void tic() { tic_toc.Clear(); tic_toc.Start(); }
int main(int argc, char *argv[]) { StopWatch chrono; // 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; ifstream imesh(mesh_file); if (!imesh) { cerr << "\nCan not open mesh file: " << mesh_file << '\n' << endl; return 2; } mesh = new Mesh(imesh, 1, 1); imesh.close(); 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 a finite element space on the mesh. Here we use the // Raviart-Thomas finite elements of the specified order. FiniteElementCollection *hdiv_coll(new RT_FECollection(order, dim)); FiniteElementCollection *l2_coll(new L2_FECollection(order, dim)); FiniteElementSpace *R_space = new FiniteElementSpace(mesh, hdiv_coll); FiniteElementSpace *W_space = new FiniteElementSpace(mesh, l2_coll); // 5. Define the BlockStructure of the problem, i.e. define the array of // offsets for each variable. The last component of the Array is the sum // of the dimensions of each block. Array<int> block_offsets(3); // number of variables + 1 block_offsets[0] = 0; block_offsets[1] = R_space->GetVSize(); block_offsets[2] = W_space->GetVSize(); block_offsets.PartialSum(); std::cout << "***********************************************************\n"; std::cout << "dim(R) = " << block_offsets[1] - block_offsets[0] << "\n"; std::cout << "dim(W) = " << block_offsets[2] - block_offsets[1] << "\n"; std::cout << "dim(R+W) = " << block_offsets.Last() << "\n"; std::cout << "***********************************************************\n"; // 6. Define the coefficients, analytical solution, and rhs of the PDE. ConstantCoefficient k(1.0); VectorFunctionCoefficient fcoeff(dim, fFun); FunctionCoefficient fnatcoeff(f_natural); FunctionCoefficient gcoeff(gFun); VectorFunctionCoefficient ucoeff(dim, uFun_ex); FunctionCoefficient pcoeff(pFun_ex); // 7. Allocate memory (x, rhs) for the analytical solution and the right hand // side. Define the GridFunction u,p for the finite element solution and // linear forms fform and gform for the right hand side. The data // allocated by x and rhs are passed as a reference to the grid functions // (u,p) and the linear forms (fform, gform). BlockVector x(block_offsets), rhs(block_offsets); LinearForm *fform(new LinearForm); fform->Update(R_space, rhs.GetBlock(0), 0); fform->AddDomainIntegrator(new VectorFEDomainLFIntegrator(fcoeff)); fform->AddBoundaryIntegrator(new VectorFEBoundaryFluxLFIntegrator(fnatcoeff)); fform->Assemble(); LinearForm *gform(new LinearForm); gform->Update(W_space, rhs.GetBlock(1), 0); gform->AddDomainIntegrator(new DomainLFIntegrator(gcoeff)); gform->Assemble(); // 8. Assemble the finite element matrices for the Darcy operator // // D = [ M B^T ] // [ B 0 ] // where: // // M = \int_\Omega k u_h \cdot v_h d\Omega u_h, v_h \in R_h // B = -\int_\Omega \div u_h q_h d\Omega u_h \in R_h, q_h \in W_h BilinearForm *mVarf(new BilinearForm(R_space)); MixedBilinearForm *bVarf(new MixedBilinearForm(R_space, W_space)); mVarf->AddDomainIntegrator(new VectorFEMassIntegrator(k)); mVarf->Assemble(); mVarf->Finalize(); SparseMatrix &M(mVarf->SpMat()); bVarf->AddDomainIntegrator(new VectorFEDivergenceIntegrator); bVarf->Assemble(); bVarf->Finalize(); SparseMatrix & B(bVarf->SpMat()); B *= -1.; SparseMatrix *BT = Transpose(B); BlockMatrix darcyMatrix(block_offsets); darcyMatrix.SetBlock(0,0, &M); darcyMatrix.SetBlock(0,1, BT); darcyMatrix.SetBlock(1,0, &B); // 9. Construct the operators for preconditioner // // P = [ diag(M) 0 ] // [ 0 B diag(M)^-1 B^T ] // // Here we use Symmetric Gauss-Seidel to approximate the inverse of the // pressure Schur Complement SparseMatrix *MinvBt = Transpose(B); Vector Md(M.Height()); M.GetDiag(Md); for (int i = 0; i < Md.Size(); i++) { MinvBt->ScaleRow(i, 1./Md(i)); } SparseMatrix *S = Mult(B, *MinvBt); Solver *invM, *invS; invM = new DSmoother(M); #ifndef MFEM_USE_SUITESPARSE invS = new GSSmoother(*S); #else invS = new UMFPackSolver(*S); #endif invM->iterative_mode = false; invS->iterative_mode = false; BlockDiagonalPreconditioner darcyPrec(block_offsets); darcyPrec.SetDiagonalBlock(0, invM); darcyPrec.SetDiagonalBlock(1, invS); // 10. Solve the linear system with MINRES. // Check the norm of the unpreconditioned residual. int maxIter(1000); double rtol(1.e-6); double atol(1.e-10); chrono.Clear(); chrono.Start(); MINRESSolver solver; solver.SetAbsTol(atol); solver.SetRelTol(rtol); solver.SetMaxIter(maxIter); solver.SetOperator(darcyMatrix); solver.SetPreconditioner(darcyPrec); solver.SetPrintLevel(1); x = 0.0; solver.Mult(rhs, x); chrono.Stop(); if (solver.GetConverged()) std::cout << "MINRES converged in " << solver.GetNumIterations() << " iterations with a residual norm of " << solver.GetFinalNorm() << ".\n"; else std::cout << "MINRES did not converge in " << solver.GetNumIterations() << " iterations. Residual norm is " << solver.GetFinalNorm() << ".\n"; std::cout << "MINRES solver took " << chrono.RealTime() << "s. \n"; // 11. Create the grid functions u and p. Compute the L2 error norms. GridFunction u, p; u.Update(R_space, x.GetBlock(0), 0); p.Update(W_space, x.GetBlock(1), 0); int order_quad = max(2, 2*order+1); const IntegrationRule *irs[Geometry::NumGeom]; for (int i=0; i < Geometry::NumGeom; ++i) { irs[i] = &(IntRules.Get(i, order_quad)); } double err_u = u.ComputeL2Error(ucoeff, irs); double norm_u = ComputeLpNorm(2., ucoeff, *mesh, irs); double err_p = p.ComputeL2Error(pcoeff, irs); double norm_p = ComputeLpNorm(2., pcoeff, *mesh, irs); std::cout << "|| u_h - u_ex || / || u_ex || = " << err_u / norm_u << "\n"; std::cout << "|| p_h - p_ex || / || p_ex || = " << err_p / norm_p << "\n"; // 12. Save the mesh and the solution. This output can be viewed later using // GLVis: "glvis -m ex5.mesh -g sol_u.gf" or "glvis -m ex5.mesh -g // sol_p.gf". { ofstream mesh_ofs("ex5.mesh"); mesh_ofs.precision(8); mesh->Print(mesh_ofs); ofstream u_ofs("sol_u.gf"); u_ofs.precision(8); u.Save(u_ofs); ofstream p_ofs("sol_p.gf"); p_ofs.precision(8); p.Save(p_ofs); } // 13. Save data in the VisIt format VisItDataCollection visit_dc("Example5", mesh); visit_dc.RegisterField("velocity", &u); visit_dc.RegisterField("pressure", &p); visit_dc.Save(); // 14. Send the solution by socket to a GLVis server. if (visualization) { char vishost[] = "localhost"; int visport = 19916; socketstream u_sock(vishost, visport); u_sock.precision(8); u_sock << "solution\n" << *mesh << u << "window_title 'Velocity'" << endl; socketstream p_sock(vishost, visport); p_sock.precision(8); p_sock << "solution\n" << *mesh << p << "window_title 'Pressure'" << endl; } // 15. Free the used memory. delete fform; delete gform; delete invM; delete invS; delete S; delete MinvBt; delete BT; delete mVarf; delete bVarf; delete W_space; delete R_space; delete l2_coll; delete hdiv_coll; delete mesh; return 0; }