656 lines
20 KiB
C++
656 lines
20 KiB
C++
// MFEM Example 5
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//
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// Compile with: make ex5
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//
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// Sample runs: ex5 -m ../data/square-disc.mesh
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// ex5 -m ../data/star.mesh
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// ex5 -m ../data/star.mesh -pa
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// ex5 -m ../data/beam-tet.mesh
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// ex5 -m ../data/beam-hex.mesh
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// ex5 -m ../data/beam-hex.mesh -pa
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// ex5 -m ../data/escher.mesh
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// ex5 -m ../data/fichera.mesh
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//
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// Device sample runs:
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// ex5 -m ../data/star.mesh -pa -d cuda
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// ex5 -m ../data/star.mesh -pa -d raja-cuda
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// ex5 -m ../data/star.mesh -pa -d raja-omp
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// ex5 -m ../data/beam-hex.mesh -pa -d cuda
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//
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// Description: This example code solves a simple 2D/3D mixed Darcy problem
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// corresponding to the saddle point system
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//
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// k*u + grad p = f
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// - div u = g
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//
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// with natural boundary condition -p = <given pressure>.
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// Here, we use a given exact solution (u,p) and compute the
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// corresponding r.h.s. (f,g). We discretize with Raviart-Thomas
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// finite elements (velocity u) and piecewise discontinuous
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// polynomials (pressure p). Alternatively, the piecewise discontinuous
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// polynomials are used for both quantities.
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//
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// The example demonstrates the use of the DarcyForm class, as
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// well as hybridization of mixed systems and the collective saving
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// of several grid functions in VisIt (visit.llnl.gov) and ParaView
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// (paraview.org) formats.
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//
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// We recommend viewing examples 1-4 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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#include <algorithm>
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using namespace std;
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using namespace mfem;
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// Define the analytical solution and forcing terms / boundary conditions
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void uFun_ex(const Vector & x, Vector & u);
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real_t pFun_ex(const Vector & x);
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void fFun(const Vector & x, Vector & f);
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real_t gFun(const Vector & x);
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real_t f_natural(const Vector & x);
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int main(int argc, char *argv[])
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{
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StopWatch chrono;
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// 1. Initialize MPI and HYPRE.
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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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bool verbose = (myid == 0);
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// 2. Parse command-line options.
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const char *mesh_file = "";
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int nx = 0;
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int ny = 0;
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int ref_levels = -1;
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int order = 1;
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bool dg = false;
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real_t td = 0.5;
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bool hybridization = false;
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bool reduction = false;
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bool pa = false;
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const char *device_config = "cpu";
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bool visualization = 1;
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bool par_format = false;
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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(&ref_levels, "-r", "--refine",
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"Number of times to refine the mesh uniformly.");
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args.AddOption(&nx, "-nx", "--ncells-x",
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"Number of cells in x.");
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args.AddOption(&ny, "-ny", "--ncells-y",
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"Number of cells in y.");
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args.AddOption(&order, "-o", "--order",
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"Finite element order (polynomial degree).");
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args.AddOption(&dg, "-dg", "--discontinuous", "-no-dg",
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"--no-discontinuous", "Enable DG elements for fluxes.");
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args.AddOption(&td, "-td", "--stab_diff",
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"Diffusion stabilization factor (1/2=default)");
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args.AddOption(&hybridization, "-hb", "--hybridization", "-no-hb",
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"--no-hybridization", "Enable hybridization.");
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args.AddOption(&reduction, "-rd", "--reduction", "-no-rd",
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"--no-reduction", "Enable reduction of DG flux.");
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args.AddOption(&pa, "-pa", "--partial-assembly", "-no-pa",
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"--no-partial-assembly", "Enable Partial Assembly.");
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args.AddOption(&device_config, "-d", "--device",
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"Device configuration string, see Device::Configure().");
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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.AddOption(&par_format, "-pf", "--parallel-format", "-sf",
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"--serial-format",
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"Format to use when saving the results for VisIt.");
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args.Parse();
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if (!args.Good())
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{
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if (verbose)
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{
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args.PrintUsage(cout);
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}
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return 1;
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}
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if (verbose)
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{
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args.PrintOptions(cout);
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}
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// 3. Enable hardware devices such as GPUs, and programming models such as
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// CUDA, OCCA, RAJA and OpenMP based on command line options.
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Device device(device_config);
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if (myid == 0) { device.Print(); }
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// 4. 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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if (ny <= 0)
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{
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ny = nx;
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}
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Mesh *mesh = NULL;
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if (strlen(mesh_file) > 0)
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{
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mesh = new Mesh(mesh_file, 1, 1);
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}
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else
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{
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mesh = new Mesh(Mesh::MakeCartesian2D(nx, ny, Element::QUADRILATERAL));
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}
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int dim = mesh->Dimension();
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// 5. 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. We choose
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// 'ref_levels' to be the largest number that gives a final mesh with no
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// more than 10,000 elements, unless the user specifies it as input.
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if (strlen(mesh_file) > 0)
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{
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if (ref_levels == -1)
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{
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ref_levels = (int)floor(log(10000./mesh->GetNE())/log(2.)/dim);
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}
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for (int l = 0; l < ref_levels; l++)
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{
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mesh->UniformRefinement();
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}
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}
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// 6. 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. Once the
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// parallel mesh is defined, the serial mesh can be deleted.
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ParMesh *pmesh = new ParMesh(MPI_COMM_WORLD, *mesh);
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delete mesh;
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/*{
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int par_ref_levels = 2;
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for (int l = 0; l < par_ref_levels; l++)
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{
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pmesh->UniformRefinement();
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}
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}*/
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// 7. Define a finite element space on the mesh. Here we use the
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// Raviart-Thomas finite elements of the specified order.
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FiniteElementCollection *R_coll;
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if (dg)
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{
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// In the case of LDG formulation, we chose a closed basis as it
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// is customary for HDG to match trace DOFs, but an open basis can
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// be used instead.
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R_coll = new L2_FECollection(order, dim, BasisType::GaussLobatto);
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}
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else
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{
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R_coll = new RT_FECollection(order, dim);
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}
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FiniteElementCollection *W_coll = new L2_FECollection(order, dim);
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ParFiniteElementSpace *R_space = new ParFiniteElementSpace(pmesh, R_coll,
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(dg)?(dim):(1));
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ParFiniteElementSpace *W_space = new ParFiniteElementSpace(pmesh, W_coll);
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ParDarcyForm *darcy = new ParDarcyForm(R_space, W_space);
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HYPRE_BigInt dimR = R_space->GlobalTrueVSize();
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HYPRE_BigInt dimW = W_space->GlobalTrueVSize();
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if (verbose)
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{
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std::cout << "***********************************************************\n";
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std::cout << "dim(R) = " << dimR << "\n";
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std::cout << "dim(W) = " << dimW << "\n";
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std::cout << "dim(R+W) = " << dimR + dimR << "\n";
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std::cout << "***********************************************************\n";
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}
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// 8. Define the two BlockStructure of the problem. block_offsets is used
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// for Vector based on dof (like ParGridFunction or ParLinearForm),
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// block_trueOffstes is used for Vector based on trueDof (HypreParVector
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// for the rhs and solution of the linear system). The offsets computed
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// here are local to the processor.
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const Array<int> &block_offsets = darcy->GetOffsets();
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const Array<int> &block_trueOffsets = darcy->GetTrueOffsets();
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// 9. Define the coefficients, analytical solution, and rhs of the PDE.
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const double k = 1.0;
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ConstantCoefficient kcoeff(k); //acoustic resistance
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RatioCoefficient ikcoeff(1., kcoeff); //inverse acoustic resistance
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VectorFunctionCoefficient fcoeff(dim, fFun); //velocity rhs
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FunctionCoefficient fnatcoeff(f_natural); //boundary velocity rhs
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FunctionCoefficient gcoeff(gFun); //pressure rhs
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VectorFunctionCoefficient ucoeff(dim, uFun_ex); //velocity
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FunctionCoefficient pcoeff(pFun_ex); //pressure
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// 10. Define the parallel grid function and parallel linear forms, solution
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// vector and rhs.
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MemoryType mt = device.GetMemoryType();
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BlockVector x(block_offsets, mt), rhs(block_offsets, mt);
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ParLinearForm *fform(new ParLinearForm);
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fform->Update(R_space, rhs.GetBlock(0), 0);
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if (dg)
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{
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fform->AddDomainIntegrator(new VectorDomainLFIntegrator(fcoeff));
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fform->AddBdrFaceIntegrator(new VectorBoundaryFluxLFIntegrator(fnatcoeff));
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}
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else
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{
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fform->AddDomainIntegrator(new VectorFEDomainLFIntegrator(fcoeff));
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fform->AddBoundaryIntegrator(new VectorFEBoundaryFluxLFIntegrator(fnatcoeff));
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}
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fform->Assemble();
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fform->SyncAliasMemory(rhs);
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ParLinearForm *gform(new ParLinearForm);
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gform->Update(W_space, rhs.GetBlock(1), 0);
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gform->AddDomainIntegrator(new DomainLFIntegrator(gcoeff));
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gform->Assemble();
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gform->SyncAliasMemory(rhs);
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// 11. Assemble the finite element matrices for the Darcy operator
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//
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// D = [ M B^T ]
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// [ B 0 ]
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// where:
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//
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// M = \int_\Omega k u_h \cdot v_h d\Omega u_h, v_h \in R_h
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// B = -\int_\Omega \div u_h q_h d\Omega u_h \in R_h, q_h \in W_h
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ParBilinearForm *mVarf = darcy->GetParFluxMassForm();
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ParMixedBilinearForm *bVarf = darcy->GetParFluxDivForm();
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ParBilinearForm *mtVarf = (dg)?(darcy->GetParPotentialMassForm()):(NULL);
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if (dg)
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{
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mVarf->AddDomainIntegrator(new VectorMassIntegrator(kcoeff));
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bVarf->AddDomainIntegrator(new VectorDivergenceIntegrator());
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bVarf->AddInteriorFaceIntegrator(new TransposeIntegrator(
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new DGNormalTraceIntegrator(-1.)));
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mtVarf->AddInteriorFaceIntegrator(new HDGDiffusionIntegrator(ikcoeff, td));
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}
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else
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{
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mVarf->AddDomainIntegrator(new VectorFEMassIntegrator(kcoeff));
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bVarf->AddDomainIntegrator(new VectorFEDivergenceIntegrator);
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}
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//set hybridization / assembly level
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Array<int> ess_flux_tdofs_list;
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FiniteElementCollection *trace_coll = NULL;
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ParFiniteElementSpace *trace_space = NULL;
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chrono.Clear();
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chrono.Start();
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if (hybridization)
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{
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trace_coll = new DG_Interface_FECollection(order, dim);
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trace_space = new ParFiniteElementSpace(pmesh, trace_coll);
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darcy->EnableHybridization(trace_space,
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new NormalTraceJumpIntegrator(),
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ess_flux_tdofs_list);
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}
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else if (reduction && dg)
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{
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darcy->EnableFluxReduction();
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}
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if (pa) { darcy->SetAssemblyLevel(AssemblyLevel::PARTIAL); }
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darcy->Assemble();
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OperatorHandle pDarcyOp;
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Vector X, B;
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x = 0.;
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darcy->FormLinearSystem(ess_flux_tdofs_list, x, rhs,
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pDarcyOp, X, B);
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chrono.Stop();
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if (verbose)
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{
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std::cout << "Assembly took " << chrono.RealTime() << "s.\n";
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}
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int maxIter(1000);
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real_t rtol(1.e-6);
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real_t atol(1.e-10);
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if (hybridization || (reduction && dg))
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{
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// 12. Construct the preconditioner
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HypreBoomerAMG prec;
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// 13. Solve the linear system with GMRES.
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// Check the norm of the unpreconditioned residual.
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chrono.Clear();
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chrono.Start();
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GMRESSolver solver(MPI_COMM_WORLD);
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solver.SetAbsTol(atol);
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solver.SetRelTol(rtol);
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solver.SetMaxIter(maxIter);
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solver.SetPreconditioner(prec);
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solver.SetOperator(*pDarcyOp);
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solver.SetPrintLevel(verbose);
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solver.Mult(B, X);
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darcy->RecoverFEMSolution(X, rhs, x);
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chrono.Stop();
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if (verbose)
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{
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if (solver.GetConverged())
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{
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std::cout << "GMRES converged in " << solver.GetNumIterations()
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<< " iterations with a residual norm of "
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<< solver.GetFinalNorm() << ".\n";
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}
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else
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{
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std::cout << "GMRES did not converge in " << solver.GetNumIterations()
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<< " iterations. Residual norm is " << solver.GetFinalNorm()
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<< ".\n";
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}
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std::cout << "GMRES solver took " << chrono.RealTime() << "s.\n";
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}
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}
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else
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{
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// 12. Construct the operators for preconditioner
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//
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// P = [ diag(M) 0 ]
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// [ 0 B diag(M)^-1 B^T ]
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//
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// Here we use Symmetric Gauss-Seidel to approximate the inverse of the
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// pressure Schur Complement
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HypreParMatrix *MinvBt = NULL;
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HypreParVector *Md = NULL;
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HypreParMatrix *S = NULL;
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Solver *invM, *invS;
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if (pa)
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{
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Vector Md(R_space->GetTrueVSize());
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mVarf->AssembleDiagonal(Md);
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auto Md_host = Md.HostRead();
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Vector invMd(Md.Size());
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for (int i=0; i<Md.Size(); ++i)
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{
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invMd(i) = 1.0 / Md_host[i];
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}
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Vector BMBt_diag(W_space->GetTrueVSize());
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bVarf->AssembleDiagonal_ADAt(invMd, BMBt_diag);
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Array<int> ess_tdof_list; // empty
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invM = new OperatorJacobiSmoother(Md, ess_tdof_list);
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invS = new OperatorJacobiSmoother(BMBt_diag, ess_tdof_list);
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}
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else
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{
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HypreParMatrix &M = *mVarf->ParallelAssembleInternal();
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Md = new HypreParVector(MPI_COMM_WORLD, M.GetGlobalNumRows(),
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M.GetRowStarts());
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M.GetDiag(*Md);
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HypreParMatrix &B = *bVarf->ParallelAssembleInternal();
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MinvBt = B.Transpose();
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MinvBt->InvScaleRows(*Md);
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S = ParMult(&B, MinvBt);
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if (mtVarf)
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{
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HypreParMatrix &Mt = *mtVarf->ParallelAssembleInternal();
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HypreParMatrix *Snew = ParAdd(&Mt, S);
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delete S;
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S = Snew;
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}
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invM = new HypreDiagScale(M);
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invS = new HypreBoomerAMG(*S);
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}
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invM->iterative_mode = false;
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invS->iterative_mode = false;
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BlockDiagonalPreconditioner darcyPrec(block_trueOffsets);
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darcyPrec.SetDiagonalBlock(0, invM);
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darcyPrec.SetDiagonalBlock(1, invS);
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// 13. Solve the linear system with MINRES.
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// Check the norm of the unpreconditioned residual.
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chrono.Clear();
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chrono.Start();
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MINRESSolver solver(MPI_COMM_WORLD);
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solver.SetAbsTol(atol);
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solver.SetRelTol(rtol);
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solver.SetMaxIter(maxIter);
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solver.SetOperator(*pDarcyOp);
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solver.SetPreconditioner(darcyPrec);
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solver.SetPrintLevel(verbose);
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solver.Mult(B, X);
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darcy->RecoverFEMSolution(X, rhs, x);
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if (device.IsEnabled()) { x.HostRead(); }
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chrono.Stop();
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if (verbose)
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{
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if (solver.GetConverged())
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{
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std::cout << "MINRES converged in " << solver.GetNumIterations()
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<< " iterations with a residual norm of "
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<< solver.GetFinalNorm() << ".\n";
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}
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else
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{
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std::cout << "MINRES did not converge in " << solver.GetNumIterations()
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<< " iterations. Residual norm is " << solver.GetFinalNorm()
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<< ".\n";
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}
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std::cout << "MINRES solver took " << chrono.RealTime() << "s.\n";
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}
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delete invM;
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delete invS;
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delete S;
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delete Md;
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delete MinvBt;
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}
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// 14. Extract the parallel grid function corresponding to the finite element
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// approximation X. This is the local solution on each processor. Compute
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// L2 error norms.
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ParGridFunction u, p;
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u.MakeRef(R_space, x.GetBlock(0), 0);
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p.MakeRef(W_space, x.GetBlock(1), 0);
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int order_quad = max(2, 2*order+1);
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const IntegrationRule *irs[Geometry::NumGeom];
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for (int i=0; i < Geometry::NumGeom; ++i)
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{
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irs[i] = &(IntRules.Get(i, order_quad));
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}
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real_t err_u = u.ComputeL2Error(ucoeff, irs);
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real_t norm_u = ComputeGlobalLpNorm(2., ucoeff, *pmesh, irs);
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real_t err_p = p.ComputeL2Error(pcoeff, irs);
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real_t norm_p = ComputeGlobalLpNorm(2., pcoeff, *pmesh, irs);
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if (verbose)
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{
|
|
std::cout << "|| u_h - u_ex || / || u_ex || = " << err_u / norm_u << "\n";
|
|
std::cout << "|| p_h - p_ex || / || p_ex || = " << err_p / norm_p << "\n";
|
|
}
|
|
|
|
// 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, u_name, p_name;
|
|
mesh_name << "mesh." << setfill('0') << setw(6) << myid;
|
|
u_name << "sol_u." << setfill('0') << setw(6) << myid;
|
|
p_name << "sol_p." << setfill('0') << setw(6) << myid;
|
|
|
|
ofstream mesh_ofs(mesh_name.str().c_str());
|
|
mesh_ofs.precision(8);
|
|
pmesh->Print(mesh_ofs);
|
|
|
|
ofstream u_ofs(u_name.str().c_str());
|
|
u_ofs.precision(8);
|
|
u.Save(u_ofs);
|
|
|
|
ofstream p_ofs(p_name.str().c_str());
|
|
p_ofs.precision(8);
|
|
p.Save(p_ofs);
|
|
}
|
|
|
|
// 16. Save data in the VisIt format
|
|
VisItDataCollection visit_dc("Example5-Parallel", pmesh);
|
|
visit_dc.RegisterField("velocity", &u);
|
|
visit_dc.RegisterField("pressure", &p);
|
|
visit_dc.SetFormat(!par_format ?
|
|
DataCollection::SERIAL_FORMAT :
|
|
DataCollection::PARALLEL_FORMAT);
|
|
visit_dc.Save();
|
|
|
|
// 17. Save data in the ParaView format
|
|
ParaViewDataCollection paraview_dc("Example5P", pmesh);
|
|
paraview_dc.SetPrefixPath("ParaView");
|
|
paraview_dc.SetLevelsOfDetail(order);
|
|
paraview_dc.SetDataFormat(VTKFormat::BINARY);
|
|
paraview_dc.SetHighOrderOutput(true);
|
|
paraview_dc.SetCycle(0);
|
|
paraview_dc.SetTime(0.0); // set the time
|
|
paraview_dc.RegisterField("velocity",&u);
|
|
paraview_dc.RegisterField("pressure",&p);
|
|
paraview_dc.Save();
|
|
|
|
// 18. Optionally output a BP (binary pack) file using ADIOS2. This can be
|
|
// visualized with the ParaView VTX reader.
|
|
#ifdef MFEM_USE_ADIOS2
|
|
if (adios2)
|
|
{
|
|
std::string postfix(mesh_file);
|
|
postfix.erase(0, std::string("../data/").size() );
|
|
postfix += "_o" + std::to_string(order);
|
|
const std::string collection_name = "ex5-p_" + postfix + ".bp";
|
|
|
|
ADIOS2DataCollection adios2_dc(MPI_COMM_WORLD, collection_name, pmesh);
|
|
adios2_dc.SetLevelsOfDetail(1);
|
|
adios2_dc.SetCycle(1);
|
|
adios2_dc.SetTime(0.0);
|
|
adios2_dc.RegisterField("velocity",&u);
|
|
adios2_dc.RegisterField("pressure",&p);
|
|
adios2_dc.Save();
|
|
}
|
|
#endif
|
|
|
|
// 19. Send the solution by socket to a GLVis server.
|
|
if (visualization)
|
|
{
|
|
char vishost[] = "localhost";
|
|
int visport = 19916;
|
|
socketstream u_sock(vishost, visport);
|
|
u_sock << "parallel " << num_procs << " " << myid << "\n";
|
|
u_sock.precision(8);
|
|
u_sock << "solution\n" << *pmesh << u << "window_title 'Velocity'"
|
|
<< endl;
|
|
u_sock << "keys Rljvvvvvmmc" << endl;
|
|
// Make sure all ranks have sent their 'u' solution before initiating
|
|
// another set of GLVis connections (one from each rank):
|
|
MPI_Barrier(pmesh->GetComm());
|
|
socketstream p_sock(vishost, visport);
|
|
p_sock << "parallel " << num_procs << " " << myid << "\n";
|
|
p_sock.precision(8);
|
|
p_sock << "solution\n" << *pmesh << p << "window_title 'Pressure'"
|
|
<< endl;
|
|
p_sock << "keys Rljmmc" << endl;
|
|
}
|
|
|
|
// 20. Free the used memory.
|
|
delete fform;
|
|
delete gform;
|
|
delete darcy;
|
|
delete W_space;
|
|
delete R_space;
|
|
delete trace_space;
|
|
delete W_coll;
|
|
delete R_coll;
|
|
delete trace_coll;
|
|
delete pmesh;
|
|
|
|
return 0;
|
|
}
|
|
|
|
|
|
void uFun_ex(const Vector & x, Vector & u)
|
|
{
|
|
real_t xi(x(0));
|
|
real_t yi(x(1));
|
|
real_t zi(0.0);
|
|
if (x.Size() == 3)
|
|
{
|
|
zi = x(2);
|
|
}
|
|
|
|
u(0) = - exp(xi)*sin(yi)*cos(zi);
|
|
u(1) = - exp(xi)*cos(yi)*cos(zi);
|
|
|
|
if (x.Size() == 3)
|
|
{
|
|
u(2) = exp(xi)*sin(yi)*sin(zi);
|
|
}
|
|
}
|
|
|
|
// Change if needed
|
|
real_t pFun_ex(const Vector & x)
|
|
{
|
|
real_t xi(x(0));
|
|
real_t yi(x(1));
|
|
real_t zi(0.0);
|
|
|
|
if (x.Size() == 3)
|
|
{
|
|
zi = x(2);
|
|
}
|
|
|
|
return exp(xi)*sin(yi)*cos(zi);
|
|
}
|
|
|
|
void fFun(const Vector & x, Vector & f)
|
|
{
|
|
f = 0.0;
|
|
}
|
|
|
|
real_t gFun(const Vector & x)
|
|
{
|
|
if (x.Size() == 3)
|
|
{
|
|
return -pFun_ex(x);
|
|
}
|
|
else
|
|
{
|
|
return 0;
|
|
}
|
|
}
|
|
|
|
real_t f_natural(const Vector & x)
|
|
{
|
|
return (-pFun_ex(x));
|
|
}
|