// MFEM Example 8 // // Compile with: make ex8 // // Sample runs: ex8 -m ../data/square-disc.mesh // ex8 -m ../data/star.mesh // ex8 -m ../data/star-mixed.mesh // ex8 -m ../data/escher.mesh // ex8 -m ../data/fichera.mesh // ex8 -m ../data/fichera-mixed.mesh // ex8 -m ../data/square-disc-p2.vtk // ex8 -m ../data/square-disc-p3.mesh // ex8 -m ../data/star-surf.mesh -o 2 // ex8 -m ../data/mobius-strip.mesh // // Description: This example code demonstrates the use of the Discontinuous // Petrov-Galerkin (DPG) method in its primal 2x2 block form as a // simple finite element discretization of the Laplace problem // -Delta u = f with homogeneous Dirichlet boundary conditions. We // use high-order continuous trial space, a high-order interfacial // (trace) space, and a high-order discontinuous test space // defining a local dual (H^{-1}) norm. // // We use the primal form of DPG, see "A primal DPG method without // a first-order reformulation", Demkowicz and Gopalakrishnan, CAM // 2013, DOI:10.1016/j.camwa.2013.06.029. // // The example highlights the use of interfacial (trace) finite // elements and spaces, trace face integrators and the definition // of block operators and preconditioners. // // We recommend viewing examples 1-5 before viewing this example. #include "mfem.hpp" #include #include using namespace std; using namespace mfem; int main(int argc, char *argv[]) { // 1. Parse command-line options. const char *mesh_file = "../data/star.mesh"; int order = 1; bool visualization = 1; OptionsParser args(argc, argv); args.AddOption(&mesh_file, "-m", "--mesh", "Mesh file to use."); args.AddOption(&order, "-o", "--order", "Finite element order (polynomial degree)."); args.AddOption(&visualization, "-vis", "--visualization", "-no-vis", "--no-visualization", "Enable or disable GLVis visualization."); args.Parse(); if (!args.Good()) { args.PrintUsage(cout); return 1; } args.PrintOptions(cout); // 2. Read the mesh from the given mesh file. We can handle triangular, // quadrilateral, tetrahedral, hexahedral, surface and volume meshes with // the same code. Mesh *mesh = new Mesh(mesh_file, 1, 1); int dim = mesh->Dimension(); // 3. Refine the mesh to increase the resolution. In this example we do // 'ref_levels' of uniform refinement. We choose 'ref_levels' to be the // largest number that gives a final mesh with no more than 10,000 // elements. { int ref_levels = (int)floor(log(10000./mesh->GetNE())/log(2.)/dim); for (int l = 0; l < ref_levels; l++) { mesh->UniformRefinement(); } } // 4. Define the trial, interfacial (trace) and test DPG spaces: // - The trial space, x0_space, contains the non-interfacial unknowns and // has the essential BC. // - The interfacial space, xhat_space, contains the interfacial unknowns // and does not have essential BC. // - The test space, test_space, is an enriched space where the enrichment // degree may depend on the spatial dimension of the domain, the type of // the mesh and the trial space order. unsigned int trial_order = order; unsigned int trace_order = order - 1; unsigned int test_order = order; /* reduced order, full order is (order + dim - 1) */ if (dim == 2 && (order%2 == 0 || (mesh->MeshGenerator() & 2 && order > 1))) { test_order++; } if (test_order < trial_order) cerr << "Warning, test space not enriched enough to handle primal" << " trial space\n"; FiniteElementCollection *x0_fec, *xhat_fec, *test_fec; x0_fec = new H1_FECollection(trial_order, dim); xhat_fec = new RT_Trace_FECollection(trace_order, dim); test_fec = new L2_FECollection(test_order, dim); FiniteElementSpace *x0_space = new FiniteElementSpace(mesh, x0_fec); FiniteElementSpace *xhat_space = new FiniteElementSpace(mesh, xhat_fec); FiniteElementSpace *test_space = new FiniteElementSpace(mesh, test_fec); // 5. Define the block structure of the problem, by creating the offset // variables. Also allocate two BlockVector objects to store the solution // and rhs. enum {x0_var, xhat_var, NVAR}; int s0 = x0_space->GetVSize(); int s1 = xhat_space->GetVSize(); int s_test = test_space->GetVSize(); Array offsets(NVAR+1); offsets[0] = 0; offsets[1] = s0; offsets[2] = s0+s1; Array offsets_test(2); offsets_test[0] = 0; offsets_test[1] = s_test; std::cout << "\nNumber of Unknowns:\n" << " Trial space, X0 : " << s0 << " (order " << trial_order << ")\n" << " Interface space, Xhat : " << s1 << " (order " << trace_order << ")\n" << " Test space, Y : " << s_test << " (order " << test_order << ")\n\n"; BlockVector x(offsets), b(offsets); x = 0.; // 6. Set up the linear form F(.) which corresponds to the right-hand side of // the FEM linear system, which in this case is (f,phi_i) where f=1.0 and // phi_i are the basis functions in the test finite element fespace. ConstantCoefficient one(1.0); LinearForm F(test_space); F.AddDomainIntegrator(new DomainLFIntegrator(one)); F.Assemble(); // 7. Set up the mixed bilinear form for the primal trial unknowns, B0, // the mixed bilinear form for the interfacial unknowns, Bhat, // the inverse stiffness matrix on the discontinuous test space, Sinv, // and the stiffness matrix on the continuous trial space, S0. Array ess_bdr(mesh->bdr_attributes.Max()); ess_bdr = 1; MixedBilinearForm *B0 = new MixedBilinearForm(x0_space,test_space); B0->AddDomainIntegrator(new DiffusionIntegrator(one)); B0->Assemble(); B0->EliminateTrialDofs(ess_bdr, x.GetBlock(x0_var), F); B0->Finalize(); MixedBilinearForm *Bhat = new MixedBilinearForm(xhat_space,test_space); Bhat->AddTraceFaceIntegrator(new TraceJumpIntegrator()); Bhat->Assemble(); Bhat->Finalize(); BilinearForm *Sinv = new BilinearForm(test_space); SumIntegrator *Sum = new SumIntegrator; Sum->AddIntegrator(new DiffusionIntegrator(one)); Sum->AddIntegrator(new MassIntegrator(one)); Sinv->AddDomainIntegrator(new InverseIntegrator(Sum)); Sinv->Assemble(); Sinv->Finalize(); BilinearForm *S0 = new BilinearForm(x0_space); S0->AddDomainIntegrator(new DiffusionIntegrator(one)); S0->Assemble(); S0->EliminateEssentialBC(ess_bdr); S0->Finalize(); SparseMatrix &matB0 = B0->SpMat(); SparseMatrix &matBhat = Bhat->SpMat(); SparseMatrix &matSinv = Sinv->SpMat(); SparseMatrix &matS0 = S0->SpMat(); // 8. Set up the 1x2 block Least Squares DPG operator, B = [B0 Bhat], // the normal equation operator, A = B^t Sinv B, and // the normal equation right-hand-size, b = B^t Sinv F. BlockOperator B(offsets_test, offsets); B.SetBlock(0,0,&matB0); B.SetBlock(0,1,&matBhat); RAPOperator A(B, matSinv, B); { Vector SinvF(s_test); matSinv.Mult(F,SinvF); B.MultTranspose(SinvF, b); } // 9. Set up a block-diagonal preconditioner for the 2x2 normal equation // // [ S0^{-1} 0 ] // [ 0 Shat^{-1} ] Shat = (Bhat^T Sinv Bhat) // // corresponding to the primal (x0) and interfacial (xhat) unknowns. SparseMatrix * Shat = RAP(matBhat, matSinv, matBhat); #ifndef MFEM_USE_SUITESPARSE const double prec_rtol = 1e-3; const int prec_maxit = 200; CGSolver *S0inv = new CGSolver; S0inv->SetOperator(matS0); S0inv->SetPrintLevel(-1); S0inv->SetRelTol(prec_rtol); S0inv->SetMaxIter(prec_maxit); CGSolver *Shatinv = new CGSolver; Shatinv->SetOperator(*Shat); Shatinv->SetPrintLevel(-1); Shatinv->SetRelTol(prec_rtol); Shatinv->SetMaxIter(prec_maxit); // Disable 'iterative_mode' when using CGSolver (or any IterativeSolver) as // a preconditioner: S0inv->iterative_mode = false; Shatinv->iterative_mode = false; #else Operator *S0inv = new UMFPackSolver(matS0); Operator *Shatinv = new UMFPackSolver(*Shat); #endif BlockDiagonalPreconditioner P(offsets); P.SetDiagonalBlock(0, S0inv); P.SetDiagonalBlock(1, Shatinv); // 10. Solve the normal equation system using the PCG iterative solver. // Check the weighted norm of residual for the DPG least square problem. // Wrap the primal variable in a GridFunction for visualization purposes. PCG(A, P, b, x, 1, 200, 1e-12, 0.0); { Vector LSres(s_test); B.Mult(x, LSres); LSres -= F; double res = sqrt(matSinv.InnerProduct(LSres, LSres)); cout << "\n|| B0*x0 + Bhat*xhat - F ||_{S^-1} = " << res << endl; } GridFunction x0; x0.MakeRef(x0_space, x.GetBlock(x0_var), 0); // 11. Save the refined mesh and the solution. This output can be viewed // later using GLVis: "glvis -m refined.mesh -g sol.gf". { ofstream mesh_ofs("refined.mesh"); mesh_ofs.precision(8); mesh->Print(mesh_ofs); ofstream sol_ofs("sol.gf"); sol_ofs.precision(8); x0.Save(sol_ofs); } // 12. Send the solution by socket to a GLVis server. if (visualization) { char vishost[] = "localhost"; int visport = 19916; socketstream sol_sock(vishost, visport); sol_sock.precision(8); sol_sock << "solution\n" << *mesh << x0 << flush; } // 13. Free the used memory. delete S0inv; delete Shatinv; delete Shat; delete Bhat; delete B0; delete S0; delete Sinv; delete test_space; delete test_fec; delete xhat_space; delete xhat_fec; delete x0_space; delete x0_fec; delete mesh; return 0; }