Actual source code: petscdt.h
petsc-3.13.6 2020-09-29
1: /*
2: Common tools for constructing discretizations
3: */
4: #if !defined(PETSCDT_H)
5: #define PETSCDT_H
7: #include <petscsys.h>
9: PETSC_EXTERN PetscClassId PETSCQUADRATURE_CLASSID;
11: /*S
12: PetscQuadrature - Quadrature rule for integration.
14: Level: beginner
16: .seealso: PetscQuadratureCreate(), PetscQuadratureDestroy()
17: S*/
18: typedef struct _p_PetscQuadrature *PetscQuadrature;
20: /*E
21: PetscGaussLobattoLegendreCreateType - algorithm used to compute the Gauss-Lobatto-Legendre nodes and weights
23: Level: intermediate
25: $ PETSCGAUSSLOBATTOLEGENDRE_VIA_LINEAR_ALGEBRA - compute the nodes via linear algebra
26: $ PETSCGAUSSLOBATTOLEGENDRE_VIA_NEWTON - compute the nodes by solving a nonlinear equation with Newton's method
28: E*/
29: typedef enum {PETSCGAUSSLOBATTOLEGENDRE_VIA_LINEAR_ALGEBRA,PETSCGAUSSLOBATTOLEGENDRE_VIA_NEWTON} PetscGaussLobattoLegendreCreateType;
31: /*E
32: PetscDTNodeType - A description of strategies for generating nodes (both
33: quadrature nodes and nodes for Lagrange polynomials)
35: Level: intermediate
37: $ PETSCDTNODES_DEFAULT - Nodes chosen by PETSc
38: $ PETSCDTNODES_GAUSSJACOBI - Nodes at either Gauss-Jacobi or Gauss-Lobatto-Jacobi quadrature points
39: $ PETSCDTNODES_EQUISPACED - Nodes equispaced either including the endpoints or excluding them
40: $ PETSCDTNODES_TANHSINH - Nodes at Tanh-Sinh quadrature points
42: Note: a PetscDTNodeType can be paired with a PetscBool to indicate whether
43: the nodes include endpoints or not, and in the case of PETSCDT_GAUSSJACOBI
44: with exponents for the weight function.
46: E*/
47: typedef enum {PETSCDTNODES_DEFAULT=-1, PETSCDTNODES_GAUSSJACOBI, PETSCDTNODES_EQUISPACED, PETSCDTNODES_TANHSINH} PetscDTNodeType;
49: PETSC_EXTERN const char *const PetscDTNodeTypes[];
51: PETSC_EXTERN PetscErrorCode PetscQuadratureCreate(MPI_Comm, PetscQuadrature *);
52: PETSC_EXTERN PetscErrorCode PetscQuadratureDuplicate(PetscQuadrature, PetscQuadrature *);
53: PETSC_EXTERN PetscErrorCode PetscQuadratureGetOrder(PetscQuadrature, PetscInt*);
54: PETSC_EXTERN PetscErrorCode PetscQuadratureSetOrder(PetscQuadrature, PetscInt);
55: PETSC_EXTERN PetscErrorCode PetscQuadratureGetNumComponents(PetscQuadrature, PetscInt*);
56: PETSC_EXTERN PetscErrorCode PetscQuadratureSetNumComponents(PetscQuadrature, PetscInt);
57: PETSC_EXTERN PetscErrorCode PetscQuadratureGetData(PetscQuadrature, PetscInt*, PetscInt*, PetscInt*, const PetscReal *[], const PetscReal *[]);
58: PETSC_EXTERN PetscErrorCode PetscQuadratureSetData(PetscQuadrature, PetscInt, PetscInt, PetscInt, const PetscReal [], const PetscReal []);
59: PETSC_EXTERN PetscErrorCode PetscQuadratureView(PetscQuadrature, PetscViewer);
60: PETSC_EXTERN PetscErrorCode PetscQuadratureDestroy(PetscQuadrature *);
62: PETSC_EXTERN PetscErrorCode PetscQuadratureExpandComposite(PetscQuadrature, PetscInt, const PetscReal[], const PetscReal[], PetscQuadrature *);
64: PETSC_EXTERN PetscErrorCode PetscQuadraturePushForward(PetscQuadrature, PetscInt, const PetscReal[], const PetscReal[], const PetscReal[], PetscInt, PetscQuadrature *);
66: PETSC_EXTERN PetscErrorCode PetscDTLegendreEval(PetscInt,const PetscReal*,PetscInt,const PetscInt*,PetscReal*,PetscReal*,PetscReal*);
67: PETSC_EXTERN PetscErrorCode PetscDTJacobiEval(PetscInt,PetscReal,PetscReal,const PetscReal*,PetscInt,const PetscInt*,PetscReal*,PetscReal*,PetscReal*);
68: PETSC_EXTERN PetscErrorCode PetscDTGaussQuadrature(PetscInt,PetscReal,PetscReal,PetscReal*,PetscReal*);
69: PETSC_EXTERN PetscErrorCode PetscDTGaussJacobiQuadrature(PetscInt,PetscReal,PetscReal,PetscReal,PetscReal,PetscReal*,PetscReal*);
70: PETSC_EXTERN PetscErrorCode PetscDTGaussLobattoJacobiQuadrature(PetscInt,PetscReal,PetscReal,PetscReal,PetscReal,PetscReal*,PetscReal*);
71: PETSC_EXTERN PetscErrorCode PetscDTGaussLobattoLegendreQuadrature(PetscInt,PetscGaussLobattoLegendreCreateType,PetscReal*,PetscReal*);
72: PETSC_EXTERN PetscErrorCode PetscDTReconstructPoly(PetscInt,PetscInt,const PetscReal*,PetscInt,const PetscReal*,PetscReal*);
73: PETSC_EXTERN PetscErrorCode PetscDTGaussTensorQuadrature(PetscInt,PetscInt,PetscInt,PetscReal,PetscReal,PetscQuadrature*);
74: PETSC_EXTERN PetscErrorCode PetscDTStroudConicalQuadrature(PetscInt,PetscInt,PetscInt,PetscReal,PetscReal,PetscQuadrature*);
76: PETSC_EXTERN PetscErrorCode PetscDTTanhSinhTensorQuadrature(PetscInt, PetscInt, PetscReal, PetscReal, PetscQuadrature *);
77: PETSC_EXTERN PetscErrorCode PetscDTTanhSinhIntegrate(void (*)(PetscReal, PetscReal *), PetscReal, PetscReal, PetscInt, PetscReal *);
78: PETSC_EXTERN PetscErrorCode PetscDTTanhSinhIntegrateMPFR(void (*)(PetscReal, PetscReal *), PetscReal, PetscReal, PetscInt, PetscReal *);
80: PETSC_EXTERN PetscErrorCode PetscGaussLobattoLegendreIntegrate(PetscInt, PetscReal *, PetscReal *, const PetscReal *, PetscReal *);
81: PETSC_EXTERN PetscErrorCode PetscGaussLobattoLegendreElementLaplacianCreate(PetscInt, PetscReal *, PetscReal *, PetscReal ***);
82: PETSC_EXTERN PetscErrorCode PetscGaussLobattoLegendreElementLaplacianDestroy(PetscInt, PetscReal *, PetscReal *, PetscReal ***);
83: PETSC_EXTERN PetscErrorCode PetscGaussLobattoLegendreElementGradientCreate(PetscInt, PetscReal *, PetscReal *, PetscReal ***, PetscReal ***);
84: PETSC_EXTERN PetscErrorCode PetscGaussLobattoLegendreElementGradientDestroy(PetscInt, PetscReal *, PetscReal *, PetscReal ***, PetscReal ***);
85: PETSC_EXTERN PetscErrorCode PetscGaussLobattoLegendreElementAdvectionCreate(PetscInt, PetscReal *, PetscReal *, PetscReal ***);
86: PETSC_EXTERN PetscErrorCode PetscGaussLobattoLegendreElementAdvectionDestroy(PetscInt, PetscReal *, PetscReal *, PetscReal ***);
87: PETSC_EXTERN PetscErrorCode PetscGaussLobattoLegendreElementMassCreate(PetscInt, PetscReal *, PetscReal *, PetscReal ***);
88: PETSC_EXTERN PetscErrorCode PetscGaussLobattoLegendreElementMassDestroy(PetscInt, PetscReal *, PetscReal *, PetscReal ***);
90: PETSC_EXTERN PetscErrorCode PetscDTAltVApply(PetscInt, PetscInt, const PetscReal *, const PetscReal *, PetscReal *);
91: PETSC_EXTERN PetscErrorCode PetscDTAltVWedge(PetscInt, PetscInt, PetscInt, const PetscReal *, const PetscReal *, PetscReal *);
92: PETSC_EXTERN PetscErrorCode PetscDTAltVWedgeMatrix(PetscInt, PetscInt, PetscInt, const PetscReal *, PetscReal *);
93: PETSC_EXTERN PetscErrorCode PetscDTAltVPullback(PetscInt, PetscInt, const PetscReal *, PetscInt, const PetscReal *, PetscReal *);
94: PETSC_EXTERN PetscErrorCode PetscDTAltVPullbackMatrix(PetscInt, PetscInt, const PetscReal *, PetscInt, PetscReal *);
95: PETSC_EXTERN PetscErrorCode PetscDTAltVInterior(PetscInt, PetscInt, const PetscReal *, const PetscReal *, PetscReal *);
96: PETSC_EXTERN PetscErrorCode PetscDTAltVInteriorMatrix(PetscInt, PetscInt, const PetscReal *, PetscReal *);
97: PETSC_EXTERN PetscErrorCode PetscDTAltVInteriorPattern(PetscInt, PetscInt, PetscInt (*)[3]);
98: PETSC_EXTERN PetscErrorCode PetscDTAltVStar(PetscInt, PetscInt, PetscInt, const PetscReal *, PetscReal *);
100: PETSC_EXTERN PetscErrorCode PetscDTBaryToIndex(PetscInt,PetscInt,const PetscInt[],PetscInt*);
101: PETSC_EXTERN PetscErrorCode PetscDTIndexToBary(PetscInt,PetscInt,PetscInt,PetscInt[]);
103: #if defined(PETSC_USE_64BIT_INDICES)
104: #define PETSC_FACTORIAL_MAX 20
105: #define PETSC_BINOMIAL_MAX 61
106: #else
107: #define PETSC_FACTORIAL_MAX 12
108: #define PETSC_BINOMIAL_MAX 29
109: #endif
111: /*MC
112: PetscDTFactorial - Approximate n! as a real number
114: Input Arguments:
115: . n - a non-negative integer
117: Output Arguments:
118: . factorial - n!
120: Level: beginner
121: M*/
122: PETSC_STATIC_INLINE PetscErrorCode PetscDTFactorial(PetscInt n, PetscReal *factorial)
123: {
124: PetscReal f = 1.0;
125: PetscInt i;
128: *factorial = -1.0;
129: if (n < 0) SETERRQ1(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Factorial called with negative number %D\n", n);
130: for (i = 1; i < n+1; ++i) f *= (PetscReal)i;
131: *factorial = f;
132: return(0);
133: }
135: /*MC
136: PetscDTFactorialInt - Compute n! as an integer
138: Input Arguments:
139: . n - a non-negative integer
141: Output Arguments:
142: . factorial - n!
144: Level: beginner
146: Note: this is limited to n such that n! can be represented by PetscInt, which is 12 if PetscInt is a signed 32-bit integer and 20 if PetscInt is a signed 64-bit integer.
147: M*/
148: PETSC_STATIC_INLINE PetscErrorCode PetscDTFactorialInt(PetscInt n, PetscInt *factorial)
149: {
150: PetscInt facLookup[13] = {1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916800, 479001600};
153: *factorial = -1;
154: if (n < 0 || n > PETSC_FACTORIAL_MAX) SETERRQ2(PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"Number of elements %D is not in supported range [0,%D]\n",n,PETSC_FACTORIAL_MAX);
155: if (n <= 12) {
156: *factorial = facLookup[n];
157: } else {
158: PetscInt f = facLookup[12];
159: PetscInt i;
161: for (i = 13; i < n+1; ++i) f *= i;
162: *factorial = f;
163: }
164: return(0);
165: }
167: /*MC
168: PetscDTBinomial - Approximate the binomial coefficient "n choose k"
170: Input Arguments:
171: + n - a non-negative integer
172: - k - an integer between 0 and n, inclusive
174: Output Arguments:
175: . binomial - approximation of the binomial coefficient n choose k
177: Level: beginner
178: M*/
179: PETSC_STATIC_INLINE PetscErrorCode PetscDTBinomial(PetscInt n, PetscInt k, PetscReal *binomial)
180: {
182: *binomial = -1.0;
183: if (n < 0 || k < 0 || k > n) SETERRQ2(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Binomial arguments (%D %D) must be non-negative, k <= n\n", n, k);
184: if (n <= 3) {
185: PetscInt binomLookup[4][4] = {{1, 0, 0, 0}, {1, 1, 0, 0}, {1, 2, 1, 0}, {1, 3, 3, 1}};
187: *binomial = (PetscReal)binomLookup[n][k];
188: } else {
189: PetscReal binom = 1.0;
190: PetscInt i;
192: k = PetscMin(k, n - k);
193: for (i = 0; i < k; i++) binom = (binom * (PetscReal)(n - i)) / (PetscReal)(i + 1);
194: *binomial = binom;
195: }
196: return(0);
197: }
199: /*MC
200: PetscDTBinomialInt - Compute the binomial coefficient "n choose k"
202: Input Arguments:
203: + n - a non-negative integer
204: - k - an integer between 0 and n, inclusive
206: Output Arguments:
207: . binomial - the binomial coefficient n choose k
209: Note: this is limited by integers that can be represented by PetscInt
211: Level: beginner
212: M*/
213: PETSC_STATIC_INLINE PetscErrorCode PetscDTBinomialInt(PetscInt n, PetscInt k, PetscInt *binomial)
214: {
215: PetscInt bin;
218: *binomial = -1;
219: if (n < 0 || k < 0 || k > n) SETERRQ2(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Binomial arguments (%D %D) must be non-negative, k <= n\n", n, k);
220: if (n > PETSC_BINOMIAL_MAX) SETERRQ2(PETSC_COMM_SELF, PETSC_ERR_ARG_OUTOFRANGE, "Binomial elements %D is larger than max for PetscInt, %D\n", n, PETSC_BINOMIAL_MAX);
221: if (n <= 3) {
222: PetscInt binomLookup[4][4] = {{1, 0, 0, 0}, {1, 1, 0, 0}, {1, 2, 1, 0}, {1, 3, 3, 1}};
224: bin = binomLookup[n][k];
225: } else {
226: PetscInt binom = 1;
227: PetscInt i;
229: k = PetscMin(k, n - k);
230: for (i = 0; i < k; i++) binom = (binom * (n - i)) / (i + 1);
231: bin = binom;
232: }
233: *binomial = bin;
234: return(0);
235: }
237: /*MC
238: PetscDTEnumPerm - Get a permutation of n integers from its encoding into the integers [0, n!) as a sequence of swaps.
240: A permutation can be described by the operations that convert the lists [0, 1, ..., n-1] into the permutation,
241: by a sequence of swaps, where the ith step swaps whatever number is in ith position with a number that is in
242: some position j >= i. This swap is encoded as the difference (j - i). The difference d_i at step i is less than
243: (n - i). This sequence of n-1 differences [d_0, ..., d_{n-2}] is encoded as the number
244: (n-1)! * d_0 + (n-2)! * d_1 + ... + 1! * d_{n-2}.
246: Input Arguments:
247: + n - a non-negative integer (see note about limits below)
248: - k - an integer in [0, n!)
250: Output Arguments:
251: + perm - the permuted list of the integers [0, ..., n-1]
252: - isOdd - if not NULL, returns wether the permutation used an even or odd number of swaps.
254: Note: this is limited to n such that n! can be represented by PetscInt, which is 12 if PetscInt is a signed 32-bit integer and 20 if PetscInt is a signed 64-bit integer.
256: Level: beginner
257: M*/
258: PETSC_STATIC_INLINE PetscErrorCode PetscDTEnumPerm(PetscInt n, PetscInt k, PetscInt *perm, PetscBool *isOdd)
259: {
260: PetscInt odd = 0;
261: PetscInt i;
262: PetscInt work[PETSC_FACTORIAL_MAX];
263: PetscInt *w;
266: if (isOdd) *isOdd = PETSC_FALSE;
267: if (n < 0 || n > PETSC_FACTORIAL_MAX) SETERRQ2(PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"Number of elements %D is not in supported range [0,%D]\n",n,PETSC_FACTORIAL_MAX);
268: w = &work[n - 2];
269: for (i = 2; i <= n; i++) {
270: *(w--) = k % i;
271: k /= i;
272: }
273: for (i = 0; i < n; i++) perm[i] = i;
274: for (i = 0; i < n - 1; i++) {
275: PetscInt s = work[i];
276: PetscInt swap = perm[i];
278: perm[i] = perm[i + s];
279: perm[i + s] = swap;
280: odd ^= (!!s);
281: }
282: if (isOdd) *isOdd = odd ? PETSC_TRUE : PETSC_FALSE;
283: return(0);
284: }
286: /*MC
287: PetscDTPermIndex - Encode a permutation of n into an integer in [0, n!). This inverts PetscDTEnumPerm.
289: Input Arguments:
290: + n - a non-negative integer (see note about limits below)
291: - perm - the permuted list of the integers [0, ..., n-1]
293: Output Arguments:
294: + k - an integer in [0, n!)
295: - isOdd - if not NULL, returns wether the permutation used an even or odd number of swaps.
297: Note: this is limited to n such that n! can be represented by PetscInt, which is 12 if PetscInt is a signed 32-bit integer and 20 if PetscInt is a signed 64-bit integer.
299: Level: beginner
300: M*/
301: PETSC_STATIC_INLINE PetscErrorCode PetscDTPermIndex(PetscInt n, const PetscInt *perm, PetscInt *k, PetscBool *isOdd)
302: {
303: PetscInt odd = 0;
304: PetscInt i, idx;
305: PetscInt work[PETSC_FACTORIAL_MAX];
306: PetscInt iwork[PETSC_FACTORIAL_MAX];
309: *k = -1;
310: if (isOdd) *isOdd = PETSC_FALSE;
311: if (n < 0 || n > PETSC_FACTORIAL_MAX) SETERRQ2(PETSC_COMM_SELF,PETSC_ERR_ARG_OUTOFRANGE,"Number of elements %D is not in supported range [0,%D]\n",n,PETSC_FACTORIAL_MAX);
312: for (i = 0; i < n; i++) work[i] = i; /* partial permutation */
313: for (i = 0; i < n; i++) iwork[i] = i; /* partial permutation inverse */
314: for (idx = 0, i = 0; i < n - 1; i++) {
315: PetscInt j = perm[i];
316: PetscInt icur = work[i];
317: PetscInt jloc = iwork[j];
318: PetscInt diff = jloc - i;
320: idx = idx * (n - i) + diff;
321: /* swap (i, jloc) */
322: work[i] = j;
323: work[jloc] = icur;
324: iwork[j] = i;
325: iwork[icur] = jloc;
326: odd ^= (!!diff);
327: }
328: *k = idx;
329: if (isOdd) *isOdd = odd ? PETSC_TRUE : PETSC_FALSE;
330: return(0);
331: }
333: /*MC
334: PetscDTEnumSubset - Get an ordered subset of the integers [0, ..., n - 1] from its encoding as an integers in [0, n choose k).
335: The encoding is in lexicographic order.
337: Input Arguments:
338: + n - a non-negative integer (see note about limits below)
339: . k - an integer in [0, n]
340: - j - an index in [0, n choose k)
342: Output Arguments:
343: . subset - the jth subset of size k of the integers [0, ..., n - 1]
345: Note: this is limited by arguments such that n choose k can be represented by PetscInt
347: Level: beginner
349: .seealso: PetscDTSubsetIndex()
350: M*/
351: PETSC_STATIC_INLINE PetscErrorCode PetscDTEnumSubset(PetscInt n, PetscInt k, PetscInt j, PetscInt *subset)
352: {
353: PetscInt Nk, i, l;
357: PetscDTBinomialInt(n, k, &Nk);
358: for (i = 0, l = 0; i < n && l < k; i++) {
359: PetscInt Nminuskminus = (Nk * (k - l)) / (n - i);
360: PetscInt Nminusk = Nk - Nminuskminus;
362: if (j < Nminuskminus) {
363: subset[l++] = i;
364: Nk = Nminuskminus;
365: } else {
366: j -= Nminuskminus;
367: Nk = Nminusk;
368: }
369: }
370: return(0);
371: }
373: /*MC
374: PetscDTSubsetIndex - Convert an ordered subset of k integers from the set [0, ..., n - 1] to its encoding as an integers in [0, n choose k) in lexicographic order. This is the inverse of PetscDTEnumSubset.
376: Input Arguments:
377: + n - a non-negative integer (see note about limits below)
378: . k - an integer in [0, n]
379: - subset - an ordered subset of the integers [0, ..., n - 1]
381: Output Arguments:
382: . index - the rank of the subset in lexicographic order
384: Note: this is limited by arguments such that n choose k can be represented by PetscInt
386: Level: beginner
388: .seealso: PetscDTEnumSubset()
389: M*/
390: PETSC_STATIC_INLINE PetscErrorCode PetscDTSubsetIndex(PetscInt n, PetscInt k, const PetscInt *subset, PetscInt *index)
391: {
392: PetscInt i, j = 0, l, Nk;
396: *index = -1;
397: PetscDTBinomialInt(n, k, &Nk);
398: for (i = 0, l = 0; i < n && l < k; i++) {
399: PetscInt Nminuskminus = (Nk * (k - l)) / (n - i);
400: PetscInt Nminusk = Nk - Nminuskminus;
402: if (subset[l] == i) {
403: l++;
404: Nk = Nminuskminus;
405: } else {
406: j += Nminuskminus;
407: Nk = Nminusk;
408: }
409: }
410: *index = j;
411: return(0);
412: }
414: /*MC
415: PetscDTEnumSubset - Split the integers [0, ..., n - 1] into two complementary ordered subsets, the first subset of size k and being the jth subset of that size in lexicographic order.
417: Input Arguments:
418: + n - a non-negative integer (see note about limits below)
419: . k - an integer in [0, n]
420: - j - an index in [0, n choose k)
422: Output Arguments:
423: + perm - the jth subset of size k of the integers [0, ..., n - 1], followed by its complementary set.
424: - isOdd - if not NULL, return whether perm is an even or odd permutation.
426: Note: this is limited by arguments such that n choose k can be represented by PetscInt
428: Level: beginner
430: .seealso: PetscDTEnumSubset(), PetscDTSubsetIndex()
431: M*/
432: PETSC_STATIC_INLINE PetscErrorCode PetscDTEnumSplit(PetscInt n, PetscInt k, PetscInt j, PetscInt *perm, PetscBool *isOdd)
433: {
434: PetscInt i, l, m, *subcomp, Nk;
435: PetscInt odd;
439: if (isOdd) *isOdd = PETSC_FALSE;
440: PetscDTBinomialInt(n, k, &Nk);
441: odd = 0;
442: subcomp = &perm[k];
443: for (i = 0, l = 0, m = 0; i < n && l < k; i++) {
444: PetscInt Nminuskminus = (Nk * (k - l)) / (n - i);
445: PetscInt Nminusk = Nk - Nminuskminus;
447: if (j < Nminuskminus) {
448: perm[l++] = i;
449: Nk = Nminuskminus;
450: } else {
451: subcomp[m++] = i;
452: j -= Nminuskminus;
453: odd ^= ((k - l) & 1);
454: Nk = Nminusk;
455: }
456: }
457: for (; i < n; i++) {
458: subcomp[m++] = i;
459: }
460: if (isOdd) *isOdd = odd ? PETSC_TRUE : PETSC_FALSE;
461: return(0);
462: }
464: struct _p_PetscTabulation {
465: PetscInt K; /* Indicates a k-jet, namely tabulated derviatives up to order k */
466: PetscInt Nr; /* The number of tabulation replicas (often 1) */
467: PetscInt Np; /* The number of tabulation points in a replica */
468: PetscInt Nb; /* The number of functions tabulated */
469: PetscInt Nc; /* The number of function components */
470: PetscInt cdim; /* The coordinate dimension */
471: PetscReal **T; /* The tabulation T[K] of functions and their derivatives
472: T[0] = B[Nr*Np][Nb][Nc]: The basis function values at quadrature points
473: T[1] = D[Nr*Np][Nb][Nc][cdim]: The basis function derivatives at quadrature points
474: T[2] = H[Nr*Np][Nb][Nc][cdim][cdim]: The basis function second derivatives at quadrature points */
475: };
476: typedef struct _p_PetscTabulation *PetscTabulation;
478: #endif