Intrepid
test_02.cpp
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43 
49 #include "Intrepid_FieldContainer.hpp"
50 #include "Intrepid_HGRAD_TET_Cn_FEM.hpp"
51 #include "Intrepid_DefaultCubatureFactory.hpp"
52 #include "Intrepid_RealSpaceTools.hpp"
53 #include "Intrepid_ArrayTools.hpp"
54 #include "Intrepid_FunctionSpaceTools.hpp"
55 #include "Intrepid_CellTools.hpp"
56 #include "Teuchos_oblackholestream.hpp"
57 #include "Teuchos_RCP.hpp"
58 #include "Teuchos_GlobalMPISession.hpp"
59 #include "Teuchos_SerialDenseMatrix.hpp"
60 #include "Teuchos_SerialDenseVector.hpp"
61 #include "Teuchos_LAPACK.hpp"
62 
63 using namespace std;
64 using namespace Intrepid;
65 
66 void rhsFunc(FieldContainer<double> &, const FieldContainer<double> &, int, int, int);
67 void neumann(FieldContainer<double> & ,
68  const FieldContainer<double> & ,
69  const FieldContainer<double> & ,
70  const shards::CellTopology & ,
71  int, int, int, int);
72 void u_exact(FieldContainer<double> &, const FieldContainer<double> &, int, int, int);
73 
75 void rhsFunc(FieldContainer<double> & result,
76  const FieldContainer<double> & points,
77  int xd,
78  int yd,
79  int zd) {
80 
81  int x = 0, y = 1, z = 2;
82 
83  // second x-derivatives of u
84  if (xd > 1) {
85  for (int cell=0; cell<result.dimension(0); cell++) {
86  for (int pt=0; pt<result.dimension(1); pt++) {
87  result(cell,pt) = - xd*(xd-1)*std::pow(points(cell,pt,x), xd-2) *
88  std::pow(points(cell,pt,y), yd) * std::pow(points(cell,pt,z), zd);
89  }
90  }
91  }
92 
93  // second y-derivatives of u
94  if (yd > 1) {
95  for (int cell=0; cell<result.dimension(0); cell++) {
96  for (int pt=0; pt<result.dimension(1); pt++) {
97  result(cell,pt) -= yd*(yd-1)*std::pow(points(cell,pt,y), yd-2) *
98  std::pow(points(cell,pt,x), xd) * std::pow(points(cell,pt,z), zd);
99  }
100  }
101  }
102 
103  // second z-derivatives of u
104  if (zd > 1) {
105  for (int cell=0; cell<result.dimension(0); cell++) {
106  for (int pt=0; pt<result.dimension(1); pt++) {
107  result(cell,pt) -= zd*(zd-1)*std::pow(points(cell,pt,z), zd-2) *
108  std::pow(points(cell,pt,x), xd) * std::pow(points(cell,pt,y), yd);
109  }
110  }
111  }
112 
113  // add u
114  for (int cell=0; cell<result.dimension(0); cell++) {
115  for (int pt=0; pt<result.dimension(1); pt++) {
116  result(cell,pt) += std::pow(points(cell,pt,x), xd) * std::pow(points(cell,pt,y), yd) * std::pow(points(cell,pt,z), zd);
117  }
118  }
119 
120 }
121 
122 
124 void neumann(FieldContainer<double> & result,
125  const FieldContainer<double> & points,
126  const FieldContainer<double> & jacs,
127  const shards::CellTopology & parentCell,
128  int sideOrdinal, int xd, int yd, int zd) {
129 
130  int x = 0, y = 1, z = 2;
131 
132  int numCells = result.dimension(0);
133  int numPoints = result.dimension(1);
134 
135  FieldContainer<double> grad_u(numCells, numPoints, 3);
136  FieldContainer<double> side_normals(numCells, numPoints, 3);
137  FieldContainer<double> normal_lengths(numCells, numPoints);
138 
139  // first x-derivatives of u
140  if (xd > 0) {
141  for (int cell=0; cell<numCells; cell++) {
142  for (int pt=0; pt<numPoints; pt++) {
143  grad_u(cell,pt,x) = xd*std::pow(points(cell,pt,x), xd-1) *
144  std::pow(points(cell,pt,y), yd) * std::pow(points(cell,pt,z), zd);
145  }
146  }
147  }
148 
149  // first y-derivatives of u
150  if (yd > 0) {
151  for (int cell=0; cell<numCells; cell++) {
152  for (int pt=0; pt<numPoints; pt++) {
153  grad_u(cell,pt,y) = yd*std::pow(points(cell,pt,y), yd-1) *
154  std::pow(points(cell,pt,x), xd) * std::pow(points(cell,pt,z), zd);
155  }
156  }
157  }
158 
159  // first z-derivatives of u
160  if (zd > 0) {
161  for (int cell=0; cell<numCells; cell++) {
162  for (int pt=0; pt<numPoints; pt++) {
163  grad_u(cell,pt,z) = zd*std::pow(points(cell,pt,z), zd-1) *
164  std::pow(points(cell,pt,x), xd) * std::pow(points(cell,pt,y), yd);
165  }
166  }
167  }
168 
169  CellTools<double>::getPhysicalSideNormals(side_normals, jacs, sideOrdinal, parentCell);
170 
171  // scale normals
172  RealSpaceTools<double>::vectorNorm(normal_lengths, side_normals, NORM_TWO);
173  FunctionSpaceTools::scalarMultiplyDataData<double>(side_normals, normal_lengths, side_normals, true);
174 
175  FunctionSpaceTools::dotMultiplyDataData<double>(result, grad_u, side_normals);
176 
177 }
178 
180 void u_exact(FieldContainer<double> & result, const FieldContainer<double> & points, int xd, int yd, int zd) {
181  int x = 0, y = 1, z = 2;
182  for (int cell=0; cell<result.dimension(0); cell++) {
183  for (int pt=0; pt<result.dimension(1); pt++) {
184  result(cell,pt) = std::pow(points(pt,x), xd)*std::pow(points(pt,y), yd)*std::pow(points(pt,z), zd);
185  }
186  }
187 }
188 
189 
190 
191 
192 int main(int argc, char *argv[]) {
193 
194  Teuchos::GlobalMPISession mpiSession(&argc, &argv);
195 
196  // This little trick lets us print to std::cout only if
197  // a (dummy) command-line argument is provided.
198  int iprint = argc - 1;
199  Teuchos::RCP<std::ostream> outStream;
200  Teuchos::oblackholestream bhs; // outputs nothing
201  if (iprint > 0)
202  outStream = Teuchos::rcp(&std::cout, false);
203  else
204  outStream = Teuchos::rcp(&bhs, false);
205 
206  // Save the format state of the original std::cout.
207  Teuchos::oblackholestream oldFormatState;
208  oldFormatState.copyfmt(std::cout);
209 
210  *outStream \
211  << "===============================================================================\n" \
212  << "| |\n" \
213  << "| Unit Test (Basis_HGRAD_TET_Cn_FEM) |\n" \
214  << "| |\n" \
215  << "| 1) Patch test involving mass and stiffness matrices, |\n" \
216  << "| for the Neumann problem on a tetrahedral patch |\n" \
217  << "| Omega with boundary Gamma. |\n" \
218  << "| |\n" \
219  << "| - div (grad u) + u = f in Omega, (grad u) . n = g on Gamma |\n" \
220  << "| |\n" \
221  << "| Questions? Contact Pavel Bochev (pbboche@sandia.gov), |\n" \
222  << "| Denis Ridzal (dridzal@sandia.gov), |\n" \
223  << "| Kara Peterson (kjpeter@sandia.gov). |\n" \
224  << "| |\n" \
225  << "| Intrepid's website: http://trilinos.sandia.gov/packages/intrepid |\n" \
226  << "| Trilinos website: http://trilinos.sandia.gov |\n" \
227  << "| |\n" \
228  << "===============================================================================\n"\
229  << "| TEST 1: Patch test |\n"\
230  << "===============================================================================\n";
231 
232 
233  int errorFlag = 0;
234 
235  outStream -> precision(16);
236 
237 
238  try {
239 
240  int max_order = 5; // max total order of polynomial solution
241  DefaultCubatureFactory<double> cubFactory; // create factory
242  shards::CellTopology cell(shards::getCellTopologyData< shards::Tetrahedron<> >()); // create parent cell topology
243  shards::CellTopology side(shards::getCellTopologyData< shards::Triangle<> >()); // create relevant subcell (side) topology
244  int cellDim = cell.getDimension();
245  int sideDim = side.getDimension();
246 
247  // Define array containing points at which the solution is evaluated, on the reference tet.
248  int numIntervals = 10;
249  int numInterpPoints = ((numIntervals + 1)*(numIntervals + 2)*(numIntervals + 3))/6;
250  FieldContainer<double> interp_points_ref(numInterpPoints, 3);
251  int counter = 0;
252  for (int k=0; k<=numIntervals; k++) {
253  for (int j=0; j<=numIntervals; j++) {
254  for (int i=0; i<=numIntervals; i++) {
255  if (i+j+k <= numIntervals) {
256  interp_points_ref(counter,0) = i*(1.0/numIntervals);
257  interp_points_ref(counter,1) = j*(1.0/numIntervals);
258  interp_points_ref(counter,2) = k*(1.0/numIntervals);
259  counter++;
260  }
261  }
262  }
263  }
264 
265  /* Definition of parent cell. */
266  FieldContainer<double> cell_nodes(1, 4, cellDim);
267  // funky tet
268  cell_nodes(0, 0, 0) = -1.0;
269  cell_nodes(0, 0, 1) = -2.0;
270  cell_nodes(0, 0, 2) = 0.0;
271  cell_nodes(0, 1, 0) = 6.0;
272  cell_nodes(0, 1, 1) = 2.0;
273  cell_nodes(0, 1, 2) = 0.0;
274  cell_nodes(0, 2, 0) = -5.0;
275  cell_nodes(0, 2, 1) = 1.0;
276  cell_nodes(0, 2, 2) = 0.0;
277  cell_nodes(0, 3, 0) = -4.0;
278  cell_nodes(0, 3, 1) = -1.0;
279  cell_nodes(0, 3, 2) = 3.0;
280  // perturbed reference tet
281  /*cell_nodes(0, 0, 0) = 0.1;
282  cell_nodes(0, 0, 1) = -0.1;
283  cell_nodes(0, 0, 2) = 0.2;
284  cell_nodes(0, 1, 0) = 1.2;
285  cell_nodes(0, 1, 1) = -0.1;
286  cell_nodes(0, 1, 2) = 0.05;
287  cell_nodes(0, 2, 0) = 0.0;
288  cell_nodes(0, 2, 1) = 0.9;
289  cell_nodes(0, 2, 2) = 0.1;
290  cell_nodes(0, 3, 0) = 0.1;
291  cell_nodes(0, 3, 1) = -0.1;
292  cell_nodes(0, 3, 2) = 1.1;*/
293  // reference tet
294  /*cell_nodes(0, 0, 0) = 0.0;
295  cell_nodes(0, 0, 1) = 0.0;
296  cell_nodes(0, 0, 2) = 0.0;
297  cell_nodes(0, 1, 0) = 1.0;
298  cell_nodes(0, 1, 1) = 0.0;
299  cell_nodes(0, 1, 2) = 0.0;
300  cell_nodes(0, 2, 0) = 0.0;
301  cell_nodes(0, 2, 1) = 1.0;
302  cell_nodes(0, 2, 2) = 0.0;
303  cell_nodes(0, 3, 0) = 0.0;
304  cell_nodes(0, 3, 1) = 0.0;
305  cell_nodes(0, 3, 2) = 1.0;*/
306 
307  FieldContainer<double> interp_points(1, numInterpPoints, cellDim);
308  CellTools<double>::mapToPhysicalFrame(interp_points, interp_points_ref, cell_nodes, cell);
309  interp_points.resize(numInterpPoints, cellDim);
310 
311  // we test two types of bases
312  EPointType pointtype[] = {POINTTYPE_EQUISPACED, POINTTYPE_WARPBLEND};
313  for (int ptype=0; ptype < 2; ptype++) {
314 
315  *outStream << "\nTesting bases with " << EPointTypeToString(pointtype[ptype]) << ":\n";
316 
317  for (int x_order=0; x_order <= max_order; x_order++) {
318  for (int y_order=0; y_order <= max_order-x_order; y_order++) {
319  for (int z_order=0; z_order <= max_order-x_order-y_order; z_order++) {
320 
321  // evaluate exact solution
322  FieldContainer<double> exact_solution(1, numInterpPoints);
323  u_exact(exact_solution, interp_points, x_order, y_order, z_order);
324 
325  int total_order = std::max(x_order + y_order + z_order, 1);
326 
327  for (int basis_order=total_order; basis_order <= max_order; basis_order++) {
328 
329  // set test tolerance;
330  double zero = basis_order*basis_order*basis_order*100*INTREPID_TOL;
331 
332  //create basis
333  Teuchos::RCP<Basis<double,FieldContainer<double> > > basis =
334  Teuchos::rcp(new Basis_HGRAD_TET_Cn_FEM<double,FieldContainer<double> >(basis_order, pointtype[ptype]) );
335  int numFields = basis->getCardinality();
336 
337  // create cubatures
338  Teuchos::RCP<Cubature<double> > cellCub = cubFactory.create(cell, 2*basis_order);
339  Teuchos::RCP<Cubature<double> > sideCub = cubFactory.create(side, 2*basis_order);
340  int numCubPointsCell = cellCub->getNumPoints();
341  int numCubPointsSide = sideCub->getNumPoints();
342 
343  /* Computational arrays. */
344  /* Section 1: Related to parent cell integration. */
345  FieldContainer<double> cub_points_cell(numCubPointsCell, cellDim);
346  FieldContainer<double> cub_points_cell_physical(1, numCubPointsCell, cellDim);
347  FieldContainer<double> cub_weights_cell(numCubPointsCell);
348  FieldContainer<double> jacobian_cell(1, numCubPointsCell, cellDim, cellDim);
349  FieldContainer<double> jacobian_inv_cell(1, numCubPointsCell, cellDim, cellDim);
350  FieldContainer<double> jacobian_det_cell(1, numCubPointsCell);
351  FieldContainer<double> weighted_measure_cell(1, numCubPointsCell);
352 
353  FieldContainer<double> value_of_basis_at_cub_points_cell(numFields, numCubPointsCell);
354  FieldContainer<double> transformed_value_of_basis_at_cub_points_cell(1, numFields, numCubPointsCell);
355  FieldContainer<double> weighted_transformed_value_of_basis_at_cub_points_cell(1, numFields, numCubPointsCell);
356  FieldContainer<double> grad_of_basis_at_cub_points_cell(numFields, numCubPointsCell, cellDim);
357  FieldContainer<double> transformed_grad_of_basis_at_cub_points_cell(1, numFields, numCubPointsCell, cellDim);
358  FieldContainer<double> weighted_transformed_grad_of_basis_at_cub_points_cell(1, numFields, numCubPointsCell, cellDim);
359  FieldContainer<double> fe_matrix(1, numFields, numFields);
360 
361  FieldContainer<double> rhs_at_cub_points_cell_physical(1, numCubPointsCell);
362  FieldContainer<double> rhs_and_soln_vector(1, numFields);
363 
364  /* Section 2: Related to subcell (side) integration. */
365  unsigned numSides = 4;
366  FieldContainer<double> cub_points_side(numCubPointsSide, sideDim);
367  FieldContainer<double> cub_weights_side(numCubPointsSide);
368  FieldContainer<double> cub_points_side_refcell(numCubPointsSide, cellDim);
369  FieldContainer<double> cub_points_side_physical(1, numCubPointsSide, cellDim);
370  FieldContainer<double> jacobian_side_refcell(1, numCubPointsSide, cellDim, cellDim);
371  FieldContainer<double> jacobian_det_side_refcell(1, numCubPointsSide);
372  FieldContainer<double> weighted_measure_side_refcell(1, numCubPointsSide);
373 
374  FieldContainer<double> value_of_basis_at_cub_points_side_refcell(numFields, numCubPointsSide);
375  FieldContainer<double> transformed_value_of_basis_at_cub_points_side_refcell(1, numFields, numCubPointsSide);
376  FieldContainer<double> weighted_transformed_value_of_basis_at_cub_points_side_refcell(1, numFields, numCubPointsSide);
377  FieldContainer<double> neumann_data_at_cub_points_side_physical(1, numCubPointsSide);
378  FieldContainer<double> neumann_fields_per_side(1, numFields);
379 
380  /* Section 3: Related to global interpolant. */
381  FieldContainer<double> value_of_basis_at_interp_points_ref(numFields, numInterpPoints);
382  FieldContainer<double> transformed_value_of_basis_at_interp_points_ref(1, numFields, numInterpPoints);
383  FieldContainer<double> interpolant(1, numInterpPoints);
384 
385  FieldContainer<int> ipiv(numFields);
386 
387 
388 
389  /******************* START COMPUTATION ***********************/
390 
391  // get cubature points and weights
392  cellCub->getCubature(cub_points_cell, cub_weights_cell);
393 
394  // compute geometric cell information
395  CellTools<double>::setJacobian(jacobian_cell, cub_points_cell, cell_nodes, cell);
396  CellTools<double>::setJacobianInv(jacobian_inv_cell, jacobian_cell);
397  CellTools<double>::setJacobianDet(jacobian_det_cell, jacobian_cell);
398 
399  // compute weighted measure
400  FunctionSpaceTools::computeCellMeasure<double>(weighted_measure_cell, jacobian_det_cell, cub_weights_cell);
401 
403  // Computing mass matrices:
404  // tabulate values of basis functions at (reference) cubature points
405  basis->getValues(value_of_basis_at_cub_points_cell, cub_points_cell, OPERATOR_VALUE);
406 
407  // transform values of basis functions
408  FunctionSpaceTools::HGRADtransformVALUE<double>(transformed_value_of_basis_at_cub_points_cell,
409  value_of_basis_at_cub_points_cell);
410 
411  // multiply with weighted measure
412  FunctionSpaceTools::multiplyMeasure<double>(weighted_transformed_value_of_basis_at_cub_points_cell,
413  weighted_measure_cell,
414  transformed_value_of_basis_at_cub_points_cell);
415 
416  // compute mass matrices
417  FunctionSpaceTools::integrate<double>(fe_matrix,
418  transformed_value_of_basis_at_cub_points_cell,
419  weighted_transformed_value_of_basis_at_cub_points_cell,
420  COMP_BLAS);
422 
424  // Computing stiffness matrices:
425  // tabulate gradients of basis functions at (reference) cubature points
426  basis->getValues(grad_of_basis_at_cub_points_cell, cub_points_cell, OPERATOR_GRAD);
427 
428  // transform gradients of basis functions
429  FunctionSpaceTools::HGRADtransformGRAD<double>(transformed_grad_of_basis_at_cub_points_cell,
430  jacobian_inv_cell,
431  grad_of_basis_at_cub_points_cell);
432 
433  // multiply with weighted measure
434  FunctionSpaceTools::multiplyMeasure<double>(weighted_transformed_grad_of_basis_at_cub_points_cell,
435  weighted_measure_cell,
436  transformed_grad_of_basis_at_cub_points_cell);
437 
438  // compute stiffness matrices and sum into fe_matrix
439  FunctionSpaceTools::integrate<double>(fe_matrix,
440  transformed_grad_of_basis_at_cub_points_cell,
441  weighted_transformed_grad_of_basis_at_cub_points_cell,
442  COMP_BLAS,
443  true);
445 
447  // Computing RHS contributions:
448  // map cell (reference) cubature points to physical space
449  CellTools<double>::mapToPhysicalFrame(cub_points_cell_physical, cub_points_cell, cell_nodes, cell);
450 
451  // evaluate rhs function
452  rhsFunc(rhs_at_cub_points_cell_physical, cub_points_cell_physical, x_order, y_order, z_order);
453 
454  // compute rhs
455  FunctionSpaceTools::integrate<double>(rhs_and_soln_vector,
456  rhs_at_cub_points_cell_physical,
457  weighted_transformed_value_of_basis_at_cub_points_cell,
458  COMP_BLAS);
459 
460  // compute neumann b.c. contributions and adjust rhs
461  sideCub->getCubature(cub_points_side, cub_weights_side);
462  for (unsigned i=0; i<numSides; i++) {
463  // compute geometric cell information
464  CellTools<double>::mapToReferenceSubcell(cub_points_side_refcell, cub_points_side, sideDim, (int)i, cell);
465  CellTools<double>::setJacobian(jacobian_side_refcell, cub_points_side_refcell, cell_nodes, cell);
466  CellTools<double>::setJacobianDet(jacobian_det_side_refcell, jacobian_side_refcell);
467 
468  // compute weighted face measure
469  FunctionSpaceTools::computeFaceMeasure<double>(weighted_measure_side_refcell,
470  jacobian_side_refcell,
471  cub_weights_side,
472  i,
473  cell);
474 
475  // tabulate values of basis functions at side cubature points, in the reference parent cell domain
476  basis->getValues(value_of_basis_at_cub_points_side_refcell, cub_points_side_refcell, OPERATOR_VALUE);
477  // transform
478  FunctionSpaceTools::HGRADtransformVALUE<double>(transformed_value_of_basis_at_cub_points_side_refcell,
479  value_of_basis_at_cub_points_side_refcell);
480 
481  // multiply with weighted measure
482  FunctionSpaceTools::multiplyMeasure<double>(weighted_transformed_value_of_basis_at_cub_points_side_refcell,
483  weighted_measure_side_refcell,
484  transformed_value_of_basis_at_cub_points_side_refcell);
485 
486  // compute Neumann data
487  // map side cubature points in reference parent cell domain to physical space
488  CellTools<double>::mapToPhysicalFrame(cub_points_side_physical, cub_points_side_refcell, cell_nodes, cell);
489  // now compute data
490  neumann(neumann_data_at_cub_points_side_physical, cub_points_side_physical, jacobian_side_refcell,
491  cell, (int)i, x_order, y_order, z_order);
492 
493  FunctionSpaceTools::integrate<double>(neumann_fields_per_side,
494  neumann_data_at_cub_points_side_physical,
495  weighted_transformed_value_of_basis_at_cub_points_side_refcell,
496  COMP_BLAS);
497 
498  // adjust RHS
499  RealSpaceTools<double>::add(rhs_and_soln_vector, neumann_fields_per_side);;
500  }
502 
504  // Solution of linear system:
505  int info = 0;
506  Teuchos::LAPACK<int, double> solver;
507  solver.GESV(numFields, 1, &fe_matrix[0], numFields, &ipiv(0), &rhs_and_soln_vector[0], numFields, &info);
509 
511  // Building interpolant:
512  // evaluate basis at interpolation points
513  basis->getValues(value_of_basis_at_interp_points_ref, interp_points_ref, OPERATOR_VALUE);
514  // transform values of basis functions
515  FunctionSpaceTools::HGRADtransformVALUE<double>(transformed_value_of_basis_at_interp_points_ref,
516  value_of_basis_at_interp_points_ref);
517  FunctionSpaceTools::evaluate<double>(interpolant, rhs_and_soln_vector, transformed_value_of_basis_at_interp_points_ref);
519 
520  /******************* END COMPUTATION ***********************/
521 
522  RealSpaceTools<double>::subtract(interpolant, exact_solution);
523 
524  *outStream << "\nRelative norm-2 error between exact solution polynomial of order ("
525  << x_order << ", " << y_order << ", " << z_order
526  << ") and finite element interpolant of order " << basis_order << ": "
527  << RealSpaceTools<double>::vectorNorm(&interpolant[0], interpolant.dimension(1), NORM_TWO) /
528  RealSpaceTools<double>::vectorNorm(&exact_solution[0], exact_solution.dimension(1), NORM_TWO) << "\n";
529 
530  if (RealSpaceTools<double>::vectorNorm(&interpolant[0], interpolant.dimension(1), NORM_TWO) /
531  RealSpaceTools<double>::vectorNorm(&exact_solution[0], exact_solution.dimension(1), NORM_TWO) > zero) {
532  *outStream << "\n\nPatch test failed for solution polynomial order ("
533  << x_order << ", " << y_order << ", " << z_order << ") and basis order " << basis_order << "\n\n";
534  errorFlag++;
535  }
536  } // end for basis_order
537  } // end for z_order
538  } // end for y_order
539  } // end for x_order
540  } // end for ptype
541 
542  }
543  // Catch unexpected errors
544  catch (const std::logic_error & err) {
545  *outStream << err.what() << "\n\n";
546  errorFlag = -1000;
547  };
548 
549  if (errorFlag != 0)
550  std::cout << "End Result: TEST FAILED\n";
551  else
552  std::cout << "End Result: TEST PASSED\n";
553 
554  // reset format state of std::cout
555  std::cout.copyfmt(oldFormatState);
556 
557  return errorFlag;
558 }
rhsFunc
void rhsFunc(FieldContainer< double > &, const FieldContainer< double > &, int, int, int)
right-hand side function
Definition: test_02.cpp:73
u_exact
void u_exact(FieldContainer< double > &, const FieldContainer< double > &, int, int, int)
exact solution
Definition: test_02.cpp:99
neumann
void neumann(FieldContainer< double > &, const FieldContainer< double > &, const FieldContainer< double > &, const shards::CellTopology &, int, int, int, int)
neumann boundary conditions
Definition: test_02.cpp:124
main
int main(int argc, char *argv[])
outdated tests for orthogonal bases
Definition: test_02.cpp:63
Intrepid_ArrayTools.hpp
Header file for utility class to provide array tools, such as tensor contractions,...
Intrepid_CellTools.hpp
Header file for the Intrepid::CellTools class.
Intrepid_DefaultCubatureFactory.hpp
Header file for the abstract base class Intrepid::DefaultCubatureFactory.
Intrepid_FieldContainer.hpp
Header file for utility class to provide multidimensional containers.
Intrepid_FunctionSpaceTools.hpp
Header file for the Intrepid::FunctionSpaceTools class.
Intrepid_HGRAD_TET_Cn_FEM.hpp
Header file for the Intrepid::HGRAD_TET_Cn_FEM class.
Intrepid_RealSpaceTools.hpp
Header file for classes providing basic linear algebra functionality in 1D, 2D and 3D.
Intrepid::Basis_HGRAD_TET_Cn_FEM
Implementation of the default H(grad)-compatible Lagrange basis of arbitrary degree on Tetrahedron ce...
Definition: Intrepid_HGRAD_TET_Cn_FEM.hpp:88
Intrepid::CellTools
A stateless class for operations on cell data. Provides methods for:
Definition: Intrepid_CellTools.hpp:109
Intrepid::DefaultCubatureFactory
A factory class that generates specific instances of cubatures.
Definition: Intrepid_DefaultCubatureFactory.hpp:77
Intrepid::DefaultCubatureFactory::create
Teuchos::RCP< Cubature< Scalar, ArrayPoint, ArrayWeight > > create(const shards::CellTopology &cellTopology, const std::vector< int > &degree)
Factory method.
Definition: Intrepid_DefaultCubatureFactoryDef.hpp:53
Intrepid::FieldContainer::dimension
int dimension(const int whichDim) const
Returns the specified dimension.
Definition: Intrepid_FieldContainerDef.hpp:658
Intrepid::RealSpaceTools
Implementation of basic linear algebra functionality in Euclidean space.
Definition: Intrepid_RealSpaceTools.hpp:65
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