Actual source code: dspacelagrange.c

      1: #include <petsc/private/petscfeimpl.h>
      2: #include <petscdmplex.h>
      3: #include <petscblaslapack.h>

      5: PetscErrorCode DMPlexGetTransitiveClosure_Internal(DM, PetscInt, PetscInt, PetscBool, PetscInt *, PetscInt *[]);

      7: struct _n_Petsc1DNodeFamily {
      8:   PetscInt        refct;
      9:   PetscDTNodeType nodeFamily;
     10:   PetscReal       gaussJacobiExp;
     11:   PetscInt        nComputed;
     12:   PetscReal     **nodesets;
     13:   PetscBool       endpoints;
     14: };

     16: /* users set node families for PETSCDUALSPACELAGRANGE with just the inputs to this function, but internally we create
     17:  * an object that can cache the computations across multiple dual spaces */
     18: static PetscErrorCode Petsc1DNodeFamilyCreate(PetscDTNodeType family, PetscReal gaussJacobiExp, PetscBool endpoints, Petsc1DNodeFamily *nf)
     19: {
     20:   Petsc1DNodeFamily f;

     22:   PetscFunctionBegin;
     23:   PetscCall(PetscNew(&f));
     24:   switch (family) {
     25:   case PETSCDTNODES_GAUSSJACOBI:
     26:   case PETSCDTNODES_EQUISPACED:
     27:     f->nodeFamily = family;
     28:     break;
     29:   default:
     30:     SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Unknown 1D node family");
     31:   }
     32:   f->endpoints      = endpoints;
     33:   f->gaussJacobiExp = 0.;
     34:   if (family == PETSCDTNODES_GAUSSJACOBI) {
     35:     PetscCheck(gaussJacobiExp > -1., PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Gauss-Jacobi exponent must be > -1.");
     36:     f->gaussJacobiExp = gaussJacobiExp;
     37:   }
     38:   f->refct = 1;
     39:   *nf      = f;
     40:   PetscFunctionReturn(PETSC_SUCCESS);
     41: }

     43: static PetscErrorCode Petsc1DNodeFamilyReference(Petsc1DNodeFamily nf)
     44: {
     45:   PetscFunctionBegin;
     46:   if (nf) nf->refct++;
     47:   PetscFunctionReturn(PETSC_SUCCESS);
     48: }

     50: static PetscErrorCode Petsc1DNodeFamilyDestroy(Petsc1DNodeFamily *nf)
     51: {
     52:   PetscInt i, nc;

     54:   PetscFunctionBegin;
     55:   if (!*nf) PetscFunctionReturn(PETSC_SUCCESS);
     56:   if (--(*nf)->refct > 0) {
     57:     *nf = NULL;
     58:     PetscFunctionReturn(PETSC_SUCCESS);
     59:   }
     60:   nc = (*nf)->nComputed;
     61:   for (i = 0; i < nc; i++) PetscCall(PetscFree((*nf)->nodesets[i]));
     62:   PetscCall(PetscFree((*nf)->nodesets));
     63:   PetscCall(PetscFree(*nf));
     64:   *nf = NULL;
     65:   PetscFunctionReturn(PETSC_SUCCESS);
     66: }

     68: static PetscErrorCode Petsc1DNodeFamilyGetNodeSets(Petsc1DNodeFamily f, PetscInt degree, PetscReal ***nodesets)
     69: {
     70:   PetscInt nc;

     72:   PetscFunctionBegin;
     73:   nc = f->nComputed;
     74:   if (degree >= nc) {
     75:     PetscInt    i, j;
     76:     PetscReal **new_nodesets;
     77:     PetscReal  *w;

     79:     PetscCall(PetscMalloc1(degree + 1, &new_nodesets));
     80:     PetscCall(PetscArraycpy(new_nodesets, f->nodesets, nc));
     81:     PetscCall(PetscFree(f->nodesets));
     82:     f->nodesets = new_nodesets;
     83:     PetscCall(PetscMalloc1(degree + 1, &w));
     84:     for (i = nc; i < degree + 1; i++) {
     85:       PetscCall(PetscMalloc1(i + 1, &f->nodesets[i]));
     86:       if (!i) {
     87:         f->nodesets[i][0] = 0.5;
     88:       } else {
     89:         switch (f->nodeFamily) {
     90:         case PETSCDTNODES_EQUISPACED:
     91:           if (f->endpoints) {
     92:             for (j = 0; j <= i; j++) f->nodesets[i][j] = (PetscReal)j / (PetscReal)i;
     93:           } else {
     94:             /* these nodes are at the centroids of the small simplices created by the equispaced nodes that include
     95:              * the endpoints */
     96:             for (j = 0; j <= i; j++) f->nodesets[i][j] = ((PetscReal)j + 0.5) / ((PetscReal)i + 1.);
     97:           }
     98:           break;
     99:         case PETSCDTNODES_GAUSSJACOBI:
    100:           if (f->endpoints) {
    101:             PetscCall(PetscDTGaussLobattoJacobiQuadrature(i + 1, 0., 1., f->gaussJacobiExp, f->gaussJacobiExp, f->nodesets[i], w));
    102:           } else {
    103:             PetscCall(PetscDTGaussJacobiQuadrature(i + 1, 0., 1., f->gaussJacobiExp, f->gaussJacobiExp, f->nodesets[i], w));
    104:           }
    105:           break;
    106:         default:
    107:           SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Unknown 1D node family");
    108:         }
    109:       }
    110:     }
    111:     PetscCall(PetscFree(w));
    112:     f->nComputed = degree + 1;
    113:   }
    114:   *nodesets = f->nodesets;
    115:   PetscFunctionReturn(PETSC_SUCCESS);
    116: }

    118: /* http://arxiv.org/abs/2002.09421 for details */
    119: static PetscErrorCode PetscNodeRecursive_Internal(PetscInt dim, PetscInt degree, PetscReal **nodesets, PetscInt tup[], PetscReal node[])
    120: {
    121:   PetscReal w;
    122:   PetscInt  i, j;

    124:   PetscFunctionBeginHot;
    125:   w = 0.;
    126:   if (dim == 1) {
    127:     node[0] = nodesets[degree][tup[0]];
    128:     node[1] = nodesets[degree][tup[1]];
    129:   } else {
    130:     for (i = 0; i < dim + 1; i++) node[i] = 0.;
    131:     for (i = 0; i < dim + 1; i++) {
    132:       PetscReal wi = nodesets[degree][degree - tup[i]];

    134:       for (j = 0; j < dim + 1; j++) tup[dim + 1 + j] = tup[j + (j >= i)];
    135:       PetscCall(PetscNodeRecursive_Internal(dim - 1, degree - tup[i], nodesets, &tup[dim + 1], &node[dim + 1]));
    136:       for (j = 0; j < dim + 1; j++) node[j + (j >= i)] += wi * node[dim + 1 + j];
    137:       w += wi;
    138:     }
    139:     for (i = 0; i < dim + 1; i++) node[i] /= w;
    140:   }
    141:   PetscFunctionReturn(PETSC_SUCCESS);
    142: }

    144: /* compute simplex nodes for the biunit simplex from the 1D node family */
    145: static PetscErrorCode Petsc1DNodeFamilyComputeSimplexNodes(Petsc1DNodeFamily f, PetscInt dim, PetscInt degree, PetscReal points[])
    146: {
    147:   PetscInt   *tup;
    148:   PetscInt    npoints;
    149:   PetscReal **nodesets = NULL;
    150:   PetscInt    worksize;
    151:   PetscReal  *nodework;
    152:   PetscInt   *tupwork;

    154:   PetscFunctionBegin;
    155:   PetscCheck(dim >= 0, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Must have non-negative dimension");
    156:   PetscCheck(degree >= 0, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Must have non-negative degree");
    157:   if (!dim) PetscFunctionReturn(PETSC_SUCCESS);
    158:   PetscCall(PetscCalloc1(dim + 2, &tup));
    159:   PetscCall(PetscDTBinomialInt(degree + dim, dim, &npoints));
    160:   PetscCall(Petsc1DNodeFamilyGetNodeSets(f, degree, &nodesets));
    161:   worksize = ((dim + 2) * (dim + 3)) / 2;
    162:   PetscCall(PetscCalloc2(worksize, &nodework, worksize, &tupwork));
    163:   /* loop over the tuples of length dim with sum at most degree */
    164:   for (PetscInt k = 0; k < npoints; k++) {
    165:     PetscInt i;

    167:     /* turn thm into tuples of length dim + 1 with sum equal to degree (barycentric indice) */
    168:     tup[0] = degree;
    169:     for (i = 0; i < dim; i++) tup[0] -= tup[i + 1];
    170:     switch (f->nodeFamily) {
    171:     case PETSCDTNODES_EQUISPACED:
    172:       /* compute equispaces nodes on the unit reference triangle */
    173:       if (f->endpoints) {
    174:         PetscCheck(degree > 0, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Must have positive degree");
    175:         for (i = 0; i < dim; i++) points[dim * k + i] = (PetscReal)tup[i + 1] / (PetscReal)degree;
    176:       } else {
    177:         for (i = 0; i < dim; i++) {
    178:           /* these nodes are at the centroids of the small simplices created by the equispaced nodes that include
    179:            * the endpoints */
    180:           points[dim * k + i] = ((PetscReal)tup[i + 1] + 1. / (dim + 1.)) / (PetscReal)(degree + 1.);
    181:         }
    182:       }
    183:       break;
    184:     default:
    185:       /* compute equispaced nodes on the barycentric reference triangle (the trace on the first dim dimensions are the
    186:        * unit reference triangle nodes */
    187:       for (i = 0; i < dim + 1; i++) tupwork[i] = tup[i];
    188:       PetscCall(PetscNodeRecursive_Internal(dim, degree, nodesets, tupwork, nodework));
    189:       for (i = 0; i < dim; i++) points[dim * k + i] = nodework[i + 1];
    190:       break;
    191:     }
    192:     PetscCall(PetscDualSpaceLatticePointLexicographic_Internal(dim, degree, &tup[1]));
    193:   }
    194:   /* map from unit simplex to biunit simplex */
    195:   for (PetscInt k = 0; k < npoints * dim; k++) points[k] = points[k] * 2. - 1.;
    196:   PetscCall(PetscFree2(nodework, tupwork));
    197:   PetscCall(PetscFree(tup));
    198:   PetscFunctionReturn(PETSC_SUCCESS);
    199: }

    201: /* If we need to get the dofs from a mesh point, or add values into dofs at a mesh point, and there is more than one dof
    202:  * on that mesh point, we have to be careful about getting/adding everything in the right place.
    203:  *
    204:  * With nodal dofs like PETSCDUALSPACELAGRANGE makes, the general approach to calculate the value of dofs associate
    205:  * with a node A is
    206:  * - transform the node locations x(A) by the map that takes the mesh point to its reorientation, x' = phi(x(A))
    207:  * - figure out which node was originally at the location of the transformed point, A' = idx(x')
    208:  * - if the dofs are not scalars, figure out how to represent the transformed dofs in terms of the basis
    209:  *   of dofs at A' (using pushforward/pullback rules)
    210:  *
    211:  * The one sticky point with this approach is the "A' = idx(x')" step: trying to go from real valued coordinates
    212:  * back to indices.  I don't want to rely on floating point tolerances.  Additionally, PETSCDUALSPACELAGRANGE may
    213:  * eventually support quasi-Lagrangian dofs, which could involve quadrature at multiple points, so the location "x(A)"
    214:  * would be ambiguous.
    215:  *
    216:  * So each dof gets an integer value coordinate (nodeIdx in the structure below).  The choice of integer coordinates
    217:  * is somewhat arbitrary, as long as all of the relevant symmetries of the mesh point correspond to *permutations* of
    218:  * the integer coordinates, which do not depend on numerical precision.
    219:  *
    220:  * So
    221:  *
    222:  * - DMPlexGetTransitiveClosure_Internal() tells me how an orientation turns into a permutation of the vertices of a
    223:  *   mesh point
    224:  * - The permutation of the vertices, and the nodeIdx values assigned to them, tells what permutation in index space
    225:  *   is associated with the orientation
    226:  * - I uses that permutation to get xi' = phi(xi(A)), the integer coordinate of the transformed dof
    227:  * - I can without numerical issues compute A' = idx(xi')
    228:  *
    229:  * Here are some examples of how the process works
    230:  *
    231:  * - With a triangle:
    232:  *
    233:  *   The triangle has the following integer coordinates for vertices, taken from the barycentric triangle
    234:  *
    235:  *     closure order 2
    236:  *     nodeIdx (0,0,1)
    237:  *      \
    238:  *       +
    239:  *       |\
    240:  *       | \
    241:  *       |  \
    242:  *       |   \    closure order 1
    243:  *       |    \ / nodeIdx (0,1,0)
    244:  *       +-----+
    245:  *        \
    246:  *      closure order 0
    247:  *      nodeIdx (1,0,0)
    248:  *
    249:  *   If I do DMPlexGetTransitiveClosure_Internal() with orientation 1, the vertices would appear
    250:  *   in the order (1, 2, 0)
    251:  *
    252:  *   If I list the nodeIdx of each vertex in closure order for orientation 0 (0, 1, 2) and orientation 1 (1, 2, 0), I
    253:  *   see
    254:  *
    255:  *   orientation 0  | orientation 1
    256:  *
    257:  *   [0] (1,0,0)      [1] (0,1,0)
    258:  *   [1] (0,1,0)      [2] (0,0,1)
    259:  *   [2] (0,0,1)      [0] (1,0,0)
    260:  *          A                B
    261:  *
    262:  *   In other words, B is the result of a row permutation of A.  But, there is also
    263:  *   a column permutation that accomplishes the same result, (2,0,1).
    264:  *
    265:  *   So if a dof has nodeIdx coordinate (a,b,c), after the transformation its nodeIdx coordinate
    266:  *   is (c,a,b), and the transformed degree of freedom will be a linear combination of dofs
    267:  *   that originally had coordinate (c,a,b).
    268:  *
    269:  * - With a quadrilateral:
    270:  *
    271:  *   The quadrilateral has the following integer coordinates for vertices, taken from concatenating barycentric
    272:  *   coordinates for two segments:
    273:  *
    274:  *     closure order 3      closure order 2
    275:  *     nodeIdx (1,0,0,1)    nodeIdx (0,1,0,1)
    276:  *                   \      /
    277:  *                    +----+
    278:  *                    |    |
    279:  *                    |    |
    280:  *                    +----+
    281:  *                   /      \
    282:  *     closure order 0      closure order 1
    283:  *     nodeIdx (1,0,1,0)    nodeIdx (0,1,1,0)
    284:  *
    285:  *   If I do DMPlexGetTransitiveClosure_Internal() with orientation 1, the vertices would appear
    286:  *   in the order (1, 2, 3, 0)
    287:  *
    288:  *   If I list the nodeIdx of each vertex in closure order for orientation 0 (0, 1, 2, 3) and
    289:  *   orientation 1 (1, 2, 3, 0), I see
    290:  *
    291:  *   orientation 0  | orientation 1
    292:  *
    293:  *   [0] (1,0,1,0)    [1] (0,1,1,0)
    294:  *   [1] (0,1,1,0)    [2] (0,1,0,1)
    295:  *   [2] (0,1,0,1)    [3] (1,0,0,1)
    296:  *   [3] (1,0,0,1)    [0] (1,0,1,0)
    297:  *          A                B
    298:  *
    299:  *   The column permutation that accomplishes the same result is (3,2,0,1).
    300:  *
    301:  *   So if a dof has nodeIdx coordinate (a,b,c,d), after the transformation its nodeIdx coordinate
    302:  *   is (d,c,a,b), and the transformed degree of freedom will be a linear combination of dofs
    303:  *   that originally had coordinate (d,c,a,b).
    304:  *
    305:  * Previously PETSCDUALSPACELAGRANGE had hardcoded symmetries for the triangle and quadrilateral,
    306:  * but this approach will work for any polytope, such as the wedge (triangular prism).
    307:  */
    308: struct _n_PetscLagNodeIndices {
    309:   PetscInt   refct;
    310:   PetscInt   nodeIdxDim;
    311:   PetscInt   nodeVecDim;
    312:   PetscInt   nNodes;
    313:   PetscInt  *nodeIdx; /* for each node an index of size nodeIdxDim */
    314:   PetscReal *nodeVec; /* for each node a vector of size nodeVecDim */
    315:   PetscInt  *perm;    /* if these are vertices, perm takes DMPlex point index to closure order;
    316:                               if these are nodes, perm lists nodes in index revlex order */
    317: };

    319: /* this is just here so I can access the values in tests/ex1.c outside the library */
    320: PetscErrorCode PetscLagNodeIndicesGetData_Internal(PetscLagNodeIndices ni, PetscInt *nodeIdxDim, PetscInt *nodeVecDim, PetscInt *nNodes, const PetscInt *nodeIdx[], const PetscReal *nodeVec[])
    321: {
    322:   PetscFunctionBegin;
    323:   *nodeIdxDim = ni->nodeIdxDim;
    324:   *nodeVecDim = ni->nodeVecDim;
    325:   *nNodes     = ni->nNodes;
    326:   *nodeIdx    = ni->nodeIdx;
    327:   *nodeVec    = ni->nodeVec;
    328:   PetscFunctionReturn(PETSC_SUCCESS);
    329: }

    331: static PetscErrorCode PetscLagNodeIndicesReference(PetscLagNodeIndices ni)
    332: {
    333:   PetscFunctionBegin;
    334:   if (ni) ni->refct++;
    335:   PetscFunctionReturn(PETSC_SUCCESS);
    336: }

    338: static PetscErrorCode PetscLagNodeIndicesDuplicate(PetscLagNodeIndices ni, PetscLagNodeIndices *niNew)
    339: {
    340:   PetscFunctionBegin;
    341:   PetscCall(PetscNew(niNew));
    342:   (*niNew)->refct      = 1;
    343:   (*niNew)->nodeIdxDim = ni->nodeIdxDim;
    344:   (*niNew)->nodeVecDim = ni->nodeVecDim;
    345:   (*niNew)->nNodes     = ni->nNodes;
    346:   PetscCall(PetscMalloc1(ni->nNodes * ni->nodeIdxDim, &((*niNew)->nodeIdx)));
    347:   PetscCall(PetscArraycpy((*niNew)->nodeIdx, ni->nodeIdx, ni->nNodes * ni->nodeIdxDim));
    348:   PetscCall(PetscMalloc1(ni->nNodes * ni->nodeVecDim, &((*niNew)->nodeVec)));
    349:   PetscCall(PetscArraycpy((*niNew)->nodeVec, ni->nodeVec, ni->nNodes * ni->nodeVecDim));
    350:   (*niNew)->perm = NULL;
    351:   PetscFunctionReturn(PETSC_SUCCESS);
    352: }

    354: static PetscErrorCode PetscLagNodeIndicesDestroy(PetscLagNodeIndices *ni)
    355: {
    356:   PetscFunctionBegin;
    357:   if (!*ni) PetscFunctionReturn(PETSC_SUCCESS);
    358:   if (--(*ni)->refct > 0) {
    359:     *ni = NULL;
    360:     PetscFunctionReturn(PETSC_SUCCESS);
    361:   }
    362:   PetscCall(PetscFree((*ni)->nodeIdx));
    363:   PetscCall(PetscFree((*ni)->nodeVec));
    364:   PetscCall(PetscFree((*ni)->perm));
    365:   PetscCall(PetscFree(*ni));
    366:   *ni = NULL;
    367:   PetscFunctionReturn(PETSC_SUCCESS);
    368: }

    370: /* The vertices are given nodeIdx coordinates (e.g. the corners of the barycentric triangle).  Those coordinates are
    371:  * in some other order, and to understand the effect of different symmetries, we need them to be in closure order.
    372:  *
    373:  * If sortIdx is PETSC_FALSE, the coordinates are already in revlex order, otherwise we must sort them
    374:  * to that order before we do the real work of this function, which is
    375:  *
    376:  * - mark the vertices in closure order
    377:  * - sort them in revlex order
    378:  * - use the resulting permutation to list the vertex coordinates in closure order
    379:  */
    380: static PetscErrorCode PetscLagNodeIndicesComputeVertexOrder(DM dm, PetscLagNodeIndices ni, PetscBool sortIdx)
    381: {
    382:   PetscInt           v, w, vStart, vEnd, c, d;
    383:   PetscInt           nVerts;
    384:   PetscInt           closureSize = 0;
    385:   PetscInt          *closure     = NULL;
    386:   PetscInt          *closureOrder;
    387:   PetscInt          *invClosureOrder;
    388:   PetscInt          *revlexOrder;
    389:   PetscInt          *newNodeIdx;
    390:   PetscInt           dim;
    391:   Vec                coordVec;
    392:   const PetscScalar *coords;

    394:   PetscFunctionBegin;
    395:   PetscCall(DMGetDimension(dm, &dim));
    396:   PetscCall(DMPlexGetDepthStratum(dm, 0, &vStart, &vEnd));
    397:   nVerts = vEnd - vStart;
    398:   PetscCall(PetscMalloc1(nVerts, &closureOrder));
    399:   PetscCall(PetscMalloc1(nVerts, &invClosureOrder));
    400:   PetscCall(PetscMalloc1(nVerts, &revlexOrder));
    401:   if (sortIdx) { /* bubble sort nodeIdx into revlex order */
    402:     PetscInt  nodeIdxDim = ni->nodeIdxDim;
    403:     PetscInt *idxOrder;

    405:     PetscCall(PetscMalloc1(nVerts * nodeIdxDim, &newNodeIdx));
    406:     PetscCall(PetscMalloc1(nVerts, &idxOrder));
    407:     for (v = 0; v < nVerts; v++) idxOrder[v] = v;
    408:     for (v = 0; v < nVerts; v++) {
    409:       for (w = v + 1; w < nVerts; w++) {
    410:         const PetscInt *iv   = &(ni->nodeIdx[idxOrder[v] * nodeIdxDim]);
    411:         const PetscInt *iw   = &(ni->nodeIdx[idxOrder[w] * nodeIdxDim]);
    412:         PetscInt        diff = 0;

    414:         for (d = nodeIdxDim - 1; d >= 0; d--)
    415:           if ((diff = (iv[d] - iw[d]))) break;
    416:         if (diff > 0) {
    417:           PetscInt swap = idxOrder[v];

    419:           idxOrder[v] = idxOrder[w];
    420:           idxOrder[w] = swap;
    421:         }
    422:       }
    423:     }
    424:     for (v = 0; v < nVerts; v++) {
    425:       for (d = 0; d < nodeIdxDim; d++) newNodeIdx[v * ni->nodeIdxDim + d] = ni->nodeIdx[idxOrder[v] * nodeIdxDim + d];
    426:     }
    427:     PetscCall(PetscFree(ni->nodeIdx));
    428:     ni->nodeIdx = newNodeIdx;
    429:     newNodeIdx  = NULL;
    430:     PetscCall(PetscFree(idxOrder));
    431:   }
    432:   PetscCall(DMPlexGetTransitiveClosure(dm, 0, PETSC_TRUE, &closureSize, &closure));
    433:   c = closureSize - nVerts;
    434:   for (v = 0; v < nVerts; v++) closureOrder[v] = closure[2 * (c + v)] - vStart;
    435:   for (v = 0; v < nVerts; v++) invClosureOrder[closureOrder[v]] = v;
    436:   PetscCall(DMPlexRestoreTransitiveClosure(dm, 0, PETSC_TRUE, &closureSize, &closure));
    437:   PetscCall(DMGetCoordinatesLocal(dm, &coordVec));
    438:   PetscCall(VecGetArrayRead(coordVec, &coords));
    439:   /* bubble sort closure vertices by coordinates in revlex order */
    440:   for (v = 0; v < nVerts; v++) revlexOrder[v] = v;
    441:   for (v = 0; v < nVerts; v++) {
    442:     for (w = v + 1; w < nVerts; w++) {
    443:       const PetscScalar *cv   = &coords[closureOrder[revlexOrder[v]] * dim];
    444:       const PetscScalar *cw   = &coords[closureOrder[revlexOrder[w]] * dim];
    445:       PetscReal          diff = 0;

    447:       for (d = dim - 1; d >= 0; d--)
    448:         if ((diff = PetscRealPart(cv[d] - cw[d])) != 0.) break;
    449:       if (diff > 0.) {
    450:         PetscInt swap = revlexOrder[v];

    452:         revlexOrder[v] = revlexOrder[w];
    453:         revlexOrder[w] = swap;
    454:       }
    455:     }
    456:   }
    457:   PetscCall(VecRestoreArrayRead(coordVec, &coords));
    458:   PetscCall(PetscMalloc1(ni->nodeIdxDim * nVerts, &newNodeIdx));
    459:   /* reorder nodeIdx to be in closure order */
    460:   for (v = 0; v < nVerts; v++) {
    461:     for (d = 0; d < ni->nodeIdxDim; d++) newNodeIdx[revlexOrder[v] * ni->nodeIdxDim + d] = ni->nodeIdx[v * ni->nodeIdxDim + d];
    462:   }
    463:   PetscCall(PetscFree(ni->nodeIdx));
    464:   ni->nodeIdx = newNodeIdx;
    465:   ni->perm    = invClosureOrder;
    466:   PetscCall(PetscFree(revlexOrder));
    467:   PetscCall(PetscFree(closureOrder));
    468:   PetscFunctionReturn(PETSC_SUCCESS);
    469: }

    471: /* the coordinates of the simplex vertices are the corners of the barycentric simplex.
    472:  * When we stack them on top of each other in revlex order, they look like the identity matrix */
    473: static PetscErrorCode PetscLagNodeIndicesCreateSimplexVertices(DM dm, PetscLagNodeIndices *nodeIndices)
    474: {
    475:   PetscLagNodeIndices ni;
    476:   PetscInt            dim, d;

    478:   PetscFunctionBegin;
    479:   PetscCall(PetscNew(&ni));
    480:   PetscCall(DMGetDimension(dm, &dim));
    481:   ni->nodeIdxDim = dim + 1;
    482:   ni->nodeVecDim = 0;
    483:   ni->nNodes     = dim + 1;
    484:   ni->refct      = 1;
    485:   PetscCall(PetscCalloc1((dim + 1) * (dim + 1), &ni->nodeIdx));
    486:   for (d = 0; d < dim + 1; d++) ni->nodeIdx[d * (dim + 2)] = 1;
    487:   PetscCall(PetscLagNodeIndicesComputeVertexOrder(dm, ni, PETSC_FALSE));
    488:   *nodeIndices = ni;
    489:   PetscFunctionReturn(PETSC_SUCCESS);
    490: }

    492: /* A polytope that is a tensor product of a facet and a segment.
    493:  * We take whatever coordinate system was being used for the facet
    494:  * and we concatenate the barycentric coordinates for the vertices
    495:  * at the end of the segment, (1,0) and (0,1), to get a coordinate
    496:  * system for the tensor product element */
    497: static PetscErrorCode PetscLagNodeIndicesCreateTensorVertices(DM dm, PetscLagNodeIndices facetni, PetscLagNodeIndices *nodeIndices)
    498: {
    499:   PetscLagNodeIndices ni;
    500:   PetscInt            nodeIdxDim, subNodeIdxDim = facetni->nodeIdxDim;
    501:   PetscInt            nVerts, nSubVerts         = facetni->nNodes;
    502:   PetscInt            dim, d, e, f, g;

    504:   PetscFunctionBegin;
    505:   PetscCall(PetscNew(&ni));
    506:   PetscCall(DMGetDimension(dm, &dim));
    507:   ni->nodeIdxDim = nodeIdxDim = subNodeIdxDim + 2;
    508:   ni->nodeVecDim              = 0;
    509:   ni->nNodes = nVerts = 2 * nSubVerts;
    510:   ni->refct           = 1;
    511:   PetscCall(PetscCalloc1(nodeIdxDim * nVerts, &ni->nodeIdx));
    512:   for (f = 0, d = 0; d < 2; d++) {
    513:     for (e = 0; e < nSubVerts; e++, f++) {
    514:       for (g = 0; g < subNodeIdxDim; g++) ni->nodeIdx[f * nodeIdxDim + g] = facetni->nodeIdx[e * subNodeIdxDim + g];
    515:       ni->nodeIdx[f * nodeIdxDim + subNodeIdxDim]     = (1 - d);
    516:       ni->nodeIdx[f * nodeIdxDim + subNodeIdxDim + 1] = d;
    517:     }
    518:   }
    519:   PetscCall(PetscLagNodeIndicesComputeVertexOrder(dm, ni, PETSC_TRUE));
    520:   *nodeIndices = ni;
    521:   PetscFunctionReturn(PETSC_SUCCESS);
    522: }

    524: /* This helps us compute symmetries, and it also helps us compute coordinates for dofs that are being pushed
    525:  * forward from a boundary mesh point.
    526:  *
    527:  * Input:
    528:  *
    529:  * dm - the target reference cell where we want new coordinates and dof directions to be valid
    530:  * vert - the vertex coordinate system for the target reference cell
    531:  * p - the point in the target reference cell that the dofs are coming from
    532:  * vertp - the vertex coordinate system for p's reference cell
    533:  * ornt - the resulting coordinates and dof vectors will be for p under this orientation
    534:  * nodep - the node coordinates and dof vectors in p's reference cell
    535:  * formDegree - the form degree that the dofs transform as
    536:  *
    537:  * Output:
    538:  *
    539:  * pfNodeIdx - the node coordinates for p's dofs, in the dm reference cell, from the ornt perspective
    540:  * pfNodeVec - the node dof vectors for p's dofs, in the dm reference cell, from the ornt perspective
    541:  */
    542: static PetscErrorCode PetscLagNodeIndicesPushForward(DM dm, PetscLagNodeIndices vert, PetscInt p, PetscLagNodeIndices vertp, PetscLagNodeIndices nodep, PetscInt ornt, PetscInt formDegree, PetscInt pfNodeIdx[], PetscReal pfNodeVec[])
    543: {
    544:   PetscInt          *closureVerts;
    545:   PetscInt           closureSize = 0;
    546:   PetscInt          *closure     = NULL;
    547:   PetscInt           dim, pdim, c, i, j, k, n, v, vStart, vEnd;
    548:   PetscInt           nSubVert      = vertp->nNodes;
    549:   PetscInt           nodeIdxDim    = vert->nodeIdxDim;
    550:   PetscInt           subNodeIdxDim = vertp->nodeIdxDim;
    551:   PetscInt           nNodes        = nodep->nNodes;
    552:   const PetscInt    *vertIdx       = vert->nodeIdx;
    553:   const PetscInt    *subVertIdx    = vertp->nodeIdx;
    554:   const PetscInt    *nodeIdx       = nodep->nodeIdx;
    555:   const PetscReal   *nodeVec       = nodep->nodeVec;
    556:   PetscReal         *J, *Jstar;
    557:   PetscReal          detJ;
    558:   PetscInt           depth, pdepth, Nk, pNk;
    559:   Vec                coordVec;
    560:   PetscScalar       *newCoords = NULL;
    561:   const PetscScalar *oldCoords = NULL;

    563:   PetscFunctionBegin;
    564:   PetscCall(DMGetDimension(dm, &dim));
    565:   PetscCall(DMPlexGetDepth(dm, &depth));
    566:   PetscCall(DMGetCoordinatesLocal(dm, &coordVec));
    567:   PetscCall(DMPlexGetPointDepth(dm, p, &pdepth));
    568:   pdim = pdepth != depth ? pdepth != 0 ? pdepth : 0 : dim;
    569:   PetscCall(DMPlexGetDepthStratum(dm, 0, &vStart, &vEnd));
    570:   PetscCall(DMGetWorkArray(dm, nSubVert, MPIU_INT, &closureVerts));
    571:   PetscCall(DMPlexGetTransitiveClosure_Internal(dm, p, ornt, PETSC_TRUE, &closureSize, &closure));
    572:   c = closureSize - nSubVert;
    573:   /* we want which cell closure indices the closure of this point corresponds to */
    574:   for (v = 0; v < nSubVert; v++) closureVerts[v] = vert->perm[closure[2 * (c + v)] - vStart];
    575:   PetscCall(DMPlexRestoreTransitiveClosure(dm, p, PETSC_TRUE, &closureSize, &closure));
    576:   /* push forward indices */
    577:   for (i = 0; i < nodeIdxDim; i++) { /* for every component of the target index space */
    578:     /* check if this is a component that all vertices around this point have in common */
    579:     for (j = 1; j < nSubVert; j++) {
    580:       if (vertIdx[closureVerts[j] * nodeIdxDim + i] != vertIdx[closureVerts[0] * nodeIdxDim + i]) break;
    581:     }
    582:     if (j == nSubVert) { /* all vertices have this component in common, directly copy to output */
    583:       PetscInt val = vertIdx[closureVerts[0] * nodeIdxDim + i];
    584:       for (n = 0; n < nNodes; n++) pfNodeIdx[n * nodeIdxDim + i] = val;
    585:     } else {
    586:       PetscInt subi = -1;
    587:       /* there must be a component in vertp that looks the same */
    588:       for (k = 0; k < subNodeIdxDim; k++) {
    589:         for (j = 0; j < nSubVert; j++) {
    590:           if (vertIdx[closureVerts[j] * nodeIdxDim + i] != subVertIdx[j * subNodeIdxDim + k]) break;
    591:         }
    592:         if (j == nSubVert) {
    593:           subi = k;
    594:           break;
    595:         }
    596:       }
    597:       PetscCheck(subi >= 0, PETSC_COMM_SELF, PETSC_ERR_PLIB, "Did not find matching coordinate");
    598:       /* that component in the vertp system becomes component i in the vert system for each dof */
    599:       for (n = 0; n < nNodes; n++) pfNodeIdx[n * nodeIdxDim + i] = nodeIdx[n * subNodeIdxDim + subi];
    600:     }
    601:   }
    602:   /* push forward vectors */
    603:   PetscCall(DMGetWorkArray(dm, dim * dim, MPIU_REAL, &J));
    604:   if (ornt != 0) { /* temporarily change the coordinate vector so
    605:                       DMPlexComputeCellGeometryAffineFEM gives us the Jacobian we want */
    606:     PetscInt  closureSize2 = 0;
    607:     PetscInt *closure2     = NULL;

    609:     PetscCall(DMPlexGetTransitiveClosure_Internal(dm, p, 0, PETSC_TRUE, &closureSize2, &closure2));
    610:     PetscCall(PetscMalloc1(dim * nSubVert, &newCoords));
    611:     PetscCall(VecGetArrayRead(coordVec, &oldCoords));
    612:     for (v = 0; v < nSubVert; v++) {
    613:       PetscInt d;
    614:       for (d = 0; d < dim; d++) newCoords[(closure2[2 * (c + v)] - vStart) * dim + d] = oldCoords[closureVerts[v] * dim + d];
    615:     }
    616:     PetscCall(VecRestoreArrayRead(coordVec, &oldCoords));
    617:     PetscCall(DMPlexRestoreTransitiveClosure(dm, p, PETSC_TRUE, &closureSize2, &closure2));
    618:     PetscCall(VecPlaceArray(coordVec, newCoords));
    619:   }
    620:   PetscCall(DMPlexComputeCellGeometryAffineFEM(dm, p, NULL, J, NULL, &detJ));
    621:   if (ornt != 0) {
    622:     PetscCall(VecResetArray(coordVec));
    623:     PetscCall(PetscFree(newCoords));
    624:   }
    625:   PetscCall(DMRestoreWorkArray(dm, nSubVert, MPIU_INT, &closureVerts));
    626:   /* compactify */
    627:   for (i = 0; i < dim; i++)
    628:     for (j = 0; j < pdim; j++) J[i * pdim + j] = J[i * dim + j];
    629:   /* We have the Jacobian mapping the point's reference cell to this reference cell:
    630:    * pulling back a function to the point and applying the dof is what we want,
    631:    * so we get the pullback matrix and multiply the dof by that matrix on the right */
    632:   PetscCall(PetscDTBinomialInt(dim, PetscAbsInt(formDegree), &Nk));
    633:   PetscCall(PetscDTBinomialInt(pdim, PetscAbsInt(formDegree), &pNk));
    634:   PetscCall(DMGetWorkArray(dm, pNk * Nk, MPIU_REAL, &Jstar));
    635:   PetscCall(PetscDTAltVPullbackMatrix(pdim, dim, J, formDegree, Jstar));
    636:   for (n = 0; n < nNodes; n++) {
    637:     for (i = 0; i < Nk; i++) {
    638:       PetscReal val = 0.;
    639:       for (j = 0; j < pNk; j++) val += nodeVec[n * pNk + j] * Jstar[j * Nk + i];
    640:       pfNodeVec[n * Nk + i] = val;
    641:     }
    642:   }
    643:   PetscCall(DMRestoreWorkArray(dm, pNk * Nk, MPIU_REAL, &Jstar));
    644:   PetscCall(DMRestoreWorkArray(dm, dim * dim, MPIU_REAL, &J));
    645:   PetscFunctionReturn(PETSC_SUCCESS);
    646: }

    648: /* given to sets of nodes, take the tensor product, where the product of the dof indices is concatenation and the
    649:  * product of the dof vectors is the wedge product */
    650: static PetscErrorCode PetscLagNodeIndicesTensor(PetscLagNodeIndices tracei, PetscInt dimT, PetscInt kT, PetscLagNodeIndices fiberi, PetscInt dimF, PetscInt kF, PetscLagNodeIndices *nodeIndices)
    651: {
    652:   PetscInt            dim = dimT + dimF;
    653:   PetscInt            nodeIdxDim, nNodes;
    654:   PetscInt            formDegree = kT + kF;
    655:   PetscInt            Nk, NkT, NkF;
    656:   PetscInt            MkT, MkF;
    657:   PetscLagNodeIndices ni;
    658:   PetscInt            i, j, l;
    659:   PetscReal          *projF, *projT;
    660:   PetscReal          *projFstar, *projTstar;
    661:   PetscReal          *workF, *workF2, *workT, *workT2, *work, *work2;
    662:   PetscReal          *wedgeMat;
    663:   PetscReal           sign;

    665:   PetscFunctionBegin;
    666:   PetscCall(PetscDTBinomialInt(dim, PetscAbsInt(formDegree), &Nk));
    667:   PetscCall(PetscDTBinomialInt(dimT, PetscAbsInt(kT), &NkT));
    668:   PetscCall(PetscDTBinomialInt(dimF, PetscAbsInt(kF), &NkF));
    669:   PetscCall(PetscDTBinomialInt(dim, PetscAbsInt(kT), &MkT));
    670:   PetscCall(PetscDTBinomialInt(dim, PetscAbsInt(kF), &MkF));
    671:   PetscCall(PetscNew(&ni));
    672:   ni->nodeIdxDim = nodeIdxDim = tracei->nodeIdxDim + fiberi->nodeIdxDim;
    673:   ni->nodeVecDim              = Nk;
    674:   ni->nNodes = nNodes = tracei->nNodes * fiberi->nNodes;
    675:   ni->refct           = 1;
    676:   PetscCall(PetscMalloc1(nNodes * nodeIdxDim, &ni->nodeIdx));
    677:   /* first concatenate the indices */
    678:   for (l = 0, j = 0; j < fiberi->nNodes; j++) {
    679:     for (i = 0; i < tracei->nNodes; i++, l++) {
    680:       PetscInt m, n = 0;

    682:       for (m = 0; m < tracei->nodeIdxDim; m++) ni->nodeIdx[l * nodeIdxDim + n++] = tracei->nodeIdx[i * tracei->nodeIdxDim + m];
    683:       for (m = 0; m < fiberi->nodeIdxDim; m++) ni->nodeIdx[l * nodeIdxDim + n++] = fiberi->nodeIdx[j * fiberi->nodeIdxDim + m];
    684:     }
    685:   }

    687:   /* now wedge together the push-forward vectors */
    688:   PetscCall(PetscMalloc1(nNodes * Nk, &ni->nodeVec));
    689:   PetscCall(PetscCalloc2(dimT * dim, &projT, dimF * dim, &projF));
    690:   for (i = 0; i < dimT; i++) projT[i * (dim + 1)] = 1.;
    691:   for (i = 0; i < dimF; i++) projF[i * (dim + dimT + 1) + dimT] = 1.;
    692:   PetscCall(PetscMalloc2(MkT * NkT, &projTstar, MkF * NkF, &projFstar));
    693:   PetscCall(PetscDTAltVPullbackMatrix(dim, dimT, projT, kT, projTstar));
    694:   PetscCall(PetscDTAltVPullbackMatrix(dim, dimF, projF, kF, projFstar));
    695:   PetscCall(PetscMalloc6(MkT, &workT, MkT, &workT2, MkF, &workF, MkF, &workF2, Nk, &work, Nk, &work2));
    696:   PetscCall(PetscMalloc1(Nk * MkT, &wedgeMat));
    697:   sign = (PetscAbsInt(kT * kF) & 1) ? -1. : 1.;
    698:   for (l = 0, j = 0; j < fiberi->nNodes; j++) {
    699:     PetscInt d, e;

    701:     /* push forward fiber k-form */
    702:     for (d = 0; d < MkF; d++) {
    703:       PetscReal val = 0.;
    704:       for (e = 0; e < NkF; e++) val += projFstar[d * NkF + e] * fiberi->nodeVec[j * NkF + e];
    705:       workF[d] = val;
    706:     }
    707:     /* Hodge star to proper form if necessary */
    708:     if (kF < 0) {
    709:       for (d = 0; d < MkF; d++) workF2[d] = workF[d];
    710:       PetscCall(PetscDTAltVStar(dim, PetscAbsInt(kF), 1, workF2, workF));
    711:     }
    712:     /* Compute the matrix that wedges this form with one of the trace k-form */
    713:     PetscCall(PetscDTAltVWedgeMatrix(dim, PetscAbsInt(kF), PetscAbsInt(kT), workF, wedgeMat));
    714:     for (i = 0; i < tracei->nNodes; i++, l++) {
    715:       /* push forward trace k-form */
    716:       for (d = 0; d < MkT; d++) {
    717:         PetscReal val = 0.;
    718:         for (e = 0; e < NkT; e++) val += projTstar[d * NkT + e] * tracei->nodeVec[i * NkT + e];
    719:         workT[d] = val;
    720:       }
    721:       /* Hodge star to proper form if necessary */
    722:       if (kT < 0) {
    723:         for (d = 0; d < MkT; d++) workT2[d] = workT[d];
    724:         PetscCall(PetscDTAltVStar(dim, PetscAbsInt(kT), 1, workT2, workT));
    725:       }
    726:       /* compute the wedge product of the push-forward trace form and firer forms */
    727:       for (d = 0; d < Nk; d++) {
    728:         PetscReal val = 0.;
    729:         for (e = 0; e < MkT; e++) val += wedgeMat[d * MkT + e] * workT[e];
    730:         work[d] = val;
    731:       }
    732:       /* inverse Hodge star from proper form if necessary */
    733:       if (formDegree < 0) {
    734:         for (d = 0; d < Nk; d++) work2[d] = work[d];
    735:         PetscCall(PetscDTAltVStar(dim, PetscAbsInt(formDegree), -1, work2, work));
    736:       }
    737:       /* insert into the array (adjusting for sign) */
    738:       for (d = 0; d < Nk; d++) ni->nodeVec[l * Nk + d] = sign * work[d];
    739:     }
    740:   }
    741:   PetscCall(PetscFree(wedgeMat));
    742:   PetscCall(PetscFree6(workT, workT2, workF, workF2, work, work2));
    743:   PetscCall(PetscFree2(projTstar, projFstar));
    744:   PetscCall(PetscFree2(projT, projF));
    745:   *nodeIndices = ni;
    746:   PetscFunctionReturn(PETSC_SUCCESS);
    747: }

    749: /* simple union of two sets of nodes */
    750: static PetscErrorCode PetscLagNodeIndicesMerge(PetscLagNodeIndices niA, PetscLagNodeIndices niB, PetscLagNodeIndices *nodeIndices)
    751: {
    752:   PetscLagNodeIndices ni;
    753:   PetscInt            nodeIdxDim, nodeVecDim, nNodes;

    755:   PetscFunctionBegin;
    756:   PetscCall(PetscNew(&ni));
    757:   ni->nodeIdxDim = nodeIdxDim = niA->nodeIdxDim;
    758:   PetscCheck(niB->nodeIdxDim == nodeIdxDim, PETSC_COMM_SELF, PETSC_ERR_ARG_INCOMP, "Cannot merge PetscLagNodeIndices with different nodeIdxDim");
    759:   ni->nodeVecDim = nodeVecDim = niA->nodeVecDim;
    760:   PetscCheck(niB->nodeVecDim == nodeVecDim, PETSC_COMM_SELF, PETSC_ERR_ARG_INCOMP, "Cannot merge PetscLagNodeIndices with different nodeVecDim");
    761:   ni->nNodes = nNodes = niA->nNodes + niB->nNodes;
    762:   ni->refct           = 1;
    763:   PetscCall(PetscMalloc1(nNodes * nodeIdxDim, &ni->nodeIdx));
    764:   PetscCall(PetscMalloc1(nNodes * nodeVecDim, &ni->nodeVec));
    765:   PetscCall(PetscArraycpy(ni->nodeIdx, niA->nodeIdx, niA->nNodes * nodeIdxDim));
    766:   PetscCall(PetscArraycpy(ni->nodeVec, niA->nodeVec, niA->nNodes * nodeVecDim));
    767:   PetscCall(PetscArraycpy(&ni->nodeIdx[niA->nNodes * nodeIdxDim], niB->nodeIdx, niB->nNodes * nodeIdxDim));
    768:   PetscCall(PetscArraycpy(&ni->nodeVec[niA->nNodes * nodeVecDim], niB->nodeVec, niB->nNodes * nodeVecDim));
    769:   *nodeIndices = ni;
    770:   PetscFunctionReturn(PETSC_SUCCESS);
    771: }

    773: #define PETSCTUPINTCOMPREVLEX(N) \
    774:   static int PetscConcat_(PetscTupIntCompRevlex_, N)(const void *a, const void *b) \
    775:   { \
    776:     const PetscInt *A = (const PetscInt *)a; \
    777:     const PetscInt *B = (const PetscInt *)b; \
    778:     int             i; \
    779:     PetscInt        diff = 0; \
    780:     for (i = 0; i < N; i++) { \
    781:       diff = A[N - i] - B[N - i]; \
    782:       if (diff) break; \
    783:     } \
    784:     return (diff <= 0) ? (diff < 0) ? -1 : 0 : 1; \
    785:   }

    787: PETSCTUPINTCOMPREVLEX(3)
    788: PETSCTUPINTCOMPREVLEX(4)
    789: PETSCTUPINTCOMPREVLEX(5)
    790: PETSCTUPINTCOMPREVLEX(6)
    791: PETSCTUPINTCOMPREVLEX(7)

    793: static int PetscTupIntCompRevlex_N(const void *a, const void *b)
    794: {
    795:   const PetscInt *A = (const PetscInt *)a;
    796:   const PetscInt *B = (const PetscInt *)b;
    797:   PetscInt        i;
    798:   PetscInt        N    = A[0];
    799:   PetscInt        diff = 0;
    800:   for (i = 0; i < N; i++) {
    801:     diff = A[N - i] - B[N - i];
    802:     if (diff) break;
    803:   }
    804:   return (diff <= 0) ? (diff < 0) ? -1 : 0 : 1;
    805: }

    807: /* The nodes are not necessarily in revlex order wrt nodeIdx: get the permutation
    808:  * that puts them in that order */
    809: static PetscErrorCode PetscLagNodeIndicesGetPermutation(PetscLagNodeIndices ni, PetscInt *perm[])
    810: {
    811:   PetscFunctionBegin;
    812:   if (!ni->perm) {
    813:     PetscInt *sorter;
    814:     PetscInt  m          = ni->nNodes;
    815:     PetscInt  nodeIdxDim = ni->nodeIdxDim;
    816:     PetscInt  i, j, k, l;
    817:     PetscInt *prm;
    818:     int (*comp)(const void *, const void *);

    820:     PetscCall(PetscMalloc1((nodeIdxDim + 2) * m, &sorter));
    821:     for (k = 0, l = 0, i = 0; i < m; i++) {
    822:       sorter[k++] = nodeIdxDim + 1;
    823:       sorter[k++] = i;
    824:       for (j = 0; j < nodeIdxDim; j++) sorter[k++] = ni->nodeIdx[l++];
    825:     }
    826:     switch (nodeIdxDim) {
    827:     case 2:
    828:       comp = PetscTupIntCompRevlex_3;
    829:       break;
    830:     case 3:
    831:       comp = PetscTupIntCompRevlex_4;
    832:       break;
    833:     case 4:
    834:       comp = PetscTupIntCompRevlex_5;
    835:       break;
    836:     case 5:
    837:       comp = PetscTupIntCompRevlex_6;
    838:       break;
    839:     case 6:
    840:       comp = PetscTupIntCompRevlex_7;
    841:       break;
    842:     default:
    843:       comp = PetscTupIntCompRevlex_N;
    844:       break;
    845:     }
    846:     qsort(sorter, m, (nodeIdxDim + 2) * sizeof(PetscInt), comp);
    847:     PetscCall(PetscMalloc1(m, &prm));
    848:     for (i = 0; i < m; i++) prm[i] = sorter[(nodeIdxDim + 2) * i + 1];
    849:     ni->perm = prm;
    850:     PetscCall(PetscFree(sorter));
    851:   }
    852:   *perm = ni->perm;
    853:   PetscFunctionReturn(PETSC_SUCCESS);
    854: }

    856: static PetscErrorCode PetscDualSpaceDestroy_Lagrange(PetscDualSpace sp)
    857: {
    858:   PetscDualSpace_Lag *lag = (PetscDualSpace_Lag *)sp->data;

    860:   PetscFunctionBegin;
    861:   if (lag->symperms) {
    862:     PetscInt **selfSyms = lag->symperms[0];

    864:     if (selfSyms) {
    865:       PetscInt i, **allocated = &selfSyms[-lag->selfSymOff];

    867:       for (i = 0; i < lag->numSelfSym; i++) PetscCall(PetscFree(allocated[i]));
    868:       PetscCall(PetscFree(allocated));
    869:     }
    870:     PetscCall(PetscFree(lag->symperms));
    871:   }
    872:   if (lag->symflips) {
    873:     PetscScalar **selfSyms = lag->symflips[0];

    875:     if (selfSyms) {
    876:       PetscInt      i;
    877:       PetscScalar **allocated = &selfSyms[-lag->selfSymOff];

    879:       for (i = 0; i < lag->numSelfSym; i++) PetscCall(PetscFree(allocated[i]));
    880:       PetscCall(PetscFree(allocated));
    881:     }
    882:     PetscCall(PetscFree(lag->symflips));
    883:   }
    884:   PetscCall(Petsc1DNodeFamilyDestroy(&lag->nodeFamily));
    885:   PetscCall(PetscLagNodeIndicesDestroy(&lag->vertIndices));
    886:   PetscCall(PetscLagNodeIndicesDestroy(&lag->intNodeIndices));
    887:   PetscCall(PetscLagNodeIndicesDestroy(&lag->allNodeIndices));
    888:   PetscCall(PetscFree(lag));
    889:   PetscCall(PetscObjectComposeFunction((PetscObject)sp, "PetscDualSpaceLagrangeGetContinuity_C", NULL));
    890:   PetscCall(PetscObjectComposeFunction((PetscObject)sp, "PetscDualSpaceLagrangeSetContinuity_C", NULL));
    891:   PetscCall(PetscObjectComposeFunction((PetscObject)sp, "PetscDualSpaceLagrangeGetTensor_C", NULL));
    892:   PetscCall(PetscObjectComposeFunction((PetscObject)sp, "PetscDualSpaceLagrangeSetTensor_C", NULL));
    893:   PetscCall(PetscObjectComposeFunction((PetscObject)sp, "PetscDualSpaceLagrangeGetTrimmed_C", NULL));
    894:   PetscCall(PetscObjectComposeFunction((PetscObject)sp, "PetscDualSpaceLagrangeSetTrimmed_C", NULL));
    895:   PetscCall(PetscObjectComposeFunction((PetscObject)sp, "PetscDualSpaceLagrangeGetNodeType_C", NULL));
    896:   PetscCall(PetscObjectComposeFunction((PetscObject)sp, "PetscDualSpaceLagrangeSetNodeType_C", NULL));
    897:   PetscCall(PetscObjectComposeFunction((PetscObject)sp, "PetscDualSpaceLagrangeGetUseMoments_C", NULL));
    898:   PetscCall(PetscObjectComposeFunction((PetscObject)sp, "PetscDualSpaceLagrangeSetUseMoments_C", NULL));
    899:   PetscCall(PetscObjectComposeFunction((PetscObject)sp, "PetscDualSpaceLagrangeGetMomentOrder_C", NULL));
    900:   PetscCall(PetscObjectComposeFunction((PetscObject)sp, "PetscDualSpaceLagrangeSetMomentOrder_C", NULL));
    901:   PetscFunctionReturn(PETSC_SUCCESS);
    902: }

    904: static PetscErrorCode PetscDualSpaceLagrangeView_Ascii(PetscDualSpace sp, PetscViewer viewer)
    905: {
    906:   PetscDualSpace_Lag *lag = (PetscDualSpace_Lag *)sp->data;

    908:   PetscFunctionBegin;
    909:   PetscCall(PetscViewerASCIIPrintf(viewer, "%s %s%sLagrange dual space\n", lag->continuous ? "Continuous" : "Discontinuous", lag->tensorSpace ? "tensor " : "", lag->trimmed ? "trimmed " : ""));
    910:   PetscFunctionReturn(PETSC_SUCCESS);
    911: }

    913: static PetscErrorCode PetscDualSpaceView_Lagrange(PetscDualSpace sp, PetscViewer viewer)
    914: {
    915:   PetscBool iascii;

    917:   PetscFunctionBegin;
    920:   PetscCall(PetscObjectTypeCompare((PetscObject)viewer, PETSCVIEWERASCII, &iascii));
    921:   if (iascii) PetscCall(PetscDualSpaceLagrangeView_Ascii(sp, viewer));
    922:   PetscFunctionReturn(PETSC_SUCCESS);
    923: }

    925: static PetscErrorCode PetscDualSpaceSetFromOptions_Lagrange(PetscDualSpace sp, PetscOptionItems *PetscOptionsObject)
    926: {
    927:   PetscBool       continuous, tensor, trimmed, flg, flg2, flg3;
    928:   PetscDTNodeType nodeType;
    929:   PetscReal       nodeExponent;
    930:   PetscInt        momentOrder;
    931:   PetscBool       nodeEndpoints, useMoments;

    933:   PetscFunctionBegin;
    934:   PetscCall(PetscDualSpaceLagrangeGetContinuity(sp, &continuous));
    935:   PetscCall(PetscDualSpaceLagrangeGetTensor(sp, &tensor));
    936:   PetscCall(PetscDualSpaceLagrangeGetTrimmed(sp, &trimmed));
    937:   PetscCall(PetscDualSpaceLagrangeGetNodeType(sp, &nodeType, &nodeEndpoints, &nodeExponent));
    938:   if (nodeType == PETSCDTNODES_DEFAULT) nodeType = PETSCDTNODES_GAUSSJACOBI;
    939:   PetscCall(PetscDualSpaceLagrangeGetUseMoments(sp, &useMoments));
    940:   PetscCall(PetscDualSpaceLagrangeGetMomentOrder(sp, &momentOrder));
    941:   PetscOptionsHeadBegin(PetscOptionsObject, "PetscDualSpace Lagrange Options");
    942:   PetscCall(PetscOptionsBool("-petscdualspace_lagrange_continuity", "Flag for continuous element", "PetscDualSpaceLagrangeSetContinuity", continuous, &continuous, &flg));
    943:   if (flg) PetscCall(PetscDualSpaceLagrangeSetContinuity(sp, continuous));
    944:   PetscCall(PetscOptionsBool("-petscdualspace_lagrange_tensor", "Flag for tensor dual space", "PetscDualSpaceLagrangeSetTensor", tensor, &tensor, &flg));
    945:   if (flg) PetscCall(PetscDualSpaceLagrangeSetTensor(sp, tensor));
    946:   PetscCall(PetscOptionsBool("-petscdualspace_lagrange_trimmed", "Flag for trimmed dual space", "PetscDualSpaceLagrangeSetTrimmed", trimmed, &trimmed, &flg));
    947:   if (flg) PetscCall(PetscDualSpaceLagrangeSetTrimmed(sp, trimmed));
    948:   PetscCall(PetscOptionsEnum("-petscdualspace_lagrange_node_type", "Lagrange node location type", "PetscDualSpaceLagrangeSetNodeType", PetscDTNodeTypes, (PetscEnum)nodeType, (PetscEnum *)&nodeType, &flg));
    949:   PetscCall(PetscOptionsBool("-petscdualspace_lagrange_node_endpoints", "Flag for nodes that include endpoints", "PetscDualSpaceLagrangeSetNodeType", nodeEndpoints, &nodeEndpoints, &flg2));
    950:   flg3 = PETSC_FALSE;
    951:   if (nodeType == PETSCDTNODES_GAUSSJACOBI) PetscCall(PetscOptionsReal("-petscdualspace_lagrange_node_exponent", "Gauss-Jacobi weight function exponent", "PetscDualSpaceLagrangeSetNodeType", nodeExponent, &nodeExponent, &flg3));
    952:   if (flg || flg2 || flg3) PetscCall(PetscDualSpaceLagrangeSetNodeType(sp, nodeType, nodeEndpoints, nodeExponent));
    953:   PetscCall(PetscOptionsBool("-petscdualspace_lagrange_use_moments", "Use moments (where appropriate) for functionals", "PetscDualSpaceLagrangeSetUseMoments", useMoments, &useMoments, &flg));
    954:   if (flg) PetscCall(PetscDualSpaceLagrangeSetUseMoments(sp, useMoments));
    955:   PetscCall(PetscOptionsInt("-petscdualspace_lagrange_moment_order", "Quadrature order for moment functionals", "PetscDualSpaceLagrangeSetMomentOrder", momentOrder, &momentOrder, &flg));
    956:   if (flg) PetscCall(PetscDualSpaceLagrangeSetMomentOrder(sp, momentOrder));
    957:   PetscOptionsHeadEnd();
    958:   PetscFunctionReturn(PETSC_SUCCESS);
    959: }

    961: static PetscErrorCode PetscDualSpaceDuplicate_Lagrange(PetscDualSpace sp, PetscDualSpace spNew)
    962: {
    963:   PetscBool           cont, tensor, trimmed, boundary;
    964:   PetscDTNodeType     nodeType;
    965:   PetscReal           exponent;
    966:   PetscDualSpace_Lag *lag = (PetscDualSpace_Lag *)sp->data;

    968:   PetscFunctionBegin;
    969:   PetscCall(PetscDualSpaceLagrangeGetContinuity(sp, &cont));
    970:   PetscCall(PetscDualSpaceLagrangeSetContinuity(spNew, cont));
    971:   PetscCall(PetscDualSpaceLagrangeGetTensor(sp, &tensor));
    972:   PetscCall(PetscDualSpaceLagrangeSetTensor(spNew, tensor));
    973:   PetscCall(PetscDualSpaceLagrangeGetTrimmed(sp, &trimmed));
    974:   PetscCall(PetscDualSpaceLagrangeSetTrimmed(spNew, trimmed));
    975:   PetscCall(PetscDualSpaceLagrangeGetNodeType(sp, &nodeType, &boundary, &exponent));
    976:   PetscCall(PetscDualSpaceLagrangeSetNodeType(spNew, nodeType, boundary, exponent));
    977:   if (lag->nodeFamily) {
    978:     PetscDualSpace_Lag *lagnew = (PetscDualSpace_Lag *)spNew->data;

    980:     PetscCall(Petsc1DNodeFamilyReference(lag->nodeFamily));
    981:     lagnew->nodeFamily = lag->nodeFamily;
    982:   }
    983:   PetscFunctionReturn(PETSC_SUCCESS);
    984: }

    986: /* for making tensor product spaces: take a dual space and product a segment space that has all the same
    987:  * specifications (trimmed, continuous, order, node set), except for the form degree */
    988: static PetscErrorCode PetscDualSpaceCreateEdgeSubspace_Lagrange(PetscDualSpace sp, PetscInt order, PetscInt k, PetscInt Nc, PetscBool interiorOnly, PetscDualSpace *bdsp)
    989: {
    990:   DM                  K;
    991:   PetscDualSpace_Lag *newlag;

    993:   PetscFunctionBegin;
    994:   PetscCall(PetscDualSpaceDuplicate(sp, bdsp));
    995:   PetscCall(PetscDualSpaceSetFormDegree(*bdsp, k));
    996:   PetscCall(DMPlexCreateReferenceCell(PETSC_COMM_SELF, DMPolytopeTypeSimpleShape(1, PETSC_TRUE), &K));
    997:   PetscCall(PetscDualSpaceSetDM(*bdsp, K));
    998:   PetscCall(DMDestroy(&K));
    999:   PetscCall(PetscDualSpaceSetOrder(*bdsp, order));
    1000:   PetscCall(PetscDualSpaceSetNumComponents(*bdsp, Nc));
    1001:   newlag               = (PetscDualSpace_Lag *)(*bdsp)->data;
    1002:   newlag->interiorOnly = interiorOnly;
    1003:   PetscCall(PetscDualSpaceSetUp(*bdsp));
    1004:   PetscFunctionReturn(PETSC_SUCCESS);
    1005: }

    1007: /* just the points, weights aren't handled */
    1008: static PetscErrorCode PetscQuadratureCreateTensor(PetscQuadrature trace, PetscQuadrature fiber, PetscQuadrature *product)
    1009: {
    1010:   PetscInt         dimTrace, dimFiber;
    1011:   PetscInt         numPointsTrace, numPointsFiber;
    1012:   PetscInt         dim, numPoints;
    1013:   const PetscReal *pointsTrace;
    1014:   const PetscReal *pointsFiber;
    1015:   PetscReal       *points;
    1016:   PetscInt         i, j, k, p;

    1018:   PetscFunctionBegin;
    1019:   PetscCall(PetscQuadratureGetData(trace, &dimTrace, NULL, &numPointsTrace, &pointsTrace, NULL));
    1020:   PetscCall(PetscQuadratureGetData(fiber, &dimFiber, NULL, &numPointsFiber, &pointsFiber, NULL));
    1021:   dim       = dimTrace + dimFiber;
    1022:   numPoints = numPointsFiber * numPointsTrace;
    1023:   PetscCall(PetscMalloc1(numPoints * dim, &points));
    1024:   for (p = 0, j = 0; j < numPointsFiber; j++) {
    1025:     for (i = 0; i < numPointsTrace; i++, p++) {
    1026:       for (k = 0; k < dimTrace; k++) points[p * dim + k] = pointsTrace[i * dimTrace + k];
    1027:       for (k = 0; k < dimFiber; k++) points[p * dim + dimTrace + k] = pointsFiber[j * dimFiber + k];
    1028:     }
    1029:   }
    1030:   PetscCall(PetscQuadratureCreate(PETSC_COMM_SELF, product));
    1031:   PetscCall(PetscQuadratureSetData(*product, dim, 0, numPoints, points, NULL));
    1032:   PetscFunctionReturn(PETSC_SUCCESS);
    1033: }

    1035: /* Kronecker tensor product where matrix is considered a matrix of k-forms, so that
    1036:  * the entries in the product matrix are wedge products of the entries in the original matrices */
    1037: static PetscErrorCode MatTensorAltV(Mat trace, Mat fiber, PetscInt dimTrace, PetscInt kTrace, PetscInt dimFiber, PetscInt kFiber, Mat *product)
    1038: {
    1039:   PetscInt     mTrace, nTrace, mFiber, nFiber, m, n, k, i, j, l;
    1040:   PetscInt     dim, NkTrace, NkFiber, Nk;
    1041:   PetscInt     dT, dF;
    1042:   PetscInt    *nnzTrace, *nnzFiber, *nnz;
    1043:   PetscInt     iT, iF, jT, jF, il, jl;
    1044:   PetscReal   *workT, *workT2, *workF, *workF2, *work, *workstar;
    1045:   PetscReal   *projT, *projF;
    1046:   PetscReal   *projTstar, *projFstar;
    1047:   PetscReal   *wedgeMat;
    1048:   PetscReal    sign;
    1049:   PetscScalar *workS;
    1050:   Mat          prod;
    1051:   /* this produces dof groups that look like the identity */

    1053:   PetscFunctionBegin;
    1054:   PetscCall(MatGetSize(trace, &mTrace, &nTrace));
    1055:   PetscCall(PetscDTBinomialInt(dimTrace, PetscAbsInt(kTrace), &NkTrace));
    1056:   PetscCheck(nTrace % NkTrace == 0, PETSC_COMM_SELF, PETSC_ERR_PLIB, "point value space of trace matrix is not a multiple of k-form size");
    1057:   PetscCall(MatGetSize(fiber, &mFiber, &nFiber));
    1058:   PetscCall(PetscDTBinomialInt(dimFiber, PetscAbsInt(kFiber), &NkFiber));
    1059:   PetscCheck(nFiber % NkFiber == 0, PETSC_COMM_SELF, PETSC_ERR_PLIB, "point value space of fiber matrix is not a multiple of k-form size");
    1060:   PetscCall(PetscMalloc2(mTrace, &nnzTrace, mFiber, &nnzFiber));
    1061:   for (i = 0; i < mTrace; i++) {
    1062:     PetscCall(MatGetRow(trace, i, &nnzTrace[i], NULL, NULL));
    1063:     PetscCheck(nnzTrace[i] % NkTrace == 0, PETSC_COMM_SELF, PETSC_ERR_PLIB, "nonzeros in trace matrix are not in k-form size blocks");
    1064:   }
    1065:   for (i = 0; i < mFiber; i++) {
    1066:     PetscCall(MatGetRow(fiber, i, &nnzFiber[i], NULL, NULL));
    1067:     PetscCheck(nnzFiber[i] % NkFiber == 0, PETSC_COMM_SELF, PETSC_ERR_PLIB, "nonzeros in fiber matrix are not in k-form size blocks");
    1068:   }
    1069:   dim = dimTrace + dimFiber;
    1070:   k   = kFiber + kTrace;
    1071:   PetscCall(PetscDTBinomialInt(dim, PetscAbsInt(k), &Nk));
    1072:   m = mTrace * mFiber;
    1073:   PetscCall(PetscMalloc1(m, &nnz));
    1074:   for (l = 0, j = 0; j < mFiber; j++)
    1075:     for (i = 0; i < mTrace; i++, l++) nnz[l] = (nnzTrace[i] / NkTrace) * (nnzFiber[j] / NkFiber) * Nk;
    1076:   n = (nTrace / NkTrace) * (nFiber / NkFiber) * Nk;
    1077:   PetscCall(MatCreateSeqAIJ(PETSC_COMM_SELF, m, n, 0, nnz, &prod));
    1078:   PetscCall(PetscObjectSetOptionsPrefix((PetscObject)prod, "altv_"));
    1079:   PetscCall(PetscFree(nnz));
    1080:   PetscCall(PetscFree2(nnzTrace, nnzFiber));
    1081:   /* reasoning about which points each dof needs depends on having zeros computed at points preserved */
    1082:   PetscCall(MatSetOption(prod, MAT_IGNORE_ZERO_ENTRIES, PETSC_FALSE));
    1083:   /* compute pullbacks */
    1084:   PetscCall(PetscDTBinomialInt(dim, PetscAbsInt(kTrace), &dT));
    1085:   PetscCall(PetscDTBinomialInt(dim, PetscAbsInt(kFiber), &dF));
    1086:   PetscCall(PetscMalloc4(dimTrace * dim, &projT, dimFiber * dim, &projF, dT * NkTrace, &projTstar, dF * NkFiber, &projFstar));
    1087:   PetscCall(PetscArrayzero(projT, dimTrace * dim));
    1088:   for (i = 0; i < dimTrace; i++) projT[i * (dim + 1)] = 1.;
    1089:   PetscCall(PetscArrayzero(projF, dimFiber * dim));
    1090:   for (i = 0; i < dimFiber; i++) projF[i * (dim + 1) + dimTrace] = 1.;
    1091:   PetscCall(PetscDTAltVPullbackMatrix(dim, dimTrace, projT, kTrace, projTstar));
    1092:   PetscCall(PetscDTAltVPullbackMatrix(dim, dimFiber, projF, kFiber, projFstar));
    1093:   PetscCall(PetscMalloc5(dT, &workT, dF, &workF, Nk, &work, Nk, &workstar, Nk, &workS));
    1094:   PetscCall(PetscMalloc2(dT, &workT2, dF, &workF2));
    1095:   PetscCall(PetscMalloc1(Nk * dT, &wedgeMat));
    1096:   sign = (PetscAbsInt(kTrace * kFiber) & 1) ? -1. : 1.;
    1097:   for (i = 0, iF = 0; iF < mFiber; iF++) {
    1098:     PetscInt           ncolsF, nformsF;
    1099:     const PetscInt    *colsF;
    1100:     const PetscScalar *valsF;

    1102:     PetscCall(MatGetRow(fiber, iF, &ncolsF, &colsF, &valsF));
    1103:     nformsF = ncolsF / NkFiber;
    1104:     for (iT = 0; iT < mTrace; iT++, i++) {
    1105:       PetscInt           ncolsT, nformsT;
    1106:       const PetscInt    *colsT;
    1107:       const PetscScalar *valsT;

    1109:       PetscCall(MatGetRow(trace, iT, &ncolsT, &colsT, &valsT));
    1110:       nformsT = ncolsT / NkTrace;
    1111:       for (j = 0, jF = 0; jF < nformsF; jF++) {
    1112:         PetscInt colF = colsF[jF * NkFiber] / NkFiber;

    1114:         for (il = 0; il < dF; il++) {
    1115:           PetscReal val = 0.;
    1116:           for (jl = 0; jl < NkFiber; jl++) val += projFstar[il * NkFiber + jl] * PetscRealPart(valsF[jF * NkFiber + jl]);
    1117:           workF[il] = val;
    1118:         }
    1119:         if (kFiber < 0) {
    1120:           for (il = 0; il < dF; il++) workF2[il] = workF[il];
    1121:           PetscCall(PetscDTAltVStar(dim, PetscAbsInt(kFiber), 1, workF2, workF));
    1122:         }
    1123:         PetscCall(PetscDTAltVWedgeMatrix(dim, PetscAbsInt(kFiber), PetscAbsInt(kTrace), workF, wedgeMat));
    1124:         for (jT = 0; jT < nformsT; jT++, j++) {
    1125:           PetscInt           colT = colsT[jT * NkTrace] / NkTrace;
    1126:           PetscInt           col  = colF * (nTrace / NkTrace) + colT;
    1127:           const PetscScalar *vals;

    1129:           for (il = 0; il < dT; il++) {
    1130:             PetscReal val = 0.;
    1131:             for (jl = 0; jl < NkTrace; jl++) val += projTstar[il * NkTrace + jl] * PetscRealPart(valsT[jT * NkTrace + jl]);
    1132:             workT[il] = val;
    1133:           }
    1134:           if (kTrace < 0) {
    1135:             for (il = 0; il < dT; il++) workT2[il] = workT[il];
    1136:             PetscCall(PetscDTAltVStar(dim, PetscAbsInt(kTrace), 1, workT2, workT));
    1137:           }

    1139:           for (il = 0; il < Nk; il++) {
    1140:             PetscReal val = 0.;
    1141:             for (jl = 0; jl < dT; jl++) val += sign * wedgeMat[il * dT + jl] * workT[jl];
    1142:             work[il] = val;
    1143:           }
    1144:           if (k < 0) {
    1145:             PetscCall(PetscDTAltVStar(dim, PetscAbsInt(k), -1, work, workstar));
    1146: #if defined(PETSC_USE_COMPLEX)
    1147:             for (l = 0; l < Nk; l++) workS[l] = workstar[l];
    1148:             vals = &workS[0];
    1149: #else
    1150:             vals = &workstar[0];
    1151: #endif
    1152:           } else {
    1153: #if defined(PETSC_USE_COMPLEX)
    1154:             for (l = 0; l < Nk; l++) workS[l] = work[l];
    1155:             vals = &workS[0];
    1156: #else
    1157:             vals = &work[0];
    1158: #endif
    1159:           }
    1160:           for (l = 0; l < Nk; l++) PetscCall(MatSetValue(prod, i, col * Nk + l, vals[l], INSERT_VALUES)); /* Nk */
    1161:         } /* jT */
    1162:       } /* jF */
    1163:       PetscCall(MatRestoreRow(trace, iT, &ncolsT, &colsT, &valsT));
    1164:     } /* iT */
    1165:     PetscCall(MatRestoreRow(fiber, iF, &ncolsF, &colsF, &valsF));
    1166:   } /* iF */
    1167:   PetscCall(PetscFree(wedgeMat));
    1168:   PetscCall(PetscFree4(projT, projF, projTstar, projFstar));
    1169:   PetscCall(PetscFree2(workT2, workF2));
    1170:   PetscCall(PetscFree5(workT, workF, work, workstar, workS));
    1171:   PetscCall(MatAssemblyBegin(prod, MAT_FINAL_ASSEMBLY));
    1172:   PetscCall(MatAssemblyEnd(prod, MAT_FINAL_ASSEMBLY));
    1173:   *product = prod;
    1174:   PetscFunctionReturn(PETSC_SUCCESS);
    1175: }

    1177: /* Union of quadrature points, with an attempt to identify common points in the two sets */
    1178: static PetscErrorCode PetscQuadraturePointsMerge(PetscQuadrature quadA, PetscQuadrature quadB, PetscQuadrature *quadJoint, PetscInt *aToJoint[], PetscInt *bToJoint[])
    1179: {
    1180:   PetscInt         dimA, dimB;
    1181:   PetscInt         nA, nB, nJoint, i, j, d;
    1182:   const PetscReal *pointsA;
    1183:   const PetscReal *pointsB;
    1184:   PetscReal       *pointsJoint;
    1185:   PetscInt        *aToJ, *bToJ;
    1186:   PetscQuadrature  qJ;

    1188:   PetscFunctionBegin;
    1189:   PetscCall(PetscQuadratureGetData(quadA, &dimA, NULL, &nA, &pointsA, NULL));
    1190:   PetscCall(PetscQuadratureGetData(quadB, &dimB, NULL, &nB, &pointsB, NULL));
    1191:   PetscCheck(dimA == dimB, PETSC_COMM_SELF, PETSC_ERR_ARG_INCOMP, "Quadrature points must be in the same dimension");
    1192:   nJoint = nA;
    1193:   PetscCall(PetscMalloc1(nA, &aToJ));
    1194:   for (i = 0; i < nA; i++) aToJ[i] = i;
    1195:   PetscCall(PetscMalloc1(nB, &bToJ));
    1196:   for (i = 0; i < nB; i++) {
    1197:     for (j = 0; j < nA; j++) {
    1198:       bToJ[i] = -1;
    1199:       for (d = 0; d < dimA; d++)
    1200:         if (PetscAbsReal(pointsB[i * dimA + d] - pointsA[j * dimA + d]) > PETSC_SMALL) break;
    1201:       if (d == dimA) {
    1202:         bToJ[i] = j;
    1203:         break;
    1204:       }
    1205:     }
    1206:     if (bToJ[i] == -1) bToJ[i] = nJoint++;
    1207:   }
    1208:   *aToJoint = aToJ;
    1209:   *bToJoint = bToJ;
    1210:   PetscCall(PetscMalloc1(nJoint * dimA, &pointsJoint));
    1211:   PetscCall(PetscArraycpy(pointsJoint, pointsA, nA * dimA));
    1212:   for (i = 0; i < nB; i++) {
    1213:     if (bToJ[i] >= nA) {
    1214:       for (d = 0; d < dimA; d++) pointsJoint[bToJ[i] * dimA + d] = pointsB[i * dimA + d];
    1215:     }
    1216:   }
    1217:   PetscCall(PetscQuadratureCreate(PETSC_COMM_SELF, &qJ));
    1218:   PetscCall(PetscQuadratureSetData(qJ, dimA, 0, nJoint, pointsJoint, NULL));
    1219:   *quadJoint = qJ;
    1220:   PetscFunctionReturn(PETSC_SUCCESS);
    1221: }

    1223: /* Matrices matA and matB are both quadrature -> dof matrices: produce a matrix that is joint quadrature -> union of
    1224:  * dofs, where the joint quadrature was produced by PetscQuadraturePointsMerge */
    1225: static PetscErrorCode MatricesMerge(Mat matA, Mat matB, PetscInt dim, PetscInt k, PetscInt numMerged, const PetscInt aToMerged[], const PetscInt bToMerged[], Mat *matMerged)
    1226: {
    1227:   PetscInt  m, n, mA, nA, mB, nB, Nk, i, j, l;
    1228:   Mat       M;
    1229:   PetscInt *nnz;
    1230:   PetscInt  maxnnz;
    1231:   PetscInt *work;

    1233:   PetscFunctionBegin;
    1234:   PetscCall(PetscDTBinomialInt(dim, PetscAbsInt(k), &Nk));
    1235:   PetscCall(MatGetSize(matA, &mA, &nA));
    1236:   PetscCheck(nA % Nk == 0, PETSC_COMM_SELF, PETSC_ERR_ARG_SIZ, "matA column space not a multiple of k-form size");
    1237:   PetscCall(MatGetSize(matB, &mB, &nB));
    1238:   PetscCheck(nB % Nk == 0, PETSC_COMM_SELF, PETSC_ERR_ARG_SIZ, "matB column space not a multiple of k-form size");
    1239:   m = mA + mB;
    1240:   n = numMerged * Nk;
    1241:   PetscCall(PetscMalloc1(m, &nnz));
    1242:   maxnnz = 0;
    1243:   for (i = 0; i < mA; i++) {
    1244:     PetscCall(MatGetRow(matA, i, &nnz[i], NULL, NULL));
    1245:     PetscCheck(nnz[i] % Nk == 0, PETSC_COMM_SELF, PETSC_ERR_PLIB, "nonzeros in matA are not in k-form size blocks");
    1246:     maxnnz = PetscMax(maxnnz, nnz[i]);
    1247:   }
    1248:   for (i = 0; i < mB; i++) {
    1249:     PetscCall(MatGetRow(matB, i, &nnz[i + mA], NULL, NULL));
    1250:     PetscCheck(nnz[i + mA] % Nk == 0, PETSC_COMM_SELF, PETSC_ERR_PLIB, "nonzeros in matB are not in k-form size blocks");
    1251:     maxnnz = PetscMax(maxnnz, nnz[i + mA]);
    1252:   }
    1253:   PetscCall(MatCreateSeqAIJ(PETSC_COMM_SELF, m, n, 0, nnz, &M));
    1254:   PetscCall(PetscObjectSetOptionsPrefix((PetscObject)M, "altv_"));
    1255:   PetscCall(PetscFree(nnz));
    1256:   /* reasoning about which points each dof needs depends on having zeros computed at points preserved */
    1257:   PetscCall(MatSetOption(M, MAT_IGNORE_ZERO_ENTRIES, PETSC_FALSE));
    1258:   PetscCall(PetscMalloc1(maxnnz, &work));
    1259:   for (i = 0; i < mA; i++) {
    1260:     const PetscInt    *cols;
    1261:     const PetscScalar *vals;
    1262:     PetscInt           nCols;
    1263:     PetscCall(MatGetRow(matA, i, &nCols, &cols, &vals));
    1264:     for (j = 0; j < nCols / Nk; j++) {
    1265:       PetscInt newCol = aToMerged[cols[j * Nk] / Nk];
    1266:       for (l = 0; l < Nk; l++) work[j * Nk + l] = newCol * Nk + l;
    1267:     }
    1268:     PetscCall(MatSetValuesBlocked(M, 1, &i, nCols, work, vals, INSERT_VALUES));
    1269:     PetscCall(MatRestoreRow(matA, i, &nCols, &cols, &vals));
    1270:   }
    1271:   for (i = 0; i < mB; i++) {
    1272:     const PetscInt    *cols;
    1273:     const PetscScalar *vals;

    1275:     PetscInt row = i + mA;
    1276:     PetscInt nCols;
    1277:     PetscCall(MatGetRow(matB, i, &nCols, &cols, &vals));
    1278:     for (j = 0; j < nCols / Nk; j++) {
    1279:       PetscInt newCol = bToMerged[cols[j * Nk] / Nk];
    1280:       for (l = 0; l < Nk; l++) work[j * Nk + l] = newCol * Nk + l;
    1281:     }
    1282:     PetscCall(MatSetValuesBlocked(M, 1, &row, nCols, work, vals, INSERT_VALUES));
    1283:     PetscCall(MatRestoreRow(matB, i, &nCols, &cols, &vals));
    1284:   }
    1285:   PetscCall(PetscFree(work));
    1286:   PetscCall(MatAssemblyBegin(M, MAT_FINAL_ASSEMBLY));
    1287:   PetscCall(MatAssemblyEnd(M, MAT_FINAL_ASSEMBLY));
    1288:   *matMerged = M;
    1289:   PetscFunctionReturn(PETSC_SUCCESS);
    1290: }

    1292: /* Take a dual space and product a segment space that has all the same specifications (trimmed, continuous, order,
    1293:  * node set), except for the form degree.  For computing boundary dofs and for making tensor product spaces */
    1294: static PetscErrorCode PetscDualSpaceCreateFacetSubspace_Lagrange(PetscDualSpace sp, DM K, PetscInt f, PetscInt k, PetscInt Ncopies, PetscBool interiorOnly, PetscDualSpace *bdsp)
    1295: {
    1296:   PetscInt            Nknew, Ncnew;
    1297:   PetscInt            dim, pointDim = -1;
    1298:   PetscInt            depth;
    1299:   DM                  dm;
    1300:   PetscDualSpace_Lag *newlag;

    1302:   PetscFunctionBegin;
    1303:   PetscCall(PetscDualSpaceGetDM(sp, &dm));
    1304:   PetscCall(DMGetDimension(dm, &dim));
    1305:   PetscCall(DMPlexGetDepth(dm, &depth));
    1306:   PetscCall(PetscDualSpaceDuplicate(sp, bdsp));
    1307:   PetscCall(PetscDualSpaceSetFormDegree(*bdsp, k));
    1308:   if (!K) {
    1309:     if (depth == dim) {
    1310:       DMPolytopeType ct;

    1312:       pointDim = dim - 1;
    1313:       PetscCall(DMPlexGetCellType(dm, f, &ct));
    1314:       PetscCall(DMPlexCreateReferenceCell(PETSC_COMM_SELF, ct, &K));
    1315:     } else if (depth == 1) {
    1316:       pointDim = 0;
    1317:       PetscCall(DMPlexCreateReferenceCell(PETSC_COMM_SELF, DM_POLYTOPE_POINT, &K));
    1318:     } else SETERRQ(PETSC_COMM_SELF, PETSC_ERR_PLIB, "Unsupported interpolation state of reference element");
    1319:   } else {
    1320:     PetscCall(PetscObjectReference((PetscObject)K));
    1321:     PetscCall(DMGetDimension(K, &pointDim));
    1322:   }
    1323:   PetscCall(PetscDualSpaceSetDM(*bdsp, K));
    1324:   PetscCall(DMDestroy(&K));
    1325:   PetscCall(PetscDTBinomialInt(pointDim, PetscAbsInt(k), &Nknew));
    1326:   Ncnew = Nknew * Ncopies;
    1327:   PetscCall(PetscDualSpaceSetNumComponents(*bdsp, Ncnew));
    1328:   newlag               = (PetscDualSpace_Lag *)(*bdsp)->data;
    1329:   newlag->interiorOnly = interiorOnly;
    1330:   PetscCall(PetscDualSpaceSetUp(*bdsp));
    1331:   PetscFunctionReturn(PETSC_SUCCESS);
    1332: }

    1334: /* Construct simplex nodes from a nodefamily, add Nk dof vectors of length Nk at each node.
    1335:  * Return the (quadrature, matrix) form of the dofs and the nodeIndices form as well.
    1336:  *
    1337:  * Sometimes we want a set of nodes to be contained in the interior of the element,
    1338:  * even when the node scheme puts nodes on the boundaries.  numNodeSkip tells
    1339:  * the routine how many "layers" of nodes need to be skipped.
    1340:  * */
    1341: static PetscErrorCode PetscDualSpaceLagrangeCreateSimplexNodeMat(Petsc1DNodeFamily nodeFamily, PetscInt dim, PetscInt sum, PetscInt Nk, PetscInt numNodeSkip, PetscQuadrature *iNodes, Mat *iMat, PetscLagNodeIndices *nodeIndices)
    1342: {
    1343:   PetscReal          *extraNodeCoords, *nodeCoords;
    1344:   PetscInt            nNodes, nExtraNodes;
    1345:   PetscInt            i, j, k, extraSum = sum + numNodeSkip * (1 + dim);
    1346:   PetscQuadrature     intNodes;
    1347:   Mat                 intMat;
    1348:   PetscLagNodeIndices ni;

    1350:   PetscFunctionBegin;
    1351:   PetscCall(PetscDTBinomialInt(dim + sum, dim, &nNodes));
    1352:   PetscCall(PetscDTBinomialInt(dim + extraSum, dim, &nExtraNodes));

    1354:   PetscCall(PetscMalloc1(dim * nExtraNodes, &extraNodeCoords));
    1355:   PetscCall(PetscNew(&ni));
    1356:   ni->nodeIdxDim = dim + 1;
    1357:   ni->nodeVecDim = Nk;
    1358:   ni->nNodes     = nNodes * Nk;
    1359:   ni->refct      = 1;
    1360:   PetscCall(PetscMalloc1(nNodes * Nk * (dim + 1), &ni->nodeIdx));
    1361:   PetscCall(PetscMalloc1(nNodes * Nk * Nk, &ni->nodeVec));
    1362:   for (i = 0; i < nNodes; i++)
    1363:     for (j = 0; j < Nk; j++)
    1364:       for (k = 0; k < Nk; k++) ni->nodeVec[(i * Nk + j) * Nk + k] = (j == k) ? 1. : 0.;
    1365:   PetscCall(Petsc1DNodeFamilyComputeSimplexNodes(nodeFamily, dim, extraSum, extraNodeCoords));
    1366:   if (numNodeSkip) {
    1367:     PetscInt  k;
    1368:     PetscInt *tup;

    1370:     PetscCall(PetscMalloc1(dim * nNodes, &nodeCoords));
    1371:     PetscCall(PetscMalloc1(dim + 1, &tup));
    1372:     for (k = 0; k < nNodes; k++) {
    1373:       PetscInt j, c;
    1374:       PetscInt index;

    1376:       PetscCall(PetscDTIndexToBary(dim + 1, sum, k, tup));
    1377:       for (j = 0; j < dim + 1; j++) tup[j] += numNodeSkip;
    1378:       for (c = 0; c < Nk; c++) {
    1379:         for (j = 0; j < dim + 1; j++) ni->nodeIdx[(k * Nk + c) * (dim + 1) + j] = tup[j] + 1;
    1380:       }
    1381:       PetscCall(PetscDTBaryToIndex(dim + 1, extraSum, tup, &index));
    1382:       for (j = 0; j < dim; j++) nodeCoords[k * dim + j] = extraNodeCoords[index * dim + j];
    1383:     }
    1384:     PetscCall(PetscFree(tup));
    1385:     PetscCall(PetscFree(extraNodeCoords));
    1386:   } else {
    1387:     PetscInt  k;
    1388:     PetscInt *tup;

    1390:     nodeCoords = extraNodeCoords;
    1391:     PetscCall(PetscMalloc1(dim + 1, &tup));
    1392:     for (k = 0; k < nNodes; k++) {
    1393:       PetscInt j, c;

    1395:       PetscCall(PetscDTIndexToBary(dim + 1, sum, k, tup));
    1396:       for (c = 0; c < Nk; c++) {
    1397:         for (j = 0; j < dim + 1; j++) {
    1398:           /* barycentric indices can have zeros, but we don't want to push forward zeros because it makes it harder to
    1399:            * determine which nodes correspond to which under symmetries, so we increase by 1.  This is fine
    1400:            * because the nodeIdx coordinates don't have any meaning other than helping to identify symmetries */
    1401:           ni->nodeIdx[(k * Nk + c) * (dim + 1) + j] = tup[j] + 1;
    1402:         }
    1403:       }
    1404:     }
    1405:     PetscCall(PetscFree(tup));
    1406:   }
    1407:   PetscCall(PetscQuadratureCreate(PETSC_COMM_SELF, &intNodes));
    1408:   PetscCall(PetscQuadratureSetData(intNodes, dim, 0, nNodes, nodeCoords, NULL));
    1409:   PetscCall(MatCreateSeqAIJ(PETSC_COMM_SELF, nNodes * Nk, nNodes * Nk, Nk, NULL, &intMat));
    1410:   PetscCall(PetscObjectSetOptionsPrefix((PetscObject)intMat, "lag_"));
    1411:   PetscCall(MatSetOption(intMat, MAT_IGNORE_ZERO_ENTRIES, PETSC_FALSE));
    1412:   for (j = 0; j < nNodes * Nk; j++) {
    1413:     PetscInt rem = j % Nk;
    1414:     PetscInt a, aprev = j - rem;
    1415:     PetscInt anext = aprev + Nk;

    1417:     for (a = aprev; a < anext; a++) PetscCall(MatSetValue(intMat, j, a, (a == j) ? 1. : 0., INSERT_VALUES));
    1418:   }
    1419:   PetscCall(MatAssemblyBegin(intMat, MAT_FINAL_ASSEMBLY));
    1420:   PetscCall(MatAssemblyEnd(intMat, MAT_FINAL_ASSEMBLY));
    1421:   *iNodes      = intNodes;
    1422:   *iMat        = intMat;
    1423:   *nodeIndices = ni;
    1424:   PetscFunctionReturn(PETSC_SUCCESS);
    1425: }

    1427: /* once the nodeIndices have been created for the interior of the reference cell, and for all of the boundary cells,
    1428:  * push forward the boundary dofs and concatenate them into the full node indices for the dual space */
    1429: static PetscErrorCode PetscDualSpaceLagrangeCreateAllNodeIdx(PetscDualSpace sp)
    1430: {
    1431:   DM                  dm;
    1432:   PetscInt            dim, nDofs;
    1433:   PetscSection        section;
    1434:   PetscInt            pStart, pEnd, p;
    1435:   PetscInt            formDegree, Nk;
    1436:   PetscInt            nodeIdxDim, spintdim;
    1437:   PetscDualSpace_Lag *lag;
    1438:   PetscLagNodeIndices ni, verti;

    1440:   PetscFunctionBegin;
    1441:   lag   = (PetscDualSpace_Lag *)sp->data;
    1442:   verti = lag->vertIndices;
    1443:   PetscCall(PetscDualSpaceGetDM(sp, &dm));
    1444:   PetscCall(DMGetDimension(dm, &dim));
    1445:   PetscCall(PetscDualSpaceGetFormDegree(sp, &formDegree));
    1446:   PetscCall(PetscDTBinomialInt(dim, PetscAbsInt(formDegree), &Nk));
    1447:   PetscCall(PetscDualSpaceGetSection(sp, &section));
    1448:   PetscCall(PetscSectionGetStorageSize(section, &nDofs));
    1449:   PetscCall(PetscNew(&ni));
    1450:   ni->nodeIdxDim = nodeIdxDim = verti->nodeIdxDim;
    1451:   ni->nodeVecDim              = Nk;
    1452:   ni->nNodes                  = nDofs;
    1453:   ni->refct                   = 1;
    1454:   PetscCall(PetscMalloc1(nodeIdxDim * nDofs, &ni->nodeIdx));
    1455:   PetscCall(PetscMalloc1(Nk * nDofs, &ni->nodeVec));
    1456:   PetscCall(DMPlexGetChart(dm, &pStart, &pEnd));
    1457:   PetscCall(PetscSectionGetDof(section, 0, &spintdim));
    1458:   if (spintdim) {
    1459:     PetscCall(PetscArraycpy(ni->nodeIdx, lag->intNodeIndices->nodeIdx, spintdim * nodeIdxDim));
    1460:     PetscCall(PetscArraycpy(ni->nodeVec, lag->intNodeIndices->nodeVec, spintdim * Nk));
    1461:   }
    1462:   for (p = pStart + 1; p < pEnd; p++) {
    1463:     PetscDualSpace      psp = sp->pointSpaces[p];
    1464:     PetscDualSpace_Lag *plag;
    1465:     PetscInt            dof, off;

    1467:     PetscCall(PetscSectionGetDof(section, p, &dof));
    1468:     if (!dof) continue;
    1469:     plag = (PetscDualSpace_Lag *)psp->data;
    1470:     PetscCall(PetscSectionGetOffset(section, p, &off));
    1471:     PetscCall(PetscLagNodeIndicesPushForward(dm, verti, p, plag->vertIndices, plag->intNodeIndices, 0, formDegree, &ni->nodeIdx[off * nodeIdxDim], &ni->nodeVec[off * Nk]));
    1472:   }
    1473:   lag->allNodeIndices = ni;
    1474:   PetscFunctionReturn(PETSC_SUCCESS);
    1475: }

    1477: /* once the (quadrature, Matrix) forms of the dofs have been created for the interior of the
    1478:  * reference cell and for the boundary cells, jk
    1479:  * push forward the boundary data and concatenate them into the full (quadrature, matrix) data
    1480:  * for the dual space */
    1481: static PetscErrorCode PetscDualSpaceCreateAllDataFromInteriorData(PetscDualSpace sp)
    1482: {
    1483:   DM              dm;
    1484:   PetscSection    section;
    1485:   PetscInt        pStart, pEnd, p, k, Nk, dim, Nc;
    1486:   PetscInt        nNodes;
    1487:   PetscInt        countNodes;
    1488:   Mat             allMat;
    1489:   PetscQuadrature allNodes;
    1490:   PetscInt        nDofs;
    1491:   PetscInt        maxNzforms, j;
    1492:   PetscScalar    *work;
    1493:   PetscReal      *L, *J, *Jinv, *v0, *pv0;
    1494:   PetscInt       *iwork;
    1495:   PetscReal      *nodes;

    1497:   PetscFunctionBegin;
    1498:   PetscCall(PetscDualSpaceGetDM(sp, &dm));
    1499:   PetscCall(DMGetDimension(dm, &dim));
    1500:   PetscCall(PetscDualSpaceGetSection(sp, &section));
    1501:   PetscCall(PetscSectionGetStorageSize(section, &nDofs));
    1502:   PetscCall(DMPlexGetChart(dm, &pStart, &pEnd));
    1503:   PetscCall(PetscDualSpaceGetFormDegree(sp, &k));
    1504:   PetscCall(PetscDualSpaceGetNumComponents(sp, &Nc));
    1505:   PetscCall(PetscDTBinomialInt(dim, PetscAbsInt(k), &Nk));
    1506:   for (p = pStart, nNodes = 0, maxNzforms = 0; p < pEnd; p++) {
    1507:     PetscDualSpace  psp;
    1508:     DM              pdm;
    1509:     PetscInt        pdim, pNk;
    1510:     PetscQuadrature intNodes;
    1511:     Mat             intMat;

    1513:     PetscCall(PetscDualSpaceGetPointSubspace(sp, p, &psp));
    1514:     if (!psp) continue;
    1515:     PetscCall(PetscDualSpaceGetDM(psp, &pdm));
    1516:     PetscCall(DMGetDimension(pdm, &pdim));
    1517:     if (pdim < PetscAbsInt(k)) continue;
    1518:     PetscCall(PetscDTBinomialInt(pdim, PetscAbsInt(k), &pNk));
    1519:     PetscCall(PetscDualSpaceGetInteriorData(psp, &intNodes, &intMat));
    1520:     if (intNodes) {
    1521:       PetscInt nNodesp;

    1523:       PetscCall(PetscQuadratureGetData(intNodes, NULL, NULL, &nNodesp, NULL, NULL));
    1524:       nNodes += nNodesp;
    1525:     }
    1526:     if (intMat) {
    1527:       PetscInt maxNzsp;
    1528:       PetscInt maxNzformsp;

    1530:       PetscCall(MatSeqAIJGetMaxRowNonzeros(intMat, &maxNzsp));
    1531:       PetscCheck(maxNzsp % pNk == 0, PETSC_COMM_SELF, PETSC_ERR_PLIB, "interior matrix is not laid out as blocks of k-forms");
    1532:       maxNzformsp = maxNzsp / pNk;
    1533:       maxNzforms  = PetscMax(maxNzforms, maxNzformsp);
    1534:     }
    1535:   }
    1536:   PetscCall(MatCreateSeqAIJ(PETSC_COMM_SELF, nDofs, nNodes * Nc, maxNzforms * Nk, NULL, &allMat));
    1537:   PetscCall(PetscObjectSetOptionsPrefix((PetscObject)allMat, "ds_"));
    1538:   PetscCall(MatSetOption(allMat, MAT_IGNORE_ZERO_ENTRIES, PETSC_FALSE));
    1539:   PetscCall(PetscMalloc7(dim, &v0, dim, &pv0, dim * dim, &J, dim * dim, &Jinv, Nk * Nk, &L, maxNzforms * Nk, &work, maxNzforms * Nk, &iwork));
    1540:   for (j = 0; j < dim; j++) pv0[j] = -1.;
    1541:   PetscCall(PetscMalloc1(dim * nNodes, &nodes));
    1542:   for (p = pStart, countNodes = 0; p < pEnd; p++) {
    1543:     PetscDualSpace  psp;
    1544:     PetscQuadrature intNodes;
    1545:     DM              pdm;
    1546:     PetscInt        pdim, pNk;
    1547:     PetscInt        countNodesIn = countNodes;
    1548:     PetscReal       detJ;
    1549:     Mat             intMat;

    1551:     PetscCall(PetscDualSpaceGetPointSubspace(sp, p, &psp));
    1552:     if (!psp) continue;
    1553:     PetscCall(PetscDualSpaceGetDM(psp, &pdm));
    1554:     PetscCall(DMGetDimension(pdm, &pdim));
    1555:     if (pdim < PetscAbsInt(k)) continue;
    1556:     PetscCall(PetscDualSpaceGetInteriorData(psp, &intNodes, &intMat));
    1557:     if (intNodes == NULL && intMat == NULL) continue;
    1558:     PetscCall(PetscDTBinomialInt(pdim, PetscAbsInt(k), &pNk));
    1559:     if (p) {
    1560:       PetscCall(DMPlexComputeCellGeometryAffineFEM(dm, p, v0, J, Jinv, &detJ));
    1561:     } else { /* identity */
    1562:       PetscInt i, j;

    1564:       for (i = 0; i < dim; i++)
    1565:         for (j = 0; j < dim; j++) J[i * dim + j] = Jinv[i * dim + j] = 0.;
    1566:       for (i = 0; i < dim; i++) J[i * dim + i] = Jinv[i * dim + i] = 1.;
    1567:       for (i = 0; i < dim; i++) v0[i] = -1.;
    1568:     }
    1569:     if (pdim != dim) { /* compactify Jacobian */
    1570:       PetscInt i, j;

    1572:       for (i = 0; i < dim; i++)
    1573:         for (j = 0; j < pdim; j++) J[i * pdim + j] = J[i * dim + j];
    1574:     }
    1575:     PetscCall(PetscDTAltVPullbackMatrix(pdim, dim, J, k, L));
    1576:     if (intNodes) { /* push forward quadrature locations by the affine transformation */
    1577:       PetscInt         nNodesp;
    1578:       const PetscReal *nodesp;
    1579:       PetscInt         j;

    1581:       PetscCall(PetscQuadratureGetData(intNodes, NULL, NULL, &nNodesp, &nodesp, NULL));
    1582:       for (j = 0; j < nNodesp; j++, countNodes++) {
    1583:         PetscInt d, e;

    1585:         for (d = 0; d < dim; d++) {
    1586:           nodes[countNodes * dim + d] = v0[d];
    1587:           for (e = 0; e < pdim; e++) nodes[countNodes * dim + d] += J[d * pdim + e] * (nodesp[j * pdim + e] - pv0[e]);
    1588:         }
    1589:       }
    1590:     }
    1591:     if (intMat) {
    1592:       PetscInt nrows;
    1593:       PetscInt off;

    1595:       PetscCall(PetscSectionGetDof(section, p, &nrows));
    1596:       PetscCall(PetscSectionGetOffset(section, p, &off));
    1597:       for (j = 0; j < nrows; j++) {
    1598:         PetscInt           ncols;
    1599:         const PetscInt    *cols;
    1600:         const PetscScalar *vals;
    1601:         PetscInt           l, d, e;
    1602:         PetscInt           row = j + off;

    1604:         PetscCall(MatGetRow(intMat, j, &ncols, &cols, &vals));
    1605:         PetscCheck(ncols % pNk == 0, PETSC_COMM_SELF, PETSC_ERR_PLIB, "interior matrix is not laid out as blocks of k-forms");
    1606:         for (l = 0; l < ncols / pNk; l++) {
    1607:           PetscInt blockcol;

    1609:           for (d = 0; d < pNk; d++) PetscCheck((cols[l * pNk + d] % pNk) == d, PETSC_COMM_SELF, PETSC_ERR_PLIB, "interior matrix is not laid out as blocks of k-forms");
    1610:           blockcol = cols[l * pNk] / pNk;
    1611:           for (d = 0; d < Nk; d++) iwork[l * Nk + d] = (blockcol + countNodesIn) * Nk + d;
    1612:           for (d = 0; d < Nk; d++) work[l * Nk + d] = 0.;
    1613:           for (d = 0; d < Nk; d++) {
    1614:             for (e = 0; e < pNk; e++) {
    1615:               /* "push forward" dof by pulling back a k-form to be evaluated on the point: multiply on the right by L */
    1616:               work[l * Nk + d] += vals[l * pNk + e] * L[e * Nk + d];
    1617:             }
    1618:           }
    1619:         }
    1620:         PetscCall(MatSetValues(allMat, 1, &row, (ncols / pNk) * Nk, iwork, work, INSERT_VALUES));
    1621:         PetscCall(MatRestoreRow(intMat, j, &ncols, &cols, &vals));
    1622:       }
    1623:     }
    1624:   }
    1625:   PetscCall(MatAssemblyBegin(allMat, MAT_FINAL_ASSEMBLY));
    1626:   PetscCall(MatAssemblyEnd(allMat, MAT_FINAL_ASSEMBLY));
    1627:   PetscCall(PetscQuadratureCreate(PETSC_COMM_SELF, &allNodes));
    1628:   PetscCall(PetscQuadratureSetData(allNodes, dim, 0, nNodes, nodes, NULL));
    1629:   PetscCall(PetscFree7(v0, pv0, J, Jinv, L, work, iwork));
    1630:   PetscCall(MatDestroy(&sp->allMat));
    1631:   sp->allMat = allMat;
    1632:   PetscCall(PetscQuadratureDestroy(&sp->allNodes));
    1633:   sp->allNodes = allNodes;
    1634:   PetscFunctionReturn(PETSC_SUCCESS);
    1635: }

    1637: static PetscErrorCode PetscDualSpaceComputeFunctionalsFromAllData_Moments(PetscDualSpace sp)
    1638: {
    1639:   Mat              allMat;
    1640:   PetscInt         momentOrder, i;
    1641:   PetscBool        tensor = PETSC_FALSE;
    1642:   const PetscReal *weights;
    1643:   PetscScalar     *array;
    1644:   PetscInt         nDofs;
    1645:   PetscInt         dim, Nc;
    1646:   DM               dm;
    1647:   PetscQuadrature  allNodes;
    1648:   PetscInt         nNodes;

    1650:   PetscFunctionBegin;
    1651:   PetscCall(PetscDualSpaceGetDM(sp, &dm));
    1652:   PetscCall(DMGetDimension(dm, &dim));
    1653:   PetscCall(PetscDualSpaceGetNumComponents(sp, &Nc));
    1654:   PetscCall(PetscDualSpaceGetAllData(sp, &allNodes, &allMat));
    1655:   PetscCall(MatGetSize(allMat, &nDofs, NULL));
    1656:   PetscCheck(nDofs == 1, PETSC_COMM_SELF, PETSC_ERR_SUP, "We do not yet support moments beyond P0, nDofs == %" PetscInt_FMT, nDofs);
    1657:   PetscCall(PetscMalloc1(nDofs, &sp->functional));
    1658:   PetscCall(PetscDualSpaceLagrangeGetMomentOrder(sp, &momentOrder));
    1659:   PetscCall(PetscDualSpaceLagrangeGetTensor(sp, &tensor));
    1660:   if (!tensor) PetscCall(PetscDTStroudConicalQuadrature(dim, Nc, PetscMax(momentOrder + 1, 1), -1.0, 1.0, &sp->functional[0]));
    1661:   else PetscCall(PetscDTGaussTensorQuadrature(dim, Nc, PetscMax(momentOrder + 1, 1), -1.0, 1.0, &sp->functional[0]));
    1662:   /* Need to replace allNodes and allMat */
    1663:   PetscCall(PetscObjectReference((PetscObject)sp->functional[0]));
    1664:   PetscCall(PetscQuadratureDestroy(&sp->allNodes));
    1665:   sp->allNodes = sp->functional[0];
    1666:   PetscCall(PetscQuadratureGetData(sp->allNodes, NULL, NULL, &nNodes, NULL, &weights));
    1667:   PetscCall(MatCreateSeqDense(PETSC_COMM_SELF, nDofs, nNodes * Nc, NULL, &allMat));
    1668:   PetscCall(PetscObjectSetOptionsPrefix((PetscObject)allMat, "ds_"));
    1669:   PetscCall(MatDenseGetArrayWrite(allMat, &array));
    1670:   for (i = 0; i < nNodes * Nc; ++i) array[i] = weights[i];
    1671:   PetscCall(MatDenseRestoreArrayWrite(allMat, &array));
    1672:   PetscCall(MatAssemblyBegin(allMat, MAT_FINAL_ASSEMBLY));
    1673:   PetscCall(MatAssemblyEnd(allMat, MAT_FINAL_ASSEMBLY));
    1674:   PetscCall(MatDestroy(&sp->allMat));
    1675:   sp->allMat = allMat;
    1676:   PetscFunctionReturn(PETSC_SUCCESS);
    1677: }

    1679: /* rather than trying to get all data from the functionals, we create
    1680:  * the functionals from rows of the quadrature -> dof matrix.
    1681:  *
    1682:  * Ideally most of the uses of PetscDualSpace in PetscFE will switch
    1683:  * to using intMat and allMat, so that the individual functionals
    1684:  * don't need to be constructed at all */
    1685: PETSC_INTERN PetscErrorCode PetscDualSpaceComputeFunctionalsFromAllData(PetscDualSpace sp)
    1686: {
    1687:   PetscQuadrature  allNodes;
    1688:   Mat              allMat;
    1689:   PetscInt         nDofs;
    1690:   PetscInt         dim, Nc, f;
    1691:   DM               dm;
    1692:   PetscInt         nNodes, spdim;
    1693:   const PetscReal *nodes = NULL;
    1694:   PetscSection     section;
    1695:   PetscBool        useMoments;

    1697:   PetscFunctionBegin;
    1698:   PetscCall(PetscDualSpaceGetDM(sp, &dm));
    1699:   PetscCall(DMGetDimension(dm, &dim));
    1700:   PetscCall(PetscDualSpaceGetNumComponents(sp, &Nc));
    1701:   PetscCall(PetscDualSpaceGetAllData(sp, &allNodes, &allMat));
    1702:   nNodes = 0;
    1703:   if (allNodes) PetscCall(PetscQuadratureGetData(allNodes, NULL, NULL, &nNodes, &nodes, NULL));
    1704:   PetscCall(MatGetSize(allMat, &nDofs, NULL));
    1705:   PetscCall(PetscDualSpaceGetSection(sp, &section));
    1706:   PetscCall(PetscSectionGetStorageSize(section, &spdim));
    1707:   PetscCheck(spdim == nDofs, PETSC_COMM_SELF, PETSC_ERR_PLIB, "incompatible all matrix size");
    1708:   PetscCall(PetscMalloc1(nDofs, &sp->functional));
    1709:   PetscCall(PetscDualSpaceLagrangeGetUseMoments(sp, &useMoments));
    1710:   for (f = 0; f < nDofs; f++) {
    1711:     PetscInt           ncols, c;
    1712:     const PetscInt    *cols;
    1713:     const PetscScalar *vals;
    1714:     PetscReal         *nodesf;
    1715:     PetscReal         *weightsf;
    1716:     PetscInt           nNodesf;
    1717:     PetscInt           countNodes;

    1719:     PetscCall(MatGetRow(allMat, f, &ncols, &cols, &vals));
    1720:     for (c = 1, nNodesf = 1; c < ncols; c++) {
    1721:       if ((cols[c] / Nc) != (cols[c - 1] / Nc)) nNodesf++;
    1722:     }
    1723:     PetscCall(PetscMalloc1(dim * nNodesf, &nodesf));
    1724:     PetscCall(PetscMalloc1(Nc * nNodesf, &weightsf));
    1725:     for (c = 0, countNodes = 0; c < ncols; c++) {
    1726:       if (!c || ((cols[c] / Nc) != (cols[c - 1] / Nc))) {
    1727:         PetscInt d;

    1729:         for (d = 0; d < Nc; d++) weightsf[countNodes * Nc + d] = 0.;
    1730:         for (d = 0; d < dim; d++) nodesf[countNodes * dim + d] = nodes[(cols[c] / Nc) * dim + d];
    1731:         countNodes++;
    1732:       }
    1733:       weightsf[(countNodes - 1) * Nc + (cols[c] % Nc)] = PetscRealPart(vals[c]);
    1734:     }
    1735:     PetscCall(PetscQuadratureCreate(PETSC_COMM_SELF, &sp->functional[f]));
    1736:     PetscCall(PetscQuadratureSetData(sp->functional[f], dim, Nc, nNodesf, nodesf, weightsf));
    1737:     PetscCall(MatRestoreRow(allMat, f, &ncols, &cols, &vals));
    1738:   }
    1739:   PetscFunctionReturn(PETSC_SUCCESS);
    1740: }

    1742: /* check if a cell is a tensor product of the segment with a facet,
    1743:  * specifically checking if f and f2 can be the "endpoints" (like the triangles
    1744:  * at either end of a wedge) */
    1745: static PetscErrorCode DMPlexPointIsTensor_Internal_Given(DM dm, PetscInt p, PetscInt f, PetscInt f2, PetscBool *isTensor)
    1746: {
    1747:   PetscInt        coneSize, c;
    1748:   const PetscInt *cone;
    1749:   const PetscInt *fCone;
    1750:   const PetscInt *f2Cone;
    1751:   PetscInt        fs[2];
    1752:   PetscInt        meetSize, nmeet;
    1753:   const PetscInt *meet;

    1755:   PetscFunctionBegin;
    1756:   fs[0] = f;
    1757:   fs[1] = f2;
    1758:   PetscCall(DMPlexGetMeet(dm, 2, fs, &meetSize, &meet));
    1759:   nmeet = meetSize;
    1760:   PetscCall(DMPlexRestoreMeet(dm, 2, fs, &meetSize, &meet));
    1761:   /* two points that have a non-empty meet cannot be at opposite ends of a cell */
    1762:   if (nmeet) {
    1763:     *isTensor = PETSC_FALSE;
    1764:     PetscFunctionReturn(PETSC_SUCCESS);
    1765:   }
    1766:   PetscCall(DMPlexGetConeSize(dm, p, &coneSize));
    1767:   PetscCall(DMPlexGetCone(dm, p, &cone));
    1768:   PetscCall(DMPlexGetCone(dm, f, &fCone));
    1769:   PetscCall(DMPlexGetCone(dm, f2, &f2Cone));
    1770:   for (c = 0; c < coneSize; c++) {
    1771:     PetscInt        e, ef;
    1772:     PetscInt        d = -1, d2 = -1;
    1773:     PetscInt        dcount, d2count;
    1774:     PetscInt        t = cone[c];
    1775:     PetscInt        tConeSize;
    1776:     PetscBool       tIsTensor;
    1777:     const PetscInt *tCone;

    1779:     if (t == f || t == f2) continue;
    1780:     /* for every other facet in the cone, check that is has
    1781:      * one ridge in common with each end */
    1782:     PetscCall(DMPlexGetConeSize(dm, t, &tConeSize));
    1783:     PetscCall(DMPlexGetCone(dm, t, &tCone));

    1785:     dcount  = 0;
    1786:     d2count = 0;
    1787:     for (e = 0; e < tConeSize; e++) {
    1788:       PetscInt q = tCone[e];
    1789:       for (ef = 0; ef < coneSize - 2; ef++) {
    1790:         if (fCone[ef] == q) {
    1791:           if (dcount) {
    1792:             *isTensor = PETSC_FALSE;
    1793:             PetscFunctionReturn(PETSC_SUCCESS);
    1794:           }
    1795:           d = q;
    1796:           dcount++;
    1797:         } else if (f2Cone[ef] == q) {
    1798:           if (d2count) {
    1799:             *isTensor = PETSC_FALSE;
    1800:             PetscFunctionReturn(PETSC_SUCCESS);
    1801:           }
    1802:           d2 = q;
    1803:           d2count++;
    1804:         }
    1805:       }
    1806:     }
    1807:     /* if the whole cell is a tensor with the segment, then this
    1808:      * facet should be a tensor with the segment */
    1809:     PetscCall(DMPlexPointIsTensor_Internal_Given(dm, t, d, d2, &tIsTensor));
    1810:     if (!tIsTensor) {
    1811:       *isTensor = PETSC_FALSE;
    1812:       PetscFunctionReturn(PETSC_SUCCESS);
    1813:     }
    1814:   }
    1815:   *isTensor = PETSC_TRUE;
    1816:   PetscFunctionReturn(PETSC_SUCCESS);
    1817: }

    1819: /* determine if a cell is a tensor with a segment by looping over pairs of facets to find a pair
    1820:  * that could be the opposite ends */
    1821: static PetscErrorCode DMPlexPointIsTensor_Internal(DM dm, PetscInt p, PetscBool *isTensor, PetscInt *endA, PetscInt *endB)
    1822: {
    1823:   PetscInt        coneSize, c, c2;
    1824:   const PetscInt *cone;

    1826:   PetscFunctionBegin;
    1827:   PetscCall(DMPlexGetConeSize(dm, p, &coneSize));
    1828:   if (!coneSize) {
    1829:     if (isTensor) *isTensor = PETSC_FALSE;
    1830:     if (endA) *endA = -1;
    1831:     if (endB) *endB = -1;
    1832:   }
    1833:   PetscCall(DMPlexGetCone(dm, p, &cone));
    1834:   for (c = 0; c < coneSize; c++) {
    1835:     PetscInt f = cone[c];
    1836:     PetscInt fConeSize;

    1838:     PetscCall(DMPlexGetConeSize(dm, f, &fConeSize));
    1839:     if (fConeSize != coneSize - 2) continue;

    1841:     for (c2 = c + 1; c2 < coneSize; c2++) {
    1842:       PetscInt  f2 = cone[c2];
    1843:       PetscBool isTensorff2;
    1844:       PetscInt  f2ConeSize;

    1846:       PetscCall(DMPlexGetConeSize(dm, f2, &f2ConeSize));
    1847:       if (f2ConeSize != coneSize - 2) continue;

    1849:       PetscCall(DMPlexPointIsTensor_Internal_Given(dm, p, f, f2, &isTensorff2));
    1850:       if (isTensorff2) {
    1851:         if (isTensor) *isTensor = PETSC_TRUE;
    1852:         if (endA) *endA = f;
    1853:         if (endB) *endB = f2;
    1854:         PetscFunctionReturn(PETSC_SUCCESS);
    1855:       }
    1856:     }
    1857:   }
    1858:   if (isTensor) *isTensor = PETSC_FALSE;
    1859:   if (endA) *endA = -1;
    1860:   if (endB) *endB = -1;
    1861:   PetscFunctionReturn(PETSC_SUCCESS);
    1862: }

    1864: /* determine if a cell is a tensor with a segment by looping over pairs of facets to find a pair
    1865:  * that could be the opposite ends */
    1866: static PetscErrorCode DMPlexPointIsTensor(DM dm, PetscInt p, PetscBool *isTensor, PetscInt *endA, PetscInt *endB)
    1867: {
    1868:   DMPlexInterpolatedFlag interpolated;

    1870:   PetscFunctionBegin;
    1871:   PetscCall(DMPlexIsInterpolated(dm, &interpolated));
    1872:   PetscCheck(interpolated == DMPLEX_INTERPOLATED_FULL, PetscObjectComm((PetscObject)dm), PETSC_ERR_ARG_WRONGSTATE, "Only for interpolated DMPlex's");
    1873:   PetscCall(DMPlexPointIsTensor_Internal(dm, p, isTensor, endA, endB));
    1874:   PetscFunctionReturn(PETSC_SUCCESS);
    1875: }

    1877: /* Let k = formDegree and k' = -sign(k) * dim + k.  Transform a symmetric frame for k-forms on the biunit simplex into
    1878:  * a symmetric frame for k'-forms on the biunit simplex.
    1879:  *
    1880:  * A frame is "symmetric" if the pullback of every symmetry of the biunit simplex is a permutation of the frame.
    1881:  *
    1882:  * forms in the symmetric frame are used as dofs in the untrimmed simplex spaces.  This way, symmetries of the
    1883:  * reference cell result in permutations of dofs grouped by node.
    1884:  *
    1885:  * Use T to transform dof matrices for k'-forms into dof matrices for k-forms as a block diagonal transformation on
    1886:  * the right.
    1887:  */
    1888: static PetscErrorCode BiunitSimplexSymmetricFormTransformation(PetscInt dim, PetscInt formDegree, PetscReal T[])
    1889: {
    1890:   PetscInt   k  = formDegree;
    1891:   PetscInt   kd = k < 0 ? dim + k : k - dim;
    1892:   PetscInt   Nk;
    1893:   PetscReal *biToEq, *eqToBi, *biToEqStar, *eqToBiStar;
    1894:   PetscInt   fact;

    1896:   PetscFunctionBegin;
    1897:   PetscCall(PetscDTBinomialInt(dim, PetscAbsInt(k), &Nk));
    1898:   PetscCall(PetscCalloc4(dim * dim, &biToEq, dim * dim, &eqToBi, Nk * Nk, &biToEqStar, Nk * Nk, &eqToBiStar));
    1899:   /* fill in biToEq: Jacobian of the transformation from the biunit simplex to the equilateral simplex */
    1900:   fact = 0;
    1901:   for (PetscInt i = 0; i < dim; i++) {
    1902:     biToEq[i * dim + i] = PetscSqrtReal(((PetscReal)i + 2.) / (2. * ((PetscReal)i + 1.)));
    1903:     fact += 4 * (i + 1);
    1904:     for (PetscInt j = i + 1; j < dim; j++) biToEq[i * dim + j] = PetscSqrtReal(1. / (PetscReal)fact);
    1905:   }
    1906:   /* fill in eqToBi: Jacobian of the transformation from the equilateral simplex to the biunit simplex */
    1907:   fact = 0;
    1908:   for (PetscInt j = 0; j < dim; j++) {
    1909:     eqToBi[j * dim + j] = PetscSqrtReal(2. * ((PetscReal)j + 1.) / ((PetscReal)j + 2));
    1910:     fact += j + 1;
    1911:     for (PetscInt i = 0; i < j; i++) eqToBi[i * dim + j] = -PetscSqrtReal(1. / (PetscReal)fact);
    1912:   }
    1913:   PetscCall(PetscDTAltVPullbackMatrix(dim, dim, biToEq, kd, biToEqStar));
    1914:   PetscCall(PetscDTAltVPullbackMatrix(dim, dim, eqToBi, k, eqToBiStar));
    1915:   /* product of pullbacks simulates the following steps
    1916:    *
    1917:    * 1. start with frame W = [w_1, w_2, ..., w_m] of k forms that is symmetric on the biunit simplex:
    1918:           if J is the Jacobian of a symmetry of the biunit simplex, then J_k* W = [J_k*w_1, ..., J_k*w_m]
    1919:           is a permutation of W.
    1920:           Even though a k' form --- a (dim - k) form represented by its Hodge star --- has the same geometric
    1921:           content as a k form, W is not a symmetric frame of k' forms on the biunit simplex.  That's because,
    1922:           for general Jacobian J, J_k* != J_k'*.
    1923:    * 2. pullback W to the equilateral triangle using the k pullback, W_eq = eqToBi_k* W.  All symmetries of the
    1924:           equilateral simplex have orthonormal Jacobians.  For an orthonormal Jacobian O, J_k* = J_k'*, so W_eq is
    1925:           also a symmetric frame for k' forms on the equilateral simplex.
    1926:      3. pullback W_eq back to the biunit simplex using the k' pulback, V = biToEq_k'* W_eq = biToEq_k'* eqToBi_k* W.
    1927:           V is a symmetric frame for k' forms on the biunit simplex.
    1928:    */
    1929:   for (PetscInt i = 0; i < Nk; i++) {
    1930:     for (PetscInt j = 0; j < Nk; j++) {
    1931:       PetscReal val = 0.;
    1932:       for (PetscInt k = 0; k < Nk; k++) val += biToEqStar[i * Nk + k] * eqToBiStar[k * Nk + j];
    1933:       T[i * Nk + j] = val;
    1934:     }
    1935:   }
    1936:   PetscCall(PetscFree4(biToEq, eqToBi, biToEqStar, eqToBiStar));
    1937:   PetscFunctionReturn(PETSC_SUCCESS);
    1938: }

    1940: /* permute a quadrature -> dof matrix so that its rows are in revlex order by nodeIdx */
    1941: static PetscErrorCode MatPermuteByNodeIdx(Mat A, PetscLagNodeIndices ni, Mat *Aperm)
    1942: {
    1943:   PetscInt   m, n, i, j;
    1944:   PetscInt   nodeIdxDim = ni->nodeIdxDim;
    1945:   PetscInt   nodeVecDim = ni->nodeVecDim;
    1946:   PetscInt  *perm;
    1947:   IS         permIS;
    1948:   IS         id;
    1949:   PetscInt  *nIdxPerm;
    1950:   PetscReal *nVecPerm;

    1952:   PetscFunctionBegin;
    1953:   PetscCall(PetscLagNodeIndicesGetPermutation(ni, &perm));
    1954:   PetscCall(MatGetSize(A, &m, &n));
    1955:   PetscCall(PetscMalloc1(nodeIdxDim * m, &nIdxPerm));
    1956:   PetscCall(PetscMalloc1(nodeVecDim * m, &nVecPerm));
    1957:   for (i = 0; i < m; i++)
    1958:     for (j = 0; j < nodeIdxDim; j++) nIdxPerm[i * nodeIdxDim + j] = ni->nodeIdx[perm[i] * nodeIdxDim + j];
    1959:   for (i = 0; i < m; i++)
    1960:     for (j = 0; j < nodeVecDim; j++) nVecPerm[i * nodeVecDim + j] = ni->nodeVec[perm[i] * nodeVecDim + j];
    1961:   PetscCall(ISCreateGeneral(PETSC_COMM_SELF, m, perm, PETSC_USE_POINTER, &permIS));
    1962:   PetscCall(ISSetPermutation(permIS));
    1963:   PetscCall(ISCreateStride(PETSC_COMM_SELF, n, 0, 1, &id));
    1964:   PetscCall(ISSetPermutation(id));
    1965:   PetscCall(MatPermute(A, permIS, id, Aperm));
    1966:   PetscCall(ISDestroy(&permIS));
    1967:   PetscCall(ISDestroy(&id));
    1968:   for (i = 0; i < m; i++) perm[i] = i;
    1969:   PetscCall(PetscFree(ni->nodeIdx));
    1970:   PetscCall(PetscFree(ni->nodeVec));
    1971:   ni->nodeIdx = nIdxPerm;
    1972:   ni->nodeVec = nVecPerm;
    1973:   PetscFunctionReturn(PETSC_SUCCESS);
    1974: }

    1976: static PetscErrorCode PetscDualSpaceSetUp_Lagrange(PetscDualSpace sp)
    1977: {
    1978:   PetscDualSpace_Lag    *lag   = (PetscDualSpace_Lag *)sp->data;
    1979:   DM                     dm    = sp->dm;
    1980:   DM                     dmint = NULL;
    1981:   PetscInt               order;
    1982:   PetscInt               Nc;
    1983:   MPI_Comm               comm;
    1984:   PetscBool              continuous;
    1985:   PetscSection           section;
    1986:   PetscInt               depth, dim, pStart, pEnd, cStart, cEnd, p, *pStratStart, *pStratEnd, d;
    1987:   PetscInt               formDegree, Nk, Ncopies;
    1988:   PetscInt               tensorf = -1, tensorf2 = -1;
    1989:   PetscBool              tensorCell, tensorSpace;
    1990:   PetscBool              uniform, trimmed;
    1991:   Petsc1DNodeFamily      nodeFamily;
    1992:   PetscInt               numNodeSkip;
    1993:   DMPlexInterpolatedFlag interpolated;
    1994:   PetscBool              isbdm;

    1996:   PetscFunctionBegin;
    1997:   /* step 1: sanitize input */
    1998:   PetscCall(PetscObjectGetComm((PetscObject)sp, &comm));
    1999:   PetscCall(DMGetDimension(dm, &dim));
    2000:   PetscCall(PetscObjectTypeCompare((PetscObject)sp, PETSCDUALSPACEBDM, &isbdm));
    2001:   if (isbdm) {
    2002:     sp->k = -(dim - 1); /* form degree of H-div */
    2003:     PetscCall(PetscObjectChangeTypeName((PetscObject)sp, PETSCDUALSPACELAGRANGE));
    2004:   }
    2005:   PetscCall(PetscDualSpaceGetFormDegree(sp, &formDegree));
    2006:   PetscCheck(PetscAbsInt(formDegree) <= dim, comm, PETSC_ERR_ARG_OUTOFRANGE, "Form degree must be bounded by dimension");
    2007:   PetscCall(PetscDTBinomialInt(dim, PetscAbsInt(formDegree), &Nk));
    2008:   if (sp->Nc <= 0 && lag->numCopies > 0) sp->Nc = Nk * lag->numCopies;
    2009:   Nc = sp->Nc;
    2010:   PetscCheck(Nc % Nk == 0, comm, PETSC_ERR_ARG_INCOMP, "Number of components is not a multiple of form degree size");
    2011:   if (lag->numCopies <= 0) lag->numCopies = Nc / Nk;
    2012:   Ncopies = lag->numCopies;
    2013:   PetscCheck(Nc / Nk == Ncopies, comm, PETSC_ERR_ARG_INCOMP, "Number of copies * (dim choose k) != Nc");
    2014:   if (!dim) sp->order = 0;
    2015:   order   = sp->order;
    2016:   uniform = sp->uniform;
    2017:   PetscCheck(uniform, PETSC_COMM_SELF, PETSC_ERR_SUP, "Variable order not supported yet");
    2018:   if (lag->trimmed && !formDegree) lag->trimmed = PETSC_FALSE; /* trimmed spaces are the same as full spaces for 0-forms */
    2019:   if (lag->nodeType == PETSCDTNODES_DEFAULT) {
    2020:     lag->nodeType     = PETSCDTNODES_GAUSSJACOBI;
    2021:     lag->nodeExponent = 0.;
    2022:     /* trimmed spaces don't include corner vertices, so don't use end nodes by default */
    2023:     lag->endNodes = lag->trimmed ? PETSC_FALSE : PETSC_TRUE;
    2024:   }
    2025:   /* If a trimmed space and the user did choose nodes with endpoints, skip them by default */
    2026:   if (lag->numNodeSkip < 0) lag->numNodeSkip = (lag->trimmed && lag->endNodes) ? 1 : 0;
    2027:   numNodeSkip = lag->numNodeSkip;
    2028:   PetscCheck(!lag->trimmed || order, PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Cannot have zeroth order trimmed elements");
    2029:   if (lag->trimmed && PetscAbsInt(formDegree) == dim) { /* convert trimmed n-forms to untrimmed of one polynomial order less */
    2030:     sp->order--;
    2031:     order--;
    2032:     lag->trimmed = PETSC_FALSE;
    2033:   }
    2034:   trimmed = lag->trimmed;
    2035:   if (!order || PetscAbsInt(formDegree) == dim) lag->continuous = PETSC_FALSE;
    2036:   continuous = lag->continuous;
    2037:   PetscCall(DMPlexGetDepth(dm, &depth));
    2038:   PetscCall(DMPlexGetChart(dm, &pStart, &pEnd));
    2039:   PetscCall(DMPlexGetHeightStratum(dm, 0, &cStart, &cEnd));
    2040:   PetscCheck(pStart == 0 && cStart == 0, PetscObjectComm((PetscObject)sp), PETSC_ERR_ARG_WRONGSTATE, "Expect DM with chart starting at zero and cells first");
    2041:   PetscCheck(cEnd == 1, PetscObjectComm((PetscObject)sp), PETSC_ERR_ARG_WRONGSTATE, "Use PETSCDUALSPACEREFINED for multi-cell reference meshes");
    2042:   PetscCall(DMPlexIsInterpolated(dm, &interpolated));
    2043:   if (interpolated != DMPLEX_INTERPOLATED_FULL) {
    2044:     PetscCall(DMPlexInterpolate(dm, &dmint));
    2045:   } else {
    2046:     PetscCall(PetscObjectReference((PetscObject)dm));
    2047:     dmint = dm;
    2048:   }
    2049:   tensorCell = PETSC_FALSE;
    2050:   if (dim > 1) PetscCall(DMPlexPointIsTensor(dmint, 0, &tensorCell, &tensorf, &tensorf2));
    2051:   lag->tensorCell = tensorCell;
    2052:   if (dim < 2 || !lag->tensorCell) lag->tensorSpace = PETSC_FALSE;
    2053:   tensorSpace = lag->tensorSpace;
    2054:   if (!lag->nodeFamily) PetscCall(Petsc1DNodeFamilyCreate(lag->nodeType, lag->nodeExponent, lag->endNodes, &lag->nodeFamily));
    2055:   nodeFamily = lag->nodeFamily;
    2056:   PetscCheck(interpolated == DMPLEX_INTERPOLATED_FULL || !continuous || (PetscAbsInt(formDegree) <= 0 && order <= 1), PETSC_COMM_SELF, PETSC_ERR_PLIB, "Reference element won't support all boundary nodes");

    2058:   if (Ncopies > 1) {
    2059:     PetscDualSpace scalarsp;

    2061:     PetscCall(PetscDualSpaceDuplicate(sp, &scalarsp));
    2062:     /* Setting the number of components to Nk is a space with 1 copy of each k-form */
    2063:     sp->setupcalled = PETSC_FALSE;
    2064:     PetscCall(PetscDualSpaceSetNumComponents(scalarsp, Nk));
    2065:     PetscCall(PetscDualSpaceSetUp(scalarsp));
    2066:     PetscCall(PetscDualSpaceSetType(sp, PETSCDUALSPACESUM));
    2067:     PetscCall(PetscDualSpaceSumSetNumSubspaces(sp, Ncopies));
    2068:     PetscCall(PetscDualSpaceSumSetConcatenate(sp, PETSC_TRUE));
    2069:     PetscCall(PetscDualSpaceSumSetInterleave(sp, PETSC_TRUE, PETSC_FALSE));
    2070:     for (PetscInt i = 0; i < Ncopies; i++) PetscCall(PetscDualSpaceSumSetSubspace(sp, i, scalarsp));
    2071:     PetscCall(PetscDualSpaceSetUp(sp));
    2072:     PetscCall(PetscDualSpaceDestroy(&scalarsp));
    2073:     PetscCall(DMDestroy(&dmint));
    2074:     PetscFunctionReturn(PETSC_SUCCESS);
    2075:   }

    2077:   /* step 2: construct the boundary spaces */
    2078:   PetscCall(PetscMalloc2(depth + 1, &pStratStart, depth + 1, &pStratEnd));
    2079:   PetscCall(PetscCalloc1(pEnd, &sp->pointSpaces));
    2080:   for (d = 0; d <= depth; ++d) PetscCall(DMPlexGetDepthStratum(dm, d, &pStratStart[d], &pStratEnd[d]));
    2081:   PetscCall(PetscDualSpaceSectionCreate_Internal(sp, &section));
    2082:   sp->pointSection = section;
    2083:   if (continuous && !lag->interiorOnly) {
    2084:     PetscInt h;

    2086:     for (p = pStratStart[depth - 1]; p < pStratEnd[depth - 1]; p++) { /* calculate the facet dual spaces */
    2087:       PetscReal      v0[3];
    2088:       DMPolytopeType ptype;
    2089:       PetscReal      J[9], detJ;
    2090:       PetscInt       q;

    2092:       PetscCall(DMPlexComputeCellGeometryAffineFEM(dm, p, v0, J, NULL, &detJ));
    2093:       PetscCall(DMPlexGetCellType(dm, p, &ptype));

    2095:       /* compare to previous facets: if computed, reference that dualspace */
    2096:       for (q = pStratStart[depth - 1]; q < p; q++) {
    2097:         DMPolytopeType qtype;

    2099:         PetscCall(DMPlexGetCellType(dm, q, &qtype));
    2100:         if (qtype == ptype) break;
    2101:       }
    2102:       if (q < p) { /* this facet has the same dual space as that one */
    2103:         PetscCall(PetscObjectReference((PetscObject)sp->pointSpaces[q]));
    2104:         sp->pointSpaces[p] = sp->pointSpaces[q];
    2105:         continue;
    2106:       }
    2107:       /* if not, recursively compute this dual space */
    2108:       PetscCall(PetscDualSpaceCreateFacetSubspace_Lagrange(sp, NULL, p, formDegree, Ncopies, PETSC_FALSE, &sp->pointSpaces[p]));
    2109:     }
    2110:     for (h = 2; h <= depth; h++) { /* get the higher subspaces from the facet subspaces */
    2111:       PetscInt hd   = depth - h;
    2112:       PetscInt hdim = dim - h;

    2114:       if (hdim < PetscAbsInt(formDegree)) break;
    2115:       for (p = pStratStart[hd]; p < pStratEnd[hd]; p++) {
    2116:         PetscInt        suppSize, s;
    2117:         const PetscInt *supp;

    2119:         PetscCall(DMPlexGetSupportSize(dm, p, &suppSize));
    2120:         PetscCall(DMPlexGetSupport(dm, p, &supp));
    2121:         for (s = 0; s < suppSize; s++) {
    2122:           DM              qdm;
    2123:           PetscDualSpace  qsp, psp;
    2124:           PetscInt        c, coneSize, q;
    2125:           const PetscInt *cone;
    2126:           const PetscInt *refCone;

    2128:           q   = supp[s];
    2129:           qsp = sp->pointSpaces[q];
    2130:           PetscCall(DMPlexGetConeSize(dm, q, &coneSize));
    2131:           PetscCall(DMPlexGetCone(dm, q, &cone));
    2132:           for (c = 0; c < coneSize; c++)
    2133:             if (cone[c] == p) break;
    2134:           PetscCheck(c != coneSize, PetscObjectComm((PetscObject)dm), PETSC_ERR_PLIB, "cone/support mismatch");
    2135:           PetscCall(PetscDualSpaceGetDM(qsp, &qdm));
    2136:           PetscCall(DMPlexGetCone(qdm, 0, &refCone));
    2137:           /* get the equivalent dual space from the support dual space */
    2138:           PetscCall(PetscDualSpaceGetPointSubspace(qsp, refCone[c], &psp));
    2139:           if (!s) {
    2140:             PetscCall(PetscObjectReference((PetscObject)psp));
    2141:             sp->pointSpaces[p] = psp;
    2142:           }
    2143:         }
    2144:       }
    2145:     }
    2146:     for (p = 1; p < pEnd; p++) {
    2147:       PetscInt pspdim;
    2148:       if (!sp->pointSpaces[p]) continue;
    2149:       PetscCall(PetscDualSpaceGetInteriorDimension(sp->pointSpaces[p], &pspdim));
    2150:       PetscCall(PetscSectionSetDof(section, p, pspdim));
    2151:     }
    2152:   }

    2154:   if (trimmed && !continuous) {
    2155:     /* the dofs of a trimmed space don't have a nice tensor/lattice structure:
    2156:      * just construct the continuous dual space and copy all of the data over,
    2157:      * allocating it all to the cell instead of splitting it up between the boundaries */
    2158:     PetscDualSpace      spcont;
    2159:     PetscInt            spdim, f;
    2160:     PetscQuadrature     allNodes;
    2161:     PetscDualSpace_Lag *lagc;
    2162:     Mat                 allMat;

    2164:     PetscCall(PetscDualSpaceDuplicate(sp, &spcont));
    2165:     PetscCall(PetscDualSpaceLagrangeSetContinuity(spcont, PETSC_TRUE));
    2166:     PetscCall(PetscDualSpaceSetUp(spcont));
    2167:     PetscCall(PetscDualSpaceGetDimension(spcont, &spdim));
    2168:     sp->spdim = sp->spintdim = spdim;
    2169:     PetscCall(PetscSectionSetDof(section, 0, spdim));
    2170:     PetscCall(PetscDualSpaceSectionSetUp_Internal(sp, section));
    2171:     PetscCall(PetscMalloc1(spdim, &sp->functional));
    2172:     for (f = 0; f < spdim; f++) {
    2173:       PetscQuadrature fn;

    2175:       PetscCall(PetscDualSpaceGetFunctional(spcont, f, &fn));
    2176:       PetscCall(PetscObjectReference((PetscObject)fn));
    2177:       sp->functional[f] = fn;
    2178:     }
    2179:     PetscCall(PetscDualSpaceGetAllData(spcont, &allNodes, &allMat));
    2180:     PetscCall(PetscObjectReference((PetscObject)allNodes));
    2181:     PetscCall(PetscObjectReference((PetscObject)allNodes));
    2182:     sp->allNodes = sp->intNodes = allNodes;
    2183:     PetscCall(PetscObjectReference((PetscObject)allMat));
    2184:     PetscCall(PetscObjectReference((PetscObject)allMat));
    2185:     sp->allMat = sp->intMat = allMat;
    2186:     lagc                    = (PetscDualSpace_Lag *)spcont->data;
    2187:     PetscCall(PetscLagNodeIndicesReference(lagc->vertIndices));
    2188:     lag->vertIndices = lagc->vertIndices;
    2189:     PetscCall(PetscLagNodeIndicesReference(lagc->allNodeIndices));
    2190:     PetscCall(PetscLagNodeIndicesReference(lagc->allNodeIndices));
    2191:     lag->intNodeIndices = lagc->allNodeIndices;
    2192:     lag->allNodeIndices = lagc->allNodeIndices;
    2193:     PetscCall(PetscDualSpaceDestroy(&spcont));
    2194:     PetscCall(PetscFree2(pStratStart, pStratEnd));
    2195:     PetscCall(DMDestroy(&dmint));
    2196:     PetscFunctionReturn(PETSC_SUCCESS);
    2197:   }

    2199:   /* step 3: construct intNodes, and intMat, and combine it with boundray data to make allNodes and allMat */
    2200:   if (!tensorSpace) {
    2201:     if (!tensorCell) PetscCall(PetscLagNodeIndicesCreateSimplexVertices(dm, &lag->vertIndices));

    2203:     if (trimmed) {
    2204:       /* there is one dof in the interior of the a trimmed element for each full polynomial of with degree at most
    2205:        * order + k - dim - 1 */
    2206:       if (order + PetscAbsInt(formDegree) > dim) {
    2207:         PetscInt sum = order + PetscAbsInt(formDegree) - dim - 1;
    2208:         PetscInt nDofs;

    2210:         PetscCall(PetscDualSpaceLagrangeCreateSimplexNodeMat(nodeFamily, dim, sum, Nk, numNodeSkip, &sp->intNodes, &sp->intMat, &lag->intNodeIndices));
    2211:         PetscCall(MatGetSize(sp->intMat, &nDofs, NULL));
    2212:         PetscCall(PetscSectionSetDof(section, 0, nDofs));
    2213:       }
    2214:       PetscCall(PetscDualSpaceSectionSetUp_Internal(sp, section));
    2215:       PetscCall(PetscDualSpaceCreateAllDataFromInteriorData(sp));
    2216:       PetscCall(PetscDualSpaceLagrangeCreateAllNodeIdx(sp));
    2217:     } else {
    2218:       if (!continuous) {
    2219:         /* if discontinuous just construct one node for each set of dofs (a set of dofs is a basis for the k-form
    2220:          * space) */
    2221:         PetscInt sum = order;
    2222:         PetscInt nDofs;

    2224:         PetscCall(PetscDualSpaceLagrangeCreateSimplexNodeMat(nodeFamily, dim, sum, Nk, numNodeSkip, &sp->intNodes, &sp->intMat, &lag->intNodeIndices));
    2225:         PetscCall(MatGetSize(sp->intMat, &nDofs, NULL));
    2226:         PetscCall(PetscSectionSetDof(section, 0, nDofs));
    2227:         PetscCall(PetscDualSpaceSectionSetUp_Internal(sp, section));
    2228:         PetscCall(PetscObjectReference((PetscObject)sp->intNodes));
    2229:         sp->allNodes = sp->intNodes;
    2230:         PetscCall(PetscObjectReference((PetscObject)sp->intMat));
    2231:         sp->allMat = sp->intMat;
    2232:         PetscCall(PetscLagNodeIndicesReference(lag->intNodeIndices));
    2233:         lag->allNodeIndices = lag->intNodeIndices;
    2234:       } else {
    2235:         /* there is one dof in the interior of the a full element for each trimmed polynomial of with degree at most
    2236:          * order + k - dim, but with complementary form degree */
    2237:         if (order + PetscAbsInt(formDegree) > dim) {
    2238:           PetscDualSpace      trimmedsp;
    2239:           PetscDualSpace_Lag *trimmedlag;
    2240:           PetscQuadrature     intNodes;
    2241:           PetscInt            trFormDegree = formDegree >= 0 ? formDegree - dim : dim - PetscAbsInt(formDegree);
    2242:           PetscInt            nDofs;
    2243:           Mat                 intMat;

    2245:           PetscCall(PetscDualSpaceDuplicate(sp, &trimmedsp));
    2246:           PetscCall(PetscDualSpaceLagrangeSetTrimmed(trimmedsp, PETSC_TRUE));
    2247:           PetscCall(PetscDualSpaceSetOrder(trimmedsp, order + PetscAbsInt(formDegree) - dim));
    2248:           PetscCall(PetscDualSpaceSetFormDegree(trimmedsp, trFormDegree));
    2249:           trimmedlag              = (PetscDualSpace_Lag *)trimmedsp->data;
    2250:           trimmedlag->numNodeSkip = numNodeSkip + 1;
    2251:           PetscCall(PetscDualSpaceSetUp(trimmedsp));
    2252:           PetscCall(PetscDualSpaceGetAllData(trimmedsp, &intNodes, &intMat));
    2253:           PetscCall(PetscObjectReference((PetscObject)intNodes));
    2254:           sp->intNodes = intNodes;
    2255:           PetscCall(PetscLagNodeIndicesReference(trimmedlag->allNodeIndices));
    2256:           lag->intNodeIndices = trimmedlag->allNodeIndices;
    2257:           PetscCall(PetscObjectReference((PetscObject)intMat));
    2258:           if (PetscAbsInt(formDegree) > 0 && PetscAbsInt(formDegree) < dim) {
    2259:             PetscReal   *T;
    2260:             PetscScalar *work;
    2261:             PetscInt     nCols, nRows;
    2262:             Mat          intMatT;

    2264:             PetscCall(MatDuplicate(intMat, MAT_COPY_VALUES, &intMatT));
    2265:             PetscCall(MatGetSize(intMat, &nRows, &nCols));
    2266:             PetscCall(PetscMalloc2(Nk * Nk, &T, nCols, &work));
    2267:             PetscCall(BiunitSimplexSymmetricFormTransformation(dim, formDegree, T));
    2268:             for (PetscInt row = 0; row < nRows; row++) {
    2269:               PetscInt           nrCols;
    2270:               const PetscInt    *rCols;
    2271:               const PetscScalar *rVals;

    2273:               PetscCall(MatGetRow(intMat, row, &nrCols, &rCols, &rVals));
    2274:               PetscCheck(nrCols % Nk == 0, PETSC_COMM_SELF, PETSC_ERR_PLIB, "nonzeros in intMat matrix are not in k-form size blocks");
    2275:               for (PetscInt b = 0; b < nrCols; b += Nk) {
    2276:                 const PetscScalar *v = &rVals[b];
    2277:                 PetscScalar       *w = &work[b];
    2278:                 for (PetscInt j = 0; j < Nk; j++) {
    2279:                   w[j] = 0.;
    2280:                   for (PetscInt i = 0; i < Nk; i++) w[j] += v[i] * T[i * Nk + j];
    2281:                 }
    2282:               }
    2283:               PetscCall(MatSetValuesBlocked(intMatT, 1, &row, nrCols, rCols, work, INSERT_VALUES));
    2284:               PetscCall(MatRestoreRow(intMat, row, &nrCols, &rCols, &rVals));
    2285:             }
    2286:             PetscCall(MatAssemblyBegin(intMatT, MAT_FINAL_ASSEMBLY));
    2287:             PetscCall(MatAssemblyEnd(intMatT, MAT_FINAL_ASSEMBLY));
    2288:             PetscCall(MatDestroy(&intMat));
    2289:             intMat = intMatT;
    2290:             PetscCall(PetscLagNodeIndicesDestroy(&lag->intNodeIndices));
    2291:             PetscCall(PetscLagNodeIndicesDuplicate(trimmedlag->allNodeIndices, &lag->intNodeIndices));
    2292:             {
    2293:               PetscInt         nNodes     = lag->intNodeIndices->nNodes;
    2294:               PetscReal       *newNodeVec = lag->intNodeIndices->nodeVec;
    2295:               const PetscReal *oldNodeVec = trimmedlag->allNodeIndices->nodeVec;

    2297:               for (PetscInt n = 0; n < nNodes; n++) {
    2298:                 PetscReal       *w = &newNodeVec[n * Nk];
    2299:                 const PetscReal *v = &oldNodeVec[n * Nk];

    2301:                 for (PetscInt j = 0; j < Nk; j++) {
    2302:                   w[j] = 0.;
    2303:                   for (PetscInt i = 0; i < Nk; i++) w[j] += v[i] * T[i * Nk + j];
    2304:                 }
    2305:               }
    2306:             }
    2307:             PetscCall(PetscFree2(T, work));
    2308:           }
    2309:           sp->intMat = intMat;
    2310:           PetscCall(MatGetSize(sp->intMat, &nDofs, NULL));
    2311:           PetscCall(PetscDualSpaceDestroy(&trimmedsp));
    2312:           PetscCall(PetscSectionSetDof(section, 0, nDofs));
    2313:         }
    2314:         PetscCall(PetscDualSpaceSectionSetUp_Internal(sp, section));
    2315:         PetscCall(PetscDualSpaceCreateAllDataFromInteriorData(sp));
    2316:         PetscCall(PetscDualSpaceLagrangeCreateAllNodeIdx(sp));
    2317:       }
    2318:     }
    2319:   } else {
    2320:     PetscQuadrature     intNodesTrace  = NULL;
    2321:     PetscQuadrature     intNodesFiber  = NULL;
    2322:     PetscQuadrature     intNodes       = NULL;
    2323:     PetscLagNodeIndices intNodeIndices = NULL;
    2324:     Mat                 intMat         = NULL;

    2326:     if (PetscAbsInt(formDegree) < dim) { /* get the trace k-forms on the first facet, and the 0-forms on the edge,
    2327:                                             and wedge them together to create some of the k-form dofs */
    2328:       PetscDualSpace      trace, fiber;
    2329:       PetscDualSpace_Lag *tracel, *fiberl;
    2330:       Mat                 intMatTrace, intMatFiber;

    2332:       if (sp->pointSpaces[tensorf]) {
    2333:         PetscCall(PetscObjectReference((PetscObject)sp->pointSpaces[tensorf]));
    2334:         trace = sp->pointSpaces[tensorf];
    2335:       } else {
    2336:         PetscCall(PetscDualSpaceCreateFacetSubspace_Lagrange(sp, NULL, tensorf, formDegree, Ncopies, PETSC_TRUE, &trace));
    2337:       }
    2338:       PetscCall(PetscDualSpaceCreateEdgeSubspace_Lagrange(sp, order, 0, 1, PETSC_TRUE, &fiber));
    2339:       tracel = (PetscDualSpace_Lag *)trace->data;
    2340:       fiberl = (PetscDualSpace_Lag *)fiber->data;
    2341:       PetscCall(PetscLagNodeIndicesCreateTensorVertices(dm, tracel->vertIndices, &lag->vertIndices));
    2342:       PetscCall(PetscDualSpaceGetInteriorData(trace, &intNodesTrace, &intMatTrace));
    2343:       PetscCall(PetscDualSpaceGetInteriorData(fiber, &intNodesFiber, &intMatFiber));
    2344:       if (intNodesTrace && intNodesFiber) {
    2345:         PetscCall(PetscQuadratureCreateTensor(intNodesTrace, intNodesFiber, &intNodes));
    2346:         PetscCall(MatTensorAltV(intMatTrace, intMatFiber, dim - 1, formDegree, 1, 0, &intMat));
    2347:         PetscCall(PetscLagNodeIndicesTensor(tracel->intNodeIndices, dim - 1, formDegree, fiberl->intNodeIndices, 1, 0, &intNodeIndices));
    2348:       }
    2349:       PetscCall(PetscObjectReference((PetscObject)intNodesTrace));
    2350:       PetscCall(PetscObjectReference((PetscObject)intNodesFiber));
    2351:       PetscCall(PetscDualSpaceDestroy(&fiber));
    2352:       PetscCall(PetscDualSpaceDestroy(&trace));
    2353:     }
    2354:     if (PetscAbsInt(formDegree) > 0) { /* get the trace (k-1)-forms on the first facet, and the 1-forms on the edge,
    2355:                                           and wedge them together to create the remaining k-form dofs */
    2356:       PetscDualSpace      trace, fiber;
    2357:       PetscDualSpace_Lag *tracel, *fiberl;
    2358:       PetscQuadrature     intNodesTrace2, intNodesFiber2, intNodes2;
    2359:       PetscLagNodeIndices intNodeIndices2;
    2360:       Mat                 intMatTrace, intMatFiber, intMat2;
    2361:       PetscInt            traceDegree = formDegree > 0 ? formDegree - 1 : formDegree + 1;
    2362:       PetscInt            fiberDegree = formDegree > 0 ? 1 : -1;

    2364:       PetscCall(PetscDualSpaceCreateFacetSubspace_Lagrange(sp, NULL, tensorf, traceDegree, Ncopies, PETSC_TRUE, &trace));
    2365:       PetscCall(PetscDualSpaceCreateEdgeSubspace_Lagrange(sp, order, fiberDegree, 1, PETSC_TRUE, &fiber));
    2366:       tracel = (PetscDualSpace_Lag *)trace->data;
    2367:       fiberl = (PetscDualSpace_Lag *)fiber->data;
    2368:       if (!lag->vertIndices) PetscCall(PetscLagNodeIndicesCreateTensorVertices(dm, tracel->vertIndices, &lag->vertIndices));
    2369:       PetscCall(PetscDualSpaceGetInteriorData(trace, &intNodesTrace2, &intMatTrace));
    2370:       PetscCall(PetscDualSpaceGetInteriorData(fiber, &intNodesFiber2, &intMatFiber));
    2371:       if (intNodesTrace2 && intNodesFiber2) {
    2372:         PetscCall(PetscQuadratureCreateTensor(intNodesTrace2, intNodesFiber2, &intNodes2));
    2373:         PetscCall(MatTensorAltV(intMatTrace, intMatFiber, dim - 1, traceDegree, 1, fiberDegree, &intMat2));
    2374:         PetscCall(PetscLagNodeIndicesTensor(tracel->intNodeIndices, dim - 1, traceDegree, fiberl->intNodeIndices, 1, fiberDegree, &intNodeIndices2));
    2375:         if (!intMat) {
    2376:           intMat         = intMat2;
    2377:           intNodes       = intNodes2;
    2378:           intNodeIndices = intNodeIndices2;
    2379:         } else {
    2380:           /* merge the matrices, quadrature points, and nodes */
    2381:           PetscInt            nM;
    2382:           PetscInt            nDof, nDof2;
    2383:           PetscInt           *toMerged = NULL, *toMerged2 = NULL;
    2384:           PetscQuadrature     merged               = NULL;
    2385:           PetscLagNodeIndices intNodeIndicesMerged = NULL;
    2386:           Mat                 matMerged            = NULL;

    2388:           PetscCall(MatGetSize(intMat, &nDof, NULL));
    2389:           PetscCall(MatGetSize(intMat2, &nDof2, NULL));
    2390:           PetscCall(PetscQuadraturePointsMerge(intNodes, intNodes2, &merged, &toMerged, &toMerged2));
    2391:           PetscCall(PetscQuadratureGetData(merged, NULL, NULL, &nM, NULL, NULL));
    2392:           PetscCall(MatricesMerge(intMat, intMat2, dim, formDegree, nM, toMerged, toMerged2, &matMerged));
    2393:           PetscCall(PetscLagNodeIndicesMerge(intNodeIndices, intNodeIndices2, &intNodeIndicesMerged));
    2394:           PetscCall(PetscFree(toMerged));
    2395:           PetscCall(PetscFree(toMerged2));
    2396:           PetscCall(MatDestroy(&intMat));
    2397:           PetscCall(MatDestroy(&intMat2));
    2398:           PetscCall(PetscQuadratureDestroy(&intNodes));
    2399:           PetscCall(PetscQuadratureDestroy(&intNodes2));
    2400:           PetscCall(PetscLagNodeIndicesDestroy(&intNodeIndices));
    2401:           PetscCall(PetscLagNodeIndicesDestroy(&intNodeIndices2));
    2402:           intNodes       = merged;
    2403:           intMat         = matMerged;
    2404:           intNodeIndices = intNodeIndicesMerged;
    2405:           if (!trimmed) {
    2406:             /* I think users expect that, when a node has a full basis for the k-forms,
    2407:              * they should be consecutive dofs.  That isn't the case for trimmed spaces,
    2408:              * but is for some of the nodes in untrimmed spaces, so in that case we
    2409:              * sort them to group them by node */
    2410:             Mat intMatPerm;

    2412:             PetscCall(MatPermuteByNodeIdx(intMat, intNodeIndices, &intMatPerm));
    2413:             PetscCall(MatDestroy(&intMat));
    2414:             intMat = intMatPerm;
    2415:           }
    2416:         }
    2417:       }
    2418:       PetscCall(PetscDualSpaceDestroy(&fiber));
    2419:       PetscCall(PetscDualSpaceDestroy(&trace));
    2420:     }
    2421:     PetscCall(PetscQuadratureDestroy(&intNodesTrace));
    2422:     PetscCall(PetscQuadratureDestroy(&intNodesFiber));
    2423:     sp->intNodes        = intNodes;
    2424:     sp->intMat          = intMat;
    2425:     lag->intNodeIndices = intNodeIndices;
    2426:     {
    2427:       PetscInt nDofs = 0;

    2429:       if (intMat) PetscCall(MatGetSize(intMat, &nDofs, NULL));
    2430:       PetscCall(PetscSectionSetDof(section, 0, nDofs));
    2431:     }
    2432:     PetscCall(PetscDualSpaceSectionSetUp_Internal(sp, section));
    2433:     if (continuous) {
    2434:       PetscCall(PetscDualSpaceCreateAllDataFromInteriorData(sp));
    2435:       PetscCall(PetscDualSpaceLagrangeCreateAllNodeIdx(sp));
    2436:     } else {
    2437:       PetscCall(PetscObjectReference((PetscObject)intNodes));
    2438:       sp->allNodes = intNodes;
    2439:       PetscCall(PetscObjectReference((PetscObject)intMat));
    2440:       sp->allMat = intMat;
    2441:       PetscCall(PetscLagNodeIndicesReference(intNodeIndices));
    2442:       lag->allNodeIndices = intNodeIndices;
    2443:     }
    2444:   }
    2445:   PetscCall(PetscSectionGetStorageSize(section, &sp->spdim));
    2446:   PetscCall(PetscSectionGetConstrainedStorageSize(section, &sp->spintdim));
    2447:   // TODO: fix this, computing functionals from moments should be no different for nodal vs modal
    2448:   if (lag->useMoments) {
    2449:     PetscCall(PetscDualSpaceComputeFunctionalsFromAllData_Moments(sp));
    2450:   } else {
    2451:     PetscCall(PetscDualSpaceComputeFunctionalsFromAllData(sp));
    2452:   }
    2453:   PetscCall(PetscFree2(pStratStart, pStratEnd));
    2454:   PetscCall(DMDestroy(&dmint));
    2455:   PetscFunctionReturn(PETSC_SUCCESS);
    2456: }

    2458: /* Create a matrix that represents the transformation that DMPlexVecGetClosure() would need
    2459:  * to get the representation of the dofs for a mesh point if the mesh point had this orientation
    2460:  * relative to the cell */
    2461: PetscErrorCode PetscDualSpaceCreateInteriorSymmetryMatrix_Lagrange(PetscDualSpace sp, PetscInt ornt, Mat *symMat)
    2462: {
    2463:   PetscDualSpace_Lag *lag;
    2464:   DM                  dm;
    2465:   PetscLagNodeIndices vertIndices, intNodeIndices;
    2466:   PetscLagNodeIndices ni;
    2467:   PetscInt            nodeIdxDim, nodeVecDim, nNodes;
    2468:   PetscInt            formDegree;
    2469:   PetscInt           *perm, *permOrnt;
    2470:   PetscInt           *nnz;
    2471:   PetscInt            n;
    2472:   PetscInt            maxGroupSize;
    2473:   PetscScalar        *V, *W, *work;
    2474:   Mat                 A;

    2476:   PetscFunctionBegin;
    2477:   if (!sp->spintdim) {
    2478:     *symMat = NULL;
    2479:     PetscFunctionReturn(PETSC_SUCCESS);
    2480:   }
    2481:   lag            = (PetscDualSpace_Lag *)sp->data;
    2482:   vertIndices    = lag->vertIndices;
    2483:   intNodeIndices = lag->intNodeIndices;
    2484:   PetscCall(PetscDualSpaceGetDM(sp, &dm));
    2485:   PetscCall(PetscDualSpaceGetFormDegree(sp, &formDegree));
    2486:   PetscCall(PetscNew(&ni));
    2487:   ni->refct      = 1;
    2488:   ni->nodeIdxDim = nodeIdxDim = intNodeIndices->nodeIdxDim;
    2489:   ni->nodeVecDim = nodeVecDim = intNodeIndices->nodeVecDim;
    2490:   ni->nNodes = nNodes = intNodeIndices->nNodes;
    2491:   PetscCall(PetscMalloc1(nNodes * nodeIdxDim, &ni->nodeIdx));
    2492:   PetscCall(PetscMalloc1(nNodes * nodeVecDim, &ni->nodeVec));
    2493:   /* push forward the dofs by the symmetry of the reference element induced by ornt */
    2494:   PetscCall(PetscLagNodeIndicesPushForward(dm, vertIndices, 0, vertIndices, intNodeIndices, ornt, formDegree, ni->nodeIdx, ni->nodeVec));
    2495:   /* get the revlex order for both the original and transformed dofs */
    2496:   PetscCall(PetscLagNodeIndicesGetPermutation(intNodeIndices, &perm));
    2497:   PetscCall(PetscLagNodeIndicesGetPermutation(ni, &permOrnt));
    2498:   PetscCall(PetscMalloc1(nNodes, &nnz));
    2499:   for (n = 0, maxGroupSize = 0; n < nNodes;) { /* incremented in the loop */
    2500:     PetscInt *nind = &(ni->nodeIdx[permOrnt[n] * nodeIdxDim]);
    2501:     PetscInt  m, nEnd;
    2502:     PetscInt  groupSize;
    2503:     /* for each group of dofs that have the same nodeIdx coordinate */
    2504:     for (nEnd = n + 1; nEnd < nNodes; nEnd++) {
    2505:       PetscInt *mind = &(ni->nodeIdx[permOrnt[nEnd] * nodeIdxDim]);
    2506:       PetscInt  d;

    2508:       /* compare the oriented permutation indices */
    2509:       for (d = 0; d < nodeIdxDim; d++)
    2510:         if (mind[d] != nind[d]) break;
    2511:       if (d < nodeIdxDim) break;
    2512:     }
    2513:     /* permOrnt[[n, nEnd)] is a group of dofs that, under the symmetry are at the same location */

    2515:     /* the symmetry had better map the group of dofs with the same permuted nodeIdx
    2516:      * to a group of dofs with the same size, otherwise we messed up */
    2517:     if (PetscDefined(USE_DEBUG)) {
    2518:       PetscInt  m;
    2519:       PetscInt *nind = &(intNodeIndices->nodeIdx[perm[n] * nodeIdxDim]);

    2521:       for (m = n + 1; m < nEnd; m++) {
    2522:         PetscInt *mind = &(intNodeIndices->nodeIdx[perm[m] * nodeIdxDim]);
    2523:         PetscInt  d;

    2525:         /* compare the oriented permutation indices */
    2526:         for (d = 0; d < nodeIdxDim; d++)
    2527:           if (mind[d] != nind[d]) break;
    2528:         if (d < nodeIdxDim) break;
    2529:       }
    2530:       PetscCheck(m >= nEnd, PETSC_COMM_SELF, PETSC_ERR_PLIB, "Dofs with same index after symmetry not same block size");
    2531:     }
    2532:     groupSize = nEnd - n;
    2533:     /* each pushforward dof vector will be expressed in a basis of the unpermuted dofs */
    2534:     for (m = n; m < nEnd; m++) nnz[permOrnt[m]] = groupSize;

    2536:     maxGroupSize = PetscMax(maxGroupSize, nEnd - n);
    2537:     n            = nEnd;
    2538:   }
    2539:   PetscCheck(maxGroupSize <= nodeVecDim, PETSC_COMM_SELF, PETSC_ERR_PLIB, "Dofs are not in blocks that can be solved");
    2540:   PetscCall(MatCreateSeqAIJ(PETSC_COMM_SELF, nNodes, nNodes, 0, nnz, &A));
    2541:   PetscCall(PetscObjectSetOptionsPrefix((PetscObject)A, "lag_"));
    2542:   PetscCall(PetscFree(nnz));
    2543:   PetscCall(PetscMalloc3(maxGroupSize * nodeVecDim, &V, maxGroupSize * nodeVecDim, &W, nodeVecDim * 2, &work));
    2544:   for (n = 0; n < nNodes;) { /* incremented in the loop */
    2545:     PetscInt *nind = &(ni->nodeIdx[permOrnt[n] * nodeIdxDim]);
    2546:     PetscInt  nEnd;
    2547:     PetscInt  m;
    2548:     PetscInt  groupSize;
    2549:     for (nEnd = n + 1; nEnd < nNodes; nEnd++) {
    2550:       PetscInt *mind = &(ni->nodeIdx[permOrnt[nEnd] * nodeIdxDim]);
    2551:       PetscInt  d;

    2553:       /* compare the oriented permutation indices */
    2554:       for (d = 0; d < nodeIdxDim; d++)
    2555:         if (mind[d] != nind[d]) break;
    2556:       if (d < nodeIdxDim) break;
    2557:     }
    2558:     groupSize = nEnd - n;
    2559:     /* get all of the vectors from the original and all of the pushforward vectors */
    2560:     for (m = n; m < nEnd; m++) {
    2561:       PetscInt d;

    2563:       for (d = 0; d < nodeVecDim; d++) {
    2564:         V[(m - n) * nodeVecDim + d] = intNodeIndices->nodeVec[perm[m] * nodeVecDim + d];
    2565:         W[(m - n) * nodeVecDim + d] = ni->nodeVec[permOrnt[m] * nodeVecDim + d];
    2566:       }
    2567:     }
    2568:     /* now we have to solve for W in terms of V: the systems isn't always square, but the span
    2569:      * of V and W should always be the same, so the solution of the normal equations works */
    2570:     {
    2571:       char         transpose = 'N';
    2572:       PetscBLASInt bm, bn, bnrhs, blda, bldb, blwork, info;

    2574:       PetscCall(PetscBLASIntCast(nodeVecDim, &bm));
    2575:       PetscCall(PetscBLASIntCast(groupSize, &bn));
    2576:       PetscCall(PetscBLASIntCast(groupSize, &bnrhs));
    2577:       PetscCall(PetscBLASIntCast(bm, &blda));
    2578:       PetscCall(PetscBLASIntCast(bm, &bldb));
    2579:       PetscCall(PetscBLASIntCast(2 * nodeVecDim, &blwork));
    2580:       PetscCallBLAS("LAPACKgels", LAPACKgels_(&transpose, &bm, &bn, &bnrhs, V, &blda, W, &bldb, work, &blwork, &info));
    2581:       PetscCheck(info == 0, PETSC_COMM_SELF, PETSC_ERR_LIB, "Bad argument to GELS");
    2582:       /* repack */
    2583:       {
    2584:         PetscInt i, j;

    2586:         for (i = 0; i < groupSize; i++) {
    2587:           for (j = 0; j < groupSize; j++) {
    2588:             /* notice the different leading dimension */
    2589:             V[i * groupSize + j] = W[i * nodeVecDim + j];
    2590:           }
    2591:         }
    2592:       }
    2593:       if (PetscDefined(USE_DEBUG)) {
    2594:         PetscReal res;

    2596:         /* check that the normal error is 0 */
    2597:         for (m = n; m < nEnd; m++) {
    2598:           PetscInt d;

    2600:           for (d = 0; d < nodeVecDim; d++) W[(m - n) * nodeVecDim + d] = ni->nodeVec[permOrnt[m] * nodeVecDim + d];
    2601:         }
    2602:         res = 0.;
    2603:         for (PetscInt i = 0; i < groupSize; i++) {
    2604:           for (PetscInt j = 0; j < nodeVecDim; j++) {
    2605:             for (PetscInt k = 0; k < groupSize; k++) W[i * nodeVecDim + j] -= V[i * groupSize + k] * intNodeIndices->nodeVec[perm[n + k] * nodeVecDim + j];
    2606:             res += PetscAbsScalar(W[i * nodeVecDim + j]);
    2607:           }
    2608:         }
    2609:         PetscCheck(res <= PETSC_SMALL, PETSC_COMM_SELF, PETSC_ERR_LIB, "Dof block did not solve");
    2610:       }
    2611:     }
    2612:     PetscCall(MatSetValues(A, groupSize, &permOrnt[n], groupSize, &perm[n], V, INSERT_VALUES));
    2613:     n = nEnd;
    2614:   }
    2615:   PetscCall(MatAssemblyBegin(A, MAT_FINAL_ASSEMBLY));
    2616:   PetscCall(MatAssemblyEnd(A, MAT_FINAL_ASSEMBLY));
    2617:   *symMat = A;
    2618:   PetscCall(PetscFree3(V, W, work));
    2619:   PetscCall(PetscLagNodeIndicesDestroy(&ni));
    2620:   PetscFunctionReturn(PETSC_SUCCESS);
    2621: }

    2623: // get the symmetries of closure points
    2624: PETSC_INTERN PetscErrorCode PetscDualSpaceGetBoundarySymmetries_Internal(PetscDualSpace sp, PetscInt ***symperms, PetscScalar ***symflips)
    2625: {
    2626:   PetscInt  closureSize = 0;
    2627:   PetscInt *closure     = NULL;
    2628:   PetscInt  r;

    2630:   PetscFunctionBegin;
    2631:   PetscCall(DMPlexGetTransitiveClosure(sp->dm, 0, PETSC_TRUE, &closureSize, &closure));
    2632:   for (r = 0; r < closureSize; r++) {
    2633:     PetscDualSpace       psp;
    2634:     PetscInt             point = closure[2 * r];
    2635:     PetscInt             pspintdim;
    2636:     const PetscInt    ***psymperms = NULL;
    2637:     const PetscScalar ***psymflips = NULL;

    2639:     if (!point) continue;
    2640:     PetscCall(PetscDualSpaceGetPointSubspace(sp, point, &psp));
    2641:     if (!psp) continue;
    2642:     PetscCall(PetscDualSpaceGetInteriorDimension(psp, &pspintdim));
    2643:     if (!pspintdim) continue;
    2644:     PetscCall(PetscDualSpaceGetSymmetries(psp, &psymperms, &psymflips));
    2645:     symperms[r] = (PetscInt **)(psymperms ? psymperms[0] : NULL);
    2646:     symflips[r] = (PetscScalar **)(psymflips ? psymflips[0] : NULL);
    2647:   }
    2648:   PetscCall(DMPlexRestoreTransitiveClosure(sp->dm, 0, PETSC_TRUE, &closureSize, &closure));
    2649:   PetscFunctionReturn(PETSC_SUCCESS);
    2650: }

    2652: #define BaryIndex(perEdge, a, b, c) (((b) * (2 * perEdge + 1 - (b))) / 2) + (c)

    2654: #define CartIndex(perEdge, a, b) (perEdge * (a) + b)

    2656: /* the existing interface for symmetries is insufficient for all cases:
    2657:  * - it should be sufficient for form degrees that are scalar (0 and n)
    2658:  * - it should be sufficient for hypercube dofs
    2659:  * - it isn't sufficient for simplex cells with non-scalar form degrees if
    2660:  *   there are any dofs in the interior
    2661:  *
    2662:  * We compute the general transformation matrices, and if they fit, we return them,
    2663:  * otherwise we error (but we should probably change the interface to allow for
    2664:  * these symmetries)
    2665:  */
    2666: static PetscErrorCode PetscDualSpaceGetSymmetries_Lagrange(PetscDualSpace sp, const PetscInt ****perms, const PetscScalar ****flips)
    2667: {
    2668:   PetscDualSpace_Lag *lag = (PetscDualSpace_Lag *)sp->data;
    2669:   PetscInt            dim, order, Nc;

    2671:   PetscFunctionBegin;
    2672:   PetscCall(PetscDualSpaceGetOrder(sp, &order));
    2673:   PetscCall(PetscDualSpaceGetNumComponents(sp, &Nc));
    2674:   PetscCall(DMGetDimension(sp->dm, &dim));
    2675:   if (!lag->symComputed) { /* store symmetries */
    2676:     PetscInt       pStart, pEnd, p;
    2677:     PetscInt       numPoints;
    2678:     PetscInt       numFaces;
    2679:     PetscInt       spintdim;
    2680:     PetscInt    ***symperms;
    2681:     PetscScalar ***symflips;

    2683:     PetscCall(DMPlexGetChart(sp->dm, &pStart, &pEnd));
    2684:     numPoints = pEnd - pStart;
    2685:     {
    2686:       DMPolytopeType ct;
    2687:       /* The number of arrangements is no longer based on the number of faces */
    2688:       PetscCall(DMPlexGetCellType(sp->dm, 0, &ct));
    2689:       numFaces = DMPolytopeTypeGetNumArrangements(ct) / 2;
    2690:     }
    2691:     PetscCall(PetscCalloc1(numPoints, &symperms));
    2692:     PetscCall(PetscCalloc1(numPoints, &symflips));
    2693:     spintdim = sp->spintdim;
    2694:     /* The nodal symmetry behavior is not present when tensorSpace != tensorCell: someone might want this for the "S"
    2695:      * family of FEEC spaces.  Most used in particular are discontinuous polynomial L2 spaces in tensor cells, where
    2696:      * the symmetries are not necessary for FE assembly.  So for now we assume this is the case and don't return
    2697:      * symmetries if tensorSpace != tensorCell */
    2698:     if (spintdim && 0 < dim && dim < 3 && (lag->tensorSpace == lag->tensorCell)) { /* compute self symmetries */
    2699:       PetscInt    **cellSymperms;
    2700:       PetscScalar **cellSymflips;
    2701:       PetscInt      ornt;
    2702:       PetscInt      nCopies = Nc / lag->intNodeIndices->nodeVecDim;
    2703:       PetscInt      nNodes  = lag->intNodeIndices->nNodes;

    2705:       lag->numSelfSym = 2 * numFaces;
    2706:       lag->selfSymOff = numFaces;
    2707:       PetscCall(PetscCalloc1(2 * numFaces, &cellSymperms));
    2708:       PetscCall(PetscCalloc1(2 * numFaces, &cellSymflips));
    2709:       /* we want to be able to index symmetries directly with the orientations, which range from [-numFaces,numFaces) */
    2710:       symperms[0] = &cellSymperms[numFaces];
    2711:       symflips[0] = &cellSymflips[numFaces];
    2712:       PetscCheck(lag->intNodeIndices->nodeVecDim * nCopies == Nc, PETSC_COMM_SELF, PETSC_ERR_PLIB, "Node indices incompatible with dofs");
    2713:       PetscCheck(nNodes * nCopies == spintdim, PETSC_COMM_SELF, PETSC_ERR_PLIB, "Node indices incompatible with dofs");
    2714:       for (ornt = -numFaces; ornt < numFaces; ornt++) { /* for every symmetry, compute the symmetry matrix, and extract rows to see if it fits in the perm + flip framework */
    2715:         Mat          symMat;
    2716:         PetscInt    *perm;
    2717:         PetscScalar *flips;
    2718:         PetscInt     i;

    2720:         if (!ornt) continue;
    2721:         PetscCall(PetscMalloc1(spintdim, &perm));
    2722:         PetscCall(PetscCalloc1(spintdim, &flips));
    2723:         for (i = 0; i < spintdim; i++) perm[i] = -1;
    2724:         PetscCall(PetscDualSpaceCreateInteriorSymmetryMatrix_Lagrange(sp, ornt, &symMat));
    2725:         for (i = 0; i < nNodes; i++) {
    2726:           PetscInt           ncols;
    2727:           PetscInt           j, k;
    2728:           const PetscInt    *cols;
    2729:           const PetscScalar *vals;
    2730:           PetscBool          nz_seen = PETSC_FALSE;

    2732:           PetscCall(MatGetRow(symMat, i, &ncols, &cols, &vals));
    2733:           for (j = 0; j < ncols; j++) {
    2734:             if (PetscAbsScalar(vals[j]) > PETSC_SMALL) {
    2735:               PetscCheck(!nz_seen, PETSC_COMM_SELF, PETSC_ERR_ARG_INCOMP, "This dual space has symmetries that can't be described as a permutation + sign flips");
    2736:               nz_seen = PETSC_TRUE;
    2737:               PetscCheck(PetscAbsReal(PetscAbsScalar(vals[j]) - PetscRealConstant(1.)) <= PETSC_SMALL, PETSC_COMM_SELF, PETSC_ERR_ARG_INCOMP, "This dual space has symmetries that can't be described as a permutation + sign flips");
    2738:               PetscCheck(PetscAbsReal(PetscImaginaryPart(vals[j])) <= PETSC_SMALL, PETSC_COMM_SELF, PETSC_ERR_ARG_INCOMP, "This dual space has symmetries that can't be described as a permutation + sign flips");
    2739:               PetscCheck(perm[cols[j] * nCopies] < 0, PETSC_COMM_SELF, PETSC_ERR_ARG_INCOMP, "This dual space has symmetries that can't be described as a permutation + sign flips");
    2740:               for (k = 0; k < nCopies; k++) perm[cols[j] * nCopies + k] = i * nCopies + k;
    2741:               if (PetscRealPart(vals[j]) < 0.) {
    2742:                 for (k = 0; k < nCopies; k++) flips[i * nCopies + k] = -1.;
    2743:               } else {
    2744:                 for (k = 0; k < nCopies; k++) flips[i * nCopies + k] = 1.;
    2745:               }
    2746:             }
    2747:           }
    2748:           PetscCall(MatRestoreRow(symMat, i, &ncols, &cols, &vals));
    2749:         }
    2750:         PetscCall(MatDestroy(&symMat));
    2751:         /* if there were no sign flips, keep NULL */
    2752:         for (i = 0; i < spintdim; i++)
    2753:           if (flips[i] != 1.) break;
    2754:         if (i == spintdim) {
    2755:           PetscCall(PetscFree(flips));
    2756:           flips = NULL;
    2757:         }
    2758:         /* if the permutation is identity, keep NULL */
    2759:         for (i = 0; i < spintdim; i++)
    2760:           if (perm[i] != i) break;
    2761:         if (i == spintdim) {
    2762:           PetscCall(PetscFree(perm));
    2763:           perm = NULL;
    2764:         }
    2765:         symperms[0][ornt] = perm;
    2766:         symflips[0][ornt] = flips;
    2767:       }
    2768:       /* if no orientations produced non-identity permutations, keep NULL */
    2769:       for (ornt = -numFaces; ornt < numFaces; ornt++)
    2770:         if (symperms[0][ornt]) break;
    2771:       if (ornt == numFaces) {
    2772:         PetscCall(PetscFree(cellSymperms));
    2773:         symperms[0] = NULL;
    2774:       }
    2775:       /* if no orientations produced sign flips, keep NULL */
    2776:       for (ornt = -numFaces; ornt < numFaces; ornt++)
    2777:         if (symflips[0][ornt]) break;
    2778:       if (ornt == numFaces) {
    2779:         PetscCall(PetscFree(cellSymflips));
    2780:         symflips[0] = NULL;
    2781:       }
    2782:     }
    2783:     PetscCall(PetscDualSpaceGetBoundarySymmetries_Internal(sp, symperms, symflips));
    2784:     for (p = 0; p < pEnd; p++)
    2785:       if (symperms[p]) break;
    2786:     if (p == pEnd) {
    2787:       PetscCall(PetscFree(symperms));
    2788:       symperms = NULL;
    2789:     }
    2790:     for (p = 0; p < pEnd; p++)
    2791:       if (symflips[p]) break;
    2792:     if (p == pEnd) {
    2793:       PetscCall(PetscFree(symflips));
    2794:       symflips = NULL;
    2795:     }
    2796:     lag->symperms    = symperms;
    2797:     lag->symflips    = symflips;
    2798:     lag->symComputed = PETSC_TRUE;
    2799:   }
    2800:   if (perms) *perms = (const PetscInt ***)lag->symperms;
    2801:   if (flips) *flips = (const PetscScalar ***)lag->symflips;
    2802:   PetscFunctionReturn(PETSC_SUCCESS);
    2803: }

    2805: static PetscErrorCode PetscDualSpaceLagrangeGetContinuity_Lagrange(PetscDualSpace sp, PetscBool *continuous)
    2806: {
    2807:   PetscDualSpace_Lag *lag = (PetscDualSpace_Lag *)sp->data;

    2809:   PetscFunctionBegin;
    2811:   PetscAssertPointer(continuous, 2);
    2812:   *continuous = lag->continuous;
    2813:   PetscFunctionReturn(PETSC_SUCCESS);
    2814: }

    2816: static PetscErrorCode PetscDualSpaceLagrangeSetContinuity_Lagrange(PetscDualSpace sp, PetscBool continuous)
    2817: {
    2818:   PetscDualSpace_Lag *lag = (PetscDualSpace_Lag *)sp->data;

    2820:   PetscFunctionBegin;
    2822:   lag->continuous = continuous;
    2823:   PetscFunctionReturn(PETSC_SUCCESS);
    2824: }

    2826: /*@
    2827:   PetscDualSpaceLagrangeGetContinuity - Retrieves the flag for element continuity

    2829:   Not Collective

    2831:   Input Parameter:
    2832: . sp - the `PetscDualSpace`

    2834:   Output Parameter:
    2835: . continuous - flag for element continuity

    2837:   Level: intermediate

    2839: .seealso: `PETSCDUALSPACELAGRANGE`, `PetscDualSpace`, `PetscDualSpaceLagrangeSetContinuity()`
    2840: @*/
    2841: PetscErrorCode PetscDualSpaceLagrangeGetContinuity(PetscDualSpace sp, PetscBool *continuous)
    2842: {
    2843:   PetscFunctionBegin;
    2845:   PetscAssertPointer(continuous, 2);
    2846:   PetscTryMethod(sp, "PetscDualSpaceLagrangeGetContinuity_C", (PetscDualSpace, PetscBool *), (sp, continuous));
    2847:   PetscFunctionReturn(PETSC_SUCCESS);
    2848: }

    2850: /*@
    2851:   PetscDualSpaceLagrangeSetContinuity - Indicate whether the element is continuous

    2853:   Logically Collective

    2855:   Input Parameters:
    2856: + sp         - the `PetscDualSpace`
    2857: - continuous - flag for element continuity

    2859:   Options Database Key:
    2860: . -petscdualspace_lagrange_continuity <bool> - use a continuous element

    2862:   Level: intermediate

    2864: .seealso: `PETSCDUALSPACELAGRANGE`, `PetscDualSpace`, `PetscDualSpaceLagrangeGetContinuity()`
    2865: @*/
    2866: PetscErrorCode PetscDualSpaceLagrangeSetContinuity(PetscDualSpace sp, PetscBool continuous)
    2867: {
    2868:   PetscFunctionBegin;
    2871:   PetscTryMethod(sp, "PetscDualSpaceLagrangeSetContinuity_C", (PetscDualSpace, PetscBool), (sp, continuous));
    2872:   PetscFunctionReturn(PETSC_SUCCESS);
    2873: }

    2875: static PetscErrorCode PetscDualSpaceLagrangeGetTensor_Lagrange(PetscDualSpace sp, PetscBool *tensor)
    2876: {
    2877:   PetscDualSpace_Lag *lag = (PetscDualSpace_Lag *)sp->data;

    2879:   PetscFunctionBegin;
    2880:   *tensor = lag->tensorSpace;
    2881:   PetscFunctionReturn(PETSC_SUCCESS);
    2882: }

    2884: static PetscErrorCode PetscDualSpaceLagrangeSetTensor_Lagrange(PetscDualSpace sp, PetscBool tensor)
    2885: {
    2886:   PetscDualSpace_Lag *lag = (PetscDualSpace_Lag *)sp->data;

    2888:   PetscFunctionBegin;
    2889:   lag->tensorSpace = tensor;
    2890:   PetscFunctionReturn(PETSC_SUCCESS);
    2891: }

    2893: static PetscErrorCode PetscDualSpaceLagrangeGetTrimmed_Lagrange(PetscDualSpace sp, PetscBool *trimmed)
    2894: {
    2895:   PetscDualSpace_Lag *lag = (PetscDualSpace_Lag *)sp->data;

    2897:   PetscFunctionBegin;
    2898:   *trimmed = lag->trimmed;
    2899:   PetscFunctionReturn(PETSC_SUCCESS);
    2900: }

    2902: static PetscErrorCode PetscDualSpaceLagrangeSetTrimmed_Lagrange(PetscDualSpace sp, PetscBool trimmed)
    2903: {
    2904:   PetscDualSpace_Lag *lag = (PetscDualSpace_Lag *)sp->data;

    2906:   PetscFunctionBegin;
    2907:   lag->trimmed = trimmed;
    2908:   PetscFunctionReturn(PETSC_SUCCESS);
    2909: }

    2911: static PetscErrorCode PetscDualSpaceLagrangeGetNodeType_Lagrange(PetscDualSpace sp, PetscDTNodeType *nodeType, PetscBool *boundary, PetscReal *exponent)
    2912: {
    2913:   PetscDualSpace_Lag *lag = (PetscDualSpace_Lag *)sp->data;

    2915:   PetscFunctionBegin;
    2916:   if (nodeType) *nodeType = lag->nodeType;
    2917:   if (boundary) *boundary = lag->endNodes;
    2918:   if (exponent) *exponent = lag->nodeExponent;
    2919:   PetscFunctionReturn(PETSC_SUCCESS);
    2920: }

    2922: static PetscErrorCode PetscDualSpaceLagrangeSetNodeType_Lagrange(PetscDualSpace sp, PetscDTNodeType nodeType, PetscBool boundary, PetscReal exponent)
    2923: {
    2924:   PetscDualSpace_Lag *lag = (PetscDualSpace_Lag *)sp->data;

    2926:   PetscFunctionBegin;
    2927:   PetscCheck(nodeType != PETSCDTNODES_GAUSSJACOBI || exponent > -1., PetscObjectComm((PetscObject)sp), PETSC_ERR_ARG_OUTOFRANGE, "Exponent must be > -1");
    2928:   lag->nodeType     = nodeType;
    2929:   lag->endNodes     = boundary;
    2930:   lag->nodeExponent = exponent;
    2931:   PetscFunctionReturn(PETSC_SUCCESS);
    2932: }

    2934: static PetscErrorCode PetscDualSpaceLagrangeGetUseMoments_Lagrange(PetscDualSpace sp, PetscBool *useMoments)
    2935: {
    2936:   PetscDualSpace_Lag *lag = (PetscDualSpace_Lag *)sp->data;

    2938:   PetscFunctionBegin;
    2939:   *useMoments = lag->useMoments;
    2940:   PetscFunctionReturn(PETSC_SUCCESS);
    2941: }

    2943: static PetscErrorCode PetscDualSpaceLagrangeSetUseMoments_Lagrange(PetscDualSpace sp, PetscBool useMoments)
    2944: {
    2945:   PetscDualSpace_Lag *lag = (PetscDualSpace_Lag *)sp->data;

    2947:   PetscFunctionBegin;
    2948:   lag->useMoments = useMoments;
    2949:   PetscFunctionReturn(PETSC_SUCCESS);
    2950: }

    2952: static PetscErrorCode PetscDualSpaceLagrangeGetMomentOrder_Lagrange(PetscDualSpace sp, PetscInt *momentOrder)
    2953: {
    2954:   PetscDualSpace_Lag *lag = (PetscDualSpace_Lag *)sp->data;

    2956:   PetscFunctionBegin;
    2957:   *momentOrder = lag->momentOrder;
    2958:   PetscFunctionReturn(PETSC_SUCCESS);
    2959: }

    2961: static PetscErrorCode PetscDualSpaceLagrangeSetMomentOrder_Lagrange(PetscDualSpace sp, PetscInt momentOrder)
    2962: {
    2963:   PetscDualSpace_Lag *lag = (PetscDualSpace_Lag *)sp->data;

    2965:   PetscFunctionBegin;
    2966:   lag->momentOrder = momentOrder;
    2967:   PetscFunctionReturn(PETSC_SUCCESS);
    2968: }

    2970: /*@
    2971:   PetscDualSpaceLagrangeGetTensor - Get the tensor nature of the dual space

    2973:   Not Collective

    2975:   Input Parameter:
    2976: . sp - The `PetscDualSpace`

    2978:   Output Parameter:
    2979: . tensor - Whether the dual space has tensor layout (vs. simplicial)

    2981:   Level: intermediate

    2983: .seealso: `PETSCDUALSPACELAGRANGE`, `PetscDualSpace`, `PetscDualSpaceLagrangeSetTensor()`, `PetscDualSpaceCreate()`
    2984: @*/
    2985: PetscErrorCode PetscDualSpaceLagrangeGetTensor(PetscDualSpace sp, PetscBool *tensor)
    2986: {
    2987:   PetscFunctionBegin;
    2989:   PetscAssertPointer(tensor, 2);
    2990:   PetscTryMethod(sp, "PetscDualSpaceLagrangeGetTensor_C", (PetscDualSpace, PetscBool *), (sp, tensor));
    2991:   PetscFunctionReturn(PETSC_SUCCESS);
    2992: }

    2994: /*@
    2995:   PetscDualSpaceLagrangeSetTensor - Set the tensor nature of the dual space

    2997:   Not Collective

    2999:   Input Parameters:
    3000: + sp     - The `PetscDualSpace`
    3001: - tensor - Whether the dual space has tensor layout (vs. simplicial)

    3003:   Level: intermediate

    3005: .seealso: `PETSCDUALSPACELAGRANGE`, `PetscDualSpace`, `PetscDualSpaceLagrangeGetTensor()`, `PetscDualSpaceCreate()`
    3006: @*/
    3007: PetscErrorCode PetscDualSpaceLagrangeSetTensor(PetscDualSpace sp, PetscBool tensor)
    3008: {
    3009:   PetscFunctionBegin;
    3011:   PetscTryMethod(sp, "PetscDualSpaceLagrangeSetTensor_C", (PetscDualSpace, PetscBool), (sp, tensor));
    3012:   PetscFunctionReturn(PETSC_SUCCESS);
    3013: }

    3015: /*@
    3016:   PetscDualSpaceLagrangeGetTrimmed - Get the trimmed nature of the dual space

    3018:   Not Collective

    3020:   Input Parameter:
    3021: . sp - The `PetscDualSpace`

    3023:   Output Parameter:
    3024: . trimmed - Whether the dual space represents to dual basis of a trimmed polynomial space (e.g. Raviart-Thomas and higher order / other form degree variants)

    3026:   Level: intermediate

    3028: .seealso: `PETSCDUALSPACELAGRANGE`, `PetscDualSpace`, `PetscDualSpaceLagrangeSetTrimmed()`, `PetscDualSpaceCreate()`
    3029: @*/
    3030: PetscErrorCode PetscDualSpaceLagrangeGetTrimmed(PetscDualSpace sp, PetscBool *trimmed)
    3031: {
    3032:   PetscFunctionBegin;
    3034:   PetscAssertPointer(trimmed, 2);
    3035:   PetscTryMethod(sp, "PetscDualSpaceLagrangeGetTrimmed_C", (PetscDualSpace, PetscBool *), (sp, trimmed));
    3036:   PetscFunctionReturn(PETSC_SUCCESS);
    3037: }

    3039: /*@
    3040:   PetscDualSpaceLagrangeSetTrimmed - Set the trimmed nature of the dual space

    3042:   Not Collective

    3044:   Input Parameters:
    3045: + sp      - The `PetscDualSpace`
    3046: - trimmed - Whether the dual space represents to dual basis of a trimmed polynomial space (e.g. Raviart-Thomas and higher order / other form degree variants)

    3048:   Level: intermediate

    3050: .seealso: `PETSCDUALSPACELAGRANGE`, `PetscDualSpace`, `PetscDualSpaceLagrangeGetTrimmed()`, `PetscDualSpaceCreate()`
    3051: @*/
    3052: PetscErrorCode PetscDualSpaceLagrangeSetTrimmed(PetscDualSpace sp, PetscBool trimmed)
    3053: {
    3054:   PetscFunctionBegin;
    3056:   PetscTryMethod(sp, "PetscDualSpaceLagrangeSetTrimmed_C", (PetscDualSpace, PetscBool), (sp, trimmed));
    3057:   PetscFunctionReturn(PETSC_SUCCESS);
    3058: }

    3060: /*@
    3061:   PetscDualSpaceLagrangeGetNodeType - Get a description of how nodes are laid out for Lagrange polynomials in this
    3062:   dual space

    3064:   Not Collective

    3066:   Input Parameter:
    3067: . sp - The `PetscDualSpace`

    3069:   Output Parameters:
    3070: + nodeType - The type of nodes
    3071: . boundary - Whether the node type is one that includes endpoints (if nodeType is `PETSCDTNODES_GAUSSJACOBI`, nodes that
    3072:              include the boundary are Gauss-Lobatto-Jacobi nodes)
    3073: - exponent - If nodeType is `PETSCDTNODES_GAUSJACOBI`, indicates the exponent used for both ends of the 1D Jacobi weight function
    3074:              '0' is Gauss-Legendre, '-0.5' is Gauss-Chebyshev of the first type, '0.5' is Gauss-Chebyshev of the second type

    3076:   Level: advanced

    3078: .seealso: `PETSCDUALSPACELAGRANGE`, `PetscDualSpace`, `PetscDTNodeType`, `PetscDualSpaceLagrangeSetNodeType()`
    3079: @*/
    3080: PetscErrorCode PetscDualSpaceLagrangeGetNodeType(PetscDualSpace sp, PetscDTNodeType *nodeType, PetscBool *boundary, PetscReal *exponent)
    3081: {
    3082:   PetscFunctionBegin;
    3084:   if (nodeType) PetscAssertPointer(nodeType, 2);
    3085:   if (boundary) PetscAssertPointer(boundary, 3);
    3086:   if (exponent) PetscAssertPointer(exponent, 4);
    3087:   PetscTryMethod(sp, "PetscDualSpaceLagrangeGetNodeType_C", (PetscDualSpace, PetscDTNodeType *, PetscBool *, PetscReal *), (sp, nodeType, boundary, exponent));
    3088:   PetscFunctionReturn(PETSC_SUCCESS);
    3089: }

    3091: /*@
    3092:   PetscDualSpaceLagrangeSetNodeType - Set a description of how nodes are laid out for Lagrange polynomials in this
    3093:   dual space

    3095:   Logically Collective

    3097:   Input Parameters:
    3098: + sp       - The `PetscDualSpace`
    3099: . nodeType - The type of nodes
    3100: . boundary - Whether the node type is one that includes endpoints (if nodeType is `PETSCDTNODES_GAUSSJACOBI`, nodes that
    3101:              include the boundary are Gauss-Lobatto-Jacobi nodes)
    3102: - exponent - If nodeType is `PETSCDTNODES_GAUSJACOBI`, indicates the exponent used for both ends of the 1D Jacobi weight function
    3103:              '0' is Gauss-Legendre, '-0.5' is Gauss-Chebyshev of the first type, '0.5' is Gauss-Chebyshev of the second type

    3105:   Level: advanced

    3107: .seealso: `PETSCDUALSPACELAGRANGE`, `PetscDualSpace`, `PetscDTNodeType`, `PetscDualSpaceLagrangeGetNodeType()`
    3108: @*/
    3109: PetscErrorCode PetscDualSpaceLagrangeSetNodeType(PetscDualSpace sp, PetscDTNodeType nodeType, PetscBool boundary, PetscReal exponent)
    3110: {
    3111:   PetscFunctionBegin;
    3113:   PetscTryMethod(sp, "PetscDualSpaceLagrangeSetNodeType_C", (PetscDualSpace, PetscDTNodeType, PetscBool, PetscReal), (sp, nodeType, boundary, exponent));
    3114:   PetscFunctionReturn(PETSC_SUCCESS);
    3115: }

    3117: /*@
    3118:   PetscDualSpaceLagrangeGetUseMoments - Get the flag for using moment functionals

    3120:   Not Collective

    3122:   Input Parameter:
    3123: . sp - The `PetscDualSpace`

    3125:   Output Parameter:
    3126: . useMoments - Moment flag

    3128:   Level: advanced

    3130: .seealso: `PETSCDUALSPACELAGRANGE`, `PetscDualSpace`, `PetscDualSpaceLagrangeSetUseMoments()`
    3131: @*/
    3132: PetscErrorCode PetscDualSpaceLagrangeGetUseMoments(PetscDualSpace sp, PetscBool *useMoments)
    3133: {
    3134:   PetscFunctionBegin;
    3136:   PetscAssertPointer(useMoments, 2);
    3137:   PetscUseMethod(sp, "PetscDualSpaceLagrangeGetUseMoments_C", (PetscDualSpace, PetscBool *), (sp, useMoments));
    3138:   PetscFunctionReturn(PETSC_SUCCESS);
    3139: }

    3141: /*@
    3142:   PetscDualSpaceLagrangeSetUseMoments - Set the flag for moment functionals

    3144:   Logically Collective

    3146:   Input Parameters:
    3147: + sp         - The `PetscDualSpace`
    3148: - useMoments - The flag for moment functionals

    3150:   Level: advanced

    3152: .seealso: `PETSCDUALSPACELAGRANGE`, `PetscDualSpace`, `PetscDualSpaceLagrangeGetUseMoments()`
    3153: @*/
    3154: PetscErrorCode PetscDualSpaceLagrangeSetUseMoments(PetscDualSpace sp, PetscBool useMoments)
    3155: {
    3156:   PetscFunctionBegin;
    3158:   PetscTryMethod(sp, "PetscDualSpaceLagrangeSetUseMoments_C", (PetscDualSpace, PetscBool), (sp, useMoments));
    3159:   PetscFunctionReturn(PETSC_SUCCESS);
    3160: }

    3162: /*@
    3163:   PetscDualSpaceLagrangeGetMomentOrder - Get the order for moment integration

    3165:   Not Collective

    3167:   Input Parameter:
    3168: . sp - The `PetscDualSpace`

    3170:   Output Parameter:
    3171: . order - Moment integration order

    3173:   Level: advanced

    3175: .seealso: `PETSCDUALSPACELAGRANGE`, `PetscDualSpace`, `PetscDualSpaceLagrangeSetMomentOrder()`
    3176: @*/
    3177: PetscErrorCode PetscDualSpaceLagrangeGetMomentOrder(PetscDualSpace sp, PetscInt *order)
    3178: {
    3179:   PetscFunctionBegin;
    3181:   PetscAssertPointer(order, 2);
    3182:   PetscUseMethod(sp, "PetscDualSpaceLagrangeGetMomentOrder_C", (PetscDualSpace, PetscInt *), (sp, order));
    3183:   PetscFunctionReturn(PETSC_SUCCESS);
    3184: }

    3186: /*@
    3187:   PetscDualSpaceLagrangeSetMomentOrder - Set the order for moment integration

    3189:   Logically Collective

    3191:   Input Parameters:
    3192: + sp    - The `PetscDualSpace`
    3193: - order - The order for moment integration

    3195:   Level: advanced

    3197: .seealso: `PETSCDUALSPACELAGRANGE`, `PetscDualSpace`, `PetscDualSpaceLagrangeGetMomentOrder()`
    3198: @*/
    3199: PetscErrorCode PetscDualSpaceLagrangeSetMomentOrder(PetscDualSpace sp, PetscInt order)
    3200: {
    3201:   PetscFunctionBegin;
    3203:   PetscTryMethod(sp, "PetscDualSpaceLagrangeSetMomentOrder_C", (PetscDualSpace, PetscInt), (sp, order));
    3204:   PetscFunctionReturn(PETSC_SUCCESS);
    3205: }

    3207: static PetscErrorCode PetscDualSpaceInitialize_Lagrange(PetscDualSpace sp)
    3208: {
    3209:   PetscFunctionBegin;
    3210:   sp->ops->destroy              = PetscDualSpaceDestroy_Lagrange;
    3211:   sp->ops->view                 = PetscDualSpaceView_Lagrange;
    3212:   sp->ops->setfromoptions       = PetscDualSpaceSetFromOptions_Lagrange;
    3213:   sp->ops->duplicate            = PetscDualSpaceDuplicate_Lagrange;
    3214:   sp->ops->setup                = PetscDualSpaceSetUp_Lagrange;
    3215:   sp->ops->createheightsubspace = NULL;
    3216:   sp->ops->createpointsubspace  = NULL;
    3217:   sp->ops->getsymmetries        = PetscDualSpaceGetSymmetries_Lagrange;
    3218:   sp->ops->apply                = PetscDualSpaceApplyDefault;
    3219:   sp->ops->applyall             = PetscDualSpaceApplyAllDefault;
    3220:   sp->ops->applyint             = PetscDualSpaceApplyInteriorDefault;
    3221:   sp->ops->createalldata        = PetscDualSpaceCreateAllDataDefault;
    3222:   sp->ops->createintdata        = PetscDualSpaceCreateInteriorDataDefault;
    3223:   PetscFunctionReturn(PETSC_SUCCESS);
    3224: }

    3226: /*MC
    3227:   PETSCDUALSPACELAGRANGE = "lagrange" - A `PetscDualSpaceType` that encapsulates a dual space of pointwise evaluation functionals

    3229:   Level: intermediate

    3231:   Developer Note:
    3232:   This `PetscDualSpace` seems to manage directly trimmed and untrimmed polynomials as well as tensor and non-tensor polynomials while for `PetscSpace` there seems to
    3233:   be different `PetscSpaceType` for them.

    3235: .seealso: `PetscDualSpace`, `PetscDualSpaceType`, `PetscDualSpaceCreate()`, `PetscDualSpaceSetType()`,
    3236:           `PetscDualSpaceLagrangeSetMomentOrder()`, `PetscDualSpaceLagrangeGetMomentOrder()`, `PetscDualSpaceLagrangeSetUseMoments()`, `PetscDualSpaceLagrangeGetUseMoments()`,
    3237:           `PetscDualSpaceLagrangeSetNodeType, PetscDualSpaceLagrangeGetNodeType, PetscDualSpaceLagrangeGetContinuity, PetscDualSpaceLagrangeSetContinuity,
    3238:           `PetscDualSpaceLagrangeGetTensor()`, `PetscDualSpaceLagrangeSetTensor()`, `PetscDualSpaceLagrangeGetTrimmed()`, `PetscDualSpaceLagrangeSetTrimmed()`
    3239: M*/
    3240: PETSC_EXTERN PetscErrorCode PetscDualSpaceCreate_Lagrange(PetscDualSpace sp)
    3241: {
    3242:   PetscDualSpace_Lag *lag;

    3244:   PetscFunctionBegin;
    3246:   PetscCall(PetscNew(&lag));
    3247:   sp->data = lag;

    3249:   lag->tensorCell  = PETSC_FALSE;
    3250:   lag->tensorSpace = PETSC_FALSE;
    3251:   lag->continuous  = PETSC_TRUE;
    3252:   lag->numCopies   = PETSC_DEFAULT;
    3253:   lag->numNodeSkip = PETSC_DEFAULT;
    3254:   lag->nodeType    = PETSCDTNODES_DEFAULT;
    3255:   lag->useMoments  = PETSC_FALSE;
    3256:   lag->momentOrder = 0;

    3258:   PetscCall(PetscDualSpaceInitialize_Lagrange(sp));
    3259:   PetscCall(PetscObjectComposeFunction((PetscObject)sp, "PetscDualSpaceLagrangeGetContinuity_C", PetscDualSpaceLagrangeGetContinuity_Lagrange));
    3260:   PetscCall(PetscObjectComposeFunction((PetscObject)sp, "PetscDualSpaceLagrangeSetContinuity_C", PetscDualSpaceLagrangeSetContinuity_Lagrange));
    3261:   PetscCall(PetscObjectComposeFunction((PetscObject)sp, "PetscDualSpaceLagrangeGetTensor_C", PetscDualSpaceLagrangeGetTensor_Lagrange));
    3262:   PetscCall(PetscObjectComposeFunction((PetscObject)sp, "PetscDualSpaceLagrangeSetTensor_C", PetscDualSpaceLagrangeSetTensor_Lagrange));
    3263:   PetscCall(PetscObjectComposeFunction((PetscObject)sp, "PetscDualSpaceLagrangeGetTrimmed_C", PetscDualSpaceLagrangeGetTrimmed_Lagrange));
    3264:   PetscCall(PetscObjectComposeFunction((PetscObject)sp, "PetscDualSpaceLagrangeSetTrimmed_C", PetscDualSpaceLagrangeSetTrimmed_Lagrange));
    3265:   PetscCall(PetscObjectComposeFunction((PetscObject)sp, "PetscDualSpaceLagrangeGetNodeType_C", PetscDualSpaceLagrangeGetNodeType_Lagrange));
    3266:   PetscCall(PetscObjectComposeFunction((PetscObject)sp, "PetscDualSpaceLagrangeSetNodeType_C", PetscDualSpaceLagrangeSetNodeType_Lagrange));
    3267:   PetscCall(PetscObjectComposeFunction((PetscObject)sp, "PetscDualSpaceLagrangeGetUseMoments_C", PetscDualSpaceLagrangeGetUseMoments_Lagrange));
    3268:   PetscCall(PetscObjectComposeFunction((PetscObject)sp, "PetscDualSpaceLagrangeSetUseMoments_C", PetscDualSpaceLagrangeSetUseMoments_Lagrange));
    3269:   PetscCall(PetscObjectComposeFunction((PetscObject)sp, "PetscDualSpaceLagrangeGetMomentOrder_C", PetscDualSpaceLagrangeGetMomentOrder_Lagrange));
    3270:   PetscCall(PetscObjectComposeFunction((PetscObject)sp, "PetscDualSpaceLagrangeSetMomentOrder_C", PetscDualSpaceLagrangeSetMomentOrder_Lagrange));
    3271:   PetscFunctionReturn(PETSC_SUCCESS);
    3272: }