629 lines
19 KiB
C++
629 lines
19 KiB
C++
// Copyright (c) 2010-2025, Lawrence Livermore National Security, LLC. Produced
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// at the Lawrence Livermore National Laboratory. All Rights reserved. See files
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// LICENSE and NOTICE for details. LLNL-CODE-806117.
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//
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// This file is part of the MFEM library. For more information and source code
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// availability visit https://mfem.org.
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//
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// MFEM is free software; you can redistribute it and/or modify it under the
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// terms of the BSD-3 license. We welcome feedback and contributions, see file
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// CONTRIBUTING.md for details.
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//
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// MFEM ultraweak DPG parallel example for diffusion
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//
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// Compile with: make pdiffusion
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//
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// Sample runs
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// mpirun -np 4 pdiffusion -m ../../data/inline-quad.mesh -o 3 -sref 1 -pref 2 -theta 0.0 -prob 0
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// mpirun -np 4 pdiffusion -m ../../data/inline-hex.mesh -o 2 -sref 0 -pref 1 -theta 0.0 -prob 0 -sc
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// mpirun -np 4 pdiffusion -m ../../data/beam-tet.mesh -o 3 -sref 0 -pref 2 -theta 0.0 -prob 0 -sc
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// L-shape runs
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// Note: uniform ref are expected to give sub-optimal rate for the L-shape problem (rate = 2/3)
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// mpirun -np 4 pdiffusion -o 2 -sref 1 -pref 5 -theta 0.0 -prob 1
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// L-shape AMR runs
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// mpirun -np 4 pdiffusion -o 1 -sref 1 -pref 10 -theta 0.8 -prob 1
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// mpirun -np 4 pdiffusion -o 2 -sref 1 -pref 8 -theta 0.75 -prob 1 -sc
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// mpirun -np 4 pdiffusion -o 3 -sref 1 -pref 6 -theta 0.75 -prob 1 -sc -do 2
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// Description:
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// This example code demonstrates the use of MFEM to define and solve
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// the "ultraweak" (UW) DPG formulation for the Poisson problem in parallel
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// - Δ u = f, in Ω
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// u = u₀, on ∂Ω
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//
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// It solves two kinds of problems
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// a) A manufactured solution problem where u_exact = sin(π * (x + y + z)).
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// This example computes and prints out convergence rates for the L2 error.
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// b) The L-shape benchmark problem with AMR. The AMR process is driven by the
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// DPG built-in residual indicator.
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// The DPG UW deals with the First Order System
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// ∇ u - σ = 0, in Ω
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// - ∇⋅σ = f, in Ω
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// u = u₀, in ∂Ω
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// Ultraweak-DPG is obtained by integration by parts of both equations and the
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// introduction of trace unknowns on the mesh skeleton
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// u ∈ L²(Ω), σ ∈ (L²(Ω))ᵈⁱᵐ
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// û ∈ H^1/2, σ̂ ∈ H^-1/2
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// -(u , ∇⋅τ) + < û, τ⋅n> - (σ , τ) = 0, ∀ τ ∈ H(div,Ω)
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// (σ , ∇ v) - < σ̂, v > = (f,v) ∀ v ∈ H¹(Ω)
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// û = u₀ on ∂Ω
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// Note:
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// û := u and σ̂ := -σ on the mesh skeleton
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// -------------------------------------------------------------
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// | | u | σ | û | σ̂ | RHS |
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// -------------------------------------------------------------
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// | τ | -(u,∇⋅τ) | -(σ,τ) | < û, τ⋅n> | | 0 |
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// | | | | | | |
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// | v | | (σ,∇ v) | | -<σ̂,v> | (f,v) |
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// where (τ,v) ∈ H(div,Ω) × H¹(Ω)
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// For more information see https://doi.org/10.1007/978-3-319-01818-8_6
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#include "mfem.hpp"
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#include "util/pweakform.hpp"
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#include "../common/mfem-common.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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using namespace mfem::common;
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enum prob_type
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{
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manufactured,
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lshape
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};
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static const char *enum_str[] =
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{
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"manufactured",
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"lshape"
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};
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prob_type prob;
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real_t exact_u(const Vector & X);
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void exact_gradu(const Vector & X, Vector &gradu);
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real_t exact_laplacian_u(const Vector & X);
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void exact_sigma(const Vector & X, Vector & sigma);
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real_t exact_hatu(const Vector & X);
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void exact_hatsigma(const Vector & X, Vector & hatsigma);
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real_t f_exact(const Vector & X);
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int main(int argc, char *argv[])
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{
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// 0. Initialize MPI and HYPRE.
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Mpi::Init();
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int myid = Mpi::WorldRank();
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Hypre::Init();
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// 1. Parse command-line options.
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const char *mesh_file = "../../data/inline-quad.mesh";
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int order = 1;
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int delta_order = 1;
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int sref = 0; // initial uniform mesh refinements
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int pref = 0; // parallel mesh refinements for AMR
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int iprob = 0;
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bool static_cond = false;
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real_t theta = 0.7;
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bool visualization = true;
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int visport = 19916;
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bool paraview = 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(&order, "-o", "--order",
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"Finite element order (polynomial degree).");
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args.AddOption(&delta_order, "-do", "--delta_order",
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"Order enrichment for DPG test space.");
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args.AddOption(&sref, "-sref", "--num-serial-refinements",
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"Number of initial serial uniform refinements");
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args.AddOption(&pref, "-pref", "--num-parallel-refinements",
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"Number of AMR refinements");
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args.AddOption(&theta, "-theta", "--theta-factor",
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"Refinement factor (0 indicates uniform refinements) ");
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args.AddOption(&iprob, "-prob", "--problem", "Problem case"
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" 0: manufactured, 1: L-shape");
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args.AddOption(&static_cond, "-sc", "--static-condensation", "-no-sc",
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"--no-static-condensation", "Enable static condensation.");
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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(¶view, "-paraview", "--paraview", "-no-paraview",
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"--no-paraview",
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"Enable or disable ParaView visualization.");
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args.AddOption(&visport, "-p", "--send-port", "Socket for GLVis.");
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args.Parse();
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if (!args.Good())
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{
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if (myid == 0)
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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 (iprob > 1) { iprob = 1; }
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prob = (prob_type)iprob;
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if (prob == prob_type::lshape)
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{
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mesh_file = "../../data/l-shape.mesh";
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}
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if (myid == 0)
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{
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args.PrintOptions(cout);
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}
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Mesh mesh(mesh_file, 1, 1);
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int dim = mesh.Dimension();
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MFEM_VERIFY(dim > 1, "Dimension = 1 is not supported in this example");
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if (prob == prob_type::lshape)
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{
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/** rotate mesh to be consistent with l-shape benchmark problem
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See https://doi.org/10.1016/j.amc.2013.05.068 */
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mesh.EnsureNodes();
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GridFunction *nodes = mesh.GetNodes();
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int size = nodes->Size()/2;
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for (int i = 0; i<size; i++)
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{
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real_t x = (*nodes)[2*i];
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(*nodes)[2*i] = 2*(*nodes)[2*i+1]-1;
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(*nodes)[2*i+1] = -2*x+1;
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}
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}
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for (int i = 0; i<sref; i++)
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{
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mesh.UniformRefinement();
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}
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mesh.EnsureNCMesh();
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ParMesh pmesh(MPI_COMM_WORLD, mesh);
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mesh.Clear();
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// Define spaces
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enum TrialSpace
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{
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u_space = 0,
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sigma_space = 1,
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hatu_space = 2,
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hatsigma_space = 3
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};
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enum TestSpace
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{
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tau_space = 0,
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v_space = 1
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};
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// L2 space for u
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FiniteElementCollection *u_fec = new L2_FECollection(order-1,dim);
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ParFiniteElementSpace *u_fes = new ParFiniteElementSpace(&pmesh,u_fec);
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// Vector L2 space for σ
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FiniteElementCollection *sigma_fec = new L2_FECollection(order-1,dim);
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ParFiniteElementSpace *sigma_fes = new ParFiniteElementSpace(&pmesh,sigma_fec,
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dim);
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// H^1/2 space for û
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FiniteElementCollection * hatu_fec = new H1_Trace_FECollection(order,dim);
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ParFiniteElementSpace *hatu_fes = new ParFiniteElementSpace(&pmesh,hatu_fec);
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// H^-1/2 space for σ̂
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FiniteElementCollection * hatsigma_fec = new RT_Trace_FECollection(order-1,dim);
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ParFiniteElementSpace *hatsigma_fes = new ParFiniteElementSpace(&pmesh,
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hatsigma_fec);
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// testspace fe collections
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int test_order = order+delta_order;
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FiniteElementCollection * tau_fec = new RT_FECollection(test_order-1, dim);
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FiniteElementCollection * v_fec = new H1_FECollection(test_order, dim);
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Array<ParFiniteElementSpace * > trial_fes;
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Array<FiniteElementCollection * > test_fec;
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trial_fes.Append(u_fes);
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trial_fes.Append(sigma_fes);
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trial_fes.Append(hatu_fes);
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trial_fes.Append(hatsigma_fes);
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test_fec.Append(tau_fec);
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test_fec.Append(v_fec);
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// Required coefficients for the weak formulation
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ConstantCoefficient one(1.0);
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ConstantCoefficient negone(-1.0);
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FunctionCoefficient f(f_exact); // rhs for the manufactured solution problem
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// Required coefficients for the exact solutions
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FunctionCoefficient uex(exact_u);
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VectorFunctionCoefficient sigmaex(dim,exact_sigma);
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FunctionCoefficient hatuex(exact_hatu);
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ParDPGWeakForm * a = new ParDPGWeakForm(trial_fes,test_fec);
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a->StoreMatrices(true); // this is needed for estimation of residual
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// -(u,∇⋅τ)
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a->AddTrialIntegrator(new MixedScalarWeakGradientIntegrator(one),
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TrialSpace::u_space,TestSpace::tau_space);
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// -(σ,τ)
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a->AddTrialIntegrator(new TransposeIntegrator(new VectorFEMassIntegrator(
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negone)), TrialSpace::sigma_space, TestSpace::tau_space);
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// (σ,∇ v)
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a->AddTrialIntegrator(new TransposeIntegrator(new GradientIntegrator(one)),
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TrialSpace::sigma_space,TestSpace::v_space);
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// <û,τ⋅n>
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a->AddTrialIntegrator(new NormalTraceIntegrator,
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TrialSpace::hatu_space,TestSpace::tau_space);
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// -<σ̂,v> (sign is included in σ̂)
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a->AddTrialIntegrator(new TraceIntegrator,
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TrialSpace::hatsigma_space, TestSpace::v_space);
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// test integrators (space-induced norm for H(div) × H1)
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// (∇⋅τ,∇⋅δτ)
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a->AddTestIntegrator(new DivDivIntegrator(one),
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TestSpace::tau_space, TestSpace::tau_space);
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// (τ,δτ)
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a->AddTestIntegrator(new VectorFEMassIntegrator(one),
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TestSpace::tau_space, TestSpace::tau_space);
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// (∇v,∇δv)
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a->AddTestIntegrator(new DiffusionIntegrator(one),
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TestSpace::v_space, TestSpace::v_space);
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// (v,δv)
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a->AddTestIntegrator(new MassIntegrator(one),
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TestSpace::v_space, TestSpace::v_space);
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// RHS
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if (prob == prob_type::manufactured)
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{
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a->AddDomainLFIntegrator(new DomainLFIntegrator(f),TestSpace::v_space);
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}
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// GridFunction for Dirichlet bdr data
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ParGridFunction hatu_gf;
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// Visualization streams
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socketstream u_out;
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socketstream sigma_out;
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if (myid == 0)
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{
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std::cout << "\n Ref |"
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<< " Dofs |"
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<< " L2 Error |"
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<< " Rate |"
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<< " Residual |"
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<< " Rate |"
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<< " PCG it |" << endl;
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std::cout << std::string(72,'-') << endl;
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}
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Array<int> elements_to_refine; // for AMR
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real_t err0 = 0.;
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int dof0=0.;
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real_t res0=0.0;
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ParGridFunction u_gf(u_fes);
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ParGridFunction sigma_gf(sigma_fes);
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u_gf = 0.0;
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sigma_gf = 0.0;
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ParaViewDataCollection * paraview_dc = nullptr;
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if (paraview)
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{
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paraview_dc = new ParaViewDataCollection(enum_str[prob], &pmesh);
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paraview_dc->SetPrefixPath("ParaView/Diffusion");
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paraview_dc->SetLevelsOfDetail(order);
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paraview_dc->SetCycle(0);
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paraview_dc->SetDataFormat(VTKFormat::BINARY);
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paraview_dc->SetHighOrderOutput(true);
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paraview_dc->SetTime(0.0); // set the time
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paraview_dc->RegisterField("u",&u_gf);
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paraview_dc->RegisterField("sigma",&sigma_gf);
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}
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if (static_cond) { a->EnableStaticCondensation(); }
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for (int it = 0; it<=pref; it++)
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{
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a->Assemble();
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Array<int> ess_tdof_list;
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Array<int> ess_bdr;
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if (pmesh.bdr_attributes.Size())
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{
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ess_bdr.SetSize(pmesh.bdr_attributes.Max());
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ess_bdr = 1;
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hatu_fes->GetEssentialTrueDofs(ess_bdr, ess_tdof_list);
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}
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// shift the ess_tdofs
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for (int i = 0; i < ess_tdof_list.Size(); i++)
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{
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ess_tdof_list[i] += u_fes->GetTrueVSize() + sigma_fes->GetTrueVSize();
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}
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Array<int> offsets(5);
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offsets[0] = 0;
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offsets[1] = u_fes->GetVSize();
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offsets[2] = sigma_fes->GetVSize();
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offsets[3] = hatu_fes->GetVSize();
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offsets[4] = hatsigma_fes->GetVSize();
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offsets.PartialSum();
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BlockVector x(offsets);
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x = 0.0;
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hatu_gf.MakeRef(hatu_fes,x.GetBlock(2),0);
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hatu_gf.ProjectBdrCoefficient(uex,ess_bdr);
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Vector X,B;
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OperatorPtr Ah;
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a->FormLinearSystem(ess_tdof_list,x,Ah,X,B);
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BlockOperator * A = Ah.As<BlockOperator>();
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BlockDiagonalPreconditioner M(A->RowOffsets());
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M.owns_blocks = 1;
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int skip = 0;
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if (!static_cond)
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{
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HypreBoomerAMG * amg0 = new HypreBoomerAMG((HypreParMatrix &)A->GetBlock(0,0));
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HypreBoomerAMG * amg1 = new HypreBoomerAMG((HypreParMatrix &)A->GetBlock(1,1));
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amg0->SetPrintLevel(0);
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amg1->SetPrintLevel(0);
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M.SetDiagonalBlock(0,amg0);
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M.SetDiagonalBlock(1,amg1);
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skip=2;
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}
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HypreBoomerAMG * amg2 = new HypreBoomerAMG((HypreParMatrix &)A->GetBlock(skip,
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skip));
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amg2->SetPrintLevel(0);
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M.SetDiagonalBlock(skip,amg2);
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HypreSolver * prec;
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if (dim == 2)
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{
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// AMS preconditioner for 2D H(div) (trace) space
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prec = new HypreAMS((HypreParMatrix &)A->GetBlock(skip+1,skip+1), hatsigma_fes);
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}
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else
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{
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// ADS preconditioner for 3D H(div) (trace) space
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prec = new HypreADS((HypreParMatrix &)A->GetBlock(skip+1,skip+1), hatsigma_fes);
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}
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M.SetDiagonalBlock(skip+1,prec);
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CGSolver cg(MPI_COMM_WORLD);
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cg.SetRelTol(1e-12);
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cg.SetMaxIter(2000);
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cg.SetPrintLevel(0);
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cg.SetPreconditioner(M);
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cg.SetOperator(*A);
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cg.Mult(B, X);
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a->RecoverFEMSolution(X,x);
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Vector & residuals = a->ComputeResidual(x);
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real_t residual = residuals.Norml2();
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real_t maxresidual = residuals.Max();
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real_t globalresidual = residual * residual;
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MPI_Allreduce(MPI_IN_PLACE, &maxresidual, 1, MPITypeMap<real_t>::mpi_type,
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MPI_MAX, MPI_COMM_WORLD);
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MPI_Allreduce(MPI_IN_PLACE, &globalresidual, 1,
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MPITypeMap<real_t>::mpi_type, MPI_SUM, MPI_COMM_WORLD);
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globalresidual = sqrt(globalresidual);
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u_gf.MakeRef(u_fes,x.GetBlock(0),0);
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sigma_gf.MakeRef(sigma_fes,x.GetBlock(1),0);
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int dofs = u_fes->GlobalTrueVSize() + sigma_fes->GlobalTrueVSize()
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+ hatu_fes->GlobalTrueVSize() + hatsigma_fes->GlobalTrueVSize();
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real_t u_err = u_gf.ComputeL2Error(uex);
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real_t sigma_err = sigma_gf.ComputeL2Error(sigmaex);
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real_t L2Error = sqrt(u_err*u_err + sigma_err*sigma_err);
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real_t rate_err = (it) ? dim*log(err0/L2Error)/log((real_t)dof0/dofs) : 0.0;
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real_t rate_res = (it) ? dim*log(res0/globalresidual)/log((
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real_t)dof0/dofs) : 0.0;
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err0 = L2Error;
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res0 = globalresidual;
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dof0 = dofs;
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if (myid == 0)
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{
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std::ios oldState(nullptr);
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oldState.copyfmt(std::cout);
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std::cout << std::right << std::setw(5) << it << " | "
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<< std::setw(10) << dof0 << " | "
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<< std::setprecision(3)
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<< std::setw(10) << std::scientific << err0 << " | "
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<< std::setprecision(2)
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<< std::setw(6) << std::fixed << rate_err << " | "
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<< std::setprecision(3)
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||
<< std::setw(10) << std::scientific << res0 << " | "
|
||
<< std::setprecision(2)
|
||
<< std::setw(6) << std::fixed << rate_res << " | "
|
||
<< std::setw(6) << std::fixed << cg.GetNumIterations() << " | "
|
||
<< std::endl;
|
||
std::cout.copyfmt(oldState);
|
||
}
|
||
|
||
if (visualization)
|
||
{
|
||
const char * keys = (it == 0 && dim == 2) ? "jRcm\n" : nullptr;
|
||
char vishost[] = "localhost";
|
||
|
||
VisualizeField(u_out,vishost,visport,u_gf,
|
||
"Numerical u", 0,0,500,500,keys);
|
||
VisualizeField(sigma_out,vishost,visport,sigma_gf,
|
||
"Numerical flux", 500,0,500,500,keys);
|
||
}
|
||
|
||
if (paraview)
|
||
{
|
||
paraview_dc->SetCycle(it);
|
||
paraview_dc->SetTime((real_t)it);
|
||
paraview_dc->Save();
|
||
}
|
||
|
||
if (it == pref) { break; }
|
||
|
||
elements_to_refine.SetSize(0);
|
||
for (int iel = 0; iel<pmesh.GetNE(); iel++)
|
||
{
|
||
if (residuals[iel] >= theta * maxresidual)
|
||
{
|
||
elements_to_refine.Append(iel);
|
||
}
|
||
}
|
||
|
||
pmesh.GeneralRefinement(elements_to_refine);
|
||
|
||
for (int i =0; i<trial_fes.Size(); i++)
|
||
{
|
||
trial_fes[i]->Update(false);
|
||
}
|
||
a->Update();
|
||
}
|
||
|
||
if (paraview)
|
||
{
|
||
delete paraview_dc;
|
||
}
|
||
|
||
delete a;
|
||
delete tau_fec;
|
||
delete v_fec;
|
||
delete hatsigma_fes;
|
||
delete hatsigma_fec;
|
||
delete hatu_fes;
|
||
delete hatu_fec;
|
||
delete sigma_fec;
|
||
delete sigma_fes;
|
||
delete u_fec;
|
||
delete u_fes;
|
||
|
||
return 0;
|
||
}
|
||
|
||
real_t exact_u(const Vector & X)
|
||
{
|
||
switch (prob)
|
||
{
|
||
case prob_type::lshape:
|
||
{
|
||
real_t x = X[0];
|
||
real_t y = X[1];
|
||
real_t r = sqrt(x*x + y*y);
|
||
real_t alpha = 2./3.;
|
||
real_t phi = atan2(y,x);
|
||
if (phi < 0) { phi += 2*M_PI; }
|
||
return pow(r,alpha) * sin(alpha * phi);
|
||
}
|
||
break;
|
||
default:
|
||
{
|
||
real_t alpha = M_PI * (X.Sum());
|
||
return sin(alpha);
|
||
}
|
||
break;
|
||
}
|
||
}
|
||
|
||
void exact_gradu(const Vector & X, Vector & du)
|
||
{
|
||
du.SetSize(X.Size());
|
||
switch (prob)
|
||
{
|
||
case prob_type::lshape:
|
||
{
|
||
real_t x = X[0];
|
||
real_t y = X[1];
|
||
real_t r = sqrt(x*x + y*y);
|
||
real_t alpha = 2./3.;
|
||
real_t phi = atan2(y,x);
|
||
if (phi < 0) { phi += 2*M_PI; }
|
||
|
||
real_t r_x = x/r;
|
||
real_t r_y = y/r;
|
||
real_t phi_x = - y / (r*r);
|
||
real_t phi_y = x / (r*r);
|
||
real_t beta = alpha * pow(r,alpha - 1.);
|
||
du[0] = beta*(r_x * sin(alpha*phi) + r * phi_x * cos(alpha*phi));
|
||
du[1] = beta*(r_y * sin(alpha*phi) + r * phi_y * cos(alpha*phi));
|
||
}
|
||
break;
|
||
default:
|
||
{
|
||
real_t alpha = M_PI * (X.Sum());
|
||
du.SetSize(X.Size());
|
||
for (int i = 0; i<du.Size(); i++)
|
||
{
|
||
du[i] = M_PI * cos(alpha);
|
||
}
|
||
}
|
||
break;
|
||
}
|
||
}
|
||
|
||
real_t exact_laplacian_u(const Vector & X)
|
||
{
|
||
switch (prob)
|
||
{
|
||
case prob_type::manufactured:
|
||
{
|
||
real_t alpha = M_PI * (X.Sum());
|
||
real_t u = sin(alpha);
|
||
return - M_PI*M_PI * u * X.Size();
|
||
}
|
||
break;
|
||
default:
|
||
MFEM_ABORT("Should be unreachable");
|
||
return 1;
|
||
break;
|
||
}
|
||
}
|
||
|
||
void exact_sigma(const Vector & X, Vector & sigma)
|
||
{
|
||
// σ = ∇ u
|
||
exact_gradu(X,sigma);
|
||
}
|
||
|
||
real_t exact_hatu(const Vector & X)
|
||
{
|
||
return exact_u(X);
|
||
}
|
||
|
||
void exact_hatsigma(const Vector & X, Vector & hatsigma)
|
||
{
|
||
exact_sigma(X,hatsigma);
|
||
hatsigma *= -1.;
|
||
}
|
||
|
||
real_t f_exact(const Vector & X)
|
||
{
|
||
MFEM_VERIFY(prob!=prob_type::lshape,
|
||
"f_exact should not be called for l-shape benchmark problem, i.e., f = 0")
|
||
return -exact_laplacian_u(X);
|
||
}
|