628 lines
19 KiB
C++
628 lines
19 KiB
C++
// Copyright (c) 2010-2023, Lawrence Livermore National Security, LLC. Produced
|
||
// at the Lawrence Livermore National Laboratory. All Rights reserved. See files
|
||
// LICENSE and NOTICE for details. LLNL-CODE-806117.
|
||
//
|
||
// This file is part of the MFEM library. For more information and source code
|
||
// availability visit https://mfem.org.
|
||
//
|
||
// MFEM is free software; you can redistribute it and/or modify it under the
|
||
// terms of the BSD-3 license. We welcome feedback and contributions, see file
|
||
// CONTRIBUTING.md for details.
|
||
//
|
||
// MFEM ultraweak DPG parallel example for diffusion
|
||
//
|
||
// Compile with: make pdiffusion
|
||
//
|
||
// Sample runs
|
||
// mpirun -np 4 pdiffusion -m ../../data/inline-quad.mesh -o 3 -sref 1 -pref 2 -theta 0.0 -prob 0
|
||
// mpirun -np 4 pdiffusion -m ../../data/inline-hex.mesh -o 2 -sref 0 -pref 1 -theta 0.0 -prob 0 -sc
|
||
// mpirun -np 4 pdiffusion -m ../../data/beam-tet.mesh -o 3 -sref 0 -pref 2 -theta 0.0 -prob 0 -sc
|
||
|
||
// L-shape runs
|
||
// Note: uniform ref are expected to give sub-optimal rate for the L-shape problem (rate = 2/3)
|
||
// mpirun -np 4 pdiffusion -o 2 -sref 1 -pref 5 -theta 0.0 -prob 1
|
||
|
||
// L-shape AMR runs
|
||
// mpirun -np 4 pdiffusion -o 1 -sref 1 -pref 20 -theta 0.8 -prob 1
|
||
// mpirun -np 4 pdiffusion -o 2 -sref 1 -pref 20 -theta 0.75 -prob 1 -sc
|
||
// mpirun -np 4 pdiffusion -o 3 -sref 1 -pref 20 -theta 0.75 -prob 1 -sc -do 2
|
||
|
||
// Description:
|
||
// This example code demonstrates the use of MFEM to define and solve
|
||
// the "ultraweak" (UW) DPG formulation for the Poisson problem in parallel
|
||
|
||
// - Δ u = f, in Ω
|
||
// u = u₀, on ∂Ω
|
||
//
|
||
// It solves two kinds of problems
|
||
// a) A manufactured solution problem where u_exact = sin(π * (x + y + z)).
|
||
// This example computes and prints out convergence rates for the L2 error.
|
||
// b) The L-shape benchmark problem with AMR. The AMR process is driven by the
|
||
// DPG built-in residual indicator.
|
||
|
||
// The DPG UW deals with the First Order System
|
||
// ∇ u - σ = 0, in Ω
|
||
// - ∇⋅σ = f, in Ω
|
||
// u = u₀, in ∂Ω
|
||
|
||
// Ultraweak-DPG is obtained by integration by parts of both equations and the
|
||
// introduction of trace unknowns on the mesh skeleton
|
||
|
||
// u ∈ L²(Ω), σ ∈ (L²(Ω))ᵈⁱᵐ
|
||
// û ∈ H^1/2, σ̂ ∈ H^-1/2
|
||
// -(u , ∇⋅τ) + < û, τ⋅n> - (σ , τ) = 0, ∀ τ ∈ H(div,Ω)
|
||
// (σ , ∇ v) - < σ̂, v > = (f,v) ∀ v ∈ H¹(Ω)
|
||
// û = u₀ on ∂Ω
|
||
|
||
// Note:
|
||
// û := u and σ̂ := -σ on the mesh skeleton
|
||
|
||
// -------------------------------------------------------------
|
||
// | | u | σ | û | σ̂ | RHS |
|
||
// -------------------------------------------------------------
|
||
// | τ | -(u,∇⋅τ) | -(σ,τ) | < û, τ⋅n> | | 0 |
|
||
// | | | | | | |
|
||
// | v | | (σ,∇ v) | | -<σ̂,v> | (f,v) |
|
||
|
||
// where (τ,v) ∈ H(div,Ω) × H¹(Ω)
|
||
|
||
// For more information see https://doi.org/10.1007/978-3-319-01818-8_6
|
||
|
||
#include "mfem.hpp"
|
||
#include "util/pweakform.hpp"
|
||
#include "../common/mfem-common.hpp"
|
||
#include <fstream>
|
||
#include <iostream>
|
||
|
||
using namespace std;
|
||
using namespace mfem;
|
||
using namespace mfem::common;
|
||
|
||
enum prob_type
|
||
{
|
||
manufactured,
|
||
lshape
|
||
};
|
||
|
||
static const char *enum_str[] =
|
||
{
|
||
"manufactured",
|
||
"lshape"
|
||
};
|
||
|
||
prob_type prob;
|
||
|
||
fptype exact_u(const Vector & X);
|
||
void exact_gradu(const Vector & X, Vector &gradu);
|
||
fptype exact_laplacian_u(const Vector & X);
|
||
void exact_sigma(const Vector & X, Vector & sigma);
|
||
fptype exact_hatu(const Vector & X);
|
||
void exact_hatsigma(const Vector & X, Vector & hatsigma);
|
||
fptype f_exact(const Vector & X);
|
||
|
||
int main(int argc, char *argv[])
|
||
{
|
||
// 0. Initialize MPI and HYPRE.
|
||
Mpi::Init();
|
||
int myid = Mpi::WorldRank();
|
||
Hypre::Init();
|
||
|
||
// 1. Parse command-line options.
|
||
const char *mesh_file = "../../data/inline-quad.mesh";
|
||
int order = 1;
|
||
int delta_order = 1;
|
||
int sref = 0; // initial uniform mesh refinements
|
||
int pref = 0; // parallel mesh refinements for AMR
|
||
int iprob = 0;
|
||
bool static_cond = false;
|
||
fptype theta = 0.7;
|
||
bool visualization = true;
|
||
bool paraview = false;
|
||
|
||
OptionsParser args(argc, argv);
|
||
args.AddOption(&mesh_file, "-m", "--mesh",
|
||
"Mesh file to use.");
|
||
args.AddOption(&order, "-o", "--order",
|
||
"Finite element order (polynomial degree).");
|
||
args.AddOption(&delta_order, "-do", "--delta_order",
|
||
"Order enrichment for DPG test space.");
|
||
args.AddOption(&sref, "-sref", "--num-serial-refinements",
|
||
"Number of initial serial uniform refinements");
|
||
args.AddOption(&pref, "-pref", "--num-parallel-refinements",
|
||
"Number of AMR refinements");
|
||
args.AddOption(&theta, "-theta", "--theta-factor",
|
||
"Refinement factor (0 indicates uniform refinements) ");
|
||
args.AddOption(&iprob, "-prob", "--problem", "Problem case"
|
||
" 0: manufactured, 1: L-shape");
|
||
args.AddOption(&static_cond, "-sc", "--static-condensation", "-no-sc",
|
||
"--no-static-condensation", "Enable static condensation.");
|
||
args.AddOption(&visualization, "-vis", "--visualization", "-no-vis",
|
||
"--no-visualization",
|
||
"Enable or disable GLVis visualization.");
|
||
args.AddOption(¶view, "-paraview", "--paraview", "-no-paraview",
|
||
"--no-paraview",
|
||
"Enable or disable ParaView visualization.");
|
||
args.Parse();
|
||
if (!args.Good())
|
||
{
|
||
if (myid == 0)
|
||
{
|
||
args.PrintUsage(cout);
|
||
}
|
||
return 1;
|
||
}
|
||
|
||
if (iprob > 1) { iprob = 1; }
|
||
prob = (prob_type)iprob;
|
||
|
||
if (prob == prob_type::lshape)
|
||
{
|
||
mesh_file = "../../data/l-shape.mesh";
|
||
}
|
||
|
||
if (myid == 0)
|
||
{
|
||
args.PrintOptions(cout);
|
||
}
|
||
|
||
Mesh mesh(mesh_file, 1, 1);
|
||
int dim = mesh.Dimension();
|
||
MFEM_VERIFY(dim > 1, "Dimension = 1 is not supported in this example");
|
||
|
||
if (prob == prob_type::lshape)
|
||
{
|
||
/** rotate mesh to be consistent with l-shape benchmark problem
|
||
See https://doi.org/10.1016/j.amc.2013.05.068 */
|
||
mesh.EnsureNodes();
|
||
GridFunction *nodes = mesh.GetNodes();
|
||
int size = nodes->Size()/2;
|
||
for (int i = 0; i<size; i++)
|
||
{
|
||
fptype x = (*nodes)[2*i];
|
||
(*nodes)[2*i] = 2*(*nodes)[2*i+1]-1;
|
||
(*nodes)[2*i+1] = -2*x+1;
|
||
}
|
||
}
|
||
|
||
|
||
for (int i = 0; i<sref; i++)
|
||
{
|
||
mesh.UniformRefinement();
|
||
}
|
||
|
||
mesh.EnsureNCMesh();
|
||
|
||
ParMesh pmesh(MPI_COMM_WORLD, mesh);
|
||
mesh.Clear();
|
||
|
||
// Define spaces
|
||
enum TrialSpace
|
||
{
|
||
u_space = 0,
|
||
sigma_space = 1,
|
||
hatu_space = 2,
|
||
hatsigma_space = 3
|
||
};
|
||
enum TestSpace
|
||
{
|
||
tau_space = 0,
|
||
v_space = 1
|
||
};
|
||
// L2 space for u
|
||
FiniteElementCollection *u_fec = new L2_FECollection(order-1,dim);
|
||
ParFiniteElementSpace *u_fes = new ParFiniteElementSpace(&pmesh,u_fec);
|
||
|
||
// Vector L2 space for σ
|
||
FiniteElementCollection *sigma_fec = new L2_FECollection(order-1,dim);
|
||
ParFiniteElementSpace *sigma_fes = new ParFiniteElementSpace(&pmesh,sigma_fec,
|
||
dim);
|
||
|
||
// H^1/2 space for û
|
||
FiniteElementCollection * hatu_fec = new H1_Trace_FECollection(order,dim);
|
||
ParFiniteElementSpace *hatu_fes = new ParFiniteElementSpace(&pmesh,hatu_fec);
|
||
|
||
// H^-1/2 space for σ̂
|
||
FiniteElementCollection * hatsigma_fec = new RT_Trace_FECollection(order-1,dim);
|
||
ParFiniteElementSpace *hatsigma_fes = new ParFiniteElementSpace(&pmesh,
|
||
hatsigma_fec);
|
||
|
||
// testspace fe collections
|
||
int test_order = order+delta_order;
|
||
FiniteElementCollection * tau_fec = new RT_FECollection(test_order-1, dim);
|
||
FiniteElementCollection * v_fec = new H1_FECollection(test_order, dim);
|
||
|
||
Array<ParFiniteElementSpace * > trial_fes;
|
||
Array<FiniteElementCollection * > test_fec;
|
||
|
||
trial_fes.Append(u_fes);
|
||
trial_fes.Append(sigma_fes);
|
||
trial_fes.Append(hatu_fes);
|
||
trial_fes.Append(hatsigma_fes);
|
||
test_fec.Append(tau_fec);
|
||
test_fec.Append(v_fec);
|
||
|
||
// Required coefficients for the weak formulation
|
||
ConstantCoefficient one(1.0);
|
||
ConstantCoefficient negone(-1.0);
|
||
FunctionCoefficient f(f_exact); // rhs for the manufactured solution problem
|
||
|
||
// Required coefficients for the exact solutions
|
||
FunctionCoefficient uex(exact_u);
|
||
VectorFunctionCoefficient sigmaex(dim,exact_sigma);
|
||
FunctionCoefficient hatuex(exact_hatu);
|
||
|
||
ParDPGWeakForm * a = new ParDPGWeakForm(trial_fes,test_fec);
|
||
a->StoreMatrices(true); // this is needed for estimation of residual
|
||
|
||
// -(u,∇⋅τ)
|
||
a->AddTrialIntegrator(new MixedScalarWeakGradientIntegrator(one),
|
||
TrialSpace::u_space,TestSpace::tau_space);
|
||
|
||
// -(σ,τ)
|
||
a->AddTrialIntegrator(new TransposeIntegrator(new VectorFEMassIntegrator(
|
||
negone)), TrialSpace::sigma_space, TestSpace::tau_space);
|
||
|
||
// (σ,∇ v)
|
||
a->AddTrialIntegrator(new TransposeIntegrator(new GradientIntegrator(one)),
|
||
TrialSpace::sigma_space,TestSpace::v_space);
|
||
|
||
// <û,τ⋅n>
|
||
a->AddTrialIntegrator(new NormalTraceIntegrator,
|
||
TrialSpace::hatu_space,TestSpace::tau_space);
|
||
|
||
// -<σ̂,v> (sign is included in σ̂)
|
||
a->AddTrialIntegrator(new TraceIntegrator,
|
||
TrialSpace::hatsigma_space, TestSpace::v_space);
|
||
|
||
// test integrators (space-induced norm for H(div) × H1)
|
||
// (∇⋅τ,∇⋅δτ)
|
||
a->AddTestIntegrator(new DivDivIntegrator(one),
|
||
TestSpace::tau_space, TestSpace::tau_space);
|
||
// (τ,δτ)
|
||
a->AddTestIntegrator(new VectorFEMassIntegrator(one),
|
||
TestSpace::tau_space, TestSpace::tau_space);
|
||
// (∇v,∇δv)
|
||
a->AddTestIntegrator(new DiffusionIntegrator(one),
|
||
TestSpace::v_space, TestSpace::v_space);
|
||
// (v,δv)
|
||
a->AddTestIntegrator(new MassIntegrator(one),
|
||
TestSpace::v_space, TestSpace::v_space);
|
||
|
||
// RHS
|
||
if (prob == prob_type::manufactured)
|
||
{
|
||
a->AddDomainLFIntegrator(new DomainLFIntegrator(f),TestSpace::v_space);
|
||
}
|
||
|
||
// GridFunction for Dirichlet bdr data
|
||
ParGridFunction hatu_gf;
|
||
|
||
// Visualization streams
|
||
socketstream u_out;
|
||
socketstream sigma_out;
|
||
|
||
if (myid == 0)
|
||
{
|
||
std::cout << "\n Ref |"
|
||
<< " Dofs |"
|
||
<< " L2 Error |"
|
||
<< " Rate |"
|
||
<< " Residual |"
|
||
<< " Rate |"
|
||
<< " PCG it |" << endl;
|
||
std::cout << std::string(72,'-') << endl;
|
||
}
|
||
|
||
Array<int> elements_to_refine; // for AMR
|
||
fptype err0 = 0.;
|
||
int dof0=0.;
|
||
fptype res0=0.0;
|
||
|
||
ParGridFunction u_gf(u_fes);
|
||
ParGridFunction sigma_gf(sigma_fes);
|
||
u_gf = 0.0;
|
||
sigma_gf = 0.0;
|
||
|
||
ParaViewDataCollection * paraview_dc = nullptr;
|
||
|
||
if (paraview)
|
||
{
|
||
paraview_dc = new ParaViewDataCollection(enum_str[prob], &pmesh);
|
||
paraview_dc->SetPrefixPath("ParaView/Diffusion");
|
||
paraview_dc->SetLevelsOfDetail(order);
|
||
paraview_dc->SetCycle(0);
|
||
paraview_dc->SetDataFormat(VTKFormat::BINARY);
|
||
paraview_dc->SetHighOrderOutput(true);
|
||
paraview_dc->SetTime(0.0); // set the time
|
||
paraview_dc->RegisterField("u",&u_gf);
|
||
paraview_dc->RegisterField("sigma",&sigma_gf);
|
||
}
|
||
|
||
if (static_cond) { a->EnableStaticCondensation(); }
|
||
for (int it = 0; it<=pref; it++)
|
||
{
|
||
a->Assemble();
|
||
|
||
Array<int> ess_tdof_list;
|
||
Array<int> ess_bdr;
|
||
if (pmesh.bdr_attributes.Size())
|
||
{
|
||
ess_bdr.SetSize(pmesh.bdr_attributes.Max());
|
||
ess_bdr = 1;
|
||
hatu_fes->GetEssentialTrueDofs(ess_bdr, ess_tdof_list);
|
||
}
|
||
|
||
// shift the ess_tdofs
|
||
for (int i = 0; i < ess_tdof_list.Size(); i++)
|
||
{
|
||
ess_tdof_list[i] += u_fes->GetTrueVSize() + sigma_fes->GetTrueVSize();
|
||
}
|
||
|
||
Array<int> offsets(5);
|
||
offsets[0] = 0;
|
||
offsets[1] = u_fes->GetVSize();
|
||
offsets[2] = sigma_fes->GetVSize();
|
||
offsets[3] = hatu_fes->GetVSize();
|
||
offsets[4] = hatsigma_fes->GetVSize();
|
||
offsets.PartialSum();
|
||
BlockVector x(offsets);
|
||
x = 0.0;
|
||
hatu_gf.MakeRef(hatu_fes,x.GetBlock(2),0);
|
||
hatu_gf.ProjectBdrCoefficient(uex,ess_bdr);
|
||
|
||
Vector X,B;
|
||
OperatorPtr Ah;
|
||
a->FormLinearSystem(ess_tdof_list,x,Ah,X,B);
|
||
|
||
BlockOperator * A = Ah.As<BlockOperator>();
|
||
|
||
BlockDiagonalPreconditioner M(A->RowOffsets());
|
||
M.owns_blocks = 1;
|
||
int skip = 0;
|
||
if (!static_cond)
|
||
{
|
||
HypreBoomerAMG * amg0 = new HypreBoomerAMG((HypreParMatrix &)A->GetBlock(0,0));
|
||
HypreBoomerAMG * amg1 = new HypreBoomerAMG((HypreParMatrix &)A->GetBlock(1,1));
|
||
amg0->SetPrintLevel(0);
|
||
amg1->SetPrintLevel(0);
|
||
M.SetDiagonalBlock(0,amg0);
|
||
M.SetDiagonalBlock(1,amg1);
|
||
skip=2;
|
||
}
|
||
HypreBoomerAMG * amg2 = new HypreBoomerAMG((HypreParMatrix &)A->GetBlock(skip,
|
||
skip));
|
||
amg2->SetPrintLevel(0);
|
||
M.SetDiagonalBlock(skip,amg2);
|
||
HypreSolver * prec;
|
||
if (dim == 2)
|
||
{
|
||
// AMS preconditioner for 2D H(div) (trace) space
|
||
prec = new HypreAMS((HypreParMatrix &)A->GetBlock(skip+1,skip+1), hatsigma_fes);
|
||
}
|
||
else
|
||
{
|
||
// ADS preconditioner for 3D H(div) (trace) space
|
||
prec = new HypreADS((HypreParMatrix &)A->GetBlock(skip+1,skip+1), hatsigma_fes);
|
||
}
|
||
M.SetDiagonalBlock(skip+1,prec);
|
||
|
||
CGSolver cg(MPI_COMM_WORLD);
|
||
cg.SetRelTol(1e-12);
|
||
cg.SetMaxIter(2000);
|
||
cg.SetPrintLevel(0);
|
||
cg.SetPreconditioner(M);
|
||
cg.SetOperator(*A);
|
||
cg.Mult(B, X);
|
||
|
||
a->RecoverFEMSolution(X,x);
|
||
|
||
Vector & residuals = a->ComputeResidual(x);
|
||
|
||
fptype residual = residuals.Norml2();
|
||
|
||
fptype maxresidual = residuals.Max();
|
||
fptype globalresidual = residual * residual;
|
||
|
||
MPI_Allreduce(MPI_IN_PLACE, &maxresidual, 1, MPITypeMap<fptype>::mpi_type,
|
||
MPI_MAX, MPI_COMM_WORLD);
|
||
MPI_Allreduce(MPI_IN_PLACE, &globalresidual, 1,
|
||
MPITypeMap<fptype>::mpi_type, MPI_SUM, MPI_COMM_WORLD);
|
||
|
||
globalresidual = sqrt(globalresidual);
|
||
|
||
u_gf.MakeRef(u_fes,x.GetBlock(0),0);
|
||
sigma_gf.MakeRef(sigma_fes,x.GetBlock(1),0);
|
||
|
||
int dofs = u_fes->GlobalTrueVSize() + sigma_fes->GlobalTrueVSize()
|
||
+ hatu_fes->GlobalTrueVSize() + hatsigma_fes->GlobalTrueVSize();
|
||
|
||
fptype u_err = u_gf.ComputeL2Error(uex);
|
||
fptype sigma_err = sigma_gf.ComputeL2Error(sigmaex);
|
||
fptype L2Error = sqrt(u_err*u_err + sigma_err*sigma_err);
|
||
fptype rate_err = (it) ? dim*log(err0/L2Error)/log((fptype)dof0/dofs) : 0.0;
|
||
fptype rate_res = (it) ? dim*log(res0/globalresidual)/log((
|
||
fptype)dof0/dofs) : 0.0;
|
||
err0 = L2Error;
|
||
res0 = globalresidual;
|
||
dof0 = dofs;
|
||
|
||
if (myid == 0)
|
||
{
|
||
std::ios oldState(nullptr);
|
||
oldState.copyfmt(std::cout);
|
||
std::cout << std::right << std::setw(5) << it << " | "
|
||
<< std::setw(10) << dof0 << " | "
|
||
<< std::setprecision(3)
|
||
<< std::setw(10) << std::scientific << err0 << " | "
|
||
<< std::setprecision(2)
|
||
<< std::setw(6) << std::fixed << rate_err << " | "
|
||
<< std::setprecision(3)
|
||
<< 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";
|
||
int visport = 19916;
|
||
|
||
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((fptype)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;
|
||
}
|
||
|
||
fptype exact_u(const Vector & X)
|
||
{
|
||
switch (prob)
|
||
{
|
||
case prob_type::lshape:
|
||
{
|
||
fptype x = X[0];
|
||
fptype y = X[1];
|
||
fptype r = sqrt(x*x + y*y);
|
||
fptype alpha = 2./3.;
|
||
fptype phi = atan2(y,x);
|
||
if (phi < 0) { phi += 2*M_PI; }
|
||
return pow(r,alpha) * sin(alpha * phi);
|
||
}
|
||
break;
|
||
default:
|
||
{
|
||
fptype 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:
|
||
{
|
||
fptype x = X[0];
|
||
fptype y = X[1];
|
||
fptype r = sqrt(x*x + y*y);
|
||
fptype alpha = 2./3.;
|
||
fptype phi = atan2(y,x);
|
||
if (phi < 0) { phi += 2*M_PI; }
|
||
|
||
fptype r_x = x/r;
|
||
fptype r_y = y/r;
|
||
fptype phi_x = - y / (r*r);
|
||
fptype phi_y = x / (r*r);
|
||
fptype 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:
|
||
{
|
||
fptype alpha = M_PI * (X.Sum());
|
||
du.SetSize(X.Size());
|
||
for (int i = 0; i<du.Size(); i++)
|
||
{
|
||
du[i] = M_PI * cos(alpha);
|
||
}
|
||
}
|
||
break;
|
||
}
|
||
}
|
||
|
||
fptype exact_laplacian_u(const Vector & X)
|
||
{
|
||
switch (prob)
|
||
{
|
||
case prob_type::manufactured:
|
||
{
|
||
fptype alpha = M_PI * (X.Sum());
|
||
fptype 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);
|
||
}
|
||
|
||
fptype exact_hatu(const Vector & X)
|
||
{
|
||
return exact_u(X);
|
||
}
|
||
|
||
void exact_hatsigma(const Vector & X, Vector & hatsigma)
|
||
{
|
||
exact_sigma(X,hatsigma);
|
||
hatsigma *= -1.;
|
||
}
|
||
|
||
fptype 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);
|
||
}
|