v0.14.0
base_functions.c
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1 /** \file base_functions.c
2 
3 */
4 
5 #include <cblas.h>
6 #include <petscsys.h>
7 #include <phg-quadrule/quad.h>
8 
9 #include <definitions.h>
10 
11 #include <base_functions.h>
12 
13 static PetscErrorCode ierr;
14 
15 PetscErrorCode Legendre_polynomials(int p, double s, double *diff_s, double *L,
16  double *diffL, const int dim) {
17  MoFEMFunctionBeginHot;
18 #ifndef NDEBUG
19  if (dim < 1)
20  SETERRQ(PETSC_COMM_SELF, MOFEM_INVALID_DATA, "dim < 1");
21  if (dim > 3)
22  SETERRQ(PETSC_COMM_SELF, MOFEM_INVALID_DATA, "dim > 3");
23  if (p < 0)
24  SETERRQ(PETSC_COMM_SELF, MOFEM_INVALID_DATA, "p < 0");
25 #endif // NDEBUG
26 
27  L[0] = 1;
28  if (diffL != NULL) {
29  for (int d = 0; d != dim; ++d) {
30  diffL[d * (p + 1) + 0] = 0;
31  }
32  }
33  if (p == 0)
34  MoFEMFunctionReturnHot(0);
35 
36  L[1] = s;
37  if (diffL != NULL) {
38 #ifndef NDEBUG
39  if (diff_s == NULL) {
40  SETERRQ(PETSC_COMM_SELF, MOFEM_INVALID_DATA, "diff_s == NULL");
41  }
42 #endif // NDEBUG
43  for (int d = 0; d != dim; ++d) {
44  diffL[d * (p + 1) + 1] = diff_s[d];
45  }
46  }
47  if (p == 1)
48  MoFEMFunctionReturnHot(0);
49 
50  int l = 1;
51  for (; l < p; l++) {
52  double A = ((2 * (double)l + 1) / ((double)l + 1));
53  double B = ((double)l / ((double)l + 1));
54  L[l + 1] = A * s * L[l] - B * L[l - 1];
55  if (diffL != NULL) {
56  for (int d = 0; d != dim; ++d) {
57  diffL[d * (p + 1) + l + 1] =
58  A * (diff_s[d] * L[l] + s * diffL[d * (p + 1) + l]) -
59  B * diffL[d * (p + 1) + l - 1];
60  }
61  }
62  }
63 
64  MoFEMFunctionReturnHot(0);
65 }
66 
67 PetscErrorCode Jacobi_polynomials(int p, double alpha, double x, double t,
68  double *diff_x, double *diff_t, double *L,
69  double *diffL, const int dim) {
70  MoFEMFunctionBeginHot;
71 #ifndef NDEBUG
72  if (dim < 1)
73  SETERRQ(PETSC_COMM_SELF, MOFEM_INVALID_DATA, "dim < 1");
74  if (dim > 3)
75  SETERRQ(PETSC_COMM_SELF, MOFEM_INVALID_DATA, "dim > 3");
76  if (p < 0)
77  SETERRQ(PETSC_COMM_SELF, MOFEM_INVALID_DATA, "p < 0");
78 
79  if (diffL != NULL) {
80  if (diff_x == NULL) {
81  SETERRQ(PETSC_COMM_SELF, MOFEM_INVALID_DATA, "diff_s == NULL");
82  }
83  }
84 
85 #endif // NDEBUG
86  L[0] = 1;
87  if (diffL != NULL) {
88  diffL[0 * (p + 1) + 0] = 0;
89  if (dim >= 2) {
90  diffL[1 * (p + 1) + 0] = 0;
91  if (dim == 3) {
92  diffL[2 * (p + 1) + 0] = 0;
93  }
94  }
95  }
96  if (p == 0)
97  MoFEMFunctionReturnHot(0);
98  L[1] = 2 * x - t + alpha * x;
99  if (diffL != NULL) {
100  int d = 0;
101  for (; d < dim; ++d) {
102  double d_t = (diff_t) ? diff_t[d] : 0;
103  diffL[d * (p + 1) + 1] = (2 + alpha) * diff_x[d] - d_t;
104  }
105  }
106  if (p == 1)
107  MoFEMFunctionReturnHot(0);
108  int l = 1;
109  for (; l < p; l++) {
110  int lp1 = l + 1;
111  double a = 2 * lp1 * (lp1 + alpha) * (2 * lp1 + alpha - 2);
112  double b = 2 * lp1 + alpha - 1;
113  double c = (2 * lp1 + alpha) * (2 * lp1 + alpha - 2);
114  double d = 2 * (lp1 + alpha - 1) * (lp1 - 1) * (2 * lp1 + alpha);
115  double A = b * (c * (2 * x - t) + alpha * alpha * t) / a;
116  double B = d * t * t / a;
117  L[lp1] = A * L[l] - B * L[l - 1];
118  if (diffL != NULL) {
119  int z = 0;
120  for (; z < dim; ++z) {
121  double d_t = (diff_t) ? diff_t[z] : 0;
122  double diffA =
123  b * (c * (2 * diff_x[z] - d_t) + alpha * alpha * d_t) / a;
124  double diffB = d * 2 * t * d_t / a;
125  diffL[z * (p + 1) + lp1] = A * diffL[z * (p + 1) + l] -
126  B * diffL[z * (p + 1) + l - 1] +
127  diffA * L[l] - diffB * L[l - 1];
128  }
129  }
130  }
131  MoFEMFunctionReturnHot(0);
132 }
133 
134 PetscErrorCode IntegratedJacobi_polynomials(int p, double alpha, double x,
135  double t, double *diff_x,
136  double *diff_t, double *L,
137  double *diffL, const int dim) {
138  MoFEMFunctionBeginHot;
139 #ifndef NDEBUG
140  if (dim < 1)
141  SETERRQ(PETSC_COMM_SELF, MOFEM_INVALID_DATA, "dim < 1");
142  if (dim > 3)
143  SETERRQ(PETSC_COMM_SELF, MOFEM_INVALID_DATA, "dim > 3");
144  if (p < 1)
145  SETERRQ(PETSC_COMM_SELF, MOFEM_INVALID_DATA, "p < 1");
146 #endif // NDEBUG
147  L[0] = x;
148  if (diffL != NULL) {
149  int d = 0;
150  for (; d != dim; ++d) {
151  diffL[d * p + 0] = diff_x[d];
152  }
153  }
154  if (p == 0)
155  MoFEMFunctionReturnHot(0);
156  double jacobi[(p + 1)];
157  double diff_jacobi[(p + 1) * dim];
158  ierr = Jacobi_polynomials(p, alpha, x, t, diff_x, diff_t, jacobi, diff_jacobi,
159  dim);
160  CHKERRQ(ierr);
161  int l = 1;
162  for (; l < p; l++) {
163  int i = l + 1;
164  const double a = (i + alpha) / ((2 * i + alpha - 1) * (2 * i + alpha));
165  const double b = alpha / ((2 * i + alpha - 2) * (2 * i + alpha));
166  const double c = (i - 1) / ((2 * i + alpha - 2) * (2 * i + alpha - 1));
167  L[l] = a * jacobi[i] + b * t * jacobi[i - 1] - c * t * t * jacobi[i - 2];
168  if (diffL != NULL) {
169  int d = 0;
170  for (; d != dim; ++d) {
171  diffL[d * p + l] = a * diff_jacobi[d * (p + 1) + i] +
172  b * (t * diff_jacobi[d * (p + 1) + i - 1] +
173  diff_t[d] * jacobi[i - 1]) -
174  c * (t * t * diff_jacobi[d * (p + 1) + i - 2] +
175  2 * t * diff_t[d] * jacobi[i - 2]);
176  }
177  }
178  }
179  MoFEMFunctionReturnHot(0);
180 }
181 
182 PetscErrorCode Lobatto_polynomials(int p, double s, double *diff_s, double *L,
183  double *diffL, const int dim) {
184 
185  MoFEMFunctionBeginHot;
186 #ifndef NDEBUG
187  if (dim < 1)
188  SETERRQ(PETSC_COMM_SELF, MOFEM_INVALID_DATA, "dim < 1");
189  if (dim > 3)
190  SETERRQ(PETSC_COMM_SELF, MOFEM_INVALID_DATA, "dim > 3");
191  if (p < 2)
192  SETERRQ(PETSC_COMM_SELF, MOFEM_INVALID_DATA, "p < 2");
193 #endif // NDEBUG
194  double l[p + 1];
195  ierr = Legendre_polynomials(p, s, NULL, l, NULL, 1);
196  CHKERRQ(ierr);
197 
198  L[0] = 1;
199  if (diffL != NULL) {
200  for (int d = 0; d != dim; ++d) {
201  diffL[d * (p + 1) + 0] = 0;
202  }
203  }
204  L[1] = s;
205  if (diffL != NULL) {
206 #ifndef NDEBUG
207  if (diff_s == NULL) {
208  SETERRQ(PETSC_COMM_SELF, MOFEM_INVALID_DATA, "diff_s == NULL");
209  }
210 #endif // NDEBUG
211  for (int d = 0; d != dim; ++d) {
212  diffL[d * (p + 1) + 1] = diff_s[d];
213  }
214  }
215 
216  // Integrated Legendre
217  for (int k = 2; k <= p; k++) {
218  const double factor = 2 * (2 * k - 1);
219  L[k] = 1.0 / factor * (l[k] - l[k - 2]);
220  }
221 
222  if (diffL != NULL) {
223  for (int k = 2; k <= p; k++) {
224  double a = l[k - 1] / 2.;
225  for (int d = 0; d != dim; ++d) {
226  diffL[d * (p + 1) + k] = a * diff_s[d];
227  }
228  }
229  }
230  MoFEMFunctionReturnHot(0);
231 }
232 
233 static double f_phi0(double x) { return LOBATTO_PHI0(x); }
234 static double f_phi1(double x) { return LOBATTO_PHI1(x); }
235 static double f_phi2(double x) { return LOBATTO_PHI2(x); }
236 static double f_phi3(double x) { return LOBATTO_PHI3(x); }
237 static double f_phi4(double x) { return LOBATTO_PHI4(x); }
238 static double f_phi5(double x) { return LOBATTO_PHI5(x); }
239 static double f_phi6(double x) { return LOBATTO_PHI6(x); }
240 static double f_phi7(double x) { return LOBATTO_PHI7(x); }
241 static double f_phi8(double x) { return LOBATTO_PHI8(x); }
242 static double f_phi9(double x) { return LOBATTO_PHI9(x); }
243 
244 static double (*f_phi[])(double x) = {f_phi0, f_phi1, f_phi2, f_phi3, f_phi4,
245  f_phi5, f_phi6, f_phi7, f_phi8, f_phi9};
246 
247 static double f_phi0x(double x) { return LOBATTO_PHI0X(x); }
248 static double f_phi1x(double x) { return LOBATTO_PHI1X(x); }
249 static double f_phi2x(double x) { return LOBATTO_PHI2X(x); }
250 static double f_phi3x(double x) { return LOBATTO_PHI3X(x); }
251 static double f_phi4x(double x) { return LOBATTO_PHI4X(x); }
252 static double f_phi5x(double x) { return LOBATTO_PHI5X(x); }
253 static double f_phi6x(double x) { return LOBATTO_PHI6X(x); }
254 static double f_phi7x(double x) { return LOBATTO_PHI7X(x); }
255 static double f_phi8x(double x) { return LOBATTO_PHI8X(x); }
256 static double f_phi9x(double x) { return LOBATTO_PHI9X(x); }
257 
258 static double (*f_phix[])(double x) = {f_phi0x, f_phi1x, f_phi2x, f_phi3x,
259  f_phi4x, f_phi5x, f_phi6x, f_phi7x,
260  f_phi8x, f_phi9x};
261 
262 // /// Legendre polynomials
263 // #define Legendre0(x) (1.0)
264 // #define Legendre1(x) (x)
265 // #define Legendre2(x) (1.0 / 2.0 * (3 * (x) * (x) - 1))
266 // #define Legendre3(x) (1.0 / 2.0 * (5 * (x) * (x) - 3) * (x))
267 // #define Legendre4(x) (1.0 / 8.0 * ((35 * (x) * (x) - 30) * (x) * (x) + 3))
268 // #define Legendre5(x) (1.0 / 8.0 * ((63 * (x) * (x) - 70) * (x) * (x) + 15) *
269 // (x)) #define Legendre6(x) (1.0 / 16.0 * (((231 * (x) * (x) - 315) * (x) * (x)
270 // + 105) * (x) * (x) - 5)) #define Legendre7(x) (1.0 / 16.0 * (((429 * (x) *
271 // (x) - 693) * (x) * (x) + 315) * (x) * (x) - 35) * (x)) #define Legendre8(x)
272 // (1.0 / 128.0 * ((((6435 * (x) * (x) - 12012) * (x) * (x) + 6930) * (x) * (x)
273 // - 1260) * (x) * (x) + 35)) #define Legendre9(x) (1.0 / 128.0 * ((((12155 *
274 // (x) * (x) - 25740) * (x) * (x) + 18018) * (x) * (x) - 4620) * (x) * (x) +
275 // 315) * (x)) #define Legendre10(x) (1.0 / 256.0 * (((((46189 * (x) * (x) -
276 // 109395) * (x) * (x) + 90090) * (x) * (x) - 30030) * (x) * (x) + 3465) * (x) *
277 // (x) - 63))
278 //
279 // /// derivatives of Legendre polynomials
280 // #define Legendre0x(x) (0.0)
281 // #define Legendre1x(x) (1.0)
282 // #define Legendre2x(x) (3.0 * (x))
283 // #define Legendre3x(x) (15.0 / 2.0 * (x) * (x) - 3.0 / 2.0)
284 // #define Legendre4x(x) (5.0 / 2.0 * (x) * (7.0 * (x) * (x) - 3.0))
285 // #define Legendre5x(x) ((315.0 / 8.0 * (x) * (x) - 105.0 / 4.0) * (x) * (x)
286 // + 15.0 / 8.0) #define Legendre6x(x) (21.0 / 8.0 * (x) * ((33.0 * (x) * (x)
287 // - 30.0) * (x) * (x) + 5.0)) #define Legendre7x(x) (((3003.0 / 16.0 * (x) *
288 // (x) - 3465.0 / 16.0) * (x) * (x) + 945.0 / 16.0) * (x) * (x) - 35.0 / 16.0)
289 // #define Legendre8x(x) (9.0 / 16.0 * (x) * (((715.0 * (x) * (x) - 1001.0) *
290 // (x) * (x) + 385.0) * (x) * (x) - 35.0)) #define Legendre9x(x) ((((109395.0 /
291 // 128.0 * (x) * (x) - 45045.0 / 32.0) * (x) * (x) + 45045.0 / 64.0) * (x) * (x)
292 // - 3465.0 / 32.0) * (x) * (x) + 315.0 / 128.0) #define Legendre10x(x) (2.0 /
293 // 256.0 * (x) * ((((230945.0 * (x) * (x) - 437580.0) * (x) * (x) + 270270.0) *
294 // (x) * (x) - 60060.0) * (x) * (x) + 3465.0))
295 //
296 // /// first two Lobatto shape functions
297 // #define l0(x) ((1.0 - (x)) * 0.5)
298 // #define l1(x) ((1.0 + (x)) * 0.5)
299 //
300 // #define l0l1(x) ((1.0 - (x)*(x)) * 0.25)
301 //
302 // /// other Lobatto shape functions
303 // #define l2(x) (phi0(x) * l0l1(x))
304 // #define l3(x) (phi1(x) * l0l1(x))
305 // #define l4(x) (phi2(x) * l0l1(x))
306 // #define l5(x) (phi3(x) * l0l1(x))
307 // #define l6(x) (phi4(x) * l0l1(x))
308 // #define l7(x) (phi5(x) * l0l1(x))
309 // #define l8(x) (phi6(x) * l0l1(x))
310 // #define l9(x) (phi7(x) * l0l1(x))
311 // #define l10(x) (phi8(x) * l0l1(x))
312 // #define l11(x) (phi9(x) * l0l1(x))
313 //
314 // /// derivatives of Lobatto functions
315 // #define dl0(x) (-0.5)
316 // #define dl1(x) (0.5)
317 // #define dl2(x) (sqrt(3.0/2.0) * Legendre1(x))
318 // #define dl3(x) (sqrt(5.0/2.0) * Legendre2(x))
319 // #define dl4(x) (sqrt(7.0/2.0) * Legendre3(x))
320 // #define dl5(x) (sqrt(9.0/2.0) * Legendre4(x))
321 // #define dl6(x) (sqrt(11.0/2.0) * Legendre5(x))
322 // #define dl7(x) (sqrt(13.0/2.0) * Legendre6(x))
323 // #define dl8(x) (sqrt(15.0/2.0) * Legendre7(x))
324 // #define dl9(x) (sqrt(17.0/2.0) * Legendre8(x))
325 // #define dl10(x) (sqrt(19.0/2.0) * Legendre9(x))
326 // #define dl11(x) (sqrt(21.0/2.0) * Legendre10(x))
327 
328 PetscErrorCode LobattoKernel_polynomials(int p, double s, double *diff_s,
329  double *L, double *diffL,
330  const int dim) {
331  MoFEMFunctionBeginHot;
332 #ifndef NDEBUG
333  if (dim < 1)
334  SETERRQ(PETSC_COMM_SELF, MOFEM_INVALID_DATA, "dim < 1");
335  if (dim > 3)
336  SETERRQ(PETSC_COMM_SELF, MOFEM_INVALID_DATA, "dim > 3");
337  if (p < 0)
338  SETERRQ(PETSC_COMM_SELF, MOFEM_INVALID_DATA, "p < 0");
339  if (p > 9)
340  SETERRQ(PETSC_COMM_SELF, MOFEM_NOT_IMPLEMENTED,
341  "Polynomial beyond order 9 is not implemented");
342  if (diffL != NULL) {
343  if (diff_s == NULL) {
344  SETERRQ(PETSC_COMM_SELF, MOFEM_INVALID_DATA, "diff_s == NULL");
345  }
346  }
347 
348 #endif // NDEBUG
349  if (L) {
350  int l = 0;
351  for (; l != p + 1; l++) {
352  L[l] = f_phi[l](s);
353  }
354  }
355  if (diffL != NULL) {
356  int l = 0;
357  for (; l != p + 1; l++) {
358  double a = f_phix[l](s);
359  int d = 0;
360  for (; d < dim; ++d) {
361  diffL[d * (p + 1) + l] = diff_s[d] * a;
362  }
363  }
364  }
365  MoFEMFunctionReturnHot(0);
366 }
MoFEMFunctionReturnHot
#define MoFEMFunctionReturnHot(a)
Last executable line of each PETSc function used for error handling. Replaces return()
Definition: definitions.h:460
LOBATTO_PHI4
#define LOBATTO_PHI4(x)
Definition: base_functions.h:132
f_phi5x
static double f_phi5x(double x)
Definition: base_functions.c:252
LOBATTO_PHI1X
#define LOBATTO_PHI1X(x)
Definition: base_functions.h:158
f_phi7
static double f_phi7(double x)
Definition: base_functions.c:240
IntegratedJacobi_polynomials
PetscErrorCode IntegratedJacobi_polynomials(int p, double alpha, double x, double t, double *diff_x, double *diff_t, double *L, double *diffL, const int dim)
Calculate integrated Jacobi approximation basis.
Definition: base_functions.c:134
LOBATTO_PHI7
#define LOBATTO_PHI7(x)
Definition: base_functions.h:141
LOBATTO_PHI9X
#define LOBATTO_PHI9X(x)
Definition: base_functions.h:179
f_phi7x
static double f_phi7x(double x)
Definition: base_functions.c:254
ierr
static PetscErrorCode ierr
Definition: base_functions.c:13
LOBATTO_PHI8
#define LOBATTO_PHI8(x)
Definition: base_functions.h:144
f_phi1x
static double f_phi1x(double x)
Definition: base_functions.c:248
f_phi4x
static double f_phi4x(double x)
Definition: base_functions.c:251
A
constexpr AssemblyType A
Definition: operators_tests.cpp:30
base_functions.h
LOBATTO_PHI6X
#define LOBATTO_PHI6X(x)
Definition: base_functions.h:169
LOBATTO_PHI1
#define LOBATTO_PHI1(x)
Definition: base_functions.h:127
LOBATTO_PHI2X
#define LOBATTO_PHI2X(x)
Definition: base_functions.h:159
f_phi8x
static double f_phi8x(double x)
Definition: base_functions.c:255
Lobatto_polynomials
PetscErrorCode Lobatto_polynomials(int p, double s, double *diff_s, double *L, double *diffL, const int dim)
Calculate Lobatto base functions .
Definition: base_functions.c:182
c
const double c
speed of light (cm/ns)
Definition: initial_diffusion.cpp:39
f_phi3
static double f_phi3(double x)
Definition: base_functions.c:236
Jacobi_polynomials
PetscErrorCode Jacobi_polynomials(int p, double alpha, double x, double t, double *diff_x, double *diff_t, double *L, double *diffL, const int dim)
Calculate Jacobi approximation basis.
Definition: base_functions.c:67
f_phi4
static double f_phi4(double x)
Definition: base_functions.c:237
f_phi
static double(* f_phi[])(double x)
Definition: base_functions.c:244
LOBATTO_PHI9
#define LOBATTO_PHI9(x)
Definition: base_functions.h:149
a
constexpr double a
Definition: approx_sphere.cpp:30
double
f_phi8
static double f_phi8(double x)
Definition: base_functions.c:241
f_phi2x
static double f_phi2x(double x)
Definition: base_functions.c:249
LOBATTO_PHI5X
#define LOBATTO_PHI5X(x)
Definition: base_functions.h:166
f_phi9
static double f_phi9(double x)
Definition: base_functions.c:242
LOBATTO_PHI8X
#define LOBATTO_PHI8X(x)
Definition: base_functions.h:175
MoFEM::L
VectorDouble L
Definition: Projection10NodeCoordsOnField.cpp:124
f_phi9x
static double f_phi9x(double x)
Definition: base_functions.c:256
f_phi0
static double f_phi0(double x)
Definition: base_functions.c:233
Legendre_polynomials
PetscErrorCode Legendre_polynomials(int p, double s, double *diff_s, double *L, double *diffL, const int dim)
Calculate Legendre approximation basis.
Definition: base_functions.c:15
definitions.h
useful compiler directives and definitions
t
constexpr double t
plate stiffness
Definition: plate.cpp:59
f_phi6x
static double f_phi6x(double x)
Definition: base_functions.c:253
i
FTensor::Index< 'i', SPACE_DIM > i
Definition: hcurl_divergence_operator_2d.cpp:27
f_phi3x
static double f_phi3x(double x)
Definition: base_functions.c:250
LobattoKernel_polynomials
PetscErrorCode LobattoKernel_polynomials(int p, double s, double *diff_s, double *L, double *diffL, const int dim)
Calculate Kernel Lobatto base functions.
Definition: base_functions.c:328
f_phi6
static double f_phi6(double x)
Definition: base_functions.c:239
LOBATTO_PHI0
#define LOBATTO_PHI0(x)
Definitions taken from Hermes2d code.
Definition: base_functions.h:126
LOBATTO_PHI3
#define LOBATTO_PHI3(x)
Definition: base_functions.h:130
f_phi1
static double f_phi1(double x)
Definition: base_functions.c:234
LOBATTO_PHI4X
#define LOBATTO_PHI4X(x)
Definition: base_functions.h:163
sdf_hertz_2d_axisymm_plane.d
float d
Definition: sdf_hertz_2d_axisymm_plane.py:4
f_phi0x
static double f_phi0x(double x)
Definition: base_functions.c:247
LOBATTO_PHI2
#define LOBATTO_PHI2(x)
Definition: base_functions.h:128
LOBATTO_PHI7X
#define LOBATTO_PHI7X(x)
Definition: base_functions.h:172
f_phi5
static double f_phi5(double x)
Definition: base_functions.c:238
f_phi2
static double f_phi2(double x)
Definition: base_functions.c:235
MoFEMFunctionBeginHot
#define MoFEMFunctionBeginHot
First executable line of each MoFEM function, used for error handling. Final line of MoFEM functions ...
Definition: definitions.h:453
LOBATTO_PHI5
#define LOBATTO_PHI5(x)
Definition: base_functions.h:135
k
FTensor::Index< 'k', 3 > k
Definition: matrix_function.cpp:20
f_phix
static double(* f_phix[])(double x)
Definition: base_functions.c:258
MOFEM_NOT_IMPLEMENTED
@ MOFEM_NOT_IMPLEMENTED
Definition: definitions.h:32
LOBATTO_PHI6
#define LOBATTO_PHI6(x)
Definition: base_functions.h:138
l
FTensor::Index< 'l', 3 > l
Definition: matrix_function.cpp:21
MOFEM_INVALID_DATA
@ MOFEM_INVALID_DATA
Definition: definitions.h:36
LOBATTO_PHI3X
#define LOBATTO_PHI3X(x)
Definition: base_functions.h:161
LOBATTO_PHI0X
#define LOBATTO_PHI0X(x)
Derivatives of kernel functions for Lobbatto base.
Definition: base_functions.h:157
quad.h
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