532 lines
19 KiB
C++
532 lines
19 KiB
C++
// MFEM Example 31 - Parallel Version
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//
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// Compile with: make ex31p
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//
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// Sample runs: mpirun -np 4 ex31p -m ../data/hexagon.mesh -o 2
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// mpirun -np 4 ex31p -m ../data/star.mesh
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// mpirun -np 4 ex31p -m ../data/square-disc.mesh -o 2
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// mpirun -np 4 ex31p -m ../data/fichera.mesh -o 3 -rs 1 -rp 0
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// mpirun -np 4 ex31p -m ../data/square-disc-nurbs.mesh -o 3
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// mpirun -np 4 ex31p -m ../data/amr-quad.mesh -o 2 -rs 1
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// mpirun -np 4 ex31p -m ../data/amr-hex.mesh -rs 1
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//
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// Description: This example code solves a simple electromagnetic diffusion
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// problem corresponding to the second order definite Maxwell
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// equation curl curl E + sigma E = f with boundary condition
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// E x n = <given tangential field>. In this example sigma is an
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// anisotropic 3x3 tensor. Here, we use a given exact solution E
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// and compute the corresponding r.h.s. f. We discretize with
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// Nedelec finite elements in 1D, 2D, or 3D.
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//
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// The example demonstrates the use of restricted H(curl) finite
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// element spaces with the curl-curl and the (vector finite
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// element) mass bilinear form, as well as the computation of
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// discretization error when the exact solution is known. These
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// restricted spaces allow the solution of 1D or 2D
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// electromagnetic problems which involve 3D field vectors. Such
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// problems arise in plasma physics and crystallography.
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//
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// We recommend viewing example 3 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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// Exact solution, E, and r.h.s., f. See below for implementation.
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void E_exact(const Vector &, Vector &);
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void CurlE_exact(const Vector &, Vector &);
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void f_exact(const Vector &, Vector &);
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real_t freq = 1.0, kappa;
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int dim;
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int main(int argc, char *argv[])
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{
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// 1. Initialize MPI.
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Mpi::Init(argc, argv);
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int num_procs = Mpi::WorldSize();
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int myid = Mpi::WorldRank();
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Hypre::Init();
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// 2. Parse command-line options.
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const char *mesh_file = "../data/inline-quad.mesh";
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int ser_ref_levels = 2;
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int par_ref_levels = 1;
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int order = 1;
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bool use_ams = true;
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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(&ser_ref_levels, "-rs", "--refine-serial",
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"Number of times to refine the mesh uniformly in serial.");
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args.AddOption(&par_ref_levels, "-rp", "--refine-parallel",
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"Number of times to refine the mesh uniformly in parallel.");
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args.AddOption(&order, "-o", "--order",
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"Finite element order (polynomial degree).");
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args.AddOption(&freq, "-f", "--frequency", "Set the frequency for the exact"
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" solution.");
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args.AddOption(&use_ams, "-ams", "--hypre-ams", "-slu",
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"--superlu", "Use AMS or SuperLU solver.");
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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.ParseCheck();
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kappa = freq * M_PI;
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// 3. Read the (serial) mesh from the given mesh file on all processors. We
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// can handle triangular, quadrilateral, tetrahedral, hexahedral, surface
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// and volume meshes with the same code.
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Mesh *mesh = new Mesh(mesh_file, 1, 1);
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dim = mesh->Dimension();
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// 4. Refine the serial mesh on all processors to increase the resolution. In
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// this example we do 'ref_levels' of uniform refinement (2 by default, or
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// specified on the command line with -rs).
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for (int lev = 0; lev < ser_ref_levels; lev++)
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{
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mesh->UniformRefinement();
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}
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// 5. Define a parallel mesh by a partitioning of the serial mesh. Refine
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// this mesh further in parallel to increase the resolution (1 time by
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// default, or specified on the command line with -rp). Once the parallel
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// mesh is defined, the serial mesh can be deleted.
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ParMesh pmesh(MPI_COMM_WORLD, *mesh);
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delete mesh;
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for (int lev = 0; lev < par_ref_levels; lev++)
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{
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pmesh.UniformRefinement();
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}
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// 6. Define a parallel finite element space on the parallel mesh. Here we
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// use the Nedelec finite elements of the specified order.
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FiniteElementCollection *fec = NULL;
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if (dim == 1)
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{
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fec = new ND_R1D_FECollection(order, dim);
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}
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else if (dim == 2)
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{
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fec = new ND_R2D_FECollection(order, dim);
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}
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else
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{
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fec = new ND_FECollection(order, dim);
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}
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ParFiniteElementSpace fespace(&pmesh, fec);
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HYPRE_Int size = fespace.GlobalTrueVSize();
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if (Mpi::Root()) { cout << "Number of H(Curl) unknowns: " << size << endl; }
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// 7. Determine the list of true (i.e. parallel conforming) essential
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// boundary dofs. In this example, the boundary conditions are defined
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// by marking all the boundary attributes from the mesh as essential
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// (Dirichlet) and converting them to a list of true dofs.
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Array<int> ess_tdof_list;
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if (pmesh.bdr_attributes.Size())
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{
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Array<int> ess_bdr(pmesh.bdr_attributes.Max());
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ess_bdr = 1;
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fespace.GetEssentialTrueDofs(ess_bdr, ess_tdof_list);
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}
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// 8. Set up the parallel linear form b(.) which corresponds to the
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// right-hand side of the FEM linear system, which in this case is
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// (f,phi_i) where f is given by the function f_exact and phi_i are the
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// basis functions in the finite element fespace.
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VectorFunctionCoefficient f(3, f_exact);
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ParLinearForm b(&fespace);
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b.AddDomainIntegrator(new VectorFEDomainLFIntegrator(f));
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b.Assemble();
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// 9. Define the solution vector x as a parallel finite element grid function
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// corresponding to fespace. Initialize x by projecting the exact
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// solution. Note that only values from the boundary edges will be used
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// when eliminating the non-homogeneous boundary condition to modify the
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// r.h.s. vector b.
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ParGridFunction sol(&fespace);
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VectorFunctionCoefficient E(3, E_exact);
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VectorFunctionCoefficient CurlE(3, CurlE_exact);
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sol.ProjectCoefficient(E);
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// 10. Set up the parallel bilinear form corresponding to the EM diffusion
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// operator curl muinv curl + sigma I, by adding the curl-curl and the
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// mass domain integrators.
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DenseMatrix sigmaMat(3);
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sigmaMat(0,0) = 2.0; sigmaMat(1,1) = 2.0; sigmaMat(2,2) = 2.0;
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sigmaMat(0,2) = 0.0; sigmaMat(2,0) = 0.0;
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sigmaMat(0,1) = M_SQRT1_2; sigmaMat(1,0) = M_SQRT1_2; // 1/sqrt(2) in cmath
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sigmaMat(1,2) = M_SQRT1_2; sigmaMat(2,1) = M_SQRT1_2;
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ConstantCoefficient muinv(1.0);
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MatrixConstantCoefficient sigma(sigmaMat);
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ParBilinearForm a(&fespace);
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a.AddDomainIntegrator(new CurlCurlIntegrator(muinv));
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a.AddDomainIntegrator(new VectorFEMassIntegrator(sigma));
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// 11. Assemble the parallel bilinear form and the corresponding linear
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// system, applying any necessary transformations such as: parallel
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// assembly, eliminating boundary conditions, applying conforming
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// constraints for non-conforming AMR, etc.
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a.Assemble();
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OperatorPtr A;
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Vector B, X;
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a.FormLinearSystem(ess_tdof_list, sol, b, A, X, B);
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// 12. Solve the system AX=B using PCG with the AMS preconditioner from hypre
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if (use_ams)
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{
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if (Mpi::Root())
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{
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cout << "Size of linear system: "
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<< A.As<HypreParMatrix>()->GetGlobalNumRows() << endl;
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}
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HypreAMS ams(*A.As<HypreParMatrix>(), &fespace);
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HyprePCG pcg(*A.As<HypreParMatrix>());
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pcg.SetTol(1e-12);
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pcg.SetMaxIter(1000);
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pcg.SetPrintLevel(2);
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pcg.SetPreconditioner(ams);
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pcg.Mult(B, X);
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}
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else
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#ifdef MFEM_USE_SUPERLU
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{
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if (Mpi::Root())
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{
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cout << "Size of linear system: "
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<< A.As<HypreParMatrix>()->GetGlobalNumRows() << endl;
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}
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SuperLURowLocMatrix A_SuperLU(*A.As<HypreParMatrix>());
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SuperLUSolver AInv(MPI_COMM_WORLD);
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AInv.SetOperator(A_SuperLU);
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AInv.Mult(B,X);
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}
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#else
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{
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if (Mpi::Root()) { cout << "No solvers available." << endl; }
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return 1;
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}
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#endif
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// 13. Recover the parallel grid function corresponding to X. This is the
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// local finite element solution on each processor.
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a.RecoverFEMSolution(X, b, sol);
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// 14. Compute and print the H(Curl) norm of the error.
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{
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real_t error = sol.ComputeHCurlError(&E, &CurlE);
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if (Mpi::Root())
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{
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cout << "\n|| E_h - E ||_{H(Curl)} = " << error << '\n' << endl;
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}
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}
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// 15. Save the refined mesh and the solution in parallel. This output can
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// be viewed later using GLVis: "glvis -np <np> -m mesh -g sol".
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{
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ostringstream mesh_name, sol_name;
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mesh_name << "mesh." << setfill('0') << setw(6) << myid;
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sol_name << "sol." << setfill('0') << setw(6) << myid;
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ofstream mesh_ofs(mesh_name.str().c_str());
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mesh_ofs.precision(8);
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pmesh.Print(mesh_ofs);
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ofstream sol_ofs(sol_name.str().c_str());
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sol_ofs.precision(8);
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sol.Save(sol_ofs);
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}
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// 16. 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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VectorGridFunctionCoefficient solCoef(&sol);
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CurlGridFunctionCoefficient dsolCoef(&sol);
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if (dim ==1)
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{
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socketstream x_sock(vishost, visport);
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socketstream y_sock(vishost, visport);
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socketstream z_sock(vishost, visport);
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socketstream dy_sock(vishost, visport);
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socketstream dz_sock(vishost, visport);
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x_sock.precision(8);
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y_sock.precision(8);
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z_sock.precision(8);
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dy_sock.precision(8);
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dz_sock.precision(8);
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Vector xVec(3); xVec = 0.0; xVec(0) = 1;
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Vector yVec(3); yVec = 0.0; yVec(1) = 1;
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Vector zVec(3); zVec = 0.0; zVec(2) = 1;
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VectorConstantCoefficient xVecCoef(xVec);
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VectorConstantCoefficient yVecCoef(yVec);
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VectorConstantCoefficient zVecCoef(zVec);
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H1_FECollection fec_h1(order, dim);
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L2_FECollection fec_l2(order-1, dim);
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ParFiniteElementSpace fes_h1(&pmesh, &fec_h1);
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ParFiniteElementSpace fes_l2(&pmesh, &fec_l2);
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ParGridFunction xComp(&fes_l2);
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ParGridFunction yComp(&fes_h1);
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ParGridFunction zComp(&fes_h1);
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ParGridFunction dyComp(&fes_l2);
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ParGridFunction dzComp(&fes_l2);
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InnerProductCoefficient xCoef(xVecCoef, solCoef);
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InnerProductCoefficient yCoef(yVecCoef, solCoef);
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InnerProductCoefficient zCoef(zVecCoef, solCoef);
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xComp.ProjectCoefficient(xCoef);
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yComp.ProjectCoefficient(yCoef);
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zComp.ProjectCoefficient(zCoef);
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x_sock << "parallel " << num_procs << " " << myid << "\n"
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<< "solution\n" << pmesh << xComp << flush
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<< "window_title 'X component'" << endl;
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y_sock << "parallel " << num_procs << " " << myid << "\n"
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<< "solution\n" << pmesh << yComp << flush
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<< "window_geometry 403 0 400 350 "
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<< "window_title 'Y component'" << endl;
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z_sock << "parallel " << num_procs << " " << myid << "\n"
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<< "solution\n" << pmesh << zComp << flush
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<< "window_geometry 806 0 400 350 "
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<< "window_title 'Z component'" << endl;
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InnerProductCoefficient dyCoef(yVecCoef, dsolCoef);
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InnerProductCoefficient dzCoef(zVecCoef, dsolCoef);
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dyComp.ProjectCoefficient(dyCoef);
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dzComp.ProjectCoefficient(dzCoef);
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dy_sock << "parallel " << num_procs << " " << myid << "\n"
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<< "solution\n" << pmesh << dyComp << flush
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<< "window_geometry 403 375 400 350 "
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<< "window_title 'Y component of Curl'" << endl;
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dz_sock << "parallel " << num_procs << " " << myid << "\n"
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<< "solution\n" << pmesh << dzComp << flush
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<< "window_geometry 806 375 400 350 "
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<< "window_title 'Z component of Curl'" << endl;
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}
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else if (dim == 2)
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{
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socketstream xy_sock(vishost, visport);
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socketstream z_sock(vishost, visport);
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socketstream dxy_sock(vishost, visport);
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socketstream dz_sock(vishost, visport);
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DenseMatrix xyMat(2,3); xyMat = 0.0;
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xyMat(0,0) = 1.0; xyMat(1,1) = 1.0;
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MatrixConstantCoefficient xyMatCoef(xyMat);
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Vector zVec(3); zVec = 0.0; zVec(2) = 1;
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VectorConstantCoefficient zVecCoef(zVec);
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MatrixVectorProductCoefficient xyCoef(xyMatCoef, solCoef);
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InnerProductCoefficient zCoef(zVecCoef, solCoef);
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H1_FECollection fec_h1(order, dim);
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ND_FECollection fec_nd(order, dim);
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RT_FECollection fec_rt(order-1, dim);
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L2_FECollection fec_l2(order-1, dim);
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ParFiniteElementSpace fes_h1(&pmesh, &fec_h1);
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ParFiniteElementSpace fes_nd(&pmesh, &fec_nd);
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ParFiniteElementSpace fes_rt(&pmesh, &fec_rt);
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ParFiniteElementSpace fes_l2(&pmesh, &fec_l2);
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ParGridFunction xyComp(&fes_nd);
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ParGridFunction zComp(&fes_h1);
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ParGridFunction dxyComp(&fes_rt);
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ParGridFunction dzComp(&fes_l2);
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xyComp.ProjectCoefficient(xyCoef);
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zComp.ProjectCoefficient(zCoef);
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xy_sock << "parallel " << num_procs << " " << myid << "\n";
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xy_sock.precision(8);
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xy_sock << "solution\n" << pmesh << xyComp
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<< "window_title 'XY components'\n" << flush;
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z_sock << "parallel " << num_procs << " " << myid << "\n"
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<< "solution\n" << pmesh << zComp << flush
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<< "window_geometry 403 0 400 350 "
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<< "window_title 'Z component'" << endl;
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MatrixVectorProductCoefficient dxyCoef(xyMatCoef, dsolCoef);
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InnerProductCoefficient dzCoef(zVecCoef, dsolCoef);
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dxyComp.ProjectCoefficient(dxyCoef);
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dzComp.ProjectCoefficient(dzCoef);
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dxy_sock << "parallel " << num_procs << " " << myid << "\n"
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<< "solution\n" << pmesh << dxyComp << flush
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<< "window_geometry 0 375 400 350 "
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<< "window_title 'XY components of Curl'" << endl;
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dz_sock << "parallel " << num_procs << " " << myid << "\n"
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<< "solution\n" << pmesh << dzComp << flush
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<< "window_geometry 403 375 400 350 "
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<< "window_title 'Z component of Curl'" << endl;
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}
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else
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{
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socketstream sol_sock(vishost, visport);
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socketstream dsol_sock(vishost, visport);
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RT_FECollection fec_rt(order-1, dim);
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ParFiniteElementSpace fes_rt(&pmesh, &fec_rt);
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ParGridFunction dsol(&fes_rt);
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dsol.ProjectCoefficient(dsolCoef);
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sol_sock << "parallel " << num_procs << " " << myid << "\n";
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sol_sock.precision(8);
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sol_sock << "solution\n" << pmesh << sol
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<< "window_title 'Solution'" << flush << endl;
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dsol_sock << "parallel " << num_procs << " " << myid << "\n"
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<< "solution\n" << pmesh << dsol << flush
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<< "window_geometry 0 375 400 350 "
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<< "window_title 'Curl of solution'" << endl;
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}
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}
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// 17. Free the used memory.
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delete fec;
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return 0;
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}
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void E_exact(const Vector &x, Vector &E)
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{
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if (dim == 1)
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{
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E(0) = 1.1 * sin(kappa * x(0) + 0.0 * M_PI);
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E(1) = 1.2 * sin(kappa * x(0) + 0.4 * M_PI);
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E(2) = 1.3 * sin(kappa * x(0) + 0.9 * M_PI);
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}
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else if (dim == 2)
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{
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E(0) = 1.1 * sin(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.0 * M_PI);
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E(1) = 1.2 * sin(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.4 * M_PI);
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E(2) = 1.3 * sin(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.9 * M_PI);
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}
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else
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{
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E(0) = 1.1 * sin(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.0 * M_PI);
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E(1) = 1.2 * sin(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.4 * M_PI);
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E(2) = 1.3 * sin(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.9 * M_PI);
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E *= cos(kappa * x(2));
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}
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}
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void CurlE_exact(const Vector &x, Vector &dE)
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{
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if (dim == 1)
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{
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real_t c4 = cos(kappa * x(0) + 0.4 * M_PI);
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real_t c9 = cos(kappa * x(0) + 0.9 * M_PI);
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|
|
|
dE(0) = 0.0;
|
|
dE(1) = -1.3 * c9;
|
|
dE(2) = 1.2 * c4;
|
|
dE *= kappa;
|
|
}
|
|
else if (dim == 2)
|
|
{
|
|
real_t c0 = cos(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.0 * M_PI);
|
|
real_t c4 = cos(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.4 * M_PI);
|
|
real_t c9 = cos(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.9 * M_PI);
|
|
|
|
dE(0) = 1.3 * c9;
|
|
dE(1) = -1.3 * c9;
|
|
dE(2) = 1.2 * c4 - 1.1 * c0;
|
|
dE *= kappa * M_SQRT1_2;
|
|
}
|
|
else
|
|
{
|
|
real_t s0 = sin(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.0 * M_PI);
|
|
real_t c0 = cos(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.0 * M_PI);
|
|
real_t s4 = sin(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.4 * M_PI);
|
|
real_t c4 = cos(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.4 * M_PI);
|
|
real_t c9 = cos(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.9 * M_PI);
|
|
real_t sk = sin(kappa * x(2));
|
|
real_t ck = cos(kappa * x(2));
|
|
|
|
dE(0) = 1.2 * s4 * sk + 1.3 * M_SQRT1_2 * c9 * ck;
|
|
dE(1) = -1.1 * s0 * sk - 1.3 * M_SQRT1_2 * c9 * ck;
|
|
dE(2) = -M_SQRT1_2 * (1.1 * c0 - 1.2 * c4) * ck;
|
|
dE *= kappa;
|
|
}
|
|
}
|
|
|
|
void f_exact(const Vector &x, Vector &f)
|
|
{
|
|
if (dim == 1)
|
|
{
|
|
real_t s0 = sin(kappa * x(0) + 0.0 * M_PI);
|
|
real_t s4 = sin(kappa * x(0) + 0.4 * M_PI);
|
|
real_t s9 = sin(kappa * x(0) + 0.9 * M_PI);
|
|
|
|
f(0) = 2.2 * s0 + 1.2 * M_SQRT1_2 * s4;
|
|
f(1) = 1.2 * (2.0 + kappa * kappa) * s4 +
|
|
M_SQRT1_2 * (1.1 * s0 + 1.3 * s9);
|
|
f(2) = 1.3 * (2.0 + kappa * kappa) * s9 + 1.2 * M_SQRT1_2 * s4;
|
|
}
|
|
else if (dim == 2)
|
|
{
|
|
real_t s0 = sin(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.0 * M_PI);
|
|
real_t s4 = sin(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.4 * M_PI);
|
|
real_t s9 = sin(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.9 * M_PI);
|
|
|
|
f(0) = 0.55 * (4.0 + kappa * kappa) * s0 +
|
|
0.6 * (M_SQRT2 - kappa * kappa) * s4;
|
|
f(1) = 0.55 * (M_SQRT2 - kappa * kappa) * s0 +
|
|
0.6 * (4.0 + kappa * kappa) * s4 +
|
|
0.65 * M_SQRT2 * s9;
|
|
f(2) = 0.6 * M_SQRT2 * s4 + 1.3 * (2.0 + kappa * kappa) * s9;
|
|
}
|
|
else
|
|
{
|
|
real_t s0 = sin(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.0 * M_PI);
|
|
real_t c0 = cos(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.0 * M_PI);
|
|
real_t s4 = sin(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.4 * M_PI);
|
|
real_t c4 = cos(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.4 * M_PI);
|
|
real_t s9 = sin(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.9 * M_PI);
|
|
real_t c9 = cos(kappa * M_SQRT1_2 * (x(0) + x(1)) + 0.9 * M_PI);
|
|
real_t sk = sin(kappa * x(2));
|
|
real_t ck = cos(kappa * x(2));
|
|
|
|
f(0) = 0.55 * (4.0 + 3.0 * kappa * kappa) * s0 * ck +
|
|
0.6 * (M_SQRT2 - kappa * kappa) * s4 * ck -
|
|
0.65 * M_SQRT2 * kappa * kappa * c9 * sk;
|
|
|
|
f(1) = 0.55 * (M_SQRT2 - kappa * kappa) * s0 * ck +
|
|
0.6 * (4.0 + 3.0 * kappa * kappa) * s4 * ck +
|
|
0.65 * M_SQRT2 * s9 * ck -
|
|
0.65 * M_SQRT2 * kappa * kappa * c9 * sk;
|
|
|
|
f(2) = 0.6 * M_SQRT2 * s4 * ck -
|
|
M_SQRT2 * kappa * kappa * (0.55 * c0 + 0.6 * c4) * sk
|
|
+ 1.3 * (2.0 + kappa * kappa) * s9 * ck;
|
|
}
|
|
}
|