section/interface/section.c

/*
   This file contains routines for basic section object implementation.
*/

#include <petsc/private/sectionimpl.h> /*I  "petscsection.h"   I*/
#include <petscsf.h>

PetscClassId PETSC_SECTION_CLASSID;

/*@
  PetscSectionCreate - Allocates a `PetscSection` and sets the map contents to the default.

  Collective

  Input Parameters:
+ comm - the MPI communicator
- s    - pointer to the section

  Level: beginner

  Notes:
  Typical calling sequence
.vb
       PetscSectionCreate(MPI_Comm,PetscSection *);!
       PetscSectionSetNumFields(PetscSection, numFields);
       PetscSectionSetChart(PetscSection,low,high);
       PetscSectionSetDof(PetscSection,point,numdof);
       PetscSectionSetUp(PetscSection);
       PetscSectionGetOffset(PetscSection,point,PetscInt *);
       PetscSectionDestroy(PetscSection);
.ve

  The `PetscSection` object and methods are intended to be used in the PETSc `Vec` and `Mat` implementations. The indices returned by the `PetscSection` are appropriate for the kind of `Vec` it is associated with. For example, if the vector being indexed is a local vector, we call the section a local section. If the section indexes a global vector, we call it a global section. For parallel vectors, like global vectors, we use negative indices to indicate dofs owned by other processes.

.seealso: [PetscSection](sec_petscsection), `PetscSection`, `PetscSectionSetChart()`, `PetscSectionDestroy()`, `PetscSectionCreateGlobalSection()`
@*/
PetscErrorCode PetscSectionCreate(MPI_Comm comm, PetscSection *s)
{
  PetscFunctionBegin;
  PetscAssertPointer(s, 2);
  PetscCall(ISInitializePackage());

  PetscCall(PetscHeaderCreate(*s, PETSC_SECTION_CLASSID, "PetscSection", "Section", "IS", comm, PetscSectionDestroy, PetscSectionView));

  (*s)->pStart              = -1;
  (*s)->pEnd                = -1;
  (*s)->perm                = NULL;
  (*s)->pointMajor          = PETSC_TRUE;
  (*s)->includesConstraints = PETSC_TRUE;
  (*s)->atlasDof            = NULL;
  (*s)->atlasOff            = NULL;
  (*s)->bc                  = NULL;
  (*s)->bcIndices           = NULL;
  (*s)->setup               = PETSC_FALSE;
  (*s)->numFields           = 0;
  (*s)->fieldNames          = NULL;
  (*s)->field               = NULL;
  (*s)->useFieldOff         = PETSC_FALSE;
  (*s)->compNames           = NULL;
  (*s)->clObj               = NULL;
  (*s)->clHash              = NULL;
  (*s)->clSection           = NULL;
  (*s)->clPoints            = NULL;
  PetscCall(PetscSectionInvalidateMaxDof_Internal(*s));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PetscSectionCopy - Creates a shallow (if possible) copy of the `PetscSection`

  Collective

  Input Parameter:
. section - the `PetscSection`

  Output Parameter:
. newSection - the copy

  Level: intermediate

  Developer Notes:
  What exactly does shallow mean in this context?

.seealso: [PetscSection](sec_petscsection), `PetscSection`, `PetscSectionCreate()`, `PetscSectionDestroy()`
@*/
PetscErrorCode PetscSectionCopy(PetscSection section, PetscSection newSection)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(section, PETSC_SECTION_CLASSID, 1);
  PetscValidHeaderSpecific(newSection, PETSC_SECTION_CLASSID, 2);
  PetscCall(PetscSectionCopy_Internal(section, newSection, NULL));
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode PetscSectionCopy_Internal(PetscSection section, PetscSection newSection, PetscBT constrained_dofs)
{
  PetscSectionSym sym;
  IS              perm;
  PetscInt        numFields, f, c, pStart, pEnd, p;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(section, PETSC_SECTION_CLASSID, 1);
  PetscValidHeaderSpecific(newSection, PETSC_SECTION_CLASSID, 2);
  PetscCall(PetscSectionReset(newSection));
  PetscCall(PetscSectionGetNumFields(section, &numFields));
  if (numFields) PetscCall(PetscSectionSetNumFields(newSection, numFields));
  for (f = 0; f < numFields; ++f) {
    const char *fieldName = NULL, *compName = NULL;
    PetscInt    numComp = 0;

    PetscCall(PetscSectionGetFieldName(section, f, &fieldName));
    PetscCall(PetscSectionSetFieldName(newSection, f, fieldName));
    PetscCall(PetscSectionGetFieldComponents(section, f, &numComp));
    PetscCall(PetscSectionSetFieldComponents(newSection, f, numComp));
    for (c = 0; c < numComp; ++c) {
      PetscCall(PetscSectionGetComponentName(section, f, c, &compName));
      PetscCall(PetscSectionSetComponentName(newSection, f, c, compName));
    }
    PetscCall(PetscSectionGetFieldSym(section, f, &sym));
    PetscCall(PetscSectionSetFieldSym(newSection, f, sym));
  }
  PetscCall(PetscSectionGetChart(section, &pStart, &pEnd));
  PetscCall(PetscSectionSetChart(newSection, pStart, pEnd));
  PetscCall(PetscSectionGetPermutation(section, &perm));
  PetscCall(PetscSectionSetPermutation(newSection, perm));
  PetscCall(PetscSectionGetSym(section, &sym));
  PetscCall(PetscSectionSetSym(newSection, sym));
  for (p = pStart; p < pEnd; ++p) {
    PetscInt  dof, cdof, fcdof = 0;
    PetscBool force_constrained = (PetscBool)(constrained_dofs && PetscBTLookup(constrained_dofs, p - pStart));

    PetscCall(PetscSectionGetDof(section, p, &dof));
    PetscCall(PetscSectionSetDof(newSection, p, dof));
    if (force_constrained) cdof = dof;
    else PetscCall(PetscSectionGetConstraintDof(section, p, &cdof));
    if (cdof) PetscCall(PetscSectionSetConstraintDof(newSection, p, cdof));
    for (f = 0; f < numFields; ++f) {
      PetscCall(PetscSectionGetFieldDof(section, p, f, &dof));
      PetscCall(PetscSectionSetFieldDof(newSection, p, f, dof));
      if (cdof) {
        if (force_constrained) fcdof = dof;
        else PetscCall(PetscSectionGetFieldConstraintDof(section, p, f, &fcdof));
        if (fcdof) PetscCall(PetscSectionSetFieldConstraintDof(newSection, p, f, fcdof));
      }
    }
  }
  PetscCall(PetscSectionSetUp(newSection));
  for (p = pStart; p < pEnd; ++p) {
    PetscInt        off, cdof, fcdof = 0;
    const PetscInt *cInd;
    PetscBool       force_constrained = (PetscBool)(constrained_dofs && PetscBTLookup(constrained_dofs, p - pStart));

    /* Must set offsets in case they do not agree with the prefix sums */
    PetscCall(PetscSectionGetOffset(section, p, &off));
    PetscCall(PetscSectionSetOffset(newSection, p, off));
    PetscCall(PetscSectionGetConstraintDof(newSection, p, &cdof));
    if (cdof) {
      if (force_constrained) cInd = NULL;
      else PetscCall(PetscSectionGetConstraintIndices(section, p, &cInd));
      PetscCall(PetscSectionSetConstraintIndices(newSection, p, cInd));
      for (f = 0; f < numFields; ++f) {
        PetscCall(PetscSectionGetFieldOffset(section, p, f, &off));
        PetscCall(PetscSectionSetFieldOffset(newSection, p, f, off));
        PetscCall(PetscSectionGetFieldConstraintDof(newSection, p, f, &fcdof));
        if (fcdof) {
          if (force_constrained) cInd = NULL;
          else PetscCall(PetscSectionGetFieldConstraintIndices(section, p, f, &cInd));
          PetscCall(PetscSectionSetFieldConstraintIndices(newSection, p, f, cInd));
        }
      }
    }
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PetscSectionClone - Creates a shallow (if possible) copy of the `PetscSection`

  Collective

  Input Parameter:
. section - the `PetscSection`

  Output Parameter:
. newSection - the copy

  Level: beginner

  Developer Notes:
  With standard PETSc terminology this should be called `PetscSectionDuplicate()`

.seealso: [PetscSection](sec_petscsection), `PetscSection`, `PetscSectionCreate()`, `PetscSectionDestroy()`, `PetscSectionCopy()`
@*/
PetscErrorCode PetscSectionClone(PetscSection section, PetscSection *newSection)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(section, PETSC_SECTION_CLASSID, 1);
  PetscAssertPointer(newSection, 2);
  PetscCall(PetscSectionCreate(PetscObjectComm((PetscObject)section), newSection));
  PetscCall(PetscSectionCopy(section, *newSection));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PetscSectionSetFromOptions - sets parameters in a `PetscSection` from the options database

  Collective

  Input Parameter:
. s - the `PetscSection`

  Options Database Key:
. -petscsection_point_major - `PETSC_TRUE` for point-major order

  Level: intermediate

.seealso: [PetscSection](sec_petscsection), `PetscSection`, `PetscSectionCreate()`, `PetscSectionDestroy()`
@*/
PetscErrorCode PetscSectionSetFromOptions(PetscSection s)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(s, PETSC_SECTION_CLASSID, 1);
  PetscObjectOptionsBegin((PetscObject)s);
  PetscCall(PetscOptionsBool("-petscsection_point_major", "The for ordering, either point major or field major", "PetscSectionSetPointMajor", s->pointMajor, &s->pointMajor, NULL));
  /* process any options handlers added with PetscObjectAddOptionsHandler() */
  PetscCall(PetscObjectProcessOptionsHandlers((PetscObject)s, PetscOptionsObject));
  PetscOptionsEnd();
  PetscCall(PetscObjectViewFromOptions((PetscObject)s, NULL, "-petscsection_view"));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PetscSectionCompare - Compares two sections

  Collective

  Input Parameters:
+ s1 - the first `PetscSection`
- s2 - the second `PetscSection`

  Output Parameter:
. congruent - `PETSC_TRUE` if the two sections are congruent, `PETSC_FALSE` otherwise

  Level: intermediate

  Note:
  Field names are disregarded.

.seealso: [PetscSection](sec_petscsection), `PetscSection`, `PetscSectionCreate()`, `PetscSectionCopy()`, `PetscSectionClone()`
@*/
PetscErrorCode PetscSectionCompare(PetscSection s1, PetscSection s2, PetscBool *congruent)
{
  PetscInt        pStart, pEnd, nfields, ncdof, nfcdof, p, f, n1, n2;
  const PetscInt *idx1, *idx2;
  IS              perm1, perm2;
  PetscBool       flg;
  PetscMPIInt     mflg;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(s1, PETSC_SECTION_CLASSID, 1);
  PetscValidHeaderSpecific(s2, PETSC_SECTION_CLASSID, 2);
  PetscAssertPointer(congruent, 3);
  flg = PETSC_FALSE;

  PetscCallMPI(MPI_Comm_compare(PetscObjectComm((PetscObject)s1), PetscObjectComm((PetscObject)s2), &mflg));
  if (mflg != MPI_CONGRUENT && mflg != MPI_IDENT) {
    *congruent = PETSC_FALSE;
    PetscFunctionReturn(PETSC_SUCCESS);
  }

  PetscCall(PetscSectionGetChart(s1, &pStart, &pEnd));
  PetscCall(PetscSectionGetChart(s2, &n1, &n2));
  if (pStart != n1 || pEnd != n2) goto not_congruent;

  PetscCall(PetscSectionGetPermutation(s1, &perm1));
  PetscCall(PetscSectionGetPermutation(s2, &perm2));
  if (perm1 && perm2) {
    PetscCall(ISEqual(perm1, perm2, congruent));
    if (!(*congruent)) goto not_congruent;
  } else if (perm1 != perm2) goto not_congruent;

  for (p = pStart; p < pEnd; ++p) {
    PetscCall(PetscSectionGetOffset(s1, p, &n1));
    PetscCall(PetscSectionGetOffset(s2, p, &n2));
    if (n1 != n2) goto not_congruent;

    PetscCall(PetscSectionGetDof(s1, p, &n1));
    PetscCall(PetscSectionGetDof(s2, p, &n2));
    if (n1 != n2) goto not_congruent;

    PetscCall(PetscSectionGetConstraintDof(s1, p, &ncdof));
    PetscCall(PetscSectionGetConstraintDof(s2, p, &n2));
    if (ncdof != n2) goto not_congruent;

    PetscCall(PetscSectionGetConstraintIndices(s1, p, &idx1));
    PetscCall(PetscSectionGetConstraintIndices(s2, p, &idx2));
    PetscCall(PetscArraycmp(idx1, idx2, ncdof, congruent));
    if (!(*congruent)) goto not_congruent;
  }

  PetscCall(PetscSectionGetNumFields(s1, &nfields));
  PetscCall(PetscSectionGetNumFields(s2, &n2));
  if (nfields != n2) goto not_congruent;

  for (f = 0; f < nfields; ++f) {
    PetscCall(PetscSectionGetFieldComponents(s1, f, &n1));
    PetscCall(PetscSectionGetFieldComponents(s2, f, &n2));
    if (n1 != n2) goto not_congruent;

    for (p = pStart; p < pEnd; ++p) {
      PetscCall(PetscSectionGetFieldOffset(s1, p, f, &n1));
      PetscCall(PetscSectionGetFieldOffset(s2, p, f, &n2));
      if (n1 != n2) goto not_congruent;

      PetscCall(PetscSectionGetFieldDof(s1, p, f, &n1));
      PetscCall(PetscSectionGetFieldDof(s2, p, f, &n2));
      if (n1 != n2) goto not_congruent;

      PetscCall(PetscSectionGetFieldConstraintDof(s1, p, f, &nfcdof));
      PetscCall(PetscSectionGetFieldConstraintDof(s2, p, f, &n2));
      if (nfcdof != n2) goto not_congruent;

      PetscCall(PetscSectionGetFieldConstraintIndices(s1, p, f, &idx1));
      PetscCall(PetscSectionGetFieldConstraintIndices(s2, p, f, &idx2));
      PetscCall(PetscArraycmp(idx1, idx2, nfcdof, congruent));
      if (!(*congruent)) goto not_congruent;
    }
  }

  flg = PETSC_TRUE;
not_congruent:
  PetscCall(MPIU_Allreduce(&flg, congruent, 1, MPIU_BOOL, MPI_LAND, PetscObjectComm((PetscObject)s1)));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PetscSectionGetNumFields - Returns the number of fields in a `PetscSection`, or 0 if no fields were defined.

  Not Collective

  Input Parameter:
. s - the `PetscSection`

  Output Parameter:
. numFields - the number of fields defined, or 0 if none were defined

  Level: intermediate

.seealso: [PetscSection](sec_petscsection), `PetscSection`, `PetscSectionSetNumFields()`
@*/
PetscErrorCode PetscSectionGetNumFields(PetscSection s, PetscInt *numFields)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(s, PETSC_SECTION_CLASSID, 1);
  PetscAssertPointer(numFields, 2);
  *numFields = s->numFields;
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PetscSectionSetNumFields - Sets the number of fields in a `PetscSection`

  Not Collective

  Input Parameters:
+ s         - the `PetscSection`
- numFields - the number of fields

  Level: intermediate

  Notes:
  Calling this destroys all the information in the `PetscSection` including the chart.

  You must call `PetscSectionSetChart()` after calling this.

.seealso: [PetscSection](sec_petscsection), `PetscSection`, `PetscSectionGetNumFields()`, `PetscSectionSetChart()`, `PetscSectionReset()`
@*/
PetscErrorCode PetscSectionSetNumFields(PetscSection s, PetscInt numFields)
{
  PetscInt f;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(s, PETSC_SECTION_CLASSID, 1);
  PetscCheck(numFields > 0, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "The number of fields %" PetscInt_FMT " must be positive", numFields);
  PetscCall(PetscSectionReset(s));

  s->numFields = numFields;
  PetscCall(PetscMalloc1(s->numFields, &s->numFieldComponents));
  PetscCall(PetscMalloc1(s->numFields, &s->fieldNames));
  PetscCall(PetscMalloc1(s->numFields, &s->compNames));
  PetscCall(PetscMalloc1(s->numFields, &s->field));
  for (f = 0; f < s->numFields; ++f) {
    char name[64];

    s->numFieldComponents[f] = 1;

    PetscCall(PetscSectionCreate(PetscObjectComm((PetscObject)s), &s->field[f]));
    PetscCall(PetscSNPrintf(name, 64, "Field_%" PetscInt_FMT, f));
    PetscCall(PetscStrallocpy(name, (char **)&s->fieldNames[f]));
    PetscCall(PetscSNPrintf(name, 64, "Component_0"));
    PetscCall(PetscMalloc1(s->numFieldComponents[f], &s->compNames[f]));
    PetscCall(PetscStrallocpy(name, (char **)&s->compNames[f][0]));
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@C
  PetscSectionGetFieldName - Returns the name of a field in the `PetscSection`

  Not Collective

  Input Parameters:
+ s     - the `PetscSection`
- field - the field number

  Output Parameter:
. fieldName - the field name

  Level: intermediate

  Note:
  Will error if the field number is out of range

.seealso: [PetscSection](sec_petscsection), `PetscSection`, `PetscSectionSetFieldName()`, `PetscSectionSetNumFields()`, `PetscSectionGetNumFields()`
@*/
PetscErrorCode PetscSectionGetFieldName(PetscSection s, PetscInt field, const char *fieldName[])
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(s, PETSC_SECTION_CLASSID, 1);
  PetscAssertPointer(fieldName, 3);
  PetscSectionCheckValidField(field, s->numFields);
  *fieldName = s->fieldNames[field];
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@C
  PetscSectionSetFieldName - Sets the name of a field in the `PetscSection`

  Not Collective

  Input Parameters:
+ s         - the `PetscSection`
. field     - the field number
- fieldName - the field name

  Level: intermediate

  Note:
  Will error if the field number is out of range

.seealso: [PetscSection](sec_petscsection), `PetscSectionGetFieldName()`, `PetscSectionSetNumFields()`, `PetscSectionGetNumFields()`
@*/
PetscErrorCode PetscSectionSetFieldName(PetscSection s, PetscInt field, const char fieldName[])
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(s, PETSC_SECTION_CLASSID, 1);
  if (fieldName) PetscAssertPointer(fieldName, 3);
  PetscSectionCheckValidField(field, s->numFields);
  PetscCall(PetscFree(s->fieldNames[field]));
  PetscCall(PetscStrallocpy(fieldName, (char **)&s->fieldNames[field]));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@C
  PetscSectionGetComponentName - Gets the name of a field component in the `PetscSection`

  Not Collective

  Input Parameters:
+ s     - the `PetscSection`
. field - the field number
- comp  - the component number

  Output Parameter:
. compName - the component name

  Level: intermediate

  Note:
  Will error if the field or component number do not exist

  Developer Notes:
  The function name should have Field in it since they are field components.

.seealso: [PetscSection](sec_petscsection), `PetscSection`, `PetscSectionGetFieldName()`, `PetscSectionSetNumFields()`, `PetscSectionGetNumFields()`,
          `PetscSectionSetComponentName()`, `PetscSectionSetFieldName()`, `PetscSectionGetFieldComponents()`, `PetscSectionSetFieldComponents()`
@*/
PetscErrorCode PetscSectionGetComponentName(PetscSection s, PetscInt field, PetscInt comp, const char *compName[])
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(s, PETSC_SECTION_CLASSID, 1);
  PetscAssertPointer(compName, 4);
  PetscSectionCheckValidField(field, s->numFields);
  PetscSectionCheckValidFieldComponent(comp, s->numFieldComponents[field]);
  *compName = s->compNames[field][comp];
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@C
  PetscSectionSetComponentName - Sets the name of a field component in the `PetscSection`

  Not Collective

  Input Parameters:
+ s        - the `PetscSection`
. field    - the field number
. comp     - the component number
- compName - the component name

  Level: advanced

  Note:
  Will error if the field or component number do not exist

  Developer Notes:
  The function name should have Field in it since they are field components.

.seealso: [PetscSection](sec_petscsection), `PetscSection`, `PetscSectionGetComponentName()`, `PetscSectionSetNumFields()`, `PetscSectionGetNumFields()`,
          `PetscSectionSetFieldName()`, `PetscSectionGetFieldComponents()`, `PetscSectionSetFieldComponents()`
@*/
PetscErrorCode PetscSectionSetComponentName(PetscSection s, PetscInt field, PetscInt comp, const char compName[])
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(s, PETSC_SECTION_CLASSID, 1);
  if (compName) PetscAssertPointer(compName, 4);
  PetscSectionCheckValidField(field, s->numFields);
  PetscSectionCheckValidFieldComponent(comp, s->numFieldComponents[field]);
  PetscCall(PetscFree(s->compNames[field][comp]));
  PetscCall(PetscStrallocpy(compName, (char **)&s->compNames[field][comp]));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PetscSectionGetFieldComponents - Returns the number of field components for the given field.

  Not Collective

  Input Parameters:
+ s     - the `PetscSection`
- field - the field number

  Output Parameter:
. numComp - the number of field components

  Level: advanced

  Developer Notes:
  This function is misnamed. There is a Num in `PetscSectionGetNumFields()` but not in this name

.seealso: [PetscSection](sec_petscsection), `PetscSection`, `PetscSectionSetFieldComponents()`, `PetscSectionGetNumFields()`,
          `PetscSectionSetComponentName()`, `PetscSectionGetComponentName()`
@*/
PetscErrorCode PetscSectionGetFieldComponents(PetscSection s, PetscInt field, PetscInt *numComp)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(s, PETSC_SECTION_CLASSID, 1);
  PetscAssertPointer(numComp, 3);
  PetscSectionCheckValidField(field, s->numFields);
  *numComp = s->numFieldComponents[field];
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PetscSectionSetFieldComponents - Sets the number of field components for the given field.

  Not Collective

  Input Parameters:
+ s       - the `PetscSection`
. field   - the field number
- numComp - the number of field components

  Level: advanced

  Note:
  This number can be different than the values set with `PetscSectionSetFieldDof()`. It can be used to indicate the number of
  components in the field of the underlying physical model which may be different than the number of degrees of freedom needed
  at a point in a discretization. For example, if in three dimensions the field is velocity, it will have 3 components, u, v, and w but
  an face based model for velocity (where the velocity normal to the face is stored) there is only 1 dof for each face point.

  The value set with this function are not needed or used in `PetscSectionSetUp()`.

  Developer Notes:
  This function is misnamed. There is a Num in `PetscSectionSetNumFields()` but not in this name

.seealso: [PetscSection](sec_petscsection), `PetscSection`, `PetscSectionGetFieldComponents()`, `PetscSectionSetComponentName()`,
          `PetscSectionGetComponentName()`, `PetscSectionGetNumFields()`
@*/
PetscErrorCode PetscSectionSetFieldComponents(PetscSection s, PetscInt field, PetscInt numComp)
{
  PetscInt c;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(s, PETSC_SECTION_CLASSID, 1);
  PetscSectionCheckValidField(field, s->numFields);
  if (s->compNames) {
    for (c = 0; c < s->numFieldComponents[field]; ++c) PetscCall(PetscFree(s->compNames[field][c]));
    PetscCall(PetscFree(s->compNames[field]));
  }

  s->numFieldComponents[field] = numComp;
  if (numComp) {
    PetscCall(PetscMalloc1(numComp, (char ***)&s->compNames[field]));
    for (c = 0; c < numComp; ++c) {
      char name[64];

      PetscCall(PetscSNPrintf(name, 64, "%" PetscInt_FMT, c));
      PetscCall(PetscStrallocpy(name, (char **)&s->compNames[field][c]));
    }
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PetscSectionGetChart - Returns the range [`pStart`, `pEnd`) in which points (indices) lie for this `PetscSection` on this MPI process

  Not Collective

  Input Parameter:
. s - the `PetscSection`

  Output Parameters:
+ pStart - the first point
- pEnd   - one past the last point

  Level: intermediate

.seealso: [PetscSection](sec_petscsection), `PetscSection`, `PetscSectionSetChart()`, `PetscSectionCreate()`
@*/
PetscErrorCode PetscSectionGetChart(PetscSection s, PetscInt *pStart, PetscInt *pEnd)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(s, PETSC_SECTION_CLASSID, 1);
  if (pStart) *pStart = s->pStart;
  if (pEnd) *pEnd = s->pEnd;
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PetscSectionSetChart - Sets the range [`pStart`, `pEnd`) in which points (indices) lie for this `PetscSection` on this MPI process

  Not Collective

  Input Parameters:
+ s      - the `PetscSection`
. pStart - the first point
- pEnd   - one past the last point, `pStart` $ \le $ `pEnd`

  Level: intermediate

  Notes:
  The charts on different MPI processes may (and often do) overlap

  If you intend to use `PetscSectionSetNumFields()` it must be called before this call.

  The chart for all fields created with `PetscSectionSetNumFields()` is the same as this chart.

.seealso: [PetscSection](sec_petscsection), `PetscSection`, `PetscSectionGetChart()`, `PetscSectionCreate()`, `PetscSectionSetNumFields()`
@*/
PetscErrorCode PetscSectionSetChart(PetscSection s, PetscInt pStart, PetscInt pEnd)
{
  PetscInt f;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(s, PETSC_SECTION_CLASSID, 1);
  PetscCheck(pEnd >= pStart, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Chart pEnd %" PetscInt_FMT " cannot be smaller than chart pStart %" PetscInt_FMT, pEnd, pStart);
  if (pStart == s->pStart && pEnd == s->pEnd) PetscFunctionReturn(PETSC_SUCCESS);
  /* Cannot Reset() because it destroys field information */
  s->setup = PETSC_FALSE;
  PetscCall(PetscSectionDestroy(&s->bc));
  PetscCall(PetscFree(s->bcIndices));
  PetscCall(PetscFree2(s->atlasDof, s->atlasOff));

  s->pStart = pStart;
  s->pEnd   = pEnd;
  PetscCall(PetscMalloc2((pEnd - pStart), &s->atlasDof, (pEnd - pStart), &s->atlasOff));
  PetscCall(PetscArrayzero(s->atlasDof, pEnd - pStart));
  for (f = 0; f < s->numFields; ++f) PetscCall(PetscSectionSetChart(s->field[f], pStart, pEnd));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PetscSectionGetPermutation - Returns the permutation of [0, `pEnd` - `pStart`) or `NULL` that was set with `PetscSectionSetPermutation()`

  Not Collective

  Input Parameter:
. s - the `PetscSection`

  Output Parameter:
. perm - The permutation as an `IS`

  Level: intermediate

.seealso: [](sec_scatter), `IS`, `PetscSection`, `PetscSectionSetPermutation()`, `PetscSectionCreate()`
@*/
PetscErrorCode PetscSectionGetPermutation(PetscSection s, IS *perm)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(s, PETSC_SECTION_CLASSID, 1);
  if (perm) {
    PetscAssertPointer(perm, 2);
    *perm = s->perm;
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PetscSectionSetPermutation - Sets a permutation of the chart for this section, [0, `pEnd` - `pStart`), which determines the order to store the `PetscSection` information

  Not Collective

  Input Parameters:
+ s    - the `PetscSection`
- perm - the permutation of points

  Level: intermediate

  Notes:
  The permutation must be provided before `PetscSectionSetUp()`.

  The data in the `PetscSection` are permuted but the access via `PetscSectionGetFieldOffset()` and `PetscSectionGetOffset()` is not changed

  Compare to `PetscSectionPermute()`

.seealso: [](sec_scatter), `IS`, `PetscSection`, `PetscSectionSetUp()`, `PetscSectionGetPermutation()`, `PetscSectionPermute()`, `PetscSectionCreate()`
@*/
PetscErrorCode PetscSectionSetPermutation(PetscSection s, IS perm)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(s, PETSC_SECTION_CLASSID, 1);
  if (perm) PetscValidHeaderSpecific(perm, IS_CLASSID, 2);
  PetscCheck(!s->setup, PetscObjectComm((PetscObject)s), PETSC_ERR_ARG_WRONGSTATE, "Cannot set a permutation after the section is setup");
  if (s->perm != perm) {
    PetscCall(ISDestroy(&s->perm));
    if (perm) {
      s->perm = perm;
      PetscCall(PetscObjectReference((PetscObject)s->perm));
    }
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@C
  PetscSectionGetBlockStarts - Returns a table indicating which points start new blocks

  Not Collective

  Input Parameter:
. s - the `PetscSection`

  Output Parameter:
. blockStarts - The `PetscBT` with a 1 for each point that begins a block

  Notes:
  The table is on [0, `pEnd` - `pStart`).

  This information is used by `DMCreateMatrix()` to create a variable block size description which is set using `MatSetVariableBlockSizes()`.

  Level: intermediate

.seealso: [](sec_scatter), `IS`, `PetscSection`, `PetscSectionSetBlockStarts()`, `PetscSectionCreate()`, `DMCreateMatrix()`, `MatSetVariableBlockSizes()`
@*/
PetscErrorCode PetscSectionGetBlockStarts(PetscSection s, PetscBT *blockStarts)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(s, PETSC_SECTION_CLASSID, 1);
  if (blockStarts) {
    PetscAssertPointer(blockStarts, 2);
    *blockStarts = s->blockStarts;
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@C
  PetscSectionSetBlockStarts - Sets a table indicating which points start new blocks

  Not Collective

  Input Parameters:
+ s           - the `PetscSection`
- blockStarts - The `PetscBT` with a 1 for each point that begins a block

  Level: intermediate

  Notes:
  The table is on [0, `pEnd` - `pStart`). PETSc takes ownership of the `PetscBT` when it is passed in and will destroy it. The user should not destroy it.

  This information is used by `DMCreateMatrix()` to create a variable block size description which is set using `MatSetVariableBlockSizes()`.

.seealso: [](sec_scatter), `IS`, `PetscSection`, `PetscSectionGetBlockStarts()`, `PetscSectionCreate()`, `DMCreateMatrix()`, `MatSetVariableBlockSizes()`
@*/
PetscErrorCode PetscSectionSetBlockStarts(PetscSection s, PetscBT blockStarts)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(s, PETSC_SECTION_CLASSID, 1);
  if (s->blockStarts != blockStarts) {
    PetscCall(PetscBTDestroy(&s->blockStarts));
    s->blockStarts = blockStarts;
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PetscSectionGetPointMajor - Returns the flag for dof ordering, `PETSC_TRUE` if it is point major, `PETSC_FALSE` if it is field major

  Not Collective

  Input Parameter:
. s - the `PetscSection`

  Output Parameter:
. pm - the flag for point major ordering

  Level: intermediate

.seealso: [PetscSection](sec_petscsection), `PetscSection`, `PetscSectionSetPointMajor()`
@*/
PetscErrorCode PetscSectionGetPointMajor(PetscSection s, PetscBool *pm)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(s, PETSC_SECTION_CLASSID, 1);
  PetscAssertPointer(pm, 2);
  *pm = s->pointMajor;
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PetscSectionSetPointMajor - Sets the flag for dof ordering, `PETSC_TRUE` for point major, otherwise it will be field major

  Not Collective

  Input Parameters:
+ s  - the `PetscSection`
- pm - the flag for point major ordering

  Level: intermediate

  Note:
  Field-major order is not recommended unless you are managing the entire problem yourself, since many higher-level functions in PETSc depend on point-major order.

  Point major order means the degrees of freedom are stored as follows
.vb
    all the degrees of freedom for each point are stored contiguously, one point after another (respecting a permutation set with `PetscSectionSetPermutation()`)
    for each point
       the degrees of freedom for each field (starting with the unnamed default field) are listed in order by field
.ve

  Field major order means the degrees of freedom are stored as follows
.vb
    all degrees of freedom for each field (including the unnamed default field) are stored contiguously, one field after another
    for each field (started with unnamed default field)
      the degrees of freedom for each point are listed in order by point (respecting a permutation set with `PetscSectionSetPermutation()`)
.ve

.seealso: [PetscSection](sec_petscsection), `PetscSection`, `PetscSectionGetPointMajor()`, `PetscSectionSetPermutation()`
@*/
PetscErrorCode PetscSectionSetPointMajor(PetscSection s, PetscBool pm)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(s, PETSC_SECTION_CLASSID, 1);
  PetscCheck(!s->setup, PetscObjectComm((PetscObject)s), PETSC_ERR_ARG_WRONGSTATE, "Cannot set the dof ordering after the section is setup");
  s->pointMajor = pm;
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PetscSectionGetIncludesConstraints - Returns the flag indicating if constrained dofs were included when computing offsets in the `PetscSection`.
  The value is set with `PetscSectionSetIncludesConstraints()`

  Not Collective

  Input Parameter:
. s - the `PetscSection`

  Output Parameter:
. includesConstraints - the flag indicating if constrained dofs were included when computing offsets

  Level: intermediate

.seealso: [PetscSection](sec_petscsection), `PetscSection`, `PetscSectionSetIncludesConstraints()`
@*/
PetscErrorCode PetscSectionGetIncludesConstraints(PetscSection s, PetscBool *includesConstraints)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(s, PETSC_SECTION_CLASSID, 1);
  PetscAssertPointer(includesConstraints, 2);
  *includesConstraints = s->includesConstraints;
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PetscSectionSetIncludesConstraints - Sets the flag indicating if constrained dofs are to be included when computing offsets

  Not Collective

  Input Parameters:
+ s                   - the `PetscSection`
- includesConstraints - the flag indicating if constrained dofs are to be included when computing offsets

  Level: intermediate

.seealso: [PetscSection](sec_petscsection), `PetscSection`, `PetscSectionGetIncludesConstraints()`
@*/
PetscErrorCode PetscSectionSetIncludesConstraints(PetscSection s, PetscBool includesConstraints)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(s, PETSC_SECTION_CLASSID, 1);
  PetscCheck(!s->setup, PetscObjectComm((PetscObject)s), PETSC_ERR_ARG_WRONGSTATE, "Cannot set includesConstraints after the section is set up");
  s->includesConstraints = includesConstraints;
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PetscSectionGetDof - Return the total number of degrees of freedom associated with a given point.

  Not Collective

  Input Parameters:
+ s     - the `PetscSection`
- point - the point

  Output Parameter:
. numDof - the number of dof

  Level: intermediate

  Notes:
  In a global section, this size will be negative for points not owned by this process.

  This number is for the unnamed default field at the given point plus all degrees of freedom associated with all fields at that point

.seealso: [PetscSection](sec_petscsection), `PetscSection`, `PetscSectionSetDof()`, `PetscSectionCreate()`
@*/
PetscErrorCode PetscSectionGetDof(PetscSection s, PetscInt point, PetscInt *numDof)
{
  PetscFunctionBeginHot;
  PetscValidHeaderSpecific(s, PETSC_SECTION_CLASSID, 1);
  PetscAssertPointer(numDof, 3);
  PetscAssert(point >= s->pStart && point < s->pEnd, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Section point %" PetscInt_FMT " should be in [%" PetscInt_FMT ", %" PetscInt_FMT ")", point, s->pStart, s->pEnd);
  *numDof = s->atlasDof[point - s->pStart];
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PetscSectionSetDof - Sets the total number of degrees of freedom associated with a given point.

  Not Collective

  Input Parameters:
+ s      - the `PetscSection`
. point  - the point
- numDof - the number of dof, these values may be negative -(dof+1) to indicate they are off process

  Level: intermediate

  Note:
  This number is for the unnamed default field at the given point plus all degrees of freedom associated with all fields at that point

.seealso: [PetscSection](sec_petscsection), `PetscSection`, `PetscSectionGetDof()`, `PetscSectionAddDof()`, `PetscSectionCreate()`
@*/
PetscErrorCode PetscSectionSetDof(PetscSection s, PetscInt point, PetscInt numDof)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(s, PETSC_SECTION_CLASSID, 1);
  PetscAssert(point >= s->pStart && point < s->pEnd, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Section point %" PetscInt_FMT " should be in [%" PetscInt_FMT ", %" PetscInt_FMT ")", point, s->pStart, s->pEnd);
  s->atlasDof[point - s->pStart] = numDof;
  PetscCall(PetscSectionInvalidateMaxDof_Internal(s));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PetscSectionAddDof - Adds to the total number of degrees of freedom associated with a given point.

  Not Collective

  Input Parameters:
+ s      - the `PetscSection`
. point  - the point
- numDof - the number of additional dof

  Level: intermediate

  Note:
  This number is for the unnamed default field at the given point plus all degrees of freedom associated with all fields at that point

.seealso: [PetscSection](sec_petscsection), `PetscSection`, `PetscSectionGetDof()`, `PetscSectionSetDof()`, `PetscSectionCreate()`
@*/
PetscErrorCode PetscSectionAddDof(PetscSection s, PetscInt point, PetscInt numDof)
{
  PetscFunctionBeginHot;
  PetscValidHeaderSpecific(s, PETSC_SECTION_CLASSID, 1);
  PetscAssert(point >= s->pStart && point < s->pEnd, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Section point %" PetscInt_FMT " should be in [%" PetscInt_FMT ", %" PetscInt_FMT ")", point, s->pStart, s->pEnd);
  PetscCheck(numDof >= 0, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "numDof %" PetscInt_FMT " should not be negative", numDof);
  s->atlasDof[point - s->pStart] += numDof;
  PetscCall(PetscSectionInvalidateMaxDof_Internal(s));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PetscSectionGetFieldDof - Return the number of degrees of freedom associated with a field on a given point.

  Not Collective

  Input Parameters:
+ s     - the `PetscSection`
. point - the point
- field - the field

  Output Parameter:
. numDof - the number of dof

  Level: intermediate

.seealso: [PetscSection](sec_petscsection), `PetscSection`, `PetscSectionSetFieldDof()`, `PetscSectionCreate()`
@*/
PetscErrorCode PetscSectionGetFieldDof(PetscSection s, PetscInt point, PetscInt field, PetscInt *numDof)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(s, PETSC_SECTION_CLASSID, 1);
  PetscAssertPointer(numDof, 4);
  PetscSectionCheckValidField(field, s->numFields);
  PetscCall(PetscSectionGetDof(s->field[field], point, numDof));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PetscSectionSetFieldDof - Sets the number of degrees of freedom associated with a field on a given point.

  Not Collective

  Input Parameters:
+ s      - the `PetscSection`
. point  - the point
. field  - the field
- numDof - the number of dof, these values may be negative -(dof+1) to indicate they are off process

  Level: intermediate

  Note:
  When setting the number of dof for a field at a point one must also ensure the count of the total number of dof at the point (summed over
  the fields and the unnamed default field) is correct by also calling `PetscSectionAddDof()` or `PetscSectionSetDof()`

  This is equivalent to
.vb
     PetscSection fs;
     PetscSectionGetField(s,field,&fs)
     PetscSectionSetDof(fs,numDof)
.ve

.seealso: [PetscSection](sec_petscsection), `PetscSection`, `PetscSectionGetFieldDof()`, `PetscSectionCreate()`, `PetscSectionAddDof()`, `PetscSectionSetDof()`
@*/
PetscErrorCode PetscSectionSetFieldDof(PetscSection s, PetscInt point, PetscInt field, PetscInt numDof)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(s, PETSC_SECTION_CLASSID, 1);
  PetscSectionCheckValidField(field, s->numFields);
  PetscCall(PetscSectionSetDof(s->field[field], point, numDof));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PetscSectionAddFieldDof - Adds a number of degrees of freedom associated with a field on a given point.

  Not Collective

  Input Parameters:
+ s      - the `PetscSection`
. point  - the point
. field  - the field
- numDof - the number of dof

  Level: intermediate

  Notes:
  When adding to the number of dof for a field at a point one must also ensure the count of the total number of dof at the point (summed over
  the fields and the unnamed default field) is correct by also calling `PetscSectionAddDof()` or `PetscSectionSetDof()`

  This is equivalent to
.vb
     PetscSection fs;
     PetscSectionGetField(s,field,&fs)
     PetscSectionAddDof(fs,numDof)
.ve

.seealso: [PetscSection](sec_petscsection), `PetscSection`, `PetscSectionSetFieldDof()`, `PetscSectionGetFieldDof()`, `PetscSectionCreate()`
@*/
PetscErrorCode PetscSectionAddFieldDof(PetscSection s, PetscInt point, PetscInt field, PetscInt numDof)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(s, PETSC_SECTION_CLASSID, 1);
  PetscSectionCheckValidField(field, s->numFields);
  PetscCall(PetscSectionAddDof(s->field[field], point, numDof));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PetscSectionGetConstraintDof - Return the number of constrained degrees of freedom associated with a given point.

  Not Collective

  Input Parameters:
+ s     - the `PetscSection`
- point - the point

  Output Parameter:
. numDof - the number of dof which are fixed by constraints

  Level: intermediate

.seealso: [PetscSection](sec_petscsection), `PetscSection`, `PetscSectionGetDof()`, `PetscSectionSetConstraintDof()`, `PetscSectionCreate()`
@*/
PetscErrorCode PetscSectionGetConstraintDof(PetscSection s, PetscInt point, PetscInt *numDof)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(s, PETSC_SECTION_CLASSID, 1);
  PetscAssertPointer(numDof, 3);
  if (s->bc) {
    PetscCall(PetscSectionGetDof(s->bc, point, numDof));
  } else *numDof = 0;
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PetscSectionSetConstraintDof - Set the number of constrained degrees of freedom associated with a given point.

  Not Collective

  Input Parameters:
+ s      - the `PetscSection`
. point  - the point
- numDof - the number of dof which are fixed by constraints

  Level: intermediate

.seealso: [PetscSection](sec_petscsection), `PetscSection`, `PetscSectionSetDof()`, `PetscSectionGetConstraintDof()`, `PetscSectionCreate()`
@*/
PetscErrorCode PetscSectionSetConstraintDof(PetscSection s, PetscInt point, PetscInt numDof)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(s, PETSC_SECTION_CLASSID, 1);
  if (numDof) {
    PetscCall(PetscSectionCheckConstraints_Private(s));
    PetscCall(PetscSectionSetDof(s->bc, point, numDof));
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PetscSectionAddConstraintDof - Increment the number of constrained degrees of freedom associated with a given point.

  Not Collective

  Input Parameters:
+ s      - the `PetscSection`
. point  - the point
- numDof - the number of additional dof which are fixed by constraints

  Level: intermediate

.seealso: [PetscSection](sec_petscsection), `PetscSection`, `PetscSectionAddDof()`, `PetscSectionGetConstraintDof()`, `PetscSectionCreate()`
@*/
PetscErrorCode PetscSectionAddConstraintDof(PetscSection s, PetscInt point, PetscInt numDof)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(s, PETSC_SECTION_CLASSID, 1);
  if (numDof) {
    PetscCall(PetscSectionCheckConstraints_Private(s));
    PetscCall(PetscSectionAddDof(s->bc, point, numDof));
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PetscSectionGetFieldConstraintDof - Return the number of constrained degrees of freedom associated with a given field on a point.

  Not Collective

  Input Parameters:
+ s     - the `PetscSection`
. point - the point
- field - the field

  Output Parameter:
. numDof - the number of dof which are fixed by constraints

  Level: intermediate

.seealso: [PetscSection](sec_petscsection), `PetscSection`, `PetscSectionGetDof()`, `PetscSectionSetFieldConstraintDof()`, `PetscSectionCreate()`
@*/
PetscErrorCode PetscSectionGetFieldConstraintDof(PetscSection s, PetscInt point, PetscInt field, PetscInt *numDof)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(s, PETSC_SECTION_CLASSID, 1);
  PetscAssertPointer(numDof, 4);
  PetscSectionCheckValidField(field, s->numFields);
  PetscCall(PetscSectionGetConstraintDof(s->field[field], point, numDof));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PetscSectionSetFieldConstraintDof - Set the number of constrained degrees of freedom associated with a given field on a point.

  Not Collective

  Input Parameters:
+ s      - the `PetscSection`
. point  - the point
. field  - the field
- numDof - the number of dof which are fixed by constraints

  Level: intermediate

.seealso: [PetscSection](sec_petscsection), `PetscSection`, `PetscSectionSetDof()`, `PetscSectionGetFieldConstraintDof()`, `PetscSectionCreate()`
@*/
PetscErrorCode PetscSectionSetFieldConstraintDof(PetscSection s, PetscInt point, PetscInt field, PetscInt numDof)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(s, PETSC_SECTION_CLASSID, 1);
  PetscSectionCheckValidField(field, s->numFields);
  PetscCall(PetscSectionSetConstraintDof(s->field[field], point, numDof));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PetscSectionAddFieldConstraintDof - Increment the number of constrained degrees of freedom associated with a given field on a point.

  Not Collective

  Input Parameters:
+ s      - the `PetscSection`
. point  - the point
. field  - the field
- numDof - the number of additional dof which are fixed by constraints

  Level: intermediate

.seealso: [PetscSection](sec_petscsection), `PetscSection`, `PetscSectionAddDof()`, `PetscSectionGetFieldConstraintDof()`, `PetscSectionCreate()`
@*/
PetscErrorCode PetscSectionAddFieldConstraintDof(PetscSection s, PetscInt point, PetscInt field, PetscInt numDof)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(s, PETSC_SECTION_CLASSID, 1);
  PetscSectionCheckValidField(field, s->numFields);
  PetscCall(PetscSectionAddConstraintDof(s->field[field], point, numDof));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PetscSectionSetUpBC - Setup the subsections describing boundary conditions.

  Not Collective

  Input Parameter:
. s - the `PetscSection`

  Level: advanced

.seealso: [PetscSection](sec_petscsection), `PetscSection`, `PetscSectionSetUp()`, `PetscSectionCreate()`
@*/
PetscErrorCode PetscSectionSetUpBC(PetscSection s)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(s, PETSC_SECTION_CLASSID, 1);
  if (s->bc) {
    const PetscInt last = (s->bc->pEnd - s->bc->pStart) - 1;

    PetscCall(PetscSectionSetUp(s->bc));
    PetscCall(PetscMalloc1((last >= 0 ? s->bc->atlasOff[last] + s->bc->atlasDof[last] : 0), &s->bcIndices));
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PetscSectionSetUp - Calculate offsets based upon the number of degrees of freedom for each point in preparation for use of the `PetscSection`

  Not Collective

  Input Parameter:
. s - the `PetscSection`

  Level: intermediate

  Notes:
  If used, `PetscSectionSetPermutation()` must be called before this routine.

  `PetscSectionSetPointMajor()`, cannot be called after this routine.

.seealso: [PetscSection](sec_petscsection), `PetscSection`, `PetscSectionCreate()`, `PetscSectionSetPermutation()`
@*/
PetscErrorCode PetscSectionSetUp(PetscSection s)
{
  const PetscInt *pind   = NULL;
  PetscInt        offset = 0, foff, p, f;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(s, PETSC_SECTION_CLASSID, 1);
  if (s->setup) PetscFunctionReturn(PETSC_SUCCESS);
  s->setup = PETSC_TRUE;
  /* Set offsets and field offsets for all points */
  /*   Assume that all fields have the same chart */
  PetscCheck(s->includesConstraints, PETSC_COMM_SELF, PETSC_ERR_SUP, "PetscSectionSetUp is currently unsupported for includesConstraints = PETSC_TRUE");
  if (s->perm) PetscCall(ISGetIndices(s->perm, &pind));
  if (s->pointMajor) {
    for (p = 0; p < s->pEnd - s->pStart; ++p) {
      const PetscInt q = pind ? pind[p] : p;

      /* Set point offset */
      s->atlasOff[q] = offset;
      offset += s->atlasDof[q];
      /* Set field offset */
      for (f = 0, foff = s->atlasOff[q]; f < s->numFields; ++f) {
        PetscSection sf = s->field[f];

        sf->atlasOff[q] = foff;
        foff += sf->atlasDof[q];
      }
    }
  } else {
    /* Set field offsets for all points */
    for (f = 0; f < s->numFields; ++f) {
      PetscSection sf = s->field[f];

      for (p = 0; p < s->pEnd - s->pStart; ++p) {
        const PetscInt q = pind ? pind[p] : p;

        sf->atlasOff[q] = offset;
        offset += sf->atlasDof[q];
      }
    }
    /* Disable point offsets since these are unused */
    for (p = 0; p < s->pEnd - s->pStart; ++p) s->atlasOff[p] = -1;
  }
  if (s->perm) PetscCall(ISRestoreIndices(s->perm, &pind));
  /* Setup BC sections */
  PetscCall(PetscSectionSetUpBC(s));
  for (f = 0; f < s->numFields; ++f) PetscCall(PetscSectionSetUpBC(s->field[f]));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PetscSectionGetMaxDof - Return the maximum number of degrees of freedom on any point in the `PetscSection`

  Not Collective

  Input Parameter:
. s - the `PetscSection`

  Output Parameter:
. maxDof - the maximum dof

  Level: intermediate

  Notes:
  The returned number is up-to-date without need for `PetscSectionSetUp()`.

  This is the maximum over all points of the sum of the number of dof in the unnamed default field plus all named fields. This is equivalent to
  the maximum over all points of the value returned by `PetscSectionGetDof()` on this MPI process

  Developer Notes:
  The returned number is calculated lazily and stashed.

  A call to `PetscSectionInvalidateMaxDof_Internal()` invalidates the stashed value.

  `PetscSectionInvalidateMaxDof_Internal()` is called in `PetscSectionSetDof()`, `PetscSectionAddDof()` and `PetscSectionReset()`

  It should also be called every time `atlasDof` is modified directly.

.seealso: [PetscSection](sec_petscsection), `PetscSection`, `PetscSectionGetDof()`, `PetscSectionSetDof()`, `PetscSectionAddDof()`, `PetscSectionCreate()`
@*/
PetscErrorCode PetscSectionGetMaxDof(PetscSection s, PetscInt *maxDof)
{
  PetscInt p;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(s, PETSC_SECTION_CLASSID, 1);
  PetscAssertPointer(maxDof, 2);
  if (s->maxDof == PETSC_MIN_INT) {
    s->maxDof = 0;
    for (p = 0; p < s->pEnd - s->pStart; ++p) s->maxDof = PetscMax(s->maxDof, s->atlasDof[p]);
  }
  *maxDof = s->maxDof;
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PetscSectionGetStorageSize - Return the size of an array or local `Vec` capable of holding all the degrees of freedom defined in a `PetscSection`

  Not Collective

  Input Parameter:
. s - the `PetscSection`

  Output Parameter:
. size - the size of an array which can hold all the dofs

  Level: intermediate

.seealso: [PetscSection](sec_petscsection), `PetscSection`, `PetscSectionGetOffset()`, `PetscSectionGetConstrainedStorageSize()`, `PetscSectionCreate()`
@*/
PetscErrorCode PetscSectionGetStorageSize(PetscSection s, PetscInt *size)
{
  PetscInt p, n = 0;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(s, PETSC_SECTION_CLASSID, 1);
  PetscAssertPointer(size, 2);
  for (p = 0; p < s->pEnd - s->pStart; ++p) n += s->atlasDof[p] > 0 ? s->atlasDof[p] : 0;
  *size = n;
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PetscSectionGetConstrainedStorageSize - Return the size of an array or local `Vec` capable of holding all unconstrained degrees of freedom in a `PetscSection`

  Not Collective

  Input Parameter:
. s - the `PetscSection`

  Output Parameter:
. size - the size of an array which can hold all unconstrained dofs

  Level: intermediate

.seealso: [PetscSection](sec_petscsection), `PetscSection`, `PetscSectionGetStorageSize()`, `PetscSectionGetOffset()`, `PetscSectionCreate()`
@*/
PetscErrorCode PetscSectionGetConstrainedStorageSize(PetscSection s, PetscInt *size)
{
  PetscInt p, n = 0;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(s, PETSC_SECTION_CLASSID, 1);
  PetscAssertPointer(size, 2);
  for (p = 0; p < s->pEnd - s->pStart; ++p) {
    const PetscInt cdof = s->bc ? s->bc->atlasDof[p] : 0;
    n += s->atlasDof[p] > 0 ? s->atlasDof[p] - cdof : 0;
  }
  *size = n;
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PetscSectionCreateGlobalSection - Create a parallel section describing the global layout using
  a local (sequential) `PetscSection` on each MPI process and a `PetscSF` describing the section point overlap.

  Input Parameters:
+ s                  - The `PetscSection` for the local field layout
. sf                 - The `PetscSF` describing parallel layout of the section points (leaves are unowned local points)
. includeConstraints - By default this is `PETSC_FALSE`, meaning that the global field vector will not possess constrained dofs
- localOffsets       - If `PETSC_TRUE`, use local rather than global offsets for the points

  Output Parameter:
. gsection - The `PetscSection` for the global field layout

  Level: intermediate

  Notes:
  On each MPI process `gsection` inherits the chart of the `s` on that process.

  This sets negative sizes and offsets to points not owned by this process as defined by `sf` but that are within the local value of the chart of `gsection`.
  In those locations the value of size is -(size+1) and the value of the offset on the remote process is -(off+1).

.seealso: [PetscSection](sec_petscsection), `PetscSection`, `PetscSectionCreate()`, `PetscSectionCreateGlobalSectionCensored()`
@*/
PetscErrorCode PetscSectionCreateGlobalSection(PetscSection s, PetscSF sf, PetscBool includeConstraints, PetscBool localOffsets, PetscSection *gsection)
{
  PetscSection    gs;
  const PetscInt *pind = NULL;
  PetscInt       *recv = NULL, *neg = NULL;
  PetscInt        pStart, pEnd, p, dof, cdof, off, foff, globalOff = 0, nroots, nlocal, maxleaf;
  PetscInt        numFields, f, numComponents;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(s, PETSC_SECTION_CLASSID, 1);
  PetscValidHeaderSpecific(sf, PETSCSF_CLASSID, 2);
  PetscValidLogicalCollectiveBool(s, includeConstraints, 3);
  PetscValidLogicalCollectiveBool(s, localOffsets, 4);
  PetscAssertPointer(gsection, 5);
  PetscCheck(s->pointMajor, PETSC_COMM_SELF, PETSC_ERR_SUP, "No support for field major ordering");
  PetscCall(PetscSectionCreate(PetscObjectComm((PetscObject)s), &gs));
  PetscCall(PetscSectionGetNumFields(s, &numFields));
  if (numFields > 0) PetscCall(PetscSectionSetNumFields(gs, numFields));
  PetscCall(PetscSectionGetChart(s, &pStart, &pEnd));
  PetscCall(PetscSectionSetChart(gs, pStart, pEnd));
  gs->includesConstraints = includeConstraints;
  PetscCall(PetscSFGetGraph(sf, &nroots, NULL, NULL, NULL));
  nlocal = nroots; /* The local/leaf space matches global/root space */
  /* Must allocate for all points visible to SF, which may be more than this section */
  if (nroots >= 0) { /* nroots < 0 means that the graph has not been set, only happens in serial */
    PetscCall(PetscSFGetLeafRange(sf, NULL, &maxleaf));
    PetscCheck(nroots >= pEnd, PETSC_COMM_SELF, PETSC_ERR_ARG_SIZ, "SF roots %" PetscInt_FMT " < pEnd %" PetscInt_FMT, nroots, pEnd);
    PetscCheck(maxleaf < nroots, PETSC_COMM_SELF, PETSC_ERR_ARG_SIZ, "Max local leaf %" PetscInt_FMT " >= nroots %" PetscInt_FMT, maxleaf, nroots);
    PetscCall(PetscMalloc2(nroots, &neg, nlocal, &recv));
    PetscCall(PetscArrayzero(neg, nroots));
  }
  /* Mark all local points with negative dof */
  for (p = pStart; p < pEnd; ++p) {
    PetscCall(PetscSectionGetDof(s, p, &dof));
    PetscCall(PetscSectionSetDof(gs, p, dof));
    PetscCall(PetscSectionGetConstraintDof(s, p, &cdof));
    if (!includeConstraints && cdof > 0) PetscCall(PetscSectionSetConstraintDof(gs, p, cdof));
    if (neg) neg[p] = -(dof + 1);
  }
  PetscCall(PetscSectionSetUpBC(gs));
  if (gs->bcIndices) PetscCall(PetscArraycpy(gs->bcIndices, s->bcIndices, gs->bc->atlasOff[gs->bc->pEnd - gs->bc->pStart - 1] + gs->bc->atlasDof[gs->bc->pEnd - gs->bc->pStart - 1]));
  if (nroots >= 0) {
    PetscCall(PetscArrayzero(recv, nlocal));
    PetscCall(PetscSFBcastBegin(sf, MPIU_INT, neg, recv, MPI_REPLACE));
    PetscCall(PetscSFBcastEnd(sf, MPIU_INT, neg, recv, MPI_REPLACE));
    for (p = pStart; p < pEnd; ++p) {
      if (recv[p] < 0) {
        gs->atlasDof[p - pStart] = recv[p];
        PetscCall(PetscSectionGetDof(s, p, &dof));
        PetscCheck(-(recv[p] + 1) == dof, PETSC_COMM_SELF, PETSC_ERR_ARG_WRONG, "Global dof %" PetscInt_FMT " for point %" PetscInt_FMT " is not the unconstrained %" PetscInt_FMT, -(recv[p] + 1), p, dof);
      }
    }
  }
  /* Calculate new sizes, get process offset, and calculate point offsets */
  if (s->perm) PetscCall(ISGetIndices(s->perm, &pind));
  for (p = 0, off = 0; p < pEnd - pStart; ++p) {
    const PetscInt q = pind ? pind[p] : p;

    cdof            = (!includeConstraints && s->bc) ? s->bc->atlasDof[q] : 0;
    gs->atlasOff[q] = off;
    off += gs->atlasDof[q] > 0 ? gs->atlasDof[q] - cdof : 0;
  }
  if (!localOffsets) {
    PetscCallMPI(MPI_Scan(&off, &globalOff, 1, MPIU_INT, MPI_SUM, PetscObjectComm((PetscObject)s)));
    globalOff -= off;
  }
  for (p = pStart, off = 0; p < pEnd; ++p) {
    gs->atlasOff[p - pStart] += globalOff;
    if (neg) neg[p] = -(gs->atlasOff[p - pStart] + 1);
  }
  if (s->perm) PetscCall(ISRestoreIndices(s->perm, &pind));
  /* Put in negative offsets for ghost points */
  if (nroots >= 0) {
    PetscCall(PetscArrayzero(recv, nlocal));
    PetscCall(PetscSFBcastBegin(sf, MPIU_INT, neg, recv, MPI_REPLACE));
    PetscCall(PetscSFBcastEnd(sf, MPIU_INT, neg, recv, MPI_REPLACE));
    for (p = pStart; p < pEnd; ++p) {
      if (recv[p] < 0) gs->atlasOff[p - pStart] = recv[p];
    }
  }
  PetscCall(PetscFree2(neg, recv));
  /* Set field dofs/offsets/constraints */
  for (f = 0; f < numFields; ++f) {
    gs->field[f]->includesConstraints = includeConstraints;
    PetscCall(PetscSectionGetFieldComponents(s, f, &numComponents));
    PetscCall(PetscSectionSetFieldComponents(gs, f, numComponents));
  }
  for (p = pStart; p < pEnd; ++p) {
    PetscCall(PetscSectionGetOffset(gs, p, &off));
    for (f = 0, foff = off; f < numFields; ++f) {
      PetscCall(PetscSectionGetFieldConstraintDof(s, p, f, &cdof));
      if (!includeConstraints && cdof > 0) PetscCall(PetscSectionSetFieldConstraintDof(gs, p, f, cdof));
      PetscCall(PetscSectionGetFieldDof(s, p, f, &dof));
      PetscCall(PetscSectionSetFieldDof(gs, p, f, off < 0 ? -(dof + 1) : dof));
      PetscCall(PetscSectionSetFieldOffset(gs, p, f, foff));
      PetscCall(PetscSectionGetFieldConstraintDof(gs, p, f, &cdof));
      foff = off < 0 ? foff - (dof - cdof) : foff + (dof - cdof);
    }
  }
  for (f = 0; f < numFields; ++f) {
    PetscSection gfs = gs->field[f];

    PetscCall(PetscSectionSetUpBC(gfs));
    if (gfs->bcIndices) PetscCall(PetscArraycpy(gfs->bcIndices, s->field[f]->bcIndices, gfs->bc->atlasOff[gfs->bc->pEnd - gfs->bc->pStart - 1] + gfs->bc->atlasDof[gfs->bc->pEnd - gfs->bc->pStart - 1]));
  }
  gs->setup = PETSC_TRUE;
  PetscCall(PetscSectionViewFromOptions(gs, NULL, "-global_section_view"));
  *gsection = gs;
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PetscSectionCreateGlobalSectionCensored - Create a `PetscSection` describing the globallayout using
  a local (sequential) `PetscSection` on each MPI process and an `PetscSF` describing the section point overlap.

  Input Parameters:
+ s                  - The `PetscSection` for the local field layout
. sf                 - The `PetscSF` describing parallel layout of the section points
. includeConstraints - By default this is `PETSC_FALSE`, meaning that the global vector will not possess constrained dofs
. numExcludes        - The number of exclusion ranges, this must have the same value on all MPI processes
- excludes           - An array [start_0, end_0, start_1, end_1, ...] where there are `numExcludes` pairs and must have the same values on all MPI processes

  Output Parameter:
. gsection - The `PetscSection` for the global field layout

  Level: advanced

  Notes:
  On each MPI process `gsection` inherits the chart of the `s` on that process.

  This sets negative sizes and offsets to points not owned by this process as defined by `sf` but that are within the local value of the chart of `gsection`.
  In those locations the value of size is -(size+1) and the value of the offset on the remote process is -(off+1).

  This routine augments `PetscSectionCreateGlobalSection()` by allowing one to exclude certain ranges in the chart of the `PetscSection`

  Developer Notes:
  This is a terrible function name

.seealso: [PetscSection](sec_petscsection), `PetscSection`, `PetscSectionCreate()`
@*/
PetscErrorCode PetscSectionCreateGlobalSectionCensored(PetscSection s, PetscSF sf, PetscBool includeConstraints, PetscInt numExcludes, const PetscInt excludes[], PetscSection *gsection)
{
  const PetscInt *pind = NULL;
  PetscInt       *neg = NULL, *tmpOff = NULL;
  PetscInt        pStart, pEnd, p, e, dof, cdof, off, globalOff = 0, nroots;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(s, PETSC_SECTION_CLASSID, 1);
  PetscValidHeaderSpecific(sf, PETSCSF_CLASSID, 2);
  PetscAssertPointer(gsection, 6);
  PetscCall(PetscSectionCreate(PetscObjectComm((PetscObject)s), gsection));
  PetscCall(PetscSectionGetChart(s, &pStart, &pEnd));
  PetscCall(PetscSectionSetChart(*gsection, pStart, pEnd));
  PetscCall(PetscSFGetGraph(sf, &nroots, NULL, NULL, NULL));
  if (nroots >= 0) {
    PetscCheck(nroots >= pEnd - pStart, PETSC_COMM_SELF, PETSC_ERR_ARG_SIZ, "PetscSF nroots %" PetscInt_FMT " < %" PetscInt_FMT " section size", nroots, pEnd - pStart);
    PetscCall(PetscCalloc1(nroots, &neg));
    if (nroots > pEnd - pStart) {
      PetscCall(PetscCalloc1(nroots, &tmpOff));
    } else {
      tmpOff = &(*gsection)->atlasDof[-pStart];
    }
  }
  /* Mark ghost points with negative dof */
  for (p = pStart; p < pEnd; ++p) {
    for (e = 0; e < numExcludes; ++e) {
      if ((p >= excludes[e * 2 + 0]) && (p < excludes[e * 2 + 1])) {
        PetscCall(PetscSectionSetDof(*gsection, p, 0));
        break;
      }
    }
    if (e < numExcludes) continue;
    PetscCall(PetscSectionGetDof(s, p, &dof));
    PetscCall(PetscSectionSetDof(*gsection, p, dof));
    PetscCall(PetscSectionGetConstraintDof(s, p, &cdof));
    if (!includeConstraints && cdof > 0) PetscCall(PetscSectionSetConstraintDof(*gsection, p, cdof));
    if (neg) neg[p] = -(dof + 1);
  }
  PetscCall(PetscSectionSetUpBC(*gsection));
  if (nroots >= 0) {
    PetscCall(PetscSFBcastBegin(sf, MPIU_INT, neg, tmpOff, MPI_REPLACE));
    PetscCall(PetscSFBcastEnd(sf, MPIU_INT, neg, tmpOff, MPI_REPLACE));
    if (nroots > pEnd - pStart) {
      for (p = pStart; p < pEnd; ++p) {
        if (tmpOff[p] < 0) (*gsection)->atlasDof[p - pStart] = tmpOff[p];
      }
    }
  }
  /* Calculate new sizes, get process offset, and calculate point offsets */
  if (s->perm) PetscCall(ISGetIndices(s->perm, &pind));
  for (p = 0, off = 0; p < pEnd - pStart; ++p) {
    const PetscInt q = pind ? pind[p] : p;

    cdof                     = (!includeConstraints && s->bc) ? s->bc->atlasDof[q] : 0;
    (*gsection)->atlasOff[q] = off;
    off += (*gsection)->atlasDof[q] > 0 ? (*gsection)->atlasDof[q] - cdof : 0;
  }
  PetscCallMPI(MPI_Scan(&off, &globalOff, 1, MPIU_INT, MPI_SUM, PetscObjectComm((PetscObject)s)));
  globalOff -= off;
  for (p = 0, off = 0; p < pEnd - pStart; ++p) {
    (*gsection)->atlasOff[p] += globalOff;
    if (neg) neg[p + pStart] = -((*gsection)->atlasOff[p] + 1);
  }
  if (s->perm) PetscCall(ISRestoreIndices(s->perm, &pind));
  /* Put in negative offsets for ghost points */
  if (nroots >= 0) {
    if (nroots == pEnd - pStart) tmpOff = &(*gsection)->atlasOff[-pStart];
    PetscCall(PetscSFBcastBegin(sf, MPIU_INT, neg, tmpOff, MPI_REPLACE));
    PetscCall(PetscSFBcastEnd(sf, MPIU_INT, neg, tmpOff, MPI_REPLACE));
    if (nroots > pEnd - pStart) {
      for (p = pStart; p < pEnd; ++p) {
        if (tmpOff[p] < 0) (*gsection)->atlasOff[p - pStart] = tmpOff[p];
      }
    }
  }
  if (nroots >= 0 && nroots > pEnd - pStart) PetscCall(PetscFree(tmpOff));
  PetscCall(PetscFree(neg));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PetscSectionGetPointLayout - Get a `PetscLayout` for the points with nonzero dof counts of the unnamed default field within this `PetscSection`s local chart

  Collective

  Input Parameters:
+ comm - The `MPI_Comm`
- s    - The `PetscSection`

  Output Parameter:
. layout - The point layout for the data that defines the section

  Level: advanced

  Notes:
  `PetscSectionGetValueLayout()` provides similar information but counting the total number of degrees of freedom on the MPI process (excluding constrained
  degrees of freedom).

  This count includes constrained degrees of freedom

  This is usually called on the default global section.

  Example:
.vb
     The chart is [2,5), point 2 has 2 dof, point 3 has 0 dof, point 4 has 1 dof
     The local size of the `PetscLayout` is 2 since 2 points have a non-zero number of dof
.ve

  Developer Notes:
  I find the names of these two functions extremely non-informative

.seealso: [PetscSection](sec_petscsection), `PetscSection`, `PetscSectionGetValueLayout()`, `PetscSectionCreate()`
@*/
PetscErrorCode PetscSectionGetPointLayout(MPI_Comm comm, PetscSection s, PetscLayout *layout)
{
  PetscInt pStart, pEnd, p, localSize = 0;

  PetscFunctionBegin;
  PetscCall(PetscSectionGetChart(s, &pStart, &pEnd));
  for (p = pStart; p < pEnd; ++p) {
    PetscInt dof;

    PetscCall(PetscSectionGetDof(s, p, &dof));
    if (dof >= 0) ++localSize;
  }
  PetscCall(PetscLayoutCreate(comm, layout));
  PetscCall(PetscLayoutSetLocalSize(*layout, localSize));
  PetscCall(PetscLayoutSetBlockSize(*layout, 1));
  PetscCall(PetscLayoutSetUp(*layout));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PetscSectionGetValueLayout - Get the `PetscLayout` associated with the section dofs of a `PetscSection`

  Collective

  Input Parameters:
+ comm - The `MPI_Comm`
- s    - The `PetscSection`

  Output Parameter:
. layout - The dof layout for the section

  Level: advanced

  Notes:
  `PetscSectionGetPointLayout()` provides similar information but only counting the number of points with nonzero degrees of freedom and
  including the constrained degrees of freedom

  This is usually called for the default global section.

  Example:
.vb
     The chart is [2,5), point 2 has 4 dof (2 constrained), point 3 has 0 dof, point 4 has 1 dof (not constrained)
     The local size of the `PetscLayout` is 3 since there are 3 unconstrained degrees of freedom on this MPI process
.ve

.seealso: [PetscSection](sec_petscsection), `PetscSection`, `PetscSectionGetPointLayout()`, `PetscSectionCreate()`
@*/
PetscErrorCode PetscSectionGetValueLayout(MPI_Comm comm, PetscSection s, PetscLayout *layout)
{
  PetscInt pStart, pEnd, p, localSize = 0;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(s, PETSC_SECTION_CLASSID, 2);
  PetscAssertPointer(layout, 3);
  PetscCall(PetscSectionGetChart(s, &pStart, &pEnd));
  for (p = pStart; p < pEnd; ++p) {
    PetscInt dof, cdof;

    PetscCall(PetscSectionGetDof(s, p, &dof));
    PetscCall(PetscSectionGetConstraintDof(s, p, &cdof));
    if (dof - cdof > 0) localSize += dof - cdof;
  }
  PetscCall(PetscLayoutCreate(comm, layout));
  PetscCall(PetscLayoutSetLocalSize(*layout, localSize));
  PetscCall(PetscLayoutSetBlockSize(*layout, 1));
  PetscCall(PetscLayoutSetUp(*layout));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PetscSectionGetOffset - Return the offset into an array or `Vec` for the dof associated with the given point.

  Not Collective

  Input Parameters:
+ s     - the `PetscSection`
- point - the point

  Output Parameter:
. offset - the offset

  Level: intermediate

  Notes:
  In a global section, `offset` will be negative for points not owned by this process.

  This is for the unnamed default field in the `PetscSection` not the named fields

  The `offset` values are different depending on a value set with `PetscSectionSetPointMajor()`

.seealso: [PetscSection](sec_petscsection), `PetscSection`, `PetscSectionGetFieldOffset()`, `PetscSectionCreate()`, `PetscSectionSetPointMajor()`
@*/
PetscErrorCode PetscSectionGetOffset(PetscSection s, PetscInt point, PetscInt *offset)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(s, PETSC_SECTION_CLASSID, 1);
  PetscAssertPointer(offset, 3);
  PetscAssert(!(point < s->pStart) && !(point >= s->pEnd), PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Section point %" PetscInt_FMT " should be in [%" PetscInt_FMT ", %" PetscInt_FMT ")", point, s->pStart, s->pEnd);
  *offset = s->atlasOff[point - s->pStart];
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PetscSectionSetOffset - Set the offset into an array or `Vec` for the dof associated with the given point.

  Not Collective

  Input Parameters:
+ s      - the `PetscSection`
. point  - the point
- offset - the offset, these values may be negative indicating the values are off process

  Level: developer

  Note:
  The user usually does not call this function, but uses `PetscSectionSetUp()`

.seealso: [PetscSection](sec_petscsection), `PetscSection`, `PetscSectionGetFieldOffset()`, `PetscSectionCreate()`, `PetscSectionSetUp()`
@*/
PetscErrorCode PetscSectionSetOffset(PetscSection s, PetscInt point, PetscInt offset)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(s, PETSC_SECTION_CLASSID, 1);
  PetscCheck(!(point < s->pStart) && !(point >= s->pEnd), PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Section point %" PetscInt_FMT " should be in [%" PetscInt_FMT ", %" PetscInt_FMT ")", point, s->pStart, s->pEnd);
  s->atlasOff[point - s->pStart] = offset;
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PetscSectionGetFieldOffset - Return the offset into an array or `Vec` for the field dof associated with the given point.

  Not Collective

  Input Parameters:
+ s     - the `PetscSection`
. point - the point
- field - the field

  Output Parameter:
. offset - the offset

  Level: intermediate

  Notes:
  In a global section, `offset` will be negative for points not owned by this process.

  The `offset` values are different depending on a value set with `PetscSectionSetPointMajor()`

.seealso: [PetscSection](sec_petscsection), `PetscSection`, `PetscSectionGetOffset()`, `PetscSectionCreate()`, `PetscSectionGetFieldPointOffset()`
@*/
PetscErrorCode PetscSectionGetFieldOffset(PetscSection s, PetscInt point, PetscInt field, PetscInt *offset)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(s, PETSC_SECTION_CLASSID, 1);
  PetscAssertPointer(offset, 4);
  PetscSectionCheckValidField(field, s->numFields);
  PetscCall(PetscSectionGetOffset(s->field[field], point, offset));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PetscSectionSetFieldOffset - Set the offset into an array or `Vec` for the dof associated with the given field at a point.

  Not Collective

  Input Parameters:
+ s      - the `PetscSection`
. point  - the point
. field  - the field
- offset - the offset, these values may be negative indicating the values are off process

  Level: developer

  Note:
  The user usually does not call this function, but uses `PetscSectionSetUp()`

.seealso: [PetscSection](sec_petscsection), `PetscSection`, `PetscSectionGetFieldOffset()`, `PetscSectionSetOffset()`, `PetscSectionCreate()`, `PetscSectionSetUp()`
@*/
PetscErrorCode PetscSectionSetFieldOffset(PetscSection s, PetscInt point, PetscInt field, PetscInt offset)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(s, PETSC_SECTION_CLASSID, 1);
  PetscSectionCheckValidField(field, s->numFields);
  PetscCall(PetscSectionSetOffset(s->field[field], point, offset));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PetscSectionGetFieldPointOffset - Return the offset for the first field dof associated with the given point relative to the offset for that point for the
  unnamed default field's first dof

  Not Collective

  Input Parameters:
+ s     - the `PetscSection`
. point - the point
- field - the field

  Output Parameter:
. offset - the offset

  Level: advanced

  Note:
  This ignores constraints

  Example:
.vb
  if PetscSectionSetPointMajor(s,PETSC_TRUE)
  The unnamed default field has 3 dof at `point`
  Field 0 has 2 dof at `point`
  Then PetscSectionGetFieldPointOffset(s,point,1,&offset) returns and offset of 5
.ve

.seealso: [PetscSection](sec_petscsection), `PetscSection`, `PetscSectionGetOffset()`, `PetscSectionCreate()`, `PetscSectionGetFieldOffset()`
@*/
PetscErrorCode PetscSectionGetFieldPointOffset(PetscSection s, PetscInt point, PetscInt field, PetscInt *offset)
{
  PetscInt off, foff;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(s, PETSC_SECTION_CLASSID, 1);
  PetscAssertPointer(offset, 4);
  PetscSectionCheckValidField(field, s->numFields);
  PetscCall(PetscSectionGetOffset(s, point, &off));
  PetscCall(PetscSectionGetOffset(s->field[field], point, &foff));
  *offset = foff - off;
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PetscSectionGetOffsetRange - Return the full range of offsets [`start`, `end`) for a `PetscSection`

  Not Collective

  Input Parameter:
. s - the `PetscSection`

  Output Parameters:
+ start - the minimum offset
- end   - one more than the maximum offset

  Level: intermediate

.seealso: [PetscSection](sec_petscsection), `PetscSection`, `PetscSectionGetOffset()`, `PetscSectionCreate()`
@*/
PetscErrorCode PetscSectionGetOffsetRange(PetscSection s, PetscInt *start, PetscInt *end)
{
  PetscInt os = 0, oe = 0, pStart, pEnd, p;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(s, PETSC_SECTION_CLASSID, 1);
  if (s->atlasOff) {
    os = s->atlasOff[0];
    oe = s->atlasOff[0];
  }
  PetscCall(PetscSectionGetChart(s, &pStart, &pEnd));
  for (p = 0; p < pEnd - pStart; ++p) {
    PetscInt dof = s->atlasDof[p], off = s->atlasOff[p];

    if (off >= 0) {
      os = PetscMin(os, off);
      oe = PetscMax(oe, off + dof);
    }
  }
  if (start) *start = os;
  if (end) *end = oe;
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PetscSectionCreateSubsection - Create a new, smaller `PetscSection` composed of only selected fields

  Collective

  Input Parameters:
+ s      - the `PetscSection`
. len    - the number of subfields
- fields - the subfield numbers

  Output Parameter:
. subs - the subsection

  Level: advanced

  Notes:
  The chart of `subs` is the same as the chart of `s`

  This will error if a fieldnumber is out of range

.seealso: [PetscSection](sec_petscsection), `PetscSection`, `PetscSectionCreateSupersection()`, `PetscSectionCreate()`
@*/
PetscErrorCode PetscSectionCreateSubsection(PetscSection s, PetscInt len, const PetscInt fields[], PetscSection *subs)
{
  PetscInt nF, f, c, pStart, pEnd, p, maxCdof = 0;

  PetscFunctionBegin;
  if (!len) PetscFunctionReturn(PETSC_SUCCESS);
  PetscValidHeaderSpecific(s, PETSC_SECTION_CLASSID, 1);
  PetscAssertPointer(fields, 3);
  PetscAssertPointer(subs, 4);
  PetscCall(PetscSectionGetNumFields(s, &nF));
  PetscCheck(len <= nF, PetscObjectComm((PetscObject)s), PETSC_ERR_ARG_WRONG, "Number of requested fields %" PetscInt_FMT " greater than number of fields %" PetscInt_FMT, len, nF);
  PetscCall(PetscSectionCreate(PetscObjectComm((PetscObject)s), subs));
  PetscCall(PetscSectionSetNumFields(*subs, len));
  for (f = 0; f < len; ++f) {
    const char     *name    = NULL;
    PetscInt        numComp = 0;
    PetscSectionSym sym;

    PetscCall(PetscSectionGetFieldName(s, fields[f], &name));
    PetscCall(PetscSectionSetFieldName(*subs, f, name));
    PetscCall(PetscSectionGetFieldComponents(s, fields[f], &numComp));
    PetscCall(PetscSectionSetFieldComponents(*subs, f, numComp));
    for (c = 0; c < s->numFieldComponents[fields[f]]; ++c) {
      PetscCall(PetscSectionGetComponentName(s, fields[f], c, &name));
      PetscCall(PetscSectionSetComponentName(*subs, f, c, name));
    }
    PetscCall(PetscSectionGetFieldSym(s, fields[f], &sym));
    PetscCall(PetscSectionSetFieldSym(*subs, f, sym));
  }
  PetscCall(PetscSectionGetChart(s, &pStart, &pEnd));
  PetscCall(PetscSectionSetChart(*subs, pStart, pEnd));
  for (p = pStart; p < pEnd; ++p) {
    PetscInt dof = 0, cdof = 0, fdof = 0, cfdof = 0;

    for (f = 0; f < len; ++f) {
      PetscCall(PetscSectionGetFieldDof(s, p, fields[f], &fdof));
      PetscCall(PetscSectionSetFieldDof(*subs, p, f, fdof));
      PetscCall(PetscSectionGetFieldConstraintDof(s, p, fields[f], &cfdof));
      if (cfdof) PetscCall(PetscSectionSetFieldConstraintDof(*subs, p, f, cfdof));
      dof += fdof;
      cdof += cfdof;
    }
    PetscCall(PetscSectionSetDof(*subs, p, dof));
    if (cdof) PetscCall(PetscSectionSetConstraintDof(*subs, p, cdof));
    maxCdof = PetscMax(cdof, maxCdof);
  }
  PetscCall(PetscSectionSetUp(*subs));
  if (maxCdof) {
    PetscInt *indices;

    PetscCall(PetscMalloc1(maxCdof, &indices));
    for (p = pStart; p < pEnd; ++p) {
      PetscInt cdof;

      PetscCall(PetscSectionGetConstraintDof(*subs, p, &cdof));
      if (cdof) {
        const PetscInt *oldIndices = NULL;
        PetscInt        fdof = 0, cfdof = 0, fc, numConst = 0, fOff = 0;

        for (f = 0; f < len; ++f) {
          PetscCall(PetscSectionGetFieldDof(s, p, fields[f], &fdof));
          PetscCall(PetscSectionGetFieldConstraintDof(s, p, fields[f], &cfdof));
          PetscCall(PetscSectionGetFieldConstraintIndices(s, p, fields[f], &oldIndices));
          PetscCall(PetscSectionSetFieldConstraintIndices(*subs, p, f, oldIndices));
          for (fc = 0; fc < cfdof; ++fc) indices[numConst + fc] = oldIndices[fc] + fOff;
          numConst += cfdof;
          fOff += fdof;
        }
        PetscCall(PetscSectionSetConstraintIndices(*subs, p, indices));
      }
    }
    PetscCall(PetscFree(indices));
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PetscSectionCreateSupersection - Create a new, larger section composed of multiple `PetscSection`s

  Collective

  Input Parameters:
+ s   - the input sections
- len - the number of input sections

  Output Parameter:
. supers - the supersection

  Level: advanced

  Notes:
  The section offsets now refer to a new, larger vector.

  Developer Notes:
  Needs to explain how the sections are composed

.seealso: [PetscSection](sec_petscsection), `PetscSection`, `PetscSectionCreateSubsection()`, `PetscSectionCreate()`
@*/
PetscErrorCode PetscSectionCreateSupersection(PetscSection s[], PetscInt len, PetscSection *supers)
{
  PetscInt Nf = 0, f, pStart = PETSC_MAX_INT, pEnd = 0, p, maxCdof = 0, i;

  PetscFunctionBegin;
  if (!len) PetscFunctionReturn(PETSC_SUCCESS);
  for (i = 0; i < len; ++i) {
    PetscInt nf, pStarti, pEndi;

    PetscCall(PetscSectionGetNumFields(s[i], &nf));
    PetscCall(PetscSectionGetChart(s[i], &pStarti, &pEndi));
    pStart = PetscMin(pStart, pStarti);
    pEnd   = PetscMax(pEnd, pEndi);
    Nf += nf;
  }
  PetscCall(PetscSectionCreate(PetscObjectComm((PetscObject)s[0]), supers));
  PetscCall(PetscSectionSetNumFields(*supers, Nf));
  for (i = 0, f = 0; i < len; ++i) {
    PetscInt nf, fi, ci;

    PetscCall(PetscSectionGetNumFields(s[i], &nf));
    for (fi = 0; fi < nf; ++fi, ++f) {
      const char *name    = NULL;
      PetscInt    numComp = 0;

      PetscCall(PetscSectionGetFieldName(s[i], fi, &name));
      PetscCall(PetscSectionSetFieldName(*supers, f, name));
      PetscCall(PetscSectionGetFieldComponents(s[i], fi, &numComp));
      PetscCall(PetscSectionSetFieldComponents(*supers, f, numComp));
      for (ci = 0; ci < s[i]->numFieldComponents[fi]; ++ci) {
        PetscCall(PetscSectionGetComponentName(s[i], fi, ci, &name));
        PetscCall(PetscSectionSetComponentName(*supers, f, ci, name));
      }
    }
  }
  PetscCall(PetscSectionSetChart(*supers, pStart, pEnd));
  for (p = pStart; p < pEnd; ++p) {
    PetscInt dof = 0, cdof = 0;

    for (i = 0, f = 0; i < len; ++i) {
      PetscInt nf, fi, pStarti, pEndi;
      PetscInt fdof = 0, cfdof = 0;

      PetscCall(PetscSectionGetNumFields(s[i], &nf));
      PetscCall(PetscSectionGetChart(s[i], &pStarti, &pEndi));
      if ((p < pStarti) || (p >= pEndi)) continue;
      for (fi = 0; fi < nf; ++fi, ++f) {
        PetscCall(PetscSectionGetFieldDof(s[i], p, fi, &fdof));
        PetscCall(PetscSectionAddFieldDof(*supers, p, f, fdof));
        PetscCall(PetscSectionGetFieldConstraintDof(s[i], p, fi, &cfdof));
        if (cfdof) PetscCall(PetscSectionAddFieldConstraintDof(*supers, p, f, cfdof));
        dof += fdof;
        cdof += cfdof;
      }
    }
    PetscCall(PetscSectionSetDof(*supers, p, dof));
    if (cdof) PetscCall(PetscSectionSetConstraintDof(*supers, p, cdof));
    maxCdof = PetscMax(cdof, maxCdof);
  }
  PetscCall(PetscSectionSetUp(*supers));
  if (maxCdof) {
    PetscInt *indices;

    PetscCall(PetscMalloc1(maxCdof, &indices));
    for (p = pStart; p < pEnd; ++p) {
      PetscInt cdof;

      PetscCall(PetscSectionGetConstraintDof(*supers, p, &cdof));
      if (cdof) {
        PetscInt dof, numConst = 0, fOff = 0;

        for (i = 0, f = 0; i < len; ++i) {
          const PetscInt *oldIndices = NULL;
          PetscInt        nf, fi, pStarti, pEndi, fdof, cfdof, fc;

          PetscCall(PetscSectionGetNumFields(s[i], &nf));
          PetscCall(PetscSectionGetChart(s[i], &pStarti, &pEndi));
          if ((p < pStarti) || (p >= pEndi)) continue;
          for (fi = 0; fi < nf; ++fi, ++f) {
            PetscCall(PetscSectionGetFieldDof(s[i], p, fi, &fdof));
            PetscCall(PetscSectionGetFieldConstraintDof(s[i], p, fi, &cfdof));
            PetscCall(PetscSectionGetFieldConstraintIndices(s[i], p, fi, &oldIndices));
            for (fc = 0; fc < cfdof; ++fc) indices[numConst + fc] = oldIndices[fc];
            PetscCall(PetscSectionSetFieldConstraintIndices(*supers, p, f, &indices[numConst]));
            for (fc = 0; fc < cfdof; ++fc) indices[numConst + fc] += fOff;
            numConst += cfdof;
          }
          PetscCall(PetscSectionGetDof(s[i], p, &dof));
          fOff += dof;
        }
        PetscCall(PetscSectionSetConstraintIndices(*supers, p, indices));
      }
    }
    PetscCall(PetscFree(indices));
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode PetscSectionCreateSubplexSection_Private(PetscSection s, IS subpointMap, PetscBool renumberPoints, PetscSection *subs)
{
  const PetscInt *points = NULL, *indices = NULL;
  PetscInt        numFields, f, c, numSubpoints = 0, pStart, pEnd, p, spStart, spEnd, subp;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(s, PETSC_SECTION_CLASSID, 1);
  PetscValidHeaderSpecific(subpointMap, IS_CLASSID, 2);
  PetscAssertPointer(subs, 4);
  PetscCall(PetscSectionGetNumFields(s, &numFields));
  PetscCall(PetscSectionCreate(PetscObjectComm((PetscObject)s), subs));
  if (numFields) PetscCall(PetscSectionSetNumFields(*subs, numFields));
  for (f = 0; f < numFields; ++f) {
    const char *name    = NULL;
    PetscInt    numComp = 0;

    PetscCall(PetscSectionGetFieldName(s, f, &name));
    PetscCall(PetscSectionSetFieldName(*subs, f, name));
    PetscCall(PetscSectionGetFieldComponents(s, f, &numComp));
    PetscCall(PetscSectionSetFieldComponents(*subs, f, numComp));
    for (c = 0; c < s->numFieldComponents[f]; ++c) {
      PetscCall(PetscSectionGetComponentName(s, f, c, &name));
      PetscCall(PetscSectionSetComponentName(*subs, f, c, name));
    }
  }
  /* For right now, we do not try to squeeze the subchart */
  if (subpointMap) {
    PetscCall(ISGetSize(subpointMap, &numSubpoints));
    PetscCall(ISGetIndices(subpointMap, &points));
  }
  PetscCall(PetscSectionGetChart(s, &pStart, &pEnd));
  if (renumberPoints) {
    spStart = 0;
    spEnd   = numSubpoints;
  } else {
    PetscCall(ISGetMinMax(subpointMap, &spStart, &spEnd));
    ++spEnd;
  }
  PetscCall(PetscSectionSetChart(*subs, spStart, spEnd));
  for (p = pStart; p < pEnd; ++p) {
    PetscInt dof, cdof, fdof = 0, cfdof = 0;

    PetscCall(PetscFindInt(p, numSubpoints, points, &subp));
    if (subp < 0) continue;
    if (!renumberPoints) subp = p;
    for (f = 0; f < numFields; ++f) {
      PetscCall(PetscSectionGetFieldDof(s, p, f, &fdof));
      PetscCall(PetscSectionSetFieldDof(*subs, subp, f, fdof));
      PetscCall(PetscSectionGetFieldConstraintDof(s, p, f, &cfdof));
      if (cfdof) PetscCall(PetscSectionSetFieldConstraintDof(*subs, subp, f, cfdof));
    }
    PetscCall(PetscSectionGetDof(s, p, &dof));
    PetscCall(PetscSectionSetDof(*subs, subp, dof));
    PetscCall(PetscSectionGetConstraintDof(s, p, &cdof));
    if (cdof) PetscCall(PetscSectionSetConstraintDof(*subs, subp, cdof));
  }
  PetscCall(PetscSectionSetUp(*subs));
  /* Change offsets to original offsets */
  for (p = pStart; p < pEnd; ++p) {
    PetscInt off, foff = 0;

    PetscCall(PetscFindInt(p, numSubpoints, points, &subp));
    if (subp < 0) continue;
    if (!renumberPoints) subp = p;
    for (f = 0; f < numFields; ++f) {
      PetscCall(PetscSectionGetFieldOffset(s, p, f, &foff));
      PetscCall(PetscSectionSetFieldOffset(*subs, subp, f, foff));
    }
    PetscCall(PetscSectionGetOffset(s, p, &off));
    PetscCall(PetscSectionSetOffset(*subs, subp, off));
  }
  /* Copy constraint indices */
  for (subp = spStart; subp < spEnd; ++subp) {
    PetscInt cdof;

    PetscCall(PetscSectionGetConstraintDof(*subs, subp, &cdof));
    if (cdof) {
      for (f = 0; f < numFields; ++f) {
        PetscCall(PetscSectionGetFieldConstraintIndices(s, points[subp - spStart], f, &indices));
        PetscCall(PetscSectionSetFieldConstraintIndices(*subs, subp, f, indices));
      }
      PetscCall(PetscSectionGetConstraintIndices(s, points[subp - spStart], &indices));
      PetscCall(PetscSectionSetConstraintIndices(*subs, subp, indices));
    }
  }
  if (subpointMap) PetscCall(ISRestoreIndices(subpointMap, &points));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PetscSectionCreateSubmeshSection - Create a new, smaller section with support on the submesh

  Collective

  Input Parameters:
+ s           - the `PetscSection`
- subpointMap - a sorted list of points in the original mesh which are in the submesh

  Output Parameter:
. subs - the subsection

  Level: advanced

  Notes:
  The points are renumbered from 0, and the section offsets now refer to a new, smaller vector. That is the chart of `subs` is `[0,sizeof(subpointmap))`

  Compare this with `PetscSectionCreateSubdomainSection()` that does not map the points numbers to start at zero but leaves them as before

  Developer Notes:
  The use of the term Submesh is confusing and needs clarification, it is not specific to meshes. It appears to be just a subset of the chart of the original `PetscSection`

.seealso: [PetscSection](sec_petscsection), `PetscSection`, `PetscSectionCreateSubdomainSection()`, `PetscSectionCreateSubsection()`, `DMPlexGetSubpointMap()`, `PetscSectionCreate()`
@*/
PetscErrorCode PetscSectionCreateSubmeshSection(PetscSection s, IS subpointMap, PetscSection *subs)
{
  PetscFunctionBegin;
  PetscCall(PetscSectionCreateSubplexSection_Private(s, subpointMap, PETSC_TRUE, subs));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PetscSectionCreateSubdomainSection - Create a new, smaller section with support on a subdomain of the mesh

  Collective

  Input Parameters:
+ s           - the `PetscSection`
- subpointMap - a sorted list of points in the original mesh which are in the subdomain

  Output Parameter:
. subs - the subsection

  Level: advanced

  Notes:
  The point numbers remain the same as in the larger `PetscSection`, but the section offsets now refer to a new, smaller vector. The chart of `subs`
  is `[min(subpointMap),max(subpointMap)+1)`

  Compare this with `PetscSectionCreateSubmeshSection()` that maps the point numbers to start at zero

  Developer Notes:
  The use of the term Subdomain is unneeded and needs clarification, it is not specific to meshes. It appears to be just a subset of the chart of the original `PetscSection`

.seealso: [PetscSection](sec_petscsection), `PetscSection`, `PetscSectionCreateSubmeshSection()`, `PetscSectionCreateSubsection()`, `DMPlexGetSubpointMap()`, `PetscSectionCreate()`
@*/
PetscErrorCode PetscSectionCreateSubdomainSection(PetscSection s, IS subpointMap, PetscSection *subs)
{
  PetscFunctionBegin;
  PetscCall(PetscSectionCreateSubplexSection_Private(s, subpointMap, PETSC_FALSE, subs));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode PetscSectionView_ASCII(PetscSection s, PetscViewer viewer)
{
  PetscInt    p;
  PetscMPIInt rank;

  PetscFunctionBegin;
  PetscCallMPI(MPI_Comm_rank(PetscObjectComm((PetscObject)viewer), &rank));
  PetscCall(PetscViewerASCIIPushSynchronized(viewer));
  PetscCall(PetscViewerASCIISynchronizedPrintf(viewer, "Process %d:\n", rank));
  for (p = 0; p < s->pEnd - s->pStart; ++p) {
    if (s->bc && s->bc->atlasDof[p] > 0) {
      PetscInt b;
      PetscCall(PetscViewerASCIISynchronizedPrintf(viewer, "  (%4" PetscInt_FMT ") dim %2" PetscInt_FMT " offset %3" PetscInt_FMT " constrained", p + s->pStart, s->atlasDof[p], s->atlasOff[p]));
      if (s->bcIndices) {
        for (b = 0; b < s->bc->atlasDof[p]; ++b) PetscCall(PetscViewerASCIISynchronizedPrintf(viewer, " %" PetscInt_FMT, s->bcIndices[s->bc->atlasOff[p] + b]));
      }
      PetscCall(PetscViewerASCIISynchronizedPrintf(viewer, "\n"));
    } else {
      PetscCall(PetscViewerASCIISynchronizedPrintf(viewer, "  (%4" PetscInt_FMT ") dim %2" PetscInt_FMT " offset %3" PetscInt_FMT "\n", p + s->pStart, s->atlasDof[p], s->atlasOff[p]));
    }
  }
  PetscCall(PetscViewerFlush(viewer));
  PetscCall(PetscViewerASCIIPopSynchronized(viewer));
  if (s->sym) {
    PetscCall(PetscViewerASCIIPushTab(viewer));
    PetscCall(PetscSectionSymView(s->sym, viewer));
    PetscCall(PetscViewerASCIIPopTab(viewer));
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@C
  PetscSectionViewFromOptions - View the `PetscSection` based on values in the options database

  Collective

  Input Parameters:
+ A    - the `PetscSection` object to view
. obj  - Optional object that provides the options prefix used for the options
- name - command line option

  Level: intermediate

  Note:
  See `PetscObjectViewFromOptions()` for available values of `PetscViewer` and `PetscViewerFormat`

.seealso: [PetscSection](sec_petscsection), `PetscSection`, `PetscSectionView`, `PetscObjectViewFromOptions()`, `PetscSectionCreate()`, `PetscSectionView()`
@*/
PetscErrorCode PetscSectionViewFromOptions(PetscSection A, PetscObject obj, const char name[])
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(A, PETSC_SECTION_CLASSID, 1);
  PetscCall(PetscObjectViewFromOptions((PetscObject)A, obj, name));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@C
  PetscSectionView - Views a `PetscSection`

  Collective

  Input Parameters:
+ s      - the `PetscSection` object to view
- viewer - the viewer

  Level: beginner

  Note:
  `PetscSectionView()`, when viewer is of type `PETSCVIEWERHDF5`, only saves
  distribution independent data, such as dofs, offsets, constraint dofs,
  and constraint indices. Points that have negative dofs, for instance,
  are not saved as they represent points owned by other processes.
  Point numbering and rank assignment is currently not stored.
  The saved section can be loaded with `PetscSectionLoad()`.

.seealso: [PetscSection](sec_petscsection), `PetscSection`, `PetscSectionCreate()`, `PetscSectionDestroy()`, `PetscSectionLoad()`, `PetscViewer`
@*/
PetscErrorCode PetscSectionView(PetscSection s, PetscViewer viewer)
{
  PetscBool isascii, ishdf5;
  PetscInt  f;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(s, PETSC_SECTION_CLASSID, 1);
  if (!viewer) PetscCall(PetscViewerASCIIGetStdout(PetscObjectComm((PetscObject)s), &viewer));
  PetscValidHeaderSpecific(viewer, PETSC_VIEWER_CLASSID, 2);
  PetscCall(PetscObjectTypeCompare((PetscObject)viewer, PETSCVIEWERASCII, &isascii));
  PetscCall(PetscObjectTypeCompare((PetscObject)viewer, PETSCVIEWERHDF5, &ishdf5));
  if (isascii) {
    PetscCall(PetscObjectPrintClassNamePrefixType((PetscObject)s, viewer));
    if (s->numFields) {
      PetscCall(PetscViewerASCIIPrintf(viewer, "%" PetscInt_FMT " fields\n", s->numFields));
      for (f = 0; f < s->numFields; ++f) {
        PetscCall(PetscViewerASCIIPrintf(viewer, "  field %" PetscInt_FMT " with %" PetscInt_FMT " components\n", f, s->numFieldComponents[f]));
        PetscCall(PetscSectionView_ASCII(s->field[f], viewer));
      }
    } else {
      PetscCall(PetscSectionView_ASCII(s, viewer));
    }
  } else if (ishdf5) {
#if PetscDefined(HAVE_HDF5)
    PetscCall(PetscSectionView_HDF5_Internal(s, viewer));
#else
    SETERRQ(PetscObjectComm((PetscObject)s), PETSC_ERR_SUP, "HDF5 not supported in this build.\nPlease reconfigure using --download-hdf5");
#endif
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@C
  PetscSectionLoad - Loads a `PetscSection`

  Collective

  Input Parameters:
+ s      - the `PetscSection` object to load
- viewer - the viewer

  Level: beginner

  Note:
  `PetscSectionLoad()`, when viewer is of type `PETSCVIEWERHDF5`, loads
  a section saved with `PetscSectionView()`. The number of processes
  used here (N) does not need to be the same as that used when saving.
  After calling this function, the chart of s on rank i will be set
  to [0, E_i), where \sum_{i=0}^{N-1}E_i equals to the total number of
  saved section points.

.seealso: [PetscSection](sec_petscsection), `PetscSection`, `PetscSectionCreate()`, `PetscSectionDestroy()`, `PetscSectionView()`
@*/
PetscErrorCode PetscSectionLoad(PetscSection s, PetscViewer viewer)
{
  PetscBool ishdf5;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(s, PETSC_SECTION_CLASSID, 1);
  PetscValidHeaderSpecific(viewer, PETSC_VIEWER_CLASSID, 2);
  PetscCall(PetscObjectTypeCompare((PetscObject)viewer, PETSCVIEWERHDF5, &ishdf5));
  if (ishdf5) {
#if PetscDefined(HAVE_HDF5)
    PetscCall(PetscSectionLoad_HDF5_Internal(s, viewer));
    PetscFunctionReturn(PETSC_SUCCESS);
#else
    SETERRQ(PetscObjectComm((PetscObject)s), PETSC_ERR_SUP, "HDF5 not supported in this build.\nPlease reconfigure using --download-hdf5");
#endif
  } else SETERRQ(PetscObjectComm((PetscObject)s), PETSC_ERR_SUP, "Viewer type %s not yet supported for PetscSection loading", ((PetscObject)viewer)->type_name);
}

/*@
  PetscSectionResetClosurePermutation - Remove any existing closure permutation

  Input Parameter:
. section - The `PetscSection`

  Level: intermediate

.seealso: `PetscSectionSetClosurePermutation()`, `PetscSectionSetClosureIndex()`, `PetscSectionReset()`
@*/
PetscErrorCode PetscSectionResetClosurePermutation(PetscSection section)
{
  PetscSectionClosurePermVal clVal;

  PetscFunctionBegin;
  if (!section->clHash) PetscFunctionReturn(PETSC_SUCCESS);
  kh_foreach_value(section->clHash, clVal, {
    PetscCall(PetscFree(clVal.perm));
    PetscCall(PetscFree(clVal.invPerm));
  });
  kh_destroy(ClPerm, section->clHash);
  section->clHash = NULL;
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PetscSectionReset - Frees all section data, the section is then as if `PetscSectionCreate()` had just been called.

  Not Collective

  Input Parameter:
. s - the `PetscSection`

  Level: beginner

.seealso: [PetscSection](sec_petscsection), `PetscSection`, `PetscSectionCreate()`
@*/
PetscErrorCode PetscSectionReset(PetscSection s)
{
  PetscInt f, c;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(s, PETSC_SECTION_CLASSID, 1);
  for (f = 0; f < s->numFields; ++f) {
    PetscCall(PetscSectionDestroy(&s->field[f]));
    PetscCall(PetscFree(s->fieldNames[f]));
    for (c = 0; c < s->numFieldComponents[f]; ++c) PetscCall(PetscFree(s->compNames[f][c]));
    PetscCall(PetscFree(s->compNames[f]));
  }
  PetscCall(PetscFree(s->numFieldComponents));
  PetscCall(PetscFree(s->fieldNames));
  PetscCall(PetscFree(s->compNames));
  PetscCall(PetscFree(s->field));
  PetscCall(PetscSectionDestroy(&s->bc));
  PetscCall(PetscFree(s->bcIndices));
  PetscCall(PetscFree2(s->atlasDof, s->atlasOff));
  PetscCall(PetscSectionDestroy(&s->clSection));
  PetscCall(ISDestroy(&s->clPoints));
  PetscCall(ISDestroy(&s->perm));
  PetscCall(PetscBTDestroy(&s->blockStarts));
  PetscCall(PetscSectionResetClosurePermutation(s));
  PetscCall(PetscSectionSymDestroy(&s->sym));
  PetscCall(PetscSectionDestroy(&s->clSection));
  PetscCall(ISDestroy(&s->clPoints));
  PetscCall(PetscSectionInvalidateMaxDof_Internal(s));
  s->pStart    = -1;
  s->pEnd      = -1;
  s->maxDof    = 0;
  s->setup     = PETSC_FALSE;
  s->numFields = 0;
  s->clObj     = NULL;
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PetscSectionDestroy - Frees a `PetscSection`

  Not Collective

  Input Parameter:
. s - the `PetscSection`

  Level: beginner

.seealso: [PetscSection](sec_petscsection), `PetscSection`, `PetscSectionCreate()`, `PetscSectionReset()`
@*/
PetscErrorCode PetscSectionDestroy(PetscSection *s)
{
  PetscFunctionBegin;
  if (!*s) PetscFunctionReturn(PETSC_SUCCESS);
  PetscValidHeaderSpecific(*s, PETSC_SECTION_CLASSID, 1);
  if (--((PetscObject)(*s))->refct > 0) {
    *s = NULL;
    PetscFunctionReturn(PETSC_SUCCESS);
  }
  PetscCall(PetscSectionReset(*s));
  PetscCall(PetscHeaderDestroy(s));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode VecIntGetValuesSection_Private(const PetscInt *baseArray, PetscSection s, PetscInt point, const PetscInt **values)
{
  const PetscInt p = point - s->pStart;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(s, PETSC_SECTION_CLASSID, 2);
  *values = &baseArray[s->atlasOff[p]];
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode VecIntSetValuesSection_Private(PetscInt *baseArray, PetscSection s, PetscInt point, const PetscInt values[], InsertMode mode)
{
  PetscInt      *array;
  const PetscInt p           = point - s->pStart;
  const PetscInt orientation = 0; /* Needs to be included for use in closure operations */
  PetscInt       cDim        = 0;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(s, PETSC_SECTION_CLASSID, 2);
  PetscCall(PetscSectionGetConstraintDof(s, p, &cDim));
  array = &baseArray[s->atlasOff[p]];
  if (!cDim) {
    if (orientation >= 0) {
      const PetscInt dim = s->atlasDof[p];
      PetscInt       i;

      if (mode == INSERT_VALUES) {
        for (i = 0; i < dim; ++i) array[i] = values ? values[i] : i;
      } else {
        for (i = 0; i < dim; ++i) array[i] += values[i];
      }
    } else {
      PetscInt offset = 0;
      PetscInt j      = -1, field, i;

      for (field = 0; field < s->numFields; ++field) {
        const PetscInt dim = s->field[field]->atlasDof[p];

        for (i = dim - 1; i >= 0; --i) array[++j] = values ? values[i + offset] : i + offset;
        offset += dim;
      }
    }
  } else {
    if (orientation >= 0) {
      const PetscInt  dim  = s->atlasDof[p];
      PetscInt        cInd = 0, i;
      const PetscInt *cDof;

      PetscCall(PetscSectionGetConstraintIndices(s, point, &cDof));
      if (mode == INSERT_VALUES) {
        for (i = 0; i < dim; ++i) {
          if ((cInd < cDim) && (i == cDof[cInd])) {
            ++cInd;
            continue;
          }
          array[i] = values ? values[i] : i;
        }
      } else {
        for (i = 0; i < dim; ++i) {
          if ((cInd < cDim) && (i == cDof[cInd])) {
            ++cInd;
            continue;
          }
          array[i] += values[i];
        }
      }
    } else {
      const PetscInt *cDof;
      PetscInt        offset  = 0;
      PetscInt        cOffset = 0;
      PetscInt        j       = 0, field;

      PetscCall(PetscSectionGetConstraintIndices(s, point, &cDof));
      for (field = 0; field < s->numFields; ++field) {
        const PetscInt dim  = s->field[field]->atlasDof[p];     /* PetscSectionGetFieldDof() */
        const PetscInt tDim = s->field[field]->bc->atlasDof[p]; /* PetscSectionGetFieldConstraintDof() */
        const PetscInt sDim = dim - tDim;
        PetscInt       cInd = 0, i, k;

        for (i = 0, k = dim + offset - 1; i < dim; ++i, ++j, --k) {
          if ((cInd < sDim) && (j == cDof[cInd + cOffset])) {
            ++cInd;
            continue;
          }
          array[j] = values ? values[k] : k;
        }
        offset += dim;
        cOffset += dim - tDim;
      }
    }
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@C
  PetscSectionHasConstraints - Determine whether a `PetscSection` has constrained dofs

  Not Collective

  Input Parameter:
. s - The `PetscSection`

  Output Parameter:
. hasConstraints - flag indicating that the section has constrained dofs

  Level: intermediate

.seealso: [PetscSection](sec_petscsection), `PetscSectionSetConstraintIndices()`, `PetscSectionGetConstraintDof()`, `PetscSection`
@*/
PetscErrorCode PetscSectionHasConstraints(PetscSection s, PetscBool *hasConstraints)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(s, PETSC_SECTION_CLASSID, 1);
  PetscAssertPointer(hasConstraints, 2);
  *hasConstraints = s->bc ? PETSC_TRUE : PETSC_FALSE;
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@C
  PetscSectionGetConstraintIndices - Get the point dof numbers, in [0, dof), which are constrained for a given point

  Not Collective

  Input Parameters:
+ s     - The `PetscSection`
- point - The point

  Output Parameter:
. indices - The constrained dofs

  Level: intermediate

  Fortran Notes:
  Use `PetscSectionGetConstraintIndicesF90()` and `PetscSectionRestoreConstraintIndicesF90()`

.seealso: [PetscSection](sec_petscsection), `PetscSectionSetConstraintIndices()`, `PetscSectionGetConstraintDof()`, `PetscSection`
@*/
PetscErrorCode PetscSectionGetConstraintIndices(PetscSection s, PetscInt point, const PetscInt **indices)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(s, PETSC_SECTION_CLASSID, 1);
  if (s->bc) {
    PetscCall(VecIntGetValuesSection_Private(s->bcIndices, s->bc, point, indices));
  } else *indices = NULL;
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@C
  PetscSectionSetConstraintIndices - Set the point dof numbers, in [0, dof), which are constrained

  Not Collective

  Input Parameters:
+ s       - The `PetscSection`
. point   - The point
- indices - The constrained dofs

  Level: intermediate

  Fortran Notes:
  Use `PetscSectionSetConstraintIndicesF90()`

.seealso: [PetscSection](sec_petscsection), `PetscSectionGetConstraintIndices()`, `PetscSectionGetConstraintDof()`, `PetscSection`
@*/
PetscErrorCode PetscSectionSetConstraintIndices(PetscSection s, PetscInt point, const PetscInt indices[])
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(s, PETSC_SECTION_CLASSID, 1);
  if (s->bc) {
    const PetscInt dof  = s->atlasDof[point];
    const PetscInt cdof = s->bc->atlasDof[point];
    PetscInt       d;

    if (indices)
      for (d = 0; d < cdof; ++d) PetscCheck(indices[d] < dof, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Point %" PetscInt_FMT " dof %" PetscInt_FMT ", invalid constraint index[%" PetscInt_FMT "]: %" PetscInt_FMT, point, dof, d, indices[d]);
    PetscCall(VecIntSetValuesSection_Private(s->bcIndices, s->bc, point, indices, INSERT_VALUES));
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@C
  PetscSectionGetFieldConstraintIndices - Get the field dof numbers, in [0, fdof), which are constrained

  Not Collective

  Input Parameters:
+ s     - The `PetscSection`
. field - The field number
- point - The point

  Output Parameter:
. indices - The constrained dofs sorted in ascending order

  Level: intermediate

  Note:
  The indices array, which is provided by the caller, must have capacity to hold the number of constrained dofs, e.g., as returned by `PetscSectionGetConstraintDof()`.

  Fortran Notes:
  Use `PetscSectionGetFieldConstraintIndicesF90()` and `PetscSectionRestoreFieldConstraintIndicesF90()`

.seealso: [PetscSection](sec_petscsection), `PetscSectionSetFieldConstraintIndices()`, `PetscSectionGetConstraintIndices()`, `PetscSectionGetConstraintDof()`, `PetscSection`
@*/
PetscErrorCode PetscSectionGetFieldConstraintIndices(PetscSection s, PetscInt point, PetscInt field, const PetscInt **indices)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(s, PETSC_SECTION_CLASSID, 1);
  PetscAssertPointer(indices, 4);
  PetscSectionCheckValidField(field, s->numFields);
  PetscCall(PetscSectionGetConstraintIndices(s->field[field], point, indices));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@C
  PetscSectionSetFieldConstraintIndices - Set the field dof numbers, in [0, fdof), which are constrained

  Not Collective

  Input Parameters:
+ s       - The `PetscSection`
. point   - The point
. field   - The field number
- indices - The constrained dofs

  Level: intermediate

  Fortran Notes:
  Use `PetscSectionSetFieldConstraintIndicesF90()`

.seealso: [PetscSection](sec_petscsection), `PetscSectionSetConstraintIndices()`, `PetscSectionGetFieldConstraintIndices()`, `PetscSectionGetConstraintDof()`, `PetscSection`
@*/
PetscErrorCode PetscSectionSetFieldConstraintIndices(PetscSection s, PetscInt point, PetscInt field, const PetscInt indices[])
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(s, PETSC_SECTION_CLASSID, 1);
  PetscSectionCheckValidField(field, s->numFields);
  PetscCall(PetscSectionSetConstraintIndices(s->field[field], point, indices));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PetscSectionPermute - Reorder the section according to the input point permutation

  Collective

  Input Parameters:
+ section     - The `PetscSection` object
- permutation - The point permutation, old point p becomes new point perm[p]

  Output Parameter:
. sectionNew - The permuted `PetscSection`

  Level: intermediate

  Note:
  The data and the access to the data via `PetscSectionGetFieldOffset()` and `PetscSectionGetOffset()` are both changed in `sectionNew`

  Compare to `PetscSectionSetPermutation()`

.seealso: [PetscSection](sec_petscsection), `IS`, `PetscSection`, `MatPermute()`, `PetscSectionSetPermutation()`
@*/
PetscErrorCode PetscSectionPermute(PetscSection section, IS permutation, PetscSection *sectionNew)
{
  PetscSection    s = section, sNew;
  const PetscInt *perm;
  PetscInt        numFields, f, c, numPoints, pStart, pEnd, p;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(section, PETSC_SECTION_CLASSID, 1);
  PetscValidHeaderSpecific(permutation, IS_CLASSID, 2);
  PetscAssertPointer(sectionNew, 3);
  PetscCall(PetscSectionCreate(PetscObjectComm((PetscObject)s), &sNew));
  PetscCall(PetscSectionGetNumFields(s, &numFields));
  if (numFields) PetscCall(PetscSectionSetNumFields(sNew, numFields));
  for (f = 0; f < numFields; ++f) {
    const char *name;
    PetscInt    numComp;

    PetscCall(PetscSectionGetFieldName(s, f, &name));
    PetscCall(PetscSectionSetFieldName(sNew, f, name));
    PetscCall(PetscSectionGetFieldComponents(s, f, &numComp));
    PetscCall(PetscSectionSetFieldComponents(sNew, f, numComp));
    for (c = 0; c < s->numFieldComponents[f]; ++c) {
      PetscCall(PetscSectionGetComponentName(s, f, c, &name));
      PetscCall(PetscSectionSetComponentName(sNew, f, c, name));
    }
  }
  PetscCall(ISGetLocalSize(permutation, &numPoints));
  PetscCall(ISGetIndices(permutation, &perm));
  PetscCall(PetscSectionGetChart(s, &pStart, &pEnd));
  PetscCall(PetscSectionSetChart(sNew, pStart, pEnd));
  PetscCheck(numPoints >= pEnd, PETSC_COMM_SELF, PETSC_ERR_ARG_SIZ, "Permutation size %" PetscInt_FMT " is less than largest Section point %" PetscInt_FMT, numPoints, pEnd);
  for (p = pStart; p < pEnd; ++p) {
    PetscInt dof, cdof;

    PetscCall(PetscSectionGetDof(s, p, &dof));
    PetscCall(PetscSectionSetDof(sNew, perm[p], dof));
    PetscCall(PetscSectionGetConstraintDof(s, p, &cdof));
    if (cdof) PetscCall(PetscSectionSetConstraintDof(sNew, perm[p], cdof));
    for (f = 0; f < numFields; ++f) {
      PetscCall(PetscSectionGetFieldDof(s, p, f, &dof));
      PetscCall(PetscSectionSetFieldDof(sNew, perm[p], f, dof));
      PetscCall(PetscSectionGetFieldConstraintDof(s, p, f, &cdof));
      if (cdof) PetscCall(PetscSectionSetFieldConstraintDof(sNew, perm[p], f, cdof));
    }
  }
  PetscCall(PetscSectionSetUp(sNew));
  for (p = pStart; p < pEnd; ++p) {
    const PetscInt *cind;
    PetscInt        cdof;

    PetscCall(PetscSectionGetConstraintDof(s, p, &cdof));
    if (cdof) {
      PetscCall(PetscSectionGetConstraintIndices(s, p, &cind));
      PetscCall(PetscSectionSetConstraintIndices(sNew, perm[p], cind));
    }
    for (f = 0; f < numFields; ++f) {
      PetscCall(PetscSectionGetFieldConstraintDof(s, p, f, &cdof));
      if (cdof) {
        PetscCall(PetscSectionGetFieldConstraintIndices(s, p, f, &cind));
        PetscCall(PetscSectionSetFieldConstraintIndices(sNew, perm[p], f, cind));
      }
    }
  }
  PetscCall(ISRestoreIndices(permutation, &perm));
  *sectionNew = sNew;
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PetscSectionSetClosureIndex - Create an internal data structure to speed up closure queries.

  Collective

  Input Parameters:
+ section   - The `PetscSection`
. obj       - A `PetscObject` which serves as the key for this index
. clSection - `PetscSection` giving the size of the closure of each point
- clPoints  - `IS` giving the points in each closure

  Level: advanced

  Note:
  This function creates an internal map from each point to its closure. We compress out closure points with no dofs in this section.

  Developer Notes:
  The information provided here is completely opaque

.seealso: [PetscSection](sec_petscsection), `PetscSection`, `PetscSectionGetClosureIndex()`, `DMPlexCreateClosureIndex()`
@*/
PetscErrorCode PetscSectionSetClosureIndex(PetscSection section, PetscObject obj, PetscSection clSection, IS clPoints)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(section, PETSC_SECTION_CLASSID, 1);
  PetscValidHeaderSpecific(clSection, PETSC_SECTION_CLASSID, 3);
  PetscValidHeaderSpecific(clPoints, IS_CLASSID, 4);
  if (section->clObj != obj) PetscCall(PetscSectionResetClosurePermutation(section));
  section->clObj = obj;
  PetscCall(PetscObjectReference((PetscObject)clSection));
  PetscCall(PetscObjectReference((PetscObject)clPoints));
  PetscCall(PetscSectionDestroy(&section->clSection));
  PetscCall(ISDestroy(&section->clPoints));
  section->clSection = clSection;
  section->clPoints  = clPoints;
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PetscSectionGetClosureIndex - Get the cache of points in the closure of each point in the section set with `PetscSectionSetClosureIndex()`

  Collective

  Input Parameters:
+ section - The `PetscSection`
- obj     - A `PetscObject` which serves as the key for this index

  Output Parameters:
+ clSection - `PetscSection` giving the size of the closure of each point
- clPoints  - `IS` giving the points in each closure

  Level: advanced

.seealso: [PetscSection](sec_petscsection), `PetscSectionSetClosureIndex()`, `DMPlexCreateClosureIndex()`
@*/
PetscErrorCode PetscSectionGetClosureIndex(PetscSection section, PetscObject obj, PetscSection *clSection, IS *clPoints)
{
  PetscFunctionBegin;
  if (section->clObj == obj) {
    if (clSection) *clSection = section->clSection;
    if (clPoints) *clPoints = section->clPoints;
  } else {
    if (clSection) *clSection = NULL;
    if (clPoints) *clPoints = NULL;
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode PetscSectionSetClosurePermutation_Internal(PetscSection section, PetscObject obj, PetscInt depth, PetscInt clSize, PetscCopyMode mode, PetscInt *clPerm)
{
  PetscInt                    i;
  khiter_t                    iter;
  int                         new_entry;
  PetscSectionClosurePermKey  key = {depth, clSize};
  PetscSectionClosurePermVal *val;

  PetscFunctionBegin;
  if (section->clObj != obj) {
    PetscCall(PetscSectionDestroy(&section->clSection));
    PetscCall(ISDestroy(&section->clPoints));
  }
  section->clObj = obj;
  if (!section->clHash) PetscCall(PetscClPermCreate(&section->clHash));
  iter = kh_put(ClPerm, section->clHash, key, &new_entry);
  val  = &kh_val(section->clHash, iter);
  if (!new_entry) {
    PetscCall(PetscFree(val->perm));
    PetscCall(PetscFree(val->invPerm));
  }
  if (mode == PETSC_COPY_VALUES) {
    PetscCall(PetscMalloc1(clSize, &val->perm));
    PetscCall(PetscArraycpy(val->perm, clPerm, clSize));
  } else if (mode == PETSC_OWN_POINTER) {
    val->perm = clPerm;
  } else SETERRQ(PetscObjectComm(obj), PETSC_ERR_SUP, "Do not support borrowed arrays");
  PetscCall(PetscMalloc1(clSize, &val->invPerm));
  for (i = 0; i < clSize; ++i) val->invPerm[clPerm[i]] = i;
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PetscSectionSetClosurePermutation - Set the dof permutation for the closure of each cell in the section, meaning clPerm[newIndex] = oldIndex.

  Not Collective

  Input Parameters:
+ section - The `PetscSection`
. obj     - A `PetscObject` which serves as the key for this index (usually a `DM`)
. depth   - Depth of points on which to apply the given permutation
- perm    - Permutation of the cell dof closure

  Level: intermediate

  Notes:
  The specified permutation will only be applied to points at depth whose closure size matches the length of perm.  In a
  mixed-topology or variable-degree finite element space, this function can be called multiple times at each depth for
  each topology and degree.

  This approach assumes that (depth, len(perm)) uniquely identifies the desired permutation; this might not be true for
  exotic/enriched spaces on mixed topology meshes.

.seealso: [PetscSection](sec_petscsection), `PetscSection`, `IS`, `PetscSectionGetClosurePermutation()`, `PetscSectionGetClosureIndex()`, `DMPlexCreateClosureIndex()`, `PetscCopyMode`
@*/
PetscErrorCode PetscSectionSetClosurePermutation(PetscSection section, PetscObject obj, PetscInt depth, IS perm)
{
  const PetscInt *clPerm = NULL;
  PetscInt        clSize = 0;

  PetscFunctionBegin;
  if (perm) {
    PetscCall(ISGetLocalSize(perm, &clSize));
    PetscCall(ISGetIndices(perm, &clPerm));
  }
  PetscCall(PetscSectionSetClosurePermutation_Internal(section, obj, depth, clSize, PETSC_COPY_VALUES, (PetscInt *)clPerm));
  if (perm) PetscCall(ISRestoreIndices(perm, &clPerm));
  PetscFunctionReturn(PETSC_SUCCESS);
}

static PetscErrorCode PetscSectionGetClosurePermutation_Private(PetscSection section, PetscObject obj, PetscInt depth, PetscInt size, const PetscInt *perm[])
{
  PetscFunctionBegin;
  if (section->clObj == obj) {
    PetscSectionClosurePermKey k = {depth, size};
    PetscSectionClosurePermVal v;

    PetscCall(PetscClPermGet(section->clHash, k, &v));
    if (perm) *perm = v.perm;
  } else {
    if (perm) *perm = NULL;
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PetscSectionGetClosurePermutation - Get the dof permutation for the closure of each cell in the section, meaning clPerm[newIndex] = oldIndex.

  Not Collective

  Input Parameters:
+ section - The `PetscSection`
. obj     - A `PetscObject` which serves as the key for this index (usually a DM)
. depth   - Depth stratum on which to obtain closure permutation
- clSize  - Closure size to be permuted (e.g., may vary with element topology and degree)

  Output Parameter:
. perm - The dof closure permutation

  Level: intermediate

  Note:
  The user must destroy the `IS` that is returned.

.seealso: [PetscSection](sec_petscsection), `PetscSection`, `IS`, `PetscSectionSetClosurePermutation()`, `PetscSectionGetClosureInversePermutation()`, `PetscSectionGetClosureIndex()`, `PetscSectionSetClosureIndex()`, `DMPlexCreateClosureIndex()`
@*/
PetscErrorCode PetscSectionGetClosurePermutation(PetscSection section, PetscObject obj, PetscInt depth, PetscInt clSize, IS *perm)
{
  const PetscInt *clPerm = NULL;

  PetscFunctionBegin;
  PetscCall(PetscSectionGetClosurePermutation_Private(section, obj, depth, clSize, &clPerm));
  PetscCheck(clPerm, PetscObjectComm((PetscObject)obj), PETSC_ERR_ARG_WRONG, "There is no closure permutation associated with this object for depth %" PetscInt_FMT " of size %" PetscInt_FMT, depth, clSize);
  PetscCall(ISCreateGeneral(PETSC_COMM_SELF, clSize, clPerm, PETSC_USE_POINTER, perm));
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscErrorCode PetscSectionGetClosureInversePermutation_Internal(PetscSection section, PetscObject obj, PetscInt depth, PetscInt size, const PetscInt *perm[])
{
  PetscFunctionBegin;
  if (section->clObj == obj && section->clHash) {
    PetscSectionClosurePermKey k = {depth, size};
    PetscSectionClosurePermVal v;
    PetscCall(PetscClPermGet(section->clHash, k, &v));
    if (perm) *perm = v.invPerm;
  } else {
    if (perm) *perm = NULL;
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PetscSectionGetClosureInversePermutation - Get the inverse dof permutation for the closure of each cell in the section, meaning clPerm[oldIndex] = newIndex.

  Not Collective

  Input Parameters:
+ section - The `PetscSection`
. obj     - A `PetscObject` which serves as the key for this index (usually a `DM`)
. depth   - Depth stratum on which to obtain closure permutation
- clSize  - Closure size to be permuted (e.g., may vary with element topology and degree)

  Output Parameter:
. perm - The dof closure permutation

  Level: intermediate

  Note:
  The user must destroy the `IS` that is returned.

.seealso: [PetscSection](sec_petscsection), `PetscSection`, `IS`, `PetscSectionSetClosurePermutation()`, `PetscSectionGetClosureIndex()`, `PetscSectionSetClosureIndex()`, `DMPlexCreateClosureIndex()`
@*/
PetscErrorCode PetscSectionGetClosureInversePermutation(PetscSection section, PetscObject obj, PetscInt depth, PetscInt clSize, IS *perm)
{
  const PetscInt *clPerm = NULL;

  PetscFunctionBegin;
  PetscCall(PetscSectionGetClosureInversePermutation_Internal(section, obj, depth, clSize, &clPerm));
  PetscCall(ISCreateGeneral(PETSC_COMM_SELF, clSize, clPerm, PETSC_USE_POINTER, perm));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PetscSectionGetField - Get the `PetscSection` associated with a single field

  Input Parameters:
+ s     - The `PetscSection`
- field - The field number

  Output Parameter:
. subs - The `PetscSection` for the given field, note the chart of `subs` is not set

  Level: intermediate

  Note:
  Does not increase the reference count of the selected sub-section. There is no matching `PetscSectionRestoreField()`

.seealso: [PetscSection](sec_petscsection), `PetscSection`, `IS`, `PetscSectionSetNumFields()`
@*/
PetscErrorCode PetscSectionGetField(PetscSection s, PetscInt field, PetscSection *subs)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(s, PETSC_SECTION_CLASSID, 1);
  PetscAssertPointer(subs, 3);
  PetscSectionCheckValidField(field, s->numFields);
  *subs = s->field[field];
  PetscFunctionReturn(PETSC_SUCCESS);
}

PetscClassId      PETSC_SECTION_SYM_CLASSID;
PetscFunctionList PetscSectionSymList = NULL;

/*@
  PetscSectionSymCreate - Creates an empty `PetscSectionSym` object.

  Collective

  Input Parameter:
. comm - the MPI communicator

  Output Parameter:
. sym - pointer to the new set of symmetries

  Level: developer

.seealso: [PetscSection](sec_petscsection), `PetscSection`, `PetscSectionSym`, `PetscSectionSymDestroy()`
@*/
PetscErrorCode PetscSectionSymCreate(MPI_Comm comm, PetscSectionSym *sym)
{
  PetscFunctionBegin;
  PetscAssertPointer(sym, 2);
  PetscCall(ISInitializePackage());
  PetscCall(PetscHeaderCreate(*sym, PETSC_SECTION_SYM_CLASSID, "PetscSectionSym", "Section Symmetry", "IS", comm, PetscSectionSymDestroy, PetscSectionSymView));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@C
  PetscSectionSymSetType - Builds a `PetscSectionSym`, for a particular implementation.

  Collective

  Input Parameters:
+ sym    - The section symmetry object
- method - The name of the section symmetry type

  Level: developer

.seealso: [PetscSection](sec_petscsection), `PetscSectionSym`, `PetscSectionSymType`, `PetscSectionSymGetType()`, `PetscSectionSymCreate()`
@*/
PetscErrorCode PetscSectionSymSetType(PetscSectionSym sym, PetscSectionSymType method)
{
  PetscErrorCode (*r)(PetscSectionSym);
  PetscBool match;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(sym, PETSC_SECTION_SYM_CLASSID, 1);
  PetscCall(PetscObjectTypeCompare((PetscObject)sym, method, &match));
  if (match) PetscFunctionReturn(PETSC_SUCCESS);

  PetscCall(PetscFunctionListFind(PetscSectionSymList, method, &r));
  PetscCheck(r, PetscObjectComm((PetscObject)sym), PETSC_ERR_ARG_UNKNOWN_TYPE, "Unknown PetscSectionSym type: %s", method);
  PetscTryTypeMethod(sym, destroy);
  sym->ops->destroy = NULL;

  PetscCall((*r)(sym));
  PetscCall(PetscObjectChangeTypeName((PetscObject)sym, method));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@C
  PetscSectionSymGetType - Gets the section symmetry type name (as a string) from the `PetscSectionSym`.

  Not Collective

  Input Parameter:
. sym - The section symmetry

  Output Parameter:
. type - The index set type name

  Level: developer

.seealso: [PetscSection](sec_petscsection), `PetscSectionSym`, `PetscSectionSymType`, `PetscSectionSymSetType()`, `PetscSectionSymCreate()`
@*/
PetscErrorCode PetscSectionSymGetType(PetscSectionSym sym, PetscSectionSymType *type)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(sym, PETSC_SECTION_SYM_CLASSID, 1);
  PetscAssertPointer(type, 2);
  *type = ((PetscObject)sym)->type_name;
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@C
  PetscSectionSymRegister - Registers a new section symmetry implementation

  Not Collective

  Input Parameters:
+ sname    - The name of a new user-defined creation routine
- function - The creation routine itself

  Level: developer

  Notes:
  `PetscSectionSymRegister()` may be called multiple times to add several user-defined vectors

.seealso: [PetscSection](sec_petscsection), `PetscSectionSym`, `PetscSectionSymType`, `PetscSectionSymCreate()`, `PetscSectionSymSetType()`
@*/
PetscErrorCode PetscSectionSymRegister(const char sname[], PetscErrorCode (*function)(PetscSectionSym))
{
  PetscFunctionBegin;
  PetscCall(ISInitializePackage());
  PetscCall(PetscFunctionListAdd(&PetscSectionSymList, sname, function));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PetscSectionSymDestroy - Destroys a section symmetry.

  Collective

  Input Parameter:
. sym - the section symmetry

  Level: developer

.seealso: [PetscSection](sec_petscsection), `PetscSectionSym`, `PetscSectionSymCreate()`
@*/
PetscErrorCode PetscSectionSymDestroy(PetscSectionSym *sym)
{
  SymWorkLink link, next;

  PetscFunctionBegin;
  if (!*sym) PetscFunctionReturn(PETSC_SUCCESS);
  PetscValidHeaderSpecific((*sym), PETSC_SECTION_SYM_CLASSID, 1);
  if (--((PetscObject)(*sym))->refct > 0) {
    *sym = NULL;
    PetscFunctionReturn(PETSC_SUCCESS);
  }
  if ((*sym)->ops->destroy) PetscCall((*(*sym)->ops->destroy)(*sym));
  PetscCheck(!(*sym)->workout, PETSC_COMM_SELF, PETSC_ERR_ARG_WRONGSTATE, "Work array still checked out");
  for (link = (*sym)->workin; link; link = next) {
    PetscInt    **perms = (PetscInt **)link->perms;
    PetscScalar **rots  = (PetscScalar **)link->rots;
    PetscCall(PetscFree2(perms, rots));
    next = link->next;
    PetscCall(PetscFree(link));
  }
  (*sym)->workin = NULL;
  PetscCall(PetscHeaderDestroy(sym));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@C
  PetscSectionSymView - Displays a section symmetry

  Collective

  Input Parameters:
+ sym    - the index set
- viewer - viewer used to display the set, for example `PETSC_VIEWER_STDOUT_SELF`.

  Level: developer

.seealso: `PetscSectionSym`, `PetscViewer`, `PetscViewerASCIIOpen()`
@*/
PetscErrorCode PetscSectionSymView(PetscSectionSym sym, PetscViewer viewer)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(sym, PETSC_SECTION_SYM_CLASSID, 1);
  if (!viewer) PetscCall(PetscViewerASCIIGetStdout(PetscObjectComm((PetscObject)sym), &viewer));
  PetscValidHeaderSpecific(viewer, PETSC_VIEWER_CLASSID, 2);
  PetscCheckSameComm(sym, 1, viewer, 2);
  PetscCall(PetscObjectPrintClassNamePrefixType((PetscObject)sym, viewer));
  PetscTryTypeMethod(sym, view, viewer);
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PetscSectionSetSym - Set the symmetries for the data referred to by the section

  Collective

  Input Parameters:
+ section - the section describing data layout
- sym     - the symmetry describing the affect of orientation on the access of the data

  Level: developer

.seealso: [PetscSection](sec_petscsection), `PetscSectionSym`, `PetscSectionGetSym()`, `PetscSectionSymCreate()`
@*/
PetscErrorCode PetscSectionSetSym(PetscSection section, PetscSectionSym sym)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(section, PETSC_SECTION_CLASSID, 1);
  PetscCall(PetscSectionSymDestroy(&(section->sym)));
  if (sym) {
    PetscValidHeaderSpecific(sym, PETSC_SECTION_SYM_CLASSID, 2);
    PetscCheckSameComm(section, 1, sym, 2);
    PetscCall(PetscObjectReference((PetscObject)sym));
  }
  section->sym = sym;
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PetscSectionGetSym - Get the symmetries for the data referred to by the section

  Not Collective

  Input Parameter:
. section - the section describing data layout

  Output Parameter:
. sym - the symmetry describing the affect of orientation on the access of the data, provided previously by `PetscSectionSetSym()`

  Level: developer

.seealso: [PetscSection](sec_petscsection), `PetscSectionSym`, `PetscSectionSetSym()`, `PetscSectionSymCreate()`
@*/
PetscErrorCode PetscSectionGetSym(PetscSection section, PetscSectionSym *sym)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(section, PETSC_SECTION_CLASSID, 1);
  *sym = section->sym;
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PetscSectionSetFieldSym - Set the symmetries for the data referred to by a field of the section

  Collective

  Input Parameters:
+ section - the section describing data layout
. field   - the field number
- sym     - the symmetry describing the affect of orientation on the access of the data

  Level: developer

.seealso: [PetscSection](sec_petscsection), `PetscSectionSym`, `PetscSectionGetFieldSym()`, `PetscSectionSymCreate()`
@*/
PetscErrorCode PetscSectionSetFieldSym(PetscSection section, PetscInt field, PetscSectionSym sym)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(section, PETSC_SECTION_CLASSID, 1);
  PetscSectionCheckValidField(field, section->numFields);
  PetscCall(PetscSectionSetSym(section->field[field], sym));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PetscSectionGetFieldSym - Get the symmetries for the data referred to by a field of the section

  Collective

  Input Parameters:
+ section - the section describing data layout
- field   - the field number

  Output Parameter:
. sym - the symmetry describing the affect of orientation on the access of the data

  Level: developer

.seealso: [PetscSection](sec_petscsection), `PetscSectionSym`, `PetscSectionSetFieldSym()`, `PetscSectionSymCreate()`
@*/
PetscErrorCode PetscSectionGetFieldSym(PetscSection section, PetscInt field, PetscSectionSym *sym)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(section, PETSC_SECTION_CLASSID, 1);
  PetscSectionCheckValidField(field, section->numFields);
  *sym = section->field[field]->sym;
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@C
  PetscSectionGetPointSyms - Get the symmetries for a set of points in a `PetscSection` under specific orientations.

  Not Collective

  Input Parameters:
+ section   - the section
. numPoints - the number of points
- points    - an array of size 2 * `numPoints`, containing a list of (point, orientation) pairs. (An orientation is an
    arbitrary integer: its interpretation is up to sym.  Orientations are used by `DM`: for their interpretation in that
    context, see `DMPlexGetConeOrientation()`).

  Output Parameters:
+ perms - The permutations for the given orientations (or `NULL` if there is no symmetry or the permutation is the identity).
- rots  - The field rotations symmetries for the given orientations (or `NULL` if there is no symmetry or the rotations are all
    identity).

  Example of usage, gathering dofs into a local array (lArray) from a section array (sArray):
.vb
     const PetscInt    **perms;
     const PetscScalar **rots;
     PetscInt            lOffset;

     PetscSectionGetPointSyms(section,numPoints,points,&perms,&rots);
     for (i = 0, lOffset = 0; i < numPoints; i++) {
       PetscInt           point = points[2*i], dof, sOffset;
       const PetscInt    *perm  = perms ? perms[i] : NULL;
       const PetscScalar *rot   = rots  ? rots[i]  : NULL;

       PetscSectionGetDof(section,point,&dof);
       PetscSectionGetOffset(section,point,&sOffset);

       if (perm) {for (j = 0; j < dof; j++) {lArray[lOffset + perm[j]]  = sArray[sOffset + j];}}
       else      {for (j = 0; j < dof; j++) {lArray[lOffset +      j ]  = sArray[sOffset + j];}}
       if (rot)  {for (j = 0; j < dof; j++) {lArray[lOffset +      j ] *= rot[j];             }}
       lOffset += dof;
     }
     PetscSectionRestorePointSyms(section,numPoints,points,&perms,&rots);
.ve

  Example of usage, adding dofs into a section array (sArray) from a local array (lArray):
.vb
     const PetscInt    **perms;
     const PetscScalar **rots;
     PetscInt            lOffset;

     PetscSectionGetPointSyms(section,numPoints,points,&perms,&rots);
     for (i = 0, lOffset = 0; i < numPoints; i++) {
       PetscInt           point = points[2*i], dof, sOffset;
       const PetscInt    *perm  = perms ? perms[i] : NULL;
       const PetscScalar *rot   = rots  ? rots[i]  : NULL;

       PetscSectionGetDof(section,point,&dof);
       PetscSectionGetOffset(section,point,&sOff);

       if (perm) {for (j = 0; j < dof; j++) {sArray[sOffset + j] += lArray[lOffset + perm[j]] * (rot ? PetscConj(rot[perm[j]]) : 1.);}}
       else      {for (j = 0; j < dof; j++) {sArray[sOffset + j] += lArray[lOffset +      j ] * (rot ? PetscConj(rot[     j ]) : 1.);}}
       offset += dof;
     }
     PetscSectionRestorePointSyms(section,numPoints,points,&perms,&rots);
.ve

  Level: developer

  Notes:
  `PetscSectionSetSym()` must have been previously called to provide the symmetries to the `PetscSection`

  Use `PetscSectionRestorePointSyms()` when finished with the data

.seealso: [PetscSection](sec_petscsection), `PetscSectionSym`, `PetscSectionRestorePointSyms()`, `PetscSectionSymCreate()`, `PetscSectionSetSym()`, `PetscSectionGetSym()`
@*/
PetscErrorCode PetscSectionGetPointSyms(PetscSection section, PetscInt numPoints, const PetscInt *points, const PetscInt ***perms, const PetscScalar ***rots)
{
  PetscSectionSym sym;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(section, PETSC_SECTION_CLASSID, 1);
  if (numPoints) PetscAssertPointer(points, 3);
  if (perms) *perms = NULL;
  if (rots) *rots = NULL;
  sym = section->sym;
  if (sym && (perms || rots)) {
    SymWorkLink link;

    if (sym->workin) {
      link        = sym->workin;
      sym->workin = sym->workin->next;
    } else {
      PetscCall(PetscNew(&link));
    }
    if (numPoints > link->numPoints) {
      PetscInt    **perms = (PetscInt **)link->perms;
      PetscScalar **rots  = (PetscScalar **)link->rots;
      PetscCall(PetscFree2(perms, rots));
      PetscCall(PetscMalloc2(numPoints, (PetscInt ***)&link->perms, numPoints, (PetscScalar ***)&link->rots));
      link->numPoints = numPoints;
    }
    link->next   = sym->workout;
    sym->workout = link;
    PetscCall(PetscArrayzero((PetscInt **)link->perms, numPoints));
    PetscCall(PetscArrayzero((PetscInt **)link->rots, numPoints));
    PetscCall((*sym->ops->getpoints)(sym, section, numPoints, points, link->perms, link->rots));
    if (perms) *perms = link->perms;
    if (rots) *rots = link->rots;
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@C
  PetscSectionRestorePointSyms - Restore the symmetries returned by `PetscSectionGetPointSyms()`

  Not Collective

  Input Parameters:
+ section   - the section
. numPoints - the number of points
. points    - an array of size 2 * `numPoints`, containing a list of (point, orientation) pairs. (An orientation is an
    arbitrary integer: its interpretation is up to sym.  Orientations are used by `DM`: for their interpretation in that
    context, see `DMPlexGetConeOrientation()`).
. perms     - The permutations for the given orientations: set to `NULL` at conclusion
- rots      - The field rotations symmetries for the given orientations: set to `NULL` at conclusion

  Level: developer

.seealso: [PetscSection](sec_petscsection), `PetscSectionSym`, `PetscSectionGetPointSyms()`, `PetscSectionSymCreate()`, `PetscSectionSetSym()`, `PetscSectionGetSym()`
@*/
PetscErrorCode PetscSectionRestorePointSyms(PetscSection section, PetscInt numPoints, const PetscInt *points, const PetscInt ***perms, const PetscScalar ***rots)
{
  PetscSectionSym sym;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(section, PETSC_SECTION_CLASSID, 1);
  sym = section->sym;
  if (sym && (perms || rots)) {
    SymWorkLink *p, link;

    for (p = &sym->workout; (link = *p); p = &link->next) {
      if ((perms && link->perms == *perms) || (rots && link->rots == *rots)) {
        *p          = link->next;
        link->next  = sym->workin;
        sym->workin = link;
        if (perms) *perms = NULL;
        if (rots) *rots = NULL;
        PetscFunctionReturn(PETSC_SUCCESS);
      }
    }
    SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_WRONGSTATE, "Array was not checked out");
  }
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@C
  PetscSectionGetFieldPointSyms - Get the symmetries for a set of points in a field of a `PetscSection` under specific orientations.

  Not Collective

  Input Parameters:
+ section   - the section
. field     - the field of the section
. numPoints - the number of points
- points    - an array of size 2 * `numPoints`, containing a list of (point, orientation) pairs. (An orientation is an
    arbitrary integer: its interpretation is up to sym.  Orientations are used by `DM`: for their interpretation in that
    context, see `DMPlexGetConeOrientation()`).

  Output Parameters:
+ perms - The permutations for the given orientations (or `NULL` if there is no symmetry or the permutation is the identity).
- rots  - The field rotations symmetries for the given orientations (or `NULL` if there is no symmetry or the rotations are all
    identity).

  Level: developer

  Notes:
  `PetscSectionSetFieldSym()` must have been previously called to provide the symmetries to the `PetscSection`

  Use `PetscSectionRestoreFieldPointSyms()` when finished with the data

.seealso: [PetscSection](sec_petscsection), `PetscSectionSym`, `PetscSectionGetPointSyms()`, `PetscSectionRestoreFieldPointSyms()`
@*/
PetscErrorCode PetscSectionGetFieldPointSyms(PetscSection section, PetscInt field, PetscInt numPoints, const PetscInt *points, const PetscInt ***perms, const PetscScalar ***rots)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(section, PETSC_SECTION_CLASSID, 1);
  PetscCheck(field <= section->numFields, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "field %" PetscInt_FMT " greater than number of fields (%" PetscInt_FMT ") in section", field, section->numFields);
  PetscCall(PetscSectionGetPointSyms(section->field[field], numPoints, points, perms, rots));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@C
  PetscSectionRestoreFieldPointSyms - Restore the symmetries returned by `PetscSectionGetFieldPointSyms()`

  Not Collective

  Input Parameters:
+ section   - the section
. field     - the field number
. numPoints - the number of points
. points    - an array of size 2 * `numPoints`, containing a list of (point, orientation) pairs. (An orientation is an
    arbitrary integer: its interpretation is up to sym.  Orientations are used by `DM`: for their interpretation in that
    context, see `DMPlexGetConeOrientation()`).
. perms     - The permutations for the given orientations: set to NULL at conclusion
- rots      - The field rotations symmetries for the given orientations: set to NULL at conclusion

  Level: developer

.seealso: [PetscSection](sec_petscsection), `PetscSectionSym`, `PetscSectionRestorePointSyms()`, `petscSectionGetFieldPointSyms()`, `PetscSectionSymCreate()`, `PetscSectionSetSym()`, `PetscSectionGetSym()`
@*/
PetscErrorCode PetscSectionRestoreFieldPointSyms(PetscSection section, PetscInt field, PetscInt numPoints, const PetscInt *points, const PetscInt ***perms, const PetscScalar ***rots)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(section, PETSC_SECTION_CLASSID, 1);
  PetscCheck(field <= section->numFields, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "field %" PetscInt_FMT " greater than number of fields (%" PetscInt_FMT ") in section", field, section->numFields);
  PetscCall(PetscSectionRestorePointSyms(section->field[field], numPoints, points, perms, rots));
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PetscSectionSymCopy - Copy the symmetries, assuming that the point structure is compatible

  Not Collective

  Input Parameter:
. sym - the `PetscSectionSym`

  Output Parameter:
. nsym - the equivalent symmetries

  Level: developer

.seealso: [PetscSection](sec_petscsection), `PetscSectionSym`, `PetscSectionSymCreate()`, `PetscSectionSetSym()`, `PetscSectionGetSym()`, `PetscSectionSymLabelSetStratum()`, `PetscSectionGetPointSyms()`
@*/
PetscErrorCode PetscSectionSymCopy(PetscSectionSym sym, PetscSectionSym nsym)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(sym, PETSC_SECTION_SYM_CLASSID, 1);
  PetscValidHeaderSpecific(nsym, PETSC_SECTION_SYM_CLASSID, 2);
  PetscTryTypeMethod(sym, copy, nsym);
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PetscSectionSymDistribute - Distribute the symmetries in accordance with the input `PetscSF`

  Collective

  Input Parameters:
+ sym         - the `PetscSectionSym`
- migrationSF - the distribution map from roots to leaves

  Output Parameter:
. dsym - the redistributed symmetries

  Level: developer

.seealso: [PetscSection](sec_petscsection), `PetscSectionSym`, `PetscSectionSymCreate()`, `PetscSectionSetSym()`, `PetscSectionGetSym()`, `PetscSectionSymLabelSetStratum()`, `PetscSectionGetPointSyms()`
@*/
PetscErrorCode PetscSectionSymDistribute(PetscSectionSym sym, PetscSF migrationSF, PetscSectionSym *dsym)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(sym, PETSC_SECTION_SYM_CLASSID, 1);
  PetscValidHeaderSpecific(migrationSF, PETSCSF_CLASSID, 2);
  PetscAssertPointer(dsym, 3);
  PetscTryTypeMethod(sym, distribute, migrationSF, dsym);
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PetscSectionGetUseFieldOffsets - Get the flag indicating if field offsets are used directly in a global section, rather than just the point offset

  Not Collective

  Input Parameter:
. s - the global `PetscSection`

  Output Parameter:
. flg - the flag

  Level: developer

.seealso: [PetscSection](sec_petscsection), `PetscSectionSym`, `PetscSectionSetChart()`, `PetscSectionCreate()`
@*/
PetscErrorCode PetscSectionGetUseFieldOffsets(PetscSection s, PetscBool *flg)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(s, PETSC_SECTION_CLASSID, 1);
  *flg = s->useFieldOff;
  PetscFunctionReturn(PETSC_SUCCESS);
}

/*@
  PetscSectionSetUseFieldOffsets - Set the flag to use field offsets directly in a global section, rather than just the point offset

  Not Collective

  Input Parameters:
+ s   - the global `PetscSection`
- flg - the flag

  Level: developer

.seealso: [PetscSection](sec_petscsection), `PetscSectionSym`, `PetscSectionGetUseFieldOffsets()`, `PetscSectionSetChart()`, `PetscSectionCreate()`
@*/
PetscErrorCode PetscSectionSetUseFieldOffsets(PetscSection s, PetscBool flg)
{
  PetscFunctionBegin;
  PetscValidHeaderSpecific(s, PETSC_SECTION_CLASSID, 1);
  s->useFieldOff = flg;
  PetscFunctionReturn(PETSC_SUCCESS);
}

#define PetscSectionExpandPoints_Loop(TYPE) \
  do { \
    PetscInt i, n, o0, o1, size; \
    TYPE    *a0 = (TYPE *)origArray, *a1; \
    PetscCall(PetscSectionGetStorageSize(s, &size)); \
    PetscCall(PetscMalloc1(size, &a1)); \
    for (i = 0; i < npoints; i++) { \
      PetscCall(PetscSectionGetOffset(origSection, points_[i], &o0)); \
      PetscCall(PetscSectionGetOffset(s, i, &o1)); \
      PetscCall(PetscSectionGetDof(s, i, &n)); \
      PetscCall(PetscMemcpy(&a1[o1], &a0[o0], n *unitsize)); \
    } \
    *newArray = (void *)a1; \
  } while (0)

/*@
  PetscSectionExtractDofsFromArray - Extracts elements of an array corresponding to DOFs of specified points.

  Not Collective

  Input Parameters:
+ origSection - the `PetscSection` describing the layout of the array
. dataType    - `MPI_Datatype` describing the data type of the array (currently only `MPIU_INT`, `MPIU_SCALAR`, `MPIU_REAL`)
. origArray   - the array; its size must be equal to the storage size of `origSection`
- points      - `IS` with points to extract; its indices must lie in the chart of `origSection`

  Output Parameters:
+ newSection - the new `PetscSection` describing the layout of the new array (with points renumbered 0,1,... but preserving numbers of DOFs)
- newArray   - the array of the extracted DOFs; its size is the storage size of `newSection`

  Level: developer

.seealso: [PetscSection](sec_petscsection), `PetscSectionSym`, `PetscSectionGetChart()`, `PetscSectionGetDof()`, `PetscSectionGetStorageSize()`, `PetscSectionCreate()`
@*/
PetscErrorCode PetscSectionExtractDofsFromArray(PetscSection origSection, MPI_Datatype dataType, const void *origArray, IS points, PetscSection *newSection, void *newArray[])
{
  PetscSection    s;
  const PetscInt *points_;
  PetscInt        i, n, npoints, pStart, pEnd;
  PetscMPIInt     unitsize;

  PetscFunctionBegin;
  PetscValidHeaderSpecific(origSection, PETSC_SECTION_CLASSID, 1);
  PetscAssertPointer(origArray, 3);
  PetscValidHeaderSpecific(points, IS_CLASSID, 4);
  if (newSection) PetscAssertPointer(newSection, 5);
  if (newArray) PetscAssertPointer(newArray, 6);
  PetscCallMPI(MPI_Type_size(dataType, &unitsize));
  PetscCall(ISGetLocalSize(points, &npoints));
  PetscCall(ISGetIndices(points, &points_));
  PetscCall(PetscSectionGetChart(origSection, &pStart, &pEnd));
  PetscCall(PetscSectionCreate(PETSC_COMM_SELF, &s));
  PetscCall(PetscSectionSetChart(s, 0, npoints));
  for (i = 0; i < npoints; i++) {
    PetscCheck(points_[i] >= pStart && points_[i] < pEnd, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "point %" PetscInt_FMT " (index %" PetscInt_FMT ") in input IS out of input section's chart", points_[i], i);
    PetscCall(PetscSectionGetDof(origSection, points_[i], &n));
    PetscCall(PetscSectionSetDof(s, i, n));
  }
  PetscCall(PetscSectionSetUp(s));
  if (newArray) {
    if (dataType == MPIU_INT) {
      PetscSectionExpandPoints_Loop(PetscInt);
    } else if (dataType == MPIU_SCALAR) {
      PetscSectionExpandPoints_Loop(PetscScalar);
    } else if (dataType == MPIU_REAL) {
      PetscSectionExpandPoints_Loop(PetscReal);
    } else SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP, "not implemented for this MPI_Datatype");
  }
  if (newSection) {
    *newSection = s;
  } else {
    PetscCall(PetscSectionDestroy(&s));
  }
  PetscCall(ISRestoreIndices(points, &points_));
  PetscFunctionReturn(PETSC_SUCCESS);
}