
      }
    }
  }
  fill[w] = nz;

  *rfill = fill;
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode DMDASetBlockFillsSparse_Private(const PetscInt *dfillsparse, PetscInt w, PetscInt **rfill)
{
  PetscInt nz;

  PetscFunctionBegin;
  if (!dfillsparse) PetscFunctionReturn(PETSC_SUCCESS);

  /* Determine number of non-zeros */
  nz = (dfillsparse[w] - w - 1);

  /* Allocate space for our copy of the given sparse matrix representation. */
  PetscCall(PetscMalloc1(nz + w + 1, rfill));
  PetscCall(PetscArraycpy(*rfill, dfillsparse, nz + w + 1));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode DMDASetBlockFills_Private2(DM_DA *dd)
{
  PetscInt i, k, cnt = 1;

  PetscFunctionBegin;
  /* ofillcount tracks the columns of ofill that have any nonzero in them; the value in each location is the number of
   columns to the left with any nonzeros in them plus 1 */
  PetscCall(PetscCalloc1(dd->w, &dd->ofillcols));
  for (i = 0; i < dd->w; i++) {
    for (k = dd->ofill[i]; k < dd->ofill[i + 1]; k++) dd->ofillcols[dd->ofill[k]] = 1;
  }
  for (i = 0; i < dd->w; i++) {
    if (dd->ofillcols[i]) dd->ofillcols[i] = cnt++;
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  DMDASetBlockFills - Sets the fill pattern in each block for a multi-component problem
  of the matrix returned by `DMCreateMatrix()`.

  Logically Collective

  Input Parameters:
+ da    - the `DMDA`
. dfill - the fill pattern in the diagonal block (may be `NULL`, means use dense block)
- ofill - the fill pattern in the off-diagonal blocks

  Level: developer

  Notes:
  This only makes sense when you are doing multicomponent problems but using the
  `MATMPIAIJ` matrix format

  The format for `dfill` and `ofill` is a 2 dimensional dof by dof matrix with 1 entries
  representing coupling and 0 entries for missing coupling. For example
.vb
            dfill[9] = {1, 0, 0,
                        1, 1, 0,
                        0, 1, 1}
.ve
  means that row 0 is coupled with only itself in the diagonal block, row 1 is coupled with
  itself and row 0 (in the diagonal block) and row 2 is coupled with itself and row 1 (in the
  diagonal block).

  `DMDASetGetMatrix()` allows you to provide general code for those more complicated nonzero patterns then
  can be represented in the `dfill`, `ofill` format

  Contributed by\:
  Glenn Hammond

.seealso: [](sec_struct), `DM`, `DMDA`, `DMCreateMatrix()`, `DMDASetGetMatrix()`, `DMSetMatrixPreallocateOnly()`, `DMDASetBlockFillsSparse()`
@*/
PetscErrorCode DMDASetBlockFills(DM da, const PetscInt *dfill, const PetscInt *ofill)
{
  DM_DA *dd = (DM_DA *)da->data;

  PetscFunctionBegin;
  /* save the given dfill and ofill information */
  PetscCall(DMDASetBlockFills_Private(dfill, dd->w, &dd->dfill));
  PetscCall(DMDASetBlockFills_Private(ofill, dd->w, &dd->ofill));

  /* count nonzeros in ofill columns */
  PetscCall(DMDASetBlockFills_Private2(dd));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  DMDASetBlockFillsSparse - Sets the fill pattern in each block for a multi-component problem
  of the matrix returned by `DMCreateMatrix()`, using sparse representations
  of fill patterns.

  Logically Collective

  Input Parameters:
+ da          - the `DMDA`
. dfillsparse - the sparse fill pattern in the diagonal block (may be `NULL`, means use dense block)
- ofillsparse - the sparse fill pattern in the off-diagonal blocks

  Level: developer

  Notes:
  This only makes sense when you are doing multicomponent problems but using the
  `MATMPIAIJ` matrix format

  The format for `dfill` and `ofill` is a sparse representation of a
  dof-by-dof matrix with 1 entries representing coupling and 0 entries
  for missing coupling.  The sparse representation is a 1 dimensional
  array of length nz + dof + 1, where nz is the number of non-zeros in
  the matrix.  The first dof entries in the array give the
  starting array indices of each row's items in the rest of the array,
  the dof+1st item contains the value nz + dof + 1 (i.e. the entire length of the array)
  and the remaining nz items give the column indices of each of
  the 1s within the logical 2D matrix.  Each row's items within
  the array are the column indices of the 1s within that row
  of the 2D matrix.  PETSc developers may recognize that this is the
  same format as that computed by the `DMDASetBlockFills_Private()`
  function from a dense 2D matrix representation.

  `DMDASetGetMatrix()` allows you to provide general code for those more complicated nonzero patterns then
  can be represented in the `dfill`, `ofill` format

  Contributed by\:
  Philip C. Roth

.seealso: [](sec_struct), `DM`, `DMDA`, `DMDASetBlockFills()`, `DMCreateMatrix()`, `DMDASetGetMatrix()`, `DMSetMatrixPreallocateOnly()`
@*/
PetscErrorCode DMDASetBlockFillsSparse(DM da, const PetscInt *dfillsparse, const PetscInt *ofillsparse)
{
  DM_DA *dd = (DM_DA *)da->data;

  PetscFunctionBegin;
  /* save the given dfill and ofill information */
  PetscCall(DMDASetBlockFillsSparse_Private(dfillsparse, dd->w, &dd->dfill));
  PetscCall(DMDASetBlockFillsSparse_Private(ofillsparse, dd->w, &dd->ofill));

  /* count nonzeros in ofill columns */
  PetscCall(DMDASetBlockFills_Private2(dd));
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode DMCreateColoring_DA(DM da, ISColoringType ctype, ISColoring *coloring)
{
  PetscInt       dim, m, n, p, nc;
  DMBoundaryType bx, by, bz;
  MPI_Comm       comm;
  PetscMPIInt    size;
  PetscBool      isBAIJ;
  DM_DA         *dd = (DM_DA *)da->data;

  PetscFunctionBegin;
  /*
                                  m
          ------------------------------------------------------
         |                                                     |
         |                                                     |
         |               ----------------------                |
         |               |                    |                |
      n  |           yn  |                    |                |
         |               |                    |                |
         |               .---------------------                |
         |             (xs,ys)     xn                          |
         |            .                                        |
         |         (gxs,gys)                                   |
         |                                                     |
          -----------------------------------------------------
  */

  /*
         nc - number of components per grid point
         col - number of colors needed in one direction for single component problem

  */
  PetscCall(DMDAGetInfo(da, &dim, NULL, NULL, NULL, &m, &n, &p, &nc, NULL, &bx, &by, &bz, NULL));

  PetscCall(PetscObjectGetComm((PetscObject)da, &comm));
  PetscCallMPI(MPI_Comm_size(comm, &size));
  if (ctype == IS_COLORING_LOCAL) {
    PetscCheck(!((m == 1 && bx == DM_BOUNDARY_PERIODIC) || (n == 1 && by == DM_BOUNDARY_PERIODIC) || (p == 1 && bz == DM_BOUNDARY_PERIODIC)), PetscObjectComm((PetscObject)da), PETSC_ERR_SUP, "IS_COLORING_LOCAL cannot be used for periodic boundary condition having both sides of the domain on the same process");
  }

  /* Tell the DMDA it has 1 degree of freedom per grid point so that the coloring for BAIJ
     matrices is for the blocks, not the individual matrix elements  */
  PetscCall(PetscStrbeginswith(da->mattype, MATBAIJ, &isBAIJ));
  if (!isBAIJ) PetscCall(PetscStrbeginswith(da->mattype, MATMPIBAIJ, &isBAIJ));
  if (!isBAIJ) PetscCall(PetscStrbeginswith(da->mattype, MATSEQBAIJ, &isBAIJ));
  if (isBAIJ) {
    dd->w  = 1;
    dd->xs = dd->xs / nc;
    dd->xe = dd->xe / nc;
    dd->Xs = dd->Xs / nc;
    dd->Xe = dd->Xe / nc;
  }

  /*
     We do not provide a getcoloring function in the DMDA operations because
   the basic DMDA does not know about matrices. We think of DMDA as being
   more low-level then matrices.
  */
  if (dim == 1) PetscCall(DMCreateColoring_DA_1d_MPIAIJ(da, ctype, coloring));
  else if (dim == 2) PetscCall(DMCreateColoring_DA_2d_MPIAIJ(da, ctype, coloring));
  else if (dim == 3) PetscCall(DMCreateColoring_DA_3d_MPIAIJ(da, ctype, coloring));
  else SETERRQ(PetscObjectComm((PetscObject)da), PETSC_ERR_SUP, "Not done for %" PetscInt_FMT " dimension, send us mail petsc-maint@mcs.anl.gov for code", dim);
  if (isBAIJ) {
    dd->w  = nc;
    dd->xs = dd->xs * nc;
    dd->xe = dd->xe * nc;
    dd->Xs = dd->Xs * nc;
    dd->Xe = dd->Xe * nc;
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode DMCreateColoring_DA_2d_MPIAIJ(DM da, ISColoringType ctype, ISColoring *coloring)
{
  PetscInt         xs, ys, nx, ny, i, j, ii, gxs, gys, gnx, gny, m, n, M, N, dim, s, k, nc, col;
  PetscInt         ncolors = 0;
  MPI_Comm         comm;
  DMBoundaryType   bx, by;
  DMDAStencilType  st;
  ISColoringValue *colors;
  DM_DA           *dd = (DM_DA *)da->data;

  PetscFunctionBegin;
  /*
         nc - number of components per grid point
         col - number of colors needed in one direction for single component problem

  */
  PetscCall(DMDAGetInfo(da, &dim, &m, &n, NULL, &M, &N, NULL, &nc, &s, &bx, &by, NULL, &st));
  col = 2 * s + 1;
  PetscCall(DMDAGetCorners(da, &xs, &ys, NULL, &nx, &ny, NULL));
  PetscCall(DMDAGetGhostCorners(da, &gxs, &gys, NULL, &gnx, &gny, NULL));
  PetscCall(PetscObjectGetComm((PetscObject)da, &comm));

  /* special case as taught to us by Paul Hovland */
  if (st == DMDA_STENCIL_STAR && s == 1) {
    PetscCall(DMCreateColoring_DA_2d_5pt_MPIAIJ(da, ctype, coloring));
  } else {
    if (ctype == IS_COLORING_GLOBAL) {
      if (!dd->localcoloring) {
        PetscCall(PetscMalloc1(nc * nx * ny, &colors));
        ii = 0;
        for (j = ys; j < ys + ny; j++) {
          for (i = xs; i < xs + nx; i++) {
            for (k = 0; k < nc; k++) PetscCall(ISColoringValueCast(k + nc * ((i % col) + col * (j % col)), colors + ii++));
          }
        }
        ncolors = nc + nc * (col - 1 + col * (col - 1));
        PetscCall(ISColoringCreate(comm, ncolors, nc * nx * ny, colors, PETSC_OWN_POINTER, &dd->localcoloring));
      }
      *coloring = dd->localcoloring;
    } else if (ctype == IS_COLORING_LOCAL) {
      if (!dd->ghostedcoloring) {
        PetscCall(PetscMalloc1(nc * gnx * gny, &colors));
        ii = 0;
        for (j = gys; j < gys + gny; j++) {
          for (i = gxs; i < gxs + gnx; i++) {
            for (k = 0; k < nc; k++) {
              /* the complicated stuff is to handle periodic boundaries */
              PetscCall(ISColoringValueCast(k + nc * ((SetInRange(i, m) % col) + col * (SetInRange(j, n) % col)), colors + ii++));
            }
          }
        }
        ncolors = nc + nc * (col - 1 + col * (col - 1));
        PetscCall(ISColoringCreate(comm, ncolors, nc * gnx * gny, colors, PETSC_OWN_POINTER, &dd->ghostedcoloring));
        /* PetscIntView(ncolors,(PetscInt*)colors,0); */

        PetscCall(ISColoringSetType(dd->ghostedcoloring, IS_COLORING_LOCAL));
      }
      *coloring = dd->ghostedcoloring;
    } else SETERRQ(PetscObjectComm((PetscObject)da), PETSC_ERR_ARG_WRONG, "Unknown ISColoringType %d", (int)ctype);
  }
  PetscCall(ISColoringReference(*coloring));
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode DMCreateColoring_DA_3d_MPIAIJ(DM da, ISColoringType ctype, ISColoring *coloring)
{
  PetscInt         xs, ys, nx, ny, i, j, gxs, gys, gnx, gny, m, n, p, dim, s, k, nc, col, zs, gzs, ii, l, nz, gnz, M, N, P;
  PetscInt         ncolors;
  MPI_Comm         comm;
  DMBoundaryType   bx, by, bz;
  DMDAStencilType  st;
  ISColoringValue *colors;
  DM_DA           *dd = (DM_DA *)da->data;

  PetscFunctionBegin;
  /*
         nc - number of components per grid point
         col - number of colors needed in one direction for single component problem
  */
  PetscCall(DMDAGetInfo(da, &dim, &m, &n, &p, &M, &N, &P, &nc, &s, &bx, &by, &bz, &st));
  col = 2 * s + 1;
  PetscCheck(bx != DM_BOUNDARY_PERIODIC || (m % col) == 0, PetscObjectComm((PetscObject)da), PETSC_ERR_SUP, "For coloring efficiency ensure number of grid points in X is divisible by 2*stencil_width + 1");
  PetscCheck(by != DM_BOUNDARY_PERIODIC || (n % col) == 0, PetscObjectComm((PetscObject)da), PETSC_ERR_SUP, "For coloring efficiency ensure number of grid points in Y is divisible by 2*stencil_width + 1");
  PetscCheck(bz != DM_BOUNDARY_PERIODIC || (p % col) == 0, PetscObjectComm((PetscObject)da), PETSC_ERR_SUP, "For coloring efficiency ensure number of grid points in Z is divisible by 2*stencil_width + 1");

  PetscCall(DMDAGetCorners(da, &xs, &ys, &zs, &nx, &ny, &nz));
  PetscCall(DMDAGetGhostCorners(da, &gxs, &gys, &gzs, &gnx, &gny, &gnz));
  PetscCall(PetscObjectGetComm((PetscObject)da, &comm));

  /* create the coloring */
  if (ctype == IS_COLORING_GLOBAL) {
    if (!dd->localcoloring) {
      PetscCall(PetscMalloc1(nc * nx * ny * nz, &colors));
      ii = 0;
      for (k = zs; k < zs + nz; k++) {
        for (j = ys; j < ys + ny; j++) {
          for (i = xs; i < xs + nx; i++) {
            for (l = 0; l < nc; l++) PetscCall(ISColoringValueCast(l + nc * ((i % col) + col * (j % col) + col * col * (k % col)), colors + ii++));
          }
        }
      }
      ncolors = nc + nc * (col - 1 + col * (col - 1) + col * col * (col - 1));
      PetscCall(ISColoringCreate(comm, ncolors, nc * nx * ny * nz, colors, PETSC_OWN_POINTER, &dd->localcoloring));
    }
    *coloring = dd->localcoloring;
  } else if (ctype == IS_COLORING_LOCAL) {
    if (!dd->ghostedcoloring) {
      PetscCall(PetscMalloc1(nc * gnx * gny * gnz, &colors));
      ii = 0;
      for (k = gzs; k < gzs + gnz; k++) {
        for (j = gys; j < gys + gny; j++) {
          for (i = gxs; i < gxs + gnx; i++) {
            for (l = 0; l < nc; l++) {
              /* the complicated stuff is to handle periodic boundaries */
              PetscCall(ISColoringValueCast(l + nc * ((SetInRange(i, m) % col) + col * (SetInRange(j, n) % col) + col * col * (SetInRange(k, p) % col)), colors + ii++));
            }
          }
        }
      }
      ncolors = nc + nc * (col - 1 + col * (col - 1) + col * col * (col - 1));
      PetscCall(ISColoringCreate(comm, ncolors, nc * gnx * gny * gnz, colors, PETSC_OWN_POINTER, &dd->ghostedcoloring));
      PetscCall(ISColoringSetType(dd->ghostedcoloring, IS_COLORING_LOCAL));
    }
    *coloring = dd->ghostedcoloring;
  } else SETERRQ(PetscObjectComm((PetscObject)da), PETSC_ERR_ARG_WRONG, "Unknown ISColoringType %d", (int)ctype);
  PetscCall(ISColoringReference(*coloring));
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode DMCreateColoring_DA_1d_MPIAIJ(DM da, ISColoringType ctype, ISColoring *coloring)
{
  PetscInt         xs, nx, i, i1, gxs, gnx, l, m, M, dim, s, nc, col;
  PetscInt         ncolors;
  MPI_Comm         comm;
  DMBoundaryType   bx;
  ISColoringValue *colors;
  DM_DA           *dd = (DM_DA *)da->data;

  PetscFunctionBegin;
  /*
         nc - number of components per grid point
         col - number of colors needed in one direction for single component problem
  */
  PetscCall(DMDAGetInfo(da, &dim, &m, NULL, NULL, &M, NULL, NULL, &nc, &s, &bx, NULL, NULL, NULL));
  col = 2 * s + 1;
  PetscCheck(bx != DM_BOUNDARY_PERIODIC || !(m % col), PetscObjectComm((PetscObject)da), PETSC_ERR_SUP, "IS_COLORING_GLOBAL can only be used for periodic boundary conditions if the number of grid points %" PetscInt_FMT " is divisible by the number of colors %" PetscInt_FMT, m, col);
  PetscCall(DMDAGetCorners(da, &xs, NULL, NULL, &nx, NULL, NULL));
  PetscCall(DMDAGetGhostCorners(da, &gxs, NULL, NULL, &gnx, NULL, NULL));
  PetscCall(PetscObjectGetComm((PetscObject)da, &comm));

  /* create the coloring */
  if (ctype == IS_COLORING_GLOBAL) {
    if (!dd->localcoloring) {
      PetscCall(PetscMalloc1(nc * nx, &colors));
      if (dd->ofillcols) {
        PetscInt tc = 0;
        for (i = 0; i < nc; i++) tc += (PetscInt)(dd->ofillcols[i] > 0);
        i1 = 0;
        for (i = xs; i < xs + nx; i++) {
          for (l = 0; l < nc; l++) {
            if (dd->ofillcols[l] && (i % col)) {
              PetscCall(ISColoringValueCast(nc - 1 + tc * ((i % col) - 1) + dd->ofillcols[l], colors + i1++));
            } else {
              PetscCall(ISColoringValueCast(l, colors + i1++));
            }
          }
        }
        ncolors = nc + 2 * s * tc;
      } else {
        i1 = 0;
        for (i = xs; i < xs + nx; i++) {
          for (l = 0; l < nc; l++) PetscCall(ISColoringValueCast(l + nc * (i % col), colors + i1++));
        }
        ncolors = nc + nc * (col - 1);
      }
      PetscCall(ISColoringCreate(comm, ncolors, nc * nx, colors, PETSC_OWN_POINTER, &dd->localcoloring));
    }
    *coloring = dd->localcoloring;
  } else if (ctype == IS_COLORING_LOCAL) {
    if (!dd->ghostedcoloring) {
      PetscCall(PetscMalloc1(nc * gnx, &colors));
      i1 = 0;
      for (i = gxs; i < gxs + gnx; i++) {
        for (l = 0; l < nc; l++) {
          /* the complicated stuff is to handle periodic boundaries */
          PetscCall(ISColoringValueCast(l + nc * (SetInRange(i, m) % col), colors + i1++));
        }
      }
      ncolors = nc + nc * (col - 1);
      PetscCall(ISColoringCreate(comm, ncolors, nc * gnx, colors, PETSC_OWN_POINTER, &dd->ghostedcoloring));
      PetscCall(ISColoringSetType(dd->ghostedcoloring, IS_COLORING_LOCAL));
    }
    *coloring = dd->ghostedcoloring;
  } else SETERRQ(PetscObjectComm((PetscObject)da), PETSC_ERR_ARG_WRONG, "Unknown ISColoringType %d", (int)ctype);
  PetscCall(ISColoringReference(*coloring));
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode DMCreateColoring_DA_2d_5pt_MPIAIJ(DM da, ISColoringType ctype, ISColoring *coloring)
{
  PetscInt         xs, ys, nx, ny, i, j, ii, gxs, gys, gnx, gny, m, n, dim, s, k, nc;
  PetscInt         ncolors;
  MPI_Comm         comm;
  DMBoundaryType   bx, by;
  ISColoringValue *colors;
  DM_DA           *dd = (DM_DA *)da->data;

  PetscFunctionBegin;
  /*
         nc - number of components per grid point
         col - number of colors needed in one direction for single component problem
  */
  PetscCall(DMDAGetInfo(da, &dim, &m, &n, NULL, NULL, NULL, NULL, &nc, &s, &bx, &by, NULL, NULL));
  PetscCall(DMDAGetCorners(da, &xs, &ys, NULL, &nx, &ny, NULL));
  PetscCall(DMDAGetGhostCorners(da, &gxs, &gys, NULL, &gnx, &gny, NULL));
  PetscCall(PetscObjectGetComm((PetscObject)da, &comm));
  /* create the coloring */
  if (ctype == IS_COLORING_GLOBAL) {
    if (!dd->localcoloring) {
      PetscCall(PetscMalloc1(nc * nx * ny, &colors));
      ii = 0;
      for (j = ys; j < ys + ny; j++) {
        for (i = xs; i < xs + nx; i++) {
          for (k = 0; k < nc; k++) PetscCall(ISColoringValueCast(k + nc * ((3 * j + i) % 5), colors + ii++));
        }
      }
      ncolors = 5 * nc;
      PetscCall(ISColoringCreate(comm, ncolors, nc * nx * ny, colors, PETSC_OWN_POINTER, &dd->localcoloring));
    }
    *coloring = dd->localcoloring;
  } else if (ctype == IS_COLORING_LOCAL) {
    if (!dd->ghostedcoloring) {
      PetscCall(PetscMalloc1(nc * gnx * gny, &colors));
      ii = 0;
      for (j = gys; j < gys + gny; j++) {
        for (i = gxs; i < gxs + gnx; i++) {
          for (k = 0; k < nc; k++) PetscCall(ISColoringValueCast(k + nc * ((3 * SetInRange(j, n) + SetInRange(i, m)) % 5), colors + ii++));
        }
      }
      ncolors = 5 * nc;
      PetscCall(ISColoringCreate(comm, ncolors, nc * gnx * gny, colors, PETSC_OWN_POINTER, &dd->ghostedcoloring));
      PetscCall(ISColoringSetType(dd->ghostedcoloring, IS_COLORING_LOCAL));
    }
    *coloring = dd->ghostedcoloring;
  } else SETERRQ(PetscObjectComm((PetscObject)da), PETSC_ERR_ARG_WRONG, "Unknown ISColoringType %d", (int)ctype);
  PetscFunctionReturn(PETSC_SUCCESS);
}

/* =========================================================================== */
extern PetscErrorCode DMCreateMatrix_DA_1d_MPIAIJ(DM, Mat);
extern PetscErrorCode DMCreateMatrix_DA_1d_MPIAIJ_Fill(DM, Mat);
extern PetscErrorCode DMCreateMatrix_DA_1d_SeqAIJ_NoPreallocation(DM, Mat);
extern PetscErrorCode DMCreateMatrix_DA_2d_MPIAIJ(DM, Mat);
extern PetscErrorCode DMCreateMatrix_DA_2d_MPIAIJ_Fill(DM, Mat);
extern PetscErrorCode DMCreateMatrix_DA_3d_MPIAIJ(DM, Mat);
extern PetscErrorCode DMCreateMatrix_DA_3d_MPIAIJ_Fill(DM, Mat);
extern PetscErrorCode DMCreateMatrix_DA_2d_MPIBAIJ(DM, Mat);
extern PetscErrorCode DMCreateMatrix_DA_3d_MPIBAIJ(DM, Mat);
extern PetscErrorCode DMCreateMatrix_DA_2d_MPISBAIJ(DM, Mat);
extern PetscErrorCode DMCreateMatrix_DA_3d_MPISBAIJ(DM, Mat);
extern PetscErrorCode DMCreateMatrix_DA_2d_MPISELL(DM, Mat);
extern PetscErrorCode DMCreateMatrix_DA_3d_MPISELL(DM, Mat);
extern PetscErrorCode DMCreateMatrix_DA_IS(DM, Mat);

static PetscErrorCode MatView_MPI_DA(Mat A, PetscViewer viewer)
{
  DM                da;
  const char       *prefix;
  Mat               AA, Anatural;
  AO                ao;
  PetscInt          rstart, rend, *petsc, i;
  IS                is;
  MPI_Comm          comm;
  PetscViewerFormat format;
  PetscBool         flag;

  PetscFunctionBegin;
  /* Check whether we are just printing info, in which case MatView() already viewed everything we wanted to view */
  PetscCall(PetscViewerGetFormat(viewer, &format));
  if (format == PETSC_VIEWER_ASCII_INFO || format == PETSC_VIEWER_ASCII_INFO_DETAIL) PetscFunctionReturn(PETSC_SUCCESS);

  PetscCall(PetscObjectGetComm((PetscObject)A, &comm));
  PetscCall(MatGetDM(A, &da));
  PetscCheck(da, PetscObjectComm((PetscObject)A), PETSC_ERR_ARG_WRONG, "Matrix not generated from a DMDA");

  PetscCall(DMDAGetAO(da, &ao));
  PetscCall(MatGetOwnershipRange(A, &rstart, &rend));
  PetscCall(PetscMalloc1(rend - rstart, &petsc));
  for (i = rstart; i < rend; i++) petsc[i - rstart] = i;
  PetscCall(AOApplicationToPetsc(ao, rend - rstart, petsc));
  PetscCall(ISCreateGeneral(comm, rend - rstart, petsc, PETSC_OWN_POINTER, &is));

  /* call viewer on natural ordering */
  PetscCall(PetscObjectBaseTypeCompare((PetscObject)A, MATMPISELL, &flag));
  if (flag) {
    PetscCall(MatConvert(A, MATAIJ, MAT_INITIAL_MATRIX, &AA));
    A = AA;
  }
  PetscCall(MatCreateSubMatrix(A, is, is, MAT_INITIAL_MATRIX, &Anatural));
  PetscCall(ISDestroy(&is));
  PetscCall(PetscObjectGetOptionsPrefix((PetscObject)A, &prefix));
  PetscCall(PetscObjectSetOptionsPrefix((PetscObject)Anatural, prefix));
  PetscCall(PetscObjectSetName((PetscObject)Anatural, ((PetscObject)A)->name));
  ((PetscObject)Anatural)->donotPetscObjectPrintClassNamePrefixType = PETSC_TRUE;
  PetscCall(MatView(Anatural, viewer));
  ((PetscObject)Anatural)->donotPetscObjectPrintClassNamePrefixType = PETSC_FALSE;
  PetscCall(MatDestroy(&Anatural));
  if (flag) PetscCall(MatDestroy(&AA));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode MatLoad_MPI_DA(Mat A, PetscViewer viewer)
{
  DM       da;
  Mat      Anatural, Aapp;
  AO       ao;
  PetscInt rstart, rend, *app, i, m, n, M, N;
  IS       is;
  MPI_Comm comm;

  PetscFunctionBegin;
  PetscCall(PetscObjectGetComm((PetscObject)A, &comm));
  PetscCall(MatGetDM(A, &da));
  PetscCheck(da, PetscObjectComm((PetscObject)A), PETSC_ERR_ARG_WRONG, "Matrix not generated from a DMDA");

  /* Load the matrix in natural ordering */
  PetscCall(MatCreate(PetscObjectComm((PetscObject)A), &Anatural));
  PetscCall(MatSetType(Anatural, ((PetscObject)A)->type_name));
  PetscCall(MatGetSize(A, &M, &N));
  PetscCall(MatGetLocalSize(A, &m, &n));
  PetscCall(MatSetSizes(Anatural, m, n, M, N));
  PetscCall(MatLoad(Anatural, viewer));

  /* Map natural ordering to application ordering and create IS */
  PetscCall(DMDAGetAO(da, &ao));
  PetscCall(MatGetOwnershipRange(Anatural, &rstart, &rend));
  PetscCall(PetscMalloc1(rend - rstart, &app));
  for (i = rstart; i < rend; i++) app[i - rstart] = i;
  PetscCall(AOPetscToApplication(ao, rend - rstart, app));
  PetscCall(ISCreateGeneral(comm, rend - rstart, app, PETSC_OWN_POINTER, &is));

  /* Do permutation and replace header */
  PetscCall(MatCreateSubMatrix(Anatural, is, is, MAT_INITIAL_MATRIX, &Aapp));
  PetscCall(MatHeaderReplace(A, &Aapp));
  PetscCall(ISDestroy(&is));
  PetscCall(MatDestroy(&Anatural));
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode DMCreateMatrix_DA(DM da, Mat *J)
{
  PetscInt    dim, dof, nx, ny, nz, dims[3], starts[3], M, N, P;
  Mat         A;
  MPI_Comm    comm;
  MatType     Atype;
  MatType     mtype;
  PetscMPIInt size;
  DM_DA      *dd  = (DM_DA *)da->data;
  PetscBool   aij = PETSC_FALSE, baij = PETSC_FALSE, sbaij = PETSC_FALSE, sell = PETSC_FALSE, is = PETSC_FALSE;

  PetscFunctionBegin;
  PetscCall(MatInitializePackage());
  mtype = da->mattype;

  /*
                                  m
          ------------------------------------------------------
         |                                                     |
         |                                                     |
         |               ----------------------                |
         |               |                    |                |
      n  |           ny  |                    |                |
         |               |                    |                |
         |               .---------------------                |
         |             (xs,ys)     nx                          |
         |            .                                        |
         |         (gxs,gys)                                   |
         |                                                     |
          -----------------------------------------------------
  */

  /*
         nc - number of components per grid point
         col - number of colors needed in one direction for single component problem
  */
  M   = dd->M;
  N   = dd->N;
  P   = dd->P;
  dim = da->dim;
  dof = dd->w;
  /* PetscCall(DMDAGetInfo(da,&dim,&M,&N,&P,NULL,NULL,NULL,&dof,NULL,NULL,NULL,NULL,NULL)); */
  PetscCall(DMDAGetCorners(da, NULL, NULL, NULL, &nx, &ny, &nz));
  PetscCall(PetscObjectGetComm((PetscObject)da, &comm));
  PetscCall(MatCreate(comm, &A));
  PetscCall(MatSetSizes(A, dof * nx * ny * nz, dof * nx * ny * nz, dof * M * N * P, dof * M * N * P));
  PetscCall(MatSetType(A, mtype));
  PetscCall(MatSetFromOptions(A));
  if (dof * nx * ny * nz < da->bind_below) {
    PetscCall(MatSetBindingPropagates(A, PETSC_TRUE));
    PetscCall(MatBindToCPU(A, PETSC_TRUE));
  }
  PetscCall(MatSetDM(A, da));
  if (da->structure_only) PetscCall(MatSetOption(A, MAT_STRUCTURE_ONLY, PETSC_TRUE));
  PetscCall(MatGetType(A, &Atype));
  /*
     We do not provide a getmatrix function in the DMDA operations because
   the basic DMDA does not know about matrices. We think of DMDA as being more
   more low-level than matrices. This is kind of cheating but, cause sometimes
   we think of DMDA has higher level than matrices.

     We could switch based on Atype (or mtype), but we do not since the
   specialized setting routines depend only on the particular preallocation
   details of the matrix, not the type itself.
  */
  if (!da->prealloc_skip) { // Flag is likely set when user intends to use MatSetPreallocationCOO()
    PetscCall(PetscObjectHasFunction((PetscObject)A, "MatMPIAIJSetPreallocation_C", &aij));
    if (!aij) PetscCall(PetscObjectHasFunction((PetscObject)A, "MatSeqAIJSetPreallocation_C", &aij));
    if (!aij) {
      PetscCall(PetscObjectHasFunction((PetscObject)A, "MatMPIBAIJSetPreallocation_C", &baij));
      if (!baij) PetscCall(PetscObjectHasFunction((PetscObject)A, "MatSeqBAIJSetPreallocation_C", &baij));
      if (!baij) {
        PetscCall(PetscObjectHasFunction((PetscObject)A, "MatMPISBAIJSetPreallocation_C", &sbaij));
        if (!sbaij) PetscCall(PetscObjectHasFunction((PetscObject)A, "MatSeqSBAIJSetPreallocation_C", &sbaij));
        if (!sbaij) {
          PetscCall(PetscObjectHasFunction((PetscObject)A, "MatMPISELLSetPreallocation_C", &sell));
          if (!sell) PetscCall(PetscObjectHasFunction((PetscObject)A, "MatSeqSELLSetPreallocation_C", &sell));
        }
        if (!sell) PetscCall(PetscObjectHasFunction((PetscObject)A, "MatISSetPreallocation_C", &is));
      }
    }
  }
  if (aij) {
    if (dim == 1) {
      if (dd->ofill) {
        PetscCall(DMCreateMatrix_DA_1d_MPIAIJ_Fill(da, A));
      } else {
        DMBoundaryType bx;
        PetscMPIInt    size;
        PetscCall(DMDAGetInfo(da, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL, NULL, &bx, NULL, NULL, NULL));
        PetscCallMPI(MPI_Comm_size(PetscObjectComm((PetscObject)da), &size));
        if (size == 1 && bx == DM_BOUNDARY_NONE) {
          PetscCall(DMCreateMatrix_DA_1d_SeqAIJ_NoPreallocation(da, A));
        } else {
          PetscCall(DMCreateMatrix_DA_1d_MPIAIJ(da, A));
        }
      }
    } else if (dim == 2) {
      if (dd->ofill) {
        PetscCall(DMCreateMatrix_DA_2d_MPIAIJ_Fill(da, A));
      } else {
        PetscCall(DMCreateMatrix_DA_2d_MPIAIJ(da, A));
      }
    } else if (dim == 3) {
      if (dd->ofill) {
        PetscCall(DMCreateMatrix_DA_3d_MPIAIJ_Fill(da, A));
      } else {
        PetscCall(DMCreateMatrix_DA_3d_MPIAIJ(da, A));
      }
    }
  } else if (baij) {
    if (dim == 2) {
      PetscCall(DMCreateMatrix_DA_2d_MPIBAIJ(da, A));
    } else if (dim == 3) {
      PetscCall(DMCreateMatrix_DA_3d_MPIBAIJ(da, A));
    } else SETERRQ(PetscObjectComm((PetscObject)da), PETSC_ERR_SUP, "Not implemented for %" PetscInt_FMT " dimension and Matrix Type: %s in %" PetscInt_FMT " dimension! Send mail to petsc-maint@mcs.anl.gov for code", dim, Atype, dim);
  } else if (sbaij) {
    if (dim == 2) {
      PetscCall(DMCreateMatrix_DA_2d_MPISBAIJ(da, A));
    } else if (dim == 3) {
      PetscCall(DMCreateMatrix_DA_3d_MPISBAIJ(da, A));
    } else SETERRQ(PetscObjectComm((PetscObject)da), PETSC_ERR_SUP, "Not implemented for %" PetscInt_FMT " dimension and Matrix Type: %s in %" PetscInt_FMT " dimension! Send mail to petsc-maint@mcs.anl.gov for code", dim, Atype, dim);
  } else if (sell) {
    if (dim == 2) {
      PetscCall(DMCreateMatrix_DA_2d_MPISELL(da, A));
    } else if (dim == 3) {
      PetscCall(DMCreateMatrix_DA_3d_MPISELL(da, A));
    } else SETERRQ(PetscObjectComm((PetscObject)da), PETSC_ERR_SUP, "Not implemented for %" PetscInt_FMT " dimension and Matrix Type: %s in %" PetscInt_FMT " dimension! Send mail to petsc-maint@mcs.anl.gov for code", dim, Atype, dim);
  } else if (is) {
    PetscCall(DMCreateMatrix_DA_IS(da, A));
  } else { // unknown type or da->prealloc_skip so structural information may be needed, but no prealloc
    ISLocalToGlobalMapping ltog;

    PetscCall(MatSetBlockSize(A, dof));
    PetscCall(MatSetUp(A));
    PetscCall(DMGetLocalToGlobalMapping(da, &ltog));
    PetscCall(MatSetLocalToGlobalMapping(A, ltog, ltog));
  }
  PetscCall(DMDAGetGhostCorners(da, &starts[0], &starts[1], &starts[2], &dims[0], &dims[1], &dims[2]));
  PetscCall(MatSetStencil(A, dim, dims, starts, dof));
  PetscCall(MatSetDM(A, da));
  PetscCallMPI(MPI_Comm_size(comm, &size));
  if (size > 1) {
    /* change viewer to display matrix in natural ordering */
    PetscCall(MatSetOperation(A, MATOP_VIEW, (PetscErrorCodeFn *)MatView_MPI_DA));
    PetscCall(MatSetOperation(A, MATOP_LOAD, (PetscErrorCodeFn *)MatLoad_MPI_DA));
  }
  *J = A;
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode DMCreateMatrix_DA_IS(DM dm, Mat J)
{
  DM_DA                 *da = (DM_DA *)dm->data;
  Mat                    lJ, P;
  ISLocalToGlobalMapping ltog;
  IS                     is;
  PetscBT                bt;
  const PetscInt        *e_loc, *idx;
  PetscInt               i, nel, nen, nv, dof, *gidx, n, N;

  /* The l2g map of DMDA has all ghosted nodes, and e_loc is a subset of all the local nodes (including the ghosted)
     We need to filter out the local indices that are not represented through the DMDAGetElements decomposition */
  PetscFunctionBegin;
  dof = da->w;
  PetscCall(MatSetBlockSize(J, dof));
  PetscCall(DMGetLocalToGlobalMapping(dm, &ltog));

  /* flag local elements indices in local DMDA numbering */
  PetscCall(ISLocalToGlobalMappingGetSize(ltog, &nv));
  PetscCall(PetscBTCreate(nv / dof, &bt));
  PetscCall(DMDAGetElements(dm, &nel, &nen, &e_loc)); /* this will throw an error if the stencil type is not DMDA_STENCIL_BOX */
  for (i = 0; i < nel * nen; i++) PetscCall(PetscBTSet(bt, e_loc[i]));

  /* filter out (set to -1) the global indices not used by the local elements */
  PetscCall(PetscMalloc1(nv / dof, &gidx));
  PetscCall(ISLocalToGlobalMappingGetBlockIndices(ltog, &idx));
  PetscCall(PetscArraycpy(gidx, idx, nv / dof));
  PetscCall(ISLocalToGlobalMappingRestoreBlockIndices(ltog, &idx));
  for (i = 0; i < nv / dof; i++)
    if (!PetscBTLookup(bt, i)) gidx[i] = -1;
  PetscCall(PetscBTDestroy(&bt));
  PetscCall(ISCreateBlock(PetscObjectComm((PetscObject)dm), dof, nv / dof, gidx, PETSC_OWN_POINTER, &is));
  PetscCall(ISLocalToGlobalMappingCreateIS(is, &ltog));
  PetscCall(MatSetLocalToGlobalMapping(J, ltog, ltog));
  PetscCall(ISLocalToGlobalMappingDestroy(&ltog));
  PetscCall(ISDestroy(&is));

  /* Preallocation */
  if (dm->prealloc_skip) {
    PetscCall(MatSetUp(J));
  } else {
    PetscCall(MatISGetLocalMat(J, &lJ));
    PetscCall(MatGetLocalToGlobalMapping(lJ, &ltog, NULL));
    PetscCall(MatCreate(PetscObjectComm((PetscObject)lJ), &P));
    PetscCall(MatSetType(P, MATPREALLOCATOR));
    PetscCall(MatSetLocalToGlobalMapping(P, ltog, ltog));
    PetscCall(MatGetSize(lJ, &N, NULL));
    PetscCall(MatGetLocalSize(lJ, &n, NULL));
    PetscCall(MatSetSizes(P, n, n, N, N));
    PetscCall(MatSetBlockSize(P, dof));
    PetscCall(MatSetUp(P));
    for (i = 0; i < nel; i++) PetscCall(MatSetValuesBlockedLocal(P, nen, e_loc + i * nen, nen, e_loc + i * nen, NULL, INSERT_VALUES));
    PetscCall(MatPreallocatorPreallocate(P, (PetscBool)!da->prealloc_only, lJ));
    PetscCall(MatISRestoreLocalMat(J, &lJ));
    PetscCall(DMDARestoreElements(dm, &nel, &nen, &e_loc));
    PetscCall(MatDestroy(&P));

    PetscCall(MatAssemblyBegin(J, MAT_FINAL_ASSEMBLY));
    PetscCall(MatAssemblyEnd(J, MAT_FINAL_ASSEMBLY));
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode DMCreateMatrix_DA_2d_MPISELL(DM da, Mat J)
{
  PetscInt               xs, ys, nx, ny, i, j, slot, gxs, gys, gnx, gny, m, n, dim, s, *cols = NULL, k, nc, *rows = NULL, col, cnt, l, p;
  PetscInt               lstart, lend, pstart, pend, *dnz, *onz;
  MPI_Comm               comm;
  PetscScalar           *values;
  DMBoundaryType         bx, by;
  ISLocalToGlobalMapping ltog;
  DMDAStencilType        st;

  PetscFunctionBegin;
  /*
         nc - number of components per grid point
         col - number of colors needed in one direction for single component problem
  */
  PetscCall(DMDAGetInfo(da, &dim, &m, &n, NULL, NULL, NULL, NULL, &nc, &s, &bx, &by, NULL, &st));
  col = 2 * s + 1;
  PetscCall(DMDAGetCorners(da, &xs, &ys, NULL, &nx, &ny, NULL));
  PetscCall(DMDAGetGhostCorners(da, &gxs, &gys, NULL, &gnx, &gny, NULL));
  PetscCall(PetscObjectGetComm((PetscObject)da, &comm));

  PetscCall(PetscMalloc2(nc, &rows, col * col * nc * nc, &cols));
  PetscCall(DMGetLocalToGlobalMapping(da, &ltog));

  PetscCall(MatSetBlockSize(J, nc));
  /* determine the matrix preallocation information */
  MatPreallocateBegin(comm, nc * nx * ny, nc * nx * ny, dnz, onz);
  for (i = xs; i < xs + nx; i++) {
    pstart = (bx == DM_BOUNDARY_PERIODIC) ? -s : (PetscMax(-s, -i));
    pend   = (bx == DM_BOUNDARY_PERIODIC) ? s : (PetscMin(s, m - i - 1));

    for (j = ys; j < ys + ny; j++) {
      slot = i - gxs + gnx * (j - gys);

      lstart = (by == DM_BOUNDARY_PERIODIC) ? -s : (PetscMax(-s, -j));
      lend   = (by == DM_BOUNDARY_PERIODIC) ? s : (PetscMin(s, n - j - 1));

      cnt = 0;
      for (k = 0; k < nc; k++) {
        for (l = lstart; l < lend + 1; l++) {
          for (p = pstart; p < pend + 1; p++) {
            if (st == DMDA_STENCIL_BOX || !l || !p) { /* entries on star have either l = 0 or p = 0 */
              cols[cnt++] = k + nc * (slot + gnx * l + p);
            }
          }
        }
        rows[k] = k + nc * (slot);
      }
      PetscCall(MatPreallocateSetLocal(ltog, nc, rows, ltog, cnt, cols, dnz, onz));
    }
  }
  PetscCall(MatSetBlockSize(J, nc));
  PetscCall(MatSeqSELLSetPreallocation(J, 0, dnz));
  PetscCall(MatMPISELLSetPreallocation(J, 0, dnz, 0, onz));
  MatPreallocateEnd(dnz, onz);

  PetscCall(MatSetLocalToGlobalMapping(J, ltog, ltog));

  /*
    For each node in the grid: we get the neighbors in the local (on processor ordering
    that includes the ghost points) then MatSetValuesLocal() maps those indices to the global
    PETSc ordering.
  */
  if (!da->prealloc_only) {
    PetscCall(PetscCalloc1(col * col * nc * nc, &values));
    for (i = xs; i < xs + nx; i++) {
      pstart = (bx == DM_BOUNDARY_PERIODIC) ? -s : (PetscMax(-s, -i));
      pend   = (bx == DM_BOUNDARY_PERIODIC) ? s : (PetscMin(s, m - i - 1));

      for (j = ys; j < ys + ny; j++) {
        slot = i - gxs + gnx * (j - gys);

        lstart = (by == DM_BOUNDARY_PERIODIC) ? -s : (PetscMax(-s, -j));
        lend   = (by == DM_BOUNDARY_PERIODIC) ? s : (PetscMin(s, n - j - 1));

        cnt = 0;
        for (k = 0; k < nc; k++) {
          for (l = lstart; l < lend + 1; l++) {
            for (p = pstart; p < pend + 1; p++) {
              if (st == DMDA_STENCIL_BOX || !l || !p) { /* entries on star have either l = 0 or p = 0 */
                cols[cnt++] = k + nc * (slot + gnx * l + p);
              }
            }
          }
          rows[k] = k + nc * (slot);
        }
        PetscCall(MatSetValuesLocal(J, nc, rows, cnt, cols, values, INSERT_VALUES));
      }
    }
    PetscCall(PetscFree(values));
    /* do not copy values to GPU since they are all zero and not yet needed there */
    PetscCall(MatBindToCPU(J, PETSC_TRUE));
    PetscCall(MatAssemblyBegin(J, MAT_FINAL_ASSEMBLY));
    PetscCall(MatAssemblyEnd(J, MAT_FINAL_ASSEMBLY));
    PetscCall(MatBindToCPU(J, PETSC_FALSE));
    PetscCall(MatSetOption(J, MAT_NEW_NONZERO_LOCATION_ERR, PETSC_TRUE));
  }
  PetscCall(PetscFree2(rows, cols));
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode DMCreateMatrix_DA_3d_MPISELL(DM da, Mat J)
{
  PetscInt               xs, ys, nx, ny, i, j, slot, gxs, gys, gnx, gny;
  PetscInt               m, n, dim, s, *cols = NULL, k, nc, *rows = NULL, col, cnt, l, p, *dnz = NULL, *onz = NULL;
  PetscInt               istart, iend, jstart, jend, kstart, kend, zs, nz, gzs, gnz, ii, jj, kk, M, N, P;
  MPI_Comm               comm;
  PetscScalar           *values;
  DMBoundaryType         bx, by, bz;
  ISLocalToGlobalMapping ltog;
  DMDAStencilType        st;

  PetscFunctionBegin;
  /*
         nc - number of components per grid point
         col - number of colors needed in one direction for single component problem
  */
  PetscCall(DMDAGetInfo(da, &dim, &m, &n, &p, &M, &N, &P, &nc, &s, &bx, &by, &bz, &st));
  col = 2 * s + 1;
  PetscCall(DMDAGetCorners(da, &xs, &ys, &zs, &nx, &ny, &nz));
  PetscCall(DMDAGetGhostCorners(da, &gxs, &gys, &gzs, &gnx, &gny, &gnz));
  PetscCall(PetscObjectGetComm((PetscObject)da, &comm));

  PetscCall(PetscMalloc2(nc, &rows, col * col * col * nc * nc, &cols));
  PetscCall(DMGetLocalToGlobalMapping(da, &ltog));

  PetscCall(MatSetBlockSize(J, nc));
  /* determine the matrix preallocation information */
  MatPreallocateBegin(comm, nc * nx * ny * nz, nc * nx * ny * nz, dnz, onz);
  for (i = xs; i < xs + nx; i++) {
    istart = (bx == DM_BOUNDARY_PERIODIC) ? -s : (PetscMax(-s, -i));
    iend   = (bx == DM_BOUNDARY_PERIODIC) ? s : (PetscMin(s, m - i - 1));
    for (j = ys; j < ys + ny; j++) {
      jstart = (by == DM_BOUNDARY_PERIODIC) ? -s : (PetscMax(-s, -j));
      jend   = (by == DM_BOUNDARY_PERIODIC) ? s : (PetscMin(s, n - j - 1));
      for (k = zs; k < zs + nz; k++) {
        kstart = (bz == DM_BOUNDARY_PERIODIC) ? -s : (PetscMax(-s, -k));
        kend   = (bz == DM_BOUNDARY_PERIODIC) ? s : (PetscMin(s, p - k - 1));

        slot = i - gxs + gnx * (j - gys) + gnx * gny * (k - gzs);

        cnt = 0;
        for (l = 0; l < nc; l++) {
          for (ii = istart; ii < iend + 1; ii++) {
            for (jj = jstart; jj < jend + 1; jj++) {
              for (kk = kstart; kk < kend + 1; kk++) {
                if (st == DMDA_STENCIL_BOX || ((!ii && !jj) || (!jj && !kk) || (!ii && !kk))) { /* entries on star*/
                  cols[cnt++] = l + nc * (slot + ii + gnx * jj + gnx * gny * kk);
                }
              }
            }
          }
          rows[l] = l + nc * (slot);
        }
        PetscCall(MatPreallocateSetLocal(ltog, nc, rows, ltog, cnt, cols, dnz, onz));
      }
    }
  }
  PetscCall(MatSetBlockSize(J, nc));
  PetscCall(MatSeqSELLSetPreallocation(J, 0, dnz));
  PetscCall(MatMPISELLSetPreallocation(J, 0, dnz, 0, onz));
  MatPreallocateEnd(dnz, onz);
  PetscCall(MatSetLocalToGlobalMapping(J, ltog, ltog));

  /*
    For each node in the grid: we get the neighbors in the local (on processor ordering
    that includes the ghost points) then MatSetValuesLocal() maps those indices to the global
    PETSc ordering.
  */
  if (!da->prealloc_only) {
    PetscCall(PetscCalloc1(col * col * col * nc * nc * nc, &values));
    for (i = xs; i < xs + nx; i++) {
      istart = (bx == DM_BOUNDARY_PERIODIC) ? -s : (PetscMax(-s, -i));
      iend   = (bx == DM_BOUNDARY_PERIODIC) ? s : (PetscMin(s, m - i - 1));
      for (j = ys; j < ys + ny; j++) {
        jstart = (by == DM_BOUNDARY_PERIODIC) ? -s : (PetscMax(-s, -j));
        jend   = (by == DM_BOUNDARY_PERIODIC) ? s : (PetscMin(s, n - j - 1));
        for (k = zs; k < zs + nz; k++) {
          kstart = (bz == DM_BOUNDARY_PERIODIC) ? -s : (PetscMax(-s, -k));
          kend   = (bz == DM_BOUNDARY_PERIODIC) ? s : (PetscMin(s, p - k - 1));

          slot = i - gxs + gnx * (j - gys) + gnx * gny * (k - gzs);

          cnt = 0;
          for (l = 0; l < nc; l++) {
            for (ii = istart; ii < iend + 1; ii++) {
              for (jj = jstart; jj < jend + 1; jj++) {
                for (kk = kstart; kk < kend + 1; kk++) {
                  if (st == DMDA_STENCIL_BOX || ((!ii && !jj) || (!jj && !kk) || (!ii && !kk))) { /* entries on star*/
                    cols[cnt++] = l + nc * (slot + ii + gnx * jj + gnx * gny * kk);
                  }
                }
              }
            }
            rows[l] = l + nc * (slot);
          }
          PetscCall(MatSetValuesLocal(J, nc, rows, cnt, cols, values, INSERT_VALUES));
        }
      }
    }
    PetscCall(PetscFree(values));
    /* do not copy values to GPU since they are all zero and not yet needed there */
    PetscCall(MatBindToCPU(J, PETSC_TRUE));
    PetscCall(MatAssemblyBegin(J, MAT_FINAL_ASSEMBLY));
    PetscCall(MatAssemblyEnd(J, MAT_FINAL_ASSEMBLY));
    PetscCall(MatBindToCPU(J, PETSC_FALSE));
    PetscCall(MatSetOption(J, MAT_NEW_NONZERO_LOCATION_ERR, PETSC_TRUE));
  }
  PetscCall(PetscFree2(rows, cols));
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode DMCreateMatrix_DA_2d_MPIAIJ(DM da, Mat J)
{
  PetscInt               xs, ys, nx, ny, i, j, slot, gxs, gys, gnx, gny, m, n, dim, s, *cols = NULL, k, nc, *rows = NULL, col, cnt, l, p, M, N;
  PetscInt               lstart, lend, pstart, pend, *dnz, *onz;
  MPI_Comm               comm;
  DMBoundaryType         bx, by;
  ISLocalToGlobalMapping ltog, mltog;
  DMDAStencilType        st;
  PetscBool              removedups = PETSC_FALSE, alreadyboundtocpu = PETSC_TRUE;

  PetscFunctionBegin;
  /*
         nc - number of components per grid point
         col - number of colors needed in one direction for single component problem
  */
  PetscCall(DMDAGetInfo(da, &dim, &m, &n, NULL, &M, &N, NULL, &nc, &s, &bx, &by, NULL, &st));
  if (bx == DM_BOUNDARY_NONE && by == DM_BOUNDARY_NONE) PetscCall(MatSetOption(J, MAT_SORTED_FULL, PETSC_TRUE));
  col = 2 * s + 1;
  /*
       With one processor in periodic domains in a skinny dimension the code will label nonzero columns multiple times
       because of "wrapping" around the end of the domain hitting an entry already counted in the other direction.
  */
  if (M == 1 && 2 * s >= m) removedups = PETSC_TRUE;
  if (N == 1 && 2 * s >= n) removedups = PETSC_TRUE;
  PetscCall(DMDAGetCorners(da, &xs, &ys, NULL, &nx, &ny, NULL));
  PetscCall(DMDAGetGhostCorners(da, &gxs, &gys, NULL, &gnx, &gny, NULL));
  PetscCall(PetscObjectGetComm((PetscObject)da, &comm));

  PetscCall(PetscMalloc2(nc, &rows, col * col * nc * nc, &cols));
  PetscCall(DMGetLocalToGlobalMapping(da, &ltog));

  PetscCall(MatSetBlockSize(J, nc));
  /* determine the matrix preallocation information */
  MatPreallocateBegin(comm, nc * nx * ny, nc * nx * ny, dnz, onz);
  for (i = xs; i < xs + nx; i++) {
    pstart = (bx == DM_BOUNDARY_PERIODIC) ? -s : (PetscMax(-s, -i));
    pend   = (bx == DM_BOUNDARY_PERIODIC) ? s : (PetscMin(s, m - i - 1));

    for (j = ys; j < ys + ny; j++) {
      slot = i - gxs + gnx * (j - gys);

      lstart = (by == DM_BOUNDARY_PERIODIC) ? -s : (PetscMax(-s, -j));
      lend   = (by == DM_BOUNDARY_PERIODIC) ? s : (PetscMin(s, n - j - 1));

      cnt = 0;
      for (k = 0; k < nc; k++) {
        for (l = lstart; l < lend + 1; l++) {
          for (p = pstart; p < pend + 1; p++) {
            if (st == DMDA_STENCIL_BOX || !l || !p) { /* entries on star have either l = 0 or p = 0 */
              cols[cnt++] = k + nc * (slot + gnx * l + p);
            }
          }
        }
        rows[k] = k + nc * (slot);
      }
      if (removedups) PetscCall(MatPreallocateSetLocalRemoveDups(ltog, nc, rows, ltog, cnt, cols, dnz, onz));
      else PetscCall(MatPreallocateSetLocal(ltog, nc, rows, ltog, cnt, cols, dnz, onz));
    }
  }
  PetscCall(MatSetBlockSize(J, nc));
  PetscCall(MatSeqAIJSetPreallocation(J, 0, dnz));
  PetscCall(MatMPIAIJSetPreallocation(J, 0, dnz, 0, onz));
  MatPreallocateEnd(dnz, onz);
  PetscCall(MatGetLocalToGlobalMapping(J, &mltog, NULL));
  if (!mltog) PetscCall(MatSetLocalToGlobalMapping(J, ltog, ltog));

  /*
    For each node in the grid: we get the neighbors in the local (on processor ordering
    that includes the ghost points) then MatSetValuesLocal() maps those indices to the global
    PETSc ordering.
  */
  if (!da->prealloc_only) {
    for (i = xs; i < xs + nx; i++) {
      pstart = (bx == DM_BOUNDARY_PERIODIC) ? -s : (PetscMax(-s, -i));
      pend   = (bx == DM_BOUNDARY_PERIODIC) ? s : (PetscMin(s, m - i - 1));

      for (j = ys; j < ys + ny; j++) {
        slot = i - gxs + gnx * (j - gys);

        lstart = (by == DM_BOUNDARY_PERIODIC) ? -s : (PetscMax(-s, -j));
        lend   = (by == DM_BOUNDARY_PERIODIC) ? s : (PetscMin(s, n - j - 1));

        cnt = 0;
        for (l = lstart; l < lend + 1; l++) {
          for (p = pstart; p < pend + 1; p++) {
            if (st == DMDA_STENCIL_BOX || !l || !p) { /* entries on star have either l = 0 or p = 0 */
              cols[cnt++] = nc * (slot + gnx * l + p);
              for (k = 1; k < nc; k++) {
                cols[cnt] = 1 + cols[cnt - 1];
                cnt++;
              }
            }
          }
        }
        for (k = 0; k < nc; k++) rows[k] = k + nc * (slot);
        PetscCall(MatSetValuesLocal(J, nc, rows, cnt, cols, NULL, INSERT_VALUES));
      }
    }
    /* do not copy values to GPU since they are all zero and not yet needed there */
    PetscCall(MatBoundToCPU(J, &alreadyboundtocpu));
    PetscCall(MatBindToCPU(J, PETSC_TRUE));
    PetscCall(MatAssemblyBegin(J, MAT_FINAL_ASSEMBLY));
    PetscCall(MatAssemblyEnd(J, MAT_FINAL_ASSEMBLY));
    if (!alreadyboundtocpu) PetscCall(MatBindToCPU(J, PETSC_FALSE));
    PetscCall(MatSetOption(J, MAT_NEW_NONZERO_LOCATION_ERR, PETSC_TRUE));
    if (bx == DM_BOUNDARY_NONE && by == DM_BOUNDARY_NONE) PetscCall(MatSetOption(J, MAT_SORTED_FULL, PETSC_FALSE));
  }
  PetscCall(PetscFree2(rows, cols));
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode DMCreateMatrix_DA_2d_MPIAIJ_Fill(DM da, Mat J)
{
  PetscInt               xs, ys, nx, ny, i, j, slot, gxs, gys, gnx, gny;
  PetscInt               m, n, dim, s, *cols, k, nc, row, col, cnt, maxcnt = 0, l, p, M, N;
  PetscInt               lstart, lend, pstart, pend, *dnz, *onz;
  DM_DA                 *dd = (DM_DA *)da->data;
  PetscInt               ifill_col, *ofill = dd->ofill, *dfill = dd->dfill;
  MPI_Comm               comm;
  DMBoundaryType         bx, by;
  ISLocalToGlobalMapping ltog;
  DMDAStencilType        st;
  PetscBool              removedups = PETSC_FALSE;

  PetscFunctionBegin;
  /*
         nc - number of components per grid point
         col - number of colors needed in one direction for single component problem
  */
  PetscCall(DMDAGetInfo(da, &dim, &m, &n, NULL, &M, &N, NULL, &nc, &s, &bx, &by, NULL, &st));
  col = 2 * s + 1;
  /*
       With one processor in periodic domains in a skinny dimension the code will label nonzero columns multiple times
       because of "wrapping" around the end of the domain hitting an entry already counted in the other direction.
  */
  if (M == 1 && 2 * s >= m) removedups = PETSC_TRUE;
  if (N == 1 && 2 * s >= n) removedups = PETSC_TRUE;
  PetscCall(DMDAGetCorners(da, &xs, &ys, NULL, &nx, &ny, NULL));
  PetscCall(DMDAGetGhostCorners(da, &gxs, &gys, NULL, &gnx, &gny, NULL));
  PetscCall(PetscObjectGetComm((PetscObject)da, &comm));

  PetscCall(PetscMalloc1(col * col * nc, &cols));
  PetscCall(DMGetLocalToGlobalMapping(da, &ltog));

  PetscCall(MatSetBlockSize(J, nc));
  /* determine the matrix preallocation information */
  MatPreallocateBegin(comm, nc * nx * ny, nc * nx * ny, dnz, onz);
  for (i = xs; i < xs + nx; i++) {
    pstart = (bx == DM_BOUNDARY_PERIODIC) ? -s : (PetscMax(-s, -i));
    pend   = (bx == DM_BOUNDARY_PERIODIC) ? s : (PetscMin(s, m - i - 1));

    for (j = ys; j < ys + ny; j++) {
      slot = i - gxs + gnx * (j - gys);

      lstart = (by == DM_BOUNDARY_PERIODIC) ? -s : (PetscMax(-s, -j));
      lend   = (by == DM_BOUNDARY_PERIODIC) ? s : (PetscMin(s, n - j - 1));

      for (k = 0; k < nc; k++) {
        cnt = 0;
        for (l = lstart; l < lend + 1; l++) {
          for (p = pstart; p < pend + 1; p++) {
            if (l || p) {
              if (st == DMDA_STENCIL_BOX || !l || !p) { /* entries on star */
                for (ifill_col = ofill[k]; ifill_col < ofill[k + 1]; ifill_col++) cols[cnt++] = ofill[ifill_col] + nc * (slot + gnx * l + p);
              }
            } else {
              if (dfill) {
                for (ifill_col = dfill[k]; ifill_col < dfill[k + 1]; ifill_col++) cols[cnt++] = dfill[ifill_col] + nc * (slot + gnx * l + p);
              } else {
                for (ifill_col = 0; ifill_col < nc; ifill_col++) cols[cnt++] = ifill_col + nc * (slot + gnx * l + p);
              }
            }
          }
        }
        row    = k + nc * (slot);
        maxcnt = PetscMax(maxcnt, cnt);
        if (removedups) PetscCall(MatPreallocateSetLocalRemoveDups(ltog, 1, &row, ltog, cnt, cols, dnz, onz));
        else PetscCall(MatPreallocateSetLocal(ltog, 1, &row, ltog, cnt, cols, dnz, onz));
      }
    }
  }
  PetscCall(MatSeqAIJSetPreallocation(J, 0, dnz));
  PetscCall(MatMPIAIJSetPreallocation(J, 0, dnz, 0, onz));
  MatPreallocateEnd(dnz, onz);
  PetscCall(MatSetLocalToGlobalMapping(J, ltog, ltog));

  /*
    For each node in the grid: we get the neighbors in the local (on processor ordering
    that includes the ghost points) then MatSetValuesLocal() maps those indices to the global
    PETSc ordering.
  */
  if (!da->prealloc_only) {
    for (i = xs; i < xs + nx; i++) {
      pstart = (bx == DM_BOUNDARY_PERIODIC) ? -s : (PetscMax(-s, -i));
      pend   = (bx == DM_BOUNDARY_PERIODIC) ? s : (PetscMin(s, m - i - 1));

      for (j = ys; j < ys + ny; j++) {
        slot = i - gxs + gnx * (j - gys);

        lstart = (by == DM_BOUNDARY_PERIODIC) ? -s : (PetscMax(-s, -j));
        lend   = (by == DM_BOUNDARY_PERIODIC) ? s : (PetscMin(s, n - j - 1));

        for (k = 0; k < nc; k++) {
          cnt = 0;
          for (l = lstart; l < lend + 1; l++) {
            for (p = pstart; p < pend + 1; p++) {
              if (l || p) {
                if (st == DMDA_STENCIL_BOX || !l || !p) { /* entries on star */
                  for (ifill_col = ofill[k]; ifill_col < ofill[k + 1]; ifill_col++) cols[cnt++] = ofill[ifill_col] + nc * (slot + gnx * l + p);
                }
              } else {
                if (dfill) {
                  for (ifill_col = dfill[k]; ifill_col < dfill[k + 1]; ifill_col++) cols[cnt++] = dfill[ifill_col] + nc * (slot + gnx * l + p);
                } else {
                  for (ifill_col = 0; ifill_col < nc; ifill_col++) cols[cnt++] = ifill_col + nc * (slot + gnx * l + p);
                }
              }
            }
          }
          row = k + nc * (slot);
          PetscCall(MatSetValuesLocal(J, 1, &row, cnt, cols, NULL, INSERT_VALUES));
        }
      }
    }
    /* do not copy values to GPU since they are all zero and not yet needed there */
    PetscCall(MatBindToCPU(J, PETSC_TRUE));
    PetscCall(MatAssemblyBegin(J, MAT_FINAL_ASSEMBLY));
    PetscCall(MatAssemblyEnd(J, MAT_FINAL_ASSEMBLY));
    PetscCall(MatBindToCPU(J, PETSC_FALSE));
    PetscCall(MatSetOption(J, MAT_NEW_NONZERO_LOCATION_ERR, PETSC_TRUE));
  }
  PetscCall(PetscFree(cols));
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode DMCreateMatrix_DA_3d_MPIAIJ(DM da, Mat J)
{
  PetscInt               xs, ys, nx, ny, i, j, slot, gxs, gys, gnx, gny;
  PetscInt               m, n, dim, s, *cols = NULL, k, nc, *rows = NULL, col, cnt, l, p, *dnz = NULL, *onz = NULL;
  PetscInt               istart, iend, jstart, jend, kstart, kend, zs, nz, gzs, gnz, ii, jj, kk, M, N, P;
  MPI_Comm               comm;
  DMBoundaryType         bx, by, bz;
  ISLocalToGlobalMapping ltog, mltog;
  DMDAStencilType        st;
  PetscBool              removedups = PETSC_FALSE;

  PetscFunctionBegin;
  /*
         nc - number of components per grid point
         col - number of colors needed in one direction for single component problem
  */
  PetscCall(DMDAGetInfo(da, &dim, &m, &n, &p, &M, &N, &P, &nc, &s, &bx, &by, &bz, &st));
  if (bx == DM_BOUNDARY_NONE && by == DM_BOUNDARY_NONE && bz == DM_BOUNDARY_NONE) PetscCall(MatSetOption(J, MAT_SORTED_FULL, PETSC_TRUE));
  col = 2 * s + 1;

  /*
       With one processor in periodic domains in a skinny dimension the code will label nonzero columns multiple times
       because of "wrapping" around the end of the domain hitting an entry already counted in the other direction.
  */
  if (M == 1 && 2 * s >= m) removedups = PETSC_TRUE;
  if (N == 1 && 2 * s >= n) removedups = PETSC_TRUE;
  if (P == 1 && 2 * s >= p) removedups = PETSC_TRUE;

  PetscCall(DMDAGetCorners(da, &xs, &ys, &zs, &nx, &ny, &nz));
  PetscCall(DMDAGetGhostCorners(da, &gxs, &gys, &gzs, &gnx, &gny, &gnz));
  PetscCall(PetscObjectGetComm((PetscObject)da, &comm));

  PetscCall(PetscMalloc2(nc, &rows, col * col * col * nc * nc, &cols));
  PetscCall(DMGetLocalToGlobalMapping(da, &ltog));

  PetscCall(MatSetBlockSize(J, nc));
  /* determine the matrix preallocation information */
  MatPreallocateBegin(comm, nc * nx * ny * nz, nc * nx * ny * nz, dnz, onz);
  for (i = xs; i < xs + nx; i++) {
    istart = (bx == DM_BOUNDARY_PERIODIC) ? -s : (PetscMax(-s, -i));
    iend   = (bx == DM_BOUNDARY_PERIODIC) ? s : (PetscMin(s, m - i - 1));
    for (j = ys; j < ys + ny; j++) {
      jstart = (by == DM_BOUNDARY_PERIODIC) ? -s : (PetscMax(-s, -j));
      jend   = (by == DM_BOUNDARY_PERIODIC) ? s : (PetscMin(s, n - j - 1));
      for (k = zs; k < zs + nz; k++) {
        kstart = (bz == DM_BOUNDARY_PERIODIC) ? -s : (PetscMax(-s, -k));
        kend   = (bz == DM_BOUNDARY_PERIODIC) ? s : (PetscMin(s, p - k - 1));

        slot = i - gxs + gnx * (j - gys) + gnx * gny * (k - gzs);

        cnt = 0;
        for (l = 0; l < nc; l++) {
          for (ii = istart; ii < iend + 1; ii++) {
            for (jj = jstart; jj < jend + 1; jj++) {
              for (kk = kstart; kk < kend + 1; kk++) {
                if (st == DMDA_STENCIL_BOX || ((!ii && !jj) || (!jj && !kk) || (!ii && !kk))) { /* entries on star*/
                  cols[cnt++] = l + nc * (slot + ii + gnx * jj + gnx * gny * kk);
                }
              }
            }
          }
          rows[l] = l + nc * (slot);
        }
        if (removedups) PetscCall(MatPreallocateSetLocalRemoveDups(ltog, nc, rows, ltog, cnt, cols, dnz, onz));
        else PetscCall(MatPreallocateSetLocal(ltog, nc, rows, ltog, cnt, cols, dnz, onz));
      }
    }
  }
  PetscCall(MatSetBlockSize(J, nc));
  PetscCall(MatSeqAIJSetPreallocation(J, 0, dnz));
  PetscCall(MatMPIAIJSetPreallocation(J, 0, dnz, 0, onz));
  MatPreallocateEnd(dnz, onz);
  PetscCall(MatGetLocalToGlobalMapping(J, &mltog, NULL));
  if (!mltog) PetscCall(MatSetLocalToGlobalMapping(J, ltog, ltog));

  /*
    For each node in the grid: we get the neighbors in the local (on processor ordering
    that includes the ghost points) then MatSetValuesLocal() maps those indices to the global
    PETSc ordering.
  */
  if (!da->prealloc_only) {
    for (i = xs; i < xs + nx; i++) {
      istart = (bx == DM_BOUNDARY_PERIODIC) ? -s : (PetscMax(-s, -i));
      iend   = (bx == DM_BOUNDARY_PERIODIC) ? s : (PetscMin(s, m - i - 1));
      for (j = ys; j < ys + ny; j++) {
        jstart = (by == DM_BOUNDARY_PERIODIC) ? -s : (PetscMax(-s, -j));
        jend   = (by == DM_BOUNDARY_PERIODIC) ? s : (PetscMin(s, n - j - 1));
        for (k = zs; k < zs + nz; k++) {
          kstart = (bz == DM_BOUNDARY_PERIODIC) ? -s : (PetscMax(-s, -k));
          kend   = (bz == DM_BOUNDARY_PERIODIC) ? s : (PetscMin(s, p - k - 1));

          slot = i - gxs + gnx * (j - gys) + gnx * gny * (k - gzs);

          cnt = 0;
          for (kk = kstart; kk < kend + 1; kk++) {
            for (jj = jstart; jj < jend + 1; jj++) {
              for (ii = istart; ii < iend + 1; ii++) {
                if (st == DMDA_STENCIL_BOX || ((!ii && !jj) || (!jj && !kk) || (!ii && !kk))) { /* entries on star*/
                  cols[cnt++] = nc * (slot + ii + gnx * jj + gnx * gny * kk);
                  for (l = 1; l < nc; l++) {
                    cols[cnt] = 1 + cols[cnt - 1];
                    cnt++;
                  }
                }
              }
            }
          }
          rows[0] = nc * (slot);
          for (l = 1; l < nc; l++) rows[l] = 1 + rows[l - 1];
          PetscCall(MatSetValuesLocal(J, nc, rows, cnt, cols, NULL, INSERT_VALUES));
        }
      }
    }
    /* do not copy values to GPU since they are all zero and not yet needed there */
    PetscCall(MatBindToCPU(J, PETSC_TRUE));
    PetscCall(MatAssemblyBegin(J, MAT_FINAL_ASSEMBLY));
    PetscCall(MatAssemblyEnd(J, MAT_FINAL_ASSEMBLY));
    if (bx == DM_BOUNDARY_NONE && by == DM_BOUNDARY_NONE && bz == DM_BOUNDARY_NONE) PetscCall(MatSetOption(J, MAT_SORTED_FULL, PETSC_FALSE));
    PetscCall(MatBindToCPU(J, PETSC_FALSE));
    PetscCall(MatSetOption(J, MAT_NEW_NONZERO_LOCATION_ERR, PETSC_TRUE));
  }
  PetscCall(PetscFree2(rows, cols));
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode DMCreateMatrix_DA_1d_MPIAIJ_Fill(DM da, Mat J)
{
  DM_DA                 *dd = (DM_DA *)da->data;
  PetscInt               xs, nx, i, j, gxs, gnx, row, k, l;
  PetscInt               m, dim, s, *cols = NULL, nc, cnt, maxcnt = 0, *ocols;
  PetscInt              *ofill = dd->ofill, *dfill = dd->dfill;
  DMBoundaryType         bx;
  ISLocalToGlobalMapping ltog;
  PetscMPIInt            rank, size;

  PetscFunctionBegin;
  PetscCallMPI(MPI_Comm_rank(PetscObjectComm((PetscObject)da), &rank));
  PetscCallMPI(MPI_Comm_size(PetscObjectComm((PetscObject)da), &size));

  /*
         nc - number of components per grid point
  */
  PetscCall(DMDAGetInfo(da, &dim, &m, NULL, NULL, NULL, NULL, NULL, &nc, &s, &bx, NULL, NULL, NULL));
  PetscCheck(s <= 1, PetscObjectComm((PetscObject)da), PETSC_ERR_SUP, "Matrix creation for 1d not implemented correctly for stencil width larger than 1");
  PetscCall(DMDAGetCorners(da, &xs, NULL, NULL, &nx, NULL, NULL));
  PetscCall(DMDAGetGhostCorners(da, &gxs, NULL, NULL, &gnx, NULL, NULL));

  PetscCall(MatSetBlockSize(J, nc));
  PetscCall(PetscCalloc2(nx * nc, &cols, nx * nc, &ocols));

  /*
        note should be smaller for first and last process with no periodic
        does not handle dfill
  */
  cnt = 0;
  /* coupling with process to the left */
  for (i = 0; i < s; i++) {
    for (j = 0; j < nc; j++) {
      ocols[cnt] = ((rank == 0) ? 0 : (s - i) * (ofill[j + 1] - ofill[j]));
      cols[cnt]  = dfill[j + 1] - dfill[j] + (s + i) * (ofill[j + 1] - ofill[j]);
      if (rank == 0 && (dd->bx == DM_BOUNDARY_PERIODIC)) {
        if (size > 1) ocols[cnt] += (s - i) * (ofill[j + 1] - ofill[j]);
        else cols[cnt] += (s - i) * (ofill[j + 1] - ofill[j]);
      }
      maxcnt = PetscMax(maxcnt, ocols[cnt] + cols[cnt]);
      cnt++;
    }
  }
  for (i = s; i < nx - s; i++) {
    for (j = 0; j < nc; j++) {
      cols[cnt] = dfill[j + 1] - dfill[j] + 2 * s * (ofill[j + 1] - ofill[j]);
      maxcnt    = PetscMax(maxcnt, ocols[cnt] + cols[cnt]);
      cnt++;
    }
  }
  /* coupling with process to the right */
  for (i = nx - s; i < nx; i++) {
    for (j = 0; j < nc; j++) {
      ocols[cnt] = ((rank == (size - 1)) ? 0 : (i - nx + s + 1) * (ofill[j + 1] - ofill[j]));
      cols[cnt]  = dfill[j + 1] - dfill[j] + (s + nx - i - 1) * (ofill[j + 1] - ofill[j]);
      if ((rank == size - 1) && (dd->bx == DM_BOUNDARY_PERIODIC)) {
        if (size > 1) ocols[cnt] += (i - nx + s + 1) * (ofill[j + 1] - ofill[j]);
        else cols[cnt] += (i - nx + s + 1) * (ofill[j + 1] - ofill[j]);
      }
      maxcnt = PetscMax(maxcnt, ocols[cnt] + cols[cnt]);
      cnt++;
    }
  }

  PetscCall(MatSeqAIJSetPreallocation(J, 0, cols));
  PetscCall(MatMPIAIJSetPreallocation(J, 0, cols, 0, ocols));
  PetscCall(PetscFree2(cols, ocols));

  PetscCall(DMGetLocalToGlobalMapping(da, &ltog));
  PetscCall(MatSetLocalToGlobalMapping(J, ltog, ltog));

  /*
    For each node in the grid: we get the neighbors in the local (on processor ordering
    that includes the ghost points) then MatSetValuesLocal() maps those indices to the global
    PETSc ordering.
  */
  if (!da->prealloc_only) {
    PetscCall(PetscMalloc1(maxcnt, &cols));
    row = xs * nc;
    /* coupling with process to the left */
    for (i = xs; i < xs + s; i++) {
      for (j = 0; j < nc; j++) {
        cnt = 0;
        if (rank) {
          for (l = 0; l < s; l++) {
            for (k = ofill[j]; k < ofill[j + 1]; k++) cols[cnt++] = (i - s + l) * nc + ofill[k];
          }
        }
        if (rank == 0 && (dd->bx == DM_BOUNDARY_PERIODIC)) {
          for (l = 0; l < s; l++) {
            for (k = ofill[j]; k < ofill[j + 1]; k++) cols[cnt++] = (m + i - s - l) * nc + ofill[k];
          }
        }
        if (dfill) {
          for (k = dfill[j]; k < dfill[j + 1]; k++) cols[cnt++] = i * nc + dfill[k];
        } else {
          for (k = 0; k < nc; k++) cols[cnt++] = i * nc + k;
        }
        for (l = 0; l < s; l++) {
          for (k = ofill[j]; k < ofill[j + 1]; k++) cols[cnt++] = (i + s - l) * nc + ofill[k];
        }
        PetscCall(MatSetValues(J, 1, &row, cnt, cols, NULL, INSERT_VALUES));
        row++;
      }
    }
    for (i = xs + s; i < xs + nx - s; i++) {
      for (j = 0; j < nc; j++) {
        cnt = 0;
        for (l = 0; l < s; l++) {
          for (k = ofill[j]; k < ofill[j + 1]; k++) cols[cnt++] = (i - s + l) * nc + ofill[k];
        }
        if (dfill) {
          for (k = dfill[j]; k < dfill[j + 1]; k++) cols[cnt++] = i * nc + dfill[k];
        } else {
          for (k = 0; k < nc; k++) cols[cnt++] = i * nc + k;
        }
        for (l = 0; l < s; l++) {
          for (k = ofill[j]; k < ofill[j + 1]; k++) cols[cnt++] = (i + s - l) * nc + ofill[k];
        }
        PetscCall(MatSetValues(J, 1, &row, cnt, cols, NULL, INSERT_VALUES));
        row++;
      }
    }
    /* coupling with process to the right */
    for (i = xs + nx - s; i < xs + nx; i++) {
      for (j = 0; j < nc; j++) {
        cnt = 0;
        for (l = 0; l < s; l++) {
          for (k = ofill[j]; k < ofill[j + 1]; k++) cols[cnt++] = (i - s + l) * nc + ofill[k];
        }
        if (dfill) {
          for (k = dfill[j]; k < dfill[j + 1]; k++) cols[cnt++] = i * nc + dfill[k];
        } else {
          for (k = 0; k < nc; k++) cols[cnt++] = i * nc + k;
        }
        if (rank < size - 1) {
          for (l = 0; l < s; l++) {
            for (k = ofill[j]; k < ofill[j + 1]; k++) cols[cnt++] = (i + s - l) * nc + ofill[k];
          }
        }
        if ((rank == size - 1) && (dd->bx == DM_BOUNDARY_PERIODIC)) {
          for (l = 0; l < s; l++) {
            for (k = ofill[j]; k < ofill[j + 1]; k++) cols[cnt++] = (i - s - l - m + 2) * nc + ofill[k];
          }
        }
        PetscCall(MatSetValues(J, 1, &row, cnt, cols, NULL, INSERT_VALUES));
        row++;
      }
    }
    PetscCall(PetscFree(cols));
    /* do not copy values to GPU since they are all zero and not yet needed there */
    PetscCall(MatBindToCPU(J, PETSC_TRUE));
    PetscCall(MatAssemblyBegin(J, MAT_FINAL_ASSEMBLY));
    PetscCall(MatAssemblyEnd(J, MAT_FINAL_ASSEMBLY));
    PetscCall(MatBindToCPU(J, PETSC_FALSE));
    PetscCall(MatSetOption(J, MAT_NEW_NONZERO_LOCATION_ERR, PETSC_TRUE));
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode DMCreateMatrix_DA_1d_MPIAIJ(DM da, Mat J)
{
  PetscInt               xs, nx, i, i1, slot, gxs, gnx;
  PetscInt               m, dim, s, *cols = NULL, nc, *rows = NULL, col, cnt, l;
  PetscInt               istart, iend;
  DMBoundaryType         bx;
  ISLocalToGlobalMapping ltog, mltog;

  PetscFunctionBegin;
  /*
         nc - number of components per grid point
         col - number of colors needed in one direction for single component problem
  */
  PetscCall(DMDAGetInfo(da, &dim, &m, NULL, NULL, NULL, NULL, NULL, &nc, &s, &bx, NULL, NULL, NULL));
  if (bx == DM_BOUNDARY_NONE) PetscCall(MatSetOption(J, MAT_SORTED_FULL, PETSC_TRUE));
  col = 2 * s + 1;

  PetscCall(DMDAGetCorners(da, &xs, NULL, NULL, &nx, NULL, NULL));
  PetscCall(DMDAGetGhostCorners(da, &gxs, NULL, NULL, &gnx, NULL, NULL));

  PetscCall(MatSetBlockSize(J, nc));
  PetscCall(MatSeqAIJSetPreallocation(J, col * nc, NULL));
  PetscCall(MatMPIAIJSetPreallocation(J, col * nc, NULL, col * nc, NULL));

  PetscCall(DMGetLocalToGlobalMapping(da, &ltog));
  PetscCall(MatGetLocalToGlobalMapping(J, &mltog, NULL));
  if (!mltog) PetscCall(MatSetLocalToGlobalMapping(J, ltog, ltog));

  /*
    For each node in the grid: we get the neighbors in the local (on processor ordering
    that includes the ghost points) then MatSetValuesLocal() maps those indices to the global
    PETSc ordering.
  */
  if (!da->prealloc_only) {
    PetscCall(PetscMalloc2(nc, &rows, col * nc * nc, &cols));
    for (i = xs; i < xs + nx; i++) {
      istart = PetscMax(-s, gxs - i);
      iend   = PetscMin(s, gxs + gnx - i - 1);
      slot   = i - gxs;

      cnt = 0;
      for (i1 = istart; i1 < iend + 1; i1++) {
        cols[cnt++] = nc * (slot + i1);
        for (l = 1; l < nc; l++) {
          cols[cnt] = 1 + cols[cnt - 1];
          cnt++;
        }
      }
      rows[0] = nc * (slot);
      for (l = 1; l < nc; l++) rows[l] = 1 + rows[l - 1];
      PetscCall(MatSetValuesLocal(J, nc, rows, cnt, cols, NULL, INSERT_VALUES));
    }
    /* do not copy values to GPU since they are all zero and not yet needed there */
    PetscCall(MatBindToCPU(J, PETSC_TRUE));
    PetscCall(MatAssemblyBegin(J, MAT_FINAL_ASSEMBLY));
    PetscCall(MatAssemblyEnd(J, MAT_FINAL_ASSEMBLY));
    if (bx == DM_BOUNDARY_NONE) PetscCall(MatSetOption(J, MAT_SORTED_FULL, PETSC_FALSE));
    PetscCall(MatBindToCPU(J, PETSC_FALSE));
    PetscCall(MatSetOption(J, MAT_NEW_NONZERO_LOCATION_ERR, PETSC_TRUE));
    PetscCall(PetscFree2(rows, cols));
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode DMCreateMatrix_DA_1d_SeqAIJ_NoPreallocation(DM da, Mat J)
{
  PetscInt               xs, nx, i, i1, slot, gxs, gnx;
  PetscInt               m, dim, s, *cols = NULL, nc, *rows = NULL, col, cnt, l;
  PetscInt               istart, iend;
  DMBoundaryType         bx;
  ISLocalToGlobalMapping ltog, mltog;

  PetscFunctionBegin;
  /*
         nc - number of components per grid point
         col - number of colors needed in one direction for single component problem
  */
  PetscCall(DMDAGetInfo(da, &dim, &m, NULL, NULL, NULL, NULL, NULL, &nc, &s, &bx, NULL, NULL, NULL));
  col = 2 * s + 1;

  PetscCall(DMDAGetCorners(da, &xs, NULL, NULL, &nx, NULL, NULL));
  PetscCall(DMDAGetGhostCorners(da, &gxs, NULL, NULL, &gnx, NULL, NULL));

  PetscCall(MatSetBlockSize(J, nc));
  PetscCall(MatSeqAIJSetTotalPreallocation(J, nx * nc * col * nc));

  PetscCall(DMGetLocalToGlobalMapping(da, &ltog));
  PetscCall(MatGetLocalToGlobalMapping(J, &mltog, NULL));
  if (!mltog) PetscCall(MatSetLocalToGlobalMapping(J, ltog, ltog));

  /*
    For each node in the grid: we get the neighbors in the local (on processor ordering
    that includes the ghost points) then MatSetValuesLocal() maps those indices to the global
    PETSc ordering.
  */
  if (!da->prealloc_only) {
    PetscCall(PetscMalloc2(nc, &rows, col * nc * nc, &cols));
    for (i = xs; i < xs + nx; i++) {
      istart = PetscMax(-s, gxs - i);
      iend   = PetscMin(s, gxs + gnx - i - 1);
      slot   = i - gxs;

      cnt = 0;
      for (i1 = istart; i1 < iend + 1; i1++) {
        cols[cnt++] = nc * (slot + i1);
        for (l = 1; l < nc; l++) {
          cols[cnt] = 1 + cols[cnt - 1];
          cnt++;
        }
      }
      rows[0] = nc * (slot);
      for (l = 1; l < nc; l++) rows[l] = 1 + rows[l - 1];
      PetscCall(MatSetValuesLocal(J, nc, rows, cnt, cols, NULL, INSERT_VALUES));
    }
    /* do not copy values to GPU since they are all zero and not yet needed there */
    PetscCall(MatBindToCPU(J, PETSC_TRUE));
    PetscCall(MatAssemblyBegin(J, MAT_FINAL_ASSEMBLY));
    PetscCall(MatAssemblyEnd(J, MAT_FINAL_ASSEMBLY));
    if (bx == DM_BOUNDARY_NONE) PetscCall(MatSetOption(J, MAT_SORTED_FULL, PETSC_FALSE));
    PetscCall(MatBindToCPU(J, PETSC_FALSE));
    PetscCall(MatSetOption(J, MAT_NEW_NONZERO_LOCATION_ERR, PETSC_TRUE));
    PetscCall(PetscFree2(rows, cols));
  }
  PetscCall(MatSetOption(J, MAT_SORTED_FULL, PETSC_FALSE));
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode DMCreateMatrix_DA_2d_MPIBAIJ(DM da, Mat J)
{
  PetscInt               xs, ys, nx, ny, i, j, slot, gxs, gys, gnx, gny;
  PetscInt               m, n, dim, s, *cols, nc, col, cnt, *dnz, *onz;
  PetscInt               istart, iend, jstart, jend, ii, jj;
  MPI_Comm               comm;
  PetscScalar           *values;
  DMBoundaryType         bx, by;
  DMDAStencilType        st;
  ISLocalToGlobalMapping ltog;

  PetscFunctionBegin;
  /*
     nc - number of components per grid point
     col - number of colors needed in one direction for single component problem
  */
  PetscCall(DMDAGetInfo(da, &dim, &m, &n, NULL, NULL, NULL, NULL, &nc, &s, &bx, &by, NULL, &st));
  col = 2 * s + 1;

  PetscCall(DMDAGetCorners(da, &xs, &ys, NULL, &nx, &ny, NULL));
  PetscCall(DMDAGetGhostCorners(da, &gxs, &gys, NULL, &gnx, &gny, NULL));
  PetscCall(PetscObjectGetComm((PetscObject)da, &comm));

  PetscCall(PetscMalloc1(col * col * nc * nc, &cols));

  PetscCall(DMGetLocalToGlobalMapping(da, &ltog));

  /* determine the matrix preallocation information */
  MatPreallocateBegin(comm, nx * ny, nx * ny, dnz, onz);
  for (i = xs; i < xs + nx; i++) {
    istart = (bx == DM_BOUNDARY_PERIODIC) ? -s : (PetscMax(-s, -i));
    iend   = (bx == DM_BOUNDARY_PERIODIC) ? s : (PetscMin(s, m - i - 1));
    for (j = ys; j < ys + ny; j++) {
      jstart = (by == DM_BOUNDARY_PERIODIC) ? -s : (PetscMax(-s, -j));
      jend   = (by == DM_BOUNDARY_PERIODIC) ? s : (PetscMin(s, n - j - 1));
      slot   = i - gxs + gnx * (j - gys);

      /* Find block columns in block row */
      cnt = 0;
      for (ii = istart; ii < iend + 1; ii++) {
        for (jj = jstart; jj < jend + 1; jj++) {
          if (st == DMDA_STENCIL_BOX || !ii || !jj) { /* BOX or on the STAR */
            cols[cnt++] = slot + ii + gnx * jj;
          }
        }
      }
      PetscCall(MatPreallocateSetLocalBlock(ltog, 1, &slot, ltog, cnt, cols, dnz, onz));
    }
  }
  PetscCall(MatSeqBAIJSetPreallocation(J, nc, 0, dnz));
  PetscCall(MatMPIBAIJSetPreallocation(J, nc, 0, dnz, 0, onz));
  MatPreallocateEnd(dnz, onz);

  PetscCall(MatSetLocalToGlobalMapping(J, ltog, ltog));

  /*
    For each node in the grid: we get the neighbors in the local (on processor ordering
    that includes the ghost points) then MatSetValuesLocal() maps those indices to the global
    PETSc ordering.
  */
  if (!da->prealloc_only) {
    PetscCall(PetscCalloc1(col * col * nc * nc, &values));
    for (i = xs; i < xs + nx; i++) {
      istart = (bx == DM_BOUNDARY_PERIODIC) ? -s : (PetscMax(-s, -i));
      iend   = (bx == DM_BOUNDARY_PERIODIC) ? s : (PetscMin(s, m - i - 1));
      for (j = ys; j < ys + ny; j++) {
        jstart = (by == DM_BOUNDARY_PERIODIC) ? -s : (PetscMax(-s, -j));
        jend   = (by == DM_BOUNDARY_PERIODIC) ? s : (PetscMin(s, n - j - 1));
        slot   = i - gxs + gnx * (j - gys);
        cnt    = 0;
        for (ii = istart; ii < iend + 1; ii++) {
          for (jj = jstart; jj < jend + 1; jj++) {
            if (st == DMDA_STENCIL_BOX || !ii || !jj) { /* BOX or on the STAR */
              cols[cnt++] = slot + ii + gnx * jj;
            }
          }
        }
        PetscCall(MatSetValuesBlockedLocal(J, 1, &slot, cnt, cols, values, INSERT_VALUES));
      }
    }
    PetscCall(PetscFree(values));
    /* do not copy values to GPU since they are all zero and not yet needed there */
    PetscCall(MatBindToCPU(J, PETSC_TRUE));
    PetscCall(MatAssemblyBegin(J, MAT_FINAL_ASSEMBLY));
    PetscCall(MatAssemblyEnd(J, MAT_FINAL_ASSEMBLY));
    PetscCall(MatBindToCPU(J, PETSC_FALSE));
    PetscCall(MatSetOption(J, MAT_NEW_NONZERO_LOCATION_ERR, PETSC_TRUE));
  }
  PetscCall(PetscFree(cols));
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode DMCreateMatrix_DA_3d_MPIBAIJ(DM da, Mat J)
{
  PetscInt               xs, ys, nx, ny, i, j, slot, gxs, gys, gnx, gny;
  PetscInt               m, n, dim, s, *cols, k, nc, col, cnt, p, *dnz, *onz;
  PetscInt               istart, iend, jstart, jend, kstart, kend, zs, nz, gzs, gnz, ii, jj, kk;
  MPI_Comm               comm;
  PetscScalar           *values;
  DMBoundaryType         bx, by, bz;
  DMDAStencilType        st;
  ISLocalToGlobalMapping ltog;

  PetscFunctionBegin;
  /*
         nc - number of components per grid point
         col - number of colors needed in one direction for single component problem
  */
  PetscCall(DMDAGetInfo(da, &dim, &m, &n, &p, NULL, NULL, NULL, &nc, &s, &bx, &by, &bz, &st));
  col = 2 * s + 1;

  PetscCall(DMDAGetCorners(da, &xs, &ys, &zs, &nx, &ny, &nz));
  PetscCall(DMDAGetGhostCorners(da, &gxs, &gys, &gzs, &gnx, &gny, &gnz));
  PetscCall(PetscObjectGetComm((PetscObject)da, &comm));

  PetscCall(PetscMalloc1(col * col * col, &cols));

  PetscCall(DMGetLocalToGlobalMapping(da, &ltog));

  /* determine the matrix preallocation information */
  MatPreallocateBegin(comm, nx * ny * nz, nx * ny * nz, dnz, onz);
  for (i = xs; i < xs + nx; i++) {
    istart = (bx == DM_BOUNDARY_PERIODIC) ? -s : (PetscMax(-s, -i));
    iend   = (bx == DM_BOUNDARY_PERIODIC) ? s : (PetscMin(s, m - i - 1));
    for (j = ys; j < ys + ny; j++) {
      jstart = (by == DM_BOUNDARY_PERIODIC) ? -s : (PetscMax(-s, -j));
      jend   = (by == DM_BOUNDARY_PERIODIC) ? s : (PetscMin(s, n - j - 1));
      for (k = zs; k < zs + nz; k++) {
        kstart = (bz == DM_BOUNDARY_PERIODIC) ? -s : (PetscMax(-s, -k));
        kend   = (bz == DM_BOUNDARY_PERIODIC) ? s : (PetscMin(s, p - k - 1));

        slot = i - gxs + gnx * (j - gys) + gnx * gny * (k - gzs);

        /* Find block columns in block row */
        cnt = 0;
        for (ii = istart; ii < iend + 1; ii++) {
          for (jj = jstart; jj < jend + 1; jj++) {
            for (kk = kstart; kk < kend + 1; kk++) {
              if (st == DMDA_STENCIL_BOX || ((!ii && !jj) || (!jj && !kk) || (!ii && !kk))) { /* entries on star*/
                cols[cnt++] = slot + ii + gnx * jj + gnx * gny * kk;
              }
            }
          }
        }
        PetscCall(MatPreallocateSetLocalBlock(ltog, 1, &slot, ltog, cnt, cols, dnz, onz));
      }
    }
  }
  PetscCall(MatSeqBAIJSetPreallocation(J, nc, 0, dnz));
  PetscCall(MatMPIBAIJSetPreallocation(J, nc, 0, dnz, 0, onz));
  MatPreallocateEnd(dnz, onz);

  PetscCall(MatSetLocalToGlobalMapping(J, ltog, ltog));

  /*
    For each node in the grid: we get the neighbors in the local (on processor ordering
    that includes the ghost points) then MatSetValuesLocal() maps those indices to the global
    PETSc ordering.
  */
  if (!da->prealloc_only) {
    PetscCall(PetscCalloc1(col * col * col * nc * nc, &values));
    for (i = xs; i < xs + nx; i++) {
      istart = (bx == DM_BOUNDARY_PERIODIC) ? -s : (PetscMax(-s, -i));
      iend   = (bx == DM_BOUNDARY_PERIODIC) ? s : (PetscMin(s, m - i - 1));
      for (j = ys; j < ys + ny; j++) {
        jstart = (by == DM_BOUNDARY_PERIODIC) ? -s : (PetscMax(-s, -j));
        jend   = (by == DM_BOUNDARY_PERIODIC) ? s : (PetscMin(s, n - j - 1));
        for (k = zs; k < zs + nz; k++) {
          kstart = (bz == DM_BOUNDARY_PERIODIC) ? -s : (PetscMax(-s, -k));
          kend   = (bz == DM_BOUNDARY_PERIODIC) ? s : (PetscMin(s, p - k - 1));

          slot = i - gxs + gnx * (j - gys) + gnx * gny * (k - gzs);

          cnt = 0;
          for (ii = istart; ii < iend + 1; ii++) {
            for (jj = jstart; jj < jend + 1; jj++) {
              for (kk = kstart; kk < kend + 1; kk++) {
                if (st == DMDA_STENCIL_BOX || ((!ii && !jj) || (!jj && !kk) || (!ii && !kk))) { /* entries on star*/
                  cols[cnt++] = slot + ii + gnx * jj + gnx * gny * kk;
                }
              }
            }
          }
          PetscCall(MatSetValuesBlockedLocal(J, 1, &slot, cnt, cols, values, INSERT_VALUES));
        }
      }
    }
    PetscCall(PetscFree(values));
    /* do not copy values to GPU since they are all zero and not yet needed there */
    PetscCall(MatBindToCPU(J, PETSC_TRUE));
    PetscCall(MatAssemblyBegin(J, MAT_FINAL_ASSEMBLY));
    PetscCall(MatAssemblyEnd(J, MAT_FINAL_ASSEMBLY));
    PetscCall(MatBindToCPU(J, PETSC_FALSE));
    PetscCall(MatSetOption(J, MAT_NEW_NONZERO_LOCATION_ERR, PETSC_TRUE));
  }
  PetscCall(PetscFree(cols));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*
  This helper is for of SBAIJ preallocation, to discard the lower-triangular values which are difficult to
  identify in the local ordering with periodic domain.
*/
static PetscErrorCode L2GFilterUpperTriangular(ISLocalToGlobalMapping ltog, PetscInt *row, PetscInt *cnt, PetscInt col[])
{
  PetscInt i, n;

  PetscFunctionBegin;
  PetscCall(ISLocalToGlobalMappingApplyBlock(ltog, 1, row, row));
  PetscCall(ISLocalToGlobalMappingApplyBlock(ltog, *cnt, col, col));
  for (i = 0, n = 0; i < *cnt; i++) {
    if (col[i] >= *row) col[n++] = col[i];
  }
  *cnt = n;
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode DMCreateMatrix_DA_2d_MPISBAIJ(DM da, Mat J)
{
  PetscInt               xs, ys, nx, ny, i, j, slot, gxs, gys, gnx, gny;
  PetscInt               m, n, dim, s, *cols, nc, col, cnt, *dnz, *onz;
  PetscInt               istart, iend, jstart, jend, ii, jj;
  MPI_Comm               comm;
  PetscScalar           *values;
  DMBoundaryType         bx, by;
  DMDAStencilType        st;
  ISLocalToGlobalMapping ltog;

  PetscFunctionBegin;
  /*
     nc - number of components per grid point
     col - number of colors needed in one direction for single component problem
  */
  PetscCall(DMDAGetInfo(da, &dim, &m, &n, NULL, NULL, NULL, NULL, &nc, &s, &bx, &by, NULL, &st));
  col = 2 * s + 1;

  PetscCall(DMDAGetCorners(da, &xs, &ys, NULL, &nx, &ny, NULL));
  PetscCall(DMDAGetGhostCorners(da, &gxs, &gys, NULL, &gnx, &gny, NULL));
  PetscCall(PetscObjectGetComm((PetscObject)da, &comm));

  PetscCall(PetscMalloc1(col * col * nc * nc, &cols));

  PetscCall(DMGetLocalToGlobalMapping(da, &ltog));

  /* determine the matrix preallocation information */
  MatPreallocateBegin(comm, nx * ny, nx * ny, dnz, onz);
  for (i = xs; i < xs + nx; i++) {
    istart = (bx == DM_BOUNDARY_PERIODIC) ? -s : (PetscMax(-s, -i));
    iend   = (bx == DM_BOUNDARY_PERIODIC) ? s : (PetscMin(s, m - i - 1));
    for (j = ys; j < ys + ny; j++) {
      jstart = (by == DM_BOUNDARY_PERIODIC) ? -s : (PetscMax(-s, -j));
      jend   = (by == DM_BOUNDARY_PERIODIC) ? s : (PetscMin(s, n - j - 1));
      slot   = i - gxs + gnx * (j - gys);

      /* Find block columns in block row */
      cnt = 0;
      for (ii = istart; ii < iend + 1; ii++) {
        for (jj = jstart; jj < jend + 1; jj++) {
          if (st == DMDA_STENCIL_BOX || !ii || !jj) cols[cnt++] = slot + ii + gnx * jj;
        }
      }
      PetscCall(L2GFilterUpperTriangular(ltog, &slot, &cnt, cols));
      PetscCall(MatPreallocateSymmetricSetBlock(slot, cnt, cols, dnz, onz));
    }
  }
  PetscCall(MatSeqSBAIJSetPreallocation(J, nc, 0, dnz));
  PetscCall(MatMPISBAIJSetPreallocation(J, nc, 0, dnz, 0, onz));
  MatPreallocateEnd(dnz, onz);

  PetscCall(MatSetLocalToGlobalMapping(J, ltog, ltog));

  /*
    For each node in the grid: we get the neighbors in the local (on processor ordering
    that includes the ghost points) then MatSetValuesLocal() maps those indices to the global
    PETSc ordering.
  */
  if (!da->prealloc_only) {
    PetscCall(PetscCalloc1(col * col * nc * nc, &values));
    for (i = xs; i < xs + nx; i++) {
      istart = (bx == DM_BOUNDARY_PERIODIC) ? -s : (PetscMax(-s, -i));
      iend   = (bx == DM_BOUNDARY_PERIODIC) ? s : (PetscMin(s, m - i - 1));
      for (j = ys; j < ys + ny; j++) {
        jstart = (by == DM_BOUNDARY_PERIODIC) ? -s : (PetscMax(-s, -j));
        jend   = (by == DM_BOUNDARY_PERIODIC) ? s : (PetscMin(s, n - j - 1));
        slot   = i - gxs + gnx * (j - gys);

        /* Find block columns in block row */
        cnt = 0;
        for (ii = istart; ii < iend + 1; ii++) {
          for (jj = jstart; jj < jend + 1; jj++) {
            if (st == DMDA_STENCIL_BOX || !ii || !jj) cols[cnt++] = slot + ii + gnx * jj;
          }
        }
        PetscCall(L2GFilterUpperTriangular(ltog, &slot, &cnt, cols));
        PetscCall(MatSetValuesBlocked(J, 1, &slot, cnt, cols, values, INSERT_VALUES));
      }
    }
    PetscCall(PetscFree(values));
    /* do not copy values to GPU since they are all zero and not yet needed there */
    PetscCall(MatBindToCPU(J, PETSC_TRUE));
    PetscCall(MatAssemblyBegin(J, MAT_FINAL_ASSEMBLY));
    PetscCall(MatAssemblyEnd(J, MAT_FINAL_ASSEMBLY));
    PetscCall(MatBindToCPU(J, PETSC_FALSE));
    PetscCall(MatSetOption(J, MAT_NEW_NONZERO_LOCATION_ERR, PETSC_TRUE));
  }
  PetscCall(PetscFree(cols));
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode DMCreateMatrix_DA_3d_MPISBAIJ(DM da, Mat J)
{
  PetscInt               xs, ys, nx, ny, i, j, slot, gxs, gys, gnx, gny;
  PetscInt               m, n, dim, s, *cols, k, nc, col, cnt, p, *dnz, *onz;
  PetscInt               istart, iend, jstart, jend, kstart, kend, zs, nz, gzs, gnz, ii, jj, kk;
  MPI_Comm               comm;
  PetscScalar           *values;
  DMBoundaryType         bx, by, bz;
  DMDAStencilType        st;
  ISLocalToGlobalMapping ltog;

  PetscFunctionBegin;
  /*
     nc - number of components per grid point
     col - number of colors needed in one direction for single component problem
  */
  PetscCall(DMDAGetInfo(da, &dim, &m, &n, &p, NULL, NULL, NULL, &nc, &s, &bx, &by, &bz, &st));
  col = 2 * s + 1;

  PetscCall(DMDAGetCorners(da, &xs, &ys, &zs, &nx, &ny, &nz));
  PetscCall(DMDAGetGhostCorners(da, &gxs, &gys, &gzs, &gnx, &gny, &gnz));
  PetscCall(PetscObjectGetComm((PetscObject)da, &comm));

  /* create the matrix */
  PetscCall(PetscMalloc1(col * col * col, &cols));

  PetscCall(DMGetLocalToGlobalMapping(da, &ltog));

  /* determine the matrix preallocation information */
  MatPreallocateBegin(comm, nx * ny * nz, nx * ny * nz, dnz, onz);
  for (i = xs; i < xs + nx; i++) {
    istart = (bx == DM_BOUNDARY_PERIODIC) ? -s : (PetscMax(-s, -i));
    iend   = (bx == DM_BOUNDARY_PERIODIC) ? s : (PetscMin(s, m - i - 1));
    for (j = ys; j < ys + ny; j++) {
      jstart = (by == DM_BOUNDARY_PERIODIC) ? -s : (PetscMax(-s, -j));
      jend   = (by == DM_BOUNDARY_PERIODIC) ? s : (PetscMin(s, n - j - 1));
      for (k = zs; k < zs + nz; k++) {
        kstart = (bz == DM_BOUNDARY_PERIODIC) ? -s : (PetscMax(-s, -k));
        kend   = (bz == DM_BOUNDARY_PERIODIC) ? s : (PetscMin(s, p - k - 1));

        slot = i - gxs + gnx * (j - gys) + gnx * gny * (k - gzs);

        /* Find block columns in block row */
        cnt = 0;
        for (ii = istart; ii < iend + 1; ii++) {
          for (jj = jstart; jj < jend + 1; jj++) {
            for (kk = kstart; kk < kend + 1; kk++) {
              if (st == DMDA_STENCIL_BOX || (!ii && !jj) || (!jj && !kk) || (!ii && !kk)) cols[cnt++] = slot + ii + gnx * jj + gnx * gny * kk;
            }
          }
        }
        PetscCall(L2GFilterUpperTriangular(ltog, &slot, &cnt, cols));
        PetscCall(MatPreallocateSymmetricSetBlock(slot, cnt, cols, dnz, onz));
      }
    }
  }
  PetscCall(MatSeqSBAIJSetPreallocation(J, nc, 0, dnz));
  PetscCall(MatMPISBAIJSetPreallocation(J, nc, 0, dnz, 0, onz));
  MatPreallocateEnd(dnz, onz);

  PetscCall(MatSetLocalToGlobalMapping(J, ltog, ltog));

  /*
    For each node in the grid: we get the neighbors in the local (on processor ordering
    that includes the ghost points) then MatSetValuesLocal() maps those indices to the global
    PETSc ordering.
  */
  if (!da->prealloc_only) {
    PetscCall(PetscCalloc1(col * col * col * nc * nc, &values));
    for (i = xs; i < xs + nx; i++) {
      istart = (bx == DM_BOUNDARY_PERIODIC) ? -s : (PetscMax(-s, -i));
      iend   = (bx == DM_BOUNDARY_PERIODIC) ? s : (PetscMin(s, m - i - 1));
      for (j = ys; j < ys + ny; j++) {
        jstart = (by == DM_BOUNDARY_PERIODIC) ? -s : (PetscMax(-s, -j));
        jend   = (by == DM_BOUNDARY_PERIODIC) ? s : (PetscMin(s, n - j - 1));
        for (k = zs; k < zs + nz; k++) {
          kstart = (bz == DM_BOUNDARY_PERIODIC) ? -s : (PetscMax(-s, -k));
          kend   = (bz == DM_BOUNDARY_PERIODIC) ? s : (PetscMin(s, p - k - 1));

          slot = i - gxs + gnx * (j - gys) + gnx * gny * (k - gzs);

          cnt = 0;
          for (ii = istart; ii < iend + 1; ii++) {
            for (jj = jstart; jj < jend + 1; jj++) {
              for (kk = kstart; kk < kend + 1; kk++) {
                if (st == DMDA_STENCIL_BOX || (!ii && !jj) || (!jj && !kk) || (!ii && !kk)) cols[cnt++] = slot + ii + gnx * jj + gnx * gny * kk;
              }
            }
          }
          PetscCall(L2GFilterUpperTriangular(ltog, &slot, &cnt, cols));
          PetscCall(MatSetValuesBlocked(J, 1, &slot, cnt, cols, values, INSERT_VALUES));
        }
      }
    }
    PetscCall(PetscFree(values));
    /* do not copy values to GPU since they are all zero and not yet needed there */
    PetscCall(MatBindToCPU(J, PETSC_TRUE));
    PetscCall(MatAssemblyBegin(J, MAT_FINAL_ASSEMBLY));
    PetscCall(MatAssemblyEnd(J, MAT_FINAL_ASSEMBLY));
    PetscCall(MatBindToCPU(J, PETSC_FALSE));
    PetscCall(MatSetOption(J, MAT_NEW_NONZERO_LOCATION_ERR, PETSC_TRUE));
  }
  PetscCall(PetscFree(cols));
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode DMCreateMatrix_DA_3d_MPIAIJ_Fill(DM da, Mat J)
{
  PetscInt               xs, ys, nx, ny, i, j, slot, gxs, gys, gnx, gny;
  PetscInt               m, n, dim, s, *cols, k, nc, row, col, cnt, maxcnt = 0, l, p, *dnz, *onz;
  PetscInt               istart, iend, jstart, jend, kstart, kend, zs, nz, gzs, gnz, ii, jj, kk, M, N, P;
  DM_DA                 *dd = (DM_DA *)da->data;
  PetscInt               ifill_col, *dfill = dd->dfill, *ofill = dd->ofill;
  MPI_Comm               comm;
  PetscScalar           *values;
  DMBoundaryType         bx, by, bz;
  ISLocalToGlobalMapping ltog;
  DMDAStencilType        st;
  PetscBool              removedups = PETSC_FALSE;

  PetscFunctionBegin;
  /*
         nc - number of components per grid point
         col - number of colors needed in one direction for single component problem
  */
  PetscCall(DMDAGetInfo(da, &dim, &m, &n, &p, &M, &N, &P, &nc, &s, &bx, &by, &bz, &st));
  col = 2 * s + 1;

  /*
       With one processor in periodic domains in a skinny dimension the code will label nonzero columns multiple times
       because of "wrapping" around the end of the domain hitting an entry already counted in the other direction.
  */
  if (M == 1 && 2 * s >= m) removedups = PETSC_TRUE;
  if (N == 1 && 2 * s >= n) removedups = PETSC_TRUE;
  if (P == 1 && 2 * s >= p) removedups = PETSC_TRUE;

  PetscCall(DMDAGetCorners(da, &xs, &ys, &zs, &nx, &ny, &nz));
  PetscCall(DMDAGetGhostCorners(da, &gxs, &gys, &gzs, &gnx, &gny, &gnz));
  PetscCall(PetscObjectGetComm((PetscObject)da, &comm));

  PetscCall(PetscMalloc1(col * col * col * nc, &cols));
  PetscCall(DMGetLocalToGlobalMapping(da, &ltog));

  /* determine the matrix preallocation information */
  MatPreallocateBegin(comm, nc * nx * ny * nz, nc * nx * ny * nz, dnz, onz);

  PetscCall(MatSetBlockSize(J, nc));
  for (i = xs; i < xs + nx; i++) {
    istart = (bx == DM_BOUNDARY_PERIODIC) ? -s : (PetscMax(-s, -i));
    iend   = (bx == DM_BOUNDARY_PERIODIC) ? s : (PetscMin(s, m - i - 1));
    for (j = ys; j < ys + ny; j++) {
      jstart = (by == DM_BOUNDARY_PERIODIC) ? -s : (PetscMax(-s, -j));
      jend   = (by == DM_BOUNDARY_PERIODIC) ? s : (PetscMin(s, n - j - 1));
      for (k = zs; k < zs + nz; k++) {
        kstart = (bz == DM_BOUNDARY_PERIODIC) ? -s : (PetscMax(-s, -k));
        kend   = (bz == DM_BOUNDARY_PERIODIC) ? s : (PetscMin(s, p - k - 1));

        slot = i - gxs + gnx * (j - gys) + gnx * gny * (k - gzs);

        for (l = 0; l < nc; l++) {
          cnt = 0;
          for (ii = istart; ii < iend + 1; ii++) {
            for (jj = jstart; jj < jend + 1; jj++) {
              for (kk = kstart; kk < kend + 1; kk++) {
                if (ii || jj || kk) {
                  if (st == DMDA_STENCIL_BOX || ((!ii && !jj) || (!jj && !kk) || (!ii && !kk))) { /* entries on star*/
                    for (ifill_col = ofill[l]; ifill_col < ofill[l + 1]; ifill_col++) cols[cnt++] = ofill[ifill_col] + nc * (slot + ii + gnx * jj + gnx * gny * kk);
                  }
                } else {
                  if (dfill) {
                    for (ifill_col = dfill[l]; ifill_col < dfill[l + 1]; ifill_col++) cols[cnt++] = dfill[ifill_col] + nc * (slot + ii + gnx * jj + gnx * gny * kk);
                  } else {
                    for (ifill_col = 0; ifill_col < nc; ifill_col++) cols[cnt++] = ifill_col + nc * (slot + ii + gnx * jj + gnx * gny * kk);
                  }
                }
              }
            }
          }
          row    = l + nc * (slot);
          maxcnt = PetscMax(maxcnt, cnt);
          if (removedups) PetscCall(MatPreallocateSetLocalRemoveDups(ltog, 1, &row, ltog, cnt, cols, dnz, onz));
          else PetscCall(MatPreallocateSetLocal(ltog, 1, &row, ltog, cnt, cols, dnz, onz));
        }
      }
    }
  }
  PetscCall(MatSeqAIJSetPreallocation(J, 0, dnz));
  PetscCall(MatMPIAIJSetPreallocation(J, 0, dnz, 0, onz));
  MatPreallocateEnd(dnz, onz);
  PetscCall(MatSetLocalToGlobalMapping(J, ltog, ltog));

  /*
    For each node in the grid: we get the neighbors in the local (on processor ordering
    that includes the ghost points) then MatSetValuesLocal() maps those indices to the global
    PETSc ordering.
  */
  if (!da->prealloc_only) {
    PetscCall(PetscCalloc1(maxcnt, &values));
    for (i = xs; i < xs + nx; i++) {
      istart = (bx == DM_BOUNDARY_PERIODIC) ? -s : (PetscMax(-s, -i));
      iend   = (bx == DM_BOUNDARY_PERIODIC) ? s : (PetscMin(s, m - i - 1));
      for (j = ys; j < ys + ny; j++) {
        jstart = (by == DM_BOUNDARY_PERIODIC) ? -s : (PetscMax(-s, -j));
        jend   = (by == DM_BOUNDARY_PERIODIC) ? s : (PetscMin(s, n - j - 1));
        for (k = zs; k < zs + nz; k++) {
          kstart = (bz == DM_BOUNDARY_PERIODIC) ? -s : (PetscMax(-s, -k));
          kend   = (bz == DM_BOUNDARY_PERIODIC) ? s : (PetscMin(s, p - k - 1));

          slot = i - gxs + gnx * (j - gys) + gnx * gny * (k - gzs);

          for (l = 0; l < nc; l++) {
            cnt = 0;
            for (ii = istart; ii < iend + 1; ii++) {
              for (jj = jstart; jj < jend + 1; jj++) {
                for (kk = kstart; kk < kend + 1; kk++) {
                  if (ii || jj || kk) {
                    if (st == DMDA_STENCIL_BOX || ((!ii && !jj) || (!jj && !kk) || (!ii && !kk))) { /* entries on star*/
                      for (ifill_col = ofill[l]; ifill_col < ofill[l + 1]; ifill_col++) cols[cnt++] = ofill[ifill_col] + nc * (slot + ii + gnx * jj + gnx * gny * kk);
                    }
                  } else {
                    if (dfill) {
                      for (ifill_col = dfill[l]; ifill_col < dfill[l + 1]; ifill_col++) cols[cnt++] = dfill[ifill_col] + nc * (slot + ii + gnx * jj + gnx * gny * kk);
                    } else {
                      for (ifill_col = 0; ifill_col < nc; ifill_col++) cols[cnt++] = ifill_col + nc * (slot + ii + gnx * jj + gnx * gny * kk);
                    }
                  }
                }
              }
            }
            row = l + nc * (slot);
            PetscCall(MatSetValuesLocal(J, 1, &row, cnt, cols, values, INSERT_VALUES));
          }
        }
      }
    }
    PetscCall(PetscFree(values));
    /* do not copy values to GPU since they are all zero and not yet needed there */
    PetscCall(MatBindToCPU(J, PETSC_TRUE));
    PetscCall(MatAssemblyBegin(J, MAT_FINAL_ASSEMBLY));
    PetscCall(MatAssemblyEnd(J, MAT_FINAL_ASSEMBLY));
    PetscCall(MatBindToCPU(J, PETSC_FALSE));
    PetscCall(MatSetOption(J, MAT_NEW_NONZERO_LOCATION_ERR, PETSC_TRUE));
  }
  PetscCall(PetscFree(cols));
  PetscFunctionReturn(PETSC_SUCCESS);
}