pconvection-diffusion.cpp

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// 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 convection-diffusion
//
// Compile with: make  pconvection-diffusion
//
// sample runs
// mpirun -np 4 pconvection-diffusion  -o 2 -ref 3 -prob 0 -eps 1e-1 -beta '4 2' -theta 0.0
// mpirun -np 4 pconvection-diffusion  -o 3 -ref 3 -prob 0 -eps 1e-2 -beta '2 3' -theta 0.0
// mpirun -np 4 pconvection-diffusion -m ../../data/inline-hex.mesh -o 2 -ref 1 -prob 0 -sc -eps 1e-1 -theta 0.0

// AMR runs
// mpirun -np 4 pconvection-diffusion  -o 3 -ref 15 -prob 1 -eps 1e-3 -beta '1 0' -theta 0.7 -sc
// mpirun -np 4 pconvection-diffusion  -o 3 -ref 20 -prob 2 -eps 5e-3 -theta 0.7 -sc
// mpirun -np 4 pconvection-diffusion  -o 2 -ref 20 -prob 3 -eps 1e-2 -beta '1 2' -theta 0.7 -sc

// Description:
// This example code demonstrates the use of MFEM to define and solve a parallel
// "ultraweak" (UW) DPG formulation for the convection-diffusion problem

//     - εΔu + ∇⋅(βu) = f,   in Ω
//                  u = u₀ , on ∂Ω

// It solves the following kinds of problems
// (a) A manufactured solution where u_exact = sin(π * (x + y + z)).
// (b) The 2D Erickson-Johnson problem
// (c) Internal layer problem
// (d) Boundary layer problem

// The DPG UW deals with the First Order System
//     - ∇⋅σ + ∇⋅(βu) = f,   in Ω
//        1/ε σ - ∇u  = 0,   in Ω
//                  u = u₀ , on ∂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, f̂ ∈ H^-1/2
// -(βu , ∇v)  + (σ , ∇v)     + < f̂ ,  v  > = (f,v),   ∀ v ∈ H¹(Ω)
//   (u , ∇⋅τ) + 1/ε (σ , τ)  + < û , τ⋅n > = 0,       ∀ τ ∈ H(div,Ω)
//                                        û = u₀   on ∂Ω

// Note:
// f̂ := βu - σ, û := -u on the mesh skeleton

// -------------------------------------------------------------
// |   |     u     |     σ     |   û       |     f̂    |  RHS    |
// -------------------------------------------------------------
// | v |-(βu , ∇v) | (σ , ∇v)  |           | < f̂ ,v > |  (f,v)  |
// |   |           |           |           |          |         |
// | τ | (u ,∇⋅τ)  | 1/ε(σ , τ)|  <û,τ⋅n>  |          |    0    |

// where (v,τ) ∈  H¹(Ωₕ) × H(div,Ωₕ)

// For more information see https://doi.org/10.1016/j.camwa.2013.06.010

#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
{
   sinusoidal,
   EJ, // see https://doi.org/10.1016/j.camwa.2013.06.010
   curved_streamlines, // see https://doi.org/10.1515/cmam-2018-0207
   bdr_layer // see  https://doi.org/10.1002/num.20640
};

static const char *enum_str[] =
{
   "sinusoidal",
   "EJ",
   "curved_streamlines",
   "bdr_layer"
};

prob_type prob;
Vector beta;
double epsilon;

double exact_u(const Vector & X);
void exact_gradu(const Vector & X, Vector & du);
double exact_laplacian_u(const Vector & X);
double exact_u(const Vector & X);
void exact_sigma(const Vector & X, Vector & sigma);
double exact_hatu(const Vector & X);
void exact_hatf(const Vector & X, Vector & hatf);
double f_exact(const Vector & X);
double bdr_data(const Vector &X);
void beta_function(const Vector & X, Vector & beta_val);
void setup_test_norm_coeffs(ParGridFunction & c1_gf, ParGridFunction & c2_gf);

int main(int argc, char *argv[])
{
   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 ref = 1;
   int iprob = 0;
   double theta = 0.7;
   bool static_cond = false;
   epsilon = 1e0;

   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(&epsilon, "-eps", "--epsilon",
                  "Epsilon coefficient");
   args.AddOption(&ref, "-ref", "--num-refinements",
                  "Number of uniform refinements");
   args.AddOption(&theta, "-theta", "--theta",
                  "Theta parameter for AMR");
   args.AddOption(&iprob, "-prob", "--problem", "Problem case"
                  " 0: lshape, 1: General");
   args.AddOption(&beta, "-beta", "--beta",
                  "Vector Coefficient beta");
   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(&paraview, "-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 > 3) { iprob = 3; }
   prob = (prob_type)iprob;

   if (prob == prob_type::EJ || prob == prob_type::curved_streamlines ||
       prob == prob_type::bdr_layer)
   {
      mesh_file = "../../data/inline-quad.mesh";
   }

   Mesh mesh(mesh_file, 1, 1);
   int dim = mesh.Dimension();
   MFEM_VERIFY(dim > 1, "Dimension = 1 is not supported in this example");

   bool exact_known = true;
   switch (prob)
   {
      case sinusoidal:
      case EJ:
      {
         if (beta.Size() == 0)
         {
            beta.SetSize(dim);
            beta = 0.0;
            beta[0] = 1.;
         }
         break;
      }
      case bdr_layer:
      {
         beta.SetSize(dim);
         beta[0] = 1.;
         beta[1] = 2.;
         exact_known = false;
      }
      break;
      default:
         // do nothing; beta is defined as a FunctionCoefficient
         break;
   }

   if (myid == 0)
   {
      args.PrintOptions(cout);
   }

   mesh.EnsureNCMesh(true);

   ParMesh pmesh(MPI_COMM_WORLD, mesh);
   mesh.Clear();

   // Define spaces
   enum TrialSpace
   {
      u_space     = 0,
      sigma_space = 1,
      hatu_space  = 2,
      hatf_space  = 3
   };
   enum TestSpace
   {
      v_space   = 0,
      tau_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 * hatf_fec = new RT_Trace_FECollection(order-1,dim);
   ParFiniteElementSpace *hatf_fes = new ParFiniteElementSpace(&pmesh,hatf_fec);

   // testspace fe collections
   int test_order = order+delta_order;
   FiniteElementCollection * v_fec = new H1_FECollection(test_order, dim);
   FiniteElementCollection * tau_fec = new RT_FECollection(test_order-1, dim);

   // Coefficients
   ConstantCoefficient one(1.0);
   ConstantCoefficient negone(-1.0);
   ConstantCoefficient eps(epsilon);
   ConstantCoefficient eps1(1./epsilon);
   ConstantCoefficient negeps1(-1./epsilon);
   ConstantCoefficient eps2(1/(epsilon*epsilon));
   ConstantCoefficient negeps(-epsilon);

   VectorFunctionCoefficient betacoeff(dim,beta_function);
   ScalarVectorProductCoefficient negbetacoeff(-1.0,betacoeff);
   OuterProductCoefficient bbtcoeff(betacoeff,betacoeff);

   // Normal equation weak formulation
   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(hatf_fes);
   test_fec.Append(v_fec);
   test_fec.Append(tau_fec);

   ParDPGWeakForm * a = new ParDPGWeakForm(trial_fes,test_fec);
   a->StoreMatrices(true);

   //-(βu , ∇v)
   a->AddTrialIntegrator(new MixedScalarWeakDivergenceIntegrator(betacoeff),
                         TrialSpace::u_space, TestSpace::v_space);

   // (σ,∇ v)
   a->AddTrialIntegrator(new TransposeIntegrator(new GradientIntegrator(one)),
                         TrialSpace::sigma_space, TestSpace::v_space);

   // (u ,∇⋅τ)
   a->AddTrialIntegrator(new MixedScalarWeakGradientIntegrator(negone),
                         TrialSpace::u_space, TestSpace::tau_space);

   // 1/ε (σ,τ)
   a->AddTrialIntegrator(new TransposeIntegrator(new VectorFEMassIntegrator(eps1)),
                         TrialSpace::sigma_space, TestSpace::tau_space);

   //  <û,τ⋅n>
   a->AddTrialIntegrator(new NormalTraceIntegrator,
                         TrialSpace::hatu_space, TestSpace::tau_space);

   // <f̂ ,v>
   a->AddTrialIntegrator(new TraceIntegrator,
                         TrialSpace::hatf_space, TestSpace::v_space);


   FiniteElementCollection *coeff_fec = new L2_FECollection(0,dim);
   ParFiniteElementSpace *coeff_fes = new ParFiniteElementSpace(&pmesh,coeff_fec);
   ParGridFunction c1_gf, c2_gf;
   GridFunctionCoefficient c1_coeff(&c1_gf);
   GridFunctionCoefficient c2_coeff(&c2_gf);

   c1_gf.SetSpace(coeff_fes);
   c2_gf.SetSpace(coeff_fes);
   setup_test_norm_coeffs(c1_gf,c2_gf);

   // c1 (v,δv)
   a->AddTestIntegrator(new MassIntegrator(c1_coeff),
                        TestSpace::v_space, TestSpace::v_space);
   // ε (∇v,∇δv)
   a->AddTestIntegrator(new DiffusionIntegrator(eps),
                        TestSpace::v_space, TestSpace::v_space);
   // (β⋅∇v, β⋅∇δv)
   a->AddTestIntegrator(new DiffusionIntegrator(bbtcoeff),
                        TestSpace::v_space, TestSpace::v_space);
   // c2 (τ,δτ)
   a->AddTestIntegrator(new VectorFEMassIntegrator(c2_coeff),
                        TestSpace::tau_space, TestSpace::tau_space);
   // (∇⋅τ,∇⋅δτ)
   a->AddTestIntegrator(new DivDivIntegrator(one),
                        TestSpace::tau_space, TestSpace::tau_space);

   FunctionCoefficient f(f_exact);
   if (prob == prob_type::sinusoidal ||
       prob == prob_type::curved_streamlines)
   {
      a->AddDomainLFIntegrator(new DomainLFIntegrator(f),TestSpace::v_space);
   }

   FunctionCoefficient hatuex(exact_hatu);
   VectorFunctionCoefficient hatfex(dim,exact_hatf);
   Array<int> elements_to_refine;
   FunctionCoefficient uex(exact_u);
   VectorFunctionCoefficient sigmaex(dim,exact_sigma);

   ParGridFunction hatu_gf;
   ParGridFunction hatf_gf;

   socketstream u_out;
   socketstream sigma_out;

   double res0 = 0.;
   double err0 = 0.;
   int dof0 = 0; // init to suppress gcc warning
   if (myid == 0)
   {
      std::cout << "  Ref |"
                << "    Dofs    |" ;
      if (exact_known)
      {
         mfem::out << "  L2 Error  |"
                   << "  Rate  |";
      }
      std::cout << "  Residual  |"
                << "  Rate  |"
                << " CG it  |" << endl;
      std::cout << std::string((exact_known) ? 72 : 50,'-')
                << endl;
   }

   if (static_cond) { a->EnableStaticCondensation(); }

   ParGridFunction u_gf(u_fes); u_gf = 0.0;
   ParGridFunction sigma_gf(sigma_fes); sigma_gf = 0.0;

   ParaViewDataCollection * paraview_dc = nullptr;

   if (paraview)
   {
      paraview_dc = new ParaViewDataCollection(enum_str[prob], &pmesh);
      paraview_dc->SetPrefixPath("ParaView/Convection-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);
   }

   for (int it = 0; it<=ref; it++)
   {
      a->Assemble();

      Array<int> ess_tdof_list_uhat;
      Array<int> ess_tdof_list_fhat;
      Array<int> ess_bdr_uhat;
      Array<int> ess_bdr_fhat;
      if (pmesh.bdr_attributes.Size())
      {
         ess_bdr_uhat.SetSize(pmesh.bdr_attributes.Max());
         ess_bdr_fhat.SetSize(pmesh.bdr_attributes.Max());

         if (prob == prob_type::EJ)
         {
            ess_bdr_uhat = 0;
            ess_bdr_fhat = 1;
            ess_bdr_uhat[1] = 1;
            ess_bdr_fhat[1] = 0;
         }
         else
         {
            ess_bdr_uhat = 1;
            ess_bdr_fhat = 0;
         }

         hatu_fes->GetEssentialTrueDofs(ess_bdr_uhat, ess_tdof_list_uhat);
         hatf_fes->GetEssentialTrueDofs(ess_bdr_fhat, ess_tdof_list_fhat);
      }

      // shift the ess_tdofs
      int n = ess_tdof_list_uhat.Size();
      int m = ess_tdof_list_fhat.Size();
      Array<int> ess_tdof_list(n+m);
      for (int j = 0; j < n; j++)
      {
         ess_tdof_list[j] = ess_tdof_list_uhat[j]
                            + u_fes->GetTrueVSize()
                            + sigma_fes->GetTrueVSize();
      }
      for (int j = 0; j < m; j++)
      {
         ess_tdof_list[j+n] = ess_tdof_list_fhat[j]
                              + u_fes->GetTrueVSize()
                              + sigma_fes->GetTrueVSize()
                              + hatu_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] = hatf_fes->GetVSize();
      offsets.PartialSum();
      BlockVector x(offsets);
      x = 0.0;
      hatu_gf.MakeRef(hatu_fes,x.GetBlock(2),0);
      FunctionCoefficient bdr_cf(bdr_data);
      hatu_gf.ProjectBdrCoefficient(bdr_cf,ess_bdr_uhat);

      hatf_gf.MakeRef(hatf_fes,x.GetBlock(3),0);
      hatf_gf.ProjectBdrCoefficientNormal(hatfex,ess_bdr_fhat);

      OperatorPtr Ah;
      Vector X,B;
      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), hatf_fes);
      }
      else
      {
         // ADS preconditioner for 3D H(div) (trace) space
         prec = new HypreADS((HypreParMatrix &)A->GetBlock(skip+1,skip+1), hatf_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);
      int num_iter = cg.GetNumIterations();

      a->RecoverFEMSolution(X,x);
      Vector & residuals = a->ComputeResidual(x);

      double residual = residuals.Norml2();
      double maxresidual = residuals.Max();

      double gresidual = residual * residual;

      MPI_Allreduce(MPI_IN_PLACE,&maxresidual,1,MPI_DOUBLE,MPI_MAX,MPI_COMM_WORLD);
      MPI_Allreduce(MPI_IN_PLACE,&gresidual,1,MPI_DOUBLE,MPI_SUM,MPI_COMM_WORLD);

      gresidual = sqrt(gresidual);

      elements_to_refine.SetSize(0);
      for (int iel = 0; iel<pmesh.GetNE(); iel++)
      {
         if (residuals[iel] > theta * maxresidual)
         {
            elements_to_refine.Append(iel);
         }
      }

      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()
                 + hatf_fes->GlobalTrueVSize();

      double L2Error = 0.0;
      double rate_err = 0.0;
      if (exact_known)
      {
         double u_err = u_gf.ComputeL2Error(uex);
         double sigma_err = sigma_gf.ComputeL2Error(sigmaex);
         L2Error = sqrt(u_err*u_err + sigma_err*sigma_err);
         rate_err = (it) ? dim*log(err0/L2Error)/log((double)dof0/dofs) : 0.0;
         err0 = L2Error;
      }
      double rate_res = (it) ? dim*log(res0/gresidual)/log((double)dof0/dofs) : 0.0;

      res0 = gresidual;
      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 << " | ";
         if (exact_known)
         {
            std::cout << std::setprecision(3) << std::setw(10)
                      << std::scientific <<  err0 << " | "
                      << std::setprecision(2)
                      << std::setw(6) << std::fixed << rate_err << " | " ;
         }
         std::cout << std::setprecision(3)
                   << std::setw(10) << std::scientific <<  res0 << " | "
                   << std::setprecision(2)
                   << std::setw(6) << std::fixed << rate_res << " | "
                   << std::setw(6) << std::fixed << num_iter << " | "
                   << std::endl;
         std::cout.copyfmt(oldState);
      }

      if (visualization)
      {
         const char * keys = (it == 0 && dim == 2) ? "cgRjmlk\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", 501,0,500, 500, keys);
      }

      if (paraview)
      {
         paraview_dc->SetCycle(it);
         paraview_dc->SetTime((double)it);
         paraview_dc->Save();
      }

      if (it == ref)
      {
         break;
      }

      pmesh.GeneralRefinement(elements_to_refine,1,1);
      for (int i =0; i<trial_fes.Size(); i++)
      {
         trial_fes[i]->Update(false);
      }
      a->Update();

      coeff_fes->Update();
      c1_gf.Update();
      c2_gf.Update();
      setup_test_norm_coeffs(c1_gf,c2_gf);
   }

   if (paraview)
   {
      delete paraview_dc;
   }

   delete coeff_fes;
   delete coeff_fec;
   delete a;
   delete tau_fec;
   delete v_fec;
   delete hatf_fes;
   delete hatf_fec;
   delete hatu_fes;
   delete hatu_fec;
   delete sigma_fec;
   delete sigma_fes;
   delete u_fec;
   delete u_fes;

   return 0;
}

double exact_u(const Vector & X)
{
   double x = X[0];
   double y = X[1];
   double z = 0.;
   if (X.Size() == 3) { z = X[2]; }
   switch (prob)
   {
      case sinusoidal:
      {
         double alpha = M_PI * (x + y + z);
         return sin(alpha);
      }
      break;
      case EJ:
      {
         double alpha = sqrt(1. + 4. * epsilon * epsilon * M_PI * M_PI);
         double r1 = (1. + alpha) / (2.*epsilon);
         double r2 = (1. - alpha) / (2.*epsilon);
         double denom = exp(-r2) - exp(-r1);

         double g1 = exp(r2*(x-1.));
         double g2 = exp(r1*(x-1.));
         double g = g1-g2;
         return g * cos(M_PI * y)/denom;
      }
      break;
      case curved_streamlines:
      {
         double r = sqrt(x*x+y*y);
         return atan((1.0-r)/epsilon);
      }
      break;
      default:
         MFEM_ABORT("Wrong code path");
         return 1;
         break;
   }
}

void exact_gradu(const Vector & X, Vector & du)
{
   double x = X[0];
   double y = X[1];
   double z = 0.;
   if (X.Size() == 3) { z = X[2]; }
   du.SetSize(X.Size());

   switch (prob)
   {
      case sinusoidal:
      {
         double alpha = M_PI * (x + y + z);
         for (int i = 0; i<du.Size(); i++)
         {
            du[i] = M_PI * cos(alpha);
         }
      }
      break;
      case EJ:
      {
         double alpha = sqrt(1. + 4. * epsilon * epsilon * M_PI * M_PI);
         double r1 = (1. + alpha) / (2.*epsilon);
         double r2 = (1. - alpha) / (2.*epsilon);
         double denom = exp(-r2) - exp(-r1);

         double g1 = exp(r2*(x-1.));
         double g1_x = r2*g1;
         double g2 = exp(r1*(x-1.));
         double g2_x = r1*g2;
         double g = g1-g2;
         double g_x = g1_x - g2_x;

         double u_x = g_x * cos(M_PI * y)/denom;
         double u_y = -M_PI * g * sin(M_PI*y)/denom;
         du[0] = u_x;
         du[1] = u_y;
      }
      break;
      case curved_streamlines:
      {
         double r = sqrt(x*x+y*y);
         double alpha = -2.0*r + r*r + epsilon*epsilon + 1;
         double denom = r*alpha;
         du[0] = - x* epsilon / denom;
         du[1] = - y* epsilon / denom;
      }
      break;
      default:
         MFEM_ABORT("Wrong code path");
         break;
   }
}

double exact_laplacian_u(const Vector & X)
{
   double x = X[0];
   double y = X[1];
   double z = 0.;
   if (X.Size() == 3) { z = X[2]; }
   switch (prob)
   {
      case sinusoidal:
      {
         double alpha = M_PI * (x + y + z);
         double u = sin(alpha);
         return - M_PI*M_PI * u * X.Size();
      }
      break;
      case EJ:
      {
         double alpha = sqrt(1. + 4. * epsilon * epsilon * M_PI * M_PI);
         double r1 = (1. + alpha) / (2.*epsilon);
         double r2 = (1. - alpha) / (2.*epsilon);
         double denom = exp(-r2) - exp(-r1);

         double g1 = exp(r2*(x-1.));
         double g1_x = r2*g1;
         double g1_xx = r2*g1_x;
         double g2 = exp(r1*(x-1.));
         double g2_x = r1*g2;
         double g2_xx = r1*g2_x;
         double g = g1-g2;
         double g_xx = g1_xx - g2_xx;

         double u = g * cos(M_PI * y)/denom;
         double u_xx = g_xx * cos(M_PI * y)/denom;
         double u_yy = -M_PI * M_PI * u;
         return u_xx + u_yy;
      }
      break;
      case curved_streamlines:
      {
         double r = sqrt(x*x+y*y);
         double alpha = -2.0*r + r*r + epsilon*epsilon + 1;
         return epsilon * (r*r - epsilon*epsilon - 1.0) / (r*alpha*alpha);
      }
      break;
      default:
         MFEM_ABORT("Wrong code path");
         return 1;
         break;
   }
}

void exact_sigma(const Vector & X, Vector & sigma)
{
   // σ = ε ∇ u
   exact_gradu(X,sigma);
   sigma *= epsilon;
}

double exact_hatu(const Vector & X)
{
   return -exact_u(X);
}

void exact_hatf(const Vector & X, Vector & hatf)
{
   Vector sigma;
   Vector beta_val;
   beta_function(X,beta_val);
   exact_sigma(X,sigma);
   double u = exact_u(X);
   hatf.SetSize(X.Size());
   for (int i = 0; i<hatf.Size(); i++)
   {
      hatf[i] = beta_val[i] * u - sigma[i];
   }
}

double f_exact(const Vector & X)
{
   // f = - εΔu + ∇⋅(βu)
   Vector du;
   exact_gradu(X,du);
   double d2u = exact_laplacian_u(X);

   Vector beta_val;
   beta_function(X,beta_val);

   double s = 0;
   for (int i = 0; i<du.Size(); i++)
   {
      s += beta_val[i] * du[i];
   }
   return -epsilon * d2u + s;
}

double bdr_data(const Vector &X)
{
   if (prob == prob_type::bdr_layer)
   {
      double x = X(0);
      double y = X(1);

      if (y==0.0)
      {
         return -(1.0-x);
      }
      else if (x == 0.0)
      {
         return -(1.0-y);
      }
      else
      {
         return 0.0;
      }
   }
   else
   {
      return exact_hatu(X);
   }
}

void beta_function(const Vector & X, Vector & beta_val)
{
   beta_val.SetSize(2);
   if (prob == prob_type::curved_streamlines)
   {
      double x = X(0);
      double y = X(1);
      beta_val(0) = exp(x)*sin(y);
      beta_val(1) = exp(x)*cos(y);
   }
   else
   {
      beta_val = beta;
   }
}

void setup_test_norm_coeffs(ParGridFunction & c1_gf, ParGridFunction & c2_gf)
{
   Array<int> vdofs;
   ParFiniteElementSpace * fes = c1_gf.ParFESpace();
   ParMesh * pmesh = fes->GetParMesh();
   for (int i = 0; i < pmesh->GetNE(); i++)
   {
      double volume = pmesh->GetElementVolume(i);
      double c1 = min(epsilon/volume, 1.);
      double c2 = min(1./epsilon, 1./volume);
      fes->GetElementDofs(i,vdofs);
      c1_gf.SetSubVector(vdofs,c1);
      c2_gf.SetSubVector(vdofs,c2);
   }
}