Andrew Ho
550ca6b702
something is conflicting with beta, possibly std::beta imported into the global namespace?
635 lines
19 KiB
C++
635 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 example for convection-diffusion
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//
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// Compile with: make convection-diffusion
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//
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// sample runs
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// convection-diffusion -m ../../data/star.mesh -o 2 -ref 2 -theta 0.0 -eps 1e-1 -beta '2 3'
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// convection-diffusion -m ../../data/beam-hex.mesh -o 2 -ref 2 -theta 0.0 -eps 1e0 -beta '1 0 2'
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// convection-diffusion -m ../../data/inline-tri.mesh -o 3 -ref 2 -theta 0.0 -eps 1e-2 -beta '4 2' -sc
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// AMR runs
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// convection-diffusion -o 3 -ref 5 -prob 1 -eps 1e-1 -theta 0.75
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// convection-diffusion -o 2 -ref 9 -prob 1 -eps 1e-2 -theta 0.75
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// convection-diffusion -o 3 -ref 9 -prob 1 -eps 1e-3 -theta 0.75 -sc
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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 convection-diffusion problem
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// - εΔu + ∇⋅(βu) = f, in Ω
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// u = u_0, on ∂Ω
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// It solves the following kinds of problems
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// (a) A manufactured solution where u_exact = sin(π * (x + y + z)).
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// (b) The 2D Erickson-Johnson problem
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// The DPG UW deals with the First Order System
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// - ∇⋅σ + ∇⋅(βu) = f, in Ω
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// 1/ε σ - ∇u = 0, in Ω
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// u = u_0, on ∂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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//
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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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// -(βu , ∇v) + (σ , ∇v) + < f̂ , v > = (f,v), ∀ v ∈ H¹(Ω)
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// (u , ∇⋅τ) + 1/ε (σ , τ) + < û , τ⋅n > = 0, ∀ τ ∈ H(div,Ω)
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// û = u_0 on ∂Ω
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// Note:
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// f̂ := βu - σ, û := -u on the mesh skeleton
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// -------------------------------------------------------------
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// | | u | σ | û | f̂ | RHS |
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// -------------------------------------------------------------
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// | v |-(βu , ∇v) | (σ , ∇v) | | < f̂ ,v > | (f,v) |
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// | | | | | | |
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// | τ | (u ,∇⋅τ) | 1/ε(σ , τ)| <û,τ⋅n> | | 0 |
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// where (v,τ) ∈ H¹(Ωₕ) × H(div,Ωₕ)
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// For more information see https://doi.org/10.1016/j.camwa.2013.06.010
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#include "mfem.hpp"
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#include "util/weakform.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 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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EJ // see https://doi.org/10.1016/j.camwa.2013.06.010
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};
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prob_type prob;
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Vector beta_;
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real_t epsilon;
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real_t exact_u(const Vector & X);
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void exact_gradu(const Vector & X, Vector & du);
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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_hatf(const Vector & X, Vector & hatf);
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real_t f_exact(const Vector & X);
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void setup_test_norm_coeffs(GridFunction & c1_gf, GridFunction & c2_gf);
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int main(int argc, char *argv[])
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{
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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 ref = 1;
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bool visualization = true;
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int iprob = 0;
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real_t theta = 0.0;
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bool static_cond = false;
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int visport = 19916;
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epsilon = 1e0;
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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(&epsilon, "-eps", "--epsilon",
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"Epsilon coefficient");
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args.AddOption(&ref, "-ref", "--num-refinements",
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"Number of uniform refinements");
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args.AddOption(&theta, "-theta", "--theta",
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"Theta parameter for AMR");
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args.AddOption(&iprob, "-prob", "--problem", "Problem case"
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" 0: manufactured, 1: Erickson-Johnson ");
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args.AddOption(&beta_, "-beta", "--beta",
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"Vector Coefficient beta");
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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(&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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args.PrintUsage(std::cout);
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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::EJ)
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{
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mesh_file = "../../data/inline-quad.mesh";
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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 (beta_.Size() == 0)
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{
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beta_.SetSize(dim);
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beta_ = 0.0;
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beta_[0] = 1.;
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}
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args.PrintOptions(std::cout);
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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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hatf_space = 3
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};
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enum TestSpace
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{
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v_space = 0,
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tau_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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FiniteElementSpace *u_fes = new FiniteElementSpace(&mesh,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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FiniteElementSpace *sigma_fes = new FiniteElementSpace(&mesh,sigma_fec, 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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FiniteElementSpace *hatu_fes = new FiniteElementSpace(&mesh,hatu_fec);
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// H^-1/2 space for σ̂
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FiniteElementCollection * hatf_fec = new RT_Trace_FECollection(order-1,dim);
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FiniteElementSpace *hatf_fes = new FiniteElementSpace(&mesh,hatf_fec);
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// testspace fe collections
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int test_order = order+delta_order;
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FiniteElementCollection * v_fec = new H1_FECollection(test_order, dim);
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FiniteElementCollection * tau_fec = new RT_FECollection(test_order-1, dim);
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// Coefficients
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ConstantCoefficient one(1.0);
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ConstantCoefficient negone(-1.0);
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ConstantCoefficient eps(epsilon);
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ConstantCoefficient eps1(1./epsilon);
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ConstantCoefficient negeps1(-1./epsilon);
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ConstantCoefficient eps2(1/(epsilon*epsilon));
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ConstantCoefficient negeps(-epsilon);
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VectorConstantCoefficient betacoeff(beta_);
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Vector negbeta = beta_; negbeta.Neg();
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DenseMatrix bbt(beta_.Size());
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MultVVt(beta_, bbt);
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MatrixConstantCoefficient bbtcoeff(bbt);
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VectorConstantCoefficient negbetacoeff(negbeta);
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Array<FiniteElementSpace * > 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(hatf_fes);
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test_fec.Append(v_fec);
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test_fec.Append(tau_fec);
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FiniteElementCollection *coeff_fec = new L2_FECollection(0,dim);
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FiniteElementSpace *coeff_fes = new FiniteElementSpace(&mesh,coeff_fec);
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GridFunction c1_gf, c2_gf;
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GridFunctionCoefficient c1_coeff(&c1_gf);
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GridFunctionCoefficient c2_coeff(&c2_gf);
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DPGWeakForm * a = new DPGWeakForm(trial_fes,test_fec);
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a->StoreMatrices(true); // needed for residual calculation
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//-(βu , ∇v)
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a->AddTrialIntegrator(new MixedScalarWeakDivergenceIntegrator(betacoeff),
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TrialSpace::u_space, TestSpace::v_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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// (u ,∇⋅τ)
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a->AddTrialIntegrator(new MixedScalarWeakGradientIntegrator(negone),
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TrialSpace::u_space, TestSpace::tau_space);
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// 1/ε (σ,τ)
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a->AddTrialIntegrator(new TransposeIntegrator(new VectorFEMassIntegrator(eps1)),
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TrialSpace::sigma_space, TestSpace::tau_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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// <f̂ ,v>
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a->AddTrialIntegrator(new TraceIntegrator,
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TrialSpace::hatf_space, TestSpace::v_space);
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// mesh dependent test norm
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c1_gf.SetSpace(coeff_fes);
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c2_gf.SetSpace(coeff_fes);
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setup_test_norm_coeffs(c1_gf,c2_gf);
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// c1 (v,δv)
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a->AddTestIntegrator(new MassIntegrator(c1_coeff),
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TestSpace::v_space, TestSpace::v_space);
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// ε (∇v,∇δv)
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a->AddTestIntegrator(new DiffusionIntegrator(eps),
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TestSpace::v_space, TestSpace::v_space);
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// (β⋅∇v, β⋅∇δv)
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a->AddTestIntegrator(new DiffusionIntegrator(bbtcoeff),
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TestSpace::v_space, TestSpace::v_space);
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// c2 (τ,δτ)
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a->AddTestIntegrator(new VectorFEMassIntegrator(c2_coeff),
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TestSpace::tau_space, TestSpace::tau_space);
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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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FunctionCoefficient f(f_exact);
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a->AddDomainLFIntegrator(new DomainLFIntegrator(f),TestSpace::v_space);
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FunctionCoefficient hatuex(exact_hatu);
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VectorFunctionCoefficient hatfex(dim,exact_hatf);
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Array<int> elements_to_refine;
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FunctionCoefficient uex(exact_u);
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VectorFunctionCoefficient sigmaex(dim,exact_sigma);
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GridFunction hatu_gf, hatf_gf;
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socketstream u_out;
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socketstream sigma_out;
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real_t res0 = 0.;
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real_t err0 = 0.;
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int dof0 = 0; // init to suppress gcc warning
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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 |" << std::endl;
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std::cout << std::string(64,'-') << std::endl;
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if (static_cond) { a->EnableStaticCondensation(); }
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for (int it = 0; it<=ref; it++)
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{
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a->Assemble();
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Array<int> ess_tdof_list_uhat;
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Array<int> ess_tdof_list_fhat;
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Array<int> ess_bdr_uhat;
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Array<int> ess_bdr_fhat;
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if (mesh.bdr_attributes.Size())
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{
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ess_bdr_uhat.SetSize(mesh.bdr_attributes.Max());
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ess_bdr_fhat.SetSize(mesh.bdr_attributes.Max());
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ess_bdr_uhat = 1; ess_bdr_fhat = 0;
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if (prob == prob_type::EJ)
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{
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ess_bdr_uhat = 0;
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ess_bdr_fhat = 1;
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ess_bdr_uhat[1] = 1;
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ess_bdr_fhat[1] = 0;
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}
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hatu_fes->GetEssentialTrueDofs(ess_bdr_uhat, ess_tdof_list_uhat);
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hatf_fes->GetEssentialTrueDofs(ess_bdr_fhat, ess_tdof_list_fhat);
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}
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// shift the ess_tdofs
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int n = ess_tdof_list_uhat.Size();
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int m = ess_tdof_list_fhat.Size();
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Array<int> ess_tdof_list(n+m);
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for (int j = 0; j < n; j++)
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{
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ess_tdof_list[j] = ess_tdof_list_uhat[j]
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+ u_fes->GetTrueVSize()
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+ sigma_fes->GetTrueVSize();
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}
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for (int j = 0; j < m; j++)
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{
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ess_tdof_list[j+n] = ess_tdof_list_fhat[j]
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+ u_fes->GetTrueVSize()
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+ sigma_fes->GetTrueVSize()
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+ hatu_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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int dofs = 0;
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for (int i = 0; i<trial_fes.Size(); i++)
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{
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offsets[i+1] = trial_fes[i]->GetVSize();
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dofs += trial_fes[i]->GetTrueVSize();
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}
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offsets.PartialSum();
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BlockVector x(offsets); x = 0.0;
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hatu_gf.MakeRef(hatu_fes,x.GetBlock(2),0);
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hatf_gf.MakeRef(hatf_fes,x.GetBlock(3),0);
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hatu_gf.ProjectBdrCoefficient(hatuex,ess_bdr_uhat);
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hatf_gf.ProjectBdrCoefficientNormal(hatfex,ess_bdr_fhat);
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OperatorPtr Ah;
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Vector X,B;
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a->FormLinearSystem(ess_tdof_list,x,Ah,X,B);
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BlockMatrix * A = Ah.As<BlockMatrix>();
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BlockDiagonalPreconditioner M(A->RowOffsets());
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M.owns_blocks = 1;
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for (int i = 0 ; i < A->NumRowBlocks(); i++)
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{
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M.SetDiagonalBlock(i,new DSmoother(A->GetBlock(i,i)));
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}
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CGSolver cg;
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cg.SetRelTol(1e-8);
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cg.SetMaxIter(20000);
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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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GridFunction u_gf, sigma_gf;
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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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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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Vector & residuals = a->ComputeResidual(x);
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real_t residual = residuals.Norml2();
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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/residual)/log((real_t)dof0/dofs) : 0.0;
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err0 = L2Error;
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res0 = residual;
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dof0 = dofs;
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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 << " | "
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<< std::setprecision(2)
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<< std::setw(6) << std::fixed << rate_res << " | "
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<< std::endl;
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std::cout.copyfmt(oldState);
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if (visualization)
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{
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const char * keys = (it == 0 && dim == 2) ? "jRcm\n" : nullptr;
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char vishost[] = "localhost";
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VisualizeField(u_out,vishost, visport, u_gf,
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"Numerical u", 0,0, 500, 500, keys);
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VisualizeField(sigma_out,vishost, visport, sigma_gf,
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"Numerical flux", 501,0,500, 500, keys);
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}
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if (it == ref)
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{
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break;
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}
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elements_to_refine.SetSize(0);
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real_t max_resid = residuals.Max();
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for (int iel = 0; iel<mesh.GetNE(); iel++)
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{
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if (residuals[iel] > theta * max_resid)
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{
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elements_to_refine.Append(iel);
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}
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}
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mesh.GeneralRefinement(elements_to_refine,1,1);
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for (int i =0; i<trial_fes.Size(); i++)
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{
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trial_fes[i]->Update(false);
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}
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a->Update();
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coeff_fes->Update();
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c1_gf.Update();
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c2_gf.Update();
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setup_test_norm_coeffs(c1_gf,c2_gf);
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}
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delete coeff_fes;
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delete coeff_fec;
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delete a;
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delete tau_fec;
|
||
delete v_fec;
|
||
delete hatf_fes;
|
||
delete hatf_fec;
|
||
delete hatu_fes;
|
||
delete hatu_fec;
|
||
delete sigma_fes;
|
||
delete sigma_fec;
|
||
delete u_fec;
|
||
delete u_fes;
|
||
|
||
return 0;
|
||
}
|
||
|
||
real_t exact_u(const Vector & X)
|
||
{
|
||
real_t x = X[0];
|
||
real_t y = X[1];
|
||
real_t z = 0.;
|
||
if (X.Size() == 3) { z = X[2]; }
|
||
switch (prob)
|
||
{
|
||
case EJ:
|
||
{
|
||
real_t alpha = sqrt(1. + 4. * epsilon * epsilon * M_PI * M_PI);
|
||
real_t r1 = (1. + alpha) / (2.*epsilon);
|
||
real_t r2 = (1. - alpha) / (2.*epsilon);
|
||
real_t denom = exp(-r2) - exp(-r1);
|
||
|
||
real_t g1 = exp(r2*(x-1.));
|
||
real_t g2 = exp(r1*(x-1.));
|
||
real_t g = g1-g2;
|
||
|
||
return g * cos(M_PI * y)/denom;
|
||
}
|
||
break;
|
||
default:
|
||
{
|
||
real_t alpha = M_PI * (x + y + z);
|
||
return sin(alpha);
|
||
}
|
||
break;
|
||
}
|
||
}
|
||
|
||
void exact_gradu(const Vector & X, Vector & du)
|
||
{
|
||
real_t x = X[0];
|
||
real_t y = X[1];
|
||
real_t z = 0.;
|
||
if (X.Size() == 3) { z = X[2]; }
|
||
du.SetSize(X.Size());
|
||
du = 0.;
|
||
switch (prob)
|
||
{
|
||
case EJ:
|
||
{
|
||
real_t alpha = sqrt(1. + 4. * epsilon * epsilon * M_PI * M_PI);
|
||
real_t r1 = (1. + alpha) / (2.*epsilon);
|
||
real_t r2 = (1. - alpha) / (2.*epsilon);
|
||
real_t denom = exp(-r2) - exp(-r1);
|
||
|
||
real_t g1 = exp(r2*(x-1.));
|
||
real_t g1_x = r2*g1;
|
||
real_t g2 = exp(r1*(x-1.));
|
||
real_t g2_x = r1*g2;
|
||
real_t g = g1-g2;
|
||
real_t g_x = g1_x - g2_x;
|
||
|
||
du[0] = g_x * cos(M_PI * y)/denom;
|
||
du[1] = -M_PI * g * sin(M_PI*y)/denom;
|
||
}
|
||
break;
|
||
default:
|
||
{
|
||
real_t alpha = M_PI * (x + y + z);
|
||
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)
|
||
{
|
||
real_t x = X[0];
|
||
real_t y = X[1];
|
||
real_t z = 0.;
|
||
if (X.Size() == 3) { z = X[2]; }
|
||
|
||
switch (prob)
|
||
{
|
||
case EJ:
|
||
{
|
||
real_t alpha = sqrt(1. + 4. * epsilon * epsilon * M_PI * M_PI);
|
||
real_t r1 = (1. + alpha) / (2.*epsilon);
|
||
real_t r2 = (1. - alpha) / (2.*epsilon);
|
||
real_t denom = exp(-r2) - exp(-r1);
|
||
|
||
real_t g1 = exp(r2*(x-1.));
|
||
real_t g1_x = r2*g1;
|
||
real_t g1_xx = r2*g1_x;
|
||
real_t g2 = exp(r1*(x-1.));
|
||
real_t g2_x = r1*g2;
|
||
real_t g2_xx = r1*g2_x;
|
||
real_t g = g1-g2;
|
||
real_t g_xx = g1_xx - g2_xx;
|
||
|
||
real_t u = g * cos(M_PI * y)/denom;
|
||
real_t u_xx = g_xx * cos(M_PI * y)/denom;
|
||
real_t u_yy = -M_PI * M_PI * u;
|
||
return u_xx + u_yy;
|
||
}
|
||
break;
|
||
default:
|
||
{
|
||
real_t alpha = M_PI * (x + y + z);
|
||
real_t u = sin(alpha);
|
||
return -M_PI*M_PI * u * X.Size();
|
||
}
|
||
break;
|
||
}
|
||
}
|
||
|
||
void exact_sigma(const Vector & X, Vector & sigma)
|
||
{
|
||
// σ = ε ∇ u
|
||
exact_gradu(X,sigma);
|
||
sigma *= epsilon;
|
||
}
|
||
|
||
real_t exact_hatu(const Vector & X)
|
||
{
|
||
return -exact_u(X);
|
||
}
|
||
|
||
void exact_hatf(const Vector & X, Vector & hatf)
|
||
{
|
||
Vector sigma;
|
||
exact_sigma(X,sigma);
|
||
real_t u = exact_u(X);
|
||
hatf.SetSize(X.Size());
|
||
for (int i = 0; i<hatf.Size(); i++)
|
||
{
|
||
hatf[i] = beta_[i] * u - sigma[i];
|
||
}
|
||
}
|
||
|
||
real_t f_exact(const Vector & X)
|
||
{
|
||
// f = - εΔu + ∇⋅(βu)
|
||
Vector du;
|
||
exact_gradu(X,du);
|
||
real_t d2u = exact_laplacian_u(X);
|
||
|
||
real_t s = 0;
|
||
for (int i = 0; i<du.Size(); i++)
|
||
{
|
||
s += beta_[i] * du[i];
|
||
}
|
||
return -epsilon * d2u + s;
|
||
}
|
||
|
||
void setup_test_norm_coeffs(GridFunction & c1_gf, GridFunction & c2_gf)
|
||
{
|
||
Array<int> vdofs;
|
||
FiniteElementSpace * fes = c1_gf.FESpace();
|
||
Mesh * mesh = fes->GetMesh();
|
||
for (int i = 0; i < mesh->GetNE(); i++)
|
||
{
|
||
real_t volume = mesh->GetElementVolume(i);
|
||
real_t c1 = std::min(epsilon/volume, (real_t) 1.);
|
||
real_t c2 = std::min(1./epsilon, 1./volume);
|
||
fes->GetElementDofs(i,vdofs);
|
||
c1_gf.SetSubVectorHost(vdofs,c1);
|
||
c2_gf.SetSubVectorHost(vdofs,c2);
|
||
}
|
||
}
|