xref: /petsc/src/ksp/pc/impls/gasm/gasm.c (revision 7addb90f52a7936ba144cdab1bb2cc37152af090)

/*
  This file defines an "generalized" additive Schwarz preconditioner for any Mat implementation.
  In this version, each MPI process may intersect multiple subdomains and any subdomain may
  intersect multiple MPI processes.  Intersections of subdomains with MPI processes are called *local
  subdomains*.

       N    - total number of distinct global subdomains  (set explicitly in PCGASMSetTotalSubdomains() or implicitly PCGASMSetSubdomains() and then calculated in PCSetUp_GASM())
       n    - actual number of local subdomains on this process (set in `PCGASMSetSubdomains()` or calculated in `PCGASMSetTotalSubdomains()`)
       nmax - maximum number of local subdomains per process (calculated in PCSetUp_GASM())
*/
#include <petsc/private/pcimpl.h> /*I "petscpc.h" I*/
#include <petscdm.h>

typedef struct {
  PetscInt   N, n, nmax;
  PetscInt   overlap;                /* overlap requested by user */
  PCGASMType type;                   /* use reduced interpolation, restriction or both */
  PetscBool  type_set;               /* if user set this value (so won't change it for symmetric problems) */
  PetscBool  same_subdomain_solvers; /* flag indicating whether all local solvers are same */
  PetscBool  sort_indices;           /* flag to sort subdomain indices */
  PetscBool  user_subdomains;        /* whether the user set explicit subdomain index sets -- keep them on PCReset() */
  PetscBool  dm_subdomains;          /* whether DM is allowed to define subdomains */
  PetscBool  hierarchicalpartitioning;
  IS        *ois;           /* index sets that define the outer (conceptually, overlapping) subdomains */
  IS        *iis;           /* index sets that define the inner (conceptually, nonoverlapping) subdomains */
  KSP       *ksp;           /* linear solvers for each subdomain */
  Mat       *pmat;          /* subdomain block matrices */
  Vec        gx, gy;        /* Merged work vectors */
  Vec       *x, *y;         /* Split work vectors; storage aliases pieces of storage of the above merged vectors. */
  VecScatter gorestriction; /* merged restriction to disjoint union of outer subdomains */
  VecScatter girestriction; /* merged restriction to disjoint union of inner subdomains */
  VecScatter pctoouter;
  IS         permutationIS;
  Mat        permutationP;
  Mat        pcmat;
  Vec        pcx, pcy;
} PC_GASM;

static PetscErrorCode PCGASMComputeGlobalSubdomainNumbering_Private(PC pc, PetscInt **numbering, PetscInt **permutation)
{
  PC_GASM *osm = (PC_GASM *)pc->data;
  PetscInt i;

  PetscFunctionBegin;
  /* Determine the number of globally-distinct subdomains and compute a global numbering for them. */
  PetscCall(PetscMalloc2(osm->n, numbering, osm->n, permutation));
  PetscCall(PetscObjectsListGetGlobalNumbering(PetscObjectComm((PetscObject)pc), osm->n, (PetscObject *)osm->iis, NULL, *numbering));
  for (i = 0; i < osm->n; ++i) (*permutation)[i] = i;
  PetscCall(PetscSortIntWithPermutation(osm->n, *numbering, *permutation));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode PCGASMSubdomainView_Private(PC pc, PetscInt i, PetscViewer viewer)
{
  PC_GASM        *osm = (PC_GASM *)pc->data;
  PetscInt        j, nidx;
  const PetscInt *idx;
  PetscViewer     sviewer;
  char           *cidx;

  PetscFunctionBegin;
  PetscCheck(i >= -1 && i < osm->n, PetscObjectComm((PetscObject)viewer), PETSC_ERR_ARG_WRONG, "Invalid subdomain %" PetscInt_FMT ": must nonnegative and less than %" PetscInt_FMT, i, osm->n);

  /* Inner subdomains. */
  /*
   No more than 15 characters per index plus a space.
   PetscViewerStringSPrintf requires a string of size at least 2, so use (nidx+1) instead of nidx,
   in case nidx == 0. That will take care of the space for the trailing '\0' as well.
   For nidx == 0, the whole string 16 '\0'.
   */
  PetscCall(PetscViewerASCIIPrintf(viewer, "Inner subdomain:\n"));
  PetscCall(PetscViewerFlush(viewer));
  PetscCall(PetscViewerASCIIPushSynchronized(viewer));
  if (i > -1) {
    PetscCall(ISGetLocalSize(osm->iis[i], &nidx));
    PetscCall(PetscMalloc1(16 * (nidx + 1) + 1, &cidx));
    PetscCall(PetscViewerStringOpen(PETSC_COMM_SELF, cidx, 16 * (nidx + 1) + 1, &sviewer));
    PetscCall(ISGetIndices(osm->iis[i], &idx));
    for (j = 0; j < nidx; ++j) PetscCall(PetscViewerStringSPrintf(sviewer, "%" PetscInt_FMT " ", idx[j]));
    PetscCall(ISRestoreIndices(osm->iis[i], &idx));
    PetscCall(PetscViewerDestroy(&sviewer));
    PetscCall(PetscViewerASCIISynchronizedPrintf(viewer, "%s", cidx));
    PetscCall(PetscFree(cidx));
  }
  PetscCall(PetscViewerFlush(viewer));
  PetscCall(PetscViewerASCIIPopSynchronized(viewer));
  PetscCall(PetscViewerASCIIPrintf(viewer, "\n"));
  PetscCall(PetscViewerFlush(viewer));

  /* Outer subdomains. */
  /*
   No more than 15 characters per index plus a space.
   PetscViewerStringSPrintf requires a string of size at least 2, so use (nidx+1) instead of nidx,
   in case nidx == 0. That will take care of the space for the trailing '\0' as well.
   For nidx == 0, the whole string 16 '\0'.
   */
  PetscCall(PetscViewerASCIIPrintf(viewer, "Outer subdomain:\n"));
  PetscCall(PetscViewerFlush(viewer));
  PetscCall(PetscViewerASCIIPushSynchronized(viewer));
  if (i > -1) {
    PetscCall(ISGetLocalSize(osm->ois[i], &nidx));
    PetscCall(PetscMalloc1(16 * (nidx + 1) + 1, &cidx));
    PetscCall(PetscViewerStringOpen(PETSC_COMM_SELF, cidx, 16 * (nidx + 1) + 1, &sviewer));
    PetscCall(ISGetIndices(osm->ois[i], &idx));
    for (j = 0; j < nidx; ++j) PetscCall(PetscViewerStringSPrintf(sviewer, "%" PetscInt_FMT " ", idx[j]));
    PetscCall(PetscViewerDestroy(&sviewer));
    PetscCall(ISRestoreIndices(osm->ois[i], &idx));
    PetscCall(PetscViewerASCIISynchronizedPrintf(viewer, "%s", cidx));
    PetscCall(PetscFree(cidx));
  }
  PetscCall(PetscViewerFlush(viewer));
  PetscCall(PetscViewerASCIIPopSynchronized(viewer));
  PetscCall(PetscViewerASCIIPrintf(viewer, "\n"));
  PetscCall(PetscViewerFlush(viewer));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode PCGASMPrintSubdomains(PC pc)
{
  PC_GASM    *osm = (PC_GASM *)pc->data;
  const char *prefix;
  char        fname[PETSC_MAX_PATH_LEN + 1];
  PetscInt    l, d, count;
  PetscBool   found;
  PetscViewer viewer;
  PetscInt   *numbering, *permutation; /* global numbering of locally-supported subdomains and the permutation from the local ordering */

  PetscFunctionBegin;
  PetscCall(PCGetOptionsPrefix(pc, &prefix));
  PetscCall(PetscOptionsHasName(NULL, prefix, "-pc_gasm_print_subdomains", &found));
  if (!found) PetscFunctionReturn(PETSC_SUCCESS);
  PetscCall(PetscOptionsGetString(NULL, prefix, "-pc_gasm_print_subdomains", fname, sizeof(fname), &found));
  if (!found) PetscCall(PetscStrncpy(fname, "stdout", sizeof(fname)));
  PetscCall(PetscViewerASCIIOpen(PetscObjectComm((PetscObject)pc), fname, &viewer));
  /*
   Make sure the viewer has a name. Otherwise this may cause a deadlock or other weird errors when creating a subcomm viewer:
   the subcomm viewer will attempt to inherit the viewer's name, which, if not set, will be constructed collectively on the comm.
  */
  PetscCall(PetscObjectName((PetscObject)viewer));
  l = 0;
  PetscCall(PCGASMComputeGlobalSubdomainNumbering_Private(pc, &numbering, &permutation));
  for (count = 0; count < osm->N; ++count) {
    /* Now let subdomains go one at a time in the global numbering order and print their subdomain/solver info. */
    if (l < osm->n) {
      d = permutation[l]; /* d is the local number of the l-th smallest (in the global ordering) among the locally supported subdomains */
      if (numbering[d] == count) l++;
      else d = -1;
    } else d = -1;
    PetscCall(PCGASMSubdomainView_Private(pc, d, viewer));
  }
  PetscCall(PetscFree2(numbering, permutation));
  PetscCall(PetscViewerDestroy(&viewer));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode PCView_GASM(PC pc, PetscViewer viewer)
{
  PC_GASM    *osm = (PC_GASM *)pc->data;
  const char *prefix;
  PetscMPIInt rank, size;
  PetscInt    bsz;
  PetscBool   iascii, view_subdomains = PETSC_FALSE;
  PetscViewer sviewer;
  PetscInt    count, l;
  char        overlap[256]     = "user-defined overlap";
  char        gsubdomains[256] = "unknown total number of subdomains";
  char        msubdomains[256] = "unknown max number of local subdomains";
  PetscInt   *numbering, *permutation; /* global numbering of locally-supported subdomains and the permutation from the local ordering */

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

  if (osm->overlap >= 0) PetscCall(PetscSNPrintf(overlap, sizeof(overlap), "requested amount of overlap = %" PetscInt_FMT, osm->overlap));
  if (osm->N != PETSC_DETERMINE) PetscCall(PetscSNPrintf(gsubdomains, sizeof(gsubdomains), "total number of subdomains = %" PetscInt_FMT, osm->N));
  if (osm->nmax != PETSC_DETERMINE) PetscCall(PetscSNPrintf(msubdomains, sizeof(msubdomains), "max number of local subdomains = %" PetscInt_FMT, osm->nmax));

  PetscCall(PCGetOptionsPrefix(pc, &prefix));
  PetscCall(PetscOptionsGetBool(NULL, prefix, "-pc_gasm_view_subdomains", &view_subdomains, NULL));

  PetscCall(PetscObjectTypeCompare((PetscObject)viewer, PETSCVIEWERASCII, &iascii));
  if (iascii) {
    /*
     Make sure the viewer has a name. Otherwise this may cause a deadlock when creating a subcomm viewer:
     the subcomm viewer will attempt to inherit the viewer's name, which, if not set, will be constructed
     collectively on the comm.
     */
    PetscCall(PetscObjectName((PetscObject)viewer));
    PetscCall(PetscViewerASCIIPrintf(viewer, "  Restriction/interpolation type: %s\n", PCGASMTypes[osm->type]));
    PetscCall(PetscViewerASCIIPrintf(viewer, "  %s\n", overlap));
    PetscCall(PetscViewerASCIIPrintf(viewer, "  %s\n", gsubdomains));
    PetscCall(PetscViewerASCIIPrintf(viewer, "  %s\n", msubdomains));
    PetscCall(PetscViewerASCIIPushSynchronized(viewer));
    PetscCall(PetscViewerASCIISynchronizedPrintf(viewer, "  [%d|%d] number of locally-supported subdomains = %" PetscInt_FMT "\n", rank, size, osm->n));
    PetscCall(PetscViewerFlush(viewer));
    PetscCall(PetscViewerASCIIPopSynchronized(viewer));
    /* Cannot take advantage of osm->same_subdomain_solvers without a global numbering of subdomains. */
    PetscCall(PetscViewerASCIIPrintf(viewer, "  Subdomain solver info is as follows:\n"));
    PetscCall(PetscViewerASCIIPushTab(viewer));
    PetscCall(PetscViewerASCIIPrintf(viewer, "  - - - - - - - - - - - - - - - - - -\n"));
    /* Now let subdomains go one at a time in the global numbering order and print their subdomain/solver info. */
    PetscCall(PCGASMComputeGlobalSubdomainNumbering_Private(pc, &numbering, &permutation));
    l = 0;
    for (count = 0; count < osm->N; ++count) {
      PetscMPIInt srank, ssize;
      if (l < osm->n) {
        PetscInt d = permutation[l]; /* d is the local number of the l-th smallest (in the global ordering) among the locally supported subdomains */
        if (numbering[d] == count) {
          PetscCallMPI(MPI_Comm_size(((PetscObject)osm->ois[d])->comm, &ssize));
          PetscCallMPI(MPI_Comm_rank(((PetscObject)osm->ois[d])->comm, &srank));
          PetscCall(PetscViewerGetSubViewer(viewer, ((PetscObject)osm->ois[d])->comm, &sviewer));
          PetscCall(ISGetLocalSize(osm->ois[d], &bsz));
          PetscCall(PetscViewerASCIISynchronizedPrintf(sviewer, "  [%d|%d] (subcomm [%d|%d]) local subdomain number %" PetscInt_FMT ", local size = %" PetscInt_FMT "\n", rank, size, srank, ssize, d, bsz));
          PetscCall(PetscViewerFlush(sviewer));
          PetscCall(PetscViewerASCIIPushTab(sviewer));
          if (view_subdomains) PetscCall(PCGASMSubdomainView_Private(pc, d, sviewer));
          if (!pc->setupcalled) {
            PetscCall(PetscViewerASCIISynchronizedPrintf(sviewer, "  Solver not set up yet: PCSetUp() not yet called\n"));
          } else {
            PetscCall(KSPView(osm->ksp[d], sviewer));
          }
          PetscCall(PetscViewerASCIIPopTab(sviewer));
          PetscCall(PetscViewerASCIIPrintf(sviewer, "  - - - - - - - - - - - - - - - - - -\n"));
          PetscCall(PetscViewerFlush(sviewer));
          PetscCall(PetscViewerRestoreSubViewer(viewer, ((PetscObject)osm->ois[d])->comm, &sviewer));
          ++l;
        } else {
          PetscCall(PetscViewerGetSubViewer(viewer, PETSC_COMM_SELF, &sviewer));
          PetscCall(PetscViewerRestoreSubViewer(viewer, PETSC_COMM_SELF, &sviewer));
        }
      } else {
        PetscCall(PetscViewerGetSubViewer(viewer, PETSC_COMM_SELF, &sviewer));
        PetscCall(PetscViewerRestoreSubViewer(viewer, PETSC_COMM_SELF, &sviewer));
      }
    }
    PetscCall(PetscFree2(numbering, permutation));
    PetscCall(PetscViewerASCIIPopTab(viewer));
    PetscCall(PetscViewerFlush(viewer));
    /* this line is needed to match the extra PetscViewerASCIIPushSynchronized() in PetscViewerGetSubViewer() */
    PetscCall(PetscViewerASCIIPopSynchronized(viewer));
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

PETSC_INTERN PetscErrorCode PCGASMCreateLocalSubdomains(Mat A, PetscInt nloc, IS *iis[]);

static PetscErrorCode PCGASMSetHierarchicalPartitioning(PC pc)
{
  PC_GASM        *osm = (PC_GASM *)pc->data;
  MatPartitioning part;
  MPI_Comm        comm;
  PetscMPIInt     size;
  PetscInt        nlocalsubdomains, fromrows_localsize;
  IS              partitioning, fromrows, isn;
  Vec             outervec;

  PetscFunctionBegin;
  PetscCall(PetscObjectGetComm((PetscObject)pc, &comm));
  PetscCallMPI(MPI_Comm_size(comm, &size));
  /* we do not need a hierarchical partitioning when
    * the total number of subdomains is consistent with
    * the number of MPI tasks.
    * For the following cases, we do not need to use HP
    * */
  if (osm->N == PETSC_DETERMINE || osm->N >= size || osm->N == 1) PetscFunctionReturn(PETSC_SUCCESS);
  PetscCheck(size % osm->N == 0, PETSC_COMM_WORLD, PETSC_ERR_ARG_INCOMP, "have to specify the total number of subdomains %" PetscInt_FMT " to be a factor of the number of ranks %d ", osm->N, size);
  nlocalsubdomains = size / osm->N;
  osm->n           = 1;
  PetscCall(MatPartitioningCreate(comm, &part));
  PetscCall(MatPartitioningSetAdjacency(part, pc->pmat));
  PetscCall(MatPartitioningSetType(part, MATPARTITIONINGHIERARCH));
  PetscCall(MatPartitioningHierarchicalSetNcoarseparts(part, osm->N));
  PetscCall(MatPartitioningHierarchicalSetNfineparts(part, nlocalsubdomains));
  PetscCall(MatPartitioningSetFromOptions(part));
  /* get new rank owner number of each vertex */
  PetscCall(MatPartitioningApply(part, &partitioning));
  PetscCall(ISBuildTwoSided(partitioning, NULL, &fromrows));
  PetscCall(ISPartitioningToNumbering(partitioning, &isn));
  PetscCall(ISDestroy(&isn));
  PetscCall(ISGetLocalSize(fromrows, &fromrows_localsize));
  PetscCall(MatPartitioningDestroy(&part));
  PetscCall(MatCreateVecs(pc->pmat, &outervec, NULL));
  PetscCall(VecCreateMPI(comm, fromrows_localsize, PETSC_DETERMINE, &osm->pcx));
  PetscCall(VecDuplicate(osm->pcx, &osm->pcy));
  PetscCall(VecScatterCreate(osm->pcx, NULL, outervec, fromrows, &osm->pctoouter));
  PetscCall(MatCreateSubMatrix(pc->pmat, fromrows, fromrows, MAT_INITIAL_MATRIX, &osm->permutationP));
  PetscCall(PetscObjectReference((PetscObject)fromrows));
  osm->permutationIS = fromrows;
  osm->pcmat         = pc->pmat;
  PetscCall(PetscObjectReference((PetscObject)osm->permutationP));
  pc->pmat = osm->permutationP;
  PetscCall(VecDestroy(&outervec));
  PetscCall(ISDestroy(&fromrows));
  PetscCall(ISDestroy(&partitioning));
  osm->n = PETSC_DETERMINE;
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode PCSetUp_GASM(PC pc)
{
  PC_GASM        *osm = (PC_GASM *)pc->data;
  PetscInt        i, nInnerIndices, nTotalInnerIndices;
  PetscMPIInt     rank, size;
  MatReuse        scall = MAT_REUSE_MATRIX;
  KSP             ksp;
  PC              subpc;
  const char     *prefix, *pprefix;
  Vec             x, y;
  PetscInt        oni;   /* Number of indices in the i-th local outer subdomain.               */
  const PetscInt *oidxi; /* Indices from the i-th subdomain local outer subdomain.             */
  PetscInt        on;    /* Number of indices in the disjoint union of local outer subdomains. */
  PetscInt       *oidx;  /* Indices in the disjoint union of local outer subdomains. */
  IS              gois;  /* Disjoint union the global indices of outer subdomains.             */
  IS              goid;  /* Identity IS of the size of the disjoint union of outer subdomains. */
  PetscScalar    *gxarray, *gyarray;
  PetscInt        gostart; /* Start of locally-owned indices in the vectors -- osm->gx,osm->gy -- over the disjoint union of outer subdomains. */
  PetscInt        num_subdomains  = 0;
  DM             *subdomain_dm    = NULL;
  char          **subdomain_names = NULL;
  PetscInt       *numbering;

  PetscFunctionBegin;
  PetscCallMPI(MPI_Comm_size(PetscObjectComm((PetscObject)pc), &size));
  PetscCallMPI(MPI_Comm_rank(PetscObjectComm((PetscObject)pc), &rank));
  if (!pc->setupcalled) {
    /* use a hierarchical partitioning */
    if (osm->hierarchicalpartitioning) PetscCall(PCGASMSetHierarchicalPartitioning(pc));
    if (osm->n == PETSC_DETERMINE) {
      if (osm->N != PETSC_DETERMINE) {
        /* No local subdomains given, but the desired number of total subdomains is known, so construct them accordingly. */
        PetscCall(PCGASMCreateSubdomains(pc->pmat, osm->N, &osm->n, &osm->iis));
      } else if (osm->dm_subdomains && pc->dm) {
        /* try pc->dm next, if allowed */
        PetscInt d;
        IS      *inner_subdomain_is, *outer_subdomain_is;
        PetscCall(DMCreateDomainDecomposition(pc->dm, &num_subdomains, &subdomain_names, &inner_subdomain_is, &outer_subdomain_is, &subdomain_dm));
        if (num_subdomains) PetscCall(PCGASMSetSubdomains(pc, num_subdomains, inner_subdomain_is, outer_subdomain_is));
        for (d = 0; d < num_subdomains; ++d) {
          if (inner_subdomain_is) PetscCall(ISDestroy(&inner_subdomain_is[d]));
          if (outer_subdomain_is) PetscCall(ISDestroy(&outer_subdomain_is[d]));
        }
        PetscCall(PetscFree(inner_subdomain_is));
        PetscCall(PetscFree(outer_subdomain_is));
      } else {
        /* still no subdomains; use one per rank */
        osm->nmax = osm->n = 1;
        PetscCallMPI(MPI_Comm_size(PetscObjectComm((PetscObject)pc), &size));
        osm->N = size;
        PetscCall(PCGASMCreateLocalSubdomains(pc->pmat, osm->n, &osm->iis));
      }
    }
    if (!osm->iis) {
      /*
       osm->n was set in PCGASMSetSubdomains(), but the actual subdomains have not been supplied.
       We create the requisite number of local inner subdomains and then expand them into
       out subdomains, if necessary.
       */
      PetscCall(PCGASMCreateLocalSubdomains(pc->pmat, osm->n, &osm->iis));
    }
    if (!osm->ois) {
      /*
            Initially make outer subdomains the same as inner subdomains. If nonzero additional overlap
            has been requested, copy the inner subdomains over so they can be modified.
      */
      PetscCall(PetscMalloc1(osm->n, &osm->ois));
      for (i = 0; i < osm->n; ++i) {
        if (osm->overlap > 0 && osm->N > 1) { /* With positive overlap, osm->iis[i] will be modified */
          PetscCall(ISDuplicate(osm->iis[i], (osm->ois) + i));
          PetscCall(ISCopy(osm->iis[i], osm->ois[i]));
        } else {
          PetscCall(PetscObjectReference((PetscObject)osm->iis[i]));
          osm->ois[i] = osm->iis[i];
        }
      }
      if (osm->overlap > 0 && osm->N > 1) {
        /* Extend the "overlapping" regions by a number of steps */
        PetscCall(MatIncreaseOverlapSplit(pc->pmat, osm->n, osm->ois, osm->overlap));
      }
    }

    /* Now the subdomains are defined.  Determine their global and max local numbers, if necessary. */
    if (osm->nmax == PETSC_DETERMINE) {
      PetscInt inwork, outwork;
      /* determine global number of subdomains and the max number of local subdomains */
      inwork = osm->n;
      PetscCallMPI(MPIU_Allreduce(&inwork, &outwork, 1, MPIU_INT, MPI_MAX, PetscObjectComm((PetscObject)pc)));
      osm->nmax = outwork;
    }
    if (osm->N == PETSC_DETERMINE) {
      /* Determine the number of globally-distinct subdomains and compute a global numbering for them. */
      PetscCall(PetscObjectsListGetGlobalNumbering(PetscObjectComm((PetscObject)pc), osm->n, (PetscObject *)osm->ois, &osm->N, NULL));
    }

    if (osm->sort_indices) {
      for (i = 0; i < osm->n; i++) {
        PetscCall(ISSort(osm->ois[i]));
        PetscCall(ISSort(osm->iis[i]));
      }
    }
    PetscCall(PCGetOptionsPrefix(pc, &prefix));
    PetscCall(PCGASMPrintSubdomains(pc));

    /*
       Merge the ISs, create merged vectors and restrictions.
     */
    /* Merge outer subdomain ISs and construct a restriction onto the disjoint union of local outer subdomains. */
    on = 0;
    for (i = 0; i < osm->n; i++) {
      PetscCall(ISGetLocalSize(osm->ois[i], &oni));
      on += oni;
    }
    PetscCall(PetscMalloc1(on, &oidx));
    on = 0;
    /* Merge local indices together */
    for (i = 0; i < osm->n; i++) {
      PetscCall(ISGetLocalSize(osm->ois[i], &oni));
      PetscCall(ISGetIndices(osm->ois[i], &oidxi));
      PetscCall(PetscArraycpy(oidx + on, oidxi, oni));
      PetscCall(ISRestoreIndices(osm->ois[i], &oidxi));
      on += oni;
    }
    PetscCall(ISCreateGeneral(((PetscObject)pc)->comm, on, oidx, PETSC_OWN_POINTER, &gois));
    nTotalInnerIndices = 0;
    for (i = 0; i < osm->n; i++) {
      PetscCall(ISGetLocalSize(osm->iis[i], &nInnerIndices));
      nTotalInnerIndices += nInnerIndices;
    }
    PetscCall(VecCreateMPI(((PetscObject)pc)->comm, nTotalInnerIndices, PETSC_DETERMINE, &x));
    PetscCall(VecDuplicate(x, &y));

    PetscCall(VecCreateMPI(PetscObjectComm((PetscObject)pc), on, PETSC_DECIDE, &osm->gx));
    PetscCall(VecDuplicate(osm->gx, &osm->gy));
    PetscCall(VecGetOwnershipRange(osm->gx, &gostart, NULL));
    PetscCall(ISCreateStride(PetscObjectComm((PetscObject)pc), on, gostart, 1, &goid));
    /* gois might indices not on local */
    PetscCall(VecScatterCreate(x, gois, osm->gx, goid, &osm->gorestriction));
    PetscCall(PetscMalloc1(osm->n, &numbering));
    PetscCall(PetscObjectsListGetGlobalNumbering(PetscObjectComm((PetscObject)pc), osm->n, (PetscObject *)osm->ois, NULL, numbering));
    PetscCall(VecDestroy(&x));
    PetscCall(ISDestroy(&gois));

    /* Merge inner subdomain ISs and construct a restriction onto the disjoint union of local inner subdomains. */
    {
      PetscInt        ini;   /* Number of indices the i-th a local inner subdomain. */
      PetscInt        in;    /* Number of indices in the disjoint union of local inner subdomains. */
      PetscInt       *iidx;  /* Global indices in the merged local inner subdomain. */
      PetscInt       *ioidx; /* Global indices of the disjoint union of inner subdomains within the disjoint union of outer subdomains. */
      IS              giis;  /* IS for the disjoint union of inner subdomains. */
      IS              giois; /* IS for the disjoint union of inner subdomains within the disjoint union of outer subdomains. */
      PetscScalar    *array;
      const PetscInt *indices;
      PetscInt        k;
      on = 0;
      for (i = 0; i < osm->n; i++) {
        PetscCall(ISGetLocalSize(osm->ois[i], &oni));
        on += oni;
      }
      PetscCall(PetscMalloc1(on, &iidx));
      PetscCall(PetscMalloc1(on, &ioidx));
      PetscCall(VecGetArray(y, &array));
      /* set communicator id to determine where overlap is */
      in = 0;
      for (i = 0; i < osm->n; i++) {
        PetscCall(ISGetLocalSize(osm->iis[i], &ini));
        for (k = 0; k < ini; ++k) array[in + k] = numbering[i];
        in += ini;
      }
      PetscCall(VecRestoreArray(y, &array));
      PetscCall(VecScatterBegin(osm->gorestriction, y, osm->gy, INSERT_VALUES, SCATTER_FORWARD));
      PetscCall(VecScatterEnd(osm->gorestriction, y, osm->gy, INSERT_VALUES, SCATTER_FORWARD));
      PetscCall(VecGetOwnershipRange(osm->gy, &gostart, NULL));
      PetscCall(VecGetArray(osm->gy, &array));
      on = 0;
      in = 0;
      for (i = 0; i < osm->n; i++) {
        PetscCall(ISGetLocalSize(osm->ois[i], &oni));
        PetscCall(ISGetIndices(osm->ois[i], &indices));
        for (k = 0; k < oni; k++) {
          /*  skip overlapping indices to get inner domain */
          if (PetscRealPart(array[on + k]) != numbering[i]) continue;
          iidx[in]    = indices[k];
          ioidx[in++] = gostart + on + k;
        }
        PetscCall(ISRestoreIndices(osm->ois[i], &indices));
        on += oni;
      }
      PetscCall(VecRestoreArray(osm->gy, &array));
      PetscCall(ISCreateGeneral(PetscObjectComm((PetscObject)pc), in, iidx, PETSC_OWN_POINTER, &giis));
      PetscCall(ISCreateGeneral(PetscObjectComm((PetscObject)pc), in, ioidx, PETSC_OWN_POINTER, &giois));
      PetscCall(VecScatterCreate(y, giis, osm->gy, giois, &osm->girestriction));
      PetscCall(VecDestroy(&y));
      PetscCall(ISDestroy(&giis));
      PetscCall(ISDestroy(&giois));
    }
    PetscCall(ISDestroy(&goid));
    PetscCall(PetscFree(numbering));

    /* Create the subdomain work vectors. */
    PetscCall(PetscMalloc1(osm->n, &osm->x));
    PetscCall(PetscMalloc1(osm->n, &osm->y));
    PetscCall(VecGetArray(osm->gx, &gxarray));
    PetscCall(VecGetArray(osm->gy, &gyarray));
    for (i = 0, on = 0; i < osm->n; ++i, on += oni) {
      PetscInt oNi;
      PetscCall(ISGetLocalSize(osm->ois[i], &oni));
      /* on a sub communicator */
      PetscCall(ISGetSize(osm->ois[i], &oNi));
      PetscCall(VecCreateMPIWithArray(((PetscObject)osm->ois[i])->comm, 1, oni, oNi, gxarray + on, &osm->x[i]));
      PetscCall(VecCreateMPIWithArray(((PetscObject)osm->ois[i])->comm, 1, oni, oNi, gyarray + on, &osm->y[i]));
    }
    PetscCall(VecRestoreArray(osm->gx, &gxarray));
    PetscCall(VecRestoreArray(osm->gy, &gyarray));
    /* Create the subdomain solvers */
    PetscCall(PetscMalloc1(osm->n, &osm->ksp));
    for (i = 0; i < osm->n; i++) {
      char subprefix[PETSC_MAX_PATH_LEN + 1];
      PetscCall(KSPCreate(((PetscObject)osm->ois[i])->comm, &ksp));
      PetscCall(KSPSetNestLevel(ksp, pc->kspnestlevel));
      PetscCall(KSPSetErrorIfNotConverged(ksp, pc->erroriffailure));
      PetscCall(PetscObjectIncrementTabLevel((PetscObject)ksp, (PetscObject)pc, 1));
      PetscCall(KSPSetType(ksp, KSPPREONLY));
      PetscCall(KSPGetPC(ksp, &subpc)); /* Why do we need this here? */
      if (subdomain_dm) {
        PetscCall(KSPSetDM(ksp, subdomain_dm[i]));
        PetscCall(DMDestroy(subdomain_dm + i));
      }
      PetscCall(PCGetOptionsPrefix(pc, &prefix));
      PetscCall(KSPSetOptionsPrefix(ksp, prefix));
      if (subdomain_names && subdomain_names[i]) {
        PetscCall(PetscSNPrintf(subprefix, PETSC_MAX_PATH_LEN, "sub_%s_", subdomain_names[i]));
        PetscCall(KSPAppendOptionsPrefix(ksp, subprefix));
        PetscCall(PetscFree(subdomain_names[i]));
      }
      PetscCall(KSPAppendOptionsPrefix(ksp, "sub_"));
      osm->ksp[i] = ksp;
    }
    PetscCall(PetscFree(subdomain_dm));
    PetscCall(PetscFree(subdomain_names));
    scall = MAT_INITIAL_MATRIX;
  } else { /* if (pc->setupcalled) */
    /*
       Destroy the submatrices from the previous iteration
    */
    if (pc->flag == DIFFERENT_NONZERO_PATTERN) {
      PetscCall(MatDestroyMatrices(osm->n, &osm->pmat));
      scall = MAT_INITIAL_MATRIX;
    }
    if (osm->permutationIS) {
      PetscCall(MatCreateSubMatrix(pc->pmat, osm->permutationIS, osm->permutationIS, scall, &osm->permutationP));
      PetscCall(PetscObjectReference((PetscObject)osm->permutationP));
      osm->pcmat = pc->pmat;
      pc->pmat   = osm->permutationP;
    }
  }

  /*
     Extract the submatrices.
  */
  if (size > 1) {
    PetscCall(MatCreateSubMatricesMPI(pc->pmat, osm->n, osm->ois, osm->ois, scall, &osm->pmat));
  } else {
    PetscCall(MatCreateSubMatrices(pc->pmat, osm->n, osm->ois, osm->ois, scall, &osm->pmat));
  }
  if (scall == MAT_INITIAL_MATRIX) {
    PetscCall(PetscObjectGetOptionsPrefix((PetscObject)pc->pmat, &pprefix));
    for (i = 0; i < osm->n; i++) PetscCall(PetscObjectSetOptionsPrefix((PetscObject)osm->pmat[i], pprefix));
  }

  /* Return control to the user so that the submatrices can be modified (e.g., to apply
     different boundary conditions for the submatrices than for the global problem) */
  PetscCall(PCModifySubMatrices(pc, osm->n, osm->ois, osm->ois, osm->pmat, pc->modifysubmatricesP));

  /*
     Loop over submatrices putting them into local ksps
  */
  for (i = 0; i < osm->n; i++) {
    PetscCall(KSPSetOperators(osm->ksp[i], osm->pmat[i], osm->pmat[i]));
    PetscCall(KSPGetOptionsPrefix(osm->ksp[i], &prefix));
    PetscCall(MatSetOptionsPrefix(osm->pmat[i], prefix));
    if (!pc->setupcalled) PetscCall(KSPSetFromOptions(osm->ksp[i]));
  }
  if (osm->pcmat) {
    PetscCall(MatDestroy(&pc->pmat));
    pc->pmat   = osm->pcmat;
    osm->pcmat = NULL;
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode PCSetUpOnBlocks_GASM(PC pc)
{
  PC_GASM *osm = (PC_GASM *)pc->data;
  PetscInt i;

  PetscFunctionBegin;
  for (i = 0; i < osm->n; i++) PetscCall(KSPSetUp(osm->ksp[i]));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode PCApply_GASM(PC pc, Vec xin, Vec yout)
{
  PC_GASM    *osm = (PC_GASM *)pc->data;
  PetscInt    i;
  Vec         x, y;
  ScatterMode forward = SCATTER_FORWARD, reverse = SCATTER_REVERSE;

  PetscFunctionBegin;
  if (osm->pctoouter) {
    PetscCall(VecScatterBegin(osm->pctoouter, xin, osm->pcx, INSERT_VALUES, SCATTER_REVERSE));
    PetscCall(VecScatterEnd(osm->pctoouter, xin, osm->pcx, INSERT_VALUES, SCATTER_REVERSE));
    x = osm->pcx;
    y = osm->pcy;
  } else {
    x = xin;
    y = yout;
  }
  /*
     support for limiting the restriction or interpolation only to the inner
     subdomain values (leaving the other values 0).
  */
  if (!(osm->type & PC_GASM_RESTRICT)) {
    /* have to zero the work RHS since scatter may leave some slots empty */
    PetscCall(VecZeroEntries(osm->gx));
    PetscCall(VecScatterBegin(osm->girestriction, x, osm->gx, INSERT_VALUES, forward));
  } else {
    PetscCall(VecScatterBegin(osm->gorestriction, x, osm->gx, INSERT_VALUES, forward));
  }
  PetscCall(VecZeroEntries(osm->gy));
  if (!(osm->type & PC_GASM_RESTRICT)) {
    PetscCall(VecScatterEnd(osm->girestriction, x, osm->gx, INSERT_VALUES, forward));
  } else {
    PetscCall(VecScatterEnd(osm->gorestriction, x, osm->gx, INSERT_VALUES, forward));
  }
  /* do the subdomain solves */
  for (i = 0; i < osm->n; ++i) {
    PetscCall(KSPSolve(osm->ksp[i], osm->x[i], osm->y[i]));
    PetscCall(KSPCheckSolve(osm->ksp[i], pc, osm->y[i]));
  }
  /* do we need to zero y? */
  PetscCall(VecZeroEntries(y));
  if (!(osm->type & PC_GASM_INTERPOLATE)) {
    PetscCall(VecScatterBegin(osm->girestriction, osm->gy, y, ADD_VALUES, reverse));
    PetscCall(VecScatterEnd(osm->girestriction, osm->gy, y, ADD_VALUES, reverse));
  } else {
    PetscCall(VecScatterBegin(osm->gorestriction, osm->gy, y, ADD_VALUES, reverse));
    PetscCall(VecScatterEnd(osm->gorestriction, osm->gy, y, ADD_VALUES, reverse));
  }
  if (osm->pctoouter) {
    PetscCall(VecScatterBegin(osm->pctoouter, y, yout, INSERT_VALUES, SCATTER_FORWARD));
    PetscCall(VecScatterEnd(osm->pctoouter, y, yout, INSERT_VALUES, SCATTER_FORWARD));
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode PCMatApply_GASM(PC pc, Mat Xin, Mat Yout)
{
  PC_GASM    *osm = (PC_GASM *)pc->data;
  Mat         X, Y, O = NULL, Z, W;
  Vec         x, y;
  PetscInt    i, m, M, N;
  ScatterMode forward = SCATTER_FORWARD, reverse = SCATTER_REVERSE;

  PetscFunctionBegin;
  PetscCheck(osm->n == 1, PetscObjectComm((PetscObject)pc), PETSC_ERR_SUP, "Not yet implemented");
  PetscCall(MatGetSize(Xin, NULL, &N));
  if (osm->pctoouter) {
    PetscCall(VecGetLocalSize(osm->pcx, &m));
    PetscCall(VecGetSize(osm->pcx, &M));
    PetscCall(MatCreateDense(PetscObjectComm((PetscObject)osm->ois[0]), m, PETSC_DECIDE, M, N, NULL, &O));
    for (i = 0; i < N; ++i) {
      PetscCall(MatDenseGetColumnVecRead(Xin, i, &x));
      PetscCall(MatDenseGetColumnVecWrite(O, i, &y));
      PetscCall(VecScatterBegin(osm->pctoouter, x, y, INSERT_VALUES, SCATTER_REVERSE));
      PetscCall(VecScatterEnd(osm->pctoouter, x, y, INSERT_VALUES, SCATTER_REVERSE));
      PetscCall(MatDenseRestoreColumnVecWrite(O, i, &y));
      PetscCall(MatDenseRestoreColumnVecRead(Xin, i, &x));
    }
    X = Y = O;
  } else {
    X = Xin;
    Y = Yout;
  }
  /*
     support for limiting the restriction or interpolation only to the inner
     subdomain values (leaving the other values 0).
  */
  PetscCall(VecGetLocalSize(osm->x[0], &m));
  PetscCall(VecGetSize(osm->x[0], &M));
  PetscCall(MatCreateDense(PetscObjectComm((PetscObject)osm->ois[0]), m, PETSC_DECIDE, M, N, NULL, &Z));
  for (i = 0; i < N; ++i) {
    PetscCall(MatDenseGetColumnVecRead(X, i, &x));
    PetscCall(MatDenseGetColumnVecWrite(Z, i, &y));
    if (!(osm->type & PC_GASM_RESTRICT)) {
      /* have to zero the work RHS since scatter may leave some slots empty */
      PetscCall(VecZeroEntries(y));
      PetscCall(VecScatterBegin(osm->girestriction, x, y, INSERT_VALUES, forward));
      PetscCall(VecScatterEnd(osm->girestriction, x, y, INSERT_VALUES, forward));
    } else {
      PetscCall(VecScatterBegin(osm->gorestriction, x, y, INSERT_VALUES, forward));
      PetscCall(VecScatterEnd(osm->gorestriction, x, y, INSERT_VALUES, forward));
    }
    PetscCall(MatDenseRestoreColumnVecWrite(Z, i, &y));
    PetscCall(MatDenseRestoreColumnVecRead(X, i, &x));
  }
  PetscCall(MatCreateDense(PetscObjectComm((PetscObject)osm->ois[0]), m, PETSC_DECIDE, M, N, NULL, &W));
  PetscCall(MatSetOption(Z, MAT_NO_OFF_PROC_ENTRIES, PETSC_TRUE));
  PetscCall(MatAssemblyBegin(Z, MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(Z, MAT_FINAL_ASSEMBLY));
  /* do the subdomain solve */
  PetscCall(KSPMatSolve(osm->ksp[0], Z, W));
  PetscCall(KSPCheckSolve(osm->ksp[0], pc, NULL));
  PetscCall(MatDestroy(&Z));
  /* do we need to zero y? */
  PetscCall(MatZeroEntries(Y));
  for (i = 0; i < N; ++i) {
    PetscCall(MatDenseGetColumnVecWrite(Y, i, &y));
    PetscCall(MatDenseGetColumnVecRead(W, i, &x));
    if (!(osm->type & PC_GASM_INTERPOLATE)) {
      PetscCall(VecScatterBegin(osm->girestriction, x, y, ADD_VALUES, reverse));
      PetscCall(VecScatterEnd(osm->girestriction, x, y, ADD_VALUES, reverse));
    } else {
      PetscCall(VecScatterBegin(osm->gorestriction, x, y, ADD_VALUES, reverse));
      PetscCall(VecScatterEnd(osm->gorestriction, x, y, ADD_VALUES, reverse));
    }
    PetscCall(MatDenseRestoreColumnVecRead(W, i, &x));
    if (osm->pctoouter) {
      PetscCall(MatDenseGetColumnVecWrite(Yout, i, &x));
      PetscCall(VecScatterBegin(osm->pctoouter, y, x, INSERT_VALUES, SCATTER_FORWARD));
      PetscCall(VecScatterEnd(osm->pctoouter, y, x, INSERT_VALUES, SCATTER_FORWARD));
      PetscCall(MatDenseRestoreColumnVecRead(Yout, i, &x));
    }
    PetscCall(MatDenseRestoreColumnVecWrite(Y, i, &y));
  }
  PetscCall(MatDestroy(&W));
  PetscCall(MatDestroy(&O));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode PCApplyTranspose_GASM(PC pc, Vec xin, Vec yout)
{
  PC_GASM    *osm = (PC_GASM *)pc->data;
  PetscInt    i;
  Vec         x, y;
  ScatterMode forward = SCATTER_FORWARD, reverse = SCATTER_REVERSE;

  PetscFunctionBegin;
  if (osm->pctoouter) {
    PetscCall(VecScatterBegin(osm->pctoouter, xin, osm->pcx, INSERT_VALUES, SCATTER_REVERSE));
    PetscCall(VecScatterEnd(osm->pctoouter, xin, osm->pcx, INSERT_VALUES, SCATTER_REVERSE));
    x = osm->pcx;
    y = osm->pcy;
  } else {
    x = xin;
    y = yout;
  }
  /*
     Support for limiting the restriction or interpolation to only local
     subdomain values (leaving the other values 0).

     Note: these are reversed from the PCApply_GASM() because we are applying the
     transpose of the three terms
  */
  if (!(osm->type & PC_GASM_INTERPOLATE)) {
    /* have to zero the work RHS since scatter may leave some slots empty */
    PetscCall(VecZeroEntries(osm->gx));
    PetscCall(VecScatterBegin(osm->girestriction, x, osm->gx, INSERT_VALUES, forward));
  } else {
    PetscCall(VecScatterBegin(osm->gorestriction, x, osm->gx, INSERT_VALUES, forward));
  }
  PetscCall(VecZeroEntries(osm->gy));
  if (!(osm->type & PC_GASM_INTERPOLATE)) {
    PetscCall(VecScatterEnd(osm->girestriction, x, osm->gx, INSERT_VALUES, forward));
  } else {
    PetscCall(VecScatterEnd(osm->gorestriction, x, osm->gx, INSERT_VALUES, forward));
  }
  /* do the local solves */
  for (i = 0; i < osm->n; ++i) { /* Note that the solves are local, so we can go to osm->n, rather than osm->nmax. */
    PetscCall(KSPSolveTranspose(osm->ksp[i], osm->x[i], osm->y[i]));
    PetscCall(KSPCheckSolve(osm->ksp[i], pc, osm->y[i]));
  }
  PetscCall(VecZeroEntries(y));
  if (!(osm->type & PC_GASM_RESTRICT)) {
    PetscCall(VecScatterBegin(osm->girestriction, osm->gy, y, ADD_VALUES, reverse));
    PetscCall(VecScatterEnd(osm->girestriction, osm->gy, y, ADD_VALUES, reverse));
  } else {
    PetscCall(VecScatterBegin(osm->gorestriction, osm->gy, y, ADD_VALUES, reverse));
    PetscCall(VecScatterEnd(osm->gorestriction, osm->gy, y, ADD_VALUES, reverse));
  }
  if (osm->pctoouter) {
    PetscCall(VecScatterBegin(osm->pctoouter, y, yout, INSERT_VALUES, SCATTER_FORWARD));
    PetscCall(VecScatterEnd(osm->pctoouter, y, yout, INSERT_VALUES, SCATTER_FORWARD));
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode PCReset_GASM(PC pc)
{
  PC_GASM *osm = (PC_GASM *)pc->data;
  PetscInt i;

  PetscFunctionBegin;
  if (osm->ksp) {
    for (i = 0; i < osm->n; i++) PetscCall(KSPReset(osm->ksp[i]));
  }
  if (osm->pmat) {
    if (osm->n > 0) {
      PetscMPIInt size;
      PetscCallMPI(MPI_Comm_size(PetscObjectComm((PetscObject)pc), &size));
      if (size > 1) {
        /* osm->pmat is created by MatCreateSubMatricesMPI(), cannot use MatDestroySubMatrices() */
        PetscCall(MatDestroyMatrices(osm->n, &osm->pmat));
      } else {
        PetscCall(MatDestroySubMatrices(osm->n, &osm->pmat));
      }
    }
  }
  if (osm->x) {
    for (i = 0; i < osm->n; i++) {
      PetscCall(VecDestroy(&osm->x[i]));
      PetscCall(VecDestroy(&osm->y[i]));
    }
  }
  PetscCall(VecDestroy(&osm->gx));
  PetscCall(VecDestroy(&osm->gy));

  PetscCall(VecScatterDestroy(&osm->gorestriction));
  PetscCall(VecScatterDestroy(&osm->girestriction));
  if (!osm->user_subdomains) {
    PetscCall(PCGASMDestroySubdomains(osm->n, &osm->ois, &osm->iis));
    osm->N    = PETSC_DETERMINE;
    osm->nmax = PETSC_DETERMINE;
  }
  if (osm->pctoouter) PetscCall(VecScatterDestroy(&osm->pctoouter));
  if (osm->permutationIS) PetscCall(ISDestroy(&osm->permutationIS));
  if (osm->pcx) PetscCall(VecDestroy(&osm->pcx));
  if (osm->pcy) PetscCall(VecDestroy(&osm->pcy));
  if (osm->permutationP) PetscCall(MatDestroy(&osm->permutationP));
  if (osm->pcmat) PetscCall(MatDestroy(&osm->pcmat));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode PCDestroy_GASM(PC pc)
{
  PC_GASM *osm = (PC_GASM *)pc->data;
  PetscInt i;

  PetscFunctionBegin;
  PetscCall(PCReset_GASM(pc));
  /* PCReset will not destroy subdomains, if user_subdomains is true. */
  PetscCall(PCGASMDestroySubdomains(osm->n, &osm->ois, &osm->iis));
  if (osm->ksp) {
    for (i = 0; i < osm->n; i++) PetscCall(KSPDestroy(&osm->ksp[i]));
    PetscCall(PetscFree(osm->ksp));
  }
  PetscCall(PetscFree(osm->x));
  PetscCall(PetscFree(osm->y));
  PetscCall(PetscObjectComposeFunction((PetscObject)pc, "PCGASMSetSubdomains_C", NULL));
  PetscCall(PetscObjectComposeFunction((PetscObject)pc, "PCGASMSetOverlap_C", NULL));
  PetscCall(PetscObjectComposeFunction((PetscObject)pc, "PCGASMSetType_C", NULL));
  PetscCall(PetscObjectComposeFunction((PetscObject)pc, "PCGASMSetSortIndices_C", NULL));
  PetscCall(PetscObjectComposeFunction((PetscObject)pc, "PCGASMGetSubKSP_C", NULL));
  PetscCall(PetscFree(pc->data));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode PCSetFromOptions_GASM(PC pc, PetscOptionItems PetscOptionsObject)
{
  PC_GASM   *osm = (PC_GASM *)pc->data;
  PetscInt   blocks, ovl;
  PetscBool  flg;
  PCGASMType gasmtype;

  PetscFunctionBegin;
  PetscOptionsHeadBegin(PetscOptionsObject, "Generalized additive Schwarz options");
  PetscCall(PetscOptionsBool("-pc_gasm_use_dm_subdomains", "If subdomains aren't set, use DMCreateDomainDecomposition() to define subdomains.", "PCGASMSetUseDMSubdomains", osm->dm_subdomains, &osm->dm_subdomains, &flg));
  PetscCall(PetscOptionsInt("-pc_gasm_total_subdomains", "Total number of subdomains across communicator", "PCGASMSetTotalSubdomains", osm->N, &blocks, &flg));
  if (flg) PetscCall(PCGASMSetTotalSubdomains(pc, blocks));
  PetscCall(PetscOptionsInt("-pc_gasm_overlap", "Number of overlapping degrees of freedom", "PCGASMSetOverlap", osm->overlap, &ovl, &flg));
  if (flg) {
    PetscCall(PCGASMSetOverlap(pc, ovl));
    osm->dm_subdomains = PETSC_FALSE;
  }
  flg = PETSC_FALSE;
  PetscCall(PetscOptionsEnum("-pc_gasm_type", "Type of restriction/extension", "PCGASMSetType", PCGASMTypes, (PetscEnum)osm->type, (PetscEnum *)&gasmtype, &flg));
  if (flg) PetscCall(PCGASMSetType(pc, gasmtype));
  PetscCall(PetscOptionsBool("-pc_gasm_use_hierachical_partitioning", "use hierarchical partitioning", NULL, osm->hierarchicalpartitioning, &osm->hierarchicalpartitioning, &flg));
  PetscOptionsHeadEnd();
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PCGASMSetTotalSubdomains - sets the total number of subdomains to use across the communicator for `PCGASM`

  Logically Collective

  Input Parameters:
+ pc - the preconditioner
- N  - total number of subdomains

  Level: beginner

.seealso: [](ch_ksp), `PCGASM`, `PCGASMSetSubdomains()`, `PCGASMSetOverlap()`
          `PCGASMCreateSubdomains2D()`
@*/
PetscErrorCode PCGASMSetTotalSubdomains(PC pc, PetscInt N)
{
  PC_GASM    *osm = (PC_GASM *)pc->data;
  PetscMPIInt size, rank;

  PetscFunctionBegin;
  PetscCheck(N >= 1, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Total number of subdomains must be 1 or more, got N = %" PetscInt_FMT, N);
  PetscCheck(!pc->setupcalled, PetscObjectComm((PetscObject)pc), PETSC_ERR_ARG_WRONGSTATE, "PCGASMSetTotalSubdomains() should be called before calling PCSetUp().");

  PetscCall(PCGASMDestroySubdomains(osm->n, &osm->iis, &osm->ois));
  osm->ois = osm->iis = NULL;

  PetscCallMPI(MPI_Comm_size(PetscObjectComm((PetscObject)pc), &size));
  PetscCallMPI(MPI_Comm_rank(PetscObjectComm((PetscObject)pc), &rank));
  osm->N             = N;
  osm->n             = PETSC_DETERMINE;
  osm->nmax          = PETSC_DETERMINE;
  osm->dm_subdomains = PETSC_FALSE;
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode PCGASMSetSubdomains_GASM(PC pc, PetscInt n, IS iis[], IS ois[])
{
  PC_GASM *osm = (PC_GASM *)pc->data;
  PetscInt i;

  PetscFunctionBegin;
  PetscCheck(n >= 1, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Each MPI rank must have 1 or more subdomains, got n = %" PetscInt_FMT, n);
  PetscCheck(!pc->setupcalled, PetscObjectComm((PetscObject)pc), PETSC_ERR_ARG_WRONGSTATE, "PCGASMSetSubdomains() should be called before calling PCSetUp().");

  PetscCall(PCGASMDestroySubdomains(osm->n, &osm->iis, &osm->ois));
  osm->iis = osm->ois = NULL;
  osm->n              = n;
  osm->N              = PETSC_DETERMINE;
  osm->nmax           = PETSC_DETERMINE;
  if (ois) {
    PetscCall(PetscMalloc1(n, &osm->ois));
    for (i = 0; i < n; i++) {
      PetscCall(PetscObjectReference((PetscObject)ois[i]));
      osm->ois[i] = ois[i];
    }
    /*
       Since the user set the outer subdomains, even if nontrivial overlap was requested via PCGASMSetOverlap(),
       it will be ignored.  To avoid confusion later on (e.g., when viewing the PC), the overlap size is set to -1.
    */
    osm->overlap = -1;
    /* inner subdomains must be provided  */
    PetscCheck(iis, PETSC_COMM_SELF, PETSC_ERR_ARG_NULL, "inner indices have to be provided ");
  } /* end if */
  if (iis) {
    PetscCall(PetscMalloc1(n, &osm->iis));
    for (i = 0; i < n; i++) {
      PetscCall(PetscObjectReference((PetscObject)iis[i]));
      osm->iis[i] = iis[i];
    }
    if (!ois) {
      osm->ois = NULL;
      /* if user does not provide outer indices, we will create the corresponding outer indices using  osm->overlap =1 in PCSetUp_GASM */
    }
  }
  if (PetscDefined(USE_DEBUG)) {
    PetscInt        j, rstart, rend, *covered, lsize;
    const PetscInt *indices;
    /* check if the inner indices cover and only cover the local portion of the matrix */
    PetscCall(MatGetOwnershipRange(pc->pmat, &rstart, &rend));
    PetscCall(PetscCalloc1(rend - rstart, &covered));
    /* check if the current MPI process owns indices from others */
    for (i = 0; i < n; i++) {
      PetscCall(ISGetIndices(osm->iis[i], &indices));
      PetscCall(ISGetLocalSize(osm->iis[i], &lsize));
      for (j = 0; j < lsize; j++) {
        PetscCheck(indices[j] >= rstart && indices[j] < rend, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "inner subdomains can not own an index %" PetscInt_FMT " from other ranks", indices[j]);
        PetscCheck(covered[indices[j] - rstart] != 1, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "inner subdomains can not have an overlapping index %" PetscInt_FMT " ", indices[j]);
        covered[indices[j] - rstart] = 1;
      }
      PetscCall(ISRestoreIndices(osm->iis[i], &indices));
    }
    /* check if we miss any indices */
    for (i = rstart; i < rend; i++) PetscCheck(covered[i - rstart], PETSC_COMM_SELF, PETSC_ERR_ARG_NULL, "local entity %" PetscInt_FMT " was not covered by inner subdomains", i);
    PetscCall(PetscFree(covered));
  }
  if (iis) osm->user_subdomains = PETSC_TRUE;
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode PCGASMSetOverlap_GASM(PC pc, PetscInt ovl)
{
  PC_GASM *osm = (PC_GASM *)pc->data;

  PetscFunctionBegin;
  PetscCheck(ovl >= 0, PetscObjectComm((PetscObject)pc), PETSC_ERR_ARG_OUTOFRANGE, "Negative overlap value requested");
  PetscCheck(!pc->setupcalled || ovl == osm->overlap, PetscObjectComm((PetscObject)pc), PETSC_ERR_ARG_WRONGSTATE, "PCGASMSetOverlap() should be called before PCSetUp().");
  if (!pc->setupcalled) osm->overlap = ovl;
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode PCGASMSetType_GASM(PC pc, PCGASMType type)
{
  PC_GASM *osm = (PC_GASM *)pc->data;

  PetscFunctionBegin;
  osm->type     = type;
  osm->type_set = PETSC_TRUE;
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode PCGASMSetSortIndices_GASM(PC pc, PetscBool doSort)
{
  PC_GASM *osm = (PC_GASM *)pc->data;

  PetscFunctionBegin;
  osm->sort_indices = doSort;
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*
   FIXME: This routine might need to be modified now that multiple processes per subdomain are allowed.
        In particular, it would upset the global subdomain number calculation.
*/
static PetscErrorCode PCGASMGetSubKSP_GASM(PC pc, PetscInt *n, PetscInt *first, KSP **ksp)
{
  PC_GASM *osm = (PC_GASM *)pc->data;

  PetscFunctionBegin;
  PetscCheck(osm->n >= 1, PetscObjectComm((PetscObject)pc), PETSC_ERR_ORDER, "Need to call PCSetUp() on PC (or KSPSetUp() on the outer KSP object) before calling here");

  if (n) *n = osm->n;
  if (first) {
    PetscCallMPI(MPI_Scan(&osm->n, first, 1, MPIU_INT, MPI_SUM, PetscObjectComm((PetscObject)pc)));
    *first -= osm->n;
  }
  if (ksp) {
    /* Assume that local solves are now different; not necessarily
       true, though!  This flag is used only for PCView_GASM() */
    *ksp                        = osm->ksp;
    osm->same_subdomain_solvers = PETSC_FALSE;
  }
  PetscFunctionReturn(PETSC_SUCCESS);
} /* PCGASMGetSubKSP_GASM() */

/*@
  PCGASMSetSubdomains - Sets the subdomains for this MPI process
  for the additive Schwarz preconditioner with multiple MPI processes per subdomain, `PCGASM`

  Collective

  Input Parameters:
+ pc  - the preconditioner object
. n   - the number of subdomains for this MPI process
. iis - the index sets that define the inner subdomains (or `NULL` for PETSc to determine subdomains), the `iis` array is
        copied so may be freed after this call.
- ois - the index sets that define the outer subdomains (or `NULL` to use the same as `iis`, or to construct by expanding `iis` by
        the requested overlap), the `ois` array is copied so may be freed after this call.

  Level: advanced

  Notes:
  The `IS` indices use the parallel, global numbering of the vector entries.

  Inner subdomains are those where the correction is applied.

  Outer subdomains are those where the residual necessary to obtain the
  corrections is obtained (see `PCGASMType` for the use of inner/outer subdomains).

  Both inner and outer subdomains can extend over several MPI processes.
  This process' portion of a subdomain is known as a local subdomain.

  Inner subdomains can not overlap with each other, do not have any entities from remote processes,
  and  have to cover the entire local subdomain owned by the current process. The index sets on each
  process should be ordered such that the ith local subdomain is connected to the ith remote subdomain
  on another MPI process.

  By default the `PGASM` preconditioner uses 1 (local) subdomain per MPI process.

  The `iis` and `ois` arrays may be freed after this call using `PCGASMDestroySubdomains()`

.seealso: [](ch_ksp), `PCGASM`, `PCGASMSetOverlap()`, `PCGASMGetSubKSP()`, `PCGASMDestroySubdomains()`,
          `PCGASMCreateSubdomains2D()`, `PCGASMGetSubdomains()`
@*/
PetscErrorCode PCGASMSetSubdomains(PC pc, PetscInt n, IS iis[], IS ois[])
{
  PC_GASM *osm = (PC_GASM *)pc->data;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(pc, PC_CLASSID, 1);
  PetscTryMethod(pc, "PCGASMSetSubdomains_C", (PC, PetscInt, IS[], IS[]), (pc, n, iis, ois));
  osm->dm_subdomains = PETSC_FALSE;
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PCGASMSetOverlap - Sets the overlap between a pair of subdomains for the
  additive Schwarz preconditioner `PCGASM`.  Either all or no MPI processes in the
  pc communicator must call this routine.

  Logically Collective

  Input Parameters:
+ pc  - the preconditioner context
- ovl - the amount of overlap between subdomains (ovl >= 0, default value = 0)

  Options Database Key:
. -pc_gasm_overlap <overlap> - Sets overlap

  Level: intermediate

  Notes:
  By default the `PCGASM` preconditioner uses 1 subdomain per process.  To use
  multiple subdomain per perocessor or "straddling" subdomains that intersect
  multiple processes use `PCGASMSetSubdomains()` (or option `-pc_gasm_total_subdomains` <n>).

  The overlap defaults to 0, so if one desires that no additional
  overlap be computed beyond what may have been set with a call to
  `PCGASMSetSubdomains()`, then `ovl` must be set to be 0.  In particular, if one does
  not explicitly set the subdomains in application code, then all overlap would be computed
  internally by PETSc, and using an overlap of 0 would result in an `PCGASM`
  variant that is equivalent to the block Jacobi preconditioner.

  One can define initial index sets with any overlap via
  `PCGASMSetSubdomains()`; the routine `PCGASMSetOverlap()` merely allows
  PETSc to extend that overlap further, if desired.

.seealso: [](ch_ksp), `PCGASM`, `PCGASMSetSubdomains()`, `PCGASMGetSubKSP()`,
          `PCGASMCreateSubdomains2D()`, `PCGASMGetSubdomains()`
@*/
PetscErrorCode PCGASMSetOverlap(PC pc, PetscInt ovl)
{
  PC_GASM *osm = (PC_GASM *)pc->data;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(pc, PC_CLASSID, 1);
  PetscValidLogicalCollectiveInt(pc, ovl, 2);
  PetscTryMethod(pc, "PCGASMSetOverlap_C", (PC, PetscInt), (pc, ovl));
  osm->dm_subdomains = PETSC_FALSE;
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PCGASMSetType - Sets the type of restriction and interpolation used
  for local problems in the `PCGASM` additive Schwarz method.

  Logically Collective

  Input Parameters:
+ pc   - the preconditioner context
- type - variant of `PCGASM`, one of
.vb
      `PC_GASM_BASIC`       - full interpolation and restriction
      `PC_GASM_RESTRICT`    - full restriction, local MPI process interpolation
      `PC_GASM_INTERPOLATE` - full interpolation, local MPI process restriction
      `PC_GASM_NONE`        - local MPI process restriction and interpolation
.ve

  Options Database Key:
. -pc_gasm_type [basic,restrict,interpolate,none] - Sets `PCGASM` type

  Level: intermediate

.seealso: [](ch_ksp), `PCGASM`, `PCGASMSetSubdomains()`, `PCGASMGetSubKSP()`,
          `PCGASMCreateSubdomains2D()`, `PCASM`, `PCASMSetType()`
@*/
PetscErrorCode PCGASMSetType(PC pc, PCGASMType type)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(pc, PC_CLASSID, 1);
  PetscValidLogicalCollectiveEnum(pc, type, 2);
  PetscTryMethod(pc, "PCGASMSetType_C", (PC, PCGASMType), (pc, type));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PCGASMSetSortIndices - Determines whether subdomain indices are sorted.

  Logically Collective

  Input Parameters:
+ pc     - the preconditioner context
- doSort - sort the subdomain indices

  Level: intermediate

.seealso: [](ch_ksp), `PCGASM`, `PCGASMSetSubdomains()`, `PCGASMGetSubKSP()`,
          `PCGASMCreateSubdomains2D()`
@*/
PetscErrorCode PCGASMSetSortIndices(PC pc, PetscBool doSort)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(pc, PC_CLASSID, 1);
  PetscValidLogicalCollectiveBool(pc, doSort, 2);
  PetscTryMethod(pc, "PCGASMSetSortIndices_C", (PC, PetscBool), (pc, doSort));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@C
  PCGASMGetSubKSP - Gets the local `KSP` contexts for all subdomains on this MPI process.

  Collective iff first_local is requested

  Input Parameter:
. pc - the preconditioner context

  Output Parameters:
+ n_local     - the number of blocks on this MPI process or `NULL`
. first_local - the global number of the first block on this process or `NULL`, all processes must request or all must pass `NULL`
- ksp         - the array of `KSP` contexts

  Level: advanced

  Note:
  After `PCGASMGetSubKSP()` the array of `KSP`es is not to be freed

  Currently for some matrix implementations only 1 block per MPI process
  is supported.

  You must call `KSPSetUp()` before calling `PCGASMGetSubKSP()`.

.seealso: [](ch_ksp), `PCGASM`, `PCGASMSetSubdomains()`, `PCGASMSetOverlap()`,
          `PCGASMCreateSubdomains2D()`,
@*/
PetscErrorCode PCGASMGetSubKSP(PC pc, PetscInt *n_local, PetscInt *first_local, KSP *ksp[])
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(pc, PC_CLASSID, 1);
  PetscUseMethod(pc, "PCGASMGetSubKSP_C", (PC, PetscInt *, PetscInt *, KSP **), (pc, n_local, first_local, ksp));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*MC
   PCGASM - Use the (restricted) additive Schwarz method, each block is (approximately) solved with
           its own `KSP` object on a subset of MPI processes

   Options Database Keys:
+  -pc_gasm_total_subdomains <n>                   - Sets total number of local subdomains to be distributed among the MPI processes
.  -pc_gasm_view_subdomains                        - activates the printing of subdomain indices in `PCView()`, -ksp_view or -snes_view
.  -pc_gasm_print_subdomains                       - activates the printing of subdomain indices in `PCSetUp()`
.  -pc_gasm_overlap <ovl>                          - Sets overlap by which to (automatically) extend local subdomains
-  -pc_gasm_type [basic,restrict,interpolate,none] - Sets `PCGASMType`

   Level: beginner

   Notes:
   To set options on the solvers for each block append `-sub_` to all the `KSP`, and `PC`
   options database keys. For example, `-sub_pc_type ilu -sub_pc_factor_levels 1 -sub_ksp_type preonly`

   To set the options on the solvers separate for each block call `PCGASMGetSubKSP()`
   and set the options directly on the resulting `KSP` object (you can access its `PC`
   with `KSPGetPC()`)

   See {cite}`dryja1987additive` and {cite}`1sbg` for details on additive Schwarz algorithms

.seealso: [](ch_ksp), `PCCreate()`, `PCSetType()`, `PCType`, `PC`, `PCASM`, `PCGASMType`, `PCGASMSetType()`,
          `PCBJACOBI`, `PCGASMGetSubKSP()`, `PCGASMSetSubdomains()`,
          `PCSetModifySubMatrices()`, `PCGASMSetOverlap()`, `PCGASMSetType()`
M*/

PETSC_EXTERN PetscErrorCode PCCreate_GASM(PC pc)
{
  PC_GASM *osm;

  PetscFunctionBegin;
  PetscCall(PetscNew(&osm));

  osm->N                        = PETSC_DETERMINE;
  osm->n                        = PETSC_DECIDE;
  osm->nmax                     = PETSC_DETERMINE;
  osm->overlap                  = 0;
  osm->ksp                      = NULL;
  osm->gorestriction            = NULL;
  osm->girestriction            = NULL;
  osm->pctoouter                = NULL;
  osm->gx                       = NULL;
  osm->gy                       = NULL;
  osm->x                        = NULL;
  osm->y                        = NULL;
  osm->pcx                      = NULL;
  osm->pcy                      = NULL;
  osm->permutationIS            = NULL;
  osm->permutationP             = NULL;
  osm->pcmat                    = NULL;
  osm->ois                      = NULL;
  osm->iis                      = NULL;
  osm->pmat                     = NULL;
  osm->type                     = PC_GASM_RESTRICT;
  osm->same_subdomain_solvers   = PETSC_TRUE;
  osm->sort_indices             = PETSC_TRUE;
  osm->dm_subdomains            = PETSC_FALSE;
  osm->hierarchicalpartitioning = PETSC_FALSE;

  pc->data                 = (void *)osm;
  pc->ops->apply           = PCApply_GASM;
  pc->ops->matapply        = PCMatApply_GASM;
  pc->ops->applytranspose  = PCApplyTranspose_GASM;
  pc->ops->setup           = PCSetUp_GASM;
  pc->ops->reset           = PCReset_GASM;
  pc->ops->destroy         = PCDestroy_GASM;
  pc->ops->setfromoptions  = PCSetFromOptions_GASM;
  pc->ops->setuponblocks   = PCSetUpOnBlocks_GASM;
  pc->ops->view            = PCView_GASM;
  pc->ops->applyrichardson = NULL;

  PetscCall(PetscObjectComposeFunction((PetscObject)pc, "PCGASMSetSubdomains_C", PCGASMSetSubdomains_GASM));
  PetscCall(PetscObjectComposeFunction((PetscObject)pc, "PCGASMSetOverlap_C", PCGASMSetOverlap_GASM));
  PetscCall(PetscObjectComposeFunction((PetscObject)pc, "PCGASMSetType_C", PCGASMSetType_GASM));
  PetscCall(PetscObjectComposeFunction((PetscObject)pc, "PCGASMSetSortIndices_C", PCGASMSetSortIndices_GASM));
  PetscCall(PetscObjectComposeFunction((PetscObject)pc, "PCGASMGetSubKSP_C", PCGASMGetSubKSP_GASM));
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode PCGASMCreateLocalSubdomains(Mat A, PetscInt nloc, IS *iis[])
{
  MatPartitioning mpart;
  const char     *prefix;
  PetscInt        i, j, rstart, rend, bs;
  PetscBool       hasop, isbaij = PETSC_FALSE, foundpart = PETSC_FALSE;
  Mat             Ad = NULL, adj;
  IS              ispart, isnumb, *is;

  PetscFunctionBegin;
  PetscCheck(nloc >= 1, PETSC_COMM_SELF, PETSC_ERR_ARG_WRONG, "number of local subdomains must > 0, got nloc = %" PetscInt_FMT, nloc);

  /* Get prefix, row distribution, and block size */
  PetscCall(MatGetOptionsPrefix(A, &prefix));
  PetscCall(MatGetOwnershipRange(A, &rstart, &rend));
  PetscCall(MatGetBlockSize(A, &bs));
  PetscCheck(rstart / bs * bs == rstart && rend / bs * bs == rend, PETSC_COMM_SELF, PETSC_ERR_ARG_WRONG, "bad row distribution [%" PetscInt_FMT ",%" PetscInt_FMT ") for matrix block size %" PetscInt_FMT, rstart, rend, bs);

  /* Get diagonal block from matrix if possible */
  PetscCall(MatHasOperation(A, MATOP_GET_DIAGONAL_BLOCK, &hasop));
  if (hasop) PetscCall(MatGetDiagonalBlock(A, &Ad));
  if (Ad) {
    PetscCall(PetscObjectBaseTypeCompare((PetscObject)Ad, MATSEQBAIJ, &isbaij));
    if (!isbaij) PetscCall(PetscObjectBaseTypeCompare((PetscObject)Ad, MATSEQSBAIJ, &isbaij));
  }
  if (Ad && nloc > 1) {
    PetscBool match, done;
    /* Try to setup a good matrix partitioning if available */
    PetscCall(MatPartitioningCreate(PETSC_COMM_SELF, &mpart));
    PetscCall(PetscObjectSetOptionsPrefix((PetscObject)mpart, prefix));
    PetscCall(MatPartitioningSetFromOptions(mpart));
    PetscCall(PetscObjectTypeCompare((PetscObject)mpart, MATPARTITIONINGCURRENT, &match));
    if (!match) PetscCall(PetscObjectTypeCompare((PetscObject)mpart, MATPARTITIONINGSQUARE, &match));
    if (!match) { /* assume a "good" partitioner is available */
      PetscInt        na;
      const PetscInt *ia, *ja;
      PetscCall(MatGetRowIJ(Ad, 0, PETSC_TRUE, isbaij, &na, &ia, &ja, &done));
      if (done) {
        /* Build adjacency matrix by hand. Unfortunately a call to
           MatConvert(Ad,MATMPIADJ,MAT_INITIAL_MATRIX,&adj) will
           remove the block-aij structure and we cannot expect
           MatPartitioning to split vertices as we need */
        PetscInt        i, j, len, nnz, cnt, *iia = NULL, *jja = NULL;
        const PetscInt *row;
        nnz = 0;
        for (i = 0; i < na; i++) { /* count number of nonzeros */
          len = ia[i + 1] - ia[i];
          row = ja + ia[i];
          for (j = 0; j < len; j++) {
            if (row[j] == i) { /* don't count diagonal */
              len--;
              break;
            }
          }
          nnz += len;
        }
        PetscCall(PetscMalloc1(na + 1, &iia));
        PetscCall(PetscMalloc1(nnz, &jja));
        nnz    = 0;
        iia[0] = 0;
        for (i = 0; i < na; i++) { /* fill adjacency */
          cnt = 0;
          len = ia[i + 1] - ia[i];
          row = ja + ia[i];
          for (j = 0; j < len; j++) {
            if (row[j] != i) jja[nnz + cnt++] = row[j]; /* if not diagonal */
          }
          nnz += cnt;
          iia[i + 1] = nnz;
        }
        /* Partitioning of the adjacency matrix */
        PetscCall(MatCreateMPIAdj(PETSC_COMM_SELF, na, na, iia, jja, NULL, &adj));
        PetscCall(MatPartitioningSetAdjacency(mpart, adj));
        PetscCall(MatPartitioningSetNParts(mpart, nloc));
        PetscCall(MatPartitioningApply(mpart, &ispart));
        PetscCall(ISPartitioningToNumbering(ispart, &isnumb));
        PetscCall(MatDestroy(&adj));
        foundpart = PETSC_TRUE;
      }
      PetscCall(MatRestoreRowIJ(Ad, 0, PETSC_TRUE, isbaij, &na, &ia, &ja, &done));
    }
    PetscCall(MatPartitioningDestroy(&mpart));
  }
  PetscCall(PetscMalloc1(nloc, &is));
  if (!foundpart) {
    /* Partitioning by contiguous chunks of rows */

    PetscInt mbs   = (rend - rstart) / bs;
    PetscInt start = rstart;
    for (i = 0; i < nloc; i++) {
      PetscInt count = (mbs / nloc + ((mbs % nloc) > i)) * bs;
      PetscCall(ISCreateStride(PETSC_COMM_SELF, count, start, 1, &is[i]));
      start += count;
    }

  } else {
    /* Partitioning by adjacency of diagonal block  */

    const PetscInt *numbering;
    PetscInt       *count, nidx, *indices, *newidx, start = 0;
    /* Get node count in each partition */
    PetscCall(PetscMalloc1(nloc, &count));
    PetscCall(ISPartitioningCount(ispart, nloc, count));
    if (isbaij && bs > 1) { /* adjust for the block-aij case */
      for (i = 0; i < nloc; i++) count[i] *= bs;
    }
    /* Build indices from node numbering */
    PetscCall(ISGetLocalSize(isnumb, &nidx));
    PetscCall(PetscMalloc1(nidx, &indices));
    for (i = 0; i < nidx; i++) indices[i] = i; /* needs to be initialized */
    PetscCall(ISGetIndices(isnumb, &numbering));
    PetscCall(PetscSortIntWithPermutation(nidx, numbering, indices));
    PetscCall(ISRestoreIndices(isnumb, &numbering));
    if (isbaij && bs > 1) { /* adjust for the block-aij case */
      PetscCall(PetscMalloc1(nidx * bs, &newidx));
      for (i = 0; i < nidx; i++) {
        for (j = 0; j < bs; j++) newidx[i * bs + j] = indices[i] * bs + j;
      }
      PetscCall(PetscFree(indices));
      nidx *= bs;
      indices = newidx;
    }
    /* Shift to get global indices */
    for (i = 0; i < nidx; i++) indices[i] += rstart;

    /* Build the index sets for each block */
    for (i = 0; i < nloc; i++) {
      PetscCall(ISCreateGeneral(PETSC_COMM_SELF, count[i], &indices[start], PETSC_COPY_VALUES, &is[i]));
      PetscCall(ISSort(is[i]));
      start += count[i];
    }

    PetscCall(PetscFree(count));
    PetscCall(PetscFree(indices));
    PetscCall(ISDestroy(&isnumb));
    PetscCall(ISDestroy(&ispart));
  }
  *iis = is;
  PetscFunctionReturn(PETSC_SUCCESS);
}

PETSC_INTERN PetscErrorCode PCGASMCreateStraddlingSubdomains(Mat A, PetscInt N, PetscInt *n, IS *iis[])
{
  PetscFunctionBegin;
  PetscCall(MatSubdomainsCreateCoalesce(A, N, n, iis));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@C
  PCGASMCreateSubdomains - Creates `n` index sets defining `n` nonoverlapping subdomains on this MPI process for the `PCGASM` additive
  Schwarz preconditioner for a any problem based on its matrix.

  Collective

  Input Parameters:
+ A - The global matrix operator
- N - the number of global subdomains requested

  Output Parameters:
+ n   - the number of subdomains created on this MPI process
- iis - the array of index sets defining the local inner subdomains (on which the correction is applied)

  Level: advanced

  Notes:
  When `N` >= A's communicator size, each subdomain is local -- contained within a single MPI process.
  When `N` < size, the subdomains are 'straddling' (process boundaries) and are no longer local.
  The resulting subdomains can be use in `PCGASMSetSubdomains`(pc,n,iss,`NULL`).  The overlapping
  outer subdomains will be automatically generated from these according to the requested amount of
  overlap; this is currently supported only with local subdomains.

  Use `PCGASMDestroySubdomains()` to free the array and the list of index sets.

.seealso: [](ch_ksp), `PCGASM`, `PCGASMSetSubdomains()`, `PCGASMDestroySubdomains()`
@*/
PetscErrorCode PCGASMCreateSubdomains(Mat A, PetscInt N, PetscInt *n, IS *iis[])
{
  PetscMPIInt size;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(A, MAT_CLASSID, 1);
  PetscAssertPointer(iis, 4);

  PetscCheck(N >= 1, PETSC_COMM_SELF, PETSC_ERR_ARG_WRONG, "Number of subdomains must be > 0, N = %" PetscInt_FMT, N);
  PetscCallMPI(MPI_Comm_size(PetscObjectComm((PetscObject)A), &size));
  if (N >= size) {
    *n = N / size + (N % size);
    PetscCall(PCGASMCreateLocalSubdomains(A, *n, iis));
  } else {
    PetscCall(PCGASMCreateStraddlingSubdomains(A, N, n, iis));
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@C
  PCGASMDestroySubdomains - Destroys the index sets created with
  `PCGASMCreateSubdomains()` or `PCGASMCreateSubdomains2D()`. Should be
  called after setting subdomains with `PCGASMSetSubdomains()`.

  Collective

  Input Parameters:
+ n   - the number of index sets
. iis - the array of inner subdomains
- ois - the array of outer subdomains, can be `NULL`

  Level: intermediate

  Note:
  This is a convenience subroutine that walks each list,
  destroys each `IS` on the list, and then frees the list. At the end the
  list pointers are set to `NULL`.

.seealso: [](ch_ksp), `PCGASM`, `PCGASMCreateSubdomains()`, `PCGASMSetSubdomains()`
@*/
PetscErrorCode PCGASMDestroySubdomains(PetscInt n, IS *iis[], IS *ois[])
{
  PetscInt i;

  PetscFunctionBegin;
  if (n <= 0) PetscFunctionReturn(PETSC_SUCCESS);
  if (ois) {
    PetscAssertPointer(ois, 3);
    if (*ois) {
      PetscAssertPointer(*ois, 3);
      for (i = 0; i < n; i++) PetscCall(ISDestroy(&(*ois)[i]));
      PetscCall(PetscFree(*ois));
    }
  }
  if (iis) {
    PetscAssertPointer(iis, 2);
    if (*iis) {
      PetscAssertPointer(*iis, 2);
      for (i = 0; i < n; i++) PetscCall(ISDestroy(&(*iis)[i]));
      PetscCall(PetscFree(*iis));
    }
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

#define PCGASMLocalSubdomainBounds2D(M, N, xleft, ylow, xright, yhigh, first, last, xleft_loc, ylow_loc, xright_loc, yhigh_loc, n) \
  do { \
    PetscInt first_row = first / M, last_row = last / M + 1; \
    /*                                                                                                    \
     Compute ylow_loc and yhigh_loc so that (ylow_loc,xleft) and (yhigh_loc,xright) are the corners       \
     of the bounding box of the intersection of the subdomain with the local ownership range (local       \
     subdomain).                                                                                          \
     Also compute xleft_loc and xright_loc as the lower and upper bounds on the first and last rows       \
     of the intersection.                                                                                 \
    */ \
    /* ylow_loc is the grid row containing the first element of the local sumbdomain */ \
    *ylow_loc = PetscMax(first_row, ylow); \
    /* xleft_loc is the offset of first element of the local subdomain within its grid row (might actually be outside the local subdomain) */ \
    *xleft_loc = *ylow_loc == first_row ? PetscMax(first % M, xleft) : xleft; \
    /* yhigh_loc is the grid row above the last local subdomain element */ \
    *yhigh_loc = PetscMin(last_row, yhigh); \
    /* xright is the offset of the end of the  local subdomain within its grid row (might actually be outside the local subdomain) */ \
    *xright_loc = *yhigh_loc == last_row ? PetscMin(xright, last % M) : xright; \
    /* Now compute the size of the local subdomain n. */ \
    *n = 0; \
    if (*ylow_loc < *yhigh_loc) { \
      PetscInt width = xright - xleft; \
      *n += width * (*yhigh_loc - *ylow_loc - 1); \
      *n += PetscMin(PetscMax(*xright_loc - xleft, 0), width); \
      *n -= PetscMin(PetscMax(*xleft_loc - xleft, 0), width); \
    } \
  } while (0)

/*@C
  PCGASMCreateSubdomains2D - Creates the index sets for the `PCGASM` overlapping Schwarz
  preconditioner for a two-dimensional problem on a regular grid.

  Collective

  Input Parameters:
+ pc       - the preconditioner context
. M        - the global number of grid points in the x direction
. N        - the global number of grid points in the y direction
. Mdomains - the global number of subdomains in the x direction
. Ndomains - the global number of subdomains in the y direction
. dof      - degrees of freedom per node
- overlap  - overlap in mesh lines

  Output Parameters:
+ nsub - the number of local subdomains created
. iis  - array of index sets defining inner (nonoverlapping) subdomains
- ois  - array of index sets defining outer (overlapping, if overlap > 0) subdomains

  Level: advanced

  Note:
  Use `PCGASMDestroySubdomains()` to free the index sets and the arrays

.seealso: [](ch_ksp), `PCGASM`, `PCGASMSetSubdomains()`, `PCGASMGetSubKSP()`, `PCGASMSetOverlap()`, `PCASMCreateSubdomains2D()`,
          `PCGASMDestroySubdomains()`
@*/
PetscErrorCode PCGASMCreateSubdomains2D(PC pc, PetscInt M, PetscInt N, PetscInt Mdomains, PetscInt Ndomains, PetscInt dof, PetscInt overlap, PetscInt *nsub, IS *iis[], IS *ois[])
{
  PetscMPIInt size, rank;
  PetscInt    i, j;
  PetscInt    maxheight, maxwidth;
  PetscInt    xstart, xleft, xright, xleft_loc, xright_loc;
  PetscInt    ystart, ylow, yhigh, ylow_loc, yhigh_loc;
  PetscInt    x[2][2], y[2][2], n[2];
  PetscInt    first, last;
  PetscInt    nidx, *idx;
  PetscInt    ii, jj, s, q, d;
  PetscInt    k, kk;
  PetscMPIInt color;
  MPI_Comm    comm, subcomm;
  IS        **xis = NULL, **is = ois, **is_local = iis;

  PetscFunctionBegin;
  PetscCall(PetscObjectGetComm((PetscObject)pc, &comm));
  PetscCallMPI(MPI_Comm_size(comm, &size));
  PetscCallMPI(MPI_Comm_rank(comm, &rank));
  PetscCall(MatGetOwnershipRange(pc->pmat, &first, &last));
  PetscCheck((first % dof) == 0 && (last % dof) == 0, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE,
             "Matrix row partitioning unsuitable for domain decomposition: local row range (%" PetscInt_FMT ",%" PetscInt_FMT ") "
             "does not respect the number of degrees of freedom per grid point %" PetscInt_FMT,
             first, last, dof);

  /* Determine the number of domains with nonzero intersections with the local ownership range. */
  s      = 0;
  ystart = 0;
  for (j = 0; j < Ndomains; ++j) {
    maxheight = N / Ndomains + ((N % Ndomains) > j); /* Maximal height of subdomain */
    PetscCheck(maxheight >= 2, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Too many %" PetscInt_FMT " subdomains in the vertical direction for mesh height %" PetscInt_FMT, Ndomains, N);
    /* Vertical domain limits with an overlap. */
    ylow   = PetscMax(ystart - overlap, 0);
    yhigh  = PetscMin(ystart + maxheight + overlap, N);
    xstart = 0;
    for (i = 0; i < Mdomains; ++i) {
      maxwidth = M / Mdomains + ((M % Mdomains) > i); /* Maximal width of subdomain */
      PetscCheck(maxwidth >= 2, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Too many %" PetscInt_FMT " subdomains in the horizontal direction for mesh width %" PetscInt_FMT, Mdomains, M);
      /* Horizontal domain limits with an overlap. */
      xleft  = PetscMax(xstart - overlap, 0);
      xright = PetscMin(xstart + maxwidth + overlap, M);
      /*
         Determine whether this subdomain intersects this rank's ownership range of pc->pmat.
      */
      PCGASMLocalSubdomainBounds2D(M, N, xleft, ylow, xright, yhigh, first, last, (&xleft_loc), (&ylow_loc), (&xright_loc), (&yhigh_loc), (&nidx));
      if (nidx) ++s;
      xstart += maxwidth;
    } /* for (i = 0; i < Mdomains; ++i) */
    ystart += maxheight;
  } /* for (j = 0; j < Ndomains; ++j) */

  /* Now we can allocate the necessary number of ISs. */
  *nsub = s;
  PetscCall(PetscMalloc1(*nsub, is));
  PetscCall(PetscMalloc1(*nsub, is_local));
  s      = 0;
  ystart = 0;
  for (j = 0; j < Ndomains; ++j) {
    maxheight = N / Ndomains + ((N % Ndomains) > j); /* Maximal height of subdomain */
    PetscCheck(maxheight >= 2, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Too many %" PetscInt_FMT " subdomains in the vertical direction for mesh height %" PetscInt_FMT, Ndomains, N);
    /* Vertical domain limits with an overlap. */
    y[0][0] = PetscMax(ystart - overlap, 0);
    y[0][1] = PetscMin(ystart + maxheight + overlap, N);
    /* Vertical domain limits without an overlap. */
    y[1][0] = ystart;
    y[1][1] = PetscMin(ystart + maxheight, N);
    xstart  = 0;
    for (i = 0; i < Mdomains; ++i) {
      maxwidth = M / Mdomains + ((M % Mdomains) > i); /* Maximal width of subdomain */
      PetscCheck(maxwidth >= 2, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Too many %" PetscInt_FMT " subdomains in the horizontal direction for mesh width %" PetscInt_FMT, Mdomains, M);
      /* Horizontal domain limits with an overlap. */
      x[0][0] = PetscMax(xstart - overlap, 0);
      x[0][1] = PetscMin(xstart + maxwidth + overlap, M);
      /* Horizontal domain limits without an overlap. */
      x[1][0] = xstart;
      x[1][1] = PetscMin(xstart + maxwidth, M);
      /*
         Determine whether this domain intersects this rank's ownership range of pc->pmat.
         Do this twice: first for the domains with overlaps, and once without.
         During the first pass create the subcommunicators, and use them on the second pass as well.
      */
      for (q = 0; q < 2; ++q) {
        PetscBool split = PETSC_FALSE;
        /*
          domain limits, (xleft, xright) and (ylow, yheigh) are adjusted
          according to whether the domain with an overlap or without is considered.
        */
        xleft  = x[q][0];
        xright = x[q][1];
        ylow   = y[q][0];
        yhigh  = y[q][1];
        PCGASMLocalSubdomainBounds2D(M, N, xleft, ylow, xright, yhigh, first, last, (&xleft_loc), (&ylow_loc), (&xright_loc), (&yhigh_loc), (&nidx));
        nidx *= dof;
        n[q] = nidx;
        /*
         Based on the counted number of indices in the local domain *with an overlap*,
         construct a subcommunicator of all the MPI ranks supporting this domain.
         Observe that a domain with an overlap might have nontrivial local support,
         while the domain without an overlap might not.  Hence, the decision to participate
         in the subcommunicator must be based on the domain with an overlap.
         */
        if (q == 0) {
          if (nidx) color = 1;
          else color = MPI_UNDEFINED;
          PetscCallMPI(MPI_Comm_split(comm, color, rank, &subcomm));
          split = PETSC_TRUE;
        }
        /*
         Proceed only if the number of local indices *with an overlap* is nonzero.
         */
        if (n[0]) {
          if (q == 0) xis = is;
          if (q == 1) {
            /*
             The IS for the no-overlap subdomain shares a communicator with the overlapping domain.
             Moreover, if the overlap is zero, the two ISs are identical.
             */
            if (overlap == 0) {
              (*is_local)[s] = (*is)[s];
              PetscCall(PetscObjectReference((PetscObject)(*is)[s]));
              continue;
            } else {
              xis     = is_local;
              subcomm = ((PetscObject)(*is)[s])->comm;
            }
          } /* if (q == 1) */
          idx = NULL;
          PetscCall(PetscMalloc1(nidx, &idx));
          if (nidx) {
            k = 0;
            for (jj = ylow_loc; jj < yhigh_loc; ++jj) {
              PetscInt x0 = (jj == ylow_loc) ? xleft_loc : xleft;
              PetscInt x1 = (jj == yhigh_loc - 1) ? xright_loc : xright;
              kk          = dof * (M * jj + x0);
              for (ii = x0; ii < x1; ++ii) {
                for (d = 0; d < dof; ++d) idx[k++] = kk++;
              }
            }
          }
          PetscCall(ISCreateGeneral(subcomm, nidx, idx, PETSC_OWN_POINTER, (*xis) + s));
          if (split) PetscCallMPI(MPI_Comm_free(&subcomm));
        } /* if (n[0]) */
      } /* for (q = 0; q < 2; ++q) */
      if (n[0]) ++s;
      xstart += maxwidth;
    } /* for (i = 0; i < Mdomains; ++i) */
    ystart += maxheight;
  } /* for (j = 0; j < Ndomains; ++j) */
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@C
  PCGASMGetSubdomains - Gets the subdomains supported on this MPI process
  for the `PCGASM` additive Schwarz preconditioner.

  Not Collective

  Input Parameter:
. pc - the preconditioner context

  Output Parameters:
+ n   - the number of subdomains for this MPI process (default value = 1)
. iis - the index sets that define the inner subdomains (without overlap) supported on this process (can be `NULL`)
- ois - the index sets that define the outer subdomains (with overlap) supported on this process (can be `NULL`)

  Level: advanced

  Notes:
  The user is responsible for destroying the `IS`s and freeing the returned arrays, this can be done with
  `PCGASMDestroySubdomains()`

  The `IS` numbering is in the parallel, global numbering of the vector.

.seealso: [](ch_ksp), `PCGASM`, `PCGASMSetOverlap()`, `PCGASMGetSubKSP()`, `PCGASMCreateSubdomains2D()`,
          `PCGASMSetSubdomains()`, `PCGASMGetSubmatrices()`, `PCGASMDestroySubdomains()`
@*/
PetscErrorCode PCGASMGetSubdomains(PC pc, PetscInt *n, IS *iis[], IS *ois[])
{
  PC_GASM  *osm;
  PetscBool match;
  PetscInt  i;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(pc, PC_CLASSID, 1);
  PetscCall(PetscObjectTypeCompare((PetscObject)pc, PCGASM, &match));
  PetscCheck(match, PetscObjectComm((PetscObject)pc), PETSC_ERR_ARG_WRONG, "Incorrect object type: expected %s, got %s instead", PCGASM, ((PetscObject)pc)->type_name);
  osm = (PC_GASM *)pc->data;
  if (n) *n = osm->n;
  if (iis) PetscCall(PetscMalloc1(osm->n, iis));
  if (ois) PetscCall(PetscMalloc1(osm->n, ois));
  if (iis || ois) {
    for (i = 0; i < osm->n; ++i) {
      if (iis) (*iis)[i] = osm->iis[i];
      if (ois) (*ois)[i] = osm->ois[i];
    }
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@C
  PCGASMGetSubmatrices - Gets the local submatrices (for this MPI process
  only) for the `PCGASM` additive Schwarz preconditioner.

  Not Collective

  Input Parameter:
. pc - the preconditioner context

  Output Parameters:
+ n   - the number of matrices for this MPI process (default value = 1)
- mat - the matrices

  Level: advanced

  Note:
  Matrices returned by this routine have the same communicators as the index sets (`IS`)
  used to define subdomains in `PCGASMSetSubdomains()`

.seealso: [](ch_ksp), `PCGASM`, `PCGASMSetOverlap()`, `PCGASMGetSubKSP()`,
          `PCGASMCreateSubdomains2D()`, `PCGASMSetSubdomains()`, `PCGASMGetSubdomains()`
@*/
PetscErrorCode PCGASMGetSubmatrices(PC pc, PetscInt *n, Mat *mat[])
{
  PC_GASM  *osm;
  PetscBool match;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(pc, PC_CLASSID, 1);
  PetscAssertPointer(n, 2);
  if (mat) PetscAssertPointer(mat, 3);
  PetscCheck(pc->setupcalled, PetscObjectComm((PetscObject)pc), PETSC_ERR_ARG_WRONGSTATE, "Must call after KSPSetUp() or PCSetUp().");
  PetscCall(PetscObjectTypeCompare((PetscObject)pc, PCGASM, &match));
  PetscCheck(match, PetscObjectComm((PetscObject)pc), PETSC_ERR_ARG_WRONG, "Expected %s, got %s instead", PCGASM, ((PetscObject)pc)->type_name);
  osm = (PC_GASM *)pc->data;
  if (n) *n = osm->n;
  if (mat) *mat = osm->pmat;
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PCGASMSetUseDMSubdomains - Indicates whether to use `DMCreateDomainDecomposition()` to define the subdomains, whenever possible for `PCGASM`

  Logically Collective

  Input Parameters:
+ pc  - the preconditioner
- flg - boolean indicating whether to use subdomains defined by the `DM`

  Options Database Key:
+ -pc_gasm_dm_subdomains    - configure subdomains
. -pc_gasm_overlap          - set overlap
- -pc_gasm_total_subdomains - set number of subdomains

  Level: intermediate

  Note:
  `PCGASMSetSubdomains()`, `PCGASMSetTotalSubdomains()` or `PCGASMSetOverlap()` take precedence over `PCGASMSetUseDMSubdomains()`,
  so setting `PCGASMSetSubdomains()` with nontrivial subdomain ISs or any of `PCGASMSetTotalSubdomains()` and `PCGASMSetOverlap()`
  automatically turns the latter off.

.seealso: [](ch_ksp), `PCGASM`, `PCGASMGetUseDMSubdomains()`, `PCGASMSetSubdomains()`, `PCGASMSetOverlap()`
          `PCGASMCreateSubdomains2D()`
@*/
PetscErrorCode PCGASMSetUseDMSubdomains(PC pc, PetscBool flg)
{
  PC_GASM  *osm = (PC_GASM *)pc->data;
  PetscBool match;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(pc, PC_CLASSID, 1);
  PetscValidLogicalCollectiveBool(pc, flg, 2);
  PetscCheck(!pc->setupcalled, ((PetscObject)pc)->comm, PETSC_ERR_ARG_WRONGSTATE, "Not for a setup PC.");
  PetscCall(PetscObjectTypeCompare((PetscObject)pc, PCGASM, &match));
  if (match) {
    if (!osm->user_subdomains && osm->N == PETSC_DETERMINE && osm->overlap < 0) osm->dm_subdomains = flg;
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PCGASMGetUseDMSubdomains - Returns flag indicating whether to use `DMCreateDomainDecomposition()` to define the subdomains, whenever possible with `PCGASM`

  Not Collective

  Input Parameter:
. pc - the preconditioner

  Output Parameter:
. flg - boolean indicating whether to use subdomains defined by the `DM`

  Level: intermediate

.seealso: [](ch_ksp), `PCGASM`, `PCGASMSetUseDMSubdomains()`, `PCGASMSetOverlap()`
          `PCGASMCreateSubdomains2D()`
@*/
PetscErrorCode PCGASMGetUseDMSubdomains(PC pc, PetscBool *flg)
{
  PC_GASM  *osm = (PC_GASM *)pc->data;
  PetscBool match;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(pc, PC_CLASSID, 1);
  PetscAssertPointer(flg, 2);
  PetscCall(PetscObjectTypeCompare((PetscObject)pc, PCGASM, &match));
  if (match) {
    if (flg) *flg = osm->dm_subdomains;
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}