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// MFEM Example 4 - Parallel Version
//
// Compile with: make ex4p
//
// Sample runs: mpirun -np 4 ex4p -m ../data/square-disc.mesh
// mpirun -np 4 ex4p -m ../data/star.mesh
// mpirun -np 4 ex4p -m ../data/beam-tet.mesh
// mpirun -np 4 ex4p -m ../data/beam-hex.mesh
// mpirun -np 4 ex4p -m ../data/escher.mesh -o 2 -sc
// mpirun -np 4 ex4p -m ../data/fichera.mesh -o 2 -hb
// mpirun -np 4 ex4p -m ../data/fichera-q2.vtk
// mpirun -np 4 ex4p -m ../data/fichera-q3.mesh -o 2 -sc
// mpirun -np 4 ex4p -m ../data/square-disc-nurbs.mesh -o 3
// mpirun -np 4 ex4p -m ../data/beam-hex-nurbs.mesh -o 3
// mpirun -np 4 ex4p -m ../data/periodic-square.mesh -no-bc
// mpirun -np 4 ex4p -m ../data/periodic-cube.mesh -no-bc
// mpirun -np 4 ex4p -m ../data/amr-quad.mesh
// mpirun -np 4 ex4p -m ../data/amr-hex.mesh -o 2 -sc
// mpirun -np 4 ex4p -m ../data/amr-hex.mesh -o 2 -hb
// mpirun -np 4 ex4p -m ../data/star-surf.mesh -o 3 -hb
//
// Description: This example code solves a simple 2D/3D H(div) diffusion
// problem corresponding to the second order definite equation
// -grad(alpha div F) + beta F = f with boundary condition F dot n
// = <given normal field>. Here, we use a given exact solution F
// and compute the corresponding r.h.s. f. We discretize with
// Raviart-Thomas finite elements.
//
// The example demonstrates the use of H(div) finite element
// spaces with the grad-div and H(div) vector finite element mass
// bilinear form, as well as the computation of discretization
// error when the exact solution is known. Bilinear form
// hybridization and static condensation are also illustrated.
//
// We recommend viewing examples 1-3 before viewing this example.
#include "mfem.hpp"
#include <fstream>
#include <iostream>
using namespace std;
using namespace mfem;
// Exact solution, F, and r.h.s., f. See below for implementation.
void F_exact(const Vector &, Vector &);
void f_exact(const Vector &, Vector &);
double freq = 1.0, kappa;
int main(int argc, char *argv[])
{
// 1. Initialize MPI.
int num_procs, myid;
MPI_Init(&argc, &argv);
MPI_Comm_size(MPI_COMM_WORLD, &num_procs);
MPI_Comm_rank(MPI_COMM_WORLD, &myid);
// 2. Parse command-line options.
const char *mesh_file = "../data/star.mesh";
int order = 1;
bool set_bc = true;
bool static_cond = false;
bool hybridization = false;
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(&set_bc, "-bc", "--impose-bc", "-no-bc", "--dont-impose-bc",
"Impose or not essential boundary conditions.");
args.AddOption(&freq, "-f", "--frequency", "Set the frequency for the exact"
" solution.");
args.AddOption(&static_cond, "-sc", "--static-condensation", "-no-sc",
"--no-static-condensation", "Enable static condensation.");
args.AddOption(&hybridization, "-hb", "--hybridization", "-no-hb",
"--no-hybridization", "Enable hybridization.");
args.AddOption(&visualization, "-vis", "--visualization", "-no-vis",
"--no-visualization",
"Enable or disable GLVis visualization.");
args.Parse();
if (!args.Good())
{
if (myid == 0)
{
args.PrintUsage(cout);
}
MPI_Finalize();
return 1;
}
if (myid == 0)
{
args.PrintOptions(cout);
}
kappa = freq * M_PI;
// 3. Read the (serial) mesh from the given mesh file on all processors. We
// can handle triangular, quadrilateral, tetrahedral, hexahedral, surface
// and volume, as well as periodic meshes with the same code.
Mesh *mesh = new Mesh(mesh_file, 1, 1);
int dim = mesh->Dimension();
int sdim = mesh->SpaceDimension();
// 4. Refine the serial mesh on all processors 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 1,000 elements.
{
int ref_levels =
(int)floor(log(1000./mesh->GetNE())/log(2.)/dim);
for (int l = 0; l < ref_levels; l++)
{
mesh->UniformRefinement();
}
}
// 5. Define a parallel mesh by a partitioning of the serial mesh. Refine
// this mesh further in parallel to increase the resolution. Once the
// parallel mesh is defined, the serial mesh can be deleted. Tetrahedral
// meshes need to be reoriented before we can define high-order Nedelec
// spaces on them (this is needed in the ADS solver below).
ParMesh *pmesh = new ParMesh(MPI_COMM_WORLD, *mesh);
delete mesh;
{
int par_ref_levels = 2;
for (int l = 0; l < par_ref_levels; l++)
{
pmesh->UniformRefinement();
}
}
pmesh->ReorientTetMesh();
// 6. Define a parallel finite element space on the parallel mesh. Here we
// use the Raviart-Thomas finite elements of the specified order.
FiniteElementCollection *fec = new RT_FECollection(order-1, dim);
ParFiniteElementSpace *fespace = new ParFiniteElementSpace(pmesh, fec);
HYPRE_Int size = fespace->GlobalTrueVSize();
if (myid == 0)
{
cout << "Number of finite element unknowns: " << size << endl;
}
// 7. Determine the list of true (i.e. parallel conforming) essential
// boundary dofs. In this example, the boundary conditions are defined
// by marking all the boundary attributes from the mesh as essential
// (Dirichlet) and converting them to a list of true dofs.
Array<int> ess_tdof_list;
if (pmesh->bdr_attributes.Size())
{
Array<int> ess_bdr(pmesh->bdr_attributes.Max());
ess_bdr = set_bc ? 1 : 0;
fespace->GetEssentialTrueDofs(ess_bdr, ess_tdof_list);
}
// 8. Set up the parallel linear form b(.) which corresponds to the
// right-hand side of the FEM linear system, which in this case is
// (f,phi_i) where f is given by the function f_exact and phi_i are the
// basis functions in the finite element fespace.
VectorFunctionCoefficient f(sdim, f_exact);
ParLinearForm *b = new ParLinearForm(fespace);
b->AddDomainIntegrator(new VectorFEDomainLFIntegrator(f));
b->Assemble();
// 9. Define the solution vector x as a parallel finite element grid function
// corresponding to fespace. Initialize x by projecting the exact
// solution. Note that only values from the boundary faces will be used
// when eliminating the non-homogeneous boundary condition to modify the
// r.h.s. vector b.
ParGridFunction x(fespace);
VectorFunctionCoefficient F(sdim, F_exact);
x.ProjectCoefficient(F);
// 10. Set up the parallel bilinear form corresponding to the H(div)
// diffusion operator grad alpha div + beta I, by adding the div-div and
// the mass domain integrators.
Coefficient *alpha = new ConstantCoefficient(1.0);
Coefficient *beta = new ConstantCoefficient(1.0);
ParBilinearForm *a = new ParBilinearForm(fespace);
a->AddDomainIntegrator(new DivDivIntegrator(*alpha));
a->AddDomainIntegrator(new VectorFEMassIntegrator(*beta));
// 11. Assemble the parallel bilinear form and the corresponding linear
// system, applying any necessary transformations such as: parallel
// assembly, eliminating boundary conditions, applying conforming
// constraints for non-conforming AMR, static condensation,
// hybridization, etc.
FiniteElementCollection *hfec = NULL;
ParFiniteElementSpace *hfes = NULL;
if (static_cond)
{
a->EnableStaticCondensation();
}
else if (hybridization)
{
hfec = new DG_Interface_FECollection(order-1, dim);
hfes = new ParFiniteElementSpace(pmesh, hfec);
a->EnableHybridization(hfes, new NormalTraceJumpIntegrator(),
ess_tdof_list);
}
a->Assemble();
HypreParMatrix A;
Vector B, X;
a->FormLinearSystem(ess_tdof_list, x, *b, A, X, B);
HYPRE_Int glob_size = A.GetGlobalNumRows();
if (myid == 0)
{
cout << "Size of linear system: " << glob_size << endl;
}
// 12. Define and apply a parallel PCG solver for A X = B with the 2D AMS or
// the 3D ADS preconditioners from hypre. If using hybridization, the
// system is preconditioned with hypre's BoomerAMG.
HypreSolver *prec = NULL;
CGSolver *pcg = new CGSolver(A.GetComm());
pcg->SetOperator(A);
pcg->SetRelTol(1e-12);
pcg->SetMaxIter(500);
pcg->SetPrintLevel(1);
if (hybridization) { prec = new HypreBoomerAMG(A); }
else
{
ParFiniteElementSpace *prec_fespace =
(a->StaticCondensationIsEnabled() ? a->SCParFESpace() : fespace);
if (dim == 2) { prec = new HypreAMS(A, prec_fespace); }
else { prec = new HypreADS(A, prec_fespace); }
}
pcg->SetPreconditioner(*prec);
pcg->Mult(B, X);
// 13. Recover the parallel grid function corresponding to X. This is the
// local finite element solution on each processor.
a->RecoverFEMSolution(X, *b, x);
// 14. Compute and print the L^2 norm of the error.
{
double err = x.ComputeL2Error(F);
if (myid == 0)
{
cout << "\n|| F_h - F ||_{L^2} = " << err << '\n' << endl;
}
}
// 15. Save the refined mesh and the solution in parallel. This output can
// be viewed later using GLVis: "glvis -np <np> -m mesh -g sol".
{
ostringstream mesh_name, sol_name;
mesh_name << "mesh." << setfill('0') << setw(6) << myid;
sol_name << "sol." << setfill('0') << setw(6) << myid;
ofstream mesh_ofs(mesh_name.str().c_str());
mesh_ofs.precision(8);
pmesh->Print(mesh_ofs);
ofstream sol_ofs(sol_name.str().c_str());
sol_ofs.precision(8);
x.Save(sol_ofs);
}
// 16. Send the solution by socket to a GLVis server.
if (visualization)
{
char vishost[] = "localhost";
int visport = 19916;
socketstream sol_sock(vishost, visport);
sol_sock << "parallel " << num_procs << " " << myid << "\n";
sol_sock.precision(8);
sol_sock << "solution\n" << *pmesh << x << flush;
}
// 17. Free the used memory.
delete pcg;
delete prec;
delete hfes;
delete hfec;
delete a;
delete alpha;
delete beta;
delete b;
delete fespace;
delete fec;
delete pmesh;
MPI_Finalize();
return 0;
}
// The exact solution (for non-surface meshes)
void F_exact(const Vector &p, Vector &F)
{
int dim = p.Size();
double x = p(0);
double y = p(1);
// double z = (dim == 3) ? p(2) : 0.0;
F(0) = cos(kappa*x)*sin(kappa*y);
F(1) = cos(kappa*y)*sin(kappa*x);
if (dim == 3)
{
F(2) = 0.0;
}
}
// The right hand side
void f_exact(const Vector &p, Vector &f)
{
int dim = p.Size();
double x = p(0);
double y = p(1);
// double z = (dim == 3) ? p(2) : 0.0;
double temp = 1 + 2*kappa*kappa;
f(0) = temp*cos(kappa*x)*sin(kappa*y);
f(1) = temp*cos(kappa*y)*sin(kappa*x);
if (dim == 3)
{
f(2) = 0;
}
}