ex33p.cpp

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//                       MFEM Example 33 - Parallel Version
//
// Compile with: make ex33p
//
// Sample runs:  mpirun -np 4 ex33p -m ../data/square-disc.mesh -alpha 0.33 -o 2
//               mpirun -np 4 ex33p -m ../data/square-disc.mesh -alpha 4.5 -o 3
//               mpirun -np 4 ex33p -m ../data/star.mesh -alpha 1.4 -o 3
//               mpirun -np 4 ex33p -m ../data/star.mesh -alpha 0.99 -o 3
//               mpirun -np 4 ex33p -m ../data/inline-quad.mesh -alpha 0.5 -o 3
//               mpirun -np 4 ex33p -m ../data/amr-quad.mesh -alpha 1.5 -o 3
//               mpirun -np 4 ex33p -m ../data/disc-nurbs.mesh -alpha 0.33 -o 3 -r 2
//               mpirun -np 4 ex33p -m ../data/disc-nurbs.mesh -alpha 2.4 -o 3 -r 4
//               mpirun -np 4 ex33p -m ../data/l-shape.mesh -alpha 0.33 -o 3 -r 4
//               mpirun -np 4 ex33p -m ../data/l-shape.mesh -alpha 1.7 -o 3 -r 5
//
// Verification runs:
//    mpirun -np 4 ex33p -m ../data/inline-segment.mesh -ver -alpha 1.7 -o 2 -r 2
//    mpirun -np 4 ex33p -m ../data/inline-quad.mesh -ver -alpha 1.2 -o 2 -r 2
//    mpirun -np 4 ex33p -m ../data/amr-quad.mesh -ver -alpha 2.6 -o 2 -r 2
//    mpirun -np 4 ex33p -m ../data/inline-hex.mesh -ver -alpha 0.3 -o 2 -r 1
 
//  Note: The manufactured solution used in this problem is
//
//            u = ∏_{i=0}^{dim-1} sin(π x_i) ,
//
//        regardless of the value of alpha.
//
// Description:
//
//  In this example we solve the following fractional PDE with MFEM:
//
//    ( - Δ )^α u = f  in Ω,      u = 0  on ∂Ω,      0 < α,
//
//  To solve this FPDE, we apply the operator ( - Δ )^(-N), where the integer
//  N is given by floor(α). By doing so, we obtain
//
//    ( - Δ )^(α-N) u = ( - Δ )^(-N) f  in Ω,      u = 0  on ∂Ω,      0 < α.
//
//  We first compute the right hand side by solving the integer order PDE
//
//   ( - Δ )^N g = f  in Ω, g = ( - Δ )^k g = 0 on ∂Ω, k = 1,..,N-1
//
//  The remaining FPDE is then given by
//
//  ( - Δ )^(α-N) u = g  in Ω,      u = 0  on ∂Ω.
//
//  We rely on a rational approximation [2] of the normal linear operator
//  A^{-α + N}, where A = - Δ (with associated homogeneous boundary conditions)
//  and (a-N) in (0,1). We approximate the operator
//
//    A^{-α+N} ≈ Σ_{i=0}^M c_i (A + d_i I)^{-1},      d_0 = 0,   d_i > 0,
//
//  where I is the L2-identity operator and the coefficients c_i and d_i
//  are generated offline to a prescribed accuracy in a pre-processing step.
//  We use the triple-A algorithm [1] to generate the rational approximation
//  that this partial fractional expansion derives from. We then solve M+1
//  independent integer-order PDEs,
//
//    A u_i + d_i u_i = c_i g  in Ω,      u_i = 0  on ∂Ω,      i=0,...,M,
//
//  using MFEM and sum u_i to arrive at an approximate solution of the FPDE
//
//    u ≈ Σ_{i=0}^M u_i.
//
//  (If alpha is an integer, we stop after the first PDE was solved.)
//
// References:
//
// [1] Nakatsukasa, Y., Sète, O., & Trefethen, L. N. (2018). The AAA algorithm
//     for rational approximation. SIAM Journal on Scientific Computing, 40(3),
//     A1494-A1522.
//
// [2] Harizanov, S., Lazarov, R., Margenov, S., Marinov, P., & Pasciak, J.
//     (2020). Analysis of numerical methods for spectral fractional elliptic
//     equations based on the best uniform rational approximation. Journal of
//     Computational Physics, 408, 109285.
//
 
#include "mfem.hpp"
#include <fstream>
#include <iostream>
#include <math.h>
#include <string>
 
#include "ex33.hpp"
 
using namespace std;
using namespace mfem;
 
int main(int argc, char *argv[])
{
#ifdef MFEM_USE_SINGLE
   cout << "This example is not supported in single precision.\n\n";
   return MFEM_SKIP_RETURN_VALUE;
#endif
 
   // 0. Initialize MPI.
   Mpi::Init(argc, argv);
   int num_procs = Mpi::WorldSize();
   int myid = Mpi::WorldRank();
   Hypre::Init();
 
   // 1. Parse command-line options.
   const char *mesh_file = "../data/star.mesh";
   int order = 1;
   int num_refs = 3;
   real_t alpha = 0.5;
   bool visualization = true;
   bool verification = 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) or -1 for"
                  " isoparametric space.");
   args.AddOption(&num_refs, "-r", "--refs",
                  "Number of uniform refinements");
   args.AddOption(&alpha, "-alpha", "--alpha",
                  "Fractional exponent");
   args.AddOption(&visualization, "-vis", "--visualization", "-no-vis",
                  "--no-visualization",
                  "Enable or disable GLVis visualization.");
   args.AddOption(&verification, "-ver", "--verification", "-no-ver",
                  "--no-verification",
                  "Use sinusoidal function (f) for manufactured "
                  "solution test.");
   args.Parse();
   if (!args.Good())
   {
      args.PrintUsage(cout);
      return 1;
   }
   if (Mpi::Root())
   {
      args.PrintOptions(cout);
   }
 
   Array<real_t> coeffs, poles;
   int progress_steps = 1;
 
   // 2. Compute the rational expansion coefficients that define the
   //    integer-order PDEs.
   const int power_of_laplace = floor(alpha);
   real_t exponent_to_approximate = alpha - power_of_laplace;
   bool integer_order = false;
   // Check if alpha is an integer or not.
   if (abs(exponent_to_approximate) > 1e-12)
   {
      if (Mpi::Root())
      {
         mfem::out << "Approximating the fractional exponent "
                   << exponent_to_approximate
                   << endl;
      }
      ComputePartialFractionApproximation(exponent_to_approximate, coeffs,
                                          poles);
 
      // If the example is build without LAPACK, the exponent_to_approximate
      // might be modified by the function call above.
      alpha = exponent_to_approximate + power_of_laplace;
   }
   else
   {
      integer_order = true;
      if (Mpi::Root())
      {
         mfem::out << "Treating integer order PDE." << endl;
      }
   }
 
   // 3. Read the mesh from the given mesh file.
   Mesh mesh(mesh_file, 1, 1);
   int dim = mesh.Dimension();
 
   // 4. Refine the mesh to increase the resolution.
   for (int i = 0; i < num_refs; i++)
   {
      mesh.UniformRefinement();
   }
   ParMesh pmesh(MPI_COMM_WORLD, mesh);
   mesh.Clear();
 
   // 5. Define a finite element space on the mesh.
   H1_FECollection fec(order, dim);
   ParFiniteElementSpace fespace(&pmesh, &fec);
   HYPRE_BigInt size = fespace.GlobalTrueVSize();
   if (Mpi::Root())
   {
      cout << "Number of degrees of freedom: "
           << size << endl;
   }
 
   // 6. Determine the list of true (i.e. conforming) essential boundary dofs.
   Array<int> ess_tdof_list;
   if (pmesh.bdr_attributes.Size())
   {
      Array<int> ess_bdr(pmesh.bdr_attributes.Max());
      ess_bdr = 1;
      fespace.GetEssentialTrueDofs(ess_bdr, ess_tdof_list);
   }
 
   // 7. Define diffusion coefficient, load, and solution GridFunction.
   auto func = [&alpha](const Vector &x)
   {
      real_t val = 1.0;
      for (int i=0; i<x.Size(); i++)
      {
         val *= sin(M_PI*x(i));
      }
      return pow(x.Size()*pow(M_PI,2), alpha) * val;
   };
   FunctionCoefficient f(func);
   ConstantCoefficient one(1.0);
   ParGridFunction u(&fespace);
   ParGridFunction x(&fespace);
   ParGridFunction g(&fespace);
   u = 0.0;
   x = 0.0;
   g = 0.0;
 
   // 8. Prepare for visualization.
   char vishost[] = "localhost";
   int  visport   = 19916;
 
   // 9. Set up the linear form b(.) for integer-order PDE solves.
   ParLinearForm b(&fespace);
   if (verification)
   {
      // This statement is only relevant for the verification of the code. It
      // uses a different f such that an manufactured solution is known and easy
      // to compare with the numerical one. The FPDE becomes:
      // (-Δ)^α u = (2\pi ^2)^α sin(\pi x) sin(\pi y) on [0,1]^2
      // -> u(x,y) = sin(\pi x) sin(\pi y)
      b.AddDomainIntegrator(new DomainLFIntegrator(f));
   }
   else
   {
      b.AddDomainIntegrator(new DomainLFIntegrator(one));
   }
   b.Assemble();
 
   // ------------------------------------------------------------------------
   // 10. Solve the PDE (-Δ)^N g = f, i.e. compute g = (-Δ)^{-1}^N f.
   // ------------------------------------------------------------------------
 
   if (power_of_laplace > 0)
   {
      // 10.1 Compute Stiffnes Matrix
      ParBilinearForm k(&fespace);
      k.AddDomainIntegrator(new DiffusionIntegrator(one));
      k.Assemble();
 
      // 10.2 Compute Mass Matrix
      ParBilinearForm m(&fespace);
      m.AddDomainIntegrator(new MassIntegrator(one));
      m.Assemble();
      HypreParMatrix mass;
      Array<int> empty;
      m.FormSystemMatrix(empty, mass);
 
      // 10.3 Form the system of equations
      Vector B, X;
      OperatorPtr Op;
      k.FormLinearSystem(ess_tdof_list, g, b, Op, X, B);
      HypreBoomerAMG prec;
      prec.SetPrintLevel(-1);
      CGSolver cg(MPI_COMM_WORLD);
      cg.SetRelTol(1e-12);
      cg.SetMaxIter(2000);
      cg.SetPrintLevel(3);
      cg.SetPreconditioner(prec);
      cg.SetOperator(*Op);
 
      if (Mpi::Root())
      {
         mfem::out << "\nComputing (-Δ) ^ -" << power_of_laplace
                   << " ( f ) " << endl;
      }
      for (int i = 0; i < power_of_laplace; i++)
      {
         // 10.4 Solve the linear system Op X = B (N times).
         cg.Mult(B, X);
         // 10.5 Visualize the solution g of -Δ ^ N g = f in the last step
         if (i == power_of_laplace - 1)
         {
            // Needed for visualization and solution verification.
            k.RecoverFEMSolution(X, b, g);
            if (integer_order && verification)
            {
               // For an integer order PDE, g is also our solution u.
               u+=g;
            }
            if (visualization)
            {
               socketstream fout;
               ostringstream oss_f;
               fout.open(vishost, visport);
               fout.precision(8);
               oss_f.str(""); oss_f.clear();
               oss_f << "Step " << progress_steps++ << ": Solution of PDE -Δ ^ "
                     << power_of_laplace
                     << " g = f";
               fout << "parallel " << num_procs << " " << myid << "\n"
                    << "solution\n" << pmesh << g
                    << "window_title '" << oss_f.str() << "'" << flush;
            }
         }
 
         // 10.6 Prepare for next iteration (primal / dual space)
         mass.Mult(X, B);
         X.SetSubVectorComplement(ess_tdof_list,0.0);
      }
 
      // 10.7 Extract solution for the next step. The b now corresponds to the
      //      function g in the PDE.
      const SparseMatrix* rm = fespace.GetRestrictionMatrix();
      rm->MultTranspose(B, b);
   }
 
   // ------------------------------------------------------------------------
   // 11. Solve the fractional PDE by solving M integer order PDEs and adding
   //     up the solutions.
   // ------------------------------------------------------------------------
   if (!integer_order)
   {
      // Setup visualization.
      socketstream xout, uout;
      ostringstream oss_x, oss_u;
      if (visualization)
      {
         xout.open(vishost, visport);
         xout.precision(8);
         uout.open(vishost, visport);
         uout.precision(8);
      }
      // Iterate over all expansion coefficient that contribute to the
      // solution.
      for (int i = 0; i < coeffs.Size(); i++)
      {
         if (Mpi::Root())
         {
            mfem::out << "\nSolving PDE -Δ u + " << -poles[i]
                      << " u = " << coeffs[i] << " g " << endl;
         }
 
         // 11.1 Reset GridFunction for integer-order PDE solve.
         x = 0.0;
 
         // 11.2 Set up the bilinear form a(.,.) for integer-order PDE solve.
         ParBilinearForm a(&fespace);
         a.AddDomainIntegrator(new DiffusionIntegrator(one));
         ConstantCoefficient d_i(-poles[i]);
         a.AddDomainIntegrator(new MassIntegrator(d_i));
         a.Assemble();
 
         // 11.3 Assemble the bilinear form and the corresponding linear system.
         OperatorPtr A;
         Vector B, X;
         a.FormLinearSystem(ess_tdof_list, x, b, A, X, B);
 
         // 11.4 Solve the linear system A X = B.
         HypreBoomerAMG prec;
         prec.SetPrintLevel(-1);
 
         CGSolver cg(MPI_COMM_WORLD);
         cg.SetRelTol(1e-12);
         cg.SetMaxIter(2000);
         cg.SetPrintLevel(3);
         cg.SetPreconditioner(prec);
         cg.SetOperator(*A);
         cg.Mult(B, X);
 
         // 11.5 Recover the solution as a finite element grid function.
         a.RecoverFEMSolution(X, b, x);
 
         // 11.6 Accumulate integer-order PDE solutions.
         x *= coeffs[i];
         u += x;
 
         // 11.7 Send fractional PDE solution to a GLVis server.
         if (visualization)
         {
            oss_x.str(""); oss_x.clear();
            oss_x << "Step " << progress_steps
                  << ": Solution of PDE -Δ u + " << -poles[i]
                  << " u = " << coeffs[i] << " g";
            xout << "parallel " << num_procs << " " << myid << "\n"
                 << "solution\n" << pmesh << x
                 << "window_title '" << oss_x.str() << "'" << flush;
 
            oss_u.str(""); oss_u.clear();
            oss_u << "Step " << progress_steps + 1
                  << ": Solution of fractional PDE (-Δ)^" << alpha
                  << " u = f";
            uout << "parallel " << num_procs << " " << myid << "\n"
                 << "solution\n" << pmesh << u
                 << "window_title '" << oss_u.str() << "'"
                 << flush;
         }
      }
   }
 
   // ------------------------------------------------------------------------
   // 12. (optional) Verify the solution.
   // ------------------------------------------------------------------------
   if (verification)
   {
      auto solution = [] (const Vector &x)
      {
         real_t val = 1.0;
         for (int i=0; i<x.Size(); i++)
         {
            val *= sin(M_PI*x(i));
         }
         return val;
      };
      FunctionCoefficient sol(solution);
      real_t l2_error = u.ComputeL2Error(sol);
 
      if (Mpi::Root())
      {
         string manufactured_solution,expected_mesh;
         switch (dim)
         {
            case 1:
               manufactured_solution = "sin(π x)";
               expected_mesh = "inline_segment.mesh";
               break;
            case 2:
               manufactured_solution = "sin(π x) sin(π y)";
               expected_mesh = "inline_quad.mesh";
               break;
            default:
               manufactured_solution = "sin(π x) sin(π y) sin(π z)";
               expected_mesh = "inline_hex.mesh";
               break;
         }
 
         mfem::out << "\n" << string(80,'=')
                   << "\n\nSolution Verification in "<< dim << "D \n\n"
                   << "Manufactured solution : " << manufactured_solution << "\n"
                   << "Expected mesh         : " << expected_mesh <<"\n"
                   << "Your mesh             : " << mesh_file << "\n"
                   << "L2 error              : " << l2_error << "\n\n"
                   << string(80,'=') << endl;
      }
   }
 
   return 0;
}