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// MFEM Example 8
//
// Compile with: make ex8
//
// Sample runs: ex8 -m ../data/square-disc.mesh
// ex8 -m ../data/star.mesh
// ex8 -m ../data/escher.mesh
// ex8 -m ../data/fichera.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 <fstream>
#include <iostream>
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<int> offsets(NVAR+1);
offsets[0] = 0;
offsets[1] = s0;
offsets[2] = s0+s1;
Array<int> 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<int> 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;
}