MFEM  v3.4
Finite element discretization library
ex8p.cpp
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1 // MFEM Example 8 - Parallel Version
2 //
3 // Compile with: make ex8p
4 //
5 // Sample runs: mpirun -np 4 ex8p -m ../data/square-disc.mesh
6 // mpirun -np 4 ex8p -m ../data/star.mesh
7 // mpirun -np 4 ex8p -m ../data/escher.mesh
8 // mpirun -np 4 ex8p -m ../data/fichera.mesh
9 // mpirun -np 4 ex8p -m ../data/square-disc-p2.vtk
10 // mpirun -np 4 ex8p -m ../data/square-disc-p3.mesh
11 // mpirun -np 4 ex8p -m ../data/star-surf.mesh -o 2
12 //
13 // Description: This example code demonstrates the use of the Discontinuous
14 // Petrov-Galerkin (DPG) method in its primal 2x2 block form as a
15 // simple finite element discretization of the Laplace problem
16 // -Delta u = f with homogeneous Dirichlet boundary conditions. We
17 // use high-order continuous trial space, a high-order interfacial
18 // (trace) space, and a high-order discontinuous test space
19 // defining a local dual (H^{-1}) norm.
20 //
21 // We use the primal form of DPG, see "A primal DPG method without
22 // a first-order reformulation", Demkowicz and Gopalakrishnan, CAM
23 // 2013, DOI:10.1016/j.camwa.2013.06.029.
24 //
25 // The example highlights the use of interfacial (trace) finite
26 // elements and spaces, trace face integrators and the definition
27 // of block operators and preconditioners. The use of the ADS
28 // preconditioner from hypre for interfacially-reduced H(div)
29 // problems is also illustrated.
30 //
31 // We recommend viewing examples 1-5 before viewing this example.
32 
33 #include "mfem.hpp"
34 #include <fstream>
35 #include <iostream>
36 
37 using namespace std;
38 using namespace mfem;
39 
40 int main(int argc, char *argv[])
41 {
42  // 1. Initialize MPI.
43  int num_procs, myid;
44  MPI_Init(&argc, &argv);
45  MPI_Comm_size(MPI_COMM_WORLD, &num_procs);
46  MPI_Comm_rank(MPI_COMM_WORLD, &myid);
47 
48  // 2. Parse command-line options.
49  const char *mesh_file = "../data/star.mesh";
50  int order = 1;
51  bool visualization = 1;
52 
53  OptionsParser args(argc, argv);
54  args.AddOption(&mesh_file, "-m", "--mesh",
55  "Mesh file to use.");
56  args.AddOption(&order, "-o", "--order",
57  "Finite element order (polynomial degree).");
58  args.AddOption(&visualization, "-vis", "--visualization", "-no-vis",
59  "--no-visualization",
60  "Enable or disable GLVis visualization.");
61  args.Parse();
62  if (!args.Good())
63  {
64  if (myid == 0)
65  {
66  args.PrintUsage(cout);
67  }
68  MPI_Finalize();
69  return 1;
70  }
71  if (myid == 0)
72  {
73  args.PrintOptions(cout);
74  }
75 
76  // 3. Read the (serial) mesh from the given mesh file on all processors. We
77  // can handle triangular, quadrilateral, tetrahedral, hexahedral, surface
78  // and volume meshes with the same code.
79  Mesh *mesh = new Mesh(mesh_file, 1, 1);
80  int dim = mesh->Dimension();
81 
82  // 4. Refine the serial mesh on all processors to increase the resolution. In
83  // this example we do 'ref_levels' of uniform refinement. We choose
84  // 'ref_levels' to be the largest number that gives a final mesh with no
85  // more than 10,000 elements.
86  {
87  int ref_levels =
88  (int)floor(log(10000./mesh->GetNE())/log(2.)/dim);
89  for (int l = 0; l < ref_levels; l++)
90  {
91  mesh->UniformRefinement();
92  }
93  }
94 
95  // 5. Define a parallel mesh by a partitioning of the serial mesh. Refine
96  // this mesh further in parallel to increase the resolution. Once the
97  // parallel mesh is defined, the serial mesh can be deleted.
98  ParMesh *pmesh = new ParMesh(MPI_COMM_WORLD, *mesh);
99  delete mesh;
100  {
101  int par_ref_levels = 1;
102  for (int l = 0; l < par_ref_levels; l++)
103  {
104  pmesh->UniformRefinement();
105  }
106  }
107  pmesh->ReorientTetMesh();
108 
109  // 6. Define the trial, interfacial (trace) and test DPG spaces:
110  // - The trial space, x0_space, contains the non-interfacial unknowns and
111  // has the essential BC.
112  // - The interfacial space, xhat_space, contains the interfacial unknowns
113  // and does not have essential BC.
114  // - The test space, test_space, is an enriched space where the enrichment
115  // degree may depend on the spatial dimension of the domain, the type of
116  // the mesh and the trial space order.
117  unsigned int trial_order = order;
118  unsigned int trace_order = order - 1;
119  unsigned int test_order = order; /* reduced order, full order is
120  (order + dim - 1) */
121  if (dim == 2 && (order%2 == 0 || (pmesh->MeshGenerator() & 2 && order > 1)))
122  {
123  test_order++;
124  }
125  if (test_order < trial_order)
126  if (myid == 0)
127  cerr << "Warning, test space not enriched enough to handle primal"
128  << " trial space\n";
129 
130  FiniteElementCollection *x0_fec, *xhat_fec, *test_fec;
131 
132  x0_fec = new H1_FECollection(trial_order, dim);
133  xhat_fec = new RT_Trace_FECollection(trace_order, dim);
134  test_fec = new L2_FECollection(test_order, dim);
135 
136  ParFiniteElementSpace *x0_space, *xhat_space, *test_space;
137 
138  x0_space = new ParFiniteElementSpace(pmesh, x0_fec);
139  xhat_space = new ParFiniteElementSpace(pmesh, xhat_fec);
140  test_space = new ParFiniteElementSpace(pmesh, test_fec);
141 
142  HYPRE_Int glob_true_s0 = x0_space->GlobalTrueVSize();
143  HYPRE_Int glob_true_s1 = xhat_space->GlobalTrueVSize();
144  HYPRE_Int glob_true_s_test = test_space->GlobalTrueVSize();
145  if (myid == 0)
146  {
147  cout << "\nNumber of Unknowns:\n"
148  << " Trial space, X0 : " << glob_true_s0
149  << " (order " << trial_order << ")\n"
150  << " Interface space, Xhat : " << glob_true_s1
151  << " (order " << trace_order << ")\n"
152  << " Test space, Y : " << glob_true_s_test
153  << " (order " << test_order << ")\n\n";
154  }
155 
156  // 7. Set up the linear form F(.) which corresponds to the right-hand side of
157  // the FEM linear system, which in this case is (f,phi_i) where f=1.0 and
158  // phi_i are the basis functions in the test finite element fespace.
159  ConstantCoefficient one(1.0);
160  ParLinearForm * F = new ParLinearForm(test_space);
162  F->Assemble();
163 
164  ParGridFunction * x0 = new ParGridFunction(x0_space);
165  *x0 = 0.;
166 
167  // 8. Set up the mixed bilinear form for the primal trial unknowns, B0,
168  // the mixed bilinear form for the interfacial unknowns, Bhat,
169  // the inverse stiffness matrix on the discontinuous test space, Sinv,
170  // and the stiffness matrix on the continuous trial space, S0.
171  Array<int> ess_bdr(pmesh->bdr_attributes.Max());
172  ess_bdr = 1;
173  Array<int> ess_dof;
174  x0_space->GetEssentialVDofs(ess_bdr, ess_dof);
175 
176  ParMixedBilinearForm *B0 = new ParMixedBilinearForm(x0_space,test_space);
178  B0->Assemble();
179  B0->EliminateEssentialBCFromTrialDofs(ess_dof, *x0, *F);
180  B0->Finalize();
181 
182  ParMixedBilinearForm *Bhat = new ParMixedBilinearForm(xhat_space,test_space);
184  Bhat->Assemble();
185  Bhat->Finalize();
186 
187  ParBilinearForm *Sinv = new ParBilinearForm(test_space);
188  SumIntegrator *Sum = new SumIntegrator;
189  Sum->AddIntegrator(new DiffusionIntegrator(one));
190  Sum->AddIntegrator(new MassIntegrator(one));
191  Sinv->AddDomainIntegrator(new InverseIntegrator(Sum));
192  Sinv->Assemble();
193  Sinv->Finalize();
194 
195  ParBilinearForm *S0 = new ParBilinearForm(x0_space);
197  S0->Assemble();
198  S0->EliminateEssentialBC(ess_bdr);
199  S0->Finalize();
200 
201  HypreParMatrix * matB0 = B0->ParallelAssemble(); delete B0;
202  HypreParMatrix * matBhat = Bhat->ParallelAssemble(); delete Bhat;
203  HypreParMatrix * matSinv = Sinv->ParallelAssemble(); delete Sinv;
204  HypreParMatrix * matS0 = S0->ParallelAssemble(); delete S0;
205 
206  // 9. Define the block structure of the problem, by creating the offset
207  // variables. Also allocate two BlockVector objects to store the solution
208  // and rhs.
209  enum {x0_var, xhat_var, NVAR};
210 
211  int true_s0 = x0_space->TrueVSize();
212  int true_s1 = xhat_space->TrueVSize();
213  int true_s_test = test_space->TrueVSize();
214 
215  Array<int> true_offsets(NVAR+1);
216  true_offsets[0] = 0;
217  true_offsets[1] = true_s0;
218  true_offsets[2] = true_s0+true_s1;
219 
220  Array<int> true_offsets_test(2);
221  true_offsets_test[0] = 0;
222  true_offsets_test[1] = true_s_test;
223 
224  BlockVector x(true_offsets), b(true_offsets);
225  x = 0.0;
226  b = 0.0;
227 
228  // 10. Set up the 1x2 block Least Squares DPG operator, B = [B0 Bhat],
229  // the normal equation operator, A = B^t Sinv B, and
230  // the normal equation right-hand-size, b = B^t Sinv F.
231  BlockOperator B(true_offsets_test, true_offsets);
232  B.SetBlock(0, 0, matB0);
233  B.SetBlock(0, 1, matBhat);
234 
235  RAPOperator A(B, *matSinv, B);
236 
237  HypreParVector *trueF = F->ParallelAssemble();
238  {
239  HypreParVector SinvF(test_space);
240  matSinv->Mult(*trueF, SinvF);
241  B.MultTranspose(SinvF, b);
242  }
243 
244  // 11. Set up a block-diagonal preconditioner for the 2x2 normal equation
245  //
246  // [ S0^{-1} 0 ]
247  // [ 0 Shat^{-1} ] Shat = (Bhat^T Sinv Bhat)
248  //
249  // corresponding to the primal (x0) and interfacial (xhat) unknowns.
250  // Since the Shat operator is equivalent to an H(div) matrix reduced to
251  // the interfacial skeleton, we approximate its inverse with one V-cycle
252  // of the ADS preconditioner from the hypre library (in 2D we use AMS for
253  // the rotated H(curl) problem).
254  HypreBoomerAMG *S0inv = new HypreBoomerAMG(*matS0);
255  S0inv->SetPrintLevel(0);
256 
257  HypreParMatrix *Shat = RAP(matSinv, matBhat);
258  HypreSolver *Shatinv;
259  if (dim == 2) { Shatinv = new HypreAMS(*Shat, xhat_space); }
260  else { Shatinv = new HypreADS(*Shat, xhat_space); }
261 
262  BlockDiagonalPreconditioner P(true_offsets);
263  P.SetDiagonalBlock(0, S0inv);
264  P.SetDiagonalBlock(1, Shatinv);
265 
266  // 12. Solve the normal equation system using the PCG iterative solver.
267  // Check the weighted norm of residual for the DPG least square problem.
268  // Wrap the primal variable in a GridFunction for visualization purposes.
269  CGSolver pcg(MPI_COMM_WORLD);
270  pcg.SetOperator(A);
271  pcg.SetPreconditioner(P);
272  pcg.SetRelTol(1e-6);
273  pcg.SetMaxIter(200);
274  pcg.SetPrintLevel(1);
275  pcg.Mult(b, x);
276 
277  {
278  HypreParVector LSres(test_space), tmp(test_space);
279  B.Mult(x, LSres);
280  LSres -= *trueF;
281  matSinv->Mult(LSres, tmp);
282  double res = sqrt(InnerProduct(LSres, tmp));
283  if (myid == 0)
284  {
285  cout << "\n|| B0*x0 + Bhat*xhat - F ||_{S^-1} = " << res << endl;
286  }
287  }
288 
289  x0->Distribute(x.GetBlock(x0_var));
290 
291  // 13. Save the refined mesh and the solution in parallel. This output can
292  // be viewed later using GLVis: "glvis -np <np> -m mesh -g sol".
293  {
294  ostringstream mesh_name, sol_name;
295  mesh_name << "mesh." << setfill('0') << setw(6) << myid;
296  sol_name << "sol." << setfill('0') << setw(6) << myid;
297 
298  ofstream mesh_ofs(mesh_name.str().c_str());
299  mesh_ofs.precision(8);
300  pmesh->Print(mesh_ofs);
301 
302  ofstream sol_ofs(sol_name.str().c_str());
303  sol_ofs.precision(8);
304  x0->Save(sol_ofs);
305  }
306 
307  // 14. Send the solution by socket to a GLVis server.
308  if (visualization)
309  {
310  char vishost[] = "localhost";
311  int visport = 19916;
312  socketstream sol_sock(vishost, visport);
313  sol_sock << "parallel " << num_procs << " " << myid << "\n";
314  sol_sock.precision(8);
315  sol_sock << "solution\n" << *pmesh << *x0 << flush;
316  }
317 
318  // 15. Free the used memory.
319  delete trueF;
320  delete Shatinv;
321  delete S0inv;
322  delete Shat;
323  delete matB0;
324  delete matBhat;
325  delete matSinv;
326  delete matS0;
327  delete x0;
328  delete F;
329  delete test_space;
330  delete xhat_space;
331  delete x0_space;
332  delete test_fec;
333  delete xhat_fec;
334  delete x0_fec;
335  delete pmesh;
336 
337  MPI_Finalize();
338 
339  return 0;
340 }
Class for domain integration L(v) := (f, v)
Definition: lininteg.hpp:93
Conjugate gradient method.
Definition: solvers.hpp:111
The Auxiliary-space Maxwell Solver in hypre.
Definition: hypre.hpp:859
Integrator defining a sum of multiple Integrators.
Definition: bilininteg.hpp:152
The Auxiliary-space Divergence Solver in hypre.
Definition: hypre.hpp:891
Subclass constant coefficient.
Definition: coefficient.hpp:57
HypreParMatrix * RAP(const HypreParMatrix *A, const HypreParMatrix *P)
Returns the matrix P^t * A * P.
Definition: hypre.cpp:1558
void PrintOptions(std::ostream &out) const
Definition: optparser.cpp:304
int Dimension() const
Definition: mesh.hpp:645
virtual void ReorientTetMesh()
See the remarks for the serial version in mesh.hpp.
Definition: pmesh.cpp:2070
void PrintUsage(std::ostream &out) const
Definition: optparser.cpp:434
void EliminateEssentialBCFromTrialDofs(Array< int > &marked_vdofs, const Vector &sol, Vector &rhs)
void Assemble()
Assembles the linear form i.e. sums over all domain/bdr integrators.
Definition: linearform.cpp:51
virtual void Mult(const Vector &b, Vector &x) const
Operator application: y=A(x).
Definition: solvers.cpp:295
T Max() const
Find the maximal element in the array, using the comparison operator < for class T.
Definition: array.cpp:109
bool Good() const
Definition: optparser.hpp:120
Abstract parallel finite element space.
Definition: pfespace.hpp:28
void AddIntegrator(BilinearFormIntegrator *integ)
Definition: bilininteg.hpp:162
STL namespace.
virtual void Finalize(int skip_zeros=1)
Finalizes the matrix initialization.
virtual void Print(std::ostream &out=mfem::out) const
Definition: pmesh.cpp:3427
HYPRE_Int Mult(HypreParVector &x, HypreParVector &y, double alpha=1.0, double beta=0.0)
Computes y = alpha * A * x + beta * y.
Definition: hypre.cpp:1005
HypreParMatrix * ParallelAssemble()
Returns the matrix assembled on the true dofs, i.e. P^t A P.
HypreParMatrix * ParallelAssemble()
Returns the matrix assembled on the true dofs, i.e. P_test^t A P_trial.
void AddDomainIntegrator(BilinearFormIntegrator *bfi)
Integrator that inverts the matrix assembled by another integrator.
Definition: bilininteg.hpp:134
The BoomerAMG solver in hypre.
Definition: hypre.hpp:796
virtual void GetEssentialVDofs(const Array< int > &bdr_attr_is_ess, Array< int > &ess_dofs, int component=-1) const
Determine the boundary degrees of freedom.
Definition: pfespace.cpp:612
Class for parallel linear form.
Definition: plinearform.hpp:26
int dim
Definition: ex3.cpp:47
void SetPrintLevel(int print_lvl)
Definition: solvers.cpp:72
A class to handle Block diagonal preconditioners in a matrix-free implementation. ...
HYPRE_Int GlobalTrueVSize() const
Definition: pfespace.hpp:247
void UniformRefinement(int i, const DSTable &, int *, int *, int *)
Definition: mesh.cpp:6741
void SetPrintLevel(int print_level)
Definition: hypre.hpp:837
void EliminateEssentialBC(const Array< int > &bdr_attr_is_ess, const Vector &sol, Vector &rhs, DiagonalPolicy dpolicy=DIAG_ONE)
Eliminate essential boundary DOFs from the system.
void SetMaxIter(int max_it)
Definition: solvers.hpp:63
int MeshGenerator()
Get the mesh generator/type.
Definition: mesh.hpp:611
void Assemble(int skip_zeros=1)
void Assemble(int skip_zeros=1)
Assemble the local matrix.
The operator x -> R*A*P*x constructed through the actions of R^T, A and P.
Definition: operator.hpp:349
void AddTraceFaceIntegrator(BilinearFormIntegrator *bfi)
virtual void MultTranspose(const Vector &x, Vector &y) const
Action of the transpose operator.
Wrapper for hypre&#39;s parallel vector class.
Definition: hypre.hpp:73
void AddDomainIntegrator(LinearFormIntegrator *lfi)
Adds new Domain Integrator.
Definition: linearform.cpp:19
Array< int > bdr_attributes
A list of all unique boundary attributes used by the Mesh.
Definition: mesh.hpp:174
void SetRelTol(double rtol)
Definition: solvers.hpp:61
virtual void Mult(const Vector &x, Vector &y) const
Operator application.
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)
Definition: optparser.hpp:74
void Distribute(const Vector *tv)
Definition: pgridfunc.cpp:131
virtual void Save(std::ostream &out) const
Definition: pgridfunc.cpp:398
int GetNE() const
Returns number of elements.
Definition: mesh.hpp:618
double InnerProduct(HypreParVector *x, HypreParVector *y)
Definition: hypre.cpp:252
Class for parallel bilinear form using different test and trial FE spaces.
virtual void Finalize(int skip_zeros=1)
Finalizes the matrix initialization.
void AddDomainIntegrator(BilinearFormIntegrator *bfi)
Adds new Domain Integrator.
int main(int argc, char *argv[])
Definition: ex8p.cpp:40
Class for parallel bilinear form.
Abstract class for hypre&#39;s solvers and preconditioners.
Definition: hypre.hpp:620
virtual void SetOperator(const Operator &op)
Also calls SetOperator for the preconditioner if there is one.
Definition: solvers.hpp:125
virtual void SetPreconditioner(Solver &pr)
This should be called before SetOperator.
Definition: solvers.cpp:93
Arbitrary order H1-conforming (continuous) finite elements.
Definition: fe_coll.hpp:79
Class for parallel grid function.
Definition: pgridfunc.hpp:32
Wrapper for hypre&#39;s ParCSR matrix class.
Definition: hypre.hpp:175
A class to handle Block systems in a matrix-free implementation.
Class for parallel meshes.
Definition: pmesh.hpp:32
void ParallelAssemble(Vector &tv)
Assemble the vector on the true dofs, i.e. P^t v.
Definition: plinearform.cpp:34
void SetBlock(int iRow, int iCol, Operator *op, double c=1.0)
Add a block op in the block-entry (iblock, jblock).
void SetDiagonalBlock(int iblock, Operator *op)
Add a square block op in the block-entry (iblock, iblock).
Arbitrary order "L2-conforming" discontinuous finite elements.
Definition: fe_coll.hpp:128