mfem/examples/petsc/ex4p.cpp

//                       MFEM Example 4 - Parallel Version
//                              PETSc Modification
//
// Compile with: make ex4p
//
// Sample runs:
//    mpirun -np 4 ex4p -m ../../data/klein-bottle.mesh -o 2 --petscopts rc_ex4p
//    mpirun -np 4 ex4p -m ../../data/klein-bottle.mesh -o 2 --petscopts rc_ex4p_bddc --nonoverlapping
//
// 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>

#ifndef MFEM_USE_PETSC
#error This example requires that MFEM is built with MFEM_USE_PETSC=YES
#endif

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;
   bool use_petsc = true;
   const char *petscrc_file = "";
   bool use_nonoverlapping = false;

   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.AddOption(&use_petsc, "-usepetsc", "--usepetsc", "-no-petsc",
                  "--no-petsc",
                  "Use or not PETSc to solve the linear system.");
   args.AddOption(&petscrc_file, "-petscopts", "--petscopts",
                  "PetscOptions file to use.");
   args.AddOption(&use_nonoverlapping, "-nonoverlapping", "--nonoverlapping",
                  "-no-nonoverlapping", "--no-nonoverlapping",
                  "Use or not the block diagonal PETSc's matrix format "
                  "for non-overlapping domain decomposition.");
   args.Parse();
   if (!args.Good())
   {
      if (myid == 0)
      {
         args.PrintUsage(cout);
      }
      MPI_Finalize();
      return 1;
   }
   if (myid == 0)
   {
      args.PrintOptions(cout);
   }
   // 2b. We initialize PETSc
   if (use_petsc) { MFEMInitializePetsc(NULL,NULL,petscrc_file,NULL); }
   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();

   Vector B, X;
   CGSolver *pcg = new CGSolver(MPI_COMM_WORLD);
   pcg->SetRelTol(1e-12);
   pcg->SetMaxIter(500);
   pcg->SetPrintLevel(1);

   if (!use_petsc)
   {
      HypreParMatrix A;
      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;
      pcg->SetOperator(A);
      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);
      delete prec;
   }
   else
   {
      PetscParMatrix A;
      PetscPreconditioner *prec = NULL;
      a->SetOperatorType(use_nonoverlapping ?
                         Operator::PETSC_MATIS : Operator::PETSC_MATAIJ);
      a->FormLinearSystem(ess_tdof_list, x, *b, A, X, B);

      if (myid == 0)
      {
         cout << "Size of linear system: " << A.M() << endl;
      }

      pcg->SetOperator(A);
      if (use_nonoverlapping)
      {
         ParFiniteElementSpace *prec_fespace =
            (a->StaticCondensationIsEnabled() ? a->SCParFESpace() : fespace);

         // Auxiliary class for BDDC customization
         PetscBDDCSolverParams opts;
         // Inform the solver about the finite element space
         opts.SetSpace(prec_fespace);
         // Inform the solver about essential dofs
         opts.SetEssBdrDofs(&ess_tdof_list);
         // Create a BDDC solver with parameters
         prec = new PetscBDDCSolver(A, opts);
      }
      else
      {
         // Create an empty preconditioner that can be customized at runtime.
         prec = new PetscPreconditioner(A, "solver_");
      }
      pcg->SetPreconditioner(*prec);
      pcg->Mult(B, X);
      delete prec;
   }
   delete pcg;

   // 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 hfes;
   delete hfec;
   delete a;
   delete alpha;
   delete beta;
   delete b;
   delete fespace;
   delete fec;
   delete pmesh;

   // We finalize PETSc
   if (use_petsc) { MFEMFinalizePetsc(); }

   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;
   }
}