Actual source code: dspacelagrange.c
petsc-3.13.6 2020-09-29
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      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: {
      9:   PetscInt         refct;
     10:   PetscDTNodeType  nodeFamily;
     11:   PetscReal        gaussJacobiExp;
     12:   PetscInt         nComputed;
     13:   PetscReal      **nodesets;
     14:   PetscBool        endpoints;
     15: };

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

     25:   PetscNew(&f);
     26:   switch (family) {
     27:   case PETSCDTNODES_GAUSSJACOBI:
     28:   case PETSCDTNODES_EQUISPACED:
     29:     f->nodeFamily = family;
     30:     break;
     31:   default: SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Unknown 1D node family");
     32:   }
     33:   f->endpoints = endpoints;
     34:   f->gaussJacobiExp = 0.;
     35:   if (family == PETSCDTNODES_GAUSSJACOBI) {
     36:     if (gaussJacobiExp <= -1.) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Gauss-Jacobi exponent must be > -1.\n");
     37:     f->gaussJacobiExp = gaussJacobiExp;
     38:   }
     39:   f->refct = 1;
     40:   *nf = f;
     41:   return(0);
     42: }

     44: static PetscErrorCode Petsc1DNodeFamilyReference(Petsc1DNodeFamily nf)
     45: {
     47:   if (nf) nf->refct++;
     48:   return(0);
     49: }

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

     56:   if (!(*nf)) return(0);
     57:   if (--(*nf)->refct > 0) {
     58:     *nf = NULL;
     59:     return(0);
     60:   }
     61:   nc = (*nf)->nComputed;
     62:   for (i = 0; i < nc; i++) {
     63:     PetscFree((*nf)->nodesets[i]);
     64:   }
     65:   PetscFree((*nf)->nodesets);
     66:   PetscFree(*nf);
     67:   *nf = NULL;
     68:   return(0);
     69: }

     71: static PetscErrorCode Petsc1DNodeFamilyGetNodeSets(Petsc1DNodeFamily f, PetscInt degree, PetscReal ***nodesets)
     72: {
     73:   PetscInt       nc;

     77:   nc = f->nComputed;
     78:   if (degree >= nc) {
     79:     PetscInt    i, j;
     80:     PetscReal **new_nodesets;
     81:     PetscReal  *w;

     83:     PetscMalloc1(degree + 1, &new_nodesets);
     84:     PetscArraycpy(new_nodesets, f->nodesets, nc);
     85:     PetscFree(f->nodesets);
     86:     f->nodesets = new_nodesets;
     87:     PetscMalloc1(degree + 1, &w);
     88:     for (i = nc; i < degree + 1; i++) {
     89:       PetscMalloc1(i + 1, &(f->nodesets[i]));
     90:       if (!i) {
     91:         f->nodesets[i][0] = 0.5;
     92:       } else {
     93:         switch (f->nodeFamily) {
     94:         case PETSCDTNODES_EQUISPACED:
     95:           if (f->endpoints) {
     96:             for (j = 0; j <= i; j++) f->nodesets[i][j] = (PetscReal) j / (PetscReal) i;
     97:           } else {
     98:             /* these nodes are at the centroids of the small simplices created by the equispaced nodes that include
     99:              * the endpoints */
    100:             for (j = 0; j <= i; j++) f->nodesets[i][j] = ((PetscReal) j + 0.5) / ((PetscReal) i + 1.);
    101:           }
    102:           break;
    103:         case PETSCDTNODES_GAUSSJACOBI:
    104:           if (f->endpoints) {
    105:             PetscDTGaussLobattoJacobiQuadrature(i + 1, 0., 1., f->gaussJacobiExp, f->gaussJacobiExp, f->nodesets[i], w);
    106:           } else {
    107:             PetscDTGaussJacobiQuadrature(i + 1, 0., 1., f->gaussJacobiExp, f->gaussJacobiExp, f->nodesets[i], w);
    108:           }
    109:           break;
    110:         default: SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Unknown 1D node family");
    111:         }
    112:       }
    113:     }
    114:     PetscFree(w);
    115:     f->nComputed = degree + 1;
    116:   }
    117:   *nodesets = f->nodesets;
    118:   return(0);
    119: }

    121: /* http://arxiv.org/abs/2002.09421 for details */
    122: static PetscErrorCode PetscNodeRecursive_Internal(PetscInt dim, PetscInt degree, PetscReal **nodesets, PetscInt tup[], PetscReal node[])
    123: {
    124:   PetscReal w;
    125:   PetscInt i, j;

    129:   w = 0.;
    130:   if (dim == 1) {
    131:     node[0] = nodesets[degree][tup[0]];
    132:     node[1] = nodesets[degree][tup[1]];
    133:   } else {
    134:     for (i = 0; i < dim + 1; i++) node[i] = 0.;
    135:     for (i = 0; i < dim + 1; i++) {
    136:       PetscReal wi = nodesets[degree][degree-tup[i]];

    138:       for (j = 0; j < dim+1; j++) tup[dim+1+j] = tup[j+(j>=i)];
    139:       PetscNodeRecursive_Internal(dim-1,degree-tup[i],nodesets,&tup[dim+1],&node[dim+1]);
    140:       for (j = 0; j < dim+1; j++) node[j+(j>=i)] += wi * node[dim+1+j];
    141:       w += wi;
    142:     }
    143:     for (i = 0; i < dim+1; i++) node[i] /= w;
    144:   }
    145:   return(0);
    146: }

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

    161:   if (dim < 0) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Must have non-negative dimension\n");
    162:   if (degree < 0) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Must have non-negative degree\n");
    163:   if (!dim) return(0);
    164:   PetscCalloc1(dim+2, &tup);
    165:   k = 0;
    166:   PetscDTBinomialInt(degree + dim, dim, &npoints);
    167:   Petsc1DNodeFamilyGetNodeSets(f, degree, &nodesets);
    168:   worksize = ((dim + 2) * (dim + 3)) / 2;
    169:   PetscMalloc2(worksize, &nodework, worksize, &tupwork);
    170:   /* loop over the tuples of length dim with sum at most degree */
    171:   for (k = 0; k < npoints; k++) {
    172:     PetscInt i;

    174:     /* turn thm into tuples of length dim + 1 with sum equal to degree (barycentric indice) */
    175:     tup[0] = degree;
    176:     for (i = 0; i < dim; i++) {
    177:       tup[0] -= tup[i+1];
    178:     }
    179:     switch(f->nodeFamily) {
    180:     case PETSCDTNODES_EQUISPACED:
    181:       /* compute equispaces nodes on the unit reference triangle */
    182:       if (f->endpoints) {
    183:         for (i = 0; i < dim; i++) {
    184:           points[dim*k + i] = (PetscReal) tup[i+1] / (PetscReal) degree;
    185:         }
    186:       } else {
    187:         for (i = 0; i < dim; i++) {
    188:           /* these nodes are at the centroids of the small simplices created by the equispaced nodes that include
    189:            * the endpoints */
    190:           points[dim*k + i] = ((PetscReal) tup[i+1] + 1./(dim+1.)) / (PetscReal) (degree + 1.);
    191:         }
    192:       }
    193:       break;
    194:     default:
    195:       /* compute equispaces nodes on the barycentric reference triangle (the trace on the first dim dimensions are the
    196:        * unit reference triangle nodes */
    197:       for (i = 0; i < dim + 1; i++) tupwork[i] = tup[i];
    198:       PetscNodeRecursive_Internal(dim, degree, nodesets, tupwork, nodework);
    199:       for (i = 0; i < dim; i++) points[dim*k + i] = nodework[i + 1];
    200:       break;
    201:     }
    202:     PetscDualSpaceLatticePointLexicographic_Internal(dim, degree, &tup[1]);
    203:   }
    204:   /* map from unit simplex to biunit simplex */
    205:   for (k = 0; k < npoints * dim; k++) points[k] = points[k] * 2. - 1.;
    206:   PetscFree2(nodework, tupwork);
    207:   PetscFree(tup);
    208:   return(0);
    209: }

    211: /* 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
    212:  * on that mesh point, we have to be careful about getting/adding everything in the right place.
    213:  *
    214:  * With nodal dofs like PETSCDUALSPACELAGRANGE makes, the general approach to calculate the value of dofs associate
    215:  * with a node A is
    216:  * - transform the node locations x(A) by the map that takes the mesh point to its reorientation, x' = phi(x(A))
    217:  * - figure out which node was originally at the location of the transformed point, A' = idx(x')
    218:  * - if the dofs are not scalars, figure out how to represent the transformed dofs in terms of the basis
    219:  *   of dofs at A' (using pushforward/pullback rules)
    220:  *
    221:  * The one sticky point with this approach is the "A' = idx(x')" step: trying to go from real valued coordinates
    222:  * back to indices.  I don't want to rely on floating point tolerances.  Additionally, PETSCDUALSPACELAGRANGE may
    223:  * eventually support quasi-Lagrangian dofs, which could involve quadrature at multiple points, so the location "x(A)"
    224:  * would be ambiguous.
    225:  *
    226:  * So each dof gets an integer value coordinate (nodeIdx in the structure below).  The choice of integer coordinates
    227:  * is somewhat arbitrary, as long as all of the relevant symmetries of the mesh point correspond to *permutations* of
    228:  * the integer coordinates, which do not depend on numerical precision.
    229:  *
    230:  * So
    231:  *
    232:  * - DMPlexGetTransitiveClosure_Internal() tells me how an orientation turns into a permutation of the vertices of a
    233:  *   mesh point
    234:  * - The permutation of the vertices, and the nodeIdx values assigned to them, tells what permutation in index space
    235:  *   is associated with the orientation
    236:  * - I uses that permutation to get xi' = phi(xi(A)), the integer coordinate of the transformed dof
    237:  * - I can without numerical issues compute A' = idx(xi')
    238:  *
    239:  * Here are some examples of how the process works
    240:  *
    241:  * - With a triangle:
    242:  *
    243:  *   The triangle has the following integer coordinates for vertices, taken from the barycentric triangle
    244:  *
    245:  *     closure order 2
    246:  *     nodeIdx (0,0,1)
    247:  *      \
    248:  *       +
    249:  *       |\
    250:  *       | \
    251:  *       |  \
    252:  *       |   \    closure order 1
    253:  *       |    \ / nodeIdx (0,1,0)
    254:  *       +-----+
    255:  *        \
    256:  *      closure order 0
    257:  *      nodeIdx (1,0,0)
    258:  *
    259:  *   If I do DMPlexGetTransitiveClosure_Internal() with orientation 1, the vertices would appear
    260:  *   in the order (1, 2, 0)
    261:  *
    262:  *   If I list the nodeIdx of each vertex in closure order for orientation 0 (0, 1, 2) and orientation 1 (1, 2, 0), I
    263:  *   see
    264:  *
    265:  *   orientation 0  | orientation 1
    266:  *
    267:  *   [0] (1,0,0)      [1] (0,1,0)
    268:  *   [1] (0,1,0)      [2] (0,0,1)
    269:  *   [2] (0,0,1)      [0] (1,0,0)
    270:  *          A                B
    271:  *
    272:  *   In other words, B is the result of a row permutation of A.  But, there is also
    273:  *   a column permutation that accomplishes the same result, (2,0,1).
    274:  *
    275:  *   So if a dof has nodeIdx coordinate (a,b,c), after the transformation its nodeIdx coordinate
    276:  *   is (c,a,b), and the transformed degree of freedom will be a linear combination of dofs
    277:  *   that originally had coordinate (c,a,b).
    278:  *
    279:  * - With a quadrilateral:
    280:  *
    281:  *   The quadrilateral has the following integer coordinates for vertices, taken from concatenating barycentric
    282:  *   coordinates for two segments:
    283:  *
    284:  *     closure order 3      closure order 2
    285:  *     nodeIdx (1,0,0,1)    nodeIdx (0,1,0,1)
    286:  *                   \      /
    287:  *                    +----+
    288:  *                    |    |
    289:  *                    |    |
    290:  *                    +----+
    291:  *                   /      \
    292:  *     closure order 0      closure order 1
    293:  *     nodeIdx (1,0,1,0)    nodeIdx (0,1,1,0)
    294:  *
    295:  *   If I do DMPlexGetTransitiveClosure_Internal() with orientation 1, the vertices would appear
    296:  *   in the order (1, 2, 3, 0)
    297:  *
    298:  *   If I list the nodeIdx of each vertex in closure order for orientation 0 (0, 1, 2, 3) and
    299:  *   orientation 1 (1, 2, 3, 0), I see
    300:  *
    301:  *   orientation 0  | orientation 1
    302:  *
    303:  *   [0] (1,0,1,0)    [1] (0,1,1,0)
    304:  *   [1] (0,1,1,0)    [2] (0,1,0,1)
    305:  *   [2] (0,1,0,1)    [3] (1,0,0,1)
    306:  *   [3] (1,0,0,1)    [0] (1,0,1,0)
    307:  *          A                B
    308:  *
    309:  *   The column permutation that accomplishes the same result is (3,2,0,1).
    310:  *
    311:  *   So if a dof has nodeIdx coordinate (a,b,c,d), after the transformation its nodeIdx coordinate
    312:  *   is (d,c,a,b), and the transformed degree of freedom will be a linear combination of dofs
    313:  *   that originally had coordinate (d,c,a,b).
    314:  *
    315:  * Previously PETSCDUALSPACELAGRANGE had hardcoded symmetries for the triangle and quadrilateral,
    316:  * but this approach will work for any polytope, such as the wedge (triangular prism).
    317:  */
    318: struct _n_PetscLagNodeIndices
    319: {
    320:   PetscInt   refct;
    321:   PetscInt   nodeIdxDim;
    322:   PetscInt   nodeVecDim;
    323:   PetscInt   nNodes;
    324:   PetscInt  *nodeIdx;      /* for each node an index of size nodeIdxDim */
    325:   PetscReal *nodeVec;      /* for each node a vector of size nodeVecDim */
    326:   PetscInt  *perm;         /* if these are vertices, perm takes DMPlex point index to closure order;
    327:                               if these are nodes, perm lists nodes in index revlex order */
    328: };

    330: /* this is just here so I can access the values in tests/ex1.c outside the library */
    331: PetscErrorCode PetscLagNodeIndicesGetData_Internal(PetscLagNodeIndices ni, PetscInt *nodeIdxDim, PetscInt *nodeVecDim, PetscInt *nNodes, const PetscInt *nodeIdx[], const PetscReal *nodeVec[])
    332: {
    334:   *nodeIdxDim = ni->nodeIdxDim;
    335:   *nodeVecDim = ni->nodeVecDim;
    336:   *nNodes = ni->nNodes;
    337:   *nodeIdx = ni->nodeIdx;
    338:   *nodeVec = ni->nodeVec;
    339:   return(0);
    340: }

    342: static PetscErrorCode PetscLagNodeIndicesReference(PetscLagNodeIndices ni)
    343: {
    345:   if (ni) ni->refct++;
    346:   return(0);
    347: }

    349: static PetscErrorCode PetscLagNodeIndicesDestroy(PetscLagNodeIndices *ni) {

    353:   if (!(*ni)) return(0);
    354:   if (--(*ni)->refct > 0) {
    355:     *ni = NULL;
    356:     return(0);
    357:   }
    358:   PetscFree((*ni)->nodeIdx);
    359:   PetscFree((*ni)->nodeVec);
    360:   PetscFree((*ni)->perm);
    361:   PetscFree(*ni);
    362:   *ni = NULL;
    363:   return(0);
    364: }

    366: /* The vertices are given nodeIdx coordinates (e.g. the corners of the barycentric triangle).  Those coordinates are
    367:  * in some other order, and to understand the effect of different symmetries, we need them to be in closure order.
    368:  *
    369:  * If sortIdx is PETSC_FALSE, the coordinates are already in revlex order, otherwise we must sort them
    370:  * to that order before we do the real work of this function, which is
    371:  *
    372:  * - mark the vertices in closure order
    373:  * - sort them in revlex order
    374:  * - use the resulting permutation to list the vertex coordinates in closure order
    375:  */
    376: static PetscErrorCode PetscLagNodeIndicesComputeVertexOrder(DM dm, PetscLagNodeIndices ni, PetscBool sortIdx)
    377: {
    378:   PetscInt        v, w, vStart, vEnd, c, d;
    379:   PetscInt        nVerts;
    380:   PetscInt        closureSize = 0;
    381:   PetscInt       *closure = NULL;
    382:   PetscInt       *closureOrder;
    383:   PetscInt       *invClosureOrder;
    384:   PetscInt       *revlexOrder;
    385:   PetscInt       *newNodeIdx;
    386:   PetscInt        dim;
    387:   Vec             coordVec;
    388:   const PetscScalar *coords;
    389:   PetscErrorCode  ierr;

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

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

    411:         for (d = nodeIdxDim - 1; d >= 0; d--) if ((diff = (iv[d] - iw[d]))) break;
    412:         if (diff > 0) {
    413:           PetscInt swap = idxOrder[v];

    415:           idxOrder[v] = idxOrder[w];
    416:           idxOrder[w] = swap;
    417:         }
    418:       }
    419:     }
    420:     for (v = 0; v < nVerts; v++) {
    421:       for (d = 0; d < nodeIdxDim; d++) {
    422:         newNodeIdx[v * ni->nodeIdxDim + d] = ni->nodeIdx[idxOrder[v] * nodeIdxDim + d];
    423:       }
    424:     }
    425:     PetscFree(ni->nodeIdx);
    426:     ni->nodeIdx = newNodeIdx;
    427:     newNodeIdx = NULL;
    428:     PetscFree(idxOrder);
    429:   }
    430:   DMPlexGetTransitiveClosure(dm, 0, PETSC_TRUE, &closureSize, &closure);
    431:   c = closureSize - nVerts;
    432:   for (v = 0; v < nVerts; v++) closureOrder[v] = closure[2 * (c + v)] - vStart;
    433:   for (v = 0; v < nVerts; v++) invClosureOrder[closureOrder[v]] = v;
    434:   DMPlexRestoreTransitiveClosure(dm, 0, PETSC_TRUE, &closureSize, &closure);
    435:   DMGetCoordinatesLocal(dm, &coordVec);
    436:   VecGetArrayRead(coordVec, &coords);
    437:   /* bubble sort closure vertices by coordinates in revlex order */
    438:   for (v = 0; v < nVerts; v++) revlexOrder[v] = v;
    439:   for (v = 0; v < nVerts; v++) {
    440:     for (w = v + 1; w < nVerts; w++) {
    441:       const PetscScalar *cv = &coords[closureOrder[revlexOrder[v]] * dim];
    442:       const PetscScalar *cw = &coords[closureOrder[revlexOrder[w]] * dim];
    443:       PetscReal diff = 0;

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

    449:         revlexOrder[v] = revlexOrder[w];
    450:         revlexOrder[w] = swap;
    451:       }
    452:     }
    453:   }
    454:   VecRestoreArrayRead(coordVec, &coords);
    455:   PetscMalloc1(ni->nodeIdxDim * nVerts, &newNodeIdx);
    456:   /* reorder nodeIdx to be in closure order */
    457:   for (v = 0; v < nVerts; v++) {
    458:     for (d = 0; d < ni->nodeIdxDim; d++) {
    459:       newNodeIdx[revlexOrder[v] * ni->nodeIdxDim + d] = ni->nodeIdx[v * ni->nodeIdxDim + d];
    460:     }
    461:   }
    462:   PetscFree(ni->nodeIdx);
    463:   ni->nodeIdx = newNodeIdx;
    464:   ni->perm = invClosureOrder;
    465:   PetscFree(revlexOrder);
    466:   PetscFree(closureOrder);
    467:   return(0);
    468: }

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

    480:   PetscNew(&ni);
    481:   DMGetDimension(dm, &dim);
    482:   ni->nodeIdxDim = dim + 1;
    483:   ni->nodeVecDim = 0;
    484:   ni->nNodes = dim + 1;
    485:   ni->refct = 1;
    486:   PetscCalloc1((dim + 1)*(dim + 1), &(ni->nodeIdx));
    487:   for (d = 0; d < dim + 1; d++) ni->nodeIdx[d*(dim + 2)] = 1;
    488:   PetscLagNodeIndicesComputeVertexOrder(dm, ni, PETSC_FALSE);
    489:   *nodeIndices = ni;
    490:   return(0);
    491: }

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

    508:   PetscNew(&ni);
    509:   DMGetDimension(dm, &dim);
    510:   ni->nodeIdxDim = nodeIdxDim = subNodeIdxDim + 2;
    511:   ni->nodeVecDim = 0;
    512:   ni->nNodes = nVerts = 2 * nSubVerts;
    513:   ni->refct = 1;
    514:   PetscCalloc1(nodeIdxDim * nVerts, &(ni->nodeIdx));
    515:   for (f = 0, d = 0; d < 2; d++) {
    516:     for (e = 0; e < nSubVerts; e++, f++) {
    517:       for (g = 0; g < subNodeIdxDim; g++) {
    518:         ni->nodeIdx[f * nodeIdxDim + g] = facetni->nodeIdx[e * subNodeIdxDim + g];
    519:       }
    520:       ni->nodeIdx[f * nodeIdxDim + subNodeIdxDim] = (1 - d);
    521:       ni->nodeIdx[f * nodeIdxDim + subNodeIdxDim + 1] = d;
    522:     }
    523:   }
    524:   PetscLagNodeIndicesComputeVertexOrder(dm, ni, PETSC_TRUE);
    525:   *nodeIndices = ni;
    526:   return(0);
    527: }

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

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

    615:     DMPlexGetTransitiveClosure_Internal(dm, p, 0, PETSC_TRUE, &closureSize2, &closure2);
    616:     PetscMalloc1(dim * nSubVert, &newCoords);
    617:     VecGetArrayRead(coordVec, &oldCoords);
    618:     for (v = 0; v < nSubVert; v++) {
    619:       PetscInt d;
    620:       for (d = 0; d < dim; d++) {
    621:         newCoords[(closure2[2 * (c + v)] - vStart) * dim + d] = oldCoords[closureVerts[v] * dim + d];
    622:       }
    623:     }
    624:     VecRestoreArrayRead(coordVec, &oldCoords);
    625:     DMPlexRestoreTransitiveClosure(dm, p, PETSC_TRUE, &closureSize2, &closure2);
    626:     VecPlaceArray(coordVec, newCoords);
    627:   }
    628:   DMPlexComputeCellGeometryAffineFEM(dm, p, NULL, J, NULL, &detJ);
    629:   if (ornt != 0) {
    630:     VecResetArray(coordVec);
    631:     PetscFree(newCoords);
    632:   }
    633:   DMRestoreWorkArray(dm, nSubVert, MPIU_INT, &closureVerts);
    634:   /* compactify */
    635:   for (i = 0; i < dim; i++) for (j = 0; j < pdim; j++) J[i * pdim + j] = J[i * dim + j];
    636:   /* We have the Jacobian mapping the point's reference cell to this reference cell:
    637:    * pulling back a function to the point and applying the dof is what we want,
    638:    * so we get the pullback matrix and multiply the dof by that matrix on the right */
    639:   PetscDTBinomialInt(dim, PetscAbsInt(formDegree), &Nk);
    640:   PetscDTBinomialInt(pdim, PetscAbsInt(formDegree), &pNk);
    641:   DMGetWorkArray(dm, pNk * Nk, MPIU_REAL, &Jstar);
    642:   PetscDTAltVPullbackMatrix(pdim, dim, J, formDegree, Jstar);
    643:   for (n = 0; n < nNodes; n++) {
    644:     for (i = 0; i < Nk; i++) {
    645:       PetscReal val = 0.;
    646:       for (j = 0; j < pNk; j++) val += nodeVec[n * pNk + j] * Jstar[j * pNk + i];
    647:       pfNodeVec[n * Nk + i] = val;
    648:     }
    649:   }
    650:   DMRestoreWorkArray(dm, pNk * Nk, MPIU_REAL, &Jstar);
    651:   DMRestoreWorkArray(dm, dim * dim, MPIU_REAL, &J);
    652:   return(0);
    653: }

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

    674:   PetscDTBinomialInt(dim, PetscAbsInt(formDegree), &Nk);
    675:   PetscDTBinomialInt(dimT, PetscAbsInt(kT), &NkT);
    676:   PetscDTBinomialInt(dimF, PetscAbsInt(kF), &NkF);
    677:   PetscDTBinomialInt(dim, PetscAbsInt(kT), &MkT);
    678:   PetscDTBinomialInt(dim, PetscAbsInt(kF), &MkF);
    679:   PetscNew(&ni);
    680:   ni->nodeIdxDim = nodeIdxDim = tracei->nodeIdxDim + fiberi->nodeIdxDim;
    681:   ni->nodeVecDim = Nk;
    682:   ni->nNodes = nNodes = tracei->nNodes * fiberi->nNodes;
    683:   ni->refct = 1;
    684:   PetscMalloc1(nNodes * nodeIdxDim, &(ni->nodeIdx));
    685:   /* first concatenate the indices */
    686:   for (l = 0, j = 0; j < fiberi->nNodes; j++) {
    687:     for (i = 0; i < tracei->nNodes; i++, l++) {
    688:       PetscInt m, n = 0;

    690:       for (m = 0; m < tracei->nodeIdxDim; m++) ni->nodeIdx[l * nodeIdxDim + n++] = tracei->nodeIdx[i * tracei->nodeIdxDim + m];
    691:       for (m = 0; m < fiberi->nodeIdxDim; m++) ni->nodeIdx[l * nodeIdxDim + n++] = fiberi->nodeIdx[j * fiberi->nodeIdxDim + m];
    692:     }
    693:   }

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

    709:     /* push forward fiber k-form */
    710:     for (d = 0; d < MkF; d++) {
    711:       PetscReal val = 0.;
    712:       for (e = 0; e < NkF; e++) val += projFstar[d * NkF + e] * fiberi->nodeVec[j * NkF + e];
    713:       workF[d] = val;
    714:     }
    715:     /* Hodge star to proper form if necessary */
    716:     if (kF < 0) {
    717:       for (d = 0; d < MkF; d++) workF2[d] = workF[d];
    718:       PetscDTAltVStar(dim, PetscAbsInt(kF), 1, workF2, workF);
    719:     }
    720:     /* Compute the matrix that wedges this form with one of the trace k-form */
    721:     PetscDTAltVWedgeMatrix(dim, PetscAbsInt(kF), PetscAbsInt(kT), workF, wedgeMat);
    722:     for (i = 0; i < tracei->nNodes; i++, l++) {
    723:       /* push forward trace k-form */
    724:       for (d = 0; d < MkT; d++) {
    725:         PetscReal val = 0.;
    726:         for (e = 0; e < NkT; e++) val += projTstar[d * NkT + e] * tracei->nodeVec[i * NkT + e];
    727:         workT[d] = val;
    728:       }
    729:       /* Hodge star to proper form if necessary */
    730:       if (kT < 0) {
    731:         for (d = 0; d < MkT; d++) workT2[d] = workT[d];
    732:         PetscDTAltVStar(dim, PetscAbsInt(kT), 1, workT2, workT);
    733:       }
    734:       /* compute the wedge product of the push-forward trace form and firer forms */
    735:       for (d = 0; d < Nk; d++) {
    736:         PetscReal val = 0.;
    737:         for (e = 0; e < MkT; e++) val += wedgeMat[d * MkT + e] * workT[e];
    738:         work[d] = val;
    739:       }
    740:       /* inverse Hodge star from proper form if necessary */
    741:       if (formDegree < 0) {
    742:         for (d = 0; d < Nk; d++) work2[d] = work[d];
    743:         PetscDTAltVStar(dim, PetscAbsInt(formDegree), -1, work2, work);
    744:       }
    745:       /* insert into the array (adjusting for sign) */
    746:       for (d = 0; d < Nk; d++) ni->nodeVec[l * Nk + d] = sign * work[d];
    747:     }
    748:   }
    749:   PetscFree(wedgeMat);
    750:   PetscFree6(workT, workT2, workF, workF2, work, work2);
    751:   PetscFree2(projTstar, projFstar);
    752:   PetscFree2(projT, projF);
    753:   *nodeIndices = ni;
    754:   return(0);
    755: }

    757: /* simple union of two sets of nodes */
    758: static PetscErrorCode PetscLagNodeIndicesMerge(PetscLagNodeIndices niA, PetscLagNodeIndices niB, PetscLagNodeIndices *nodeIndices)
    759: {
    760:   PetscLagNodeIndices ni;
    761:   PetscInt            nodeIdxDim, nodeVecDim, nNodes;
    762:   PetscErrorCode      ierr;

    765:   PetscNew(&ni);
    766:   ni->nodeIdxDim = nodeIdxDim = niA->nodeIdxDim;
    767:   if (niB->nodeIdxDim != nodeIdxDim) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_INCOMP, "Cannot merge PetscLagNodeIndices with different nodeIdxDim");
    768:   ni->nodeVecDim = nodeVecDim = niA->nodeVecDim;
    769:   if (niB->nodeVecDim != nodeVecDim) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_INCOMP, "Cannot merge PetscLagNodeIndices with different nodeVecDim");
    770:   ni->nNodes = nNodes = niA->nNodes + niB->nNodes;
    771:   ni->refct = 1;
    772:   PetscMalloc1(nNodes * nodeIdxDim, &(ni->nodeIdx));
    773:   PetscMalloc1(nNodes * nodeVecDim, &(ni->nodeVec));
    774:   PetscArraycpy(ni->nodeIdx, niA->nodeIdx, niA->nNodes * nodeIdxDim);
    775:   PetscArraycpy(ni->nodeVec, niA->nodeVec, niA->nNodes * nodeVecDim);
    776:   PetscArraycpy(&(ni->nodeIdx[niA->nNodes * nodeIdxDim]), niB->nodeIdx, niB->nNodes * nodeIdxDim);
    777:   PetscArraycpy(&(ni->nodeVec[niA->nNodes * nodeVecDim]), niB->nodeVec, niB->nNodes * nodeVecDim);
    778:   *nodeIndices = ni;
    779:   return(0);
    780: }

    782: #define PETSCTUPINTCOMPREVLEX(N)                                   \
    783: static int PetscTupIntCompRevlex_##N(const void *a, const void *b) \
    784: {                                                                  \
    785:   const PetscInt *A = (const PetscInt *) a;                        \
    786:   const PetscInt *B = (const PetscInt *) b;                        \
    787:   int i;                                                           \
    788:   PetscInt diff = 0;                                               \
    789:   for (i = 0; i < N; i++) {                                        \
    790:     diff = A[N - i] - B[N - i];                                    \
    791:     if (diff) break;                                               \
    792:   }                                                                \
    793:   return (diff <= 0) ? (diff < 0) ? -1 : 0 : 1;                    \
    794: }

    796: PETSCTUPINTCOMPREVLEX(3)
    797: PETSCTUPINTCOMPREVLEX(4)
    798: PETSCTUPINTCOMPREVLEX(5)
    799: PETSCTUPINTCOMPREVLEX(6)
    800: PETSCTUPINTCOMPREVLEX(7)

    802: static int PetscTupIntCompRevlex_N(const void *a, const void *b)
    803: {
    804:   const PetscInt *A = (const PetscInt *) a;
    805:   const PetscInt *B = (const PetscInt *) b;
    806:   int i;
    807:   int N = A[0];
    808:   PetscInt diff = 0;
    809:   for (i = 0; i < N; i++) {
    810:     diff = A[N - i] - B[N - i];
    811:     if (diff) break;
    812:   }
    813:   return (diff <= 0) ? (diff < 0) ? -1 : 0 : 1;
    814: }

    816: /* The nodes are not necessarily in revlex order wrt nodeIdx: get the permutation
    817:  * that puts them in that order */
    818: static PetscErrorCode PetscLagNodeIndicesGetPermutation(PetscLagNodeIndices ni, PetscInt *perm[])
    819: {

    823:   if (!(ni->perm)) {
    824:     PetscInt *sorter;
    825:     PetscInt m = ni->nNodes;
    826:     PetscInt nodeIdxDim = ni->nodeIdxDim;
    827:     PetscInt i, j, k, l;
    828:     PetscInt *prm;
    829:     int (*comp) (const void *, const void *);

    831:     PetscMalloc1((nodeIdxDim + 2) * m, &sorter);
    832:     for (k = 0, l = 0, i = 0; i < m; i++) {
    833:       sorter[k++] = nodeIdxDim + 1;
    834:       sorter[k++] = i;
    835:       for (j = 0; j < nodeIdxDim; j++) sorter[k++] = ni->nodeIdx[l++];
    836:     }
    837:     switch (nodeIdxDim) {
    838:     case 2:
    839:       comp = PetscTupIntCompRevlex_3;
    840:       break;
    841:     case 3:
    842:       comp = PetscTupIntCompRevlex_4;
    843:       break;
    844:     case 4:
    845:       comp = PetscTupIntCompRevlex_5;
    846:       break;
    847:     case 5:
    848:       comp = PetscTupIntCompRevlex_6;
    849:       break;
    850:     case 6:
    851:       comp = PetscTupIntCompRevlex_7;
    852:       break;
    853:     default:
    854:       comp = PetscTupIntCompRevlex_N;
    855:       break;
    856:     }
    857:     qsort(sorter, m, (nodeIdxDim + 2) * sizeof(PetscInt), comp);
    858:     PetscMalloc1(m, &prm);
    859:     for (i = 0; i < m; i++) prm[i] = sorter[(nodeIdxDim + 2) * i + 1];
    860:     ni->perm = prm;
    861:     PetscFree(sorter);
    862:   }
    863:   *perm = ni->perm;
    864:   return(0);
    865: }

    867: static PetscErrorCode PetscDualSpaceDestroy_Lagrange(PetscDualSpace sp)
    868: {
    869:   PetscDualSpace_Lag *lag = (PetscDualSpace_Lag *) sp->data;
    870:   PetscErrorCode      ierr;

    873:   if (lag->symperms) {
    874:     PetscInt **selfSyms = lag->symperms[0];

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

    879:       for (i = 0; i < lag->numSelfSym; i++) {
    880:         PetscFree(allocated[i]);
    881:       }
    882:       PetscFree(allocated);
    883:     }
    884:     PetscFree(lag->symperms);
    885:   }
    886:   if (lag->symflips) {
    887:     PetscScalar **selfSyms = lag->symflips[0];

    889:     if (selfSyms) {
    890:       PetscInt i;
    891:       PetscScalar **allocated = &selfSyms[-lag->selfSymOff];

    893:       for (i = 0; i < lag->numSelfSym; i++) {
    894:         PetscFree(allocated[i]);
    895:       }
    896:       PetscFree(allocated);
    897:     }
    898:     PetscFree(lag->symflips);
    899:   }
    900:   Petsc1DNodeFamilyDestroy(&(lag->nodeFamily));
    901:   PetscLagNodeIndicesDestroy(&(lag->vertIndices));
    902:   PetscLagNodeIndicesDestroy(&(lag->intNodeIndices));
    903:   PetscLagNodeIndicesDestroy(&(lag->allNodeIndices));
    904:   PetscFree(lag);
    905:   PetscObjectComposeFunction((PetscObject) sp, "PetscDualSpaceLagrangeGetContinuity_C", NULL);
    906:   PetscObjectComposeFunction((PetscObject) sp, "PetscDualSpaceLagrangeSetContinuity_C", NULL);
    907:   PetscObjectComposeFunction((PetscObject) sp, "PetscDualSpaceLagrangeGetTensor_C", NULL);
    908:   PetscObjectComposeFunction((PetscObject) sp, "PetscDualSpaceLagrangeSetTensor_C", NULL);
    909:   PetscObjectComposeFunction((PetscObject) sp, "PetscDualSpaceLagrangeGetTrimmed_C", NULL);
    910:   PetscObjectComposeFunction((PetscObject) sp, "PetscDualSpaceLagrangeSetTrimmed_C", NULL);
    911:   PetscObjectComposeFunction((PetscObject) sp, "PetscDualSpaceLagrangeGetNodeType_C", NULL);
    912:   PetscObjectComposeFunction((PetscObject) sp, "PetscDualSpaceLagrangeSetNodeType_C", NULL);
    913:   return(0);
    914: }

    916: static PetscErrorCode PetscDualSpaceLagrangeView_Ascii(PetscDualSpace sp, PetscViewer viewer)
    917: {
    918:   PetscDualSpace_Lag *lag = (PetscDualSpace_Lag *) sp->data;
    919:   PetscErrorCode      ierr;

    922:   PetscViewerASCIIPrintf(viewer, "%s %s%sLagrange dual space\n", lag->continuous ? "Continuous" : "Discontinuous", lag->tensorSpace ? "tensor " : "", lag->trimmed ? "trimmed " : "");
    923:   return(0);
    924: }

    926: static PetscErrorCode PetscDualSpaceView_Lagrange(PetscDualSpace sp, PetscViewer viewer)
    927: {
    928:   PetscBool      iascii;

    934:   PetscObjectTypeCompare((PetscObject) viewer, PETSCVIEWERASCII, &iascii);
    935:   if (iascii) {PetscDualSpaceLagrangeView_Ascii(sp, viewer);}
    936:   return(0);
    937: }

    939: static PetscErrorCode PetscDualSpaceSetFromOptions_Lagrange(PetscOptionItems *PetscOptionsObject,PetscDualSpace sp)
    940: {
    941:   PetscBool      continuous, tensor, trimmed, flg, flg2, flg3;
    942:   PetscDTNodeType nodeType;
    943:   PetscReal      nodeExponent;
    944:   PetscBool      nodeEndpoints;

    948:   PetscDualSpaceLagrangeGetContinuity(sp, &continuous);
    949:   PetscDualSpaceLagrangeGetTensor(sp, &tensor);
    950:   PetscDualSpaceLagrangeGetTrimmed(sp, &trimmed);
    951:   PetscDualSpaceLagrangeGetNodeType(sp, &nodeType, &nodeEndpoints, &nodeExponent);
    952:   if (nodeType == PETSCDTNODES_DEFAULT) nodeType = PETSCDTNODES_GAUSSJACOBI;
    953:   PetscOptionsHead(PetscOptionsObject,"PetscDualSpace Lagrange Options");
    954:   PetscOptionsBool("-petscdualspace_lagrange_continuity", "Flag for continuous element", "PetscDualSpaceLagrangeSetContinuity", continuous, &continuous, &flg);
    955:   if (flg) {PetscDualSpaceLagrangeSetContinuity(sp, continuous);}
    956:   PetscOptionsBool("-petscdualspace_lagrange_tensor", "Flag for tensor dual space", "PetscDualSpaceLagrangeSetTensor", tensor, &tensor, &flg);
    957:   if (flg) {PetscDualSpaceLagrangeSetTensor(sp, tensor);}
    958:   PetscOptionsBool("-petscdualspace_lagrange_trimmed", "Flag for trimmed dual space", "PetscDualSpaceLagrangeSetTrimmed", trimmed, &trimmed, &flg);
    959:   if (flg) {PetscDualSpaceLagrangeSetTrimmed(sp, trimmed);}
    960:   PetscOptionsEnum("-petscdualspace_lagrange_node_type", "Lagrange node location type", "PetscDualSpaceLagrangeSetNodeType", PetscDTNodeTypes, (PetscEnum)nodeType, (PetscEnum *)&nodeType, &flg);
    961:   PetscOptionsBool("-petscdualspace_lagrange_node_endpoints", "Flag for nodes that include endpoints", "PetscDualSpaceLagrangeSetNodeType", nodeEndpoints, &nodeEndpoints, &flg2);
    962:   flg3 = PETSC_FALSE;
    963:   if (nodeType == PETSCDTNODES_GAUSSJACOBI) {
    964:     PetscOptionsReal("-petscdualspace_lagrange_node_exponent", "Gauss-Jacobi weight function exponent", "PetscDualSpaceLagrangeSetNodeType", nodeExponent, &nodeExponent, &flg3);
    965:   }
    966:   if (flg || flg2 || flg3) {PetscDualSpaceLagrangeSetNodeType(sp, nodeType, nodeEndpoints, nodeExponent);}
    967:   PetscOptionsTail();
    968:   return(0);
    969: }

    971: static PetscErrorCode PetscDualSpaceDuplicate_Lagrange(PetscDualSpace sp, PetscDualSpace spNew)
    972: {
    973:   PetscBool           cont, tensor, trimmed, boundary;
    974:   PetscDTNodeType     nodeType;
    975:   PetscReal           exponent;
    976:   PetscDualSpace_Lag *lag    = (PetscDualSpace_Lag *) sp->data;
    977:   PetscErrorCode      ierr;

    980:   PetscDualSpaceLagrangeGetContinuity(sp, &cont);
    981:   PetscDualSpaceLagrangeSetContinuity(spNew, cont);
    982:   PetscDualSpaceLagrangeGetTensor(sp, &tensor);
    983:   PetscDualSpaceLagrangeSetTensor(spNew, tensor);
    984:   PetscDualSpaceLagrangeGetTrimmed(sp, &trimmed);
    985:   PetscDualSpaceLagrangeSetTrimmed(spNew, trimmed);
    986:   PetscDualSpaceLagrangeGetNodeType(sp, &nodeType, &boundary, &exponent);
    987:   PetscDualSpaceLagrangeSetNodeType(spNew, nodeType, boundary, exponent);
    988:   if (lag->nodeFamily) {
    989:     PetscDualSpace_Lag *lagnew = (PetscDualSpace_Lag *) spNew->data;

    991:     Petsc1DNodeFamilyReference(lag->nodeFamily);
    992:     lagnew->nodeFamily = lag->nodeFamily;
    993:   }
    994:   return(0);
    995: }

    997: /* for making tensor product spaces: take a dual space and product a segment space that has all the same
    998:  * specifications (trimmed, continuous, order, node set), except for the form degree */
    999: static PetscErrorCode PetscDualSpaceCreateEdgeSubspace_Lagrange(PetscDualSpace sp, PetscInt order, PetscInt k, PetscInt Nc, PetscBool interiorOnly, PetscDualSpace *bdsp)
    1000: {
    1001:   DM                 K;
    1002:   PetscDualSpace_Lag *newlag;
    1003:   PetscErrorCode     ierr;

    1006:   PetscDualSpaceDuplicate(sp,bdsp);
    1007:   PetscDualSpaceSetFormDegree(*bdsp, k);
    1008:   PetscDualSpaceCreateReferenceCell(*bdsp, 1, PETSC_TRUE, &K);
    1009:   PetscDualSpaceSetDM(*bdsp, K);
    1010:   DMDestroy(&K);
    1011:   PetscDualSpaceSetOrder(*bdsp, order);
    1012:   PetscDualSpaceSetNumComponents(*bdsp, Nc);
    1013:   newlag = (PetscDualSpace_Lag *) (*bdsp)->data;
    1014:   newlag->interiorOnly = interiorOnly;
    1015:   PetscDualSpaceSetUp(*bdsp);
    1016:   return(0);
    1017: }

    1019: /* just the points, weights aren't handled */
    1020: static PetscErrorCode PetscQuadratureCreateTensor(PetscQuadrature trace, PetscQuadrature fiber, PetscQuadrature *product)
    1021: {
    1022:   PetscInt         dimTrace, dimFiber;
    1023:   PetscInt         numPointsTrace, numPointsFiber;
    1024:   PetscInt         dim, numPoints;
    1025:   const PetscReal *pointsTrace;
    1026:   const PetscReal *pointsFiber;
    1027:   PetscReal       *points;
    1028:   PetscInt         i, j, k, p;
    1029:   PetscErrorCode   ierr;

    1032:   PetscQuadratureGetData(trace, &dimTrace, NULL, &numPointsTrace, &pointsTrace, NULL);
    1033:   PetscQuadratureGetData(fiber, &dimFiber, NULL, &numPointsFiber, &pointsFiber, NULL);
    1034:   dim = dimTrace + dimFiber;
    1035:   numPoints = numPointsFiber * numPointsTrace;
    1036:   PetscMalloc1(numPoints * dim, &points);
    1037:   for (p = 0, j = 0; j < numPointsFiber; j++) {
    1038:     for (i = 0; i < numPointsTrace; i++, p++) {
    1039:       for (k = 0; k < dimTrace; k++) points[p * dim +            k] = pointsTrace[i * dimTrace + k];
    1040:       for (k = 0; k < dimFiber; k++) points[p * dim + dimTrace + k] = pointsFiber[j * dimFiber + k];
    1041:     }
    1042:   }
    1043:   PetscQuadratureCreate(PETSC_COMM_SELF, product);
    1044:   PetscQuadratureSetData(*product, dim, 0, numPoints, points, NULL);
    1045:   return(0);
    1046: }

    1048: /* Kronecker tensor product where matrix is considered a matrix of k-forms, so that
    1049:  * the entries in the product matrix are wedge products of the entries in the original matrices */
    1050: static PetscErrorCode MatTensorAltV(Mat trace, Mat fiber, PetscInt dimTrace, PetscInt kTrace, PetscInt dimFiber, PetscInt kFiber, Mat *product)
    1051: {
    1052:   PetscInt mTrace, nTrace, mFiber, nFiber, m, n, k, i, j, l;
    1053:   PetscInt dim, NkTrace, NkFiber, Nk;
    1054:   PetscInt dT, dF;
    1055:   PetscInt *nnzTrace, *nnzFiber, *nnz;
    1056:   PetscInt iT, iF, jT, jF, il, jl;
    1057:   PetscReal *workT, *workT2, *workF, *workF2, *work, *workstar;
    1058:   PetscReal *projT, *projF;
    1059:   PetscReal *projTstar, *projFstar;
    1060:   PetscReal *wedgeMat;
    1061:   PetscReal sign;
    1062:   PetscScalar *workS;
    1063:   Mat prod;
    1064:   /* this produces dof groups that look like the identity */

    1068:   MatGetSize(trace, &mTrace, &nTrace);
    1069:   PetscDTBinomialInt(dimTrace, PetscAbsInt(kTrace), &NkTrace);
    1070:   if (nTrace % NkTrace) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_PLIB, "point value space of trace matrix is not a multiple of k-form size");
    1071:   MatGetSize(fiber, &mFiber, &nFiber);
    1072:   PetscDTBinomialInt(dimFiber, PetscAbsInt(kFiber), &NkFiber);
    1073:   if (nFiber % NkFiber) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_PLIB, "point value space of fiber matrix is not a multiple of k-form size");
    1074:   PetscMalloc2(mTrace, &nnzTrace, mFiber, &nnzFiber);
    1075:   for (i = 0; i < mTrace; i++) {
    1076:     MatGetRow(trace, i, &(nnzTrace[i]), NULL, NULL);
    1077:     if (nnzTrace[i] % NkTrace) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_PLIB, "nonzeros in trace matrix are not in k-form size blocks");
    1078:   }
    1079:   for (i = 0; i < mFiber; i++) {
    1080:     MatGetRow(fiber, i, &(nnzFiber[i]), NULL, NULL);
    1081:     if (nnzFiber[i] % NkFiber) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_PLIB, "nonzeros in fiber matrix are not in k-form size blocks");
    1082:   }
    1083:   dim = dimTrace + dimFiber;
    1084:   k = kFiber + kTrace;
    1085:   PetscDTBinomialInt(dim, PetscAbsInt(k), &Nk);
    1086:   m = mTrace * mFiber;
    1087:   PetscMalloc1(m, &nnz);
    1088:   for (l = 0, j = 0; j < mFiber; j++) for (i = 0; i < mTrace; i++, l++) nnz[l] = (nnzTrace[i] / NkTrace) * (nnzFiber[j] / NkFiber) * Nk;
    1089:   n = (nTrace / NkTrace) * (nFiber / NkFiber) * Nk;
    1090:   MatCreateSeqAIJ(PETSC_COMM_SELF, m, n, 0, nnz, &prod);
    1091:   PetscFree(nnz);
    1092:   PetscFree2(nnzTrace,nnzFiber);
    1093:   /* reasoning about which points each dof needs depends on having zeros computed at points preserved */
    1094:   MatSetOption(prod, MAT_IGNORE_ZERO_ENTRIES, PETSC_FALSE);
    1095:   /* compute pullbacks */
    1096:   PetscDTBinomialInt(dim, PetscAbsInt(kTrace), &dT);
    1097:   PetscDTBinomialInt(dim, PetscAbsInt(kFiber), &dF);
    1098:   PetscMalloc4(dimTrace * dim, &projT, dimFiber * dim, &projF, dT * NkTrace, &projTstar, dF * NkFiber, &projFstar);
    1099:   PetscArrayzero(projT, dimTrace * dim);
    1100:   for (i = 0; i < dimTrace; i++) projT[i * (dim + 1)] = 1.;
    1101:   PetscArrayzero(projF, dimFiber * dim);
    1102:   for (i = 0; i < dimFiber; i++) projF[i * (dim + 1) + dimTrace] = 1.;
    1103:   PetscDTAltVPullbackMatrix(dim, dimTrace, projT, kTrace, projTstar);
    1104:   PetscDTAltVPullbackMatrix(dim, dimFiber, projF, kFiber, projFstar);
    1105:   PetscMalloc5(dT, &workT, dF, &workF, Nk, &work, Nk, &workstar, Nk, &workS);
    1106:   PetscMalloc2(dT, &workT2, dF, &workF2);
    1107:   PetscMalloc1(Nk * dT, &wedgeMat);
    1108:   sign = (PetscAbsInt(kTrace * kFiber) & 1) ? -1. : 1.;
    1109:   for (i = 0, iF = 0; iF < mFiber; iF++) {
    1110:     PetscInt           ncolsF, nformsF;
    1111:     const PetscInt    *colsF;
    1112:     const PetscScalar *valsF;

    1114:     MatGetRow(fiber, iF, &ncolsF, &colsF, &valsF);
    1115:     nformsF = ncolsF / NkFiber;
    1116:     for (iT = 0; iT < mTrace; iT++, i++) {
    1117:       PetscInt           ncolsT, nformsT;
    1118:       const PetscInt    *colsT;
    1119:       const PetscScalar *valsT;

    1121:       MatGetRow(trace, iT, &ncolsT, &colsT, &valsT);
    1122:       nformsT = ncolsT / NkTrace;
    1123:       for (j = 0, jF = 0; jF < nformsF; jF++) {
    1124:         PetscInt colF = colsF[jF * NkFiber] / NkFiber;

    1126:         for (il = 0; il < dF; il++) {
    1127:           PetscReal val = 0.;
    1128:           for (jl = 0; jl < NkFiber; jl++) val += projFstar[il * NkFiber + jl] * PetscRealPart(valsF[jF * NkFiber + jl]);
    1129:           workF[il] = val;
    1130:         }
    1131:         if (kFiber < 0) {
    1132:           for (il = 0; il < dF; il++) workF2[il] = workF[il];
    1133:           PetscDTAltVStar(dim, PetscAbsInt(kFiber), 1, workF2, workF);
    1134:         }
    1135:         PetscDTAltVWedgeMatrix(dim, PetscAbsInt(kFiber), PetscAbsInt(kTrace), workF, wedgeMat);
    1136:         for (jT = 0; jT < nformsT; jT++, j++) {
    1137:           PetscInt colT = colsT[jT * NkTrace] / NkTrace;
    1138:           PetscInt col = colF * (nTrace / NkTrace) + colT;
    1139:           const PetscScalar *vals;

    1141:           for (il = 0; il < dT; il++) {
    1142:             PetscReal val = 0.;
    1143:             for (jl = 0; jl < NkTrace; jl++) val += projTstar[il * NkTrace + jl] * PetscRealPart(valsT[jT * NkTrace + jl]);
    1144:             workT[il] = val;
    1145:           }
    1146:           if (kTrace < 0) {
    1147:             for (il = 0; il < dT; il++) workT2[il] = workT[il];
    1148:             PetscDTAltVStar(dim, PetscAbsInt(kTrace), 1, workT2, workT);
    1149:           }

    1151:           for (il = 0; il < Nk; il++) {
    1152:             PetscReal val = 0.;
    1153:             for (jl = 0; jl < dT; jl++) val += sign * wedgeMat[il * dT + jl] * workT[jl];
    1154:             work[il] = val;
    1155:           }
    1156:           if (k < 0) {
    1157:             PetscDTAltVStar(dim, PetscAbsInt(k), -1, work, workstar);
    1158: #if defined(PETSC_USE_COMPLEX)
    1159:             for (l = 0; l < Nk; l++) workS[l] = workstar[l];
    1160:             vals = &workS[0];
    1161: #else
    1162:             vals = &workstar[0];
    1163: #endif
    1164:           } else {
    1165: #if defined(PETSC_USE_COMPLEX)
    1166:             for (l = 0; l < Nk; l++) workS[l] = work[l];
    1167:             vals = &workS[0];
    1168: #else
    1169:             vals = &work[0];
    1170: #endif
    1171:           }
    1172:           for (l = 0; l < Nk; l++) {
    1173:             MatSetValue(prod, i, col * Nk + l, vals[l], INSERT_VALUES);
    1174:           } /* Nk */
    1175:         } /* jT */
    1176:       } /* jF */
    1177:       MatRestoreRow(trace, iT, &ncolsT, &colsT, &valsT);
    1178:     } /* iT */
    1179:     MatRestoreRow(fiber, iF, &ncolsF, &colsF, &valsF);
    1180:   } /* iF */
    1181:   PetscFree(wedgeMat);
    1182:   PetscFree4(projT, projF, projTstar, projFstar);
    1183:   PetscFree2(workT2, workF2);
    1184:   PetscFree5(workT, workF, work, workstar, workS);
    1185:   MatAssemblyBegin(prod, MAT_FINAL_ASSEMBLY);
    1186:   MatAssemblyEnd(prod, MAT_FINAL_ASSEMBLY);
    1187:   *product = prod;
    1188:   return(0);
    1189: }

    1191: /* Union of quadrature points, with an attempt to identify commont points in the two sets */
    1192: static PetscErrorCode PetscQuadraturePointsMerge(PetscQuadrature quadA, PetscQuadrature quadB, PetscQuadrature *quadJoint, PetscInt *aToJoint[], PetscInt *bToJoint[])
    1193: {
    1194:   PetscInt         dimA, dimB;
    1195:   PetscInt         nA, nB, nJoint, i, j, d;
    1196:   const PetscReal *pointsA;
    1197:   const PetscReal *pointsB;
    1198:   PetscReal       *pointsJoint;
    1199:   PetscInt        *aToJ, *bToJ;
    1200:   PetscQuadrature  qJ;
    1201:   PetscErrorCode   ierr;

    1204:   PetscQuadratureGetData(quadA, &dimA, NULL, &nA, &pointsA, NULL);
    1205:   PetscQuadratureGetData(quadB, &dimB, NULL, &nB, &pointsB, NULL);
    1206:   if (dimA != dimB) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_INCOMP, "Quadrature points must be in the same dimension");
    1207:   nJoint = nA;
    1208:   PetscMalloc1(nA, &aToJ);
    1209:   for (i = 0; i < nA; i++) aToJ[i] = i;
    1210:   PetscMalloc1(nB, &bToJ);
    1211:   for (i = 0; i < nB; i++) {
    1212:     for (j = 0; j < nA; j++) {
    1213:       bToJ[i] = -1;
    1214:       for (d = 0; d < dimA; d++) if (PetscAbsReal(pointsB[i * dimA + d] - pointsA[j * dimA + d]) > PETSC_SMALL) break;
    1215:       if (d == dimA) {
    1216:         bToJ[i] = j;
    1217:         break;
    1218:       }
    1219:     }
    1220:     if (bToJ[i] == -1) {
    1221:       bToJ[i] = nJoint++;
    1222:     }
    1223:   }
    1224:   *aToJoint = aToJ;
    1225:   *bToJoint = bToJ;
    1226:   PetscMalloc1(nJoint * dimA, &pointsJoint);
    1227:   PetscArraycpy(pointsJoint, pointsA, nA * dimA);
    1228:   for (i = 0; i < nB; i++) {
    1229:     if (bToJ[i] >= nA) {
    1230:       for (d = 0; d < dimA; d++) pointsJoint[bToJ[i] * dimA + d] = pointsB[i * dimA + d];
    1231:     }
    1232:   }
    1233:   PetscQuadratureCreate(PETSC_COMM_SELF, &qJ);
    1234:   PetscQuadratureSetData(qJ, dimA, 0, nJoint, pointsJoint, NULL);
    1235:   *quadJoint = qJ;
    1236:   return(0);
    1237: }

    1239: /* Matrices matA and matB are both quadrature -> dof matrices: produce a matrix that is joint quadrature -> union of
    1240:  * dofs, where the joint quadrature was produced by PetscQuadraturePointsMerge */
    1241: static PetscErrorCode MatricesMerge(Mat matA, Mat matB, PetscInt dim, PetscInt k, PetscInt numMerged, const PetscInt aToMerged[], const PetscInt bToMerged[], Mat *matMerged)
    1242: {
    1243:   PetscInt m, n, mA, nA, mB, nB, Nk, i, j, l;
    1244:   Mat      M;
    1245:   PetscInt *nnz;
    1246:   PetscInt maxnnz;
    1247:   PetscInt *work;

    1251:   PetscDTBinomialInt(dim, PetscAbsInt(k), &Nk);
    1252:   MatGetSize(matA, &mA, &nA);
    1253:   if (nA % Nk) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_SIZ, "matA column space not a multiple of k-form size");
    1254:   MatGetSize(matB, &mB, &nB);
    1255:   if (nB % Nk) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_SIZ, "matB column space not a multiple of k-form size");
    1256:   m = mA + mB;
    1257:   n = numMerged * Nk;
    1258:   PetscMalloc1(m, &nnz);
    1259:   maxnnz = 0;
    1260:   for (i = 0; i < mA; i++) {
    1261:     MatGetRow(matA, i, &(nnz[i]), NULL, NULL);
    1262:     if (nnz[i] % Nk) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_PLIB, "nonzeros in matA are not in k-form size blocks");
    1263:     maxnnz = PetscMax(maxnnz, nnz[i]);
    1264:   }
    1265:   for (i = 0; i < mB; i++) {
    1266:     MatGetRow(matB, i, &(nnz[i+mA]), NULL, NULL);
    1267:     if (nnz[i+mA] % Nk) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_PLIB, "nonzeros in matB are not in k-form size blocks");
    1268:     maxnnz = PetscMax(maxnnz, nnz[i+mA]);
    1269:   }
    1270:   MatCreateSeqAIJ(PETSC_COMM_SELF, m, n, 0, nnz, &M);
    1271:   PetscFree(nnz);
    1272:   /* reasoning about which points each dof needs depends on having zeros computed at points preserved */
    1273:   MatSetOption(M, MAT_IGNORE_ZERO_ENTRIES, PETSC_FALSE);
    1274:   PetscMalloc1(maxnnz, &work);
    1275:   for (i = 0; i < mA; i++) {
    1276:     const PetscInt *cols;
    1277:     const PetscScalar *vals;
    1278:     PetscInt nCols;
    1279:     MatGetRow(matA, i, &nCols, &cols, &vals);
    1280:     for (j = 0; j < nCols / Nk; j++) {
    1281:       PetscInt newCol = aToMerged[cols[j * Nk] / Nk];
    1282:       for (l = 0; l < Nk; l++) work[j * Nk + l] = newCol * Nk + l;
    1283:     }
    1284:     MatSetValuesBlocked(M, 1, &i, nCols, work, vals, INSERT_VALUES);
    1285:     MatRestoreRow(matA, i, &nCols, &cols, &vals);
    1286:   }
    1287:   for (i = 0; i < mB; i++) {
    1288:     const PetscInt *cols;
    1289:     const PetscScalar *vals;

    1291:     PetscInt row = i + mA;
    1292:     PetscInt nCols;
    1293:     MatGetRow(matB, i, &nCols, &cols, &vals);
    1294:     for (j = 0; j < nCols / Nk; j++) {
    1295:       PetscInt newCol = bToMerged[cols[j * Nk] / Nk];
    1296:       for (l = 0; l < Nk; l++) work[j * Nk + l] = newCol * Nk + l;
    1297:     }
    1298:     MatSetValuesBlocked(M, 1, &row, nCols, work, vals, INSERT_VALUES);
    1299:     MatRestoreRow(matB, i, &nCols, &cols, &vals);
    1300:   }
    1301:   PetscFree(work);
    1302:   MatAssemblyBegin(M, MAT_FINAL_ASSEMBLY);
    1303:   MatAssemblyEnd(M, MAT_FINAL_ASSEMBLY);
    1304:   *matMerged = M;
    1305:   return(0);
    1306: }

    1308: /* Take a dual space and product a segment space that has all the same specifications (trimmed, continuous, order,
    1309:  * node set), except for the form degree.  For computing boundary dofs and for making tensor product spaces */
    1310: static PetscErrorCode PetscDualSpaceCreateFacetSubspace_Lagrange(PetscDualSpace sp, DM K, PetscInt f, PetscInt k, PetscInt Ncopies, PetscBool interiorOnly, PetscDualSpace *bdsp)
    1311: {
    1312:   PetscInt           Nknew, Ncnew;
    1313:   PetscInt           dim, pointDim = -1;
    1314:   PetscInt           depth;
    1315:   DM                 dm;
    1316:   PetscDualSpace_Lag *newlag;
    1317:   PetscErrorCode     ierr;

    1320:   PetscDualSpaceGetDM(sp,&dm);
    1321:   DMGetDimension(dm,&dim);
    1322:   DMPlexGetDepth(dm,&depth);
    1323:   PetscDualSpaceDuplicate(sp,bdsp);
    1324:   PetscDualSpaceSetFormDegree(*bdsp,k);
    1325:   if (!K) {
    1326:     PetscBool isSimplex;

    1329:     if (depth == dim) {
    1330:       PetscInt coneSize;

    1332:       pointDim = dim - 1;
    1333:       DMPlexGetConeSize(dm,f,&coneSize);
    1334:       isSimplex = (PetscBool) (coneSize == dim);
    1335:       PetscDualSpaceCreateReferenceCell(*bdsp, dim-1, isSimplex, &K);
    1336:     } else if (depth == 1) {
    1337:       pointDim = 0;
    1338:       PetscDualSpaceCreateReferenceCell(*bdsp, 0, PETSC_TRUE, &K);
    1339:     } else SETERRQ(PETSC_COMM_SELF, PETSC_ERR_PLIB, "Unsupported interpolation state of reference element");
    1340:   } else {
    1341:     PetscObjectReference((PetscObject)K);
    1342:     DMGetDimension(K, &pointDim);
    1343:   }
    1344:   PetscDualSpaceSetDM(*bdsp, K);
    1345:   DMDestroy(&K);
    1346:   PetscDTBinomialInt(pointDim, PetscAbsInt(k), &Nknew);
    1347:   Ncnew = Nknew * Ncopies;
    1348:   PetscDualSpaceSetNumComponents(*bdsp, Ncnew);
    1349:   newlag = (PetscDualSpace_Lag *) (*bdsp)->data;
    1350:   newlag->interiorOnly = interiorOnly;
    1351:   PetscDualSpaceSetUp(*bdsp);
    1352:   return(0);
    1353: }

    1355: /* Construct simplex nodes from a nodefamily, add Nk dof vectors of length Nk at each node.
    1356:  * Return the (quadrature, matrix) form of the dofs and the nodeIndices form as well.
    1357:  *
    1358:  * Sometimes we want a set of nodes to be contained in the interior of the element,
    1359:  * even when the node scheme puts nodes on the boundaries.  numNodeSkip tells
    1360:  * the routine how many "layers" of nodes need to be skipped.
    1361:  * */
    1362: static PetscErrorCode PetscDualSpaceLagrangeCreateSimplexNodeMat(Petsc1DNodeFamily nodeFamily, PetscInt dim, PetscInt sum, PetscInt Nk, PetscInt numNodeSkip, PetscQuadrature *iNodes, Mat *iMat, PetscLagNodeIndices *nodeIndices)
    1363: {
    1364:   PetscReal *extraNodeCoords, *nodeCoords;
    1365:   PetscInt nNodes, nExtraNodes;
    1366:   PetscInt i, j, k, extraSum = sum + numNodeSkip * (1 + dim);
    1367:   PetscQuadrature intNodes;
    1368:   Mat intMat;
    1369:   PetscLagNodeIndices ni;

    1373:   PetscDTBinomialInt(dim + sum, dim, &nNodes);
    1374:   PetscDTBinomialInt(dim + extraSum, dim, &nExtraNodes);

    1376:   PetscMalloc1(dim * nExtraNodes, &extraNodeCoords);
    1377:   PetscNew(&ni);
    1378:   ni->nodeIdxDim = dim + 1;
    1379:   ni->nodeVecDim = Nk;
    1380:   ni->nNodes = nNodes * Nk;
    1381:   ni->refct = 1;
    1382:   PetscMalloc1(nNodes * Nk * (dim + 1), &(ni->nodeIdx));
    1383:   PetscMalloc1(nNodes * Nk * Nk, &(ni->nodeVec));
    1384:   for (i = 0; i < nNodes; i++) for (j = 0; j < Nk; j++) for (k = 0; k < Nk; k++) ni->nodeVec[(i * Nk + j) * Nk + k] = (j == k) ? 1. : 0.;
    1385:   Petsc1DNodeFamilyComputeSimplexNodes(nodeFamily, dim, extraSum, extraNodeCoords);
    1386:   if (numNodeSkip) {
    1387:     PetscInt k;
    1388:     PetscInt *tup;

    1390:     PetscMalloc1(dim * nNodes, &nodeCoords);
    1391:     PetscMalloc1(dim + 1, &tup);
    1392:     for (k = 0; k < nNodes; k++) {
    1393:       PetscInt j, c;
    1394:       PetscInt index;

    1396:       PetscDTIndexToBary(dim + 1, sum, k, tup);
    1397:       for (j = 0; j < dim + 1; j++) tup[j] += numNodeSkip;
    1398:       for (c = 0; c < Nk; c++) {
    1399:         for (j = 0; j < dim + 1; j++) {
    1400:           ni->nodeIdx[(k * Nk + c) * (dim + 1) + j] = tup[j] + 1;
    1401:         }
    1402:       }
    1403:       PetscDTBaryToIndex(dim + 1, extraSum, tup, &index);
    1404:       for (j = 0; j < dim; j++) nodeCoords[k * dim + j] = extraNodeCoords[index * dim + j];
    1405:     }
    1406:     PetscFree(tup);
    1407:     PetscFree(extraNodeCoords);
    1408:   } else {
    1409:     PetscInt k;
    1410:     PetscInt *tup;

    1412:     nodeCoords = extraNodeCoords;
    1413:     PetscMalloc1(dim + 1, &tup);
    1414:     for (k = 0; k < nNodes; k++) {
    1415:       PetscInt j, c;

    1417:       PetscDTIndexToBary(dim + 1, sum, k, tup);
    1418:       for (c = 0; c < Nk; c++) {
    1419:         for (j = 0; j < dim + 1; j++) {
    1420:           /* barycentric indices can have zeros, but we don't want to push forward zeros because it makes it harder to
    1421:            * determine which nodes correspond to which under symmetries, so we increase by 1.  This is fine
    1422:            * because the nodeIdx coordinates don't have any meaning other than helping to identify symmetries */
    1423:           ni->nodeIdx[(k * Nk + c) * (dim + 1) + j] = tup[j] + 1;
    1424:         }
    1425:       }
    1426:     }
    1427:     PetscFree(tup);
    1428:   }
    1429:   PetscQuadratureCreate(PETSC_COMM_SELF, &intNodes);
    1430:   PetscQuadratureSetData(intNodes, dim, 0, nNodes, nodeCoords, NULL);
    1431:   MatCreateSeqAIJ(PETSC_COMM_SELF, nNodes * Nk, nNodes * Nk, Nk, NULL, &intMat);
    1432:   MatSetOption(intMat,MAT_IGNORE_ZERO_ENTRIES,PETSC_FALSE);
    1433:   for (j = 0; j < nNodes * Nk; j++) {
    1434:     PetscInt rem = j % Nk;
    1435:     PetscInt a, aprev = j - rem;
    1436:     PetscInt anext = aprev + Nk;

    1438:     for (a = aprev; a < anext; a++) {
    1439:       MatSetValue(intMat, j, a, (a == j) ? 1. : 0., INSERT_VALUES);
    1440:     }
    1441:   }
    1442:   MatAssemblyBegin(intMat, MAT_FINAL_ASSEMBLY);
    1443:   MatAssemblyEnd(intMat, MAT_FINAL_ASSEMBLY);
    1444:   *iNodes = intNodes;
    1445:   *iMat = intMat;
    1446:   *nodeIndices = ni;
    1447:   return(0);
    1448: }

    1450: /* once the nodeIndices have been created for the interior of the reference cell, and for all of the boundary cells,
    1451:  * push forward the boudary dofs and concatenate them into the full node indices for the dual space */
    1452: static PetscErrorCode PetscDualSpaceLagrangeCreateAllNodeIdx(PetscDualSpace sp)
    1453: {
    1454:   DM             dm;
    1455:   PetscInt       dim, nDofs;
    1456:   PetscSection   section;
    1457:   PetscInt       pStart, pEnd, p;
    1458:   PetscInt       formDegree, Nk;
    1459:   PetscInt       nodeIdxDim, spintdim;
    1460:   PetscDualSpace_Lag *lag;
    1461:   PetscLagNodeIndices ni, verti;

    1465:   lag = (PetscDualSpace_Lag *) sp->data;
    1466:   verti = lag->vertIndices;
    1467:   PetscDualSpaceGetDM(sp, &dm);
    1468:   DMGetDimension(dm, &dim);
    1469:   PetscDualSpaceGetFormDegree(sp, &formDegree);
    1470:   PetscDTBinomialInt(dim, PetscAbsInt(formDegree), &Nk);
    1471:   PetscDualSpaceGetSection(sp, &section);
    1472:   PetscSectionGetStorageSize(section, &nDofs);
    1473:   PetscNew(&ni);
    1474:   ni->nodeIdxDim = nodeIdxDim = verti->nodeIdxDim;
    1475:   ni->nodeVecDim = Nk;
    1476:   ni->nNodes = nDofs;
    1477:   ni->refct = 1;
    1478:   PetscMalloc1(nodeIdxDim * nDofs, &(ni->nodeIdx));
    1479:   PetscMalloc1(Nk * nDofs, &(ni->nodeVec));
    1480:   DMPlexGetChart(dm, &pStart, &pEnd);
    1481:   PetscSectionGetDof(section, 0, &spintdim);
    1482:   if (spintdim) {
    1483:     PetscArraycpy(ni->nodeIdx, lag->intNodeIndices->nodeIdx, spintdim * nodeIdxDim);
    1484:     PetscArraycpy(ni->nodeVec, lag->intNodeIndices->nodeVec, spintdim * Nk);
    1485:   }
    1486:   for (p = pStart + 1; p < pEnd; p++) {
    1487:     PetscDualSpace psp = sp->pointSpaces[p];
    1488:     PetscDualSpace_Lag *plag;
    1489:     PetscInt dof, off;

    1491:     PetscSectionGetDof(section, p, &dof);
    1492:     if (!dof) continue;
    1493:     plag = (PetscDualSpace_Lag *) psp->data;
    1494:     PetscSectionGetOffset(section, p, &off);
    1495:     PetscLagNodeIndicesPushForward(dm, verti, p, plag->vertIndices, plag->intNodeIndices, 0, formDegree, &(ni->nodeIdx[off * nodeIdxDim]), &(ni->nodeVec[off * Nk]));
    1496:   }
    1497:   lag->allNodeIndices = ni;
    1498:   return(0);
    1499: }

    1501: /* once the (quadrature, Matrix) forms of the dofs have been created for the interior of the
    1502:  * reference cell and for the boundary cells, jk
    1503:  * push forward the boundary data and concatenate them into the full (quadrature, matrix) data
    1504:  * for the dual space */
    1505: static PetscErrorCode PetscDualSpaceCreateAllDataFromInteriorData(PetscDualSpace sp)
    1506: {
    1507:   DM               dm;
    1508:   PetscSection     section;
    1509:   PetscInt         pStart, pEnd, p, k, Nk, dim, Nc;
    1510:   PetscInt         nNodes;
    1511:   PetscInt         countNodes;
    1512:   Mat              allMat;
    1513:   PetscQuadrature  allNodes;
    1514:   PetscInt         nDofs;
    1515:   PetscInt         maxNzforms, j;
    1516:   PetscScalar      *work;
    1517:   PetscReal        *L, *J, *Jinv, *v0, *pv0;
    1518:   PetscInt         *iwork;
    1519:   PetscReal        *nodes;
    1520:   PetscErrorCode   ierr;

    1523:   PetscDualSpaceGetDM(sp, &dm);
    1524:   DMGetDimension(dm, &dim);
    1525:   PetscDualSpaceGetSection(sp, &section);
    1526:   PetscSectionGetStorageSize(section, &nDofs);
    1527:   DMPlexGetChart(dm, &pStart, &pEnd);
    1528:   PetscDualSpaceGetFormDegree(sp, &k);
    1529:   PetscDualSpaceGetNumComponents(sp, &Nc);
    1530:   PetscDTBinomialInt(dim, PetscAbsInt(k), &Nk);
    1531:   for (p = pStart, nNodes = 0, maxNzforms = 0; p < pEnd; p++) {
    1532:     PetscDualSpace  psp;
    1533:     DM              pdm;
    1534:     PetscInt        pdim, pNk;
    1535:     PetscQuadrature intNodes;
    1536:     Mat intMat;

    1538:     PetscDualSpaceGetPointSubspace(sp, p, &psp);
    1539:     if (!psp) continue;
    1540:     PetscDualSpaceGetDM(psp, &pdm);
    1541:     DMGetDimension(pdm, &pdim);
    1542:     if (pdim < PetscAbsInt(k)) continue;
    1543:     PetscDTBinomialInt(pdim, PetscAbsInt(k), &pNk);
    1544:     PetscDualSpaceGetInteriorData(psp, &intNodes, &intMat);
    1545:     if (intNodes) {
    1546:       PetscInt nNodesp;

    1548:       PetscQuadratureGetData(intNodes, NULL, NULL, &nNodesp, NULL, NULL);
    1549:       nNodes += nNodesp;
    1550:     }
    1551:     if (intMat) {
    1552:       PetscInt maxNzsp;
    1553:       PetscInt maxNzformsp;

    1555:       MatSeqAIJGetMaxRowNonzeros(intMat, &maxNzsp);
    1556:       if (maxNzsp % pNk) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_PLIB, "interior matrix is not laid out as blocks of k-forms");
    1557:       maxNzformsp = maxNzsp / pNk;
    1558:       maxNzforms = PetscMax(maxNzforms, maxNzformsp);
    1559:     }
    1560:   }
    1561:   MatCreateSeqAIJ(PETSC_COMM_SELF, nDofs, nNodes * Nc, maxNzforms * Nk, NULL, &allMat);
    1562:   MatSetOption(allMat,MAT_IGNORE_ZERO_ENTRIES,PETSC_FALSE);
    1563:   PetscMalloc7(dim, &v0, dim, &pv0, dim * dim, &J, dim * dim, &Jinv, Nk * Nk, &L, maxNzforms * Nk, &work, maxNzforms * Nk, &iwork);
    1564:   for (j = 0; j < dim; j++) pv0[j] = -1.;
    1565:   PetscMalloc1(dim * nNodes, &nodes);
    1566:   for (p = pStart, countNodes = 0; p < pEnd; p++) {
    1567:     PetscDualSpace  psp;
    1568:     PetscQuadrature intNodes;
    1569:     DM pdm;
    1570:     PetscInt pdim, pNk;
    1571:     PetscInt countNodesIn = countNodes;
    1572:     PetscReal detJ;
    1573:     Mat intMat;

    1575:     PetscDualSpaceGetPointSubspace(sp, p, &psp);
    1576:     if (!psp) continue;
    1577:     PetscDualSpaceGetDM(psp, &pdm);
    1578:     DMGetDimension(pdm, &pdim);
    1579:     if (pdim < PetscAbsInt(k)) continue;
    1580:     PetscDualSpaceGetInteriorData(psp, &intNodes, &intMat);
    1581:     if (intNodes == NULL && intMat == NULL) continue;
    1582:     PetscDTBinomialInt(pdim, PetscAbsInt(k), &pNk);
    1583:     if (p) {
    1584:       DMPlexComputeCellGeometryAffineFEM(dm, p, v0, J, Jinv, &detJ);
    1585:     } else { /* identity */
    1586:       PetscInt i,j;

    1588:       for (i = 0; i < dim; i++) for (j = 0; j < dim; j++) J[i * dim + j] = Jinv[i * dim + j] = 0.;
    1589:       for (i = 0; i < dim; i++) J[i * dim + i] = Jinv[i * dim + i] = 1.;
    1590:       for (i = 0; i < dim; i++) v0[i] = -1.;
    1591:     }
    1592:     if (pdim != dim) { /* compactify Jacobian */
    1593:       PetscInt i, j;

    1595:       for (i = 0; i < dim; i++) for (j = 0; j < pdim; j++) J[i * pdim + j] = J[i * dim + j];
    1596:     }
    1597:     PetscDTAltVPullbackMatrix(pdim, dim, J, k, L);
    1598:     if (intNodes) { /* push forward quadrature locations by the affine transformation */
    1599:       PetscInt nNodesp;
    1600:       const PetscReal *nodesp;
    1601:       PetscInt j;

    1603:       PetscQuadratureGetData(intNodes, NULL, NULL, &nNodesp, &nodesp, NULL);
    1604:       for (j = 0; j < nNodesp; j++, countNodes++) {
    1605:         PetscInt d, e;

    1607:         for (d = 0; d < dim; d++) {
    1608:           nodes[countNodes * dim + d] = v0[d];
    1609:           for (e = 0; e < pdim; e++) {
    1610:             nodes[countNodes * dim + d] += J[d * pdim + e] * (nodesp[j * pdim + e] - pv0[e]);
    1611:           }
    1612:         }
    1613:       }
    1614:     }
    1615:     if (intMat) {
    1616:       PetscInt nrows;
    1617:       PetscInt off;

    1619:       PetscSectionGetDof(section, p, &nrows);
    1620:       PetscSectionGetOffset(section, p, &off);
    1621:       for (j = 0; j < nrows; j++) {
    1622:         PetscInt ncols;
    1623:         const PetscInt *cols;
    1624:         const PetscScalar *vals;
    1625:         PetscInt l, d, e;
    1626:         PetscInt row = j + off;

    1628:         MatGetRow(intMat, j, &ncols, &cols, &vals);
    1629:         if (ncols % pNk) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_PLIB, "interior matrix is not laid out as blocks of k-forms");
    1630:         for (l = 0; l < ncols / pNk; l++) {
    1631:           PetscInt blockcol;

    1633:           for (d = 0; d < pNk; d++) {
    1634:             if ((cols[l * pNk + d] % pNk) != d) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_PLIB, "interior matrix is not laid out as blocks of k-forms");
    1635:           }
    1636:           blockcol = cols[l * pNk] / pNk;
    1637:           for (d = 0; d < Nk; d++) {
    1638:             iwork[l * Nk + d] = (blockcol + countNodesIn) * Nk + d;
    1639:           }
    1640:           for (d = 0; d < Nk; d++) work[l * Nk + d] = 0.;
    1641:           for (d = 0; d < Nk; d++) {
    1642:             for (e = 0; e < pNk; e++) {
    1643:               /* "push forward" dof by pulling back a k-form to be evaluated on the point: multiply on the right by L */
    1644:               work[l * Nk + d] += vals[l * pNk + e] * L[e * pNk + d];
    1645:             }
    1646:           }
    1647:         }
    1648:         MatSetValues(allMat, 1, &row, (ncols / pNk) * Nk, iwork, work, INSERT_VALUES);
    1649:         MatRestoreRow(intMat, j, &ncols, &cols, &vals);
    1650:       }
    1651:     }
    1652:   }
    1653:   MatAssemblyBegin(allMat, MAT_FINAL_ASSEMBLY);
    1654:   MatAssemblyEnd(allMat, MAT_FINAL_ASSEMBLY);
    1655:   PetscQuadratureCreate(PETSC_COMM_SELF, &allNodes);
    1656:   PetscQuadratureSetData(allNodes, dim, 0, nNodes, nodes, NULL);
    1657:   PetscFree7(v0, pv0, J, Jinv, L, work, iwork);
    1658:   MatDestroy(&(sp->allMat));
    1659:   sp->allMat = allMat;
    1660:   PetscQuadratureDestroy(&(sp->allNodes));
    1661:   sp->allNodes = allNodes;
    1662:   return(0);
    1663: }

    1665: /* rather than trying to get all data from the functionals, we create
    1666:  * the functionals from rows of the quadrature -> dof matrix.
    1667:  *
    1668:  * Ideally most of the uses of PetscDualSpace in PetscFE will switch
    1669:  * to using intMat and allMat, so that the individual functionals
    1670:  * don't need to be constructed at all */
    1671: static PetscErrorCode PetscDualSpaceComputeFunctionalsFromAllData(PetscDualSpace sp)
    1672: {
    1673:   PetscQuadrature allNodes;
    1674:   Mat             allMat;
    1675:   PetscInt        nDofs;
    1676:   PetscInt        dim, k, Nk, Nc, f;
    1677:   DM              dm;
    1678:   PetscInt        nNodes, spdim;
    1679:   const PetscReal *nodes = NULL;
    1680:   PetscSection    section;
    1681:   PetscErrorCode  ierr;

    1684:   PetscDualSpaceGetDM(sp, &dm);
    1685:   DMGetDimension(dm, &dim);
    1686:   PetscDualSpaceGetNumComponents(sp, &Nc);
    1687:   PetscDualSpaceGetFormDegree(sp, &k);
    1688:   PetscDTBinomialInt(dim, PetscAbsInt(k), &Nk);
    1689:   PetscDualSpaceGetAllData(sp, &allNodes, &allMat);
    1690:   nNodes = 0;
    1691:   if (allNodes) {
    1692:     PetscQuadratureGetData(allNodes, NULL, NULL, &nNodes, &nodes, NULL);
    1693:   }
    1694:   MatGetSize(allMat, &nDofs, NULL);
    1695:   PetscDualSpaceGetSection(sp, &section);
    1696:   PetscSectionGetStorageSize(section, &spdim);
    1697:   if (spdim != nDofs) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_PLIB, "incompatible all matrix size");
    1698:   PetscMalloc1(nDofs, &(sp->functional));
    1699:   for (f = 0; f < nDofs; f++) {
    1700:     PetscInt ncols, c;
    1701:     const PetscInt *cols;
    1702:     const PetscScalar *vals;
    1703:     PetscReal *nodesf;
    1704:     PetscReal *weightsf;
    1705:     PetscInt nNodesf;
    1706:     PetscInt countNodes;

    1708:     MatGetRow(allMat, f, &ncols, &cols, &vals);
    1709:     if (ncols % Nk) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_PLIB, "all matrix is not laid out as blocks of k-forms");
    1710:     for (c = 1, nNodesf = 1; c < ncols; c++) {
    1711:       if ((cols[c] / Nc) != (cols[c-1] / Nc)) nNodesf++;
    1712:     }
    1713:     PetscMalloc1(dim * nNodesf, &nodesf);
    1714:     PetscMalloc1(Nc * nNodesf, &weightsf);
    1715:     for (c = 0, countNodes = 0; c < ncols; c++) {
    1716:       if (!c || ((cols[c] / Nc) != (cols[c-1] / Nc))) {
    1717:         PetscInt d;

    1719:         for (d = 0; d < Nc; d++) {
    1720:           weightsf[countNodes * Nc + d] = 0.;
    1721:         }
    1722:         for (d = 0; d < dim; d++) {
    1723:           nodesf[countNodes * dim + d] = nodes[(cols[c] / Nc) * dim + d];
    1724:         }
    1725:         countNodes++;
    1726:       }
    1727:       weightsf[(countNodes - 1) * Nc + (cols[c] % Nc)] = PetscRealPart(vals[c]);
    1728:     }
    1729:     PetscQuadratureCreate(PETSC_COMM_SELF, &(sp->functional[f]));
    1730:     PetscQuadratureSetData(sp->functional[f], dim, Nc, nNodesf, nodesf, weightsf);
    1731:     MatRestoreRow(allMat, f, &ncols, &cols, &vals);
    1732:   }
    1733:   return(0);
    1734: }

    1736: /* take a matrix meant for k-forms and expand it to one for Ncopies */
    1737: static PetscErrorCode PetscDualSpaceLagrangeMatrixCreateCopies(Mat A, PetscInt Nk, PetscInt Ncopies, Mat *Abs)
    1738: {
    1739:   PetscInt       m, n, i, j, k;
    1740:   PetscInt       maxnnz, *nnz, *iwork;
    1741:   Mat            Ac;

    1745:   MatGetSize(A, &m, &n);
    1746:   if (n % Nk) SETERRQ2(PETSC_COMM_SELF, PETSC_ERR_PLIB, "Number of columns in A %D is not a multiple of Nk %D", n, Nk);
    1747:   PetscMalloc1(m * Ncopies, &nnz);
    1748:   for (i = 0, maxnnz = 0; i < m; i++) {
    1749:     PetscInt innz;
    1750:     MatGetRow(A, i, &innz, NULL, NULL);
    1751:     if (innz % Nk) SETERRQ2(PETSC_COMM_SELF, PETSC_ERR_PLIB, "A row %D nnzs is not a multiple of Nk %D", innz, Nk);
    1752:     for (j = 0; j < Ncopies; j++) nnz[i * Ncopies + j] = innz;
    1753:     maxnnz = PetscMax(maxnnz, innz);
    1754:   }
    1755:   MatCreateSeqAIJ(PETSC_COMM_SELF, m * Ncopies, n * Ncopies, 0, nnz, &Ac);
    1756:   MatSetOption(Ac, MAT_IGNORE_ZERO_ENTRIES, PETSC_FALSE);
    1757:   PetscFree(nnz);
    1758:   PetscMalloc1(maxnnz, &iwork);
    1759:   for (i = 0; i < m; i++) {
    1760:     PetscInt innz;
    1761:     const PetscInt    *cols;
    1762:     const PetscScalar *vals;

    1764:     MatGetRow(A, i, &innz, &cols, &vals);
    1765:     for (j = 0; j < innz; j++) iwork[j] = (cols[j] / Nk) * (Nk * Ncopies) + (cols[j] % Nk);
    1766:     for (j = 0; j < Ncopies; j++) {
    1767:       PetscInt row = i * Ncopies + j;

    1769:       MatSetValues(Ac, 1, &row, innz, iwork, vals, INSERT_VALUES);
    1770:       for (k = 0; k < innz; k++) iwork[k] += Nk;
    1771:     }
    1772:     MatRestoreRow(A, i, &innz, &cols, &vals);
    1773:   }
    1774:   PetscFree(iwork);
    1775:   MatAssemblyBegin(Ac, MAT_FINAL_ASSEMBLY);
    1776:   MatAssemblyEnd(Ac, MAT_FINAL_ASSEMBLY);
    1777:   *Abs = Ac;
    1778:   return(0);
    1779: }

    1781: /* check if a cell is a tensor product of the segment with a facet,
    1782:  * specifically checking if f and f2 can be the "endpoints" (like the triangles
    1783:  * at either end of a wedge) */
    1784: static PetscErrorCode DMPlexPointIsTensor_Internal_Given(DM dm, PetscInt p, PetscInt f, PetscInt f2, PetscBool *isTensor)
    1785: {
    1786:   PetscInt        coneSize, c;
    1787:   const PetscInt *cone;
    1788:   const PetscInt *fCone;
    1789:   const PetscInt *f2Cone;
    1790:   PetscInt        fs[2];
    1791:   PetscInt        meetSize, nmeet;
    1792:   const PetscInt *meet;
    1793:   PetscErrorCode  ierr;

    1796:   fs[0] = f;
    1797:   fs[1] = f2;
    1798:   DMPlexGetMeet(dm, 2, fs, &meetSize, &meet);
    1799:   nmeet = meetSize;
    1800:   DMPlexRestoreMeet(dm, 2, fs, &meetSize, &meet);
    1801:   /* two points that have a non-empty meet cannot be at opposite ends of a cell */
    1802:   if (nmeet) {
    1803:     *isTensor = PETSC_FALSE;
    1804:     return(0);
    1805:   }
    1806:   DMPlexGetConeSize(dm, p, &coneSize);
    1807:   DMPlexGetCone(dm, p, &cone);
    1808:   DMPlexGetCone(dm, f, &fCone);
    1809:   DMPlexGetCone(dm, f2, &f2Cone);
    1810:   for (c = 0; c < coneSize; c++) {
    1811:     PetscInt e, ef;
    1812:     PetscInt d = -1, d2 = -1;
    1813:     PetscInt dcount, d2count;
    1814:     PetscInt t = cone[c];
    1815:     PetscInt tConeSize;
    1816:     PetscBool tIsTensor;
    1817:     const PetscInt *tCone;

    1819:     if (t == f || t == f2) continue;
    1820:     /* for every other facet in the cone, check that is has
    1821:      * one ridge in common with each end */
    1822:     DMPlexGetConeSize(dm, t, &tConeSize);
    1823:     DMPlexGetCone(dm, t, &tCone);

    1825:     dcount = 0;
    1826:     d2count = 0;
    1827:     for (e = 0; e < tConeSize; e++) {
    1828:       PetscInt q = tCone[e];
    1829:       for (ef = 0; ef < coneSize - 2; ef++) {
    1830:         if (fCone[ef] == q) {
    1831:           if (dcount) {
    1832:             *isTensor = PETSC_FALSE;
    1833:             return(0);
    1834:           }
    1835:           d = q;
    1836:           dcount++;
    1837:         } else if (f2Cone[ef] == q) {
    1838:           if (d2count) {
    1839:             *isTensor = PETSC_FALSE;
    1840:             return(0);
    1841:           }
    1842:           d2 = q;
    1843:           d2count++;
    1844:         }
    1845:       }
    1846:     }
    1847:     /* if the whole cell is a tensor with the segment, then this
    1848:      * facet should be a tensor with the segment */
    1849:     DMPlexPointIsTensor_Internal_Given(dm, t, d, d2, &tIsTensor);
    1850:     if (!tIsTensor) {
    1851:       *isTensor = PETSC_FALSE;
    1852:       return(0);
    1853:     }
    1854:   }
    1855:   *isTensor = PETSC_TRUE;
    1856:   return(0);
    1857: }

    1859: /* determine if a cell is a tensor with a segment by looping over pairs of facets to find a pair
    1860:  * that could be the opposite ends */
    1861: static PetscErrorCode DMPlexPointIsTensor_Internal(DM dm, PetscInt p, PetscBool *isTensor, PetscInt *endA, PetscInt *endB)
    1862: {
    1863:   PetscInt        coneSize, c, c2;
    1864:   const PetscInt *cone;
    1865:   PetscErrorCode  ierr;

    1868:   DMPlexGetConeSize(dm, p, &coneSize);
    1869:   if (!coneSize) {
    1870:     if (isTensor) *isTensor = PETSC_FALSE;
    1871:     if (endA) *endA = -1;
    1872:     if (endB) *endB = -1;
    1873:   }
    1874:   DMPlexGetCone(dm, p, &cone);
    1875:   for (c = 0; c < coneSize; c++) {
    1876:     PetscInt f = cone[c];
    1877:     PetscInt fConeSize;

    1879:     DMPlexGetConeSize(dm, f, &fConeSize);
    1880:     if (fConeSize != coneSize - 2) continue;

    1882:     for (c2 = c + 1; c2 < coneSize; c2++) {
    1883:       PetscInt  f2 = cone[c2];
    1884:       PetscBool isTensorff2;
    1885:       PetscInt f2ConeSize;

    1887:       DMPlexGetConeSize(dm, f2, &f2ConeSize);
    1888:       if (f2ConeSize != coneSize - 2) continue;

    1890:       DMPlexPointIsTensor_Internal_Given(dm, p, f, f2, &isTensorff2);
    1891:       if (isTensorff2) {
    1892:         if (isTensor) *isTensor = PETSC_TRUE;
    1893:         if (endA) *endA = f;
    1894:         if (endB) *endB = f2;
    1895:         return(0);
    1896:       }
    1897:     }
    1898:   }
    1899:   if (isTensor) *isTensor = PETSC_FALSE;
    1900:   if (endA) *endA = -1;
    1901:   if (endB) *endB = -1;
    1902:   return(0);
    1903: }

    1905: /* determine if a cell is a tensor with a segment by looping over pairs of facets to find a pair
    1906:  * that could be the opposite ends */
    1907: static PetscErrorCode DMPlexPointIsTensor(DM dm, PetscInt p, PetscBool *isTensor, PetscInt *endA, PetscInt *endB)
    1908: {
    1909:   DMPlexInterpolatedFlag interpolated;

    1913:   DMPlexIsInterpolated(dm, &interpolated);
    1914:   if (interpolated != DMPLEX_INTERPOLATED_FULL) SETERRQ(PetscObjectComm((PetscObject)dm), PETSC_ERR_ARG_WRONGSTATE, "Only for interpolated DMPlex's");
    1915:   DMPlexPointIsTensor_Internal(dm, p, isTensor, endA, endB);
    1916:   return(0);
    1917: }

    1919: /* permute a quadrature -> dof matrix so that its rows are in revlex order by nodeIdx */
    1920: static PetscErrorCode MatPermuteByNodeIdx(Mat A, PetscLagNodeIndices ni, Mat *Aperm)
    1921: {
    1922:   PetscInt       m, n, i, j;
    1923:   PetscInt       nodeIdxDim = ni->nodeIdxDim;
    1924:   PetscInt       nodeVecDim = ni->nodeVecDim;
    1925:   PetscInt       *perm;
    1926:   IS             permIS;
    1927:   IS             id;
    1928:   PetscInt       *nIdxPerm;
    1929:   PetscReal      *nVecPerm;

    1933:   PetscLagNodeIndicesGetPermutation(ni, &perm);
    1934:   MatGetSize(A, &m, &n);
    1935:   PetscMalloc1(nodeIdxDim * m, &nIdxPerm);
    1936:   PetscMalloc1(nodeVecDim * m, &nVecPerm);
    1937:   for (i = 0; i < m; i++) for (j = 0; j < nodeIdxDim; j++) nIdxPerm[i * nodeIdxDim + j] = ni->nodeIdx[perm[i] * nodeIdxDim + j];
    1938:   for (i = 0; i < m; i++) for (j = 0; j < nodeVecDim; j++) nVecPerm[i * nodeVecDim + j] = ni->nodeVec[perm[i] * nodeVecDim + j];
    1939:   ISCreateGeneral(PETSC_COMM_SELF, m, perm, PETSC_USE_POINTER, &permIS);
    1940:   ISSetPermutation(permIS);
    1941:   ISCreateStride(PETSC_COMM_SELF, n, 0, 1, &id);
    1942:   ISSetPermutation(id);
    1943:   MatPermute(A, permIS, id, Aperm);
    1944:   ISDestroy(&permIS);
    1945:   ISDestroy(&id);
    1946:   for (i = 0; i < m; i++) perm[i] = i;
    1947:   PetscFree(ni->nodeIdx);
    1948:   PetscFree(ni->nodeVec);
    1949:   ni->nodeIdx = nIdxPerm;
    1950:   ni->nodeVec = nVecPerm;
    1951:   return(0);
    1952: }

    1954: static PetscErrorCode PetscDualSpaceSetUp_Lagrange(PetscDualSpace sp)
    1955: {
    1956:   PetscDualSpace_Lag *lag   = (PetscDualSpace_Lag *) sp->data;
    1957:   DM                  dm    = sp->dm;
    1958:   DM                  dmint = NULL;
    1959:   PetscInt            order;
    1960:   PetscInt            Nc    = sp->Nc;
    1961:   MPI_Comm            comm;
    1962:   PetscBool           continuous;
    1963:   PetscSection        section;
    1964:   PetscInt            depth, dim, pStart, pEnd, cStart, cEnd, p, *pStratStart, *pStratEnd, d;
    1965:   PetscInt            formDegree, Nk, Ncopies;
    1966:   PetscInt            tensorf = -1, tensorf2 = -1;
    1967:   PetscBool           tensorCell, tensorSpace;
    1968:   PetscBool           uniform, trimmed;
    1969:   Petsc1DNodeFamily   nodeFamily;
    1970:   PetscInt            numNodeSkip;
    1971:   DMPlexInterpolatedFlag interpolated;
    1972:   PetscBool           isbdm;
    1973:   PetscErrorCode      ierr;

    1976:   /* step 1: sanitize input */
    1977:   PetscObjectGetComm((PetscObject) sp, &comm);
    1978:   DMGetDimension(dm, &dim);
    1979:   PetscObjectTypeCompare((PetscObject)sp, "bdm", &isbdm);
    1980:   if (isbdm) {
    1981:     sp->k = -(dim-1); /* form degree of H-div */
    1982:     PetscObjectChangeTypeName((PetscObject)sp, PETSCDUALSPACELAGRANGE);
    1983:   }
    1984:   PetscDualSpaceGetFormDegree(sp, &formDegree);
    1985:   if (PetscAbsInt(formDegree) > dim) SETERRQ(comm, PETSC_ERR_ARG_OUTOFRANGE, "Form degree must be bounded by dimension");
    1986:   PetscDTBinomialInt(dim,PetscAbsInt(formDegree),&Nk);
    1987:   if (sp->Nc <= 0 && lag->numCopies > 0) sp->Nc = Nk * lag->numCopies;
    1988:   Nc = sp->Nc;
    1989:   if (Nc % Nk) SETERRQ(comm, PETSC_ERR_ARG_INCOMP, "Number of components is not a multiple of form degree size");
    1990:   if (lag->numCopies <= 0) lag->numCopies = Nc / Nk;
    1991:   Ncopies = lag->numCopies;
    1992:   if (Nc / Nk != Ncopies) SETERRQ(comm, PETSC_ERR_ARG_INCOMP, "Number of copies * (dim choose k) != Nc");
    1993:   if (!dim) sp->order = 0;
    1994:   order = sp->order;
    1995:   uniform = sp->uniform;
    1996:   if (!uniform) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_SUP, "Variable order not supported yet");
    1997:   if (lag->trimmed && !formDegree) lag->trimmed = PETSC_FALSE; /* trimmed spaces are the same as full spaces for 0-forms */
    1998:   if (lag->nodeType == PETSCDTNODES_DEFAULT) {
    1999:     lag->nodeType = PETSCDTNODES_GAUSSJACOBI;
    2000:     lag->nodeExponent = 0.;
    2001:     /* trimmed spaces don't include corner vertices, so don't use end nodes by default */
    2002:     lag->endNodes = lag->trimmed ? PETSC_FALSE : PETSC_TRUE;
    2003:   }
    2004:   /* If a trimmed space and the user did choose nodes with endpoints, skip them by default */
    2005:   if (lag->numNodeSkip < 0) lag->numNodeSkip = (lag->trimmed && lag->endNodes) ? 1 : 0;
    2006:   numNodeSkip = lag->numNodeSkip;
    2007:   if (lag->trimmed && !order) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Cannot have zeroth order trimmed elements");
    2008:   if (lag->trimmed && PetscAbsInt(formDegree) == dim) { /* convert trimmed n-forms to untrimmed of one polynomial order less */
    2009:     sp->order--;
    2010:     order--;
    2011:     lag->trimmed = PETSC_FALSE;
    2012:   }
    2013:   trimmed = lag->trimmed;
    2014:   if (!order || PetscAbsInt(formDegree) == dim) lag->continuous = PETSC_FALSE;
    2015:   continuous = lag->continuous;
    2016:   DMPlexGetDepth(dm, &depth);
    2017:   DMPlexGetChart(dm, &pStart, &pEnd);
    2018:   DMPlexGetHeightStratum(dm, 0, &cStart, &cEnd);
    2019:   if (pStart != 0 || cStart != 0) SETERRQ(PetscObjectComm((PetscObject)sp), PETSC_ERR_ARG_WRONGSTATE, "Expect DM with chart starting at zero and cells first");
    2020:   if (cEnd != 1) SETERRQ(PetscObjectComm((PetscObject)sp), PETSC_ERR_ARG_WRONGSTATE, "Use PETSCDUALSPACEREFINED for multi-cell reference meshes");
    2021:   DMPlexIsInterpolated(dm, &interpolated);
    2022:   if (interpolated != DMPLEX_INTERPOLATED_FULL) {
    2023:     DMPlexInterpolate(dm, &dmint);
    2024:   } else {
    2025:     PetscObjectReference((PetscObject)dm);
    2026:     dmint = dm;
    2027:   }
    2028:   tensorCell = PETSC_FALSE;
    2029:   if (dim > 1) {
    2030:     DMPlexPointIsTensor(dmint, 0, &tensorCell, &tensorf, &tensorf2);
    2031:   }
    2032:   lag->tensorCell = tensorCell;
    2033:   if (dim < 2 || !lag->tensorCell) lag->tensorSpace = PETSC_FALSE;
    2034:   tensorSpace = lag->tensorSpace;
    2035:   if (!lag->nodeFamily) {
    2036:     Petsc1DNodeFamilyCreate(lag->nodeType, lag->nodeExponent, lag->endNodes, &lag->nodeFamily);
    2037:   }
    2038:   nodeFamily = lag->nodeFamily;
    2039:   if (interpolated != DMPLEX_INTERPOLATED_FULL && continuous && (PetscAbsInt(formDegree) > 0 || order > 1)) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_PLIB,"Reference element won't support all boundary nodes");

    2041:   /* step 2: construct the boundary spaces */
    2042:   PetscMalloc2(depth+1,&pStratStart,depth+1,&pStratEnd);
    2043:   PetscCalloc1(pEnd,&(sp->pointSpaces));
    2044:   for (d = 0; d <= depth; ++d) {DMPlexGetDepthStratum(dm, d, &pStratStart[d], &pStratEnd[d]);}
    2045:   PetscDualSpaceSectionCreate_Internal(sp, &section);
    2046:   sp->pointSection = section;
    2047:   if (continuous && !(lag->interiorOnly)) {
    2048:     PetscInt h;

    2050:     for (p = pStratStart[depth - 1]; p < pStratEnd[depth - 1]; p++) { /* calculate the facet dual spaces */
    2051:       PetscReal v0[3];
    2052:       DMPolytopeType ptype;
    2053:       PetscReal J[9], detJ;
    2054:       PetscInt  q;

    2056:       DMPlexComputeCellGeometryAffineFEM(dm, p, v0, J, NULL, &detJ);
    2057:       DMPlexGetCellType(dm, p, &ptype);

    2059:       /* compare to previous facets: if computed, reference that dualspace */
    2060:       for (q = pStratStart[depth - 1]; q < p; q++) {
    2061:         DMPolytopeType qtype;

    2063:         DMPlexGetCellType(dm, q, &qtype);
    2064:         if (qtype == ptype) break;
    2065:       }
    2066:       if (q < p) { /* this facet has the same dual space as that one */
    2067:         PetscObjectReference((PetscObject)sp->pointSpaces[q]);
    2068:         sp->pointSpaces[p] = sp->pointSpaces[q];
    2069:         continue;
    2070:       }
    2071:       /* if not, recursively compute this dual space */
    2072:       PetscDualSpaceCreateFacetSubspace_Lagrange(sp,NULL,p,formDegree,Ncopies,PETSC_FALSE,&sp->pointSpaces[p]);
    2073:     }
    2074:     for (h = 2; h <= depth; h++) { /* get the higher subspaces from the facet subspaces */
    2075:       PetscInt hd = depth - h;
    2076:       PetscInt hdim = dim - h;

    2078:       if (hdim < PetscAbsInt(formDegree)) break;
    2079:       for (p = pStratStart[hd]; p < pStratEnd[hd]; p++) {
    2080:         PetscInt suppSize, s;
    2081:         const PetscInt *supp;

    2083:         DMPlexGetSupportSize(dm, p, &suppSize);
    2084:         DMPlexGetSupport(dm, p, &supp);
    2085:         for (s = 0; s < suppSize; s++) {
    2086:           DM             qdm;
    2087:           PetscDualSpace qsp, psp;
    2088:           PetscInt c, coneSize, q;
    2089:           const PetscInt *cone;
    2090:           const PetscInt *refCone;

    2092:           q = supp[0];
    2093:           qsp = sp->pointSpaces[q];
    2094:           DMPlexGetConeSize(dm, q, &coneSize);
    2095:           DMPlexGetCone(dm, q, &cone);
    2096:           for (c = 0; c < coneSize; c++) if (cone[c] == p) break;
    2097:           if (c == coneSize) SETERRQ(PetscObjectComm((PetscObject)dm), PETSC_ERR_PLIB, "cone/suppport mismatch");
    2098:           PetscDualSpaceGetDM(qsp, &qdm);
    2099:           DMPlexGetCone(qdm, 0, &refCone);
    2100:           /* get the equivalent dual space from the support dual space */
    2101:           PetscDualSpaceGetPointSubspace(qsp, refCone[c], &psp);
    2102:           if (!s) {
    2103:             PetscObjectReference((PetscObject)psp);
    2104:             sp->pointSpaces[p] = psp;
    2105:           }
    2106:         }
    2107:       }
    2108:     }
    2109:     for (p = 1; p < pEnd; p++) {
    2110:       PetscInt pspdim;
    2111:       if (!sp->pointSpaces[p]) continue;
    2112:       PetscDualSpaceGetInteriorDimension(sp->pointSpaces[p], &pspdim);
    2113:       PetscSectionSetDof(section, p, pspdim);
    2114:     }
    2115:   }

    2117:   if (Ncopies > 1) {
    2118:     Mat intMatScalar, allMatScalar;
    2119:     PetscDualSpace scalarsp;
    2120:     PetscDualSpace_Lag *scalarlag;

    2122:     PetscDualSpaceDuplicate(sp, &scalarsp);
    2123:     /* Setting the number of components to Nk is a space with 1 copy of each k-form */
    2124:     PetscDualSpaceSetNumComponents(scalarsp, Nk);
    2125:     PetscDualSpaceSetUp(scalarsp);
    2126:     PetscDualSpaceGetInteriorData(scalarsp, &(sp->intNodes), &intMatScalar);
    2127:     PetscObjectReference((PetscObject)(sp->intNodes));
    2128:     if (intMatScalar) {PetscDualSpaceLagrangeMatrixCreateCopies(intMatScalar, Nk, Ncopies, &(sp->intMat));}
    2129:     PetscDualSpaceGetAllData(scalarsp, &(sp->allNodes), &allMatScalar);
    2130:     PetscObjectReference((PetscObject)(sp->allNodes));
    2131:     PetscDualSpaceLagrangeMatrixCreateCopies(allMatScalar, Nk, Ncopies, &(sp->allMat));
    2132:     sp->spdim = scalarsp->spdim * Ncopies;
    2133:     sp->spintdim = scalarsp->spintdim * Ncopies;
    2134:     scalarlag = (PetscDualSpace_Lag *) scalarsp->data;
    2135:     PetscLagNodeIndicesReference(scalarlag->vertIndices);
    2136:     lag->vertIndices = scalarlag->vertIndices;
    2137:     PetscLagNodeIndicesReference(scalarlag->intNodeIndices);
    2138:     lag->intNodeIndices = scalarlag->intNodeIndices;
    2139:     PetscLagNodeIndicesReference(scalarlag->allNodeIndices);
    2140:     lag->allNodeIndices = scalarlag->allNodeIndices;
    2141:     PetscDualSpaceDestroy(&scalarsp);
    2142:     PetscSectionSetDof(section, 0, sp->spintdim);
    2143:     PetscDualSpaceSectionSetUp_Internal(sp, section);
    2144:     PetscDualSpaceComputeFunctionalsFromAllData(sp);
    2145:     PetscFree2(pStratStart, pStratEnd);
    2146:     DMDestroy(&dmint);
    2147:     return(0);
    2148:   }

    2150:   if (trimmed && !continuous) {
    2151:     /* the dofs of a trimmed space don't have a nice tensor/lattice structure:
    2152:      * just construct the continuous dual space and copy all of the data over,
    2153:      * allocating it all to the cell instead of splitting it up between the boundaries */
    2154:     PetscDualSpace  spcont;
    2155:     PetscInt        spdim, f;
    2156:     PetscQuadrature allNodes;
    2157:     PetscDualSpace_Lag *lagc;
    2158:     Mat             allMat;

    2160:     PetscDualSpaceDuplicate(sp, &spcont);
    2161:     PetscDualSpaceLagrangeSetContinuity(spcont, PETSC_TRUE);
    2162:     PetscDualSpaceSetUp(spcont);
    2163:     PetscDualSpaceGetDimension(spcont, &spdim);
    2164:     sp->spdim = sp->spintdim = spdim;
    2165:     PetscSectionSetDof(section, 0, spdim);
    2166:     PetscDualSpaceSectionSetUp_Internal(sp, section);
    2167:     PetscMalloc1(spdim, &(sp->functional));
    2168:     for (f = 0; f < spdim; f++) {
    2169:       PetscQuadrature fn;

    2171:       PetscDualSpaceGetFunctional(spcont, f, &fn);
    2172:       PetscObjectReference((PetscObject)fn);
    2173:       sp->functional[f] = fn;
    2174:     }
    2175:     PetscDualSpaceGetAllData(spcont, &allNodes, &allMat);
    2176:     PetscObjectReference((PetscObject) allNodes);
    2177:     PetscObjectReference((PetscObject) allNodes);
    2178:     sp->allNodes = sp->intNodes = allNodes;
    2179:     PetscObjectReference((PetscObject) allMat);
    2180:     PetscObjectReference((PetscObject) allMat);
    2181:     sp->allMat = sp->intMat = allMat;
    2182:     lagc = (PetscDualSpace_Lag *) spcont->data;
    2183:     PetscLagNodeIndicesReference(lagc->vertIndices);
    2184:     lag->vertIndices = lagc->vertIndices;
    2185:     PetscLagNodeIndicesReference(lagc->allNodeIndices);
    2186:     PetscLagNodeIndicesReference(lagc->allNodeIndices);
    2187:     lag->intNodeIndices = lagc->allNodeIndices;
    2188:     lag->allNodeIndices = lagc->allNodeIndices;
    2189:     PetscDualSpaceDestroy(&spcont);
    2190:     PetscFree2(pStratStart, pStratEnd);
    2191:     DMDestroy(&dmint);
    2192:     return(0);
    2193:   }

    2195:   /* step 3: construct intNodes, and intMat, and combine it with boundray data to make allNodes and allMat */
    2196:   if (!tensorSpace) {
    2197:     if (!tensorCell) {PetscLagNodeIndicesCreateSimplexVertices(dm, &(lag->vertIndices));}

    2199:     if (trimmed) {
    2200:       /* there is one dof in the interior of the a trimmed element for each full polynomial of with degree at most
    2201:        * order + k - dim - 1 */
    2202:       if (order + PetscAbsInt(formDegree) > dim) {
    2203:         PetscInt sum = order + PetscAbsInt(formDegree) - dim - 1;
    2204:         PetscInt nDofs;

    2206:         PetscDualSpaceLagrangeCreateSimplexNodeMat(nodeFamily, dim, sum, Nk, numNodeSkip, &sp->intNodes, &sp->intMat, &(lag->intNodeIndices));
    2207:         MatGetSize(sp->intMat, &nDofs, NULL);
    2208:         PetscSectionSetDof(section, 0, nDofs);
    2209:       }
    2210:       PetscDualSpaceSectionSetUp_Internal(sp, section);
    2211:       PetscDualSpaceCreateAllDataFromInteriorData(sp);
    2212:       PetscDualSpaceLagrangeCreateAllNodeIdx(sp);
    2213:     } else {
    2214:       if (!continuous) {
    2215:         /* if discontinuous just construct one node for each set of dofs (a set of dofs is a basis for the k-form
    2216:          * space) */
    2217:         PetscInt sum = order;
    2218:         PetscInt nDofs;

    2220:         PetscDualSpaceLagrangeCreateSimplexNodeMat(nodeFamily, dim, sum, Nk, numNodeSkip, &sp->intNodes, &sp->intMat, &(lag->intNodeIndices));
    2221:         MatGetSize(sp->intMat, &nDofs, NULL);
    2222:         PetscSectionSetDof(section, 0, nDofs);
    2223:         PetscDualSpaceSectionSetUp_Internal(sp, section);
    2224:         PetscObjectReference((PetscObject)(sp->intNodes));
    2225:         sp->allNodes = sp->intNodes;
    2226:         PetscObjectReference((PetscObject)(sp->intMat));
    2227:         sp->allMat = sp->intMat;
    2228:         PetscLagNodeIndicesReference(lag->intNodeIndices);
    2229:         lag->allNodeIndices = lag->intNodeIndices;
    2230:       } else {
    2231:         /* there is one dof in the interior of the a full element for each trimmed polynomial of with degree at most
    2232:          * order + k - dim, but with complementary form degree */
    2233:         if (order + PetscAbsInt(formDegree) > dim) {
    2234:           PetscDualSpace trimmedsp;
    2235:           PetscDualSpace_Lag *trimmedlag;
    2236:           PetscQuadrature intNodes;
    2237:           PetscInt trFormDegree = formDegree >= 0 ? formDegree - dim : dim - PetscAbsInt(formDegree);
    2238:           PetscInt nDofs;
    2239:           Mat intMat;

    2241:           PetscDualSpaceDuplicate(sp, &trimmedsp);
    2242:           PetscDualSpaceLagrangeSetTrimmed(trimmedsp, PETSC_TRUE);
    2243:           PetscDualSpaceSetOrder(trimmedsp, order + PetscAbsInt(formDegree) - dim);
    2244:           PetscDualSpaceSetFormDegree(trimmedsp, trFormDegree);
    2245:           trimmedlag = (PetscDualSpace_Lag *) trimmedsp->data;
    2246:           trimmedlag->numNodeSkip = numNodeSkip + 1;
    2247:           PetscDualSpaceSetUp(trimmedsp);
    2248:           PetscDualSpaceGetAllData(trimmedsp, &intNodes, &intMat);
    2249:           PetscObjectReference((PetscObject)intNodes);
    2250:           sp->intNodes = intNodes;
    2251:           PetscObjectReference((PetscObject)intMat);
    2252:           sp->intMat = intMat;
    2253:           MatGetSize(sp->intMat, &nDofs, NULL);
    2254:           PetscLagNodeIndicesReference(trimmedlag->allNodeIndices);
    2255:           lag->intNodeIndices = trimmedlag->allNodeIndices;
    2256:           PetscDualSpaceDestroy(&trimmedsp);
    2257:           PetscSectionSetDof(section, 0, nDofs);
    2258:         }
    2259:         PetscDualSpaceSectionSetUp_Internal(sp, section);
    2260:         PetscDualSpaceCreateAllDataFromInteriorData(sp);
    2261:         PetscDualSpaceLagrangeCreateAllNodeIdx(sp);
    2262:       }
    2263:     }
    2264:   } else {
    2265:     PetscQuadrature intNodesTrace = NULL;
    2266:     PetscQuadrature intNodesFiber = NULL;
    2267:     PetscQuadrature intNodes = NULL;
    2268:     PetscLagNodeIndices intNodeIndices = NULL;
    2269:     Mat             intMat = NULL;

    2271:     if (PetscAbsInt(formDegree) < dim) { /* get the trace k-forms on the first facet, and the 0-forms on the edge,
    2272:                                             and wedge them together to create some of the k-form dofs */
    2273:       PetscDualSpace  trace, fiber;
    2274:       PetscDualSpace_Lag *tracel, *fiberl;
    2275:       Mat             intMatTrace, intMatFiber;

    2277:       if (sp->pointSpaces[tensorf]) {
    2278:         PetscObjectReference((PetscObject)(sp->pointSpaces[tensorf]));
    2279:         trace = sp->pointSpaces[tensorf];
    2280:       } else {
    2281:         PetscDualSpaceCreateFacetSubspace_Lagrange(sp,NULL,tensorf,formDegree,Ncopies,PETSC_TRUE,&trace);
    2282:       }
    2283:       PetscDualSpaceCreateEdgeSubspace_Lagrange(sp,order,0,1,PETSC_TRUE,&fiber);
    2284:       tracel = (PetscDualSpace_Lag *) trace->data;
    2285:       fiberl = (PetscDualSpace_Lag *) fiber->data;
    2286:       PetscLagNodeIndicesCreateTensorVertices(dm, tracel->vertIndices, &(lag->vertIndices));
    2287:       PetscDualSpaceGetInteriorData(trace, &intNodesTrace, &intMatTrace);
    2288:       PetscDualSpaceGetInteriorData(fiber, &intNodesFiber, &intMatFiber);
    2289:       if (intNodesTrace && intNodesFiber) {
    2290:         PetscQuadratureCreateTensor(intNodesTrace, intNodesFiber, &intNodes);
    2291:         MatTensorAltV(intMatTrace, intMatFiber, dim-1, formDegree, 1, 0, &intMat);
    2292:         PetscLagNodeIndicesTensor(tracel->intNodeIndices, dim - 1, formDegree, fiberl->intNodeIndices, 1, 0, &intNodeIndices);
    2293:       }
    2294:       PetscObjectReference((PetscObject) intNodesTrace);
    2295:       PetscObjectReference((PetscObject) intNodesFiber);
    2296:       PetscDualSpaceDestroy(&fiber);
    2297:       PetscDualSpaceDestroy(&trace);
    2298:     }
    2299:     if (PetscAbsInt(formDegree) > 0) { /* get the trace (k-1)-forms on the first facet, and the 1-forms on the edge,
    2300:                                           and wedge them together to create the remaining k-form dofs */
    2301:       PetscDualSpace  trace, fiber;
    2302:       PetscDualSpace_Lag *tracel, *fiberl;
    2303:       PetscQuadrature intNodesTrace2, intNodesFiber2, intNodes2;
    2304:       PetscLagNodeIndices intNodeIndices2;
    2305:       Mat             intMatTrace, intMatFiber, intMat2;
    2306:       PetscInt        traceDegree = formDegree > 0 ? formDegree - 1 : formDegree + 1;
    2307:       PetscInt        fiberDegree = formDegree > 0 ? 1 : -1;

    2309:       PetscDualSpaceCreateFacetSubspace_Lagrange(sp,NULL,tensorf,traceDegree,Ncopies,PETSC_TRUE,&trace);
    2310:       PetscDualSpaceCreateEdgeSubspace_Lagrange(sp,order,fiberDegree,1,PETSC_TRUE,&fiber);
    2311:       tracel = (PetscDualSpace_Lag *) trace->data;
    2312:       fiberl = (PetscDualSpace_Lag *) fiber->data;
    2313:       if (!lag->vertIndices) {
    2314:         PetscLagNodeIndicesCreateTensorVertices(dm, tracel->vertIndices, &(lag->vertIndices));
    2315:       }
    2316:       PetscDualSpaceGetInteriorData(trace, &intNodesTrace2, &intMatTrace);
    2317:       PetscDualSpaceGetInteriorData(fiber, &intNodesFiber2, &intMatFiber);
    2318:       if (intNodesTrace2 && intNodesFiber2) {
    2319:         PetscQuadratureCreateTensor(intNodesTrace2, intNodesFiber2, &intNodes2);
    2320:         MatTensorAltV(intMatTrace, intMatFiber, dim-1, traceDegree, 1, fiberDegree, &intMat2);
    2321:         PetscLagNodeIndicesTensor(tracel->intNodeIndices, dim - 1, traceDegree, fiberl->intNodeIndices, 1, fiberDegree, &intNodeIndices2);
    2322:         if (!intMat) {
    2323:           intMat = intMat2;
    2324:           intNodes = intNodes2;
    2325:           intNodeIndices = intNodeIndices2;
    2326:         } else {
    2327:           /* merge the matrices, quadrature points, and nodes */
    2328:           PetscInt         nM;
    2329:           PetscInt         nDof, nDof2;
    2330:           PetscInt        *toMerged = NULL, *toMerged2 = NULL;
    2331:           PetscQuadrature  merged = NULL;
    2332:           PetscLagNodeIndices intNodeIndicesMerged = NULL;
    2333:           Mat              matMerged = NULL;

    2335:           MatGetSize(intMat, &nDof, 0);
    2336:           MatGetSize(intMat2, &nDof2, 0);
    2337:           PetscQuadraturePointsMerge(intNodes, intNodes2, &merged, &toMerged, &toMerged2);
    2338:           PetscQuadratureGetData(merged, NULL, NULL, &nM, NULL, NULL);
    2339:           MatricesMerge(intMat, intMat2, dim, formDegree, nM, toMerged, toMerged2, &matMerged);
    2340:           PetscLagNodeIndicesMerge(intNodeIndices, intNodeIndices2, &intNodeIndicesMerged);
    2341:           PetscFree(toMerged);
    2342:           PetscFree(toMerged2);
    2343:           MatDestroy(&intMat);
    2344:           MatDestroy(&intMat2);
    2345:           PetscQuadratureDestroy(&intNodes);
    2346:           PetscQuadratureDestroy(&intNodes2);
    2347:           PetscLagNodeIndicesDestroy(&intNodeIndices);
    2348:           PetscLagNodeIndicesDestroy(&intNodeIndices2);
    2349:           intNodes = merged;
    2350:           intMat = matMerged;
    2351:           intNodeIndices = intNodeIndicesMerged;
    2352:           if (!trimmed) {
    2353:             /* I think users expect that, when a node has a full basis for the k-forms,
    2354:              * they should be consecutive dofs.  That isn't the case for trimmed spaces,
    2355:              * but is for some of the nodes in untrimmed spaces, so in that case we
    2356:              * sort them to group them by node */
    2357:             Mat intMatPerm;

    2359:             MatPermuteByNodeIdx(intMat, intNodeIndices, &intMatPerm);
    2360:             MatDestroy(&intMat);
    2361:             intMat = intMatPerm;
    2362:           }
    2363:         }
    2364:       }
    2365:       PetscDualSpaceDestroy(&fiber);
    2366:       PetscDualSpaceDestroy(&trace);
    2367:     }
    2368:     PetscQuadratureDestroy(&intNodesTrace);
    2369:     PetscQuadratureDestroy(&intNodesFiber);
    2370:     sp->intNodes = intNodes;
    2371:     sp->intMat = intMat;
    2372:     lag->intNodeIndices = intNodeIndices;
    2373:     {
    2374:       PetscInt nDofs = 0;

    2376:       if (intMat) {
    2377:         MatGetSize(intMat, &nDofs, NULL);
    2378:       }
    2379:       PetscSectionSetDof(section, 0, nDofs);
    2380:     }
    2381:     PetscDualSpaceSectionSetUp_Internal(sp, section);
    2382:     if (continuous) {
    2383:       PetscDualSpaceCreateAllDataFromInteriorData(sp);
    2384:       PetscDualSpaceLagrangeCreateAllNodeIdx(sp);
    2385:     } else {
    2386:       PetscObjectReference((PetscObject) intNodes);
    2387:       sp->allNodes = intNodes;
    2388:       PetscObjectReference((PetscObject) intMat);
    2389:       sp->allMat = intMat;
    2390:       PetscLagNodeIndicesReference(intNodeIndices);
    2391:       lag->allNodeIndices = intNodeIndices;
    2392:     }
    2393:   }
    2394:   PetscSectionGetStorageSize(section, &sp->spdim);
    2395:   PetscSectionGetConstrainedStorageSize(section, &sp->spintdim);
    2396:   PetscDualSpaceComputeFunctionalsFromAllData(sp);
    2397:   PetscFree2(pStratStart, pStratEnd);
    2398:   DMDestroy(&dmint);
    2399:   return(0);
    2400: }

    2402: /* Create a matrix that represents the transformation that DMPlexVecGetClosure() would need
    2403:  * to get the representation of the dofs for a mesh point if the mesh point had this orientation
    2404:  * relative to the cell */
    2405: PetscErrorCode PetscDualSpaceCreateInteriorSymmetryMatrix_Lagrange(PetscDualSpace sp, PetscInt ornt, Mat *symMat)
    2406: {
    2407:   PetscDualSpace_Lag *lag;
    2408:   DM dm;
    2409:   PetscLagNodeIndices vertIndices, intNodeIndices;
    2410:   PetscLagNodeIndices ni;
    2411:   PetscInt nodeIdxDim, nodeVecDim, nNodes;
    2412:   PetscInt formDegree;
    2413:   PetscInt *perm, *permOrnt;
    2414:   PetscInt *nnz;
    2415:   PetscInt n;
    2416:   PetscInt maxGroupSize;
    2417:   PetscScalar *V, *W, *work;
    2418:   Mat A;

    2422:   if (!sp->spintdim) {
    2423:     *symMat = NULL;
    2424:     return(0);
    2425:   }
    2426:   lag = (PetscDualSpace_Lag *) sp->data;
    2427:   vertIndices = lag->vertIndices;
    2428:   intNodeIndices = lag->intNodeIndices;
    2429:   PetscDualSpaceGetDM(sp, &dm);
    2430:   PetscDualSpaceGetFormDegree(sp, &formDegree);
    2431:   PetscNew(&ni);
    2432:   ni->refct = 1;
    2433:   ni->nodeIdxDim = nodeIdxDim = intNodeIndices->nodeIdxDim;
    2434:   ni->nodeVecDim = nodeVecDim = intNodeIndices->nodeVecDim;
    2435:   ni->nNodes = nNodes = intNodeIndices->nNodes;
    2436:   PetscMalloc1(nNodes * nodeIdxDim, &(ni->nodeIdx));
    2437:   PetscMalloc1(nNodes * nodeVecDim, &(ni->nodeVec));
    2438:   /* push forward the dofs by the symmetry of the reference element induced by ornt */
    2439:   PetscLagNodeIndicesPushForward(dm, vertIndices, 0, vertIndices, intNodeIndices, ornt, formDegree, ni->nodeIdx, ni->nodeVec);
    2440:   /* get the revlex order for both the original and transformed dofs */
    2441:   PetscLagNodeIndicesGetPermutation(intNodeIndices, &perm);
    2442:   PetscLagNodeIndicesGetPermutation(ni, &permOrnt);
    2443:   PetscMalloc1(nNodes, &nnz);
    2444:   for (n = 0, maxGroupSize = 0; n < nNodes;) { /* incremented in the loop */
    2445:     PetscInt *nind = &(ni->nodeIdx[permOrnt[n] * nodeIdxDim]);
    2446:     PetscInt m, nEnd;
    2447:     PetscInt groupSize;
    2448:     /* for each group of dofs that have the same nodeIdx coordinate */
    2449:     for (nEnd = n + 1; nEnd < nNodes; nEnd++) {
    2450:       PetscInt *mind = &(ni->nodeIdx[permOrnt[nEnd] * nodeIdxDim]);
    2451:       PetscInt d;

    2453:       /* compare the oriented permutation indices */
    2454:       for (d = 0; d < nodeIdxDim; d++) if (mind[d] != nind[d]) break;
    2455:       if (d < nodeIdxDim) break;
    2456:     }
    2457:     /* permOrnt[[n, nEnd)] is a group of dofs that, under the symmetry are at the same location */
    2458: #if defined(PETSC_USE_DEBUG)
    2459:     /* the symmetry had better map the group of dofs with the same permuted nodeIdx
    2460:      * to a group of dofs with the same size, otherwise we messed up */
    2461:     {
    2462:       PetscInt m;
    2463:       PetscInt *nind = &(intNodeIndices->nodeIdx[perm[n] * nodeIdxDim]);

    2465:       for (m = n + 1; m < nEnd; m++) {
    2466:         PetscInt *mind = &(intNodeIndices->nodeIdx[perm[m] * nodeIdxDim]);
    2467:         PetscInt d;

    2469:         /* compare the oriented permutation indices */
    2470:         for (d = 0; d < nodeIdxDim; d++) if (mind[d] != nind[d]) break;
    2471:         if (d < nodeIdxDim) break;
    2472:       }
    2473:       if (m < nEnd) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_PLIB, "Dofs with same index after symmetry not same block size");
    2474:     }
    2475: #endif
    2476:     groupSize = nEnd - n;
    2477:     /* each pushforward dof vector will be expressed in a basis of the unpermuted dofs */
    2478:     for (m = n; m < nEnd; m++) nnz[permOrnt[m]] = groupSize;

    2480:     maxGroupSize = PetscMax(maxGroupSize, nEnd - n);
    2481:     n = nEnd;
    2482:   }
    2483:   if (maxGroupSize > nodeVecDim) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_PLIB, "Dofs are not in blocks that can be solved");
    2484:   MatCreateSeqAIJ(PETSC_COMM_SELF, nNodes, nNodes, 0, nnz, &A);
    2485:   PetscFree(nnz);
    2486:   PetscMalloc3(maxGroupSize * nodeVecDim, &V, maxGroupSize * nodeVecDim, &W, nodeVecDim * 2, &work);
    2487:   for (n = 0; n < nNodes;) { /* incremented in the loop */
    2488:     PetscInt *nind = &(ni->nodeIdx[permOrnt[n] * nodeIdxDim]);
    2489:     PetscInt nEnd;
    2490:     PetscInt m;
    2491:     PetscInt groupSize;
    2492:     for (nEnd = n + 1; nEnd < nNodes; nEnd++) {
    2493:       PetscInt *mind = &(ni->nodeIdx[permOrnt[nEnd] * nodeIdxDim]);
    2494:       PetscInt d;

    2496:       /* compare the oriented permutation indices */
    2497:       for (d = 0; d < nodeIdxDim; d++) if (mind[d] != nind[d]) break;
    2498:       if (d < nodeIdxDim) break;
    2499:     }
    2500:     groupSize = nEnd - n;
    2501:     /* get all of the vectors from the original and all of the pushforward vectors */
    2502:     for (m = n; m < nEnd; m++) {
    2503:       PetscInt d;

    2505:       for (d = 0; d < nodeVecDim; d++) {
    2506:         V[(m - n) * nodeVecDim + d] = intNodeIndices->nodeVec[perm[m] * nodeVecDim + d];
    2507:         W[(m - n) * nodeVecDim + d] = ni->nodeVec[permOrnt[m] * nodeVecDim + d];
    2508:       }
    2509:     }
    2510:     /* now we have to solve for W in terms of V: the systems isn't always square, but the span
    2511:      * of V and W should always be the same, so the solution of the normal equations works */
    2512:     {
    2513:       char transpose = 'N';
    2514:       PetscBLASInt bm = nodeVecDim;
    2515:       PetscBLASInt bn = groupSize;
    2516:       PetscBLASInt bnrhs = groupSize;
    2517:       PetscBLASInt blda = bm;
    2518:       PetscBLASInt bldb = bm;
    2519:       PetscBLASInt blwork = 2 * nodeVecDim;
    2520:       PetscBLASInt info;

    2522:       PetscStackCallBLAS("LAPACKgels",LAPACKgels_(&transpose,&bm,&bn,&bnrhs,V,&blda,W,&bldb,work,&blwork, &info));
    2523:       if (info != 0) SETERRQ(PETSC_COMM_SELF,PETSC_ERR_LIB,"Bad argument to GELS");
    2524:       /* repack */
    2525:       {
    2526:         PetscInt i, j;

    2528:         for (i = 0; i < groupSize; i++) {
    2529:           for (j = 0; j < groupSize; j++) {
    2530:             /* notice the different leading dimension */
    2531:             V[i * groupSize + j] = W[i * nodeVecDim + j];
    2532:           }
    2533:         }
    2534:       }
    2535:     }
    2536:     MatSetValues(A, groupSize, &permOrnt[n], groupSize, &perm[n], V, INSERT_VALUES);
    2537:     n = nEnd;
    2538:   }
    2539:   MatAssemblyBegin(A, MAT_FINAL_ASSEMBLY);
    2540:   MatAssemblyEnd(A, MAT_FINAL_ASSEMBLY);
    2541:   *symMat = A;
    2542:   PetscFree3(V,W,work);
    2543:   PetscLagNodeIndicesDestroy(&ni);
    2544:   return(0);
    2545: }

    2547: #define BaryIndex(perEdge,a,b,c) (((b)*(2*perEdge+1-(b)))/2)+(c)

    2549: #define CartIndex(perEdge,a,b) (perEdge*(a)+b)

    2551: /* the existing interface for symmetries is insufficient for all cases:
    2552:  * - it should be sufficient for form degrees that are scalar (0 and n)
    2553:  * - it should be sufficient for hypercube dofs
    2554:  * - it isn't sufficient for simplex cells with non-scalar form degrees if
    2555:  *   there are any dofs in the interior
    2556:  *
    2557:  * We compute the general transformation matrices, and if they fit, we return them,
    2558:  * otherwise we error (but we should probably change the interface to allow for
    2559:  * these symmetries)
    2560:  */
    2561: static PetscErrorCode PetscDualSpaceGetSymmetries_Lagrange(PetscDualSpace sp, const PetscInt ****perms, const PetscScalar ****flips)
    2562: {
    2563:   PetscDualSpace_Lag *lag = (PetscDualSpace_Lag *) sp->data;
    2564:   PetscInt           dim, order, Nc;
    2565:   PetscErrorCode     ierr;

    2568:   PetscDualSpaceGetOrder(sp,&order);
    2569:   PetscDualSpaceGetNumComponents(sp,&Nc);
    2570:   DMGetDimension(sp->dm,&dim);
    2571:   if (!lag->symComputed) { /* store symmetries */
    2572:     PetscInt       pStart, pEnd, p;
    2573:     PetscInt       numPoints;
    2574:     PetscInt       numFaces;
    2575:     PetscInt       spintdim;
    2576:     PetscInt       ***symperms;
    2577:     PetscScalar    ***symflips;

    2579:     DMPlexGetChart(sp->dm, &pStart, &pEnd);
    2580:     numPoints = pEnd - pStart;
    2581:     DMPlexGetConeSize(sp->dm, 0, &numFaces);
    2582:     PetscCalloc1(numPoints,&symperms);
    2583:     PetscCalloc1(numPoints,&symflips);
    2584:     spintdim = sp->spintdim;
    2585:     /* The nodal symmetry behavior is not present when tensorSpace != tensorCell: someone might want this for the "S"
    2586:      * family of FEEC spaces.  Most used in particular are discontinuous polynomial L2 spaces in tensor cells, where
    2587:      * the symmetries are not necessary for FE assembly.  So for now we assume this is the case and don't return
    2588:      * symmetries if tensorSpace != tensorCell */
    2589:     if (spintdim && 0 < dim && dim < 3 && (lag->tensorSpace == lag->tensorCell)) { /* compute self symmetries */
    2590:       PetscInt **cellSymperms;
    2591:       PetscScalar **cellSymflips;
    2592:       PetscInt ornt;
    2593:       PetscInt nCopies = Nc / lag->intNodeIndices->nodeVecDim;
    2594:       PetscInt nNodes = lag->intNodeIndices->nNodes;

    2596:       lag->numSelfSym = 2 * numFaces;
    2597:       lag->selfSymOff = numFaces;
    2598:       PetscCalloc1(2*numFaces,&cellSymperms);
    2599:       PetscCalloc1(2*numFaces,&cellSymflips);
    2600:       /* we want to be able to index symmetries directly with the orientations, which range from [-numFaces,numFaces) */
    2601:       symperms[0] = &cellSymperms[numFaces];
    2602:       symflips[0] = &cellSymflips[numFaces];
    2603:       if (lag->intNodeIndices->nodeVecDim * nCopies != Nc) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_PLIB, "Node indices incompatible with dofs");
    2604:       if (nNodes * nCopies != spintdim) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_PLIB, "Node indices incompatible with dofs");
    2605:       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 */
    2606:         Mat symMat;
    2607:         PetscInt *perm;
    2608:         PetscScalar *flips;
    2609:         PetscInt i;

    2611:         if (!ornt) continue;
    2612:         PetscMalloc1(spintdim, &perm);
    2613:         PetscCalloc1(spintdim, &flips);
    2614:         for (i = 0; i < spintdim; i++) perm[i] = -1;
    2615:         PetscDualSpaceCreateInteriorSymmetryMatrix_Lagrange(sp, ornt, &symMat);
    2616:         for (i = 0; i < nNodes; i++) {
    2617:           PetscInt ncols;
    2618:           PetscInt j, k;
    2619:           const PetscInt *cols;
    2620:           const PetscScalar *vals;
    2621:           PetscBool nz_seen = PETSC_FALSE;

    2623:           MatGetRow(symMat, i, &ncols, &cols, &vals);
    2624:           for (j = 0; j < ncols; j++) {
    2625:             if (PetscAbsScalar(vals[j]) > PETSC_SMALL) {
    2626:               if (nz_seen) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_INCOMP, "This dual space has symmetries that can't be described as a permutation + sign flips");
    2627:               nz_seen = PETSC_TRUE;
    2628:               if (PetscAbsReal(PetscAbsScalar(vals[j]) - PetscRealConstant(1.)) > PETSC_SMALL) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_INCOMP, "This dual space has symmetries that can't be described as a permutation + sign flips");
    2629:               if (PetscAbsReal(PetscImaginaryPart(vals[j])) > PETSC_SMALL) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_INCOMP, "This dual space has symmetries that can't be described as a permutation + sign flips");
    2630:               if (perm[cols[j] * nCopies] >= 0) SETERRQ(PETSC_COMM_SELF, PETSC_ERR_ARG_INCOMP, "This dual space has symmetries that can't be described as a permutation + sign flips");
    2631:               for (k = 0; k < nCopies; k++) {
    2632:                 perm[cols[j] * nCopies + k] = i * nCopies + k;
    2633:               }
    2634:               if (PetscRealPart(vals[j]) < 0.) {
    2635:                 for (k = 0; k < nCopies; k++) {
    2636:                   flips[i * nCopies + k] = -1.;
    2637:                 }
    2638:               } else {
    2639:                 for (k = 0; k < nCopies; k++) {
    2640:                   flips[i * nCopies + k] = 1.;
    2641:                 }
    2642:               }
    2643:             }
    2644:           }
    2645:           MatRestoreRow(symMat, i, &ncols, &cols, &vals);
    2646:         }
    2647:         MatDestroy(&symMat);
    2648:         /* if there were no sign flips, keep NULL */
    2649:         for (i = 0; i < spintdim; i++) if (flips[i] != 1.) break;
    2650:         if (i == spintdim) {
    2651:           PetscFree(flips);
    2652:           flips = NULL;
    2653:         }
    2654:         /* if the permutation is identity, keep NULL */
    2655:         for (i = 0; i < spintdim; i++) if (perm[i] != i) break;
    2656:         if (i == spintdim) {
    2657:           PetscFree(perm);
    2658:           perm = NULL;
    2659:         }
    2660:         symperms[0][ornt] = perm;
    2661:         symflips[0][ornt] = flips;
    2662:       }
    2663:       /* if no orientations produced non-identity permutations, keep NULL */
    2664:       for (ornt = -numFaces; ornt < numFaces; ornt++) if (symperms[0][ornt]) break;
    2665:       if (ornt == numFaces) {
    2666:         PetscFree(cellSymperms);
    2667:         symperms[0] = NULL;
    2668:       }
    2669:       /* if no orientations produced sign flips, keep NULL */
    2670:       for (ornt = -numFaces; ornt < numFaces; ornt++) if (symflips[0][ornt]) break;
    2671:       if (ornt == numFaces) {
    2672:         PetscFree(cellSymflips);
    2673:         symflips[0] = NULL;
    2674:       }
    2675:     }
    2676:     { /* get the symmetries of closure points */
    2677:       PetscInt closureSize = 0;
    2678:       PetscInt *closure = NULL;
    2679:       PetscInt r;

    2681:       DMPlexGetTransitiveClosure(sp->dm,0,PETSC_TRUE,&closureSize,&closure);
    2682:       for (r = 0; r < closureSize; r++) {
    2683:         PetscDualSpace psp;
    2684:         PetscInt point = closure[2 * r];
    2685:         PetscInt pspintdim;
    2686:         const PetscInt ***psymperms = NULL;
    2687:         const PetscScalar ***psymflips = NULL;

    2689:         if (!point) continue;
    2690:         PetscDualSpaceGetPointSubspace(sp, point, &psp);
    2691:         if (!psp) continue;
    2692:         PetscDualSpaceGetInteriorDimension(psp, &pspintdim);
    2693:         if (!pspintdim) continue;
    2694:         PetscDualSpaceGetSymmetries(psp,&psymperms,&psymflips);
    2695:         symperms[r] = (PetscInt **) (psymperms ? psymperms[0] : NULL);
    2696:         symflips[r] = (PetscScalar **) (psymflips ? psymflips[0] : NULL);
    2697:       }
    2698:       DMPlexRestoreTransitiveClosure(sp->dm,0,PETSC_TRUE,&closureSize,&closure);
    2699:     }
    2700:     for (p = 0; p < pEnd; p++) if (symperms[p]) break;
    2701:     if (p == pEnd) {
    2702:       PetscFree(symperms);
    2703:       symperms = NULL;
    2704:     }
    2705:     for (p = 0; p < pEnd; p++) if (symflips[p]) break;
    2706:     if (p == pEnd) {
    2707:       PetscFree(symflips);
    2708:       symflips = NULL;
    2709:     }
    2710:     lag->symperms = symperms;
    2711:     lag->symflips = symflips;
    2712:     lag->symComputed = PETSC_TRUE;
    2713:   }
    2714:   if (perms) *perms = (const PetscInt ***) lag->symperms;
    2715:   if (flips) *flips = (const PetscScalar ***) lag->symflips;
    2716:   return(0);
    2717: }

    2719: static PetscErrorCode PetscDualSpaceLagrangeGetContinuity_Lagrange(PetscDualSpace sp, PetscBool *continuous)
    2720: {
    2721:   PetscDualSpace_Lag *lag = (PetscDualSpace_Lag *) sp->data;

    2726:   *continuous = lag->continuous;
    2727:   return(0);
    2728: }

    2730: static PetscErrorCode PetscDualSpaceLagrangeSetContinuity_Lagrange(PetscDualSpace sp, PetscBool continuous)
    2731: {
    2732:   PetscDualSpace_Lag *lag = (PetscDualSpace_Lag *) sp->data;

    2736:   lag->continuous = continuous;
    2737:   return(0);
    2738: }

    2740: /*@
    2741:   PetscDualSpaceLagrangeGetContinuity - Retrieves the flag for element continuity

    2743:   Not Collective

    2745:   Input Parameter:
    2746: . sp         - the PetscDualSpace

    2748:   Output Parameter:
    2749: . continuous - flag for element continuity

    2751:   Level: intermediate

    2753: .seealso: PetscDualSpaceLagrangeSetContinuity()
    2754: @*/
    2755: PetscErrorCode PetscDualSpaceLagrangeGetContinuity(PetscDualSpace sp, PetscBool *continuous)
    2756: {

    2762:   PetscTryMethod(sp, "PetscDualSpaceLagrangeGetContinuity_C", (PetscDualSpace,PetscBool*),(sp,continuous));
    2763:   return(0);
    2764: }

    2766: /*@
    2767:   PetscDualSpaceLagrangeSetContinuity - Indicate whether the element is continuous

    2769:   Logically Collective on sp

    2771:   Input Parameters:
    2772: + sp         - the PetscDualSpace
    2773: - continuous - flag for element continuity

    2775:   Options Database:
    2776: . -petscdualspace_lagrange_continuity <bool>

    2778:   Level: intermediate

    2780: .seealso: PetscDualSpaceLagrangeGetContinuity()
    2781: @*/
    2782: PetscErrorCode PetscDualSpaceLagrangeSetContinuity(PetscDualSpace sp, PetscBool continuous)
    2783: {

    2789:   PetscTryMethod(sp, "PetscDualSpaceLagrangeSetContinuity_C", (PetscDualSpace,PetscBool),(sp,continuous));
    2790:   return(0);
    2791: }

    2793: static PetscErrorCode PetscDualSpaceLagrangeGetTensor_Lagrange(PetscDualSpace sp, PetscBool *tensor)
    2794: {
    2795:   PetscDualSpace_Lag *lag = (PetscDualSpace_Lag *)sp->data;

    2798:   *tensor = lag->tensorSpace;
    2799:   return(0);
    2800: }

    2802: static PetscErrorCode PetscDualSpaceLagrangeSetTensor_Lagrange(PetscDualSpace sp, PetscBool tensor)
    2803: {
    2804:   PetscDualSpace_Lag *lag = (PetscDualSpace_Lag *)sp->data;

    2807:   lag->tensorSpace = tensor;
    2808:   return(0);
    2809: }

    2811: static PetscErrorCode PetscDualSpaceLagrangeGetTrimmed_Lagrange(PetscDualSpace sp, PetscBool *trimmed)
    2812: {
    2813:   PetscDualSpace_Lag *lag = (PetscDualSpace_Lag *)sp->data;

    2816:   *trimmed = lag->trimmed;
    2817:   return(0);
    2818: }

    2820: static PetscErrorCode PetscDualSpaceLagrangeSetTrimmed_Lagrange(PetscDualSpace sp, PetscBool trimmed)
    2821: {
    2822:   PetscDualSpace_Lag *lag = (PetscDualSpace_Lag *)sp->data;

    2825:   lag->trimmed = trimmed;
    2826:   return(0);
    2827: }

    2829: static PetscErrorCode PetscDualSpaceLagrangeGetNodeType_Lagrange(PetscDualSpace sp, PetscDTNodeType *nodeType, PetscBool *boundary, PetscReal *exponent)
    2830: {
    2831:   PetscDualSpace_Lag *lag = (PetscDualSpace_Lag *)sp->data;

    2834:   if (nodeType) *nodeType = lag->nodeType;
    2835:   if (boundary) *boundary = lag->endNodes;
    2836:   if (exponent) *exponent = lag->nodeExponent;
    2837:   return(0);
    2838: }

    2840: static PetscErrorCode PetscDualSpaceLagrangeSetNodeType_Lagrange(PetscDualSpace sp, PetscDTNodeType nodeType, PetscBool boundary, PetscReal exponent)
    2841: {
    2842:   PetscDualSpace_Lag *lag = (PetscDualSpace_Lag *)sp->data;

    2845:   if (nodeType == PETSCDTNODES_GAUSSJACOBI && exponent <= -1.) SETERRQ(PetscObjectComm((PetscObject) sp), PETSC_ERR_ARG_OUTOFRANGE, "Exponent must be > -1");
    2846:   lag->nodeType = nodeType;
    2847:   lag->endNodes = boundary;
    2848:   lag->nodeExponent = exponent;
    2849:   return(0);
    2850: }

    2852: /*@
    2853:   PetscDualSpaceLagrangeGetTensor - Get the tensor nature of the dual space

    2855:   Not collective

    2857:   Input Parameter:
    2858: . sp - The PetscDualSpace

    2860:   Output Parameter:
    2861: . tensor - Whether the dual space has tensor layout (vs. simplicial)

    2863:   Level: intermediate

    2865: .seealso: PetscDualSpaceLagrangeSetTensor(), PetscDualSpaceCreate()
    2866: @*/
    2867: PetscErrorCode PetscDualSpaceLagrangeGetTensor(PetscDualSpace sp, PetscBool *tensor)
    2868: {

    2874:   PetscTryMethod(sp,"PetscDualSpaceLagrangeGetTensor_C",(PetscDualSpace,PetscBool *),(sp,tensor));
    2875:   return(0);
    2876: }

    2878: /*@
    2879:   PetscDualSpaceLagrangeSetTensor - Set the tensor nature of the dual space

    2881:   Not collective

    2883:   Input Parameters:
    2884: + sp - The PetscDualSpace
    2885: - tensor - Whether the dual space has tensor layout (vs. simplicial)

    2887:   Level: intermediate

    2889: .seealso: PetscDualSpaceLagrangeGetTensor(), PetscDualSpaceCreate()
    2890: @*/
    2891: PetscErrorCode PetscDualSpaceLagrangeSetTensor(PetscDualSpace sp, PetscBool tensor)
    2892: {

    2897:   PetscTryMethod(sp,"PetscDualSpaceLagrangeSetTensor_C",(PetscDualSpace,PetscBool),(sp,tensor));
    2898:   return(0);
    2899: }

    2901: /*@
    2902:   PetscDualSpaceLagrangeGetTrimmed - Get the trimmed nature of the dual space

    2904:   Not collective

    2906:   Input Parameter:
    2907: . sp - The PetscDualSpace

    2909:   Output Parameter:
    2910: . 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)

    2912:   Level: intermediate

    2914: .seealso: PetscDualSpaceLagrangeSetTrimmed(), PetscDualSpaceCreate()
    2915: @*/
    2916: PetscErrorCode PetscDualSpaceLagrangeGetTrimmed(PetscDualSpace sp, PetscBool *trimmed)
    2917: {

    2923:   PetscTryMethod(sp,"PetscDualSpaceLagrangeGetTrimmed_C",(PetscDualSpace,PetscBool *),(sp,trimmed));
    2924:   return(0);
    2925: }

    2927: /*@
    2928:   PetscDualSpaceLagrangeSetTrimmed - Set the trimmed nature of the dual space

    2930:   Not collective

    2932:   Input Parameters:
    2933: + sp - The PetscDualSpace
    2934: - 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)

    2936:   Level: intermediate

    2938: .seealso: PetscDualSpaceLagrangeGetTrimmed(), PetscDualSpaceCreate()
    2939: @*/
    2940: PetscErrorCode PetscDualSpaceLagrangeSetTrimmed(PetscDualSpace sp, PetscBool trimmed)
    2941: {

    2946:   PetscTryMethod(sp,"PetscDualSpaceLagrangeSetTrimmed_C",(PetscDualSpace,PetscBool),(sp,trimmed));
    2947:   return(0);
    2948: }

    2950: /*@
    2951:   PetscDualSpaceLagrangeGetNodeType - Get a description of how nodes are laid out for Lagrange polynomials in this
    2952:   dual space

    2954:   Not collective

    2956:   Input Parameter:
    2957: . sp - The PetscDualSpace

    2959:   Output Parameters:
    2960: + nodeType - The type of nodes
    2961: . boundary - Whether the node type is one that includes endpoints (if nodeType is PETSCDTNODES_GAUSSJACOBI, nodes that
    2962:              include the boundary are Gauss-Lobatto-Jacobi nodes)
    2963: - exponent - If nodeType is PETSCDTNODES_GAUSJACOBI, indicates the exponent used for both ends of the 1D Jacobi weight function
    2964:              '0' is Gauss-Legendre, '-0.5' is Gauss-Chebyshev of the first type, '0.5' is Gauss-Chebyshev of the second type

    2966:   Level: advanced

    2968: .seealso: PetscDTNodeType, PetscDualSpaceLagrangeSetNodeType()
    2969: @*/
    2970: PetscErrorCode PetscDualSpaceLagrangeGetNodeType(PetscDualSpace sp, PetscDTNodeType *nodeType, PetscBool *boundary, PetscReal *exponent)
    2971: {

    2979:   PetscTryMethod(sp,"PetscDualSpaceLagrangeGetNodeType_C",(PetscDualSpace,PetscDTNodeType *,PetscBool *,PetscReal *),(sp,nodeType,boundary,exponent));
    2980:   return(0);
    2981: }

    2983: /*@
    2984:   PetscDualSpaceLagrangeSetNodeType - Set a description of how nodes are laid out for Lagrange polynomials in this
    2985:   dual space

    2987:   Logically collective

    2989:   Input Parameters:
    2990: + sp - The PetscDualSpace
    2991: . nodeType - The type of nodes
    2992: . boundary - Whether the node type is one that includes endpoints (if nodeType is PETSCDTNODES_GAUSSJACOBI, nodes that
    2993:              include the boundary are Gauss-Lobatto-Jacobi nodes)
    2994: - exponent - If nodeType is PETSCDTNODES_GAUSJACOBI, indicates the exponent used for both ends of the 1D Jacobi weight function
    2995:              '0' is Gauss-Legendre, '-0.5' is Gauss-Chebyshev of the first type, '0.5' is Gauss-Chebyshev of the second type

    2997:   Level: advanced

    2999: .seealso: PetscDTNodeType, PetscDualSpaceLagrangeGetNodeType()
    3000: @*/
    3001: PetscErrorCode PetscDualSpaceLagrangeSetNodeType(PetscDualSpace sp, PetscDTNodeType nodeType, PetscBool boundary, PetscReal exponent)
    3002: {

    3007:   PetscTryMethod(sp,"PetscDualSpaceLagrangeSetNodeType_C",(PetscDualSpace,PetscDTNodeType,PetscBool,PetscReal),(sp,nodeType,boundary,exponent));
    3008:   return(0);
    3009: }

    3012: static PetscErrorCode PetscDualSpaceInitialize_Lagrange(PetscDualSpace sp)
    3013: {
    3015:   sp->ops->destroy              = PetscDualSpaceDestroy_Lagrange;
    3016:   sp->ops->view                 = PetscDualSpaceView_Lagrange;
    3017:   sp->ops->setfromoptions       = PetscDualSpaceSetFromOptions_Lagrange;
    3018:   sp->ops->duplicate            = PetscDualSpaceDuplicate_Lagrange;
    3019:   sp->ops->setup                = PetscDualSpaceSetUp_Lagrange;
    3020:   sp->ops->createheightsubspace = NULL;
    3021:   sp->ops->createpointsubspace  = NULL;
    3022:   sp->ops->getsymmetries        = PetscDualSpaceGetSymmetries_Lagrange;
    3023:   sp->ops->apply                = PetscDualSpaceApplyDefault;
    3024:   sp->ops->applyall             = PetscDualSpaceApplyAllDefault;
    3025:   sp->ops->applyint             = PetscDualSpaceApplyInteriorDefault;
    3026:   sp->ops->createalldata        = PetscDualSpaceCreateAllDataDefault;
    3027:   sp->ops->createintdata        = PetscDualSpaceCreateInteriorDataDefault;
    3028:   return(0);
    3029: }

    3031: /*MC
    3032:   PETSCDUALSPACELAGRANGE = "lagrange" - A PetscDualSpace object that encapsulates a dual space of pointwise evaluation functionals

    3034:   Level: intermediate

    3036: .seealso: PetscDualSpaceType, PetscDualSpaceCreate(), PetscDualSpaceSetType()
    3037: M*/
    3038: PETSC_EXTERN PetscErrorCode PetscDualSpaceCreate_Lagrange(PetscDualSpace sp)
    3039: {
    3040:   PetscDualSpace_Lag *lag;
    3041:   PetscErrorCode      ierr;

    3045:   PetscNewLog(sp,&lag);
    3046:   sp->data = lag;

    3048:   lag->tensorCell  = PETSC_FALSE;
    3049:   lag->tensorSpace = PETSC_FALSE;
    3050:   lag->continuous  = PETSC_TRUE;
    3051:   lag->numCopies   = PETSC_DEFAULT;
    3052:   lag->numNodeSkip = PETSC_DEFAULT;
    3053:   lag->nodeType    = PETSCDTNODES_DEFAULT;

    3055:   PetscDualSpaceInitialize_Lagrange(sp);
    3056:   PetscObjectComposeFunction((PetscObject) sp, "PetscDualSpaceLagrangeGetContinuity_C", PetscDualSpaceLagrangeGetContinuity_Lagrange);
    3057:   PetscObjectComposeFunction((PetscObject) sp, "PetscDualSpaceLagrangeSetContinuity_C", PetscDualSpaceLagrangeSetContinuity_Lagrange);
    3058:   PetscObjectComposeFunction((PetscObject) sp, "PetscDualSpaceLagrangeGetTensor_C", PetscDualSpaceLagrangeGetTensor_Lagrange);
    3059:   PetscObjectComposeFunction((PetscObject) sp, "PetscDualSpaceLagrangeSetTensor_C", PetscDualSpaceLagrangeSetTensor_Lagrange);
    3060:   PetscObjectComposeFunction((PetscObject) sp, "PetscDualSpaceLagrangeGetTrimmed_C", PetscDualSpaceLagrangeGetTrimmed_Lagrange);
    3061:   PetscObjectComposeFunction((PetscObject) sp, "PetscDualSpaceLagrangeSetTrimmed_C", PetscDualSpaceLagrangeSetTrimmed_Lagrange);
    3062:   PetscObjectComposeFunction((PetscObject) sp, "PetscDualSpaceLagrangeGetNodeType_C", PetscDualSpaceLagrangeGetNodeType_Lagrange);
    3063:   PetscObjectComposeFunction((PetscObject) sp, "PetscDualSpaceLagrangeSetNodeType_C", PetscDualSpaceLagrangeSetNodeType_Lagrange);
    3064:   return(0);
    3065: }