xref: /libCEED/interface/ceed-preconditioning.c (revision 9bc663991d6482bcb1d60b1f116148f11db83fa1)
1 // Copyright (c) 2017-2024, Lawrence Livermore National Security, LLC and other CEED contributors.
2 // All Rights Reserved. See the top-level LICENSE and NOTICE files for details.
3 //
4 // SPDX-License-Identifier: BSD-2-Clause
5 //
6 // This file is part of CEED:  http://github.com/ceed
7 
8 #include <ceed-impl.h>
9 #include <ceed.h>
10 #include <ceed/backend.h>
11 #include <assert.h>
12 #include <math.h>
13 #include <stdbool.h>
14 #include <stdio.h>
15 #include <string.h>
16 
17 /// @file
18 /// Implementation of CeedOperator preconditioning interfaces
19 
20 /// ----------------------------------------------------------------------------
21 /// CeedOperator Library Internal Preconditioning Functions
22 /// ----------------------------------------------------------------------------
23 /// @addtogroup CeedOperatorDeveloper
24 /// @{
25 
26 /**
27   @brief Duplicate a `CeedQFunction` with a reference `Ceed` to fallback for advanced `CeedOperator` functionality
28 
29   @param[in]  fallback_ceed `Ceed` on which to create fallback `CeedQFunction`
30   @param[in]  qf            `CeedQFunction` to create fallback for
31   @param[out] qf_fallback   Fallback `CeedQFunction`
32 
33   @return An error code: 0 - success, otherwise - failure
34 
35   @ref Developer
36 **/
37 static int CeedQFunctionCreateFallback(Ceed fallback_ceed, CeedQFunction qf, CeedQFunction *qf_fallback) {
38   char               *source_path_with_name = NULL;
39   CeedInt             num_input_fields, num_output_fields;
40   CeedQFunctionField *input_fields, *output_fields;
41 
42   // Check if NULL qf passed in
43   if (!qf) return CEED_ERROR_SUCCESS;
44 
45   CeedDebug256(CeedQFunctionReturnCeed(qf), 1, "---------- CeedOperator Fallback ----------\n");
46   CeedDebug(CeedQFunctionReturnCeed(qf), "Creating fallback CeedQFunction\n");
47 
48   if (qf->source_path) {
49     size_t path_len = strlen(qf->source_path), name_len = strlen(qf->kernel_name);
50 
51     CeedCall(CeedCalloc(path_len + name_len + 2, &source_path_with_name));
52     memcpy(source_path_with_name, qf->source_path, path_len);
53     memcpy(&source_path_with_name[path_len], ":", 1);
54     memcpy(&source_path_with_name[path_len + 1], qf->kernel_name, name_len);
55   } else if (qf->user_source) {
56     CeedCall(CeedStringAllocCopy(qf->user_source, &source_path_with_name));
57   } else {
58     CeedCall(CeedCalloc(1, &source_path_with_name));
59   }
60 
61   {
62     CeedInt           vec_length;
63     CeedQFunctionUser f;
64 
65     CeedCall(CeedQFunctionGetVectorLength(qf, &vec_length));
66     CeedCall(CeedQFunctionGetUserFunction(qf, &f));
67     CeedCall(CeedQFunctionCreateInterior(fallback_ceed, vec_length, f, source_path_with_name, qf_fallback));
68   }
69   {
70     CeedQFunctionContext ctx;
71 
72     CeedCall(CeedQFunctionGetContext(qf, &ctx));
73     CeedCall(CeedQFunctionSetContext(*qf_fallback, ctx));
74   }
75   CeedCall(CeedQFunctionGetFields(qf, &num_input_fields, &input_fields, &num_output_fields, &output_fields));
76   for (CeedInt i = 0; i < num_input_fields; i++) {
77     const char  *field_name;
78     CeedInt      size;
79     CeedEvalMode eval_mode;
80 
81     CeedCall(CeedQFunctionFieldGetData(input_fields[i], &field_name, &size, &eval_mode));
82     CeedCall(CeedQFunctionAddInput(*qf_fallback, field_name, size, eval_mode));
83   }
84   for (CeedInt i = 0; i < num_output_fields; i++) {
85     const char  *field_name;
86     CeedInt      size;
87     CeedEvalMode eval_mode;
88 
89     CeedCall(CeedQFunctionFieldGetData(output_fields[i], &field_name, &size, &eval_mode));
90     CeedCall(CeedQFunctionAddOutput(*qf_fallback, field_name, size, eval_mode));
91   }
92   CeedCall(CeedFree(&source_path_with_name));
93   return CEED_ERROR_SUCCESS;
94 }
95 
96 /**
97   @brief Duplicate a `CeedOperator` with a reference `Ceed` to fallback for advanced `CeedOperator` functionality
98 
99   @param[in,out] op `CeedOperator` to create fallback for
100 
101   @return An error code: 0 - success, otherwise - failure
102 
103   @ref Developer
104 **/
105 static int CeedOperatorCreateFallback(CeedOperator op) {
106   bool         is_composite;
107   Ceed         ceed, ceed_fallback;
108   CeedOperator op_fallback;
109 
110   // Check not already created
111   if (op->op_fallback) return CEED_ERROR_SUCCESS;
112 
113   // Fallback Ceed
114   CeedCall(CeedOperatorGetCeed(op, &ceed));
115   CeedCall(CeedGetOperatorFallbackCeed(ceed, &ceed_fallback));
116   CeedCall(CeedDestroy(&ceed));
117   if (!ceed_fallback) return CEED_ERROR_SUCCESS;
118 
119   CeedDebug256(CeedOperatorReturnCeed(op), 1, "---------- CeedOperator Fallback ----------\n");
120   CeedDebug(CeedOperatorReturnCeed(op), "Creating fallback CeedOperator\n");
121 
122   // Clone Op
123   CeedCall(CeedOperatorIsComposite(op, &is_composite));
124   if (is_composite) {
125     CeedInt       num_suboperators;
126     CeedOperator *sub_operators;
127 
128     CeedCall(CeedCompositeOperatorCreate(ceed_fallback, &op_fallback));
129     CeedCall(CeedCompositeOperatorGetNumSub(op, &num_suboperators));
130     CeedCall(CeedCompositeOperatorGetSubList(op, &sub_operators));
131     for (CeedInt i = 0; i < num_suboperators; i++) {
132       CeedOperator op_sub_fallback;
133 
134       CeedCall(CeedOperatorGetFallback(sub_operators[i], &op_sub_fallback));
135       CeedCall(CeedCompositeOperatorAddSub(op_fallback, op_sub_fallback));
136     }
137   } else {
138     CeedInt            num_input_fields, num_output_fields;
139     CeedQFunction      qf_fallback = NULL, dqf_fallback = NULL, dqfT_fallback = NULL;
140     CeedOperatorField *input_fields, *output_fields;
141 
142     CeedCall(CeedQFunctionCreateFallback(ceed_fallback, op->qf, &qf_fallback));
143     CeedCall(CeedQFunctionCreateFallback(ceed_fallback, op->dqf, &dqf_fallback));
144     CeedCall(CeedQFunctionCreateFallback(ceed_fallback, op->dqfT, &dqfT_fallback));
145     CeedCall(CeedOperatorCreate(ceed_fallback, qf_fallback, dqf_fallback, dqfT_fallback, &op_fallback));
146     CeedCall(CeedOperatorGetFields(op, &num_input_fields, &input_fields, &num_output_fields, &output_fields));
147     for (CeedInt i = 0; i < num_input_fields; i++) {
148       const char         *field_name;
149       CeedVector          vec;
150       CeedElemRestriction rstr;
151       CeedBasis           basis;
152 
153       CeedCall(CeedOperatorFieldGetData(input_fields[i], &field_name, &rstr, &basis, &vec));
154       CeedCall(CeedOperatorSetField(op_fallback, field_name, rstr, basis, vec));
155       CeedCall(CeedVectorDestroy(&vec));
156       CeedCall(CeedElemRestrictionDestroy(&rstr));
157       CeedCall(CeedBasisDestroy(&basis));
158     }
159     for (CeedInt i = 0; i < num_output_fields; i++) {
160       const char         *field_name;
161       CeedVector          vec;
162       CeedElemRestriction rstr;
163       CeedBasis           basis;
164 
165       CeedCall(CeedOperatorFieldGetData(output_fields[i], &field_name, &rstr, &basis, &vec));
166       CeedCall(CeedOperatorSetField(op_fallback, field_name, rstr, basis, vec));
167       CeedCall(CeedVectorDestroy(&vec));
168       CeedCall(CeedElemRestrictionDestroy(&rstr));
169       CeedCall(CeedBasisDestroy(&basis));
170     }
171     {
172       CeedQFunctionAssemblyData data;
173 
174       CeedCall(CeedOperatorGetQFunctionAssemblyData(op, &data));
175       CeedCall(CeedQFunctionAssemblyDataReferenceCopy(data, &op_fallback->qf_assembled));
176     }
177     // Cleanup
178     CeedCall(CeedQFunctionDestroy(&qf_fallback));
179     CeedCall(CeedQFunctionDestroy(&dqf_fallback));
180     CeedCall(CeedQFunctionDestroy(&dqfT_fallback));
181   }
182   CeedCall(CeedOperatorSetName(op_fallback, op->name));
183   CeedCall(CeedOperatorCheckReady(op_fallback));
184   // Note: No ref-counting here so we don't get caught in a reference loop.
185   //       The op holds the only reference to op_fallback and is responsible for deleting itself and op_fallback.
186   op->op_fallback                 = op_fallback;
187   op_fallback->op_fallback_parent = op;
188   CeedCall(CeedDestroy(&ceed_fallback));
189   return CEED_ERROR_SUCCESS;
190 }
191 
192 /**
193   @brief Core logic for assembling operator diagonal or point block diagonal
194 
195   @param[in]  op             `CeedOperator` to assemble diagonal or point block diagonal
196   @param[in]  request        Address of @ref CeedRequest for non-blocking completion, else @ref CEED_REQUEST_IMMEDIATE
197   @param[in]  is_point_block Boolean flag to assemble diagonal or point block diagonal
198   @param[out] assembled      `CeedVector` to store assembled diagonal
199 
200   @return An error code: 0 - success, otherwise - failure
201 
202   @ref Developer
203 **/
204 static inline int CeedSingleOperatorLinearAssembleAddDiagonal_Mesh(CeedOperator op, CeedRequest *request, const bool is_point_block,
205                                                                    CeedVector assembled) {
206   bool is_composite;
207 
208   CeedCall(CeedOperatorIsComposite(op, &is_composite));
209   CeedCheck(!is_composite, CeedOperatorReturnCeed(op), CEED_ERROR_UNSUPPORTED, "Composite operator not supported");
210 
211   // Assemble QFunction
212   CeedInt             layout_qf[3];
213   const CeedScalar   *assembled_qf_array;
214   CeedVector          assembled_qf        = NULL;
215   CeedElemRestriction assembled_elem_rstr = NULL;
216 
217   CeedCall(CeedOperatorLinearAssembleQFunctionBuildOrUpdate(op, &assembled_qf, &assembled_elem_rstr, request));
218   CeedCall(CeedElemRestrictionGetELayout(assembled_elem_rstr, layout_qf));
219   CeedCall(CeedElemRestrictionDestroy(&assembled_elem_rstr));
220   CeedCall(CeedVectorGetArrayRead(assembled_qf, CEED_MEM_HOST, &assembled_qf_array));
221 
222   // Get assembly data
223   const CeedEvalMode     **eval_modes_in, **eval_modes_out;
224   CeedInt                  num_active_bases_in, *num_eval_modes_in, num_active_bases_out, *num_eval_modes_out;
225   CeedSize               **eval_mode_offsets_in, **eval_mode_offsets_out, num_output_components;
226   CeedBasis               *active_bases_in, *active_bases_out;
227   CeedElemRestriction     *active_elem_rstrs_in, *active_elem_rstrs_out;
228   CeedOperatorAssemblyData data;
229 
230   CeedCall(CeedOperatorGetOperatorAssemblyData(op, &data));
231   CeedCall(CeedOperatorAssemblyDataGetEvalModes(data, &num_active_bases_in, &num_eval_modes_in, &eval_modes_in, &eval_mode_offsets_in,
232                                                 &num_active_bases_out, &num_eval_modes_out, &eval_modes_out, &eval_mode_offsets_out,
233                                                 &num_output_components));
234   CeedCall(CeedOperatorAssemblyDataGetBases(data, NULL, &active_bases_in, NULL, NULL, &active_bases_out, NULL));
235   CeedCall(CeedOperatorAssemblyDataGetElemRestrictions(data, NULL, &active_elem_rstrs_in, NULL, &active_elem_rstrs_out));
236 
237   // Loop over all active bases (find matching input/output pairs)
238   for (CeedInt b = 0; b < CeedIntMin(num_active_bases_in, num_active_bases_out); b++) {
239     CeedInt             b_in, b_out, num_elem, num_nodes, num_qpts, num_comp;
240     bool                has_eval_none = false;
241     CeedScalar         *elem_diag_array, *identity = NULL;
242     CeedVector          elem_diag;
243     CeedElemRestriction diag_elem_rstr;
244 
245     if (num_active_bases_in <= num_active_bases_out) {
246       b_in = b;
247       for (b_out = 0; b_out < num_active_bases_out; b_out++) {
248         if (active_bases_in[b_in] == active_bases_out[b_out]) {
249           break;
250         }
251       }
252       if (b_out == num_active_bases_out) {
253         continue;
254       }  // No matching output basis found
255     } else {
256       b_out = b;
257       for (b_in = 0; b_in < num_active_bases_in; b_in++) {
258         if (active_bases_in[b_in] == active_bases_out[b_out]) {
259           break;
260         }
261       }
262       if (b_in == num_active_bases_in) {
263         continue;
264       }  // No matching output basis found
265     }
266     CeedCheck(active_elem_rstrs_in[b_in] == active_elem_rstrs_out[b_out], CeedOperatorReturnCeed(op), CEED_ERROR_UNSUPPORTED,
267               "Cannot assemble operator diagonal with different input and output active element restrictions");
268 
269     // Assemble point block diagonal restriction, if needed
270     if (is_point_block) {
271       CeedCall(CeedOperatorCreateActivePointBlockRestriction(active_elem_rstrs_in[b_in], &diag_elem_rstr));
272     } else {
273       CeedCall(CeedElemRestrictionCreateUnsignedCopy(active_elem_rstrs_in[b_in], &diag_elem_rstr));
274     }
275 
276     // Create diagonal vector
277     CeedCall(CeedElemRestrictionCreateVector(diag_elem_rstr, NULL, &elem_diag));
278 
279     // Assemble element operator diagonals
280     CeedCall(CeedVectorSetValue(elem_diag, 0.0));
281     CeedCall(CeedVectorGetArray(elem_diag, CEED_MEM_HOST, &elem_diag_array));
282     CeedCall(CeedElemRestrictionGetNumElements(diag_elem_rstr, &num_elem));
283     CeedCall(CeedBasisGetNumNodes(active_bases_in[b_in], &num_nodes));
284     CeedCall(CeedBasisGetNumComponents(active_bases_in[b_in], &num_comp));
285     if (active_bases_in[b_in] == CEED_BASIS_NONE) num_qpts = num_nodes;
286     else CeedCall(CeedBasisGetNumQuadraturePoints(active_bases_in[b_in], &num_qpts));
287 
288     // Construct identity matrix for basis if required
289     for (CeedInt i = 0; i < num_eval_modes_in[b_in]; i++) {
290       has_eval_none = has_eval_none || (eval_modes_in[b_in][i] == CEED_EVAL_NONE);
291     }
292     for (CeedInt i = 0; i < num_eval_modes_out[b_out]; i++) {
293       has_eval_none = has_eval_none || (eval_modes_out[b_out][i] == CEED_EVAL_NONE);
294     }
295     if (has_eval_none) {
296       CeedCall(CeedCalloc(num_qpts * num_nodes, &identity));
297       for (CeedInt i = 0; i < (num_nodes < num_qpts ? num_nodes : num_qpts); i++) identity[i * num_nodes + i] = 1.0;
298     }
299 
300     // Compute the diagonal of B^T D B
301     // Each element
302     for (CeedSize e = 0; e < num_elem; e++) {
303       // Each basis eval mode pair
304       CeedInt      d_out              = 0, q_comp_out;
305       CeedEvalMode eval_mode_out_prev = CEED_EVAL_NONE;
306 
307       for (CeedInt e_out = 0; e_out < num_eval_modes_out[b_out]; e_out++) {
308         CeedInt           d_in              = 0, q_comp_in;
309         const CeedScalar *B_t               = NULL;
310         CeedEvalMode      eval_mode_in_prev = CEED_EVAL_NONE;
311 
312         CeedCall(CeedOperatorGetBasisPointer(active_bases_out[b_out], eval_modes_out[b_out][e_out], identity, &B_t));
313         CeedCall(CeedBasisGetNumQuadratureComponents(active_bases_out[b_out], eval_modes_out[b_out][e_out], &q_comp_out));
314         if (q_comp_out > 1) {
315           if (e_out == 0 || eval_modes_out[b_out][e_out] != eval_mode_out_prev) d_out = 0;
316           else B_t = &B_t[(++d_out) * num_qpts * num_nodes];
317         }
318         eval_mode_out_prev = eval_modes_out[b_out][e_out];
319 
320         for (CeedInt e_in = 0; e_in < num_eval_modes_in[b_in]; e_in++) {
321           const CeedScalar *B = NULL;
322 
323           CeedCall(CeedOperatorGetBasisPointer(active_bases_in[b_in], eval_modes_in[b_in][e_in], identity, &B));
324           CeedCall(CeedBasisGetNumQuadratureComponents(active_bases_in[b_in], eval_modes_in[b_in][e_in], &q_comp_in));
325           if (q_comp_in > 1) {
326             if (e_in == 0 || eval_modes_in[b_in][e_in] != eval_mode_in_prev) d_in = 0;
327             else B = &B[(++d_in) * num_qpts * num_nodes];
328           }
329           eval_mode_in_prev = eval_modes_in[b_in][e_in];
330 
331           // Each component
332           for (CeedInt c_out = 0; c_out < num_comp; c_out++) {
333             // Each qpt/node pair
334             for (CeedInt q = 0; q < num_qpts; q++) {
335               if (is_point_block) {
336                 // Point Block Diagonal
337                 for (CeedInt c_in = 0; c_in < num_comp; c_in++) {
338                   const CeedSize c_offset =
339                       (eval_mode_offsets_in[b_in][e_in] + c_in) * num_output_components + eval_mode_offsets_out[b_out][e_out] + c_out;
340                   const CeedScalar qf_value = assembled_qf_array[q * layout_qf[0] + c_offset * layout_qf[1] + e * layout_qf[2]];
341 
342                   for (CeedInt n = 0; n < num_nodes; n++) {
343                     elem_diag_array[((e * num_comp + c_out) * num_comp + c_in) * num_nodes + n] +=
344                         B_t[q * num_nodes + n] * qf_value * B[q * num_nodes + n];
345                   }
346                 }
347               } else {
348                 // Diagonal Only
349                 const CeedInt c_offset =
350                     (eval_mode_offsets_in[b_in][e_in] + c_out) * num_output_components + eval_mode_offsets_out[b_out][e_out] + c_out;
351                 const CeedScalar qf_value = assembled_qf_array[q * layout_qf[0] + c_offset * layout_qf[1] + e * layout_qf[2]];
352 
353                 for (CeedInt n = 0; n < num_nodes; n++) {
354                   elem_diag_array[(e * num_comp + c_out) * num_nodes + n] += B_t[q * num_nodes + n] * qf_value * B[q * num_nodes + n];
355                 }
356               }
357             }
358           }
359         }
360       }
361     }
362     CeedCall(CeedVectorRestoreArray(elem_diag, &elem_diag_array));
363 
364     // Assemble local operator diagonal
365     CeedCall(CeedElemRestrictionApply(diag_elem_rstr, CEED_TRANSPOSE, elem_diag, assembled, request));
366 
367     // Cleanup
368     CeedCall(CeedElemRestrictionDestroy(&diag_elem_rstr));
369     CeedCall(CeedVectorDestroy(&elem_diag));
370     CeedCall(CeedFree(&identity));
371   }
372   CeedCall(CeedVectorRestoreArrayRead(assembled_qf, &assembled_qf_array));
373   CeedCall(CeedVectorDestroy(&assembled_qf));
374   return CEED_ERROR_SUCCESS;
375 }
376 
377 /**
378   @brief Core logic for assembling operator diagonal or point block diagonal
379 
380   @param[in]  op             `CeedOperator` to assemble diagonal or point block diagonal
381   @param[in]  request        Address of @ref CeedRequest for non-blocking completion, else @ref CEED_REQUEST_IMMEDIATE
382   @param[in]  is_point_block Boolean flag to assemble diagonal or point block diagonal
383   @param[out] assembled      `CeedVector` to store assembled diagonal
384 
385   @return An error code: 0 - success, otherwise - failure
386 
387   @ref Developer
388 **/
389 static inline int CeedSingleOperatorLinearAssembleAddDiagonal(CeedOperator op, CeedRequest *request, const bool is_point_block,
390                                                               CeedVector assembled) {
391   bool is_at_points;
392 
393   CeedCall(CeedOperatorIsAtPoints(op, &is_at_points));
394   CeedCheck(!is_at_points, CeedOperatorReturnCeed(op), CEED_ERROR_UNSUPPORTED, "AtPoints operator not supported");
395   CeedCall(CeedSingleOperatorLinearAssembleAddDiagonal_Mesh(op, request, is_point_block, assembled));
396   return CEED_ERROR_SUCCESS;
397 }
398 
399 /**
400   @brief Core logic for assembling composite operator diagonal
401 
402   @param[in]  op             `CeedOperator` to assemble point block diagonal
403   @param[in]  request        Address of @ref CeedRequest for non-blocking completion, else @ref CEED_REQUEST_IMMEDIATE
404   @param[in]  is_point_block Boolean flag to assemble diagonal or point block diagonal
405   @param[out] assembled      `CeedVector` to store assembled diagonal
406 
407   @return An error code: 0 - success, otherwise - failure
408 
409   @ref Developer
410 **/
411 static inline int CeedCompositeOperatorLinearAssembleAddDiagonal(CeedOperator op, CeedRequest *request, const bool is_point_block,
412                                                                  CeedVector assembled) {
413   CeedInt       num_sub;
414   CeedOperator *suboperators;
415 
416   CeedCall(CeedCompositeOperatorGetNumSub(op, &num_sub));
417   CeedCall(CeedCompositeOperatorGetSubList(op, &suboperators));
418   for (CeedInt i = 0; i < num_sub; i++) {
419     if (is_point_block) {
420       CeedCall(CeedOperatorLinearAssembleAddPointBlockDiagonal(suboperators[i], assembled, request));
421     } else {
422       CeedCall(CeedOperatorLinearAssembleAddDiagonal(suboperators[i], assembled, request));
423     }
424   }
425   return CEED_ERROR_SUCCESS;
426 }
427 
428 /**
429   @brief Build nonzero pattern for non-composite CeedOperator`.
430 
431   Users should generally use @ref CeedOperatorLinearAssembleSymbolic().
432 
433   @param[in]  op     `CeedOperator` to assemble nonzero pattern
434   @param[in]  offset Offset for number of entries
435   @param[out] rows   Row number for each entry
436   @param[out] cols   Column number for each entry
437 
438   @return An error code: 0 - success, otherwise - failure
439 
440   @ref Developer
441 **/
442 static int CeedSingleOperatorAssembleSymbolic(CeedOperator op, CeedInt offset, CeedInt *rows, CeedInt *cols) {
443   Ceed                ceed;
444   bool                is_composite;
445   CeedSize            num_nodes_in, num_nodes_out, local_num_entries, count = 0;
446   CeedInt             num_elem_in, elem_size_in, num_comp_in, layout_er_in[3];
447   CeedInt             num_elem_out, elem_size_out, num_comp_out, layout_er_out[3];
448   CeedScalar         *array;
449   const CeedScalar   *elem_dof_a_in, *elem_dof_a_out;
450   CeedVector          index_vec_in, index_vec_out, elem_dof_in, elem_dof_out;
451   CeedElemRestriction elem_rstr_in, elem_rstr_out, index_elem_rstr_in, index_elem_rstr_out;
452 
453   CeedCall(CeedOperatorIsComposite(op, &is_composite));
454   CeedCheck(!is_composite, ceed, CEED_ERROR_UNSUPPORTED, "Composite operator not supported");
455   CeedCall(CeedOperatorGetCeed(op, &ceed));
456 
457   CeedCall(CeedOperatorGetActiveVectorLengths(op, &num_nodes_in, &num_nodes_out));
458   CeedCall(CeedOperatorGetActiveElemRestrictions(op, &elem_rstr_in, &elem_rstr_out));
459   CeedCall(CeedElemRestrictionGetNumElements(elem_rstr_in, &num_elem_in));
460   CeedCall(CeedElemRestrictionGetElementSize(elem_rstr_in, &elem_size_in));
461   CeedCall(CeedElemRestrictionGetNumComponents(elem_rstr_in, &num_comp_in));
462   CeedCall(CeedElemRestrictionGetELayout(elem_rstr_in, layout_er_in));
463 
464   // Determine elem_dof relation for input
465   CeedCall(CeedVectorCreate(ceed, num_nodes_in, &index_vec_in));
466   CeedCall(CeedVectorGetArrayWrite(index_vec_in, CEED_MEM_HOST, &array));
467   for (CeedSize i = 0; i < num_nodes_in; i++) array[i] = i;
468   CeedCall(CeedVectorRestoreArray(index_vec_in, &array));
469   CeedCall(CeedVectorCreate(ceed, num_elem_in * elem_size_in * num_comp_in, &elem_dof_in));
470   CeedCall(CeedVectorSetValue(elem_dof_in, 0.0));
471   CeedCall(CeedElemRestrictionCreateUnorientedCopy(elem_rstr_in, &index_elem_rstr_in));
472   CeedCall(CeedElemRestrictionApply(index_elem_rstr_in, CEED_NOTRANSPOSE, index_vec_in, elem_dof_in, CEED_REQUEST_IMMEDIATE));
473   CeedCall(CeedVectorGetArrayRead(elem_dof_in, CEED_MEM_HOST, &elem_dof_a_in));
474   CeedCall(CeedVectorDestroy(&index_vec_in));
475   CeedCall(CeedElemRestrictionDestroy(&index_elem_rstr_in));
476 
477   if (elem_rstr_in != elem_rstr_out) {
478     CeedCall(CeedElemRestrictionGetNumElements(elem_rstr_out, &num_elem_out));
479     CeedCheck(num_elem_in == num_elem_out, ceed, CEED_ERROR_UNSUPPORTED,
480               "Active input and output operator restrictions must have the same number of elements."
481               " Input has %" CeedInt_FMT " elements; output has %" CeedInt_FMT "elements.",
482               num_elem_in, num_elem_out);
483     CeedCall(CeedElemRestrictionGetElementSize(elem_rstr_out, &elem_size_out));
484     CeedCall(CeedElemRestrictionGetNumComponents(elem_rstr_out, &num_comp_out));
485     CeedCall(CeedElemRestrictionGetELayout(elem_rstr_out, layout_er_out));
486 
487     // Determine elem_dof relation for output
488     CeedCall(CeedVectorCreate(ceed, num_nodes_out, &index_vec_out));
489     CeedCall(CeedVectorGetArrayWrite(index_vec_out, CEED_MEM_HOST, &array));
490     for (CeedSize i = 0; i < num_nodes_out; i++) array[i] = i;
491     CeedCall(CeedVectorRestoreArray(index_vec_out, &array));
492     CeedCall(CeedVectorCreate(ceed, num_elem_out * elem_size_out * num_comp_out, &elem_dof_out));
493     CeedCall(CeedVectorSetValue(elem_dof_out, 0.0));
494     CeedCall(CeedElemRestrictionCreateUnorientedCopy(elem_rstr_out, &index_elem_rstr_out));
495     CeedCall(CeedElemRestrictionApply(index_elem_rstr_out, CEED_NOTRANSPOSE, index_vec_out, elem_dof_out, CEED_REQUEST_IMMEDIATE));
496     CeedCall(CeedVectorGetArrayRead(elem_dof_out, CEED_MEM_HOST, &elem_dof_a_out));
497     CeedCall(CeedVectorDestroy(&index_vec_out));
498     CeedCall(CeedElemRestrictionDestroy(&index_elem_rstr_out));
499   } else {
500     num_elem_out     = num_elem_in;
501     elem_size_out    = elem_size_in;
502     num_comp_out     = num_comp_in;
503     layout_er_out[0] = layout_er_in[0];
504     layout_er_out[1] = layout_er_in[1];
505     layout_er_out[2] = layout_er_in[2];
506     elem_dof_a_out   = elem_dof_a_in;
507   }
508   local_num_entries = (CeedSize)elem_size_out * num_comp_out * elem_size_in * num_comp_in * num_elem_in;
509 
510   // Determine i, j locations for element matrices
511   for (CeedInt e = 0; e < num_elem_in; e++) {
512     for (CeedInt comp_in = 0; comp_in < num_comp_in; comp_in++) {
513       for (CeedInt comp_out = 0; comp_out < num_comp_out; comp_out++) {
514         for (CeedInt i = 0; i < elem_size_out; i++) {
515           for (CeedInt j = 0; j < elem_size_in; j++) {
516             const CeedInt elem_dof_index_row = i * layout_er_out[0] + comp_out * layout_er_out[1] + e * layout_er_out[2];
517             const CeedInt elem_dof_index_col = j * layout_er_in[0] + comp_in * layout_er_in[1] + e * layout_er_in[2];
518             const CeedInt row                = elem_dof_a_out[elem_dof_index_row];
519             const CeedInt col                = elem_dof_a_in[elem_dof_index_col];
520 
521             rows[offset + count] = row;
522             cols[offset + count] = col;
523             count++;
524           }
525         }
526       }
527     }
528   }
529   CeedCheck(count == local_num_entries, ceed, CEED_ERROR_MAJOR, "Error computing assembled entries");
530   CeedCall(CeedVectorRestoreArrayRead(elem_dof_in, &elem_dof_a_in));
531   CeedCall(CeedVectorDestroy(&elem_dof_in));
532   if (elem_rstr_in != elem_rstr_out) {
533     CeedCall(CeedVectorRestoreArrayRead(elem_dof_out, &elem_dof_a_out));
534     CeedCall(CeedVectorDestroy(&elem_dof_out));
535   }
536   CeedCall(CeedElemRestrictionDestroy(&elem_rstr_in));
537   CeedCall(CeedElemRestrictionDestroy(&elem_rstr_out));
538   CeedCall(CeedDestroy(&ceed));
539   return CEED_ERROR_SUCCESS;
540 }
541 
542 /**
543   @brief Assemble nonzero entries for non-composite `CeedOperator`.
544 
545   Users should generally use @ref CeedOperatorLinearAssemble().
546 
547   @param[in]  op     `CeedOperator` to assemble
548   @param[in]  offset Offset for number of entries
549   @param[out] values Values to assemble into matrix
550 
551   @return An error code: 0 - success, otherwise - failure
552 
553   @ref Developer
554 **/
555 static int CeedSingleOperatorAssemble(CeedOperator op, CeedInt offset, CeedVector values) {
556   bool is_composite;
557 
558   CeedCall(CeedOperatorIsComposite(op, &is_composite));
559   CeedCheck(!is_composite, CeedOperatorReturnCeed(op), CEED_ERROR_UNSUPPORTED, "Composite operator not supported");
560 
561   // Early exit for empty operator
562   {
563     CeedInt num_elem = 0;
564 
565     CeedCall(CeedOperatorGetNumElements(op, &num_elem));
566     if (num_elem == 0) return CEED_ERROR_SUCCESS;
567   }
568 
569   if (op->LinearAssembleSingle) {
570     // Backend version
571     CeedCall(op->LinearAssembleSingle(op, offset, values));
572     return CEED_ERROR_SUCCESS;
573   } else {
574     // Operator fallback
575     CeedOperator op_fallback;
576 
577     CeedCall(CeedOperatorGetFallback(op, &op_fallback));
578     if (op_fallback) {
579       CeedCall(CeedSingleOperatorAssemble(op_fallback, offset, values));
580       return CEED_ERROR_SUCCESS;
581     }
582   }
583 
584   // Assemble QFunction
585   CeedInt             layout_qf[3];
586   const CeedScalar   *assembled_qf_array;
587   CeedVector          assembled_qf        = NULL;
588   CeedElemRestriction assembled_elem_rstr = NULL;
589 
590   CeedCall(CeedOperatorLinearAssembleQFunctionBuildOrUpdate(op, &assembled_qf, &assembled_elem_rstr, CEED_REQUEST_IMMEDIATE));
591   CeedCall(CeedElemRestrictionGetELayout(assembled_elem_rstr, layout_qf));
592   CeedCall(CeedElemRestrictionDestroy(&assembled_elem_rstr));
593   CeedCall(CeedVectorGetArrayRead(assembled_qf, CEED_MEM_HOST, &assembled_qf_array));
594 
595   // Get assembly data
596   CeedInt                  num_elem_in, elem_size_in, num_comp_in, num_qpts_in;
597   CeedInt                  num_elem_out, elem_size_out, num_comp_out, num_qpts_out;
598   CeedSize                 local_num_entries, count = 0;
599   const CeedEvalMode     **eval_modes_in, **eval_modes_out;
600   CeedInt                  num_active_bases_in, *num_eval_modes_in, num_active_bases_out, *num_eval_modes_out;
601   CeedBasis               *active_bases_in, *active_bases_out, basis_in, basis_out;
602   const CeedScalar       **B_mats_in, **B_mats_out, *B_mat_in, *B_mat_out;
603   CeedElemRestriction      elem_rstr_in, elem_rstr_out;
604   CeedRestrictionType      elem_rstr_type_in, elem_rstr_type_out;
605   const bool              *elem_rstr_orients_in = NULL, *elem_rstr_orients_out = NULL;
606   const CeedInt8          *elem_rstr_curl_orients_in = NULL, *elem_rstr_curl_orients_out = NULL;
607   CeedOperatorAssemblyData data;
608 
609   CeedCall(CeedOperatorGetOperatorAssemblyData(op, &data));
610   CeedCall(CeedOperatorAssemblyDataGetEvalModes(data, &num_active_bases_in, &num_eval_modes_in, &eval_modes_in, NULL, &num_active_bases_out,
611                                                 &num_eval_modes_out, &eval_modes_out, NULL, NULL));
612 
613   CeedCheck(num_active_bases_in == 1 && num_active_bases_out == 1, CeedOperatorReturnCeed(op), CEED_ERROR_UNSUPPORTED,
614             "Cannot assemble operator with multiple active bases");
615   CeedCheck(num_eval_modes_in[0] > 0 && num_eval_modes_out[0] > 0, CeedOperatorReturnCeed(op), CEED_ERROR_UNSUPPORTED,
616             "Cannot assemble operator without inputs/outputs");
617 
618   CeedCall(CeedOperatorAssemblyDataGetBases(data, NULL, &active_bases_in, &B_mats_in, NULL, &active_bases_out, &B_mats_out));
619   CeedCall(CeedOperatorGetActiveElemRestrictions(op, &elem_rstr_in, &elem_rstr_out));
620   basis_in  = active_bases_in[0];
621   basis_out = active_bases_out[0];
622   B_mat_in  = B_mats_in[0];
623   B_mat_out = B_mats_out[0];
624 
625   CeedCall(CeedElemRestrictionGetNumElements(elem_rstr_in, &num_elem_in));
626   CeedCall(CeedElemRestrictionGetElementSize(elem_rstr_in, &elem_size_in));
627   CeedCall(CeedElemRestrictionGetNumComponents(elem_rstr_in, &num_comp_in));
628   if (basis_in == CEED_BASIS_NONE) num_qpts_in = elem_size_in;
629   else CeedCall(CeedBasisGetNumQuadraturePoints(basis_in, &num_qpts_in));
630 
631   CeedCall(CeedElemRestrictionGetType(elem_rstr_in, &elem_rstr_type_in));
632   if (elem_rstr_type_in == CEED_RESTRICTION_ORIENTED) {
633     CeedCall(CeedElemRestrictionGetOrientations(elem_rstr_in, CEED_MEM_HOST, &elem_rstr_orients_in));
634   } else if (elem_rstr_type_in == CEED_RESTRICTION_CURL_ORIENTED) {
635     CeedCall(CeedElemRestrictionGetCurlOrientations(elem_rstr_in, CEED_MEM_HOST, &elem_rstr_curl_orients_in));
636   }
637 
638   if (elem_rstr_in != elem_rstr_out) {
639     CeedCall(CeedElemRestrictionGetNumElements(elem_rstr_out, &num_elem_out));
640     CeedCheck(num_elem_in == num_elem_out, CeedOperatorReturnCeed(op), CEED_ERROR_UNSUPPORTED,
641               "Active input and output operator restrictions must have the same number of elements."
642               " Input has %" CeedInt_FMT " elements; output has %" CeedInt_FMT "elements.",
643               num_elem_in, num_elem_out);
644     CeedCall(CeedElemRestrictionGetElementSize(elem_rstr_out, &elem_size_out));
645     CeedCall(CeedElemRestrictionGetNumComponents(elem_rstr_out, &num_comp_out));
646     if (basis_out == CEED_BASIS_NONE) num_qpts_out = elem_size_out;
647     else CeedCall(CeedBasisGetNumQuadraturePoints(basis_out, &num_qpts_out));
648     CeedCheck(num_qpts_in == num_qpts_out, CeedOperatorReturnCeed(op), CEED_ERROR_UNSUPPORTED,
649               "Active input and output bases must have the same number of quadrature points."
650               " Input has %" CeedInt_FMT " points; output has %" CeedInt_FMT "points.",
651               num_qpts_in, num_qpts_out);
652 
653     CeedCall(CeedElemRestrictionGetType(elem_rstr_out, &elem_rstr_type_out));
654     if (elem_rstr_type_out == CEED_RESTRICTION_ORIENTED) {
655       CeedCall(CeedElemRestrictionGetOrientations(elem_rstr_out, CEED_MEM_HOST, &elem_rstr_orients_out));
656     } else if (elem_rstr_type_out == CEED_RESTRICTION_CURL_ORIENTED) {
657       CeedCall(CeedElemRestrictionGetCurlOrientations(elem_rstr_out, CEED_MEM_HOST, &elem_rstr_curl_orients_out));
658     }
659   } else {
660     num_elem_out  = num_elem_in;
661     elem_size_out = elem_size_in;
662     num_comp_out  = num_comp_in;
663     num_qpts_out  = num_qpts_in;
664 
665     elem_rstr_orients_out      = elem_rstr_orients_in;
666     elem_rstr_curl_orients_out = elem_rstr_curl_orients_in;
667   }
668   local_num_entries = (CeedSize)elem_size_out * num_comp_out * elem_size_in * num_comp_in * num_elem_in;
669 
670   // Loop over elements and put in data structure
671   // We store B_mat_in, B_mat_out, BTD, elem_mat in row-major order
672   CeedTensorContract contract;
673   CeedScalar        *vals, *BTD_mat = NULL, *elem_mat = NULL, *elem_mat_b = NULL;
674 
675   CeedCall(CeedBasisGetTensorContract(basis_in, &contract));
676   CeedCall(CeedCalloc(elem_size_out * num_qpts_in * num_eval_modes_in[0], &BTD_mat));
677   CeedCall(CeedCalloc(elem_size_out * elem_size_in, &elem_mat));
678   if (elem_rstr_curl_orients_in || elem_rstr_curl_orients_out) CeedCall(CeedCalloc(elem_size_out * elem_size_in, &elem_mat_b));
679 
680   CeedCall(CeedVectorGetArray(values, CEED_MEM_HOST, &vals));
681   for (CeedSize e = 0; e < num_elem_in; e++) {
682     for (CeedInt comp_in = 0; comp_in < num_comp_in; comp_in++) {
683       for (CeedInt comp_out = 0; comp_out < num_comp_out; comp_out++) {
684         // Compute B^T*D
685         for (CeedSize n = 0; n < elem_size_out; n++) {
686           for (CeedSize q = 0; q < num_qpts_in; q++) {
687             for (CeedInt e_in = 0; e_in < num_eval_modes_in[0]; e_in++) {
688               const CeedSize btd_index = n * (num_qpts_in * num_eval_modes_in[0]) + q * num_eval_modes_in[0] + e_in;
689               CeedScalar     sum       = 0.0;
690 
691               for (CeedInt e_out = 0; e_out < num_eval_modes_out[0]; e_out++) {
692                 const CeedSize b_out_index     = (q * num_eval_modes_out[0] + e_out) * elem_size_out + n;
693                 const CeedSize eval_mode_index = ((e_in * num_comp_in + comp_in) * num_eval_modes_out[0] + e_out) * num_comp_out + comp_out;
694                 const CeedSize qf_index        = q * layout_qf[0] + eval_mode_index * layout_qf[1] + e * layout_qf[2];
695 
696                 sum += B_mat_out[b_out_index] * assembled_qf_array[qf_index];
697               }
698               BTD_mat[btd_index] = sum;
699             }
700           }
701         }
702 
703         // Form element matrix itself (for each block component)
704         if (contract) {
705           CeedCall(CeedTensorContractApply(contract, 1, num_qpts_in * num_eval_modes_in[0], elem_size_in, elem_size_out, BTD_mat, CEED_NOTRANSPOSE,
706                                            false, B_mat_in, elem_mat));
707         } else {
708           Ceed ceed;
709 
710           CeedCall(CeedOperatorGetCeed(op, &ceed));
711           CeedCall(CeedMatrixMatrixMultiply(ceed, BTD_mat, B_mat_in, elem_mat, elem_size_out, elem_size_in, num_qpts_in * num_eval_modes_in[0]));
712           CeedCall(CeedDestroy(&ceed));
713         }
714 
715         // Transform the element matrix if required
716         if (elem_rstr_orients_out) {
717           const bool *elem_orients = &elem_rstr_orients_out[e * elem_size_out];
718 
719           for (CeedInt i = 0; i < elem_size_out; i++) {
720             const double orient = elem_orients[i] ? -1.0 : 1.0;
721 
722             for (CeedInt j = 0; j < elem_size_in; j++) {
723               elem_mat[i * elem_size_in + j] *= orient;
724             }
725           }
726         } else if (elem_rstr_curl_orients_out) {
727           const CeedInt8 *elem_curl_orients = &elem_rstr_curl_orients_out[e * 3 * elem_size_out];
728 
729           // T^T*(B^T*D*B)
730           memcpy(elem_mat_b, elem_mat, elem_size_out * elem_size_in * sizeof(CeedScalar));
731           for (CeedInt i = 0; i < elem_size_out; i++) {
732             for (CeedInt j = 0; j < elem_size_in; j++) {
733               elem_mat[i * elem_size_in + j] = elem_mat_b[i * elem_size_in + j] * elem_curl_orients[3 * i + 1] +
734                                                (i > 0 ? elem_mat_b[(i - 1) * elem_size_in + j] * elem_curl_orients[3 * i - 1] : 0.0) +
735                                                (i < elem_size_out - 1 ? elem_mat_b[(i + 1) * elem_size_in + j] * elem_curl_orients[3 * i + 3] : 0.0);
736             }
737           }
738         }
739         if (elem_rstr_orients_in) {
740           const bool *elem_orients = &elem_rstr_orients_in[e * elem_size_in];
741 
742           for (CeedInt i = 0; i < elem_size_out; i++) {
743             for (CeedInt j = 0; j < elem_size_in; j++) {
744               elem_mat[i * elem_size_in + j] *= elem_orients[j] ? -1.0 : 1.0;
745             }
746           }
747         } else if (elem_rstr_curl_orients_in) {
748           const CeedInt8 *elem_curl_orients = &elem_rstr_curl_orients_in[e * 3 * elem_size_in];
749 
750           // (B^T*D*B)*T
751           memcpy(elem_mat_b, elem_mat, elem_size_out * elem_size_in * sizeof(CeedScalar));
752           for (CeedInt i = 0; i < elem_size_out; i++) {
753             for (CeedInt j = 0; j < elem_size_in; j++) {
754               elem_mat[i * elem_size_in + j] = elem_mat_b[i * elem_size_in + j] * elem_curl_orients[3 * j + 1] +
755                                                (j > 0 ? elem_mat_b[i * elem_size_in + j - 1] * elem_curl_orients[3 * j - 1] : 0.0) +
756                                                (j < elem_size_in - 1 ? elem_mat_b[i * elem_size_in + j + 1] * elem_curl_orients[3 * j + 3] : 0.0);
757             }
758           }
759         }
760 
761         // Put element matrix in coordinate data structure
762         for (CeedInt i = 0; i < elem_size_out; i++) {
763           for (CeedInt j = 0; j < elem_size_in; j++) {
764             vals[offset + count] = elem_mat[i * elem_size_in + j];
765             count++;
766           }
767         }
768       }
769     }
770   }
771   CeedCheck(count == local_num_entries, CeedOperatorReturnCeed(op), CEED_ERROR_MAJOR, "Error computing entries");
772   CeedCall(CeedVectorRestoreArray(values, &vals));
773 
774   // Cleanup
775   CeedCall(CeedFree(&BTD_mat));
776   CeedCall(CeedFree(&elem_mat));
777   CeedCall(CeedFree(&elem_mat_b));
778   if (elem_rstr_type_in == CEED_RESTRICTION_ORIENTED) {
779     CeedCall(CeedElemRestrictionRestoreOrientations(elem_rstr_in, &elem_rstr_orients_in));
780   } else if (elem_rstr_type_in == CEED_RESTRICTION_CURL_ORIENTED) {
781     CeedCall(CeedElemRestrictionRestoreCurlOrientations(elem_rstr_in, &elem_rstr_curl_orients_in));
782   }
783   if (elem_rstr_in != elem_rstr_out) {
784     if (elem_rstr_type_out == CEED_RESTRICTION_ORIENTED) {
785       CeedCall(CeedElemRestrictionRestoreOrientations(elem_rstr_out, &elem_rstr_orients_out));
786     } else if (elem_rstr_type_out == CEED_RESTRICTION_CURL_ORIENTED) {
787       CeedCall(CeedElemRestrictionRestoreCurlOrientations(elem_rstr_out, &elem_rstr_curl_orients_out));
788     }
789   }
790   CeedCall(CeedVectorRestoreArrayRead(assembled_qf, &assembled_qf_array));
791   CeedCall(CeedVectorDestroy(&assembled_qf));
792   CeedCall(CeedElemRestrictionDestroy(&elem_rstr_in));
793   CeedCall(CeedElemRestrictionDestroy(&elem_rstr_out));
794   return CEED_ERROR_SUCCESS;
795 }
796 
797 /**
798   @brief Count number of entries for assembled `CeedOperator`
799 
800   @param[in]  op          `CeedOperator` to assemble
801   @param[out] num_entries Number of entries in assembled representation
802 
803   @return An error code: 0 - success, otherwise - failure
804 
805   @ref Utility
806 **/
807 static int CeedSingleOperatorAssemblyCountEntries(CeedOperator op, CeedSize *num_entries) {
808   bool                is_composite;
809   CeedInt             num_elem_in, elem_size_in, num_comp_in, num_elem_out, elem_size_out, num_comp_out;
810   CeedElemRestriction rstr_in, rstr_out;
811 
812   CeedCall(CeedOperatorIsComposite(op, &is_composite));
813   CeedCheck(!is_composite, CeedOperatorReturnCeed(op), CEED_ERROR_UNSUPPORTED, "Composite operator not supported");
814 
815   CeedCall(CeedOperatorGetActiveElemRestrictions(op, &rstr_in, &rstr_out));
816   CeedCall(CeedElemRestrictionGetNumElements(rstr_in, &num_elem_in));
817   CeedCall(CeedElemRestrictionGetElementSize(rstr_in, &elem_size_in));
818   CeedCall(CeedElemRestrictionGetNumComponents(rstr_in, &num_comp_in));
819   if (rstr_in != rstr_out) {
820     CeedCall(CeedElemRestrictionGetNumElements(rstr_out, &num_elem_out));
821     CeedCheck(num_elem_in == num_elem_out, CeedOperatorReturnCeed(op), CEED_ERROR_UNSUPPORTED,
822               "Active input and output operator restrictions must have the same number of elements."
823               " Input has %" CeedInt_FMT " elements; output has %" CeedInt_FMT "elements.",
824               num_elem_in, num_elem_out);
825     CeedCall(CeedElemRestrictionGetElementSize(rstr_out, &elem_size_out));
826     CeedCall(CeedElemRestrictionGetNumComponents(rstr_out, &num_comp_out));
827   } else {
828     num_elem_out  = num_elem_in;
829     elem_size_out = elem_size_in;
830     num_comp_out  = num_comp_in;
831   }
832   CeedCall(CeedElemRestrictionDestroy(&rstr_in));
833   CeedCall(CeedElemRestrictionDestroy(&rstr_out));
834   *num_entries = (CeedSize)elem_size_in * num_comp_in * elem_size_out * num_comp_out * num_elem_in;
835   return CEED_ERROR_SUCCESS;
836 }
837 
838 /**
839   @brief Common code for creating a multigrid coarse `CeedOperator` and level transfer `CeedOperator` for a `CeedOperator`
840 
841   @param[in]  op_fine      Fine grid `CeedOperator`
842   @param[in]  p_mult_fine  L-vector multiplicity in parallel gather/scatter, or `NULL` if not creating prolongation/restriction `CeedOperator`
843   @param[in]  rstr_coarse  Coarse grid `CeedElemRestriction`
844   @param[in]  basis_coarse Coarse grid active vector `CeedBasis`
845   @param[in]  basis_c_to_f `CeedBasis` for coarse to fine interpolation, or `NULL` if not creating prolongation/restriction operators
846   @param[out] op_coarse    Coarse grid `CeedOperator`
847   @param[out] op_prolong   Coarse to fine `CeedOperator`, or `NULL`
848   @param[out] op_restrict  Fine to coarse `CeedOperator`, or `NULL`
849 
850   @return An error code: 0 - success, otherwise - failure
851 
852   @ref Developer
853 **/
854 static int CeedSingleOperatorMultigridLevel(CeedOperator op_fine, CeedVector p_mult_fine, CeedElemRestriction rstr_coarse, CeedBasis basis_coarse,
855                                             CeedBasis basis_c_to_f, CeedOperator *op_coarse, CeedOperator *op_prolong, CeedOperator *op_restrict) {
856   bool                is_composite;
857   Ceed                ceed;
858   CeedInt             num_comp, num_input_fields, num_output_fields;
859   CeedVector          mult_vec         = NULL;
860   CeedElemRestriction rstr_p_mult_fine = NULL, rstr_fine = NULL;
861   CeedOperatorField  *input_fields, *output_fields;
862 
863   CeedCall(CeedOperatorGetCeed(op_fine, &ceed));
864 
865   // Check for composite operator
866   CeedCall(CeedOperatorIsComposite(op_fine, &is_composite));
867   CeedCheck(!is_composite, ceed, CEED_ERROR_UNSUPPORTED, "Automatic multigrid setup for composite operators not supported");
868 
869   // Coarse Grid
870   CeedCall(CeedOperatorCreate(ceed, op_fine->qf, op_fine->dqf, op_fine->dqfT, op_coarse));
871   CeedCall(CeedOperatorGetFields(op_fine, &num_input_fields, &input_fields, &num_output_fields, &output_fields));
872   // -- Clone input fields
873   for (CeedInt i = 0; i < num_input_fields; i++) {
874     const char         *field_name;
875     CeedVector          vec;
876     CeedElemRestriction rstr  = NULL;
877     CeedBasis           basis = NULL;
878 
879     CeedCall(CeedOperatorFieldGetName(input_fields[i], &field_name));
880     CeedCall(CeedOperatorFieldGetVector(input_fields[i], &vec));
881     if (vec == CEED_VECTOR_ACTIVE) {
882       CeedCall(CeedElemRestrictionReferenceCopy(rstr_coarse, &rstr));
883       CeedCall(CeedBasisReferenceCopy(basis_coarse, &basis));
884       if (!rstr_fine) CeedCall(CeedOperatorFieldGetElemRestriction(input_fields[i], &rstr_fine));
885     } else {
886       CeedCall(CeedOperatorFieldGetElemRestriction(input_fields[i], &rstr));
887       CeedCall(CeedOperatorFieldGetBasis(input_fields[i], &basis));
888     }
889     CeedCall(CeedOperatorSetField(*op_coarse, field_name, rstr, basis, vec));
890     CeedCall(CeedVectorDestroy(&vec));
891     CeedCall(CeedElemRestrictionDestroy(&rstr));
892     CeedCall(CeedBasisDestroy(&basis));
893   }
894   // -- Clone output fields
895   for (CeedInt i = 0; i < num_output_fields; i++) {
896     const char         *field_name;
897     CeedVector          vec;
898     CeedElemRestriction rstr  = NULL;
899     CeedBasis           basis = NULL;
900 
901     CeedCall(CeedOperatorFieldGetName(output_fields[i], &field_name));
902     CeedCall(CeedOperatorFieldGetVector(output_fields[i], &vec));
903     if (vec == CEED_VECTOR_ACTIVE) {
904       CeedCall(CeedElemRestrictionReferenceCopy(rstr_coarse, &rstr));
905       CeedCall(CeedBasisReferenceCopy(basis_coarse, &basis));
906       if (!rstr_fine) CeedCall(CeedOperatorFieldGetElemRestriction(output_fields[i], &rstr_fine));
907     } else {
908       CeedCall(CeedOperatorFieldGetElemRestriction(output_fields[i], &rstr));
909       CeedCall(CeedOperatorFieldGetBasis(output_fields[i], &basis));
910     }
911     CeedCall(CeedOperatorSetField(*op_coarse, field_name, rstr, basis, vec));
912     CeedCall(CeedVectorDestroy(&vec));
913     CeedCall(CeedElemRestrictionDestroy(&rstr));
914     CeedCall(CeedBasisDestroy(&basis));
915   }
916   // -- Clone QFunctionAssemblyData
917   {
918     CeedQFunctionAssemblyData fine_data;
919 
920     CeedCall(CeedOperatorGetQFunctionAssemblyData(op_fine, &fine_data));
921     CeedCall(CeedQFunctionAssemblyDataReferenceCopy(fine_data, &(*op_coarse)->qf_assembled));
922   }
923 
924   // Multiplicity vector
925   if (op_restrict || op_prolong) {
926     CeedVector          mult_e_vec;
927     CeedRestrictionType rstr_type;
928 
929     CeedCall(CeedElemRestrictionGetType(rstr_fine, &rstr_type));
930     CeedCheck(rstr_type != CEED_RESTRICTION_CURL_ORIENTED, ceed, CEED_ERROR_UNSUPPORTED,
931               "Element restrictions created with CeedElemRestrictionCreateCurlOriented are not supported");
932     CeedCheck(p_mult_fine, ceed, CEED_ERROR_INCOMPATIBLE, "Prolongation or restriction operator creation requires fine grid multiplicity vector");
933     CeedCall(CeedElemRestrictionCreateUnsignedCopy(rstr_fine, &rstr_p_mult_fine));
934     CeedCall(CeedElemRestrictionCreateVector(rstr_fine, &mult_vec, &mult_e_vec));
935     CeedCall(CeedVectorSetValue(mult_e_vec, 0.0));
936     CeedCall(CeedElemRestrictionApply(rstr_p_mult_fine, CEED_NOTRANSPOSE, p_mult_fine, mult_e_vec, CEED_REQUEST_IMMEDIATE));
937     CeedCall(CeedVectorSetValue(mult_vec, 0.0));
938     CeedCall(CeedElemRestrictionApply(rstr_p_mult_fine, CEED_TRANSPOSE, mult_e_vec, mult_vec, CEED_REQUEST_IMMEDIATE));
939     CeedCall(CeedVectorDestroy(&mult_e_vec));
940     CeedCall(CeedVectorReciprocal(mult_vec));
941   }
942 
943   // Clone name
944   bool   has_name = op_fine->name;
945   size_t name_len = op_fine->name ? strlen(op_fine->name) : 0;
946   CeedCall(CeedOperatorSetName(*op_coarse, op_fine->name));
947 
948   // Check that coarse to fine basis is provided if prolong/restrict operators are requested
949   CeedCheck(basis_c_to_f || (!op_restrict && !op_prolong), ceed, CEED_ERROR_INCOMPATIBLE,
950             "Prolongation or restriction operator creation requires coarse-to-fine basis");
951 
952   // Restriction/Prolongation Operators
953   CeedCall(CeedBasisGetNumComponents(basis_coarse, &num_comp));
954 
955   // Restriction
956   if (op_restrict) {
957     CeedInt             *num_comp_r_data;
958     CeedQFunctionContext ctx_r;
959     CeedQFunction        qf_restrict;
960 
961     CeedCall(CeedQFunctionCreateInteriorByName(ceed, "Scale", &qf_restrict));
962     CeedCall(CeedCalloc(1, &num_comp_r_data));
963     num_comp_r_data[0] = num_comp;
964     CeedCall(CeedQFunctionContextCreate(ceed, &ctx_r));
965     CeedCall(CeedQFunctionContextSetData(ctx_r, CEED_MEM_HOST, CEED_OWN_POINTER, sizeof(*num_comp_r_data), num_comp_r_data));
966     CeedCall(CeedQFunctionSetContext(qf_restrict, ctx_r));
967     CeedCall(CeedQFunctionContextDestroy(&ctx_r));
968     CeedCall(CeedQFunctionAddInput(qf_restrict, "input", num_comp, CEED_EVAL_NONE));
969     CeedCall(CeedQFunctionAddInput(qf_restrict, "scale", num_comp, CEED_EVAL_NONE));
970     CeedCall(CeedQFunctionAddOutput(qf_restrict, "output", num_comp, CEED_EVAL_INTERP));
971     CeedCall(CeedQFunctionSetUserFlopsEstimate(qf_restrict, num_comp));
972 
973     CeedCall(CeedOperatorCreate(ceed, qf_restrict, CEED_QFUNCTION_NONE, CEED_QFUNCTION_NONE, op_restrict));
974     CeedCall(CeedOperatorSetField(*op_restrict, "input", rstr_fine, CEED_BASIS_NONE, CEED_VECTOR_ACTIVE));
975     CeedCall(CeedOperatorSetField(*op_restrict, "scale", rstr_p_mult_fine, CEED_BASIS_NONE, mult_vec));
976     CeedCall(CeedOperatorSetField(*op_restrict, "output", rstr_coarse, basis_c_to_f, CEED_VECTOR_ACTIVE));
977 
978     // Set name
979     char *restriction_name;
980 
981     CeedCall(CeedCalloc(17 + name_len, &restriction_name));
982     sprintf(restriction_name, "restriction%s%s", has_name ? " for " : "", has_name ? op_fine->name : "");
983     CeedCall(CeedOperatorSetName(*op_restrict, restriction_name));
984     CeedCall(CeedFree(&restriction_name));
985 
986     // Check
987     CeedCall(CeedOperatorCheckReady(*op_restrict));
988 
989     // Cleanup
990     CeedCall(CeedQFunctionDestroy(&qf_restrict));
991   }
992 
993   // Prolongation
994   if (op_prolong) {
995     CeedInt             *num_comp_p_data;
996     CeedQFunctionContext ctx_p;
997     CeedQFunction        qf_prolong;
998 
999     CeedCall(CeedQFunctionCreateInteriorByName(ceed, "Scale", &qf_prolong));
1000     CeedCall(CeedCalloc(1, &num_comp_p_data));
1001     num_comp_p_data[0] = num_comp;
1002     CeedCall(CeedQFunctionContextCreate(ceed, &ctx_p));
1003     CeedCall(CeedQFunctionContextSetData(ctx_p, CEED_MEM_HOST, CEED_OWN_POINTER, sizeof(*num_comp_p_data), num_comp_p_data));
1004     CeedCall(CeedQFunctionSetContext(qf_prolong, ctx_p));
1005     CeedCall(CeedQFunctionContextDestroy(&ctx_p));
1006     CeedCall(CeedQFunctionAddInput(qf_prolong, "input", num_comp, CEED_EVAL_INTERP));
1007     CeedCall(CeedQFunctionAddInput(qf_prolong, "scale", num_comp, CEED_EVAL_NONE));
1008     CeedCall(CeedQFunctionAddOutput(qf_prolong, "output", num_comp, CEED_EVAL_NONE));
1009     CeedCall(CeedQFunctionSetUserFlopsEstimate(qf_prolong, num_comp));
1010 
1011     CeedCall(CeedOperatorCreate(ceed, qf_prolong, CEED_QFUNCTION_NONE, CEED_QFUNCTION_NONE, op_prolong));
1012     CeedCall(CeedOperatorSetField(*op_prolong, "input", rstr_coarse, basis_c_to_f, CEED_VECTOR_ACTIVE));
1013     CeedCall(CeedOperatorSetField(*op_prolong, "scale", rstr_p_mult_fine, CEED_BASIS_NONE, mult_vec));
1014     CeedCall(CeedOperatorSetField(*op_prolong, "output", rstr_fine, CEED_BASIS_NONE, CEED_VECTOR_ACTIVE));
1015 
1016     // Set name
1017     char *prolongation_name;
1018 
1019     CeedCall(CeedCalloc(18 + name_len, &prolongation_name));
1020     sprintf(prolongation_name, "prolongation%s%s", has_name ? " for " : "", has_name ? op_fine->name : "");
1021     CeedCall(CeedOperatorSetName(*op_prolong, prolongation_name));
1022     CeedCall(CeedFree(&prolongation_name));
1023 
1024     // Check
1025     CeedCall(CeedOperatorCheckReady(*op_prolong));
1026 
1027     // Cleanup
1028     CeedCall(CeedQFunctionDestroy(&qf_prolong));
1029   }
1030 
1031   // Check
1032   CeedCall(CeedOperatorCheckReady(*op_coarse));
1033 
1034   // Cleanup
1035   CeedCall(CeedDestroy(&ceed));
1036   CeedCall(CeedVectorDestroy(&mult_vec));
1037   CeedCall(CeedElemRestrictionDestroy(&rstr_fine));
1038   CeedCall(CeedElemRestrictionDestroy(&rstr_p_mult_fine));
1039   CeedCall(CeedBasisDestroy(&basis_c_to_f));
1040   return CEED_ERROR_SUCCESS;
1041 }
1042 
1043 /**
1044   @brief Build 1D mass matrix and Laplacian with perturbation
1045 
1046   @param[in]  interp_1d   Interpolation matrix in one dimension
1047   @param[in]  grad_1d     Gradient matrix in one dimension
1048   @param[in]  q_weight_1d Quadrature weights in one dimension
1049   @param[in]  P_1d        Number of basis nodes in one dimension
1050   @param[in]  Q_1d        Number of quadrature points in one dimension
1051   @param[in]  dim         Dimension of basis
1052   @param[out] mass        Assembled mass matrix in one dimension
1053   @param[out] laplace     Assembled perturbed Laplacian in one dimension
1054 
1055   @return An error code: 0 - success, otherwise - failure
1056 
1057   @ref Developer
1058 **/
1059 CeedPragmaOptimizeOff
1060 static int CeedBuildMassLaplace(const CeedScalar *interp_1d, const CeedScalar *grad_1d, const CeedScalar *q_weight_1d, CeedInt P_1d, CeedInt Q_1d,
1061                                 CeedInt dim, CeedScalar *mass, CeedScalar *laplace) {
1062   for (CeedInt i = 0; i < P_1d; i++) {
1063     for (CeedInt j = 0; j < P_1d; j++) {
1064       CeedScalar sum = 0.0;
1065       for (CeedInt k = 0; k < Q_1d; k++) sum += interp_1d[k * P_1d + i] * q_weight_1d[k] * interp_1d[k * P_1d + j];
1066       mass[i + j * P_1d] = sum;
1067     }
1068   }
1069   // -- Laplacian
1070   for (CeedInt i = 0; i < P_1d; i++) {
1071     for (CeedInt j = 0; j < P_1d; j++) {
1072       CeedScalar sum = 0.0;
1073 
1074       for (CeedInt k = 0; k < Q_1d; k++) sum += grad_1d[k * P_1d + i] * q_weight_1d[k] * grad_1d[k * P_1d + j];
1075       laplace[i + j * P_1d] = sum;
1076     }
1077   }
1078   CeedScalar perturbation = dim > 2 ? 1e-6 : 1e-4;
1079   for (CeedInt i = 0; i < P_1d; i++) laplace[i + P_1d * i] += perturbation;
1080   return CEED_ERROR_SUCCESS;
1081 }
1082 CeedPragmaOptimizeOn
1083 
1084 /// @}
1085 
1086 /// ----------------------------------------------------------------------------
1087 /// CeedOperator Backend API
1088 /// ----------------------------------------------------------------------------
1089 /// @addtogroup CeedOperatorBackend
1090 /// @{
1091 
1092 /**
1093   @brief Select correct basis matrix pointer based on @ref CeedEvalMode
1094 
1095   @param[in]  basis     `CeedBasis` from which to get the basis matrix
1096   @param[in]  eval_mode Current basis evaluation mode
1097   @param[in]  identity  Pointer to identity matrix
1098   @param[out] basis_ptr `CeedBasis` pointer to set
1099 
1100   @ref Backend
1101 **/
1102 int CeedOperatorGetBasisPointer(CeedBasis basis, CeedEvalMode eval_mode, const CeedScalar *identity, const CeedScalar **basis_ptr) {
1103   switch (eval_mode) {
1104     case CEED_EVAL_NONE:
1105       *basis_ptr = identity;
1106       break;
1107     case CEED_EVAL_INTERP:
1108       CeedCall(CeedBasisGetInterp(basis, basis_ptr));
1109       break;
1110     case CEED_EVAL_GRAD:
1111       CeedCall(CeedBasisGetGrad(basis, basis_ptr));
1112       break;
1113     case CEED_EVAL_DIV:
1114       CeedCall(CeedBasisGetDiv(basis, basis_ptr));
1115       break;
1116     case CEED_EVAL_CURL:
1117       CeedCall(CeedBasisGetCurl(basis, basis_ptr));
1118       break;
1119     case CEED_EVAL_WEIGHT:
1120       break;  // Caught by QF Assembly
1121   }
1122   assert(*basis_ptr != NULL);
1123   return CEED_ERROR_SUCCESS;
1124 }
1125 
1126 /**
1127   @brief Create point block restriction for active `CeedOperatorField`
1128 
1129   @param[in]  rstr             Original `CeedElemRestriction` for active field
1130   @param[out] point_block_rstr Address of the variable where the newly created `CeedElemRestriction` will be stored
1131 
1132   @return An error code: 0 - success, otherwise - failure
1133 
1134   @ref Backend
1135 **/
1136 int CeedOperatorCreateActivePointBlockRestriction(CeedElemRestriction rstr, CeedElemRestriction *point_block_rstr) {
1137   Ceed           ceed;
1138   CeedInt        num_elem, num_comp, shift, elem_size, comp_stride, *point_block_offsets;
1139   CeedSize       l_size;
1140   const CeedInt *offsets;
1141 
1142   CeedCall(CeedElemRestrictionGetCeed(rstr, &ceed));
1143   CeedCall(CeedElemRestrictionGetOffsets(rstr, CEED_MEM_HOST, &offsets));
1144 
1145   // Expand offsets
1146   CeedCall(CeedElemRestrictionGetNumElements(rstr, &num_elem));
1147   CeedCall(CeedElemRestrictionGetNumComponents(rstr, &num_comp));
1148   CeedCall(CeedElemRestrictionGetElementSize(rstr, &elem_size));
1149   CeedCall(CeedElemRestrictionGetCompStride(rstr, &comp_stride));
1150   CeedCall(CeedElemRestrictionGetLVectorSize(rstr, &l_size));
1151   shift = num_comp;
1152   if (comp_stride != 1) shift *= num_comp;
1153   CeedCall(CeedCalloc(num_elem * elem_size, &point_block_offsets));
1154   for (CeedInt i = 0; i < num_elem * elem_size; i++) {
1155     point_block_offsets[i] = offsets[i] * shift;
1156   }
1157 
1158   // Create new restriction
1159   CeedCall(CeedElemRestrictionCreate(ceed, num_elem, elem_size, num_comp * num_comp, 1, l_size * num_comp, CEED_MEM_HOST, CEED_OWN_POINTER,
1160                                      point_block_offsets, point_block_rstr));
1161 
1162   // Cleanup
1163   CeedCall(CeedElemRestrictionRestoreOffsets(rstr, &offsets));
1164   CeedCall(CeedDestroy(&ceed));
1165   return CEED_ERROR_SUCCESS;
1166 }
1167 
1168 /**
1169   @brief Get `CeedQFunctionAssemblyData`
1170 
1171   @param[in]  op   `CeedOperator` to assemble
1172   @param[out] data `CeedQFunctionAssemblyData`
1173 
1174   @return An error code: 0 - success, otherwise - failure
1175 
1176   @ref Backend
1177 **/
1178 int CeedOperatorGetQFunctionAssemblyData(CeedOperator op, CeedQFunctionAssemblyData *data) {
1179   if (!op->qf_assembled) {
1180     CeedQFunctionAssemblyData data;
1181 
1182     CeedCall(CeedQFunctionAssemblyDataCreate(op->ceed, &data));
1183     op->qf_assembled = data;
1184   }
1185   *data = op->qf_assembled;
1186   return CEED_ERROR_SUCCESS;
1187 }
1188 
1189 /**
1190   @brief Create object holding `CeedQFunction` assembly data for `CeedOperator`
1191 
1192   @param[in]  ceed `Ceed` object used to create the `CeedQFunctionAssemblyData`
1193   @param[out] data Address of the variable where the newly created `CeedQFunctionAssemblyData` will be stored
1194 
1195   @return An error code: 0 - success, otherwise - failure
1196 
1197   @ref Backend
1198 **/
1199 int CeedQFunctionAssemblyDataCreate(Ceed ceed, CeedQFunctionAssemblyData *data) {
1200   CeedCall(CeedCalloc(1, data));
1201   (*data)->ref_count = 1;
1202   (*data)->ceed      = ceed;
1203   CeedCall(CeedReference(ceed));
1204   return CEED_ERROR_SUCCESS;
1205 }
1206 
1207 /**
1208   @brief Increment the reference counter for a `CeedQFunctionAssemblyData`
1209 
1210   @param[in,out] data `CeedQFunctionAssemblyData` to increment the reference counter
1211 
1212   @return An error code: 0 - success, otherwise - failure
1213 
1214   @ref Backend
1215 **/
1216 int CeedQFunctionAssemblyDataReference(CeedQFunctionAssemblyData data) {
1217   data->ref_count++;
1218   return CEED_ERROR_SUCCESS;
1219 }
1220 
1221 /**
1222   @brief Set re-use of `CeedQFunctionAssemblyData`
1223 
1224   @param[in,out] data       `CeedQFunctionAssemblyData` to mark for reuse
1225   @param[in]     reuse_data Boolean flag indicating data re-use
1226 
1227   @return An error code: 0 - success, otherwise - failure
1228 
1229   @ref Backend
1230 **/
1231 int CeedQFunctionAssemblyDataSetReuse(CeedQFunctionAssemblyData data, bool reuse_data) {
1232   data->reuse_data        = reuse_data;
1233   data->needs_data_update = true;
1234   return CEED_ERROR_SUCCESS;
1235 }
1236 
1237 /**
1238   @brief Mark `CeedQFunctionAssemblyData` as stale
1239 
1240   @param[in,out] data              `CeedQFunctionAssemblyData` to mark as stale
1241   @param[in]     needs_data_update Boolean flag indicating if update is needed or completed
1242 
1243   @return An error code: 0 - success, otherwise - failure
1244 
1245   @ref Backend
1246 **/
1247 int CeedQFunctionAssemblyDataSetUpdateNeeded(CeedQFunctionAssemblyData data, bool needs_data_update) {
1248   data->needs_data_update = needs_data_update;
1249   return CEED_ERROR_SUCCESS;
1250 }
1251 
1252 /**
1253   @brief Determine if `CeedQFunctionAssemblyData` needs update
1254 
1255   @param[in]  data             `CeedQFunctionAssemblyData` to mark as stale
1256   @param[out] is_update_needed Boolean flag indicating if re-assembly is required
1257 
1258   @return An error code: 0 - success, otherwise - failure
1259 
1260   @ref Backend
1261 **/
1262 int CeedQFunctionAssemblyDataIsUpdateNeeded(CeedQFunctionAssemblyData data, bool *is_update_needed) {
1263   *is_update_needed = !data->reuse_data || data->needs_data_update;
1264   return CEED_ERROR_SUCCESS;
1265 }
1266 
1267 /**
1268   @brief Copy the pointer to a `CeedQFunctionAssemblyData`.
1269 
1270   Both pointers should be destroyed with @ref CeedQFunctionAssemblyDataDestroy().
1271 
1272   Note: If the value of ` *data_copy` passed to this function is non-`NULL` , then it is assumed that ` *data_copy` is a pointer to a `CeedQFunctionAssemblyData`.
1273         This `CeedQFunctionAssemblyData` will be destroyed if ` *data_copy` is the only reference to this `CeedQFunctionAssemblyData`.
1274 
1275   @param[in]     data      `CeedQFunctionAssemblyData` to copy reference to
1276   @param[in,out] data_copy Variable to store copied reference
1277 
1278   @return An error code: 0 - success, otherwise - failure
1279 
1280   @ref Backend
1281 **/
1282 int CeedQFunctionAssemblyDataReferenceCopy(CeedQFunctionAssemblyData data, CeedQFunctionAssemblyData *data_copy) {
1283   CeedCall(CeedQFunctionAssemblyDataReference(data));
1284   CeedCall(CeedQFunctionAssemblyDataDestroy(data_copy));
1285   *data_copy = data;
1286   return CEED_ERROR_SUCCESS;
1287 }
1288 
1289 /**
1290   @brief Get setup status for internal objects for `CeedQFunctionAssemblyData`
1291 
1292   @param[in]  data     `CeedQFunctionAssemblyData` to retrieve status
1293   @param[out] is_setup Boolean flag for setup status
1294 
1295   @return An error code: 0 - success, otherwise - failure
1296 
1297   @ref Backend
1298 **/
1299 int CeedQFunctionAssemblyDataIsSetup(CeedQFunctionAssemblyData data, bool *is_setup) {
1300   *is_setup = data->is_setup;
1301   return CEED_ERROR_SUCCESS;
1302 }
1303 
1304 /**
1305   @brief Set internal objects for `CeedQFunctionAssemblyData`
1306 
1307   @param[in,out] data `CeedQFunctionAssemblyData` to set objects
1308   @param[in]     vec  `CeedVector` to store assembled `CeedQFunction` at quadrature points
1309   @param[in]     rstr `CeedElemRestriction` for `CeedVector` containing assembled `CeedQFunction`
1310 
1311   @return An error code: 0 - success, otherwise - failure
1312 
1313   @ref Backend
1314 **/
1315 int CeedQFunctionAssemblyDataSetObjects(CeedQFunctionAssemblyData data, CeedVector vec, CeedElemRestriction rstr) {
1316   CeedCall(CeedVectorReferenceCopy(vec, &data->vec));
1317   CeedCall(CeedElemRestrictionReferenceCopy(rstr, &data->rstr));
1318 
1319   data->is_setup = true;
1320   return CEED_ERROR_SUCCESS;
1321 }
1322 
1323 /**
1324   @brief Get internal objects for `CeedQFunctionAssemblyData`
1325 
1326   @param[in,out] data `CeedQFunctionAssemblyData` to set objects
1327   @param[out]    vec  `CeedVector` to store assembled `CeedQFunction` at quadrature points
1328   @param[out]    rstr `CeedElemRestriction` for `CeedVector` containing assembled `CeedQFunction`
1329 
1330   @return An error code: 0 - success, otherwise - failure
1331 
1332   @ref Backend
1333 **/
1334 int CeedQFunctionAssemblyDataGetObjects(CeedQFunctionAssemblyData data, CeedVector *vec, CeedElemRestriction *rstr) {
1335   CeedCheck(data->is_setup, data->ceed, CEED_ERROR_INCOMPLETE, "Internal objects not set; must call CeedQFunctionAssemblyDataSetObjects first.");
1336 
1337   CeedCall(CeedVectorReferenceCopy(data->vec, vec));
1338   CeedCall(CeedElemRestrictionReferenceCopy(data->rstr, rstr));
1339   return CEED_ERROR_SUCCESS;
1340 }
1341 
1342 /**
1343   @brief Destroy `CeedQFunctionAssemblyData`
1344 
1345   @param[in,out] data  `CeedQFunctionAssemblyData` to destroy
1346 
1347   @return An error code: 0 - success, otherwise - failure
1348 
1349   @ref Backend
1350 **/
1351 int CeedQFunctionAssemblyDataDestroy(CeedQFunctionAssemblyData *data) {
1352   if (!*data || --(*data)->ref_count > 0) {
1353     *data = NULL;
1354     return CEED_ERROR_SUCCESS;
1355   }
1356   CeedCall(CeedDestroy(&(*data)->ceed));
1357   CeedCall(CeedVectorDestroy(&(*data)->vec));
1358   CeedCall(CeedElemRestrictionDestroy(&(*data)->rstr));
1359 
1360   CeedCall(CeedFree(data));
1361   return CEED_ERROR_SUCCESS;
1362 }
1363 
1364 /**
1365   @brief Get `CeedOperatorAssemblyData`
1366 
1367   @param[in]  op   `CeedOperator` to assemble
1368   @param[out] data `CeedOperatorAssemblyData`
1369 
1370   @return An error code: 0 - success, otherwise - failure
1371 
1372   @ref Backend
1373 **/
1374 int CeedOperatorGetOperatorAssemblyData(CeedOperator op, CeedOperatorAssemblyData *data) {
1375   if (!op->op_assembled) {
1376     CeedOperatorAssemblyData data;
1377 
1378     CeedCall(CeedOperatorAssemblyDataCreate(op->ceed, op, &data));
1379     op->op_assembled = data;
1380   }
1381   *data = op->op_assembled;
1382   return CEED_ERROR_SUCCESS;
1383 }
1384 
1385 /**
1386   @brief Create object holding `CeedOperator` assembly data.
1387 
1388   The `CeedOperatorAssemblyData` holds an array with references to every active `CeedBasis` used in the `CeedOperator`.
1389   An array with references to the corresponding active `CeedElemRestriction` is also stored.
1390   For each active `CeedBasis, the `CeedOperatorAssemblyData` holds an array of all input and output @ref CeedEvalMode for this `CeedBasis`.
1391   The `CeedOperatorAssemblyData` holds an array of offsets for indexing into the assembled `CeedQFunction` arrays to the row representing each @ref CeedEvalMode.
1392   The number of input columns across all active bases for the assembled `CeedQFunction` is also stored.
1393   Lastly, the `CeedOperatorAssembly` data holds assembled matrices representing the full action of the `CeedBasis` for all @ref CeedEvalMode.
1394 
1395   @param[in]  ceed `Ceed` object used to create the `CeedOperatorAssemblyData`
1396   @param[in]  op   `CeedOperator` to be assembled
1397   @param[out] data Address of the variable where the newly created `CeedOperatorAssemblyData` will be stored
1398 
1399   @return An error code: 0 - success, otherwise - failure
1400 
1401   @ref Backend
1402 **/
1403 int CeedOperatorAssemblyDataCreate(Ceed ceed, CeedOperator op, CeedOperatorAssemblyData *data) {
1404   CeedInt             num_active_bases_in = 0, num_active_bases_out = 0, offset = 0;
1405   CeedInt             num_input_fields, *num_eval_modes_in = NULL, num_output_fields, *num_eval_modes_out = NULL;
1406   CeedSize          **eval_mode_offsets_in = NULL, **eval_mode_offsets_out = NULL;
1407   CeedEvalMode      **eval_modes_in = NULL, **eval_modes_out = NULL;
1408   CeedQFunctionField *qf_fields;
1409   CeedQFunction       qf;
1410   CeedOperatorField  *op_fields;
1411   bool                is_composite;
1412 
1413   CeedCall(CeedOperatorIsComposite(op, &is_composite));
1414   CeedCheck(!is_composite, ceed, CEED_ERROR_INCOMPATIBLE, "Can only create CeedOperator assembly data for non-composite operators.");
1415 
1416   // Allocate
1417   CeedCall(CeedCalloc(1, data));
1418   (*data)->ceed = ceed;
1419   CeedCall(CeedReference(ceed));
1420 
1421   // Build OperatorAssembly data
1422   CeedCall(CeedOperatorGetQFunction(op, &qf));
1423 
1424   // Determine active input basis
1425   CeedCall(CeedQFunctionGetFields(qf, &num_input_fields, &qf_fields, NULL, NULL));
1426   CeedCall(CeedOperatorGetFields(op, NULL, &op_fields, NULL, NULL));
1427   for (CeedInt i = 0; i < num_input_fields; i++) {
1428     CeedVector vec;
1429 
1430     CeedCall(CeedOperatorFieldGetVector(op_fields[i], &vec));
1431     if (vec == CEED_VECTOR_ACTIVE) {
1432       CeedInt      index = -1, num_comp, q_comp;
1433       CeedEvalMode eval_mode;
1434       CeedBasis    basis_in = NULL;
1435 
1436       CeedCall(CeedOperatorFieldGetBasis(op_fields[i], &basis_in));
1437       CeedCall(CeedQFunctionFieldGetEvalMode(qf_fields[i], &eval_mode));
1438       CeedCall(CeedBasisGetNumComponents(basis_in, &num_comp));
1439       CeedCall(CeedBasisGetNumQuadratureComponents(basis_in, eval_mode, &q_comp));
1440       for (CeedInt i = 0; i < num_active_bases_in; i++) {
1441         if ((*data)->active_bases_in[i] == basis_in) index = i;
1442       }
1443       if (index == -1) {
1444         CeedElemRestriction elem_rstr_in;
1445 
1446         index = num_active_bases_in;
1447         CeedCall(CeedRealloc(num_active_bases_in + 1, &(*data)->active_bases_in));
1448         (*data)->active_bases_in[num_active_bases_in] = NULL;
1449         CeedCall(CeedBasisReferenceCopy(basis_in, &(*data)->active_bases_in[num_active_bases_in]));
1450         CeedCall(CeedRealloc(num_active_bases_in + 1, &(*data)->active_elem_rstrs_in));
1451         (*data)->active_elem_rstrs_in[num_active_bases_in] = NULL;
1452         CeedCall(CeedOperatorFieldGetElemRestriction(op_fields[i], &elem_rstr_in));
1453         CeedCall(CeedElemRestrictionReferenceCopy(elem_rstr_in, &(*data)->active_elem_rstrs_in[num_active_bases_in]));
1454         CeedCall(CeedElemRestrictionDestroy(&elem_rstr_in));
1455         CeedCall(CeedRealloc(num_active_bases_in + 1, &num_eval_modes_in));
1456         num_eval_modes_in[index] = 0;
1457         CeedCall(CeedRealloc(num_active_bases_in + 1, &eval_modes_in));
1458         eval_modes_in[index] = NULL;
1459         CeedCall(CeedRealloc(num_active_bases_in + 1, &eval_mode_offsets_in));
1460         eval_mode_offsets_in[index] = NULL;
1461         CeedCall(CeedRealloc(num_active_bases_in + 1, &(*data)->assembled_bases_in));
1462         (*data)->assembled_bases_in[index] = NULL;
1463         num_active_bases_in++;
1464       }
1465       if (eval_mode != CEED_EVAL_WEIGHT) {
1466         // q_comp = 1 if CEED_EVAL_NONE, CEED_EVAL_WEIGHT caught by QF Assembly
1467         CeedCall(CeedRealloc(num_eval_modes_in[index] + q_comp, &eval_modes_in[index]));
1468         CeedCall(CeedRealloc(num_eval_modes_in[index] + q_comp, &eval_mode_offsets_in[index]));
1469         for (CeedInt d = 0; d < q_comp; d++) {
1470           eval_modes_in[index][num_eval_modes_in[index] + d]        = eval_mode;
1471           eval_mode_offsets_in[index][num_eval_modes_in[index] + d] = offset;
1472           offset += num_comp;
1473         }
1474         num_eval_modes_in[index] += q_comp;
1475       }
1476       CeedCall(CeedBasisDestroy(&basis_in));
1477     }
1478     CeedCall(CeedVectorDestroy(&vec));
1479   }
1480 
1481   // Determine active output basis
1482   CeedCall(CeedQFunctionGetFields(qf, NULL, NULL, &num_output_fields, &qf_fields));
1483   CeedCall(CeedOperatorGetFields(op, NULL, NULL, NULL, &op_fields));
1484   offset = 0;
1485   for (CeedInt i = 0; i < num_output_fields; i++) {
1486     CeedVector vec;
1487 
1488     CeedCall(CeedOperatorFieldGetVector(op_fields[i], &vec));
1489     if (vec == CEED_VECTOR_ACTIVE) {
1490       CeedInt      index = -1, num_comp, q_comp;
1491       CeedEvalMode eval_mode;
1492       CeedBasis    basis_out = NULL;
1493 
1494       CeedCall(CeedOperatorFieldGetBasis(op_fields[i], &basis_out));
1495       CeedCall(CeedQFunctionFieldGetEvalMode(qf_fields[i], &eval_mode));
1496       CeedCall(CeedBasisGetNumComponents(basis_out, &num_comp));
1497       CeedCall(CeedBasisGetNumQuadratureComponents(basis_out, eval_mode, &q_comp));
1498       for (CeedInt i = 0; i < num_active_bases_out; i++) {
1499         if ((*data)->active_bases_out[i] == basis_out) index = i;
1500       }
1501       if (index == -1) {
1502         CeedElemRestriction elem_rstr_out;
1503 
1504         index = num_active_bases_out;
1505         CeedCall(CeedRealloc(num_active_bases_out + 1, &(*data)->active_bases_out));
1506         (*data)->active_bases_out[num_active_bases_out] = NULL;
1507         CeedCall(CeedBasisReferenceCopy(basis_out, &(*data)->active_bases_out[num_active_bases_out]));
1508         CeedCall(CeedRealloc(num_active_bases_out + 1, &(*data)->active_elem_rstrs_out));
1509         (*data)->active_elem_rstrs_out[num_active_bases_out] = NULL;
1510         CeedCall(CeedOperatorFieldGetElemRestriction(op_fields[i], &elem_rstr_out));
1511         CeedCall(CeedElemRestrictionReferenceCopy(elem_rstr_out, &(*data)->active_elem_rstrs_out[num_active_bases_out]));
1512         CeedCall(CeedElemRestrictionDestroy(&elem_rstr_out));
1513         CeedCall(CeedRealloc(num_active_bases_out + 1, &num_eval_modes_out));
1514         num_eval_modes_out[index] = 0;
1515         CeedCall(CeedRealloc(num_active_bases_out + 1, &eval_modes_out));
1516         eval_modes_out[index] = NULL;
1517         CeedCall(CeedRealloc(num_active_bases_out + 1, &eval_mode_offsets_out));
1518         eval_mode_offsets_out[index] = NULL;
1519         CeedCall(CeedRealloc(num_active_bases_out + 1, &(*data)->assembled_bases_out));
1520         (*data)->assembled_bases_out[index] = NULL;
1521         num_active_bases_out++;
1522       }
1523       if (eval_mode != CEED_EVAL_WEIGHT) {
1524         // q_comp = 1 if CEED_EVAL_NONE, CEED_EVAL_WEIGHT caught by QF Assembly
1525         CeedCall(CeedRealloc(num_eval_modes_out[index] + q_comp, &eval_modes_out[index]));
1526         CeedCall(CeedRealloc(num_eval_modes_out[index] + q_comp, &eval_mode_offsets_out[index]));
1527         for (CeedInt d = 0; d < q_comp; d++) {
1528           eval_modes_out[index][num_eval_modes_out[index] + d]        = eval_mode;
1529           eval_mode_offsets_out[index][num_eval_modes_out[index] + d] = offset;
1530           offset += num_comp;
1531         }
1532         num_eval_modes_out[index] += q_comp;
1533       }
1534       CeedCall(CeedBasisDestroy(&basis_out));
1535     }
1536     CeedCall(CeedVectorDestroy(&vec));
1537   }
1538   (*data)->num_active_bases_in   = num_active_bases_in;
1539   (*data)->num_eval_modes_in     = num_eval_modes_in;
1540   (*data)->eval_modes_in         = eval_modes_in;
1541   (*data)->eval_mode_offsets_in  = eval_mode_offsets_in;
1542   (*data)->num_active_bases_out  = num_active_bases_out;
1543   (*data)->num_eval_modes_out    = num_eval_modes_out;
1544   (*data)->eval_modes_out        = eval_modes_out;
1545   (*data)->eval_mode_offsets_out = eval_mode_offsets_out;
1546   (*data)->num_output_components = offset;
1547   return CEED_ERROR_SUCCESS;
1548 }
1549 
1550 /**
1551   @brief Get `CeedOperator` @ref CeedEvalMode for assembly.
1552 
1553   Note: See @ref CeedOperatorAssemblyDataCreate() for a full description of the data stored in this object.
1554 
1555   @param[in]  data                  `CeedOperatorAssemblyData`
1556   @param[out] num_active_bases_in   Total number of active bases for input
1557   @param[out] num_eval_modes_in     Pointer to hold array of numbers of input @ref CeedEvalMode, or `NULL`.
1558                                       `eval_modes_in[0]` holds an array of eval modes for the first active `CeedBasis`.
1559   @param[out] eval_modes_in         Pointer to hold arrays of input @ref CeedEvalMode, or `NULL`
1560   @param[out] eval_mode_offsets_in  Pointer to hold arrays of input offsets at each quadrature point
1561   @param[out] num_active_bases_out  Total number of active bases for output
1562   @param[out] num_eval_modes_out    Pointer to hold array of numbers of output @ref CeedEvalMode, or `NULL`
1563   @param[out] eval_modes_out        Pointer to hold arrays of output @ref CeedEvalMode, or `NULL`
1564   @param[out] eval_mode_offsets_out Pointer to hold arrays of output offsets at each quadrature point
1565   @param[out] num_output_components The number of columns in the assembled `CeedQFunction` matrix for each quadrature point, including contributions of all active bases
1566 
1567   @return An error code: 0 - success, otherwise - failure
1568 
1569   @ref Backend
1570 **/
1571 int CeedOperatorAssemblyDataGetEvalModes(CeedOperatorAssemblyData data, CeedInt *num_active_bases_in, CeedInt **num_eval_modes_in,
1572                                          const CeedEvalMode ***eval_modes_in, CeedSize ***eval_mode_offsets_in, CeedInt *num_active_bases_out,
1573                                          CeedInt **num_eval_modes_out, const CeedEvalMode ***eval_modes_out, CeedSize ***eval_mode_offsets_out,
1574                                          CeedSize *num_output_components) {
1575   if (num_active_bases_in) *num_active_bases_in = data->num_active_bases_in;
1576   if (num_eval_modes_in) *num_eval_modes_in = data->num_eval_modes_in;
1577   if (eval_modes_in) *eval_modes_in = (const CeedEvalMode **)data->eval_modes_in;
1578   if (eval_mode_offsets_in) *eval_mode_offsets_in = data->eval_mode_offsets_in;
1579   if (num_active_bases_out) *num_active_bases_out = data->num_active_bases_out;
1580   if (num_eval_modes_out) *num_eval_modes_out = data->num_eval_modes_out;
1581   if (eval_modes_out) *eval_modes_out = (const CeedEvalMode **)data->eval_modes_out;
1582   if (eval_mode_offsets_out) *eval_mode_offsets_out = data->eval_mode_offsets_out;
1583   if (num_output_components) *num_output_components = data->num_output_components;
1584   return CEED_ERROR_SUCCESS;
1585 }
1586 
1587 /**
1588   @brief Get `CeedOperator` `CeedBasis` data for assembly.
1589 
1590   Note: See @ref CeedOperatorAssemblyDataCreate() for a full description of the data stored in this object.
1591 
1592   @param[in]  data                 `CeedOperatorAssemblyData`
1593   @param[out] num_active_bases_in  Number of active input bases, or `NULL`
1594   @param[out] active_bases_in      Pointer to hold active input `CeedBasis`, or `NULL`
1595   @param[out] assembled_bases_in   Pointer to hold assembled active input `B` , or `NULL`
1596   @param[out] num_active_bases_out Number of active output bases, or `NULL`
1597   @param[out] active_bases_out     Pointer to hold active output `CeedBasis`, or `NULL`
1598   @param[out] assembled_bases_out  Pointer to hold assembled active output `B` , or `NULL`
1599 
1600   @return An error code: 0 - success, otherwise - failure
1601 
1602   @ref Backend
1603 **/
1604 int CeedOperatorAssemblyDataGetBases(CeedOperatorAssemblyData data, CeedInt *num_active_bases_in, CeedBasis **active_bases_in,
1605                                      const CeedScalar ***assembled_bases_in, CeedInt *num_active_bases_out, CeedBasis **active_bases_out,
1606                                      const CeedScalar ***assembled_bases_out) {
1607   // Assemble B_in, B_out if needed
1608   if (assembled_bases_in && !data->assembled_bases_in[0]) {
1609     CeedInt num_qpts;
1610 
1611     if (data->active_bases_in[0] == CEED_BASIS_NONE) CeedCall(CeedElemRestrictionGetElementSize(data->active_elem_rstrs_in[0], &num_qpts));
1612     else CeedCall(CeedBasisGetNumQuadraturePoints(data->active_bases_in[0], &num_qpts));
1613     for (CeedInt b = 0; b < data->num_active_bases_in; b++) {
1614       bool        has_eval_none = false;
1615       CeedInt     num_nodes;
1616       CeedScalar *B_in = NULL, *identity = NULL;
1617 
1618       CeedCall(CeedElemRestrictionGetElementSize(data->active_elem_rstrs_in[b], &num_nodes));
1619       CeedCall(CeedCalloc(num_qpts * num_nodes * data->num_eval_modes_in[b], &B_in));
1620 
1621       for (CeedInt i = 0; i < data->num_eval_modes_in[b]; i++) {
1622         has_eval_none = has_eval_none || (data->eval_modes_in[b][i] == CEED_EVAL_NONE);
1623       }
1624       if (has_eval_none) {
1625         CeedCall(CeedCalloc(num_qpts * num_nodes, &identity));
1626         for (CeedInt i = 0; i < (num_nodes < num_qpts ? num_nodes : num_qpts); i++) {
1627           identity[i * num_nodes + i] = 1.0;
1628         }
1629       }
1630 
1631       for (CeedInt q = 0; q < num_qpts; q++) {
1632         for (CeedInt n = 0; n < num_nodes; n++) {
1633           CeedInt      d_in              = 0, q_comp_in;
1634           CeedEvalMode eval_mode_in_prev = CEED_EVAL_NONE;
1635 
1636           for (CeedInt e_in = 0; e_in < data->num_eval_modes_in[b]; e_in++) {
1637             const CeedInt     qq = data->num_eval_modes_in[b] * q;
1638             const CeedScalar *B  = NULL;
1639 
1640             CeedCall(CeedOperatorGetBasisPointer(data->active_bases_in[b], data->eval_modes_in[b][e_in], identity, &B));
1641             CeedCall(CeedBasisGetNumQuadratureComponents(data->active_bases_in[b], data->eval_modes_in[b][e_in], &q_comp_in));
1642             if (q_comp_in > 1) {
1643               if (e_in == 0 || data->eval_modes_in[b][e_in] != eval_mode_in_prev) d_in = 0;
1644               else B = &B[(++d_in) * num_qpts * num_nodes];
1645             }
1646             eval_mode_in_prev                 = data->eval_modes_in[b][e_in];
1647             B_in[(qq + e_in) * num_nodes + n] = B[q * num_nodes + n];
1648           }
1649         }
1650       }
1651       if (identity) CeedCall(CeedFree(&identity));
1652       data->assembled_bases_in[b] = B_in;
1653     }
1654   }
1655 
1656   if (assembled_bases_out && !data->assembled_bases_out[0]) {
1657     CeedInt num_qpts;
1658 
1659     if (data->active_bases_out[0] == CEED_BASIS_NONE) CeedCall(CeedElemRestrictionGetElementSize(data->active_elem_rstrs_out[0], &num_qpts));
1660     else CeedCall(CeedBasisGetNumQuadraturePoints(data->active_bases_out[0], &num_qpts));
1661     for (CeedInt b = 0; b < data->num_active_bases_out; b++) {
1662       bool        has_eval_none = false;
1663       CeedInt     num_nodes;
1664       CeedScalar *B_out = NULL, *identity = NULL;
1665 
1666       CeedCall(CeedElemRestrictionGetElementSize(data->active_elem_rstrs_out[b], &num_nodes));
1667       CeedCall(CeedCalloc(num_qpts * num_nodes * data->num_eval_modes_out[b], &B_out));
1668 
1669       for (CeedInt i = 0; i < data->num_eval_modes_out[b]; i++) {
1670         has_eval_none = has_eval_none || (data->eval_modes_out[b][i] == CEED_EVAL_NONE);
1671       }
1672       if (has_eval_none) {
1673         CeedCall(CeedCalloc(num_qpts * num_nodes, &identity));
1674         for (CeedInt i = 0; i < (num_nodes < num_qpts ? num_nodes : num_qpts); i++) {
1675           identity[i * num_nodes + i] = 1.0;
1676         }
1677       }
1678 
1679       for (CeedInt q = 0; q < num_qpts; q++) {
1680         for (CeedInt n = 0; n < num_nodes; n++) {
1681           CeedInt      d_out              = 0, q_comp_out;
1682           CeedEvalMode eval_mode_out_prev = CEED_EVAL_NONE;
1683 
1684           for (CeedInt e_out = 0; e_out < data->num_eval_modes_out[b]; e_out++) {
1685             const CeedInt     qq = data->num_eval_modes_out[b] * q;
1686             const CeedScalar *B  = NULL;
1687 
1688             CeedCall(CeedOperatorGetBasisPointer(data->active_bases_out[b], data->eval_modes_out[b][e_out], identity, &B));
1689             CeedCall(CeedBasisGetNumQuadratureComponents(data->active_bases_out[b], data->eval_modes_out[b][e_out], &q_comp_out));
1690             if (q_comp_out > 1) {
1691               if (e_out == 0 || data->eval_modes_out[b][e_out] != eval_mode_out_prev) d_out = 0;
1692               else B = &B[(++d_out) * num_qpts * num_nodes];
1693             }
1694             eval_mode_out_prev                  = data->eval_modes_out[b][e_out];
1695             B_out[(qq + e_out) * num_nodes + n] = B[q * num_nodes + n];
1696           }
1697         }
1698       }
1699       if (identity) CeedCall(CeedFree(&identity));
1700       data->assembled_bases_out[b] = B_out;
1701     }
1702   }
1703 
1704   // Pass out assembled data
1705   if (num_active_bases_in) *num_active_bases_in = data->num_active_bases_in;
1706   if (active_bases_in) *active_bases_in = data->active_bases_in;
1707   if (assembled_bases_in) *assembled_bases_in = (const CeedScalar **)data->assembled_bases_in;
1708   if (num_active_bases_out) *num_active_bases_out = data->num_active_bases_out;
1709   if (active_bases_out) *active_bases_out = data->active_bases_out;
1710   if (assembled_bases_out) *assembled_bases_out = (const CeedScalar **)data->assembled_bases_out;
1711   return CEED_ERROR_SUCCESS;
1712 }
1713 
1714 /**
1715   @brief Get `CeedOperator` `CeedBasis` data for assembly.
1716 
1717   Note: See @ref CeedOperatorAssemblyDataCreate() for a full description of the data stored in this object.
1718 
1719   @param[in]  data                      `CeedOperatorAssemblyData`
1720   @param[out] num_active_elem_rstrs_in  Number of active input element restrictions, or `NULL`
1721   @param[out] active_elem_rstrs_in      Pointer to hold active input `CeedElemRestriction`, or `NULL`
1722   @param[out] num_active_elem_rstrs_out Number of active output element restrictions, or `NULL`
1723   @param[out] active_elem_rstrs_out     Pointer to hold active output `CeedElemRestriction`, or `NULL`
1724 
1725   @return An error code: 0 - success, otherwise - failure
1726 
1727   @ref Backend
1728 **/
1729 int CeedOperatorAssemblyDataGetElemRestrictions(CeedOperatorAssemblyData data, CeedInt *num_active_elem_rstrs_in,
1730                                                 CeedElemRestriction **active_elem_rstrs_in, CeedInt *num_active_elem_rstrs_out,
1731                                                 CeedElemRestriction **active_elem_rstrs_out) {
1732   if (num_active_elem_rstrs_in) *num_active_elem_rstrs_in = data->num_active_bases_in;
1733   if (active_elem_rstrs_in) *active_elem_rstrs_in = data->active_elem_rstrs_in;
1734   if (num_active_elem_rstrs_out) *num_active_elem_rstrs_out = data->num_active_bases_out;
1735   if (active_elem_rstrs_out) *active_elem_rstrs_out = data->active_elem_rstrs_out;
1736   return CEED_ERROR_SUCCESS;
1737 }
1738 
1739 /**
1740   @brief Destroy `CeedOperatorAssemblyData`
1741 
1742   @param[in,out] data `CeedOperatorAssemblyData` to destroy
1743 
1744   @return An error code: 0 - success, otherwise - failure
1745 
1746   @ref Backend
1747 **/
1748 int CeedOperatorAssemblyDataDestroy(CeedOperatorAssemblyData *data) {
1749   if (!*data) {
1750     *data = NULL;
1751     return CEED_ERROR_SUCCESS;
1752   }
1753   CeedCall(CeedDestroy(&(*data)->ceed));
1754   for (CeedInt b = 0; b < (*data)->num_active_bases_in; b++) {
1755     CeedCall(CeedBasisDestroy(&(*data)->active_bases_in[b]));
1756     CeedCall(CeedElemRestrictionDestroy(&(*data)->active_elem_rstrs_in[b]));
1757     CeedCall(CeedFree(&(*data)->eval_modes_in[b]));
1758     CeedCall(CeedFree(&(*data)->eval_mode_offsets_in[b]));
1759     CeedCall(CeedFree(&(*data)->assembled_bases_in[b]));
1760   }
1761   for (CeedInt b = 0; b < (*data)->num_active_bases_out; b++) {
1762     CeedCall(CeedBasisDestroy(&(*data)->active_bases_out[b]));
1763     CeedCall(CeedElemRestrictionDestroy(&(*data)->active_elem_rstrs_out[b]));
1764     CeedCall(CeedFree(&(*data)->eval_modes_out[b]));
1765     CeedCall(CeedFree(&(*data)->eval_mode_offsets_out[b]));
1766     CeedCall(CeedFree(&(*data)->assembled_bases_out[b]));
1767   }
1768   CeedCall(CeedFree(&(*data)->active_bases_in));
1769   CeedCall(CeedFree(&(*data)->active_bases_out));
1770   CeedCall(CeedFree(&(*data)->active_elem_rstrs_in));
1771   CeedCall(CeedFree(&(*data)->active_elem_rstrs_out));
1772   CeedCall(CeedFree(&(*data)->num_eval_modes_in));
1773   CeedCall(CeedFree(&(*data)->num_eval_modes_out));
1774   CeedCall(CeedFree(&(*data)->eval_modes_in));
1775   CeedCall(CeedFree(&(*data)->eval_modes_out));
1776   CeedCall(CeedFree(&(*data)->eval_mode_offsets_in));
1777   CeedCall(CeedFree(&(*data)->eval_mode_offsets_out));
1778   CeedCall(CeedFree(&(*data)->assembled_bases_in));
1779   CeedCall(CeedFree(&(*data)->assembled_bases_out));
1780 
1781   CeedCall(CeedFree(data));
1782   return CEED_ERROR_SUCCESS;
1783 }
1784 
1785 /**
1786   @brief Retrieve fallback `CeedOperator` with a reference `Ceed` for advanced `CeedOperator` functionality
1787 
1788   @param[in]  op          `CeedOperator` to retrieve fallback for
1789   @param[out] op_fallback Fallback `CeedOperator`
1790 
1791   @return An error code: 0 - success, otherwise - failure
1792 
1793   @ref Backend
1794 **/
1795 int CeedOperatorGetFallback(CeedOperator op, CeedOperator *op_fallback) {
1796   // Create if needed
1797   if (!op->op_fallback) CeedCall(CeedOperatorCreateFallback(op));
1798   if (op->op_fallback) {
1799     bool is_debug;
1800     Ceed ceed;
1801 
1802     CeedCall(CeedOperatorGetCeed(op, &ceed));
1803     CeedCall(CeedIsDebug(ceed, &is_debug));
1804     if (is_debug) {
1805       Ceed        ceed_fallback;
1806       const char *resource, *resource_fallback;
1807 
1808       CeedCall(CeedGetOperatorFallbackCeed(ceed, &ceed_fallback));
1809       CeedCall(CeedGetResource(ceed, &resource));
1810       CeedCall(CeedGetResource(ceed_fallback, &resource_fallback));
1811 
1812       CeedDebug256(ceed, CEED_DEBUG_COLOR_SUCCESS, "---------- CeedOperator Fallback ----------\n");
1813       CeedDebug(ceed, "Falling back from %s operator at address %p to %s operator at address %p\n", resource, op, resource_fallback, op->op_fallback);
1814       CeedCall(CeedDestroy(&ceed_fallback));
1815     }
1816     CeedCall(CeedDestroy(&ceed));
1817   }
1818   *op_fallback = op->op_fallback;
1819   return CEED_ERROR_SUCCESS;
1820 }
1821 
1822 /**
1823   @brief Get the parent `CeedOperator` for a fallback `CeedOperator`
1824 
1825   @param[in]  op     `CeedOperator` context
1826   @param[out] parent Variable to store parent `CeedOperator` context
1827 
1828   @return An error code: 0 - success, otherwise - failure
1829 
1830   @ref Backend
1831 **/
1832 int CeedOperatorGetFallbackParent(CeedOperator op, CeedOperator *parent) {
1833   *parent = op->op_fallback_parent ? op->op_fallback_parent : NULL;
1834   return CEED_ERROR_SUCCESS;
1835 }
1836 
1837 /**
1838   @brief Get the `Ceed` context of the parent `CeedOperator` for a fallback `CeedOperator`
1839 
1840   @param[in]  op     `CeedOperator` context
1841   @param[out] parent Variable to store parent `Ceed` context
1842 
1843   @return An error code: 0 - success, otherwise - failure
1844 
1845   @ref Backend
1846 **/
1847 int CeedOperatorGetFallbackParentCeed(CeedOperator op, Ceed *parent) {
1848   *parent = NULL;
1849   if (op->op_fallback_parent) CeedCall(CeedReferenceCopy(op->op_fallback_parent->ceed, parent));
1850   else CeedCall(CeedReferenceCopy(CeedOperatorReturnCeed(op), parent));
1851   return CEED_ERROR_SUCCESS;
1852 }
1853 
1854 /// @}
1855 
1856 /// ----------------------------------------------------------------------------
1857 /// CeedOperator Public API
1858 /// ----------------------------------------------------------------------------
1859 /// @addtogroup CeedOperatorUser
1860 /// @{
1861 
1862 /**
1863   @brief Assemble a linear `CeedQFunction` associated with a `CeedOperator`.
1864 
1865   This returns a `CeedVector` containing a matrix at each quadrature point providing the action of the `CeedQFunction` associated with the `CeedOperator`.
1866   The vector `assembled` is of shape `[num_elements, num_input_fields, num_output_fields, num_quad_points]` and contains column-major matrices representing the action of the `CeedQFunction` for a corresponding quadrature point on an element.
1867 
1868   Inputs and outputs are in the order provided by the user when adding `CeedOperator` fields.
1869   For example, a `CeedQFunction` with inputs `u` and `gradu` and outputs `gradv` and `v` , provided in that order, would result in an assembled `CeedQFunction` that consists of `(1 + dim) x (dim + 1)` matrices at each quadrature point acting on the input ` [u, du_0, du_1]` and producing the output `[dv_0, dv_1, v]`.
1870 
1871   Note: Calling this function asserts that setup is complete and sets the `CeedOperator` as immutable.
1872 
1873   @param[in]  op        `CeedOperator` to assemble `CeedQFunction`
1874   @param[out] assembled `CeedVector` to store assembled `CeedQFunction` at quadrature points
1875   @param[out] rstr      `CeedElemRestriction` for `CeedVector` containing assembled `CeedQFunction`
1876   @param[in]  request   Address of @ref CeedRequest for non-blocking completion, else @ref CEED_REQUEST_IMMEDIATE
1877 
1878   @return An error code: 0 - success, otherwise - failure
1879 
1880   @ref User
1881 **/
1882 int CeedOperatorLinearAssembleQFunction(CeedOperator op, CeedVector *assembled, CeedElemRestriction *rstr, CeedRequest *request) {
1883   CeedCall(CeedOperatorCheckReady(op));
1884 
1885   if (op->LinearAssembleQFunction) {
1886     // Backend version
1887     CeedCall(op->LinearAssembleQFunction(op, assembled, rstr, request));
1888   } else {
1889     // Operator fallback
1890     CeedOperator op_fallback;
1891 
1892     CeedCall(CeedOperatorGetFallback(op, &op_fallback));
1893     if (op_fallback) CeedCall(CeedOperatorLinearAssembleQFunction(op_fallback, assembled, rstr, request));
1894     else return CeedError(CeedOperatorReturnCeed(op), CEED_ERROR_UNSUPPORTED, "Backend does not support CeedOperatorLinearAssembleQFunction");
1895   }
1896   return CEED_ERROR_SUCCESS;
1897 }
1898 
1899 /**
1900   @brief Assemble `CeedQFunction` and store result internally.
1901 
1902   Return copied references of stored data to the caller.
1903   Caller is responsible for ownership and destruction of the copied references.
1904   See also @ref CeedOperatorLinearAssembleQFunction().
1905 
1906   Note: If the value of `assembled` or `rstr` passed to this function are non-`NULL` , then it is assumed that they hold valid pointers.
1907         These objects will be destroyed if `*assembled` or `*rstr` is the only reference to the object.
1908 
1909   @param[in]  op        `CeedOperator` to assemble `CeedQFunction`
1910   @param[out] assembled `CeedVector` to store assembled `CeedQFunction` at quadrature points
1911   @param[out] rstr      `CeedElemRestriction` for `CeedVector` containing assembled `CeedQFunction`
1912   @param[in]  request   Address of @ref CeedRequest for non-blocking completion, else @ref CEED_REQUEST_IMMEDIATE
1913 
1914   @return An error code: 0 - success, otherwise - failure
1915 
1916   @ref User
1917 **/
1918 int CeedOperatorLinearAssembleQFunctionBuildOrUpdate(CeedOperator op, CeedVector *assembled, CeedElemRestriction *rstr, CeedRequest *request) {
1919   int (*LinearAssembleQFunctionUpdate)(CeedOperator, CeedVector, CeedElemRestriction, CeedRequest *) = NULL;
1920   CeedOperator op_assemble                                                                           = NULL;
1921   CeedOperator op_fallback_parent                                                                    = NULL;
1922 
1923   CeedCall(CeedOperatorCheckReady(op));
1924 
1925   // Determine if fallback parent or operator has implementation
1926   CeedCall(CeedOperatorGetFallbackParent(op, &op_fallback_parent));
1927   if (op_fallback_parent && op_fallback_parent->LinearAssembleQFunctionUpdate) {
1928     // -- Backend version for op fallback parent is faster, if it exists
1929     LinearAssembleQFunctionUpdate = op_fallback_parent->LinearAssembleQFunctionUpdate;
1930     op_assemble                   = op_fallback_parent;
1931   } else if (op->LinearAssembleQFunctionUpdate) {
1932     // -- Backend version for op
1933     LinearAssembleQFunctionUpdate = op->LinearAssembleQFunctionUpdate;
1934     op_assemble                   = op;
1935   }
1936 
1937   // Assemble QFunction
1938   if (LinearAssembleQFunctionUpdate) {
1939     // Backend or fallback parent version
1940     CeedQFunctionAssemblyData data;
1941     bool                      data_is_setup;
1942     CeedVector                assembled_vec  = NULL;
1943     CeedElemRestriction       assembled_rstr = NULL;
1944 
1945     CeedCall(CeedOperatorGetQFunctionAssemblyData(op, &data));
1946     CeedCall(CeedQFunctionAssemblyDataIsSetup(data, &data_is_setup));
1947     if (data_is_setup) {
1948       bool update_needed;
1949 
1950       CeedCall(CeedQFunctionAssemblyDataGetObjects(data, &assembled_vec, &assembled_rstr));
1951       CeedCall(CeedQFunctionAssemblyDataIsUpdateNeeded(data, &update_needed));
1952       if (update_needed) CeedCall(LinearAssembleQFunctionUpdate(op_assemble, assembled_vec, assembled_rstr, request));
1953     } else {
1954       CeedCall(CeedOperatorLinearAssembleQFunction(op_assemble, &assembled_vec, &assembled_rstr, request));
1955       CeedCall(CeedQFunctionAssemblyDataSetObjects(data, assembled_vec, assembled_rstr));
1956     }
1957     CeedCall(CeedQFunctionAssemblyDataSetUpdateNeeded(data, false));
1958 
1959     // Copy reference from internally held copy
1960     CeedCall(CeedVectorReferenceCopy(assembled_vec, assembled));
1961     CeedCall(CeedElemRestrictionReferenceCopy(assembled_rstr, rstr));
1962     CeedCall(CeedVectorDestroy(&assembled_vec));
1963     CeedCall(CeedElemRestrictionDestroy(&assembled_rstr));
1964   } else {
1965     // Operator fallback
1966     CeedOperator op_fallback;
1967 
1968     CeedCall(CeedOperatorGetFallback(op, &op_fallback));
1969     if (op_fallback) CeedCall(CeedOperatorLinearAssembleQFunctionBuildOrUpdate(op_fallback, assembled, rstr, request));
1970     else return CeedError(CeedOperatorReturnCeed(op), CEED_ERROR_UNSUPPORTED, "Backend does not support CeedOperatorLinearAssembleQFunctionUpdate");
1971   }
1972   return CEED_ERROR_SUCCESS;
1973 }
1974 
1975 /**
1976   @brief Assemble the diagonal of a square linear `CeedOperator`
1977 
1978   This overwrites a `CeedVector` with the diagonal of a linear `CeedOperator`.
1979 
1980   Note: Currently only non-composite `CeedOperator` with a single field and composite `CeedOperator` with single field sub-operators are supported.
1981 
1982   Note: Calling this function asserts that setup is complete and sets the `CeedOperator` as immutable.
1983 
1984   @param[in]  op        `CeedOperator` to assemble `CeedQFunction`
1985   @param[out] assembled `CeedVector` to store assembled `CeedOperator` diagonal
1986   @param[in]  request   Address of @ref CeedRequest for non-blocking completion, else @ref CEED_REQUEST_IMMEDIATE
1987 
1988   @return An error code: 0 - success, otherwise - failure
1989 
1990   @ref User
1991 **/
1992 int CeedOperatorLinearAssembleDiagonal(CeedOperator op, CeedVector assembled, CeedRequest *request) {
1993   bool     is_composite;
1994   CeedSize input_size = 0, output_size = 0;
1995 
1996   CeedCall(CeedOperatorCheckReady(op));
1997   CeedCall(CeedOperatorIsComposite(op, &is_composite));
1998 
1999   CeedCall(CeedOperatorGetActiveVectorLengths(op, &input_size, &output_size));
2000   CeedCheck(input_size == output_size, CeedOperatorReturnCeed(op), CEED_ERROR_DIMENSION, "Operator must be square");
2001 
2002   // Early exit for empty operator
2003   if (!is_composite) {
2004     CeedInt num_elem = 0;
2005 
2006     CeedCall(CeedOperatorGetNumElements(op, &num_elem));
2007     if (num_elem == 0) return CEED_ERROR_SUCCESS;
2008   }
2009 
2010   if (op->LinearAssembleDiagonal) {
2011     // Backend version
2012     CeedCall(op->LinearAssembleDiagonal(op, assembled, request));
2013     return CEED_ERROR_SUCCESS;
2014   } else if (op->LinearAssembleAddDiagonal) {
2015     // Backend version with zeroing first
2016     CeedCall(CeedVectorSetValue(assembled, 0.0));
2017     CeedCall(op->LinearAssembleAddDiagonal(op, assembled, request));
2018     return CEED_ERROR_SUCCESS;
2019   } else {
2020     // Operator fallback
2021     CeedOperator op_fallback;
2022 
2023     CeedCall(CeedOperatorGetFallback(op, &op_fallback));
2024     if (op_fallback) {
2025       CeedCall(CeedOperatorLinearAssembleDiagonal(op_fallback, assembled, request));
2026       return CEED_ERROR_SUCCESS;
2027     }
2028   }
2029   // Default interface implementation
2030   CeedCall(CeedVectorSetValue(assembled, 0.0));
2031   CeedCall(CeedOperatorLinearAssembleAddDiagonal(op, assembled, request));
2032   return CEED_ERROR_SUCCESS;
2033 }
2034 
2035 /**
2036   @brief Assemble the diagonal of a square linear `CeedOperator`.
2037 
2038   This sums into a `CeedVector` the diagonal of a linear `CeedOperator`.
2039 
2040   Note: Currently only non-composite `CeedOperator` with a single field and composite `CeedOperator` with single field sub-operators are supported.
2041 
2042   Note: Calling this function asserts that setup is complete and sets the CeedOperator as immutable.
2043 
2044   @param[in]  op        `CeedOperator` to assemble `CeedQFunction`
2045   @param[out] assembled `CeedVector` to store assembled `CeedOperator` diagonal
2046   @param[in]  request   Address of @ref CeedRequest for non-blocking completion, else @ref CEED_REQUEST_IMMEDIATE
2047 
2048   @return An error code: 0 - success, otherwise - failure
2049 
2050   @ref User
2051 **/
2052 int CeedOperatorLinearAssembleAddDiagonal(CeedOperator op, CeedVector assembled, CeedRequest *request) {
2053   bool     is_composite;
2054   CeedSize input_size = 0, output_size = 0;
2055 
2056   CeedCall(CeedOperatorCheckReady(op));
2057   CeedCall(CeedOperatorIsComposite(op, &is_composite));
2058 
2059   CeedCall(CeedOperatorGetActiveVectorLengths(op, &input_size, &output_size));
2060   CeedCheck(input_size == output_size, CeedOperatorReturnCeed(op), CEED_ERROR_DIMENSION, "Operator must be square");
2061 
2062   // Early exit for empty operator
2063   if (!is_composite) {
2064     CeedInt num_elem = 0;
2065 
2066     CeedCall(CeedOperatorGetNumElements(op, &num_elem));
2067     if (num_elem == 0) return CEED_ERROR_SUCCESS;
2068   }
2069 
2070   if (op->LinearAssembleAddDiagonal) {
2071     // Backend version
2072     CeedCall(op->LinearAssembleAddDiagonal(op, assembled, request));
2073     return CEED_ERROR_SUCCESS;
2074   } else {
2075     // Operator fallback
2076     CeedOperator op_fallback;
2077 
2078     CeedCall(CeedOperatorGetFallback(op, &op_fallback));
2079     if (op_fallback) {
2080       CeedCall(CeedOperatorLinearAssembleAddDiagonal(op_fallback, assembled, request));
2081       return CEED_ERROR_SUCCESS;
2082     }
2083   }
2084   // Default interface implementation
2085   if (is_composite) {
2086     CeedCall(CeedCompositeOperatorLinearAssembleAddDiagonal(op, request, false, assembled));
2087   } else {
2088     CeedCall(CeedSingleOperatorLinearAssembleAddDiagonal(op, request, false, assembled));
2089   }
2090   return CEED_ERROR_SUCCESS;
2091 }
2092 
2093 /**
2094    @brief Fully assemble the point-block diagonal pattern of a linear `CeedOperator`.
2095 
2096    Expected to be used in conjunction with @ref CeedOperatorLinearAssemblePointBlockDiagonal().
2097 
2098    The assembly routines use coordinate format, with `num_entries` tuples of the form `(i, j, value)` which indicate that value should be added to the matrix in entry `(i, j)`.
2099    Note that the `(i, j)` pairs are unique.
2100    This function returns the number of entries and their `(i, j)` locations, while @ref CeedOperatorLinearAssemblePointBlockDiagonal() provides the values in the same ordering.
2101 
2102    This will generally be slow unless your operator is low-order.
2103 
2104    Note: Calling this function asserts that setup is complete and sets the `CeedOperator` as immutable.
2105 
2106    @param[in]  op          `CeedOperator` to assemble
2107    @param[out] num_entries Number of entries in coordinate nonzero pattern
2108    @param[out] rows        Row number for each entry
2109    @param[out] cols        Column number for each entry
2110 
2111    @ref User
2112 **/
2113 int CeedOperatorLinearAssemblePointBlockDiagonalSymbolic(CeedOperator op, CeedSize *num_entries, CeedInt **rows, CeedInt **cols) {
2114   bool          is_composite;
2115   CeedInt       num_active_components, num_sub_operators;
2116   CeedOperator *sub_operators;
2117 
2118   CeedCall(CeedOperatorIsComposite(op, &is_composite));
2119 
2120   CeedSize input_size = 0, output_size = 0;
2121   CeedCall(CeedOperatorGetActiveVectorLengths(op, &input_size, &output_size));
2122   CeedCheck(input_size == output_size, CeedOperatorReturnCeed(op), CEED_ERROR_DIMENSION, "Operator must be square");
2123 
2124   if (is_composite) {
2125     CeedCall(CeedCompositeOperatorGetNumSub(op, &num_sub_operators));
2126     CeedCall(CeedCompositeOperatorGetSubList(op, &sub_operators));
2127   } else {
2128     sub_operators     = &op;
2129     num_sub_operators = 1;
2130   }
2131 
2132   // Verify operator can be assembled correctly
2133   {
2134     CeedOperatorAssemblyData data;
2135     CeedInt                  num_active_elem_rstrs, comp_stride;
2136     CeedElemRestriction     *active_elem_rstrs;
2137 
2138     // Get initial values to check against
2139     CeedCall(CeedOperatorGetOperatorAssemblyData(sub_operators[0], &data));
2140     CeedCall(CeedOperatorAssemblyDataGetElemRestrictions(data, &num_active_elem_rstrs, &active_elem_rstrs, NULL, NULL));
2141     CeedCall(CeedElemRestrictionGetCompStride(active_elem_rstrs[0], &comp_stride));
2142     CeedCall(CeedElemRestrictionGetNumComponents(active_elem_rstrs[0], &num_active_components));
2143 
2144     // Verify that all active element restrictions have same component stride and number of components
2145     for (CeedInt k = 0; k < num_sub_operators; k++) {
2146       CeedCall(CeedOperatorGetOperatorAssemblyData(sub_operators[k], &data));
2147       CeedCall(CeedOperatorAssemblyDataGetElemRestrictions(data, &num_active_elem_rstrs, &active_elem_rstrs, NULL, NULL));
2148       for (CeedInt i = 0; i < num_active_elem_rstrs; i++) {
2149         CeedInt comp_stride_sub, num_active_components_sub;
2150 
2151         CeedCall(CeedElemRestrictionGetCompStride(active_elem_rstrs[i], &comp_stride_sub));
2152         CeedCheck(comp_stride == comp_stride_sub, CeedOperatorReturnCeed(op), CEED_ERROR_DIMENSION,
2153                   "Active element restrictions must have the same component stride: %d vs %d", comp_stride, comp_stride_sub);
2154         CeedCall(CeedElemRestrictionGetNumComponents(active_elem_rstrs[i], &num_active_components_sub));
2155         CeedCheck(num_active_components == num_active_components_sub, CeedOperatorReturnCeed(op), CEED_ERROR_INCOMPATIBLE,
2156                   "All suboperators must have the same number of output components."
2157                   " Previous: %" CeedInt_FMT " Current: %" CeedInt_FMT,
2158                   num_active_components, num_active_components_sub);
2159       }
2160     }
2161   }
2162   *num_entries = input_size * num_active_components;
2163   CeedCall(CeedCalloc(*num_entries, rows));
2164   CeedCall(CeedCalloc(*num_entries, cols));
2165 
2166   for (CeedInt o = 0; o < num_sub_operators; o++) {
2167     CeedElemRestriction active_elem_rstr, point_block_active_elem_rstr;
2168     CeedInt             comp_stride, num_elem, elem_size;
2169     const CeedInt      *offsets, *point_block_offsets;
2170 
2171     CeedCall(CeedOperatorGetActiveElemRestriction(sub_operators[o], &active_elem_rstr));
2172     CeedCall(CeedElemRestrictionGetCompStride(active_elem_rstr, &comp_stride));
2173     CeedCall(CeedElemRestrictionGetNumElements(active_elem_rstr, &num_elem));
2174     CeedCall(CeedElemRestrictionGetElementSize(active_elem_rstr, &elem_size));
2175     CeedCall(CeedElemRestrictionGetOffsets(active_elem_rstr, CEED_MEM_HOST, &offsets));
2176 
2177     CeedCall(CeedOperatorCreateActivePointBlockRestriction(active_elem_rstr, &point_block_active_elem_rstr));
2178     CeedCall(CeedElemRestrictionGetOffsets(point_block_active_elem_rstr, CEED_MEM_HOST, &point_block_offsets));
2179 
2180     for (CeedSize i = 0; i < num_elem * elem_size; i++) {
2181       for (CeedInt c_out = 0; c_out < num_active_components; c_out++) {
2182         for (CeedInt c_in = 0; c_in < num_active_components; c_in++) {
2183           (*rows)[point_block_offsets[i] + c_out * num_active_components + c_in] = offsets[i] + c_out * comp_stride;
2184           (*cols)[point_block_offsets[i] + c_out * num_active_components + c_in] = offsets[i] + c_in * comp_stride;
2185         }
2186       }
2187     }
2188 
2189     CeedCall(CeedElemRestrictionRestoreOffsets(active_elem_rstr, &offsets));
2190     CeedCall(CeedElemRestrictionRestoreOffsets(point_block_active_elem_rstr, &point_block_offsets));
2191     CeedCall(CeedElemRestrictionDestroy(&active_elem_rstr));
2192     CeedCall(CeedElemRestrictionDestroy(&point_block_active_elem_rstr));
2193   }
2194   return CEED_ERROR_SUCCESS;
2195 }
2196 
2197 /**
2198   @brief Assemble the point block diagonal of a square linear `CeedOperator`.
2199 
2200   This overwrites a `CeedVector` with the point block diagonal of a linear `CeedOperator`.
2201 
2202   Note: Currently only non-composite `CeedOperator` with a single field and composite `CeedOperator` with single field sub-operators are supported.
2203 
2204   Note: Calling this function asserts that setup is complete and sets the `CeedOperator` as immutable.
2205 
2206   @param[in]  op        `CeedOperator` to assemble `CeedQFunction`
2207   @param[out] assembled `CeedVector` to store assembled `CeedOperator` point block diagonal, provided in row-major form with an `num_comp * num_comp` block at each node.
2208                           The dimensions of this vector are derived from the active vector for the `CeedOperator`.
2209                           The array has shape `[nodes, component out, component in]`.
2210   @param[in]  request   Address of @ref CeedRequest for non-blocking completion, else @ref CEED_REQUEST_IMMEDIATE
2211 
2212   @return An error code: 0 - success, otherwise - failure
2213 
2214   @ref User
2215 **/
2216 int CeedOperatorLinearAssemblePointBlockDiagonal(CeedOperator op, CeedVector assembled, CeedRequest *request) {
2217   bool     is_composite;
2218   CeedSize input_size = 0, output_size = 0;
2219 
2220   CeedCall(CeedOperatorCheckReady(op));
2221   CeedCall(CeedOperatorIsComposite(op, &is_composite));
2222 
2223   CeedCall(CeedOperatorGetActiveVectorLengths(op, &input_size, &output_size));
2224   CeedCheck(input_size == output_size, CeedOperatorReturnCeed(op), CEED_ERROR_DIMENSION, "Operator must be square");
2225 
2226   // Early exit for empty operator
2227   if (!is_composite) {
2228     CeedInt num_elem = 0;
2229 
2230     CeedCall(CeedOperatorGetNumElements(op, &num_elem));
2231     if (num_elem == 0) return CEED_ERROR_SUCCESS;
2232   }
2233 
2234   if (op->LinearAssemblePointBlockDiagonal) {
2235     // Backend version
2236     CeedCall(op->LinearAssemblePointBlockDiagonal(op, assembled, request));
2237     return CEED_ERROR_SUCCESS;
2238   } else if (op->LinearAssembleAddPointBlockDiagonal) {
2239     // Backend version with zeroing first
2240     CeedCall(CeedVectorSetValue(assembled, 0.0));
2241     CeedCall(CeedOperatorLinearAssembleAddPointBlockDiagonal(op, assembled, request));
2242     return CEED_ERROR_SUCCESS;
2243   } else {
2244     // Operator fallback
2245     CeedOperator op_fallback;
2246 
2247     CeedCall(CeedOperatorGetFallback(op, &op_fallback));
2248     if (op_fallback) {
2249       CeedCall(CeedOperatorLinearAssemblePointBlockDiagonal(op_fallback, assembled, request));
2250       return CEED_ERROR_SUCCESS;
2251     }
2252   }
2253   // Default interface implementation
2254   CeedCall(CeedVectorSetValue(assembled, 0.0));
2255   CeedCall(CeedOperatorLinearAssembleAddPointBlockDiagonal(op, assembled, request));
2256   return CEED_ERROR_SUCCESS;
2257 }
2258 
2259 /**
2260   @brief Assemble the point block diagonal of a square linear `CeedOperator`.
2261 
2262   This sums into a `CeedVector` with the point block diagonal of a linear `CeedOperator`.
2263 
2264   Note: Currently only non-composite `CeedOperator` with a single field and composite `CeedOperator` with single field sub-operators are supported.
2265 
2266   Note: Calling this function asserts that setup is complete and sets the `CeedOperator` as immutable.
2267 
2268   @param[in]  op        `CeedOperator` to assemble `CeedQFunction`
2269   @param[out] assembled `CeedVector` to store assembled CeedOperator point block diagonal, provided in row-major form with an `num_comp * num_comp` block at each node.
2270                           The dimensions of this vector are derived from the active vector for the `CeedOperator`.
2271                           The array has shape `[nodes, component out, component in]`.
2272   @param[in]  request   Address of @ref CeedRequest for non-blocking completion, else @ref CEED_REQUEST_IMMEDIATE
2273 
2274   @return An error code: 0 - success, otherwise - failure
2275 
2276   @ref User
2277 **/
2278 int CeedOperatorLinearAssembleAddPointBlockDiagonal(CeedOperator op, CeedVector assembled, CeedRequest *request) {
2279   bool     is_composite;
2280   CeedSize input_size = 0, output_size = 0;
2281 
2282   CeedCall(CeedOperatorCheckReady(op));
2283   CeedCall(CeedOperatorIsComposite(op, &is_composite));
2284 
2285   CeedCall(CeedOperatorGetActiveVectorLengths(op, &input_size, &output_size));
2286   CeedCheck(input_size == output_size, CeedOperatorReturnCeed(op), CEED_ERROR_DIMENSION, "Operator must be square");
2287 
2288   // Early exit for empty operator
2289   if (!is_composite) {
2290     CeedInt num_elem = 0;
2291 
2292     CeedCall(CeedOperatorGetNumElements(op, &num_elem));
2293     if (num_elem == 0) return CEED_ERROR_SUCCESS;
2294   }
2295 
2296   if (op->LinearAssembleAddPointBlockDiagonal) {
2297     // Backend version
2298     CeedCall(op->LinearAssembleAddPointBlockDiagonal(op, assembled, request));
2299     return CEED_ERROR_SUCCESS;
2300   } else {
2301     // Operator fallback
2302     CeedOperator op_fallback;
2303 
2304     CeedCall(CeedOperatorGetFallback(op, &op_fallback));
2305     if (op_fallback) {
2306       CeedCall(CeedOperatorLinearAssembleAddPointBlockDiagonal(op_fallback, assembled, request));
2307       return CEED_ERROR_SUCCESS;
2308     }
2309   }
2310   // Default interface implementation
2311   if (is_composite) {
2312     CeedCall(CeedCompositeOperatorLinearAssembleAddDiagonal(op, request, true, assembled));
2313   } else {
2314     CeedCall(CeedSingleOperatorLinearAssembleAddDiagonal(op, request, true, assembled));
2315   }
2316   return CEED_ERROR_SUCCESS;
2317 }
2318 
2319 /**
2320    @brief Fully assemble the nonzero pattern of a linear `CeedOperator`.
2321 
2322    Expected to be used in conjunction with @ref CeedOperatorLinearAssemble().
2323 
2324    The assembly routines use coordinate format, with `num_entries` tuples of the form `(i, j, value)` which indicate that value should be added to the matrix in entry `(i, j)`.
2325    Note that the `(i, j)` pairs are not unique and may repeat.
2326    This function returns the number of entries and their `(i, j)` locations, while @ref CeedOperatorLinearAssemble() provides the values in the same ordering.
2327 
2328    This will generally be slow unless your operator is low-order.
2329 
2330    Note: Calling this function asserts that setup is complete and sets the `CeedOperator` as immutable.
2331 
2332    @param[in]  op          `CeedOperator` to assemble
2333    @param[out] num_entries Number of entries in coordinate nonzero pattern
2334    @param[out] rows        Row number for each entry
2335    @param[out] cols        Column number for each entry
2336 
2337    @ref User
2338 **/
2339 int CeedOperatorLinearAssembleSymbolic(CeedOperator op, CeedSize *num_entries, CeedInt **rows, CeedInt **cols) {
2340   bool          is_composite;
2341   CeedInt       num_suboperators, offset = 0;
2342   CeedSize      single_entries;
2343   CeedOperator *sub_operators;
2344 
2345   CeedCall(CeedOperatorCheckReady(op));
2346   CeedCall(CeedOperatorIsComposite(op, &is_composite));
2347 
2348   if (op->LinearAssembleSymbolic) {
2349     // Backend version
2350     CeedCall(op->LinearAssembleSymbolic(op, num_entries, rows, cols));
2351     return CEED_ERROR_SUCCESS;
2352   } else {
2353     // Operator fallback
2354     CeedOperator op_fallback;
2355 
2356     CeedCall(CeedOperatorGetFallback(op, &op_fallback));
2357     if (op_fallback) {
2358       CeedCall(CeedOperatorLinearAssembleSymbolic(op_fallback, num_entries, rows, cols));
2359       return CEED_ERROR_SUCCESS;
2360     }
2361   }
2362 
2363   // Default interface implementation
2364 
2365   // Count entries and allocate rows, cols arrays
2366   *num_entries = 0;
2367   if (is_composite) {
2368     CeedCall(CeedCompositeOperatorGetNumSub(op, &num_suboperators));
2369     CeedCall(CeedCompositeOperatorGetSubList(op, &sub_operators));
2370     for (CeedInt k = 0; k < num_suboperators; ++k) {
2371       CeedCall(CeedSingleOperatorAssemblyCountEntries(sub_operators[k], &single_entries));
2372       *num_entries += single_entries;
2373     }
2374   } else {
2375     CeedCall(CeedSingleOperatorAssemblyCountEntries(op, &single_entries));
2376     *num_entries += single_entries;
2377   }
2378   CeedCall(CeedCalloc(*num_entries, rows));
2379   CeedCall(CeedCalloc(*num_entries, cols));
2380 
2381   // Assemble nonzero locations
2382   if (is_composite) {
2383     CeedCall(CeedCompositeOperatorGetNumSub(op, &num_suboperators));
2384     CeedCall(CeedCompositeOperatorGetSubList(op, &sub_operators));
2385     for (CeedInt k = 0; k < num_suboperators; ++k) {
2386       CeedCall(CeedSingleOperatorAssembleSymbolic(sub_operators[k], offset, *rows, *cols));
2387       CeedCall(CeedSingleOperatorAssemblyCountEntries(sub_operators[k], &single_entries));
2388       offset += single_entries;
2389     }
2390   } else {
2391     CeedCall(CeedSingleOperatorAssembleSymbolic(op, offset, *rows, *cols));
2392   }
2393   return CEED_ERROR_SUCCESS;
2394 }
2395 
2396 /**
2397    @brief Fully assemble the nonzero entries of a linear operator.
2398 
2399    Expected to be used in conjunction with @ref CeedOperatorLinearAssembleSymbolic().
2400 
2401    The assembly routines use coordinate format, with `num_entries` tuples of the form `(i, j, value)` which indicate that value should be added to the matrix in entry `(i, j)`.
2402    Note that the `(i, j)` pairs are not unique and may repeat.
2403    This function returns the values of the nonzero entries to be added, their `(i, j)` locations are provided by @ref CeedOperatorLinearAssembleSymbolic().
2404 
2405    This will generally be slow unless your operator is low-order.
2406 
2407    Note: Calling this function asserts that setup is complete and sets the `CeedOperator` as immutable.
2408 
2409    @param[in]  op     `CeedOperator` to assemble
2410    @param[out] values Values to assemble into matrix
2411 
2412    @ref User
2413 **/
2414 int CeedOperatorLinearAssemble(CeedOperator op, CeedVector values) {
2415   bool          is_composite;
2416   CeedInt       num_suboperators, offset = 0;
2417   CeedSize      single_entries = 0;
2418   CeedOperator *sub_operators;
2419 
2420   CeedCall(CeedOperatorCheckReady(op));
2421   CeedCall(CeedOperatorIsComposite(op, &is_composite));
2422 
2423   // Early exit for empty operator
2424   if (!is_composite) {
2425     CeedInt num_elem = 0;
2426 
2427     CeedCall(CeedOperatorGetNumElements(op, &num_elem));
2428     if (num_elem == 0) return CEED_ERROR_SUCCESS;
2429   }
2430 
2431   if (op->LinearAssemble) {
2432     // Backend version
2433     CeedCall(op->LinearAssemble(op, values));
2434     return CEED_ERROR_SUCCESS;
2435   } else {
2436     // Operator fallback
2437     CeedOperator op_fallback;
2438 
2439     CeedCall(CeedOperatorGetFallback(op, &op_fallback));
2440     if (op_fallback) {
2441       CeedCall(CeedOperatorLinearAssemble(op_fallback, values));
2442       return CEED_ERROR_SUCCESS;
2443     }
2444   }
2445 
2446   // Default interface implementation
2447   CeedCall(CeedVectorSetValue(values, 0.0));
2448   if (is_composite) {
2449     CeedCall(CeedCompositeOperatorGetNumSub(op, &num_suboperators));
2450     CeedCall(CeedCompositeOperatorGetSubList(op, &sub_operators));
2451     for (CeedInt k = 0; k < num_suboperators; k++) {
2452       CeedCall(CeedSingleOperatorAssemble(sub_operators[k], offset, values));
2453       CeedCall(CeedSingleOperatorAssemblyCountEntries(sub_operators[k], &single_entries));
2454       offset += single_entries;
2455     }
2456   } else {
2457     CeedCall(CeedSingleOperatorAssemble(op, offset, values));
2458   }
2459   return CEED_ERROR_SUCCESS;
2460 }
2461 
2462 /**
2463   @brief Get the multiplicity of nodes across sub-operators in a composite `CeedOperator`.
2464 
2465   Note: Calling this function asserts that setup is complete and sets the `CeedOperator` as immutable.
2466 
2467   @param[in]  op               Composite `CeedOperator`
2468   @param[in]  num_skip_indices Number of sub-operators to skip
2469   @param[in]  skip_indices     Array of indices of sub-operators to skip
2470   @param[out] mult             Vector to store multiplicity (of size `l_size` )
2471 
2472   @return An error code: 0 - success, otherwise - failure
2473 
2474   @ref User
2475 **/
2476 int CeedCompositeOperatorGetMultiplicity(CeedOperator op, CeedInt num_skip_indices, CeedInt *skip_indices, CeedVector mult) {
2477   Ceed                ceed;
2478   CeedInt             num_suboperators;
2479   CeedSize            l_vec_len;
2480   CeedScalar         *mult_array;
2481   CeedVector          ones_l_vec;
2482   CeedElemRestriction elem_rstr, mult_elem_rstr;
2483   CeedOperator       *sub_operators;
2484 
2485   CeedCall(CeedOperatorCheckReady(op));
2486 
2487   // Zero mult vector
2488   CeedCall(CeedVectorSetValue(mult, 0.0));
2489 
2490   // Get suboperators
2491   CeedCall(CeedCompositeOperatorGetNumSub(op, &num_suboperators));
2492   if (num_suboperators == 0) return CEED_ERROR_SUCCESS;
2493   CeedCall(CeedCompositeOperatorGetSubList(op, &sub_operators));
2494 
2495   // Work vector
2496   CeedCall(CeedVectorGetLength(mult, &l_vec_len));
2497   CeedCall(CeedOperatorGetCeed(op, &ceed));
2498   CeedCall(CeedVectorCreate(ceed, l_vec_len, &ones_l_vec));
2499   CeedCall(CeedDestroy(&ceed));
2500   CeedCall(CeedVectorSetValue(ones_l_vec, 1.0));
2501   CeedCall(CeedVectorGetArray(mult, CEED_MEM_HOST, &mult_array));
2502 
2503   // Compute multiplicity across suboperators
2504   for (CeedInt i = 0; i < num_suboperators; i++) {
2505     const CeedScalar *sub_mult_array;
2506     CeedVector        sub_mult_l_vec, ones_e_vec;
2507 
2508     // -- Check for suboperator to skip
2509     for (CeedInt j = 0; j < num_skip_indices; j++) {
2510       if (skip_indices[j] == i) continue;
2511     }
2512 
2513     // -- Sub operator multiplicity
2514     CeedCall(CeedOperatorGetActiveElemRestriction(sub_operators[i], &elem_rstr));
2515     CeedCall(CeedElemRestrictionCreateUnorientedCopy(elem_rstr, &mult_elem_rstr));
2516     CeedCall(CeedElemRestrictionDestroy(&elem_rstr));
2517     CeedCall(CeedElemRestrictionCreateVector(mult_elem_rstr, &sub_mult_l_vec, &ones_e_vec));
2518     CeedCall(CeedVectorSetValue(sub_mult_l_vec, 0.0));
2519     CeedCall(CeedElemRestrictionApply(mult_elem_rstr, CEED_NOTRANSPOSE, ones_l_vec, ones_e_vec, CEED_REQUEST_IMMEDIATE));
2520     CeedCall(CeedElemRestrictionApply(mult_elem_rstr, CEED_TRANSPOSE, ones_e_vec, sub_mult_l_vec, CEED_REQUEST_IMMEDIATE));
2521     CeedCall(CeedVectorGetArrayRead(sub_mult_l_vec, CEED_MEM_HOST, &sub_mult_array));
2522     // ---- Flag every node present in the current suboperator
2523     for (CeedSize j = 0; j < l_vec_len; j++) {
2524       if (sub_mult_array[j] > 0.0) mult_array[j] += 1.0;
2525     }
2526     CeedCall(CeedVectorRestoreArrayRead(sub_mult_l_vec, &sub_mult_array));
2527     CeedCall(CeedVectorDestroy(&sub_mult_l_vec));
2528     CeedCall(CeedVectorDestroy(&ones_e_vec));
2529     CeedCall(CeedElemRestrictionDestroy(&mult_elem_rstr));
2530   }
2531   CeedCall(CeedVectorRestoreArray(mult, &mult_array));
2532   CeedCall(CeedVectorDestroy(&ones_l_vec));
2533   return CEED_ERROR_SUCCESS;
2534 }
2535 
2536 /**
2537   @brief Create a multigrid coarse `CeedOperator` and level transfer `CeedOperator` for a `CeedOperator`, creating the prolongation basis from the fine and coarse grid interpolation.
2538 
2539   Note: Calling this function asserts that setup is complete and sets all four `CeedOperator` as immutable.
2540 
2541   @param[in]  op_fine      Fine grid `CeedOperator`
2542   @param[in]  p_mult_fine  L-vector multiplicity in parallel gather/scatter, or `NULL` if not creating prolongation/restriction `CeedOperator`
2543   @param[in]  rstr_coarse  Coarse grid `CeedElemRestriction`
2544   @param[in]  basis_coarse Coarse grid active vector `CeedBasis`
2545   @param[out] op_coarse    Coarse grid `CeedOperator`
2546   @param[out] op_prolong   Coarse to fine `CeedOperator`, or `NULL`
2547   @param[out] op_restrict  Fine to coarse `CeedOperator`, or `NULL`
2548 
2549   @return An error code: 0 - success, otherwise - failure
2550 
2551   @ref User
2552 **/
2553 int CeedOperatorMultigridLevelCreate(CeedOperator op_fine, CeedVector p_mult_fine, CeedElemRestriction rstr_coarse, CeedBasis basis_coarse,
2554                                      CeedOperator *op_coarse, CeedOperator *op_prolong, CeedOperator *op_restrict) {
2555   CeedBasis basis_c_to_f = NULL;
2556 
2557   CeedCall(CeedOperatorCheckReady(op_fine));
2558 
2559   // Build prolongation matrix, if required
2560   if (op_prolong || op_restrict) {
2561     CeedBasis basis_fine;
2562 
2563     CeedCall(CeedOperatorGetActiveBasis(op_fine, &basis_fine));
2564     CeedCall(CeedBasisCreateProjection(basis_coarse, basis_fine, &basis_c_to_f));
2565     CeedCall(CeedBasisDestroy(&basis_fine));
2566   }
2567 
2568   // Core code
2569   CeedCall(CeedSingleOperatorMultigridLevel(op_fine, p_mult_fine, rstr_coarse, basis_coarse, basis_c_to_f, op_coarse, op_prolong, op_restrict));
2570   return CEED_ERROR_SUCCESS;
2571 }
2572 
2573 /**
2574   @brief Create a multigrid coarse `CeedOperator` and level transfer `CeedOperator` for a `CeedOperator` with a tensor basis for the active basis.
2575 
2576   Note: Calling this function asserts that setup is complete and sets all four `CeedOperator` as immutable.
2577 
2578   @param[in]  op_fine       Fine grid `CeedOperator`
2579   @param[in]  p_mult_fine   L-vector multiplicity in parallel gather/scatter, or `NULL` if not creating prolongation/restriction `CeedOperator`
2580   @param[in]  rstr_coarse   Coarse grid `CeedElemRestriction`
2581   @param[in]  basis_coarse  Coarse grid active vector `CeedBasis`
2582   @param[in]  interp_c_to_f Matrix for coarse to fine interpolation, or `NULL` if not creating prolongation/restriction `CeedOperator`
2583   @param[out] op_coarse     Coarse grid `CeedOperator`
2584   @param[out] op_prolong    Coarse to fine `CeedOperator`, or `NULL`
2585   @param[out] op_restrict   Fine to coarse `CeedOperator`, or `NULL`
2586 
2587   @return An error code: 0 - success, otherwise - failure
2588 
2589   @ref User
2590 **/
2591 int CeedOperatorMultigridLevelCreateTensorH1(CeedOperator op_fine, CeedVector p_mult_fine, CeedElemRestriction rstr_coarse, CeedBasis basis_coarse,
2592                                              const CeedScalar *interp_c_to_f, CeedOperator *op_coarse, CeedOperator *op_prolong,
2593                                              CeedOperator *op_restrict) {
2594   Ceed      ceed;
2595   CeedInt   Q_f, Q_c;
2596   CeedBasis basis_fine, basis_c_to_f = NULL;
2597 
2598   CeedCall(CeedOperatorCheckReady(op_fine));
2599   CeedCall(CeedOperatorGetCeed(op_fine, &ceed));
2600 
2601   // Check for compatible quadrature spaces
2602   CeedCall(CeedOperatorGetActiveBasis(op_fine, &basis_fine));
2603   CeedCall(CeedBasisGetNumQuadraturePoints(basis_fine, &Q_f));
2604   CeedCall(CeedBasisGetNumQuadraturePoints(basis_coarse, &Q_c));
2605   CeedCheck(Q_f == Q_c, ceed, CEED_ERROR_DIMENSION,
2606             "Bases must have compatible quadrature spaces."
2607             " Fine grid: %" CeedInt_FMT " points, Coarse grid: %" CeedInt_FMT " points",
2608             Q_f, Q_c);
2609 
2610   // Create coarse to fine basis, if required
2611   if (op_prolong || op_restrict) {
2612     CeedInt     dim, num_comp, num_nodes_c, P_1d_f, P_1d_c;
2613     CeedScalar *q_ref, *q_weight, *grad;
2614 
2615     // Check if interpolation matrix is provided
2616     CeedCheck(interp_c_to_f, ceed, CEED_ERROR_INCOMPATIBLE,
2617               "Prolongation or restriction operator creation requires coarse-to-fine interpolation matrix");
2618     CeedCall(CeedBasisGetDimension(basis_fine, &dim));
2619     CeedCall(CeedBasisGetNumComponents(basis_fine, &num_comp));
2620     CeedCall(CeedBasisGetNumNodes1D(basis_fine, &P_1d_f));
2621     CeedCall(CeedBasisDestroy(&basis_fine));
2622     CeedCall(CeedElemRestrictionGetElementSize(rstr_coarse, &num_nodes_c));
2623     P_1d_c = dim == 1 ? num_nodes_c : dim == 2 ? sqrt(num_nodes_c) : cbrt(num_nodes_c);
2624     CeedCall(CeedCalloc(P_1d_f, &q_ref));
2625     CeedCall(CeedCalloc(P_1d_f, &q_weight));
2626     CeedCall(CeedCalloc(P_1d_f * P_1d_c * dim, &grad));
2627     CeedCall(CeedBasisCreateTensorH1(ceed, dim, num_comp, P_1d_c, P_1d_f, interp_c_to_f, grad, q_ref, q_weight, &basis_c_to_f));
2628     CeedCall(CeedFree(&q_ref));
2629     CeedCall(CeedFree(&q_weight));
2630     CeedCall(CeedFree(&grad));
2631   }
2632 
2633   // Core code
2634   CeedCall(CeedSingleOperatorMultigridLevel(op_fine, p_mult_fine, rstr_coarse, basis_coarse, basis_c_to_f, op_coarse, op_prolong, op_restrict));
2635   CeedCall(CeedDestroy(&ceed));
2636   return CEED_ERROR_SUCCESS;
2637 }
2638 
2639 /**
2640   @brief Create a multigrid coarse `CeedOperator` and level transfer `CeedOperator` for a `CeedOperator` with a non-tensor basis for the active vector
2641 
2642   Note: Calling this function asserts that setup is complete and sets all four `CeedOperator` as immutable.
2643 
2644   @param[in]  op_fine       Fine grid `CeedOperator`
2645   @param[in]  p_mult_fine   L-vector multiplicity in parallel gather/scatter, or `NULL` if not creating prolongation/restriction `CeedOperator`
2646   @param[in]  rstr_coarse   Coarse grid `CeedElemRestriction`
2647   @param[in]  basis_coarse  Coarse grid active vector `CeedBasis`
2648   @param[in]  interp_c_to_f Matrix for coarse to fine interpolation, or `NULL` if not creating prolongation/restriction `CeedOperator`
2649   @param[out] op_coarse     Coarse grid `CeedOperator`
2650   @param[out] op_prolong    Coarse to fine `CeedOperator`, or `NULL`
2651   @param[out] op_restrict   Fine to coarse `CeedOperator`, or `NULL`
2652 
2653   @return An error code: 0 - success, otherwise - failure
2654 
2655   @ref User
2656 **/
2657 int CeedOperatorMultigridLevelCreateH1(CeedOperator op_fine, CeedVector p_mult_fine, CeedElemRestriction rstr_coarse, CeedBasis basis_coarse,
2658                                        const CeedScalar *interp_c_to_f, CeedOperator *op_coarse, CeedOperator *op_prolong,
2659                                        CeedOperator *op_restrict) {
2660   Ceed      ceed;
2661   CeedInt   Q_f, Q_c;
2662   CeedBasis basis_fine, basis_c_to_f = NULL;
2663 
2664   CeedCall(CeedOperatorCheckReady(op_fine));
2665   CeedCall(CeedOperatorGetCeed(op_fine, &ceed));
2666 
2667   // Check for compatible quadrature spaces
2668   CeedCall(CeedOperatorGetActiveBasis(op_fine, &basis_fine));
2669   CeedCall(CeedBasisGetNumQuadraturePoints(basis_fine, &Q_f));
2670   CeedCall(CeedBasisGetNumQuadraturePoints(basis_coarse, &Q_c));
2671   CeedCheck(Q_f == Q_c, ceed, CEED_ERROR_DIMENSION, "Bases must have compatible quadrature spaces");
2672 
2673   // Coarse to fine basis
2674   if (op_prolong || op_restrict) {
2675     CeedInt          dim, num_comp, num_nodes_c, num_nodes_f;
2676     CeedScalar      *q_ref, *q_weight, *grad;
2677     CeedElemTopology topo;
2678 
2679     // Check if interpolation matrix is provided
2680     CeedCheck(interp_c_to_f, ceed, CEED_ERROR_INCOMPATIBLE,
2681               "Prolongation or restriction operator creation requires coarse-to-fine interpolation matrix");
2682     CeedCall(CeedBasisGetTopology(basis_fine, &topo));
2683     CeedCall(CeedBasisGetDimension(basis_fine, &dim));
2684     CeedCall(CeedBasisGetNumComponents(basis_fine, &num_comp));
2685     CeedCall(CeedBasisGetNumNodes(basis_fine, &num_nodes_f));
2686     CeedCall(CeedBasisDestroy(&basis_fine));
2687     CeedCall(CeedElemRestrictionGetElementSize(rstr_coarse, &num_nodes_c));
2688     CeedCall(CeedCalloc(num_nodes_f * dim, &q_ref));
2689     CeedCall(CeedCalloc(num_nodes_f, &q_weight));
2690     CeedCall(CeedCalloc(num_nodes_f * num_nodes_c * dim, &grad));
2691     CeedCall(CeedBasisCreateH1(ceed, topo, num_comp, num_nodes_c, num_nodes_f, interp_c_to_f, grad, q_ref, q_weight, &basis_c_to_f));
2692     CeedCall(CeedFree(&q_ref));
2693     CeedCall(CeedFree(&q_weight));
2694     CeedCall(CeedFree(&grad));
2695   }
2696 
2697   // Core code
2698   CeedCall(CeedSingleOperatorMultigridLevel(op_fine, p_mult_fine, rstr_coarse, basis_coarse, basis_c_to_f, op_coarse, op_prolong, op_restrict));
2699   CeedCall(CeedDestroy(&ceed));
2700   return CEED_ERROR_SUCCESS;
2701 }
2702 
2703 /**
2704   @brief Build a FDM based approximate inverse for each element for a `CeedOperator`.
2705 
2706   This returns a `CeedOperator` and `CeedVector` to apply a Fast Diagonalization Method based approximate inverse.
2707   This function obtains the simultaneous diagonalization for the 1D mass and Laplacian operators, \f$M = V^T V, K = V^T S V\f$.
2708   The assembled `CeedQFunction` is used to modify the eigenvalues from simultaneous diagonalization and obtain an approximate inverse of the form \f$V^T \hat S V\f$.
2709   The `CeedOperator` must be linear and non-composite.
2710   The associated `CeedQFunction` must therefore also be linear.
2711 
2712   Note: Calling this function asserts that setup is complete and sets the `CeedOperator` as immutable.
2713 
2714   @param[in]  op      `CeedOperator` to create element inverses
2715   @param[out] fdm_inv `CeedOperator` to apply the action of a FDM based inverse for each element
2716   @param[in]  request Address of @ref CeedRequest for non-blocking completion, else @ref CEED_REQUEST_IMMEDIATE
2717 
2718   @return An error code: 0 - success, otherwise - failure
2719 
2720   @ref User
2721 **/
2722 int CeedOperatorCreateFDMElementInverse(CeedOperator op, CeedOperator *fdm_inv, CeedRequest *request) {
2723   Ceed                 ceed, ceed_parent;
2724   bool                 interp = false, grad = false, is_tensor_basis = true;
2725   CeedInt              num_input_fields, P_1d, Q_1d, num_nodes, num_qpts, dim, num_comp = 1, num_elem = 1;
2726   CeedScalar          *mass, *laplace, *x, *fdm_interp, *lambda, *elem_avg;
2727   const CeedScalar    *interp_1d, *grad_1d, *q_weight_1d;
2728   CeedVector           q_data;
2729   CeedElemRestriction  rstr  = NULL, rstr_qd_i;
2730   CeedBasis            basis = NULL, fdm_basis;
2731   CeedQFunctionContext ctx_fdm;
2732   CeedQFunctionField  *qf_fields;
2733   CeedQFunction        qf, qf_fdm;
2734   CeedOperatorField   *op_fields;
2735 
2736   CeedCall(CeedOperatorCheckReady(op));
2737 
2738   if (op->CreateFDMElementInverse) {
2739     // Backend version
2740     CeedCall(op->CreateFDMElementInverse(op, fdm_inv, request));
2741     return CEED_ERROR_SUCCESS;
2742   } else {
2743     // Operator fallback
2744     CeedOperator op_fallback;
2745 
2746     CeedCall(CeedOperatorGetFallback(op, &op_fallback));
2747     if (op_fallback) {
2748       CeedCall(CeedOperatorCreateFDMElementInverse(op_fallback, fdm_inv, request));
2749       return CEED_ERROR_SUCCESS;
2750     }
2751   }
2752 
2753   // Default interface implementation
2754   CeedCall(CeedOperatorGetCeed(op, &ceed));
2755   CeedCall(CeedOperatorGetFallbackParentCeed(op, &ceed_parent));
2756   CeedCall(CeedOperatorGetQFunction(op, &qf));
2757 
2758   // Determine active input basis
2759   CeedCall(CeedOperatorGetFields(op, &num_input_fields, &op_fields, NULL, NULL));
2760   CeedCall(CeedQFunctionGetFields(qf, NULL, &qf_fields, NULL, NULL));
2761   for (CeedInt i = 0; i < num_input_fields; i++) {
2762     CeedVector vec;
2763 
2764     CeedCall(CeedOperatorFieldGetVector(op_fields[i], &vec));
2765     if (vec == CEED_VECTOR_ACTIVE) {
2766       CeedEvalMode eval_mode;
2767 
2768       CeedCall(CeedQFunctionFieldGetEvalMode(qf_fields[i], &eval_mode));
2769       interp = interp || eval_mode == CEED_EVAL_INTERP;
2770       grad   = grad || eval_mode == CEED_EVAL_GRAD;
2771       if (!basis) CeedCall(CeedOperatorFieldGetBasis(op_fields[i], &basis));
2772       if (!rstr) CeedCall(CeedOperatorFieldGetElemRestriction(op_fields[i], &rstr));
2773     }
2774     CeedCall(CeedVectorDestroy(&vec));
2775   }
2776   CeedCheck(basis, ceed, CEED_ERROR_BACKEND, "No active field set");
2777   CeedCall(CeedBasisGetNumNodes1D(basis, &P_1d));
2778   CeedCall(CeedBasisGetNumNodes(basis, &num_nodes));
2779   CeedCall(CeedBasisGetNumQuadraturePoints1D(basis, &Q_1d));
2780   CeedCall(CeedBasisGetNumQuadraturePoints(basis, &num_qpts));
2781   CeedCall(CeedBasisGetDimension(basis, &dim));
2782   CeedCall(CeedBasisGetNumComponents(basis, &num_comp));
2783   CeedCall(CeedElemRestrictionGetNumElements(rstr, &num_elem));
2784 
2785   // Build and diagonalize 1D Mass and Laplacian
2786   CeedCall(CeedBasisIsTensor(basis, &is_tensor_basis));
2787   CeedCheck(is_tensor_basis, ceed, CEED_ERROR_BACKEND, "FDMElementInverse only supported for tensor bases");
2788   CeedCall(CeedCalloc(P_1d * P_1d, &mass));
2789   CeedCall(CeedCalloc(P_1d * P_1d, &laplace));
2790   CeedCall(CeedCalloc(P_1d * P_1d, &x));
2791   CeedCall(CeedCalloc(P_1d * P_1d, &fdm_interp));
2792   CeedCall(CeedCalloc(P_1d, &lambda));
2793   // -- Build matrices
2794   CeedCall(CeedBasisGetInterp1D(basis, &interp_1d));
2795   CeedCall(CeedBasisGetGrad1D(basis, &grad_1d));
2796   CeedCall(CeedBasisGetQWeights(basis, &q_weight_1d));
2797   CeedCall(CeedBuildMassLaplace(interp_1d, grad_1d, q_weight_1d, P_1d, Q_1d, dim, mass, laplace));
2798 
2799   // -- Diagonalize
2800   CeedCall(CeedSimultaneousDiagonalization(ceed, laplace, mass, x, lambda, P_1d));
2801   CeedCall(CeedFree(&mass));
2802   CeedCall(CeedFree(&laplace));
2803   for (CeedInt i = 0; i < P_1d; i++) {
2804     for (CeedInt j = 0; j < P_1d; j++) fdm_interp[i + j * P_1d] = x[j + i * P_1d];
2805   }
2806   CeedCall(CeedFree(&x));
2807 
2808   {
2809     CeedInt             layout[3], num_modes = (interp ? 1 : 0) + (grad ? dim : 0);
2810     CeedScalar          max_norm = 0;
2811     const CeedScalar   *assembled_array, *q_weight_array;
2812     CeedVector          assembled = NULL, q_weight;
2813     CeedElemRestriction rstr_qf   = NULL;
2814 
2815     // Assemble QFunction
2816     CeedCall(CeedOperatorLinearAssembleQFunctionBuildOrUpdate(op, &assembled, &rstr_qf, request));
2817     CeedCall(CeedElemRestrictionGetELayout(rstr_qf, layout));
2818     CeedCall(CeedElemRestrictionDestroy(&rstr_qf));
2819     CeedCall(CeedVectorNorm(assembled, CEED_NORM_MAX, &max_norm));
2820 
2821     // Calculate element averages
2822     CeedCall(CeedVectorCreate(ceed_parent, num_qpts, &q_weight));
2823     CeedCall(CeedBasisApply(basis, 1, CEED_NOTRANSPOSE, CEED_EVAL_WEIGHT, CEED_VECTOR_NONE, q_weight));
2824     CeedCall(CeedVectorGetArrayRead(assembled, CEED_MEM_HOST, &assembled_array));
2825     CeedCall(CeedVectorGetArrayRead(q_weight, CEED_MEM_HOST, &q_weight_array));
2826     CeedCall(CeedCalloc(num_elem, &elem_avg));
2827     const CeedScalar qf_value_bound = max_norm * 100 * CEED_EPSILON;
2828 
2829     for (CeedInt e = 0; e < num_elem; e++) {
2830       CeedInt count = 0;
2831 
2832       for (CeedInt q = 0; q < num_qpts; q++) {
2833         for (CeedInt i = 0; i < num_comp * num_comp * num_modes * num_modes; i++) {
2834           if (fabs(assembled_array[q * layout[0] + i * layout[1] + e * layout[2]]) > qf_value_bound) {
2835             elem_avg[e] += assembled_array[q * layout[0] + i * layout[1] + e * layout[2]] / q_weight_array[q];
2836             count++;
2837           }
2838         }
2839       }
2840       if (count) {
2841         elem_avg[e] /= count;
2842       } else {
2843         elem_avg[e] = 1.0;
2844       }
2845     }
2846     CeedCall(CeedVectorRestoreArrayRead(assembled, &assembled_array));
2847     CeedCall(CeedVectorDestroy(&assembled));
2848     CeedCall(CeedVectorRestoreArrayRead(q_weight, &q_weight_array));
2849     CeedCall(CeedVectorDestroy(&q_weight));
2850   }
2851 
2852   // Build FDM diagonal
2853   {
2854     CeedScalar *q_data_array, *fdm_diagonal;
2855 
2856     CeedCall(CeedCalloc(num_comp * num_nodes, &fdm_diagonal));
2857     const CeedScalar fdm_diagonal_bound = num_nodes * CEED_EPSILON;
2858     for (CeedInt c = 0; c < num_comp; c++) {
2859       for (CeedInt n = 0; n < num_nodes; n++) {
2860         if (interp) fdm_diagonal[c * num_nodes + n] = 1.0;
2861         if (grad) {
2862           for (CeedInt d = 0; d < dim; d++) {
2863             CeedInt i = (n / CeedIntPow(P_1d, d)) % P_1d;
2864             fdm_diagonal[c * num_nodes + n] += lambda[i];
2865           }
2866         }
2867         if (fabs(fdm_diagonal[c * num_nodes + n]) < fdm_diagonal_bound) fdm_diagonal[c * num_nodes + n] = fdm_diagonal_bound;
2868       }
2869     }
2870     CeedCall(CeedVectorCreate(ceed_parent, num_elem * num_comp * num_nodes, &q_data));
2871     CeedCall(CeedVectorSetValue(q_data, 0.0));
2872     CeedCall(CeedVectorGetArrayWrite(q_data, CEED_MEM_HOST, &q_data_array));
2873     for (CeedInt e = 0; e < num_elem; e++) {
2874       for (CeedInt c = 0; c < num_comp; c++) {
2875         for (CeedInt n = 0; n < num_nodes; n++) {
2876           q_data_array[(e * num_comp + c) * num_nodes + n] = 1. / (elem_avg[e] * fdm_diagonal[c * num_nodes + n]);
2877         }
2878       }
2879     }
2880     CeedCall(CeedFree(&elem_avg));
2881     CeedCall(CeedFree(&fdm_diagonal));
2882     CeedCall(CeedVectorRestoreArray(q_data, &q_data_array));
2883   }
2884 
2885   // Setup FDM operator
2886   // -- Basis
2887   {
2888     CeedScalar *grad_dummy, *q_ref_dummy, *q_weight_dummy;
2889 
2890     CeedCall(CeedCalloc(P_1d * P_1d, &grad_dummy));
2891     CeedCall(CeedCalloc(P_1d, &q_ref_dummy));
2892     CeedCall(CeedCalloc(P_1d, &q_weight_dummy));
2893     CeedCall(CeedBasisCreateTensorH1(ceed_parent, dim, num_comp, P_1d, P_1d, fdm_interp, grad_dummy, q_ref_dummy, q_weight_dummy, &fdm_basis));
2894     CeedCall(CeedFree(&fdm_interp));
2895     CeedCall(CeedFree(&grad_dummy));
2896     CeedCall(CeedFree(&q_ref_dummy));
2897     CeedCall(CeedFree(&q_weight_dummy));
2898     CeedCall(CeedFree(&lambda));
2899   }
2900 
2901   // -- Restriction
2902   {
2903     CeedInt strides[3] = {1, num_nodes, num_nodes * num_comp};
2904     CeedCall(CeedElemRestrictionCreateStrided(ceed_parent, num_elem, num_nodes, num_comp,
2905                                               (CeedSize)num_elem * (CeedSize)num_comp * (CeedSize)num_nodes, strides, &rstr_qd_i));
2906   }
2907 
2908   // -- QFunction
2909   CeedCall(CeedQFunctionCreateInteriorByName(ceed_parent, "Scale", &qf_fdm));
2910   CeedCall(CeedQFunctionAddInput(qf_fdm, "input", num_comp, CEED_EVAL_INTERP));
2911   CeedCall(CeedQFunctionAddInput(qf_fdm, "scale", num_comp, CEED_EVAL_NONE));
2912   CeedCall(CeedQFunctionAddOutput(qf_fdm, "output", num_comp, CEED_EVAL_INTERP));
2913   CeedCall(CeedQFunctionSetUserFlopsEstimate(qf_fdm, num_comp));
2914 
2915   // -- QFunction context
2916   {
2917     CeedInt *num_comp_data;
2918 
2919     CeedCall(CeedCalloc(1, &num_comp_data));
2920     num_comp_data[0] = num_comp;
2921     CeedCall(CeedQFunctionContextCreate(ceed, &ctx_fdm));
2922     CeedCall(CeedQFunctionContextSetData(ctx_fdm, CEED_MEM_HOST, CEED_OWN_POINTER, sizeof(*num_comp_data), num_comp_data));
2923   }
2924   CeedCall(CeedQFunctionSetContext(qf_fdm, ctx_fdm));
2925   CeedCall(CeedQFunctionContextDestroy(&ctx_fdm));
2926 
2927   // -- Operator
2928   CeedCall(CeedOperatorCreate(ceed_parent, qf_fdm, NULL, NULL, fdm_inv));
2929   CeedCall(CeedOperatorSetField(*fdm_inv, "input", rstr, fdm_basis, CEED_VECTOR_ACTIVE));
2930   CeedCall(CeedOperatorSetField(*fdm_inv, "scale", rstr_qd_i, CEED_BASIS_NONE, q_data));
2931   CeedCall(CeedOperatorSetField(*fdm_inv, "output", rstr, fdm_basis, CEED_VECTOR_ACTIVE));
2932 
2933   // Cleanup
2934   CeedCall(CeedDestroy(&ceed));
2935   CeedCall(CeedDestroy(&ceed_parent));
2936   CeedCall(CeedVectorDestroy(&q_data));
2937   CeedCall(CeedElemRestrictionDestroy(&rstr));
2938   CeedCall(CeedElemRestrictionDestroy(&rstr_qd_i));
2939   CeedCall(CeedBasisDestroy(&basis));
2940   CeedCall(CeedBasisDestroy(&fdm_basis));
2941   CeedCall(CeedQFunctionDestroy(&qf_fdm));
2942   return CEED_ERROR_SUCCESS;
2943 }
2944 
2945 /// @}
2946