MFEM  v4.5.1 Finite element discretization library
ex31.cpp
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1 // MFEM Example 31
2 //
3 // Compile with: make ex31
4 //
5 // Sample runs: ex31 -m ../data/inline-segment.mesh -o 2
6 // ex31 -m ../data/hexagon.mesh -o 2
7 // ex31 -m ../data/star.mesh -o 2
8 // ex31 -m ../data/fichera.mesh -o 3 -r 1
9 // ex31 -m ../data/square-disc-nurbs.mesh -o 3
10 // ex31 -m ../data/amr-quad.mesh -o 2 -r 1
11 // ex31 -m ../data/amr-hex.mesh -r 1
12 //
13 // Description: This example code solves a simple electromagnetic diffusion
14 // problem corresponding to the second order definite Maxwell
15 // equation curl curl E + sigma E = f with boundary condition
16 // E x n = <given tangential field>. In this example sigma is an
17 // anisotropic 3x3 tensor. Here, we use a given exact solution E
18 // and compute the corresponding r.h.s. f. We discretize with
19 // Nedelec finite elements in 1D, 2D, or 3D.
20 //
21 // The example demonstrates the use of restricted H(curl) finite
22 // element spaces with the curl-curl and the (vector finite
23 // element) mass bilinear form, as well as the computation of
24 // discretization error when the exact solution is known. These
25 // restricted spaces allow the solution of 1D or 2D
26 // electromagnetic problems which involve 3D field vectors. Such
27 // problems arise in plasma physics and crystallography.
28 //
29 // We recommend viewing example 3 before viewing this example.
30
31 #include "mfem.hpp"
32 #include <fstream>
33 #include <iostream>
34
35 using namespace std;
36 using namespace mfem;
37
38 // Exact solution, E, and r.h.s., f. See below for implementation.
39 void E_exact(const Vector &, Vector &);
40 void CurlE_exact(const Vector &, Vector &);
41 void f_exact(const Vector &, Vector &);
42 double freq = 1.0, kappa;
43 int dim;
44
45 int main(int argc, char *argv[])
46 {
47  // 1. Parse command-line options.
48  const char *mesh_file = "../data/inline-quad.mesh";
49  int ref_levels = 2;
50  int order = 1;
51  bool visualization = 1;
52
53  OptionsParser args(argc, argv);
55  "Mesh file to use.");
57  "Number of times to refine the mesh uniformly.");
59  "Finite element order (polynomial degree).");
60  args.AddOption(&freq, "-f", "--frequency", "Set the frequency for the exact"
61  " solution.");
63  "--no-visualization",
64  "Enable or disable GLVis visualization.");
65  args.ParseCheck();
66
67  kappa = freq * M_PI;
68
69  // 2. Read the mesh from the given mesh file. We can handle triangular,
70  // quadrilateral, or mixed meshes with the same code.
71  Mesh mesh(mesh_file, 1, 1);
72  dim = mesh.Dimension();
73
74  // 3. Refine the mesh to increase the resolution. In this example we do
75  // 'ref_levels' of uniform refinement (2 by default, or specified on
76  // the command line with -r).
77  for (int lev = 0; lev < ref_levels; lev++)
78  {
79  mesh.UniformRefinement();
80  }
81
82  // 4. Define a finite element space on the mesh. Here we use the Nedelec
83  // finite elements of the specified order restricted to 1D, 2D, or 3D
84  // depending on the dimension of the given mesh file.
85  FiniteElementCollection *fec = NULL;
86  if (dim == 1)
87  {
88  fec = new ND_R1D_FECollection(order, dim);
89  }
90  else if (dim == 2)
91  {
92  fec = new ND_R2D_FECollection(order, dim);
93  }
94  else
95  {
96  fec = new ND_FECollection(order, dim);
97  }
98  FiniteElementSpace fespace(&mesh, fec);
99  int size = fespace.GetTrueVSize();
100  cout << "Number of H(Curl) unknowns: " << size << endl;
101
102  // 5. Determine the list of true essential boundary dofs. In this example,
103  // the boundary conditions are defined by marking all the boundary
104  // attributes from the mesh as essential (Dirichlet) and converting them
105  // to a list of true dofs.
106  Array<int> ess_tdof_list;
107  if (mesh.bdr_attributes.Size())
108  {
109  Array<int> ess_bdr(mesh.bdr_attributes.Max());
110  ess_bdr = 1;
111  fespace.GetEssentialTrueDofs(ess_bdr, ess_tdof_list);
112  }
113
114  // 6. Set up the linear form b(.) which corresponds to the right-hand side
115  // of the FEM linear system, which in this case is (f,phi_i) where f is
116  // given by the function f_exact and phi_i are the basis functions in
117  // the finite element fespace.
119  LinearForm b(&fespace);
121  b.Assemble();
122
123  // 7. Define the solution vector x as a finite element grid function
124  // corresponding to fespace. Initialize x by projecting the exact
125  // solution. Note that only values from the boundary edges will be used
126  // when eliminating the non-homogeneous boundary condition to modify the
127  // r.h.s. vector b.
128  GridFunction sol(&fespace);
131  sol.ProjectCoefficient(E);
132
133  // 8. Set up the bilinear form corresponding to the EM diffusion operator
134  // curl muinv curl + sigma I, by adding the curl-curl and the mass domain
135  // integrators.
136  DenseMatrix sigmaMat(3);
137  sigmaMat(0,0) = 2.0; sigmaMat(1,1) = 2.0; sigmaMat(2,2) = 2.0;
138  sigmaMat(0,2) = 0.0; sigmaMat(2,0) = 0.0;
139  sigmaMat(0,1) = M_SQRT1_2; sigmaMat(1,0) = M_SQRT1_2; // 1/sqrt(2) in cmath
140  sigmaMat(1,2) = M_SQRT1_2; sigmaMat(2,1) = M_SQRT1_2;
141
142  ConstantCoefficient muinv(1.0);
144  BilinearForm a(&fespace);
147
148  // 9. Assemble the bilinear form and the corresponding linear system,
149  // applying any necessary transformations such as: eliminating boundary
150  // conditions, applying conforming constraints for non-conforming AMR,
151  // etc.
152  a.Assemble();
153
154  OperatorPtr A;
155  Vector B, X;
156
157  a.FormLinearSystem(ess_tdof_list, sol, b, A, X, B);
158
159  // 10. Solve the system A X = B.
160
161 #ifndef MFEM_USE_SUITESPARSE
162  // 11. Define a simple symmetric Gauss-Seidel preconditioner and use it to
163  // solve the system Ax=b with PCG.
164  GSSmoother M((SparseMatrix&)(*A));
165  PCG(*A, M, B, X, 1, 500, 1e-12, 0.0);
166 #else
167  // 11. If MFEM was compiled with SuiteSparse, use UMFPACK to solve the
168  // system.
169  UMFPackSolver umf_solver;
170  umf_solver.Control[UMFPACK_ORDERING] = UMFPACK_ORDERING_METIS;
171  umf_solver.SetOperator(*A);
172  umf_solver.Mult(B, X);
173 #endif
174
175  // 12. Recover the solution as a finite element grid function.
176  a.RecoverFEMSolution(X, b, sol);
177
178  // 13. Compute and print the H(Curl) norm of the error.
179  {
180  double error = sol.ComputeHCurlError(&E, &CurlE);
181  cout << "\n|| E_h - E ||_{H(Curl)} = " << error << '\n' << endl;
182  }
183
184
185  // 14. Save the refined mesh and the solution. This output can be viewed
186  // later using GLVis: "glvis -m refined.mesh -g sol.gf".
187  {
188  ofstream mesh_ofs("refined.mesh");
189  mesh_ofs.precision(8);
190  mesh.Print(mesh_ofs);
191  ofstream sol_ofs("sol.gf");
192  sol_ofs.precision(8);
193  sol.Save(sol_ofs);
194  }
195
196  // 15. Send the solution by socket to a GLVis server.
197  if (visualization)
198  {
199  char vishost[] = "localhost";
200  int visport = 19916;
201
202  VectorGridFunctionCoefficient solCoef(&sol);
203  CurlGridFunctionCoefficient dsolCoef(&sol);
204
205  if (dim ==1)
206  {
207  socketstream x_sock(vishost, visport);
208  socketstream y_sock(vishost, visport);
209  socketstream z_sock(vishost, visport);
210  socketstream dy_sock(vishost, visport);
211  socketstream dz_sock(vishost, visport);
212  x_sock.precision(8);
213  y_sock.precision(8);
214  z_sock.precision(8);
215  dy_sock.precision(8);
216  dz_sock.precision(8);
217
218  Vector xVec(3); xVec = 0.0; xVec(0) = 1;
219  Vector yVec(3); yVec = 0.0; yVec(1) = 1;
220  Vector zVec(3); zVec = 0.0; zVec(2) = 1;
221  VectorConstantCoefficient xVecCoef(xVec);
222  VectorConstantCoefficient yVecCoef(yVec);
223  VectorConstantCoefficient zVecCoef(zVec);
224
225  H1_FECollection fec_h1(order, dim);
226  L2_FECollection fec_l2(order-1, dim);
227
228  FiniteElementSpace fes_h1(&mesh, &fec_h1);
229  FiniteElementSpace fes_l2(&mesh, &fec_l2);
230
231  GridFunction xComp(&fes_l2);
232  GridFunction yComp(&fes_h1);
233  GridFunction zComp(&fes_h1);
234
235  GridFunction dyComp(&fes_l2);
236  GridFunction dzComp(&fes_l2);
237
238  InnerProductCoefficient xCoef(xVecCoef, solCoef);
239  InnerProductCoefficient yCoef(yVecCoef, solCoef);
240  InnerProductCoefficient zCoef(zVecCoef, solCoef);
241
242  xComp.ProjectCoefficient(xCoef);
243  yComp.ProjectCoefficient(yCoef);
244  zComp.ProjectCoefficient(zCoef);
245
246  x_sock << "solution\n" << mesh << xComp << flush
247  << "window_title 'X component'" << endl;
248  y_sock << "solution\n" << mesh << yComp << flush
249  << "window_geometry 403 0 400 350 "
250  << "window_title 'Y component'" << endl;
251  z_sock << "solution\n" << mesh << zComp << flush
252  << "window_geometry 806 0 400 350 "
253  << "window_title 'Z component'" << endl;
254
255  InnerProductCoefficient dyCoef(yVecCoef, dsolCoef);
256  InnerProductCoefficient dzCoef(zVecCoef, dsolCoef);
257
258  dyComp.ProjectCoefficient(dyCoef);
259  dzComp.ProjectCoefficient(dzCoef);
260
261  dy_sock << "solution\n" << mesh << dyComp << flush
262  << "window_geometry 403 375 400 350 "
263  << "window_title 'Y component of Curl'" << endl;
264  dz_sock << "solution\n" << mesh << dzComp << flush
265  << "window_geometry 806 375 400 350 "
266  << "window_title 'Z component of Curl'" << endl;
267  }
268  else if (dim == 2)
269  {
270  socketstream xy_sock(vishost, visport);
271  socketstream z_sock(vishost, visport);
272  socketstream dxy_sock(vishost, visport);
273  socketstream dz_sock(vishost, visport);
274
275  DenseMatrix xyMat(2,3); xyMat = 0.0;
276  xyMat(0,0) = 1.0; xyMat(1,1) = 1.0;
277  MatrixConstantCoefficient xyMatCoef(xyMat);
278  Vector zVec(3); zVec = 0.0; zVec(2) = 1;
279  VectorConstantCoefficient zVecCoef(zVec);
280
281  MatrixVectorProductCoefficient xyCoef(xyMatCoef, solCoef);
282  InnerProductCoefficient zCoef(zVecCoef, solCoef);
283
284  H1_FECollection fec_h1(order, dim);
285  ND_FECollection fec_nd(order, dim);
286  RT_FECollection fec_rt(order-1, dim);
287  L2_FECollection fec_l2(order-1, dim);
288
289  FiniteElementSpace fes_h1(&mesh, &fec_h1);
290  FiniteElementSpace fes_nd(&mesh, &fec_nd);
291  FiniteElementSpace fes_rt(&mesh, &fec_rt);
292  FiniteElementSpace fes_l2(&mesh, &fec_l2);
293
294  GridFunction xyComp(&fes_nd);
295  GridFunction zComp(&fes_h1);
296
297  GridFunction dxyComp(&fes_rt);
298  GridFunction dzComp(&fes_l2);
299
300  xyComp.ProjectCoefficient(xyCoef);
301  zComp.ProjectCoefficient(zCoef);
302
303  xy_sock.precision(8);
304  xy_sock << "solution\n" << mesh << xyComp
305  << "window_title 'XY components'\n" << flush;
306  z_sock << "solution\n" << mesh << zComp << flush
307  << "window_geometry 403 0 400 350 "
308  << "window_title 'Z component'" << endl;
309
310  MatrixVectorProductCoefficient dxyCoef(xyMatCoef, dsolCoef);
311  InnerProductCoefficient dzCoef(zVecCoef, dsolCoef);
312
313  dxyComp.ProjectCoefficient(dxyCoef);
314  dzComp.ProjectCoefficient(dzCoef);
315
316  dxy_sock << "solution\n" << mesh << dxyComp << flush
317  << "window_geometry 0 375 400 350 "
318  << "window_title 'XY components of Curl'" << endl;
319  dz_sock << "solution\n" << mesh << dzComp << flush
320  << "window_geometry 403 375 400 350 "
321  << "window_title 'Z component of Curl'" << endl;
322  }
323  else
324  {
325  socketstream sol_sock(vishost, visport);
326  socketstream dsol_sock(vishost, visport);
327
328  RT_FECollection fec_rt(order-1, dim);
329
330  FiniteElementSpace fes_rt(&mesh, &fec_rt);
331
332  GridFunction dsol(&fes_rt);
333
334  dsol.ProjectCoefficient(dsolCoef);
335
336  sol_sock.precision(8);
337  sol_sock << "solution\n" << mesh << sol
338  << "window_title 'Solution'" << flush << endl;
339  dsol_sock << "solution\n" << mesh << dsol << flush
340  << "window_geometry 0 375 400 350 "
341  << "window_title 'Curl of solution'" << endl;
342  }
343  }
344
345  // 16. Free the used memory.
346  delete fec;
347
348  return 0;
349 }
350
351
352 void E_exact(const Vector &x, Vector &E)
353 {
354  if (dim == 1)
355  {
356  E(0) = 1.1 * sin(kappa * x(0) + 0.0 * M_PI);
357  E(1) = 1.2 * sin(kappa * x(0) + 0.4 * M_PI);
358  E(2) = 1.3 * sin(kappa * x(0) + 0.9 * M_PI);
359  }
360  else if (dim == 2)
361  {
362  E(0) = 1.1 * sin(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.0 * M_PI);
363  E(1) = 1.2 * sin(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.4 * M_PI);
364  E(2) = 1.3 * sin(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.9 * M_PI);
365  }
366  else
367  {
368  E(0) = 1.1 * sin(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.0 * M_PI);
369  E(1) = 1.2 * sin(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.4 * M_PI);
370  E(2) = 1.3 * sin(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.9 * M_PI);
371  E *= cos(kappa * x(2));
372  }
373 }
374
375 void CurlE_exact(const Vector &x, Vector &dE)
376 {
377  if (dim == 1)
378  {
379  double c4 = cos(kappa * x(0) + 0.4 * M_PI);
380  double c9 = cos(kappa * x(0) + 0.9 * M_PI);
381
382  dE(0) = 0.0;
383  dE(1) = -1.3 * c9;
384  dE(2) = 1.2 * c4;
385  dE *= kappa;
386  }
387  else if (dim == 2)
388  {
389  double c0 = cos(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.0 * M_PI);
390  double c4 = cos(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.4 * M_PI);
391  double c9 = cos(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.9 * M_PI);
392
393  dE(0) = 1.3 * c9;
394  dE(1) = -1.3 * c9;
395  dE(2) = 1.2 * c4 - 1.1 * c0;
396  dE *= kappa * M_SQRT1_2;
397  }
398  else
399  {
400  double s0 = sin(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.0 * M_PI);
401  double c0 = cos(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.0 * M_PI);
402  double s4 = sin(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.4 * M_PI);
403  double c4 = cos(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.4 * M_PI);
404  double c9 = cos(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.9 * M_PI);
405  double sk = sin(kappa * x(2));
406  double ck = cos(kappa * x(2));
407
408  dE(0) = 1.2 * s4 * sk + 1.3 * M_SQRT1_2 * c9 * ck;
409  dE(1) = -1.1 * s0 * sk - 1.3 * M_SQRT1_2 * c9 * ck;
410  dE(2) = -M_SQRT1_2 * (1.1 * c0 - 1.2 * c4) * ck;
411  dE *= kappa;
412  }
413 }
414
415 void f_exact(const Vector &x, Vector &f)
416 {
417  if (dim == 1)
418  {
419  double s0 = sin(kappa * x(0) + 0.0 * M_PI);
420  double s4 = sin(kappa * x(0) + 0.4 * M_PI);
421  double s9 = sin(kappa * x(0) + 0.9 * M_PI);
422
423  f(0) = 2.2 * s0 + 1.2 * M_SQRT1_2 * s4;
424  f(1) = 1.2 * (2.0 + kappa * kappa) * s4 +
425  M_SQRT1_2 * (1.1 * s0 + 1.3 * s9);
426  f(2) = 1.3 * (2.0 + kappa * kappa) * s9 + 1.2 * M_SQRT1_2 * s4;
427  }
428  else if (dim == 2)
429  {
430  double s0 = sin(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.0 * M_PI);
431  double s4 = sin(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.4 * M_PI);
432  double s9 = sin(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.9 * M_PI);
433
434  f(0) = 0.55 * (4.0 + kappa * kappa) * s0 +
435  0.6 * (M_SQRT2 - kappa * kappa) * s4;
436  f(1) = 0.55 * (M_SQRT2 - kappa * kappa) * s0 +
437  0.6 * (4.0 + kappa * kappa) * s4 +
438  0.65 * M_SQRT2 * s9;
439  f(2) = 0.6 * M_SQRT2 * s4 + 1.3 * (2.0 + kappa * kappa) * s9;
440  }
441  else
442  {
443  double s0 = sin(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.0 * M_PI);
444  double c0 = cos(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.0 * M_PI);
445  double s4 = sin(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.4 * M_PI);
446  double c4 = cos(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.4 * M_PI);
447  double s9 = sin(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.9 * M_PI);
448  double c9 = cos(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.9 * M_PI);
449  double sk = sin(kappa * x(2));
450  double ck = cos(kappa * x(2));
451
452  f(0) = 0.55 * (4.0 + 3.0 * kappa * kappa) * s0 * ck +
453  0.6 * (M_SQRT2 - kappa * kappa) * s4 * ck -
454  0.65 * M_SQRT2 * kappa * kappa * c9 * sk;
455
456  f(1) = 0.55 * (M_SQRT2 - kappa * kappa) * s0 * ck +
457  0.6 * (4.0 + 3.0 * kappa * kappa) * s4 * ck +
458  0.65 * M_SQRT2 * s9 * ck -
459  0.65 * M_SQRT2 * kappa * kappa * c9 * sk;
460
461  f(2) = 0.6 * M_SQRT2 * s4 * ck -
462  M_SQRT2 * kappa * kappa * (0.55 * c0 + 0.6 * c4) * sk
463  + 1.3 * (2.0 + kappa * kappa) * s9 * ck;
464  }
465 }
A matrix coefficient that is constant in space and time.
int Size() const
Return the logical size of the array.
Definition: array.hpp:138
Class for grid function - Vector with associated FE space.
Definition: gridfunc.hpp:30
void Assemble(int skip_zeros=1)
Assembles the form i.e. sums over all domain/bdr integrators.
A coefficient that is constant across space and time.
Definition: coefficient.hpp:84
Scalar coefficient defined as the inner product of two vector coefficients.
Arbitrary order 3D H(curl)-conforming Nedelec finite elements in 1D.
Definition: fe_coll.hpp:464
Integrator for (curl u, curl v) for Nedelec elements.
Vector coefficient that is constant in space and time.
void Assemble()
Assembles the linear form i.e. sums over all domain/bdr integrators.
Definition: linearform.cpp:168
Pointer to an Operator of a specified type.
Definition: handle.hpp:33
virtual void FormLinearSystem(const Array< int > &ess_tdof_list, Vector &x, Vector &b, OperatorHandle &A, Vector &X, Vector &B, int copy_interior=0)
Form the linear system A X = B, corresponding to this bilinear form and the linear form b(...
virtual void RecoverFEMSolution(const Vector &X, const Vector &b, Vector &x)
Recover the solution of a linear system formed with FormLinearSystem().
Data type dense matrix using column-major storage.
Definition: densemat.hpp:23
virtual void GetEssentialTrueDofs(const Array< int > &bdr_attr_is_ess, Array< int > &ess_tdof_list, int component=-1)
Get a list of essential true dofs, ess_tdof_list, corresponding to the boundary attributes marked in ...
Definition: fespace.cpp:564
Arbitrary order 3D H(curl)-conforming Nedelec finite elements in 2D.
Definition: fe_coll.hpp:518
double kappa
Definition: ex24.cpp:54
Data type for Gauss-Seidel smoother of sparse matrix.
Direct sparse solver using UMFPACK.
Definition: solvers.hpp:1025
virtual void Save(std::ostream &out) const
Save the GridFunction to an output stream.
Definition: gridfunc.cpp:3676
Data type sparse matrix.
Definition: sparsemat.hpp:50
constexpr char vishost[]
double b
Definition: lissajous.cpp:42
void UniformRefinement(int i, const DSTable &, int *, int *, int *)
Definition: mesh.cpp:9816
constexpr int visport
Vector coefficient defined as the Curl of a vector GridFunction.
T Max() const
Find the maximal element in the array, using the comparison operator &lt; for class T.
Definition: array.cpp:68
void PCG(const Operator &A, Solver &B, const Vector &b, Vector &x, int print_iter, int max_num_iter, double RTOLERANCE, double ATOLERANCE)
Preconditioned conjugate gradient method. (tolerances are squared)
Definition: solvers.cpp:904
virtual int GetTrueVSize() const
Return the number of vector true (conforming) dofs.
Definition: fespace.hpp:590
int Dimension() const
Definition: mesh.hpp:1006
Arbitrary order H(div)-conforming Raviart-Thomas finite elements.
Definition: fe_coll.hpp:341
A general vector function coefficient.
Adds new Domain Integrator. Assumes ownership of lfi.
Definition: linearform.cpp:41
Array< int > bdr_attributes
A list of all unique boundary attributes used by the Mesh.
Definition: mesh.hpp:270
double Control[UMFPACK_CONTROL]
Definition: solvers.hpp:1036
Class FiniteElementSpace - responsible for providing FEM view of the mesh, mainly managing the set of...
Definition: fespace.hpp:96
Collection of finite elements from the same family in multiple dimensions. This class is used to matc...
Definition: fe_coll.hpp:26
void AddOption(bool *var, const char *enable_short_name, const char *enable_long_name, const char *disable_short_name, const char *disable_long_name, const char *description, bool required=false)
Add a boolean option and set &#39;var&#39; to receive the value. Enable/disable tags are used to set the bool...
Definition: optparser.hpp:82
void f_exact(const Vector &, Vector &)
Definition: ex3.cpp:258
Vector coefficient defined as a product of a matrix coefficient and a vector coefficient.
virtual double ComputeHCurlError(VectorCoefficient *exsol, VectorCoefficient *excurl, const IntegrationRule *irs[]=NULL) const
Returns the error measured in H(curl)-norm for ND elements.
Definition: gridfunc.cpp:3175
double a
Definition: lissajous.cpp:41
A &quot;square matrix&quot; operator for the associated FE space and BLFIntegrators The sum of all the BLFInteg...
virtual void Print(std::ostream &os=mfem::out) const
Definition: mesh.hpp:1671
int dim
Definition: ex24.cpp:53
double freq
Definition: ex24.cpp:54
Adds new Domain Integrator. Assumes ownership of bfi.
virtual void ProjectCoefficient(Coefficient &coeff)
Project coeff Coefficient to this GridFunction. The projection computation depends on the choice of t...
Definition: gridfunc.cpp:2399
void E_exact(const Vector &, Vector &)
Definition: ex3.cpp:242
for VectorFiniteElements (Nedelec, Raviart-Thomas)
Definition: lininteg.hpp:346
Arbitrary order H(curl)-conforming Nedelec finite elements.
Definition: fe_coll.hpp:415
Vector data type.
Definition: vector.hpp:60
Vector coefficient defined by a vector GridFunction.
void CurlE_exact(const Vector &, Vector &)
Definition: ex31.cpp:375
Arbitrary order H1-conforming (continuous) finite elements.
Definition: fe_coll.hpp:220
Vector with associated FE space and LinearFormIntegrators.
Definition: linearform.hpp:24
int main()
void ParseCheck(std::ostream &out=mfem::out)
Definition: optparser.cpp:252
virtual void Mult(const Vector &b, Vector &x) const
Operator application: y=A(x).
Definition: solvers.cpp:3189
Arbitrary order &quot;L2-conforming&quot; discontinuous finite elements.
Definition: fe_coll.hpp:288
double f(const Vector &p)
virtual void SetOperator(const Operator &op)
Factorize the given Operator op which must be a SparseMatrix.
Definition: solvers.cpp:3094
double sigma(const Vector &x)
Definition: maxwell.cpp:102