404 lines
11 KiB
C++
404 lines
11 KiB
C++
// MFEM Ultraweak DPG example
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//
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// Compile with: make uw_dpg
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//
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// sample runs
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// ./uw_dpg -m ../lshape2.mesh -o 2 -ref 20 -graph-norm -do 1 -prob 0
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// - Δ u = f, in Ω
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// u = u_0, on ∂Ω
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// 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 = 0, in ∂Ω
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// UW-DPG:
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//
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// u ∈ L^2(Ω), σ ∈ (L^2(Ω))^dim
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// û ∈ H^1/2, σ̂ ∈ H^-1/2
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// -(u , ∇⋅τ) - (σ , τ) + < û, τ⋅n> = 0, ∀ τ ∈ H(div,Ω)
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// (σ , ∇ v) + < σ̂, v > = (f,v) ∀ v ∈ H^1(Ω)
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// û = 0 on ∂Ω
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// Note:
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// û := u
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// σ̂ := -σ
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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) × H^1(Ω)
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#include "mfem.hpp"
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#include <fstream>
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#include <iostream>
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using namespace std;
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using namespace mfem;
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enum prob_type
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{
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lshape,
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general
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};
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prob_type prob;
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void solution(const Vector & X, double & u, Vector & du, double & d2u);
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double exact_u(const Vector & X)
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{
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double u, d2u;
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Vector du;
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solution(X,u,du,d2u);
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return u;
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}
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void exact_sigma(const Vector & X, Vector & sigma)
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{
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double u, d2u;
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Vector du;
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solution(X,u,du,d2u);
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// σ = ∇ u
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sigma = du;
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}
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double exact_hatu(const Vector & X)
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{
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return exact_u(X);
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}
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void exact_hatsigma(const Vector & X, Vector & hatsigma)
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{
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exact_sigma(X,hatsigma);
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hatsigma *= -1.;
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}
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double f_exact(const Vector & X)
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{
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double u, d2u;
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Vector du;
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solution(X,u,du,d2u);
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return -d2u;
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}
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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 adjoint_graph_norm = false;
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bool visualization = true;
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int iprob = 0;
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bool static_cond = 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(&ref, "-ref", "--num_refinements",
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"Number of uniform refinements");
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args.AddOption(&adjoint_graph_norm, "-graph-norm", "--adjoint-graph-norm",
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"-no-graph-norm", "--no-adjoint-graph-norm",
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"Enable or disable Adjoint Graph Norm on the test space");
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args.AddOption(&iprob, "-prob", "--problem", "Problem case"
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" 0: lshape, 1: General");
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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.Parse();
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if (!args.Good())
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{
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args.PrintUsage(cout);
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return 1;
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}
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args.PrintOptions(cout);
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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 = "../lshape2.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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mesh.UniformRefinement();
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// Define spaces
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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 * hatsigma_fec = new RT_Trace_FECollection(order-1,dim);
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FiniteElementSpace *hatsigma_fes = new FiniteElementSpace(&mesh,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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// Coefficients
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ConstantCoefficient one(1.0);
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ConstantCoefficient negone(-1.0);
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// Normal equation weak formulation
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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(hatsigma_fes);
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test_fec.Append(tau_fec);
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test_fec.Append(v_fec);
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NormalEquations * a = new NormalEquations(trial_fes,test_fec);
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a->StoreMatrices(true);
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// -(u,∇⋅τ)
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a->AddTrialIntegrator(new MixedScalarWeakGradientIntegrator(one),0,0);
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// -(σ,τ)
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a->AddTrialIntegrator(new TransposeIntegrator(new VectorFEMassIntegrator(negone)),1,0);
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// (σ,∇ v)
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a->AddTrialIntegrator(new TransposeIntegrator(new GradientIntegrator(one)),1,1);
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// <û,τ⋅n>
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a->AddTrialIntegrator(new NormalTraceIntegrator,2,0);
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// <σ̂,v>
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a->AddTrialIntegrator(new TraceIntegrator,3,1);
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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),0,0);
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// (τ,δτ)
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a->AddTestIntegrator(new VectorFEMassIntegrator(one),0,0);
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// (∇v,∇δv)
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a->AddTestIntegrator(new DiffusionIntegrator(one),1,1);
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// (v,δv)
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a->AddTestIntegrator(new MassIntegrator(one),1,1);
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// additional terms for adjoint graph norm
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if (adjoint_graph_norm)
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{
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// -(∇v,δτ)
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a->AddTestIntegrator(new MixedVectorGradientIntegrator(negone),1,0);
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// -(τ,∇δv)
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a->AddTestIntegrator(new MixedVectorWeakDivergenceIntegrator(one),0,1);
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// (τ,δτ)
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a->AddTestIntegrator(new VectorFEMassIntegrator(one),0,0);
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}
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// RHS
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FunctionCoefficient f(f_exact);
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if (prob == prob_type::general)
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{
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a->AddDomainLFIntegrator(new DomainLFIntegrator(f),1);
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}
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FunctionCoefficient hatuex(exact_hatu);
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Array<int> elements_to_refine;
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GridFunction hatu_gf;
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socketstream u_out;
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// socketstream sigma_out;
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socketstream mesh_out;
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if (visualization)
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{
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char vishost[] = "localhost";
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int visport = 19916;
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u_out.open(vishost, visport);
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// sigma_out.open(vishost, visport);
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mesh_out.open(vishost, visport);
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}
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if (static_cond) { a->EnableStaticCondensation(); }
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for (int i = 0; i<ref; i++)
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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 (mesh.bdr_attributes.Size())
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{
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ess_bdr.SetSize(mesh.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));
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hatu_gf.ProjectBdrCoefficient(hatuex,ess_bdr);
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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 = new BlockDiagonalPreconditioner(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 UMFPackSolver(A->GetBlock(i,i)));
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}
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CGSolver cg;
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cg.SetRelTol(1e-12);
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cg.SetMaxIter(2000);
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cg.SetPrintLevel(3);
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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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delete M;
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a->RecoverFEMSolution(X,x);
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Vector & residuals = a->ComputeResidual(x);
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double residual = residuals.Norml2();
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cout << "Residual = " << residual << endl;
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elements_to_refine.SetSize(0);
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double max_resid = residuals.Max();
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double theta = 0.7;
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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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GridFunction u_gf;
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u_gf.MakeRef(u_fes,x.GetBlock(0));
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GridFunction sigma_gf;
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sigma_gf.MakeRef(sigma_fes,x.GetBlock(1));
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if (visualization)
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{
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u_out.precision(8);
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string keys = (i == 0) ? "keys em\n" : "keys";
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u_out << "solution\n" << mesh << u_gf
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<< "window_title 'Numerical u' "
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<< flush;
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// sigma_out.precision(8);
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// sigma_out << "solution\n" << mesh << sigma_gf <<
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// "window_title 'Numerical flux' "
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// << flush;
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mesh_out.precision(8);
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mesh_out << "mesh\n" << mesh
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<< keys
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<< "window_title 'Mesh' "
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<< flush;
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}
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mesh.GeneralRefinement(elements_to_refine);
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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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}
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delete a;
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delete tau_fec;
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delete v_fec;
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delete hatsigma_fes;
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delete hatsigma_fec;
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delete hatu_fes;
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delete hatu_fec;
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delete sigma_fec;
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delete u_fec;
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delete u_fes;
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return 0;
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}
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void solution(const Vector & X, double & u, Vector & du, double & d2u)
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{
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double x = X[0];
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double y = X[1];
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double z = 0.;
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if (X.Size() == 3) z = X[2];
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du.SetSize(X.Size());
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du = 0.;
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d2u = 0.;
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switch(prob)
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{
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case lshape:
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{
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double r = sqrt(x*x + y*y);
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double alpha = 2./3.;
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double theta = atan2(y,x);
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if (theta < 0) theta += 2*M_PI;
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u = pow(r,alpha) * sin(alpha * theta);
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}
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break;
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default:
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{
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double alpha = M_PI * (x + y + z);
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u = sin(alpha);
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du.SetSize(X.Size());
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for (int i = 0; i<du.Size(); i++)
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{
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du[i] = M_PI * cos(alpha);
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}
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d2u = - M_PI*M_PI * u * du.Size();
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}
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break;
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}
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}
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