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