poisson2_example.cpp

The boundary value problem for the Poisson equation is defined as follows:

\begin{eqnarray*} -\Delta u &=& f \quad \text{in} \quad \Omega, \\ u &=& g \quad \text{on} \quad \Gamma_D, \\ \mathbf{n} \cdot \nabla u &=& h \quad \text{on} \quad \Gamma_N, \end{eqnarray*}

where \(\mathbf{n}\) is the outward pointing unit normal vector, \(\Gamma_D\) and \(\Gamma_N\) are the parts of the boundary where Dirichlet (essential) and Neumann (natural) boundary conditions are prescribed, respectively. The corresponding weak formulation is:

\begin{eqnarray*} \text{Find }u\in\Sigma_{h}\text{ such that}\\ \int_\Omega \nabla u \cdot \nabla \psi \text{d}\Omega = \int_\Omega f\psi \text{d}\Omega + \int_{\Gamma_N} h\psi \text{d}\Gamma\\ \forall\psi\in\Sigma_{h} \end{eqnarray*}

Where \(\Sigma_{h}\) is the function space where the Dirichlet boundary conditions according to function \(h\) are applied and \(\psi\) are the basis functions.

The system to assemble a system of equations based on this weak formulation is constructed in poisson2_example.cpp using the gsExprAssembler. This file takes as input any domain \(\Omega\), any function \(g\) and any function \(h\). In this example, we take the xml file filedata/pde/poisson2d_bvp.xml for the input.

First we include the library and use the gismo namespace.

#include <gismo.h>
 
using namespace gismo;

Then we define our command line options. For example, we use the option -f or --file to set the path to the file that contains our geometry and our boundary conditions.

    bool plot = false;
    index_t numRefine  = 5;
    index_t numElevate = 0;
    index_t numSplit = 0;
    bool last{false}, export_b64{false};
    std::string fn("pde/poisson2d_bvp.xml");
 
    gsCmdLine cmd("Tutorial on solving a Poisson problem.");
    cmd.addInt( "e", "degreeElevation",
                "Number of degree elevation steps to perform before solving (0: equalize degree in all directions)", numElevate );
    cmd.addInt( "r", "uniformRefine", "Number of Uniform h-refinement loops",  numRefine );
    cmd.addInt( "s", "uniformSplit", "Subdivide the input pathes this many time",  numSplit );
    cmd.addString( "f", "file", "Input XML file", fn );
    cmd.addSwitch("last", "Solve solely for the last level of h-refinement",
                  last);
    cmd.addSwitch(
        "plot", "Create a ParaView visualization file with the solution", plot);
    cmd.addSwitch("binary", "Use B64 encoding for Paraview", export_b64);
 
    try { cmd.getValues(argc,argv); } catch (int rv) { return rv; }

Then we read the input from the file that contains the boundary value problem. In this example, we use the function gsFileData::getId() to read input from an xml file. In this function, the first argument specifies the ID of the object in the xml file (the id flag) and the second argument is the object to which the file info is written. Note that for the boundary conditions, we immedeately set the geometry map (based on the gsMultiPatch class).

 
    gsFileData<> fd(fn);
    gsInfo << "Loaded file "<< fd.lastPath() <<"\n";
 
    gsMultiPatch<> mp;
    fd.getId(0, mp); // id=0: Multipatch domain
    for (index_t i = 0; i<numSplit;++i)
      mp = mp.uniformSplit();
 
    gsFunctionExpr<> f;
    fd.getId(1, f); // id=1: source function
    gsInfo<<"Source function "<< f << "\n";
 
    gsBoundaryConditions<> bc;
    fd.getId(2, bc); // id=2: boundary conditions
    bc.setGeoMap(mp);
    gsInfo<<"Boundary conditions:\n"<< bc <<"\n";
 
    gsFunctionExpr<> ms;
    fd.getId(3, ms); // id=3: reference solution
 
    gsOptionList Aopt;
    fd.getId(4, Aopt); // id=4: assembler options
 

Before we define our gsExprAssembler, we elevate the degree or we refine the basis. This is done in the following snippet

    gsMultiBasis<> dbasis(mp, true);//true: poly-splines (not NURBS)
 
    // Elevate and p-refine the basis to order p + numElevate
    // where p is the highest degree in the bases
    dbasis.setDegree( dbasis.maxCwiseDegree() + numElevate);
 
    // h-refine each basis
    if (last)
    {
        for (int r =0; r < numRefine; ++r)
            dbasis.uniformRefine();
        numRefine = 0;
    }
 
    gsInfo << "Patches: "<< mp.nPatches() <<", degree: "<< dbasis.minCwiseDegree() <<"\n";
#ifdef _OPENMP
    gsInfo<< "Available threads: "<< omp_get_max_threads() <<"\n";
#endif

Now it is time to look at the setup of our boundary value problem. In the code snippet below, we first define our gsExprAssembler. The inputs (1,1) here mean that we have 1 test function space and 1 trial function space. Furthermore, few typedef s can be seen. These are just there to prevent lengthy types. The assembler is fed with the basis using the gsExprAssembler::setIntegrationElements() function. When the basis is defined, a gsExprEvaluator is constructed for later use. The geometry is connected to the gsExprAssembler via the gsExprAssembler::getMap() function and we create an expression for the space via gsExprAssembler::getSpace(). Functions for the right-hand side and the manufactured solution are constructed via gsExprAssembler::getCoef() and gsExprEvaluator::getVariable() functions, respectively. Note that the manufactured solution is defined in the gsExprEvaluator because it will not be used in assembly of the system. The problem setup finishes by defining the solution object, which is constructed via gsExprAssembler::getSolution() which requires a space and a reference to a vector that will contain the solution coefficients later.

    gsExprAssembler<> A(1,1);
    A.setOptions(Aopt);
 
    gsInfo<<"Active options:\n"<< A.options() <<"\n";
 
    typedef gsExprAssembler<>::geometryMap geometryMap;
    typedef gsExprAssembler<>::variable    variable;
    typedef gsExprAssembler<>::space       space;
    typedef gsExprAssembler<>::solution    solution;
 
    // Elements used for numerical integration
    A.setIntegrationElements(dbasis);
    gsExprEvaluator<> ev(A);
 
    // Set the geometry map
    geometryMap G = A.getMap(mp);
 
    // Set the discretization space
    space u = A.getSpace(dbasis);
 
    // Set the source term
    auto ff = A.getCoeff(f, G);
 
    // Recover manufactured solution
    auto u_ex = ev.getVariable(ms, G);
 
    // Solution vector and solution variable
    gsMatrix<> solVector;
    solution u_sol = A.getSolution(u, solVector);
 

The next step is to actually solve the system at hand. In this particular example, this is done within a loop of uniform refinement (unless the last option is selected) in order to study convergence behaviour of the method.

As seen in the snippet below, we first define the solver that we will use (based on the Eigen-library) and we define the vectors to store the errors. Then, within the loop, we first refine the basis uniformly and then the gsExprAssembler<>::space::setup() function is used. This function computes the basis (boundary conditions) and sets the interface continuity. Then the gsExprAssembler is initialized using gsExprAssembler::initSystem() and using gsExprAssembler::assemble() the matrix expression (first argument) and the rhs expression (second argument) are assembled to a system. We also call the function gsExprAssembler::assembleRhsBc() which is dedicated to assemble an expression over a boundary; in this case the Neumann boundaries which are called via gsBoundaryConditions::neumannSides(). When the Neumann boundary conditions are assembled, the system is factorized using gsSparseSolver::compute() and solved via gsSparseSolver::solve(). The errors are then computed via the gsExprEvaluator, where it should be noted that the solution object is appearing, which uses the solution vector and the space as defined earlier.

#ifdef GISMO_WITH_PARDISO
    gsSparseSolver<>::PardisoLDLT solver;
#else
    gsSparseSolver<>::CGDiagonal solver;
#endif
 
    gsVector<> l2err(numRefine+1), h1err(numRefine+1);
    gsInfo<< "(dot1=assembled, dot2=solved, dot3=got_error)\n"
        "\nDoFs: ";
    double setup_time(0), ma_time(0), slv_time(0), err_time(0);
    gsStopwatch timer;
    for (int r=0; r<=numRefine; ++r)
    {
        timer.restart();
        dbasis.uniformRefine();
 
        // Setup the space \a u with strongly imposed Dirichlet part
        //u.setup(bc, dirichlet::interpolation, 0);
        u.setup(bc, dirichlet::l2Projection, 0);
 
        // Initialize the system
        A.initSystem();
        // Compute sparsity patter: this is done automatically - but
        // is needed if assemble(.) is called twice
        A.computePattern( igrad(u) * igrad(u).tr() );
        setup_time += timer.stop();
        gsInfo<< A.numDofs() <<std::flush;
 
        timer.restart();
        // Compute the system matrix and right-hand side
        A.assemble(
            igrad(u, G) * igrad(u, G).tr() * meas(G) //matrix
            ,
            u * ff * meas(G) //rhs vector
            );
 
        // Compute the Neumann terms defined on physical space
        auto g_N = A.getBdrFunction(G);
        A.assembleBdr(bc.get("Neumann"), u * g_N.tr() * nv(G) );
        ma_time += timer.stop();
        // gsDebugVar(A.matrix().toDense());
        // gsDebugVar(A.rhs().transpose()   );
        gsInfo<< "." <<std::flush;// Assemblying done
 
        timer.restart();
        solver.compute( A.matrix() );
        solVector = solver.solve(A.rhs());
        slv_time += timer.stop();
        gsInfo<< "." <<std::flush; // Linear solving done
 
        timer.restart();
        // Compute the L2 and H1 error, based on manufactured solution
        l2err[r]= math::sqrt( ev.integral( (u_ex - u_sol).sqNorm() * meas(G) ) );
        h1err[r]= l2err[r] +
            math::sqrt(ev.integral( ( igrad(u_ex) - igrad(u_sol,G) ).sqNorm() * meas(G) ));
        err_time += timer.stop();
        gsInfo<< ". " <<std::flush; // Error computations done
 
    } //for loop

Error norms are computed in the following snippet:

    gsInfo<< "\nL2 error: "<<std::scientific<<std::setprecision(3)<<l2err.transpose()<<"\n";
    gsInfo<< "H1 error: "<<std::scientific<<h1err.transpose()<<"\n";
 
    if (!last && numRefine>0)
    {
        gsInfo<< "\nEoC (L2): " << std::fixed<<std::setprecision(2)
              <<  ( l2err.head(numRefine).array()  /
                   l2err.tail(numRefine).array() ).log().transpose() / std::log(2.0)
                   <<"\n";
 
        gsInfo<<   "EoC (H1): "<< std::fixed<<std::setprecision(2)
              <<( h1err.head(numRefine).array() /
                  h1err.tail(numRefine).array() ).log().transpose() / std::log(2.0) <<"\n";
    }

Lastly, the export to ParaView is done in the snipped below. Note that this does not directly use the gsWriteParaview class, but instead the gsExprEvaluator::writeParaview() function is employed.

    if (plot)
    {
        gsInfo<<"Plotting in Paraview...\n";
 
        gsParaviewCollection collection("ParaviewOutput/solution", &ev);
        collection.options().setSwitch("plotElements", true);
        collection.options().setSwitch("base64", export_b64);
        collection.options().setInt("plotElements.resolution", 16);
        collection.newTimeStep(&mp);
        collection.addField(u_sol,"numerical solution");
        collection.addField(u_ex, "exact solution");
        collection.saveTimeStep();
        collection.save();
 
        gsFileManager::open("ParaviewOutput/solution.pvd");
    }
    else
        gsInfo << "Done. No output created, re-run with --plot to get a ParaView "
                  "file containing the solution.\n";

Annotated source file

Here is the full file examples/poisson2_example.cpp. Clicking on a function or class name will lead you to its reference documentation.

 
#include <gismo.h>
 
using namespace gismo;
 
int main(int argc, char *argv[])
{
    bool plot = false;
    index_t numRefine  = 5;
    index_t numElevate = 0;
    index_t numSplit = 0;
    bool last{false}, export_b64{false};
    std::string fn("pde/poisson2d_bvp.xml");
 
    gsCmdLine cmd("Tutorial on solving a Poisson problem.");
    cmd.addInt( "e", "degreeElevation",
                "Number of degree elevation steps to perform before solving (0: equalize degree in all directions)", numElevate );
    cmd.addInt( "r", "uniformRefine", "Number of Uniform h-refinement loops",  numRefine );
    cmd.addInt( "s", "uniformSplit", "Subdivide the input pathes this many time",  numSplit );
    cmd.addString( "f", "file", "Input XML file", fn );
    cmd.addSwitch("last", "Solve solely for the last level of h-refinement",
                  last);
    cmd.addSwitch(
        "plot", "Create a ParaView visualization file with the solution", plot);
    cmd.addSwitch("binary", "Use B64 encoding for Paraview", export_b64);
 
    try { cmd.getValues(argc,argv); } catch (int rv) { return rv; }
 
 
    gsFileData<> fd(fn);
    gsInfo << "Loaded file "<< fd.lastPath() <<"\n";
 
    gsMultiPatch<> mp;
    fd.getId(0, mp); // id=0: Multipatch domain
    for (index_t i = 0; i<numSplit;++i)
      mp = mp.uniformSplit();
 
    gsFunctionExpr<> f;
    fd.getId(1, f); // id=1: source function
    gsInfo<<"Source function "<< f << "\n";
 
    gsBoundaryConditions<> bc;
    fd.getId(2, bc); // id=2: boundary conditions
    bc.setGeoMap(mp);
    gsInfo<<"Boundary conditions:\n"<< bc <<"\n";
 
    gsFunctionExpr<> ms;
    fd.getId(3, ms); // id=3: reference solution
 
    gsOptionList Aopt;
    fd.getId(4, Aopt); // id=4: assembler options
 
 
    gsMultiBasis<> dbasis(mp, true);//true: poly-splines (not NURBS)
 
    // Elevate and p-refine the basis to order p + numElevate
    // where p is the highest degree in the bases
    dbasis.setDegree( dbasis.maxCwiseDegree() + numElevate);
 
    // h-refine each basis
    if (last)
    {
        for (int r =0; r < numRefine; ++r)
            dbasis.uniformRefine();
        numRefine = 0;
    }
 
    gsInfo << "Patches: "<< mp.nPatches() <<", degree: "<< dbasis.minCwiseDegree() <<"\n";
#ifdef _OPENMP
    gsInfo<< "Available threads: "<< omp_get_max_threads() <<"\n";
#endif
 
    gsExprAssembler<> A(1,1);
    A.setOptions(Aopt);
 
    gsInfo<<"Active options:\n"<< A.options() <<"\n";
 
    typedef gsExprAssembler<>::geometryMap geometryMap;
    typedef gsExprAssembler<>::variable    variable;
    typedef gsExprAssembler<>::space       space;
    typedef gsExprAssembler<>::solution    solution;
 
    // Elements used for numerical integration
    A.setIntegrationElements(dbasis);
    gsExprEvaluator<> ev(A);
 
    // Set the geometry map
    geometryMap G = A.getMap(mp);
 
    // Set the discretization space
    space u = A.getSpace(dbasis);
 
    // Set the source term
    auto ff = A.getCoeff(f, G);
 
    // Recover manufactured solution
    auto u_ex = ev.getVariable(ms, G);
 
    // Solution vector and solution variable
    gsMatrix<> solVector;
    solution u_sol = A.getSolution(u, solVector);
 
 
#ifdef GISMO_WITH_PARDISO
    gsSparseSolver<>::PardisoLDLT solver;
#else
    gsSparseSolver<>::CGDiagonal solver;
#endif
 
    gsVector<> l2err(numRefine+1), h1err(numRefine+1);
    gsInfo<< "(dot1=assembled, dot2=solved, dot3=got_error)\n"
        "\nDoFs: ";
    double setup_time(0), ma_time(0), slv_time(0), err_time(0);
    gsStopwatch timer;
    for (int r=0; r<=numRefine; ++r)
    {
        timer.restart();
        dbasis.uniformRefine();
 
        // Setup the space \a u with strongly imposed Dirichlet part
        //u.setup(bc, dirichlet::interpolation, 0);
        u.setup(bc, dirichlet::l2Projection, 0);
 
        // Initialize the system
        A.initSystem();
        // Compute sparsity patter: this is done automatically - but
        // is needed if assemble(.) is called twice
        A.computePattern( igrad(u) * igrad(u).tr() );
        setup_time += timer.stop();
        gsInfo<< A.numDofs() <<std::flush;
 
        timer.restart();
        // Compute the system matrix and right-hand side
        A.assemble(
            igrad(u, G) * igrad(u, G).tr() * meas(G) //matrix
            ,
            u * ff * meas(G) //rhs vector
            );
 
        // Compute the Neumann terms defined on physical space
        auto g_N = A.getBdrFunction(G);
        A.assembleBdr(bc.get("Neumann"), u * g_N.tr() * nv(G) );
        ma_time += timer.stop();
        // gsDebugVar(A.matrix().toDense());
        // gsDebugVar(A.rhs().transpose()   );
        gsInfo<< "." <<std::flush;// Assemblying done
 
        timer.restart();
        solver.compute( A.matrix() );
        solVector = solver.solve(A.rhs());
        slv_time += timer.stop();
        gsInfo<< "." <<std::flush; // Linear solving done
 
        timer.restart();
        // Compute the L2 and H1 error, based on manufactured solution
        l2err[r]= math::sqrt( ev.integral( (u_ex - u_sol).sqNorm() * meas(G) ) );
        h1err[r]= l2err[r] +
            math::sqrt(ev.integral( ( igrad(u_ex) - igrad(u_sol,G) ).sqNorm() * meas(G) ));
        err_time += timer.stop();
        gsInfo<< ". " <<std::flush; // Error computations done
 
    } //for loop
 
    gsInfo<<"\n\nTotal time: "<< setup_time+ma_time+slv_time+err_time <<"\n";
    gsInfo<<"   Setup: "<< setup_time <<"\n";
    gsInfo<<"Assembly: "<< ma_time    <<"\n";
    gsInfo<<" Solving: "<< slv_time   <<"\n";
    gsInfo<<"   Norms: "<< err_time   <<"\n";
 
    gsInfo<< "\nL2 error: "<<std::scientific<<std::setprecision(3)<<l2err.transpose()<<"\n";
    gsInfo<< "H1 error: "<<std::scientific<<h1err.transpose()<<"\n";
 
    if (!last && numRefine>0)
    {
        gsInfo<< "\nEoC (L2): " << std::fixed<<std::setprecision(2)
              <<  ( l2err.head(numRefine).array()  /
                   l2err.tail(numRefine).array() ).log().transpose() / std::log(2.0)
                   <<"\n";
 
        gsInfo<<   "EoC (H1): "<< std::fixed<<std::setprecision(2)
              <<( h1err.head(numRefine).array() /
                  h1err.tail(numRefine).array() ).log().transpose() / std::log(2.0) <<"\n";
    }
 
    if (plot)
    {
        gsInfo<<"Plotting in Paraview...\n";
 
        gsParaviewCollection collection("ParaviewOutput/solution", &ev);
        collection.options().setSwitch("plotElements", true);
        collection.options().setSwitch("base64", export_b64);
        collection.options().setInt("plotElements.resolution", 16);
        collection.newTimeStep(&mp);
        collection.addField(u_sol,"numerical solution");
        collection.addField(u_ex, "exact solution");
        collection.saveTimeStep();
        collection.save();
 
        gsFileManager::open("ParaviewOutput/solution.pvd");
    }
    else
        gsInfo << "Done. No output created, re-run with --plot to get a ParaView "
                  "file containing the solution.\n";
 
    return EXIT_SUCCESS;
 
}// end main