dgnavierstokesvelvecfem.hh

// -*- tab-width: 2; indent-tabs-mode: nil -*-
// vi: set et ts=2 sw=2 sts=2:
 
#ifndef DUNE_PDELAB_LOCALOPERATOR_DGNAVIERSTOKESVELVECFEM_HH
#define DUNE_PDELAB_LOCALOPERATOR_DGNAVIERSTOKESVELVECFEM_HH
 
#include <dune/common/exceptions.hh>
#include <dune/common/fvector.hh>
#include <dune/common/parametertree.hh>
 
#include <dune/localfunctions/common/interfaceswitch.hh>
#include <dune/pdelab/localoperator/idefault.hh>
 
#include <dune/pdelab/common/quadraturerules.hh>
#include <dune/pdelab/localoperator/defaultimp.hh>
#include <dune/pdelab/localoperator/pattern.hh>
#include <dune/pdelab/localoperator/flags.hh>
#include <dune/pdelab/localoperator/dgnavierstokesparameter.hh>
#include <dune/pdelab/localoperator/navierstokesmass.hh>
 
#ifndef VBLOCK
#define VBLOCK 0
#endif
#define PBLOCK (- VBLOCK + 1)
 
namespace Dune {
  namespace PDELab {
 
  template<class Basis, class Dummy = void>
  struct VectorBasisInterfaceSwitch {
    using DomainLocal = typename Basis::Traits::DomainLocal;
    using RangeField = typename Basis::Traits::RangeField;
    static const std::size_t dimRange = Basis::Traits::dimRange;
 
    template<typename Geometry>
    static void jacobian(const Basis& basis, const Geometry& geometry,
                         const DomainLocal& xl,
                         std::vector<FieldMatrix<RangeField, dimRange,
                                          Geometry::coorddimension> >& jac)
    {
      jac.resize(basis.size());
      basis.evaluateJacobian(xl, jac);
    }
  };
 
  template<class Basis>
  struct VectorBasisInterfaceSwitch<
    Basis, typename std::enable_if<
             Std::to_true_type<
               std::integral_constant<
                 std::size_t,
                 Basis::Traits::dimDomain
                 >
               >::value
             >::type
    >
  {
    using DomainLocal = typename Basis::Traits::DomainType;
    using RangeField = typename Basis::Traits::RangeFieldType;
    static const std::size_t dimRange = Basis::Traits::dimRange;
 
    template<typename Geometry>
    static void jacobian(const Basis& basis, const Geometry& geometry,
                         const DomainLocal& xl,
                         std::vector<FieldMatrix<RangeField, dimRange,
                                          Geometry::coorddimension> >& jac)
    {
      jac.resize(basis.size());
 
      std::vector<FieldMatrix<
      RangeField, dimRange, Geometry::coorddimension> > ljac(basis.size());
      basis.evaluateJacobian(xl, ljac);
 
      const typename Geometry::JacobianInverseTransposed& geojac =
        geometry.jacobianInverseTransposed(xl);
 
      for(std::size_t i = 0; i < basis.size(); ++i)
        for(std::size_t row=0; row < dimRange; ++row)
          geojac.mv(ljac[i][row], jac[i][row]);
    }
  };
 
    template<typename PRM>
    class DGNavierStokesVelVecFEM :
      public LocalOperatorDefaultFlags,
      public FullSkeletonPattern, public FullVolumePattern,
      public InstationaryLocalOperatorDefaultMethods<typename PRM::Traits::RangeField>
    {
      using BC = StokesBoundaryCondition;
      using RF = typename PRM::Traits::RangeField;
 
      using InstatBase = InstationaryLocalOperatorDefaultMethods<typename PRM::Traits::RangeField>;
      using Real = typename InstatBase::RealType;
 
      static const bool navier = PRM::assemble_navier;
      static const bool full_tensor = PRM::assemble_full_tensor;
 
    public :
 
      // pattern assembly flags
      enum { doPatternVolume = true };
      enum { doPatternSkeleton = true };
 
      // call the assembler for each face only once
      enum { doSkeletonTwoSided = false };
 
      // residual assembly flags
      enum { doAlphaVolume    = true };
      enum { doAlphaSkeleton  = true };
      enum { doAlphaBoundary  = true };
      enum { doLambdaVolume   = true };
 
      DGNavierStokesVelVecFEM (PRM& _prm, int _superintegration_order=0) :
        prm(_prm), superintegration_order(_superintegration_order),
        current_dt(1.0)
      {}
 
      // Store current dt
      void preStep (Real , Real dt, int )
      {
        current_dt = dt;
      }
 
      // set time in parameter class
      void setTime(Real t)
      {
        InstatBase::setTime(t);
        prm.setTime(t);
      }
 
      // volume integral depending on test and ansatz functions
      template<typename EG, typename LFSU, typename X, typename LFSV, typename R>
      void alpha_volume (const EG& eg, const LFSU& lfsu, const X& x, const LFSV& lfsv, R& r) const
      {
        const unsigned int dim = EG::Geometry::mydimension;
 
        // subspaces
        using namespace Indices;
        using LFSV_V = TypeTree::Child<LFSV,_0>;
        const auto& lfsv_v = child(lfsv,_0);
        const auto& lfsu_v = child(lfsu,_0);
 
        using LFSV_P = TypeTree::Child<LFSV,_1>;
        const auto& lfsv_p = child(lfsv,_1);
        const auto& lfsu_p = child(lfsu,_1);
 
        // domain and range field type
        using FESwitch_V = FiniteElementInterfaceSwitch<typename LFSV_V::Traits::FiniteElementType >;
        using BasisSwitch_V = BasisInterfaceSwitch<typename FESwitch_V::Basis >;
        using VectorBasisSwitch_V = VectorBasisInterfaceSwitch<typename FESwitch_V::Basis >;
        using FESwitch_P = FiniteElementInterfaceSwitch<typename LFSV_P::Traits::FiniteElementType >;
        using BasisSwitch_P = BasisInterfaceSwitch<typename FESwitch_P::Basis >;
        using RF = typename BasisSwitch_V::RangeField;
        using Range_V = typename BasisSwitch_V::Range;
        using Range_P = typename BasisSwitch_P::Range;
        using size_type = typename LFSV::Traits::SizeType;
 
        // Get geometry
        auto geo = eg.geometry();
 
        // Determine quadrature order
        const int v_order = FESwitch_V::basis(lfsv_v.finiteElement()).order();
        const int det_jac_order = geo.type().isSimplex() ? 0 : (dim-1);
        const int jac_order = geo.type().isSimplex() ? 0 : 1;
        const int qorder = 3*v_order - 1 + jac_order + det_jac_order + superintegration_order;
 
        const RF incomp_scaling = prm.incompressibilityScaling(current_dt);
 
        // loop over quadrature points
        for (const auto& ip : quadratureRule(geo,qorder))
          {
            auto local = ip.position();
            auto mu = prm.mu(eg,local);
            auto rho = prm.rho(eg,local);
 
            // compute u (if Navier term enabled)
            std::vector<Range_V> phi_v(lfsv_v.size());
            Range_V val_u(0.0);
            if(navier) {
              FESwitch_V::basis(lfsv_v.finiteElement()).evaluateFunction(local,phi_v);
              for (size_type i=0; i<lfsu_v.size(); i++)
                val_u.axpy(x(lfsu_v,i),phi_v[i]);
            }
 
            // values of pressure shape functions
            std::vector<Range_P> phi_p(lfsv_p.size());
            FESwitch_P::basis(lfsv_p.finiteElement()).evaluateFunction(local,phi_p);
 
            // compute pressure value
            Range_P val_p(0.0);
            for (size_type i=0; i<lfsu_p.size(); i++)
              val_p.axpy(x(lfsu_p,i),phi_p[i]);
 
            // evaluate jacobian of velocity shape functions on reference element
            std::vector<Dune::FieldMatrix<RF,dim,dim> > jac_phi_v(lfsu_v.size());
            VectorBasisSwitch_V::jacobian
              (FESwitch_V::basis(lfsv_v.finiteElement()), geo, local, jac_phi_v);
 
            // compute divergence of test functions
            std::vector<RF> div_phi_v(lfsv_v.size(),0.0);
            for (size_type i=0; i<lfsv_v.size(); i++)
              for (size_type d=0; d<dim; d++)
                div_phi_v[i] += jac_phi_v[i][d][d];
 
            // compute velocity jacobian and divergence
            Dune::FieldMatrix<RF,dim,dim> jac_u(0.0);
            RF div_u(0.0);
            for (size_type i=0; i<lfsu_v.size(); i++){
              jac_u.axpy(x(lfsu_v,i),jac_phi_v[i]);
              div_u += x(lfsu_v,i) * div_phi_v[i];
            }
 
            auto detj = geo.integrationElement(ip.position());
            auto weight = ip.weight() * detj;
 
            for (size_type i=0; i<lfsv_v.size(); i++) {
              //================================================//
              // \int (mu*grad_u*grad_v)
              //================================================//
              RF dvdu(0);
              contraction(jac_u,jac_phi_v[i],dvdu);
              r.accumulate(lfsv_v, i, dvdu * mu * weight);
 
              //================================================//
              // \int -p \nabla\cdot v
              //================================================//
              r.accumulate(lfsv_v, i, - div_phi_v[i] * val_p * weight);
 
              //================================================//
              // \int \rho ((u\cdot\nabla ) u )\cdot v
              //================================================//
              if(navier) {
                // compute (grad u) u (matrix-vector product)
                Range_V nabla_u_u(0.0);
                jac_u.mv(val_u,nabla_u_u);
                r.accumulate(lfsv_v, i, rho * (nabla_u_u*phi_v[i]) * weight);
              } // end navier
 
            } // end i
 
            for (size_type i=0; i<lfsv_p.size(); i++) {
              //================================================//
              // \int -q \nabla\cdot u
              //================================================//
              r.accumulate(lfsv_p, i, - div_u * phi_p[i] * incomp_scaling * weight);
            }
 
          } // end loop quadrature points
      } // end alpha_volume
 
      // jacobian of volume term
      template<typename EG, typename LFSU, typename X, typename LFSV,
               typename LocalMatrix>
      void jacobian_volume (const EG& eg, const LFSU& lfsu, const X& x, const LFSV& lfsv,
                            LocalMatrix& mat) const
      {
        const unsigned int dim = EG::Geometry::mydimension;
 
        // subspaces
        using namespace Indices;
        using LFSV_V = TypeTree::Child<LFSV,_0>;
        const auto& lfsv_v = child(lfsv,_0);
        const auto& lfsu_v = child(lfsu,_0);
 
        using LFSV_P = TypeTree::Child<LFSV,_1>;
        const auto& lfsv_p = child(lfsv,_1);
        const auto& lfsu_p = child(lfsu,_1);
 
        // domain and range field type
        using FESwitch_V = FiniteElementInterfaceSwitch<typename LFSV_V::Traits::FiniteElementType >;
        using BasisSwitch_V = BasisInterfaceSwitch<typename FESwitch_V::Basis >;
        using VectorBasisSwitch_V = VectorBasisInterfaceSwitch<typename FESwitch_V::Basis >;
        using FESwitch_P = FiniteElementInterfaceSwitch<typename LFSV_P::Traits::FiniteElementType >;
        using BasisSwitch_P = BasisInterfaceSwitch<typename FESwitch_P::Basis >;
        using RF = typename BasisSwitch_V::RangeField;
        using Range_V = typename BasisSwitch_V::Range;
        using Range_P = typename BasisSwitch_P::Range;
        using size_type = typename LFSV::Traits::SizeType;
 
        // Get geometry
        auto geo = eg.geometry();
 
        // Determine quadrature order
        const int v_order = FESwitch_V::basis(lfsv_v.finiteElement()).order();
        const int det_jac_order = geo.type().isSimplex() ? 0 : (dim-1);
        const int jac_order = geo.type().isSimplex() ? 0 : 1;
        const int qorder = 3*v_order - 1 + jac_order + det_jac_order + superintegration_order;
 
        const RF incomp_scaling = prm.incompressibilityScaling(current_dt);
 
        // loop over quadrature points
        for (const auto& ip : quadratureRule(geo,qorder))
          {
            auto local = ip.position();
            auto mu = prm.mu(eg,local);
            auto rho = prm.rho(eg,local);
 
            // compute u (if Navier term enabled)
            std::vector<Range_V> phi_v(lfsv_v.size());
            Range_V val_u(0.0);
            if(navier) {
              FESwitch_V::basis(lfsv_v.finiteElement()).evaluateFunction(local,phi_v);
              for (size_type i=0; i<lfsu_v.size(); i++)
                val_u.axpy(x(lfsu_v,i),phi_v[i]);
            }
 
            // values of pressure shape functions
            std::vector<Range_P> phi_p(lfsv_p.size());
            FESwitch_P::basis(lfsv_p.finiteElement()).evaluateFunction(local,phi_p);
 
            // evaluate jacobian of velocity shape functions on reference element
            std::vector<Dune::FieldMatrix<RF,dim,dim> > jac_phi_v(lfsu_v.size());
            VectorBasisSwitch_V::jacobian
              (FESwitch_V::basis(lfsv_v.finiteElement()), geo, local, jac_phi_v);
 
            assert(lfsu_v.size() == lfsv_v.size());
            // compute divergence of velocity shape functions
            std::vector<RF> div_phi_v(lfsv_v.size(),0.0);
            for (size_type i=0; i<lfsv_v.size(); i++)
              for (size_type d=0; d<dim; d++)
                div_phi_v[i] += jac_phi_v[i][d][d];
 
            // compute velocity jacobian (if Navier term enabled)
            Dune::FieldMatrix<RF,dim,dim> jac_u(0.0);
            if(navier) {
              for (size_type i=0; i<lfsu_v.size(); i++){
                jac_u.axpy(x(lfsu_v,i),jac_phi_v[i]);
              }
            }
 
            auto detj = geo.integrationElement(ip.position());
            auto weight = ip.weight() * detj;
 
            for(size_type i=0; i<lfsv_v.size(); i++) {
 
              for(size_type j=0; j<lfsu_v.size(); j++) {
                //================================================//
                // \int (mu*grad_u*grad_v)
                //================================================//
                RF dvdu(0.0);
                contraction(jac_phi_v[j],jac_phi_v[i],dvdu);
                mat.accumulate(lfsv_v,i,lfsu_v,j, mu * dvdu * weight);
 
                //================================================//
                // \int \rho ((u\cdot\nabla ) u )\cdot v
                //================================================//
                if(navier) {
                  // compute (grad u) phi_v_j (matrix-vector product)
                  Range_V nabla_u_phi_v_j(0.0);
                  jac_u.mv(phi_v[j],nabla_u_phi_v_j);
                  // compute (grad phi_v_j) u (matrix-vector product)
                  Range_V nabla_phi_v_j_u(0.0);
                  jac_phi_v[j].mv(val_u,nabla_phi_v_j_u);
                  mat.accumulate(lfsv_v,i,lfsu_v,j, rho * ((nabla_u_phi_v_j*phi_v[i]) + (nabla_phi_v_j_u*phi_v[i])) * weight);
                } // end navier
 
              } // end j
 
              for(size_type j=0; j<lfsv_p.size(); j++) {
                //================================================//
                // - p * div v
                // - q * div u
                //================================================//
                mat.accumulate(lfsv_v,i,lfsu_p,j, -phi_p[j] * div_phi_v[i] * weight);
                mat.accumulate(lfsv_p,j,lfsu_v,i, -phi_p[j] * div_phi_v[i] * incomp_scaling * weight);
              }
            } // end i
 
          } // end loop quadrature points
      } // end jacobian_volume
 
      // skeleton term, each face is only visited ONCE
      template<typename IG, typename LFSU, typename X, typename LFSV, typename R>
      void alpha_skeleton (const IG& ig,
                           const LFSU& lfsu_s, const X& x_s, const LFSV& lfsv_s,
                           const LFSU& lfsu_n, const X& x_n, const LFSV& lfsv_n,
                           R& r_s, R& r_n) const
      {
        // dimensions
        const unsigned int dim = IG::Entity::dimension;
        const unsigned int dimw = IG::coorddimension;
 
        // subspaces
        using namespace Indices;
        using LFSV_V = TypeTree::Child<LFSV,_0>;
        const auto& lfsv_v_s = child(lfsv_s,_0);
        const auto& lfsu_v_s = child(lfsu_s,_0);
        const auto& lfsv_v_n = child(lfsv_n,_0);
        const auto& lfsu_v_n = child(lfsu_n,_0);
 
        using LFSV_P = TypeTree::Child<LFSV,_1>;
        const auto& lfsv_p_s = child(lfsv_s,_1);
        const auto& lfsu_p_s = child(lfsu_s,_1);
        const auto& lfsv_p_n = child(lfsv_n,_1);
        const auto& lfsu_p_n = child(lfsu_n,_1);
 
        // domain and range field type
        using FESwitch_V = FiniteElementInterfaceSwitch<typename LFSV_V::Traits::FiniteElementType >;
        using BasisSwitch_V = BasisInterfaceSwitch<typename FESwitch_V::Basis >;
        using VectorBasisSwitch_V = VectorBasisInterfaceSwitch<typename FESwitch_V::Basis >;
        using FESwitch_P = FiniteElementInterfaceSwitch<typename LFSV_P::Traits::FiniteElementType >;
        using BasisSwitch_P = BasisInterfaceSwitch<typename FESwitch_P::Basis >;
        using DF = typename BasisSwitch_V::DomainField;
        using RF = typename BasisSwitch_V::RangeField;
        using Range_V = typename BasisSwitch_V::Range;
        using Range_P = typename BasisSwitch_P::Range;
        using size_type = typename LFSV::Traits::SizeType;
 
        // References to inside and outside cells
        const auto& cell_inside = ig.inside();
        const auto& cell_outside = ig.outside();
 
        // Get geometries
        auto geo = ig.geometry();
        auto geo_inside = cell_inside.geometry();
        auto geo_outside = cell_outside.geometry();
 
        // Get geometry of intersection in local coordinates of cell_inside and cell_outside
        auto geo_in_inside = ig.geometryInInside();
        auto geo_in_outside = ig.geometryInOutside();
 
        // Determine quadrature order
        const int v_order = FESwitch_V::basis(lfsv_v_s.finiteElement()).order();
        const int det_jac_order = geo.type().isSimplex() ? 0 : (dim-2);
        const int qorder = 2*v_order + det_jac_order + superintegration_order;
 
        const int epsilon = prm.epsilonIPSymmetryFactor();
        const RF incomp_scaling = prm.incompressibilityScaling(current_dt);
 
        auto penalty_factor = prm.getFaceIP(geo,geo_inside,geo_outside);
 
        // loop over quadrature points and integrate normal flux
        for (const auto& ip : quadratureRule(geo,qorder))
          {
 
            // position of quadrature point in local coordinates of element
            auto local_s = geo_in_inside.global(ip.position());
            auto local_n = geo_in_outside.global(ip.position());
 
            // values of velocity shape functions
            std::vector<Range_V> phi_v_s(lfsv_v_s.size());
            std::vector<Range_V> phi_v_n(lfsv_v_n.size());
            FESwitch_V::basis(lfsv_v_s.finiteElement()).evaluateFunction(local_s,phi_v_s);
            FESwitch_V::basis(lfsv_v_n.finiteElement()).evaluateFunction(local_n,phi_v_n);
 
            // values of pressure shape functions
            std::vector<Range_P> phi_p_s(lfsv_p_s.size());
            std::vector<Range_P> phi_p_n(lfsv_p_n.size());
            FESwitch_P::basis(lfsv_p_s.finiteElement()).evaluateFunction(local_s,phi_p_s);
            FESwitch_P::basis(lfsv_p_n.finiteElement()).evaluateFunction(local_n,phi_p_n);
 
            // compute pressure value
            Range_P val_p_s(0.0);
            Range_P val_p_n(0.0);
            for (size_type i=0; i<lfsu_p_s.size(); i++)
              val_p_s.axpy(x_s(lfsu_p_s,i),phi_p_s[i]);
            for (size_type i=0; i<lfsu_p_n.size(); i++)
              val_p_n.axpy(x_n(lfsu_p_n,i),phi_p_n[i]);
 
            // evaluate jacobian of velocity shape functions on reference element
            std::vector<Dune::FieldMatrix<RF,dim,dim> > jac_phi_v_s(lfsu_v_s.size());
            std::vector<Dune::FieldMatrix<RF,dim,dim> > jac_phi_v_n(lfsu_v_n.size());
            VectorBasisSwitch_V::jacobian
              (FESwitch_V::basis(lfsv_v_s.finiteElement()), geo_inside, local_s, jac_phi_v_s);
            VectorBasisSwitch_V::jacobian
              (FESwitch_V::basis(lfsv_v_n.finiteElement()), geo_outside, local_n, jac_phi_v_n);
 
            // compute velocity value, jacobian, and divergence
            Range_V val_u_s(0.0);
            Range_V val_u_n(0.0);
            Dune::FieldMatrix<RF,dim,dim> jac_u_s(0.0);
            Dune::FieldMatrix<RF,dim,dim> jac_u_n(0.0);
            for (size_type i=0; i<lfsu_v_s.size(); i++){
              val_u_s.axpy(x_s(lfsu_v_s,i),phi_v_s[i]);
              jac_u_s.axpy(x_s(lfsu_v_s,i),jac_phi_v_s[i]);
            }
            for (size_type i=0; i<lfsu_v_n.size(); i++){
              val_u_n.axpy(x_n(lfsu_v_n,i),phi_v_n[i]);
              jac_u_n.axpy(x_n(lfsu_v_n,i),jac_phi_v_n[i]);
            }
 
            auto normal = ig.unitOuterNormal(ip.position());
            auto weight = ip.weight()*geo.integrationElement(ip.position());
            auto mu = prm.mu(ig,ip.position());
 
            auto factor = mu * weight;
 
            // compute jump in velocity
            auto jump = val_u_s - val_u_n;
 
            // compute mean in pressure
            auto mean_p = 0.5*(val_p_s + val_p_n);
 
            // compute flux of velocity jacobian
            Dune::FieldVector<DF,dimw> flux_jac_u(0.0);
            add_compute_flux(jac_u_s,normal,flux_jac_u);
            add_compute_flux(jac_u_n,normal,flux_jac_u);
            flux_jac_u *= 0.5;
 
            // loop over test functions, same element
            for (size_t i=0; i<lfsv_v_s.size(); i++) {
              //================================================//
              // diffusion term
              //================================================//
              r_s.accumulate(lfsv_v_s, i, -(flux_jac_u * phi_v_s[i]) * factor);
 
              //================================================//
              // (non-)symmetric IP term
              //================================================//
              Dune::FieldVector<DF,dimw> flux_jac_phi(0.0);
              add_compute_flux(jac_phi_v_s[i],normal,flux_jac_phi);
              r_s.accumulate(lfsv_v_s, i, epsilon * 0.5 * (flux_jac_phi * jump) * factor);
 
              //================================================//
              // standard IP term integral
              //================================================//
              r_s.accumulate(lfsv_v_s,i, penalty_factor * (jump*phi_v_s[i]) * factor);
 
              //================================================//
              // pressure-velocity-coupling in momentum equation
              //================================================//
              r_s.accumulate(lfsv_v_s,i, mean_p * (phi_v_s[i]*normal) * weight);
            }
 
            // loop over test functions, neighbour element
            for (size_t i=0; i<lfsv_v_n.size(); i++) {
              //================================================//
              // diffusion term
              //================================================//
              r_n.accumulate(lfsv_v_n, i,  (flux_jac_u * phi_v_n[i]) * factor);
 
              //================================================//
              // (non-)symmetric IP term
              //================================================//
              Dune::FieldVector<DF,dimw> flux_jac_phi(0.0);
              add_compute_flux(jac_phi_v_n[i],normal,flux_jac_phi);
              r_n.accumulate(lfsv_v_n, i, epsilon * 0.5 * (flux_jac_phi * jump) * factor);
 
              //================================================//
              // standard IP term integral
              //================================================//
              r_n.accumulate(lfsv_v_n,i, -penalty_factor * (jump*phi_v_n[i]) * factor);
 
              //================================================//
              // pressure-velocity-coupling in momentum equation
              //================================================//
              r_n.accumulate(lfsv_v_n,i, -mean_p * (phi_v_n[i]*normal) * weight);
            }
 
            //================================================//
            // incompressibility constraint
            //================================================//
            for (size_t i=0; i<lfsv_p_s.size(); i++)
              r_s.accumulate(lfsv_p_s,i, 0.5*phi_p_s[i] * (jump*normal) * incomp_scaling * weight);
            for (size_t i=0; i<lfsv_p_n.size(); i++)
              r_n.accumulate(lfsv_p_n,i, 0.5*phi_p_n[i] * (jump*normal) * incomp_scaling * weight);
 
          } // end loop quadrature points
      } // end alpha_skeleton
 
      // jacobian of skeleton term, each face is only visited ONCE
      template<typename IG, typename LFSU, typename X, typename LFSV,
               typename LocalMatrix>
      void jacobian_skeleton (const IG& ig,
                              const LFSU& lfsu_s, const X&, const LFSV& lfsv_s,
                              const LFSU& lfsu_n, const X&, const LFSV& lfsv_n,
                              LocalMatrix& mat_ss, LocalMatrix& mat_sn,
                              LocalMatrix& mat_ns, LocalMatrix& mat_nn) const
      {
        // dimensions
        const unsigned int dim = IG::Entity::dimension;
        const unsigned int dimw = IG::coorddimension;
 
        // subspaces
        using namespace Indices;
        using LFSV_V = TypeTree::Child<LFSV,_0>;
        const auto& lfsv_v_s = child(lfsv_s,_0);
        const auto& lfsu_v_s = child(lfsu_s,_0);
        const auto& lfsv_v_n = child(lfsv_n,_0);
        const auto& lfsu_v_n = child(lfsu_n,_0);
 
        using LFSV_P = TypeTree::Child<LFSV,_1>;
        const auto& lfsv_p_s = child(lfsv_s,_1);
        const auto& lfsu_p_s = child(lfsu_s,_1);
        const auto& lfsv_p_n = child(lfsv_n,_1);
        const auto& lfsu_p_n = child(lfsu_n,_1);
 
        // domain and range field type
        using FESwitch_V = FiniteElementInterfaceSwitch<typename LFSV_V::Traits::FiniteElementType >;
        using BasisSwitch_V = BasisInterfaceSwitch<typename FESwitch_V::Basis >;
        using VectorBasisSwitch_V = VectorBasisInterfaceSwitch<typename FESwitch_V::Basis >;
        using FESwitch_P = FiniteElementInterfaceSwitch<typename LFSV_P::Traits::FiniteElementType >;
        using BasisSwitch_P = BasisInterfaceSwitch<typename FESwitch_P::Basis >;
        using DF = typename BasisSwitch_V::DomainField;
        using RF = typename BasisSwitch_V::RangeField;
        using Range_V = typename BasisSwitch_V::Range;
        using Range_P = typename BasisSwitch_P::Range;
        using size_type = typename LFSV::Traits::SizeType;
 
        // References to inside and outside cells
        auto const& cell_inside = ig.inside();
        auto const& cell_outside = ig.outside();
 
        // Get geometries
        auto geo = ig.geometry();
        auto geo_inside = cell_inside.geometry();
        auto geo_outside = cell_outside.geometry();
 
        // Get geometry of intersection in local coordinates of cell_inside and cell_outside
        auto geo_in_inside = ig.geometryInInside();
        auto geo_in_outside = ig.geometryInOutside();
 
        // Determine quadrature order
        const int v_order = FESwitch_V::basis(lfsv_v_s.finiteElement()).order();
        const int det_jac_order = geo.type().isSimplex() ? 0 : (dim-2);
        const int qorder = 2*v_order + det_jac_order + superintegration_order;
 
        auto epsilon = prm.epsilonIPSymmetryFactor();
        auto incomp_scaling = prm.incompressibilityScaling(current_dt);
 
        auto penalty_factor = prm.getFaceIP(geo,geo_inside,geo_outside);
 
        // loop over quadrature points and integrate normal flux
        for (const auto& ip : quadratureRule(geo,qorder))
          {
 
            // position of quadrature point in local coordinates of element
            auto local_s = geo_in_inside.global(ip.position());
            auto local_n = geo_in_outside.global(ip.position());
 
            // values of velocity shape functions
            std::vector<Range_V> phi_v_s(lfsv_v_s.size());
            std::vector<Range_V> phi_v_n(lfsv_v_n.size());
            FESwitch_V::basis(lfsv_v_s.finiteElement()).evaluateFunction(local_s,phi_v_s);
            FESwitch_V::basis(lfsv_v_n.finiteElement()).evaluateFunction(local_n,phi_v_n);
 
            // values of pressure shape functions
            std::vector<Range_P> phi_p_s(lfsv_p_s.size());
            std::vector<Range_P> phi_p_n(lfsv_p_n.size());
            FESwitch_P::basis(lfsv_p_s.finiteElement()).evaluateFunction(local_s,phi_p_s);
            FESwitch_P::basis(lfsv_p_n.finiteElement()).evaluateFunction(local_n,phi_p_n);
 
            // evaluate jacobian of velocity shape functions on reference element
            std::vector<Dune::FieldMatrix<RF,dim,dim> > jac_phi_v_s(lfsu_v_s.size());
            std::vector<Dune::FieldMatrix<RF,dim,dim> > jac_phi_v_n(lfsu_v_n.size());
            VectorBasisSwitch_V::jacobian
              (FESwitch_V::basis(lfsv_v_s.finiteElement()), geo_inside, local_s, jac_phi_v_s);
            VectorBasisSwitch_V::jacobian
              (FESwitch_V::basis(lfsv_v_n.finiteElement()), geo_outside, local_n, jac_phi_v_n);
 
            auto normal = ig.unitOuterNormal(ip.position());
            auto weight = ip.weight()*geo.integrationElement(ip.position());
            auto mu = prm.mu(ig,ip.position());
 
            auto factor = mu * weight;
 
            //============================================
            // loop over test functions, same element
            //============================================
            for(size_type i=0; i<lfsv_v_s.size(); i++) {
 
              // compute flux
              Dune::FieldVector<DF,dimw> flux_jac_phi_i(0.0);
              add_compute_flux(jac_phi_v_s[i],normal,flux_jac_phi_i);
 
              //============================================
              // diffusion
              // (non-)symmetric IP-Term
              // standard IP integral
              //============================================
              for(size_type j=0; j<lfsu_v_s.size(); j++) {
                Dune::FieldVector<DF,dimw> flux_jac_phi_j(0.0);
                add_compute_flux(jac_phi_v_s[j],normal,flux_jac_phi_j);
 
                mat_ss.accumulate(lfsv_v_s,i,lfsu_v_s,j, -0.5 * (flux_jac_phi_j*phi_v_s[i]) * factor);
                mat_ss.accumulate(lfsv_v_s,i,lfsu_v_s,j, epsilon * 0.5 * (flux_jac_phi_i*phi_v_s[j]) * factor);
                mat_ss.accumulate(lfsv_v_s,i,lfsu_v_s,j, penalty_factor * (phi_v_s[j]*phi_v_s[i]) * factor);
              }
 
              for(size_type j=0; j<lfsu_v_n.size(); j++) {
                Dune::FieldVector<DF,dimw> flux_jac_phi_j(0.0);
                add_compute_flux(jac_phi_v_n[j],normal,flux_jac_phi_j);
 
                mat_sn.accumulate(lfsv_v_s,i,lfsu_v_n,j, -0.5 * (flux_jac_phi_j*phi_v_s[i]) * factor);
                mat_sn.accumulate(lfsv_v_s,i,lfsu_v_n,j, -epsilon * 0.5 * (flux_jac_phi_i*phi_v_n[j]) * factor);
                mat_sn.accumulate(lfsv_v_s,i,lfsu_v_n,j, -penalty_factor * (phi_v_n[j]*phi_v_s[i]) * factor);
              }
 
              //============================================
              // pressure-velocity coupling in momentum equation
              //============================================
              for(size_type j=0; j<lfsu_p_s.size(); j++) {
                mat_ss.accumulate(lfsv_v_s,i,lfsu_p_s,j, 0.5*phi_p_s[j] * (phi_v_s[i]*normal) * weight);
              }
 
              for(size_type j=0; j<lfsu_p_n.size(); j++) {
                mat_sn.accumulate(lfsv_v_s,i,lfsu_p_n,j, 0.5*phi_p_n[j] * (phi_v_s[i]*normal) * weight);
              }
            } // end i (same)
 
            //============================================
            // loop over test functions, neighbour element
            //============================================
            for(size_type i=0; i<lfsv_v_n.size(); i++) {
 
              // compute flux
              Dune::FieldVector<DF,dimw> flux_jac_phi_i(0.0);
              add_compute_flux(jac_phi_v_n[i],normal,flux_jac_phi_i);
 
              //============================================
              // diffusion
              // (non-)symmetric IP-Term
              // standard IP integral
              //============================================
              for(size_type j=0; j<lfsu_v_s.size(); j++) {
                Dune::FieldVector<DF,dimw> flux_jac_phi_j(0.0);
                add_compute_flux(jac_phi_v_s[j],normal,flux_jac_phi_j);
 
                mat_ns.accumulate(lfsv_v_n,i,lfsu_v_s,j, 0.5 * (flux_jac_phi_j*phi_v_n[i]) * factor);
                mat_ns.accumulate(lfsv_v_n,i,lfsu_v_s,j, epsilon * 0.5 * (flux_jac_phi_i*phi_v_s[j]) * factor);
                mat_ns.accumulate(lfsv_v_n,i,lfsu_v_s,j, -penalty_factor * (phi_v_s[j]*phi_v_n[i]) * factor);
              }
 
              for(size_type j=0; j<lfsu_v_n.size(); j++) {
                Dune::FieldVector<DF,dimw> flux_jac_phi_j(0.0);
                add_compute_flux(jac_phi_v_n[j],normal,flux_jac_phi_j);
 
                mat_nn.accumulate(lfsv_v_n,i,lfsu_v_n,j, 0.5 * (flux_jac_phi_j*phi_v_n[i]) * factor);
                mat_nn.accumulate(lfsv_v_n,i,lfsu_v_n,j, -epsilon * 0.5 * (flux_jac_phi_i*phi_v_n[j]) * factor);
                mat_nn.accumulate(lfsv_v_n,i,lfsu_v_n,j, penalty_factor * (phi_v_n[j]*phi_v_n[i]) * factor);
              }
 
              //============================================
              // pressure-velocity coupling in momentum equation
              //============================================
              for(size_type j=0; j<lfsu_p_s.size(); j++) {
                mat_ns.accumulate(lfsv_v_n,i,lfsu_p_s,j, -0.5*phi_p_s[j] * (phi_v_n[i]*normal) * weight);
              }
 
              for(size_type j=0; j<lfsu_p_n.size(); j++) {
                mat_nn.accumulate(lfsv_v_n,i,lfsu_p_n,j, -0.5*phi_p_n[j] * (phi_v_n[i]*normal) * weight);
              }
            } // end i (neighbour)
 
            //================================================//
            // \int <q> [u] n
            //================================================//
            for(size_type i=0; i<lfsv_p_s.size(); i++) {
              for(size_type j=0; j<lfsu_v_s.size(); j++)
                mat_ss.accumulate(lfsv_p_s,i,lfsu_v_s,j, 0.5*phi_p_s[i] * (phi_v_s[j]*normal) * incomp_scaling * weight);
 
              for(size_type j=0; j<lfsu_v_n.size(); j++)
                mat_sn.accumulate(lfsv_p_s,i,lfsu_v_n,j, -0.5*phi_p_s[i] * (phi_v_n[j]*normal) * incomp_scaling * weight);
            }
 
            for(size_type i=0; i<lfsv_p_n.size(); i++) {
              for(size_type j=0; j<lfsu_v_s.size(); j++)
                mat_ns.accumulate(lfsv_p_n,i,lfsu_v_s,j, 0.5*phi_p_n[i] * (phi_v_s[j]*normal) * incomp_scaling * weight);
 
              for(size_type j=0; j<lfsu_v_n.size(); j++)
                mat_nn.accumulate(lfsv_p_n,i,lfsu_v_n,j, -0.5*phi_p_n[i] * (phi_v_n[j]*normal) * incomp_scaling * weight);
            }
 
          } // end loop quadrature points
      } // end jacobian_skeleton
 
      // boundary term
      template<typename IG, typename LFSU, typename X, typename LFSV, typename R>
      void alpha_boundary (const IG& ig,
                           const LFSU& lfsu, const X& x, const LFSV& lfsv,
                           R& r) const
      {
        // dimensions
        const unsigned int dim = IG::Entity::dimension;
        const unsigned int dimw = IG::coorddimension;
 
        // subspaces
        using namespace Indices;
        using LFSV_V = TypeTree::Child<LFSV,_0>;
        const auto& lfsv_v = child(lfsv,_0);
        const auto& lfsu_v = child(lfsu,_0);
 
        using LFSV_P = TypeTree::Child<LFSV,_1>;
        const auto& lfsv_p = child(lfsv,_1);
        const auto& lfsu_p = child(lfsu,_1);
 
        // domain and range field type
        using FESwitch_V = FiniteElementInterfaceSwitch<typename LFSV_V::Traits::FiniteElementType >;
        using BasisSwitch_V = BasisInterfaceSwitch<typename FESwitch_V::Basis >;
        using VectorBasisSwitch_V = VectorBasisInterfaceSwitch<typename FESwitch_V::Basis >;
        using FESwitch_P = FiniteElementInterfaceSwitch<typename LFSV_P::Traits::FiniteElementType >;
        using BasisSwitch_P = BasisInterfaceSwitch<typename FESwitch_P::Basis >;
        using DF = typename BasisSwitch_V::DomainField;
        using RF = typename BasisSwitch_V::RangeField;
        using Range_V = typename BasisSwitch_V::Range;
        using Range_P = typename BasisSwitch_P::Range;
        using size_type = typename LFSV::Traits::SizeType;
 
        // References to inside cell
        const auto& cell_inside = ig.inside();
 
        // Get geometries
        auto geo = ig.geometry();
        auto geo_inside = cell_inside.geometry();
 
        // Get geometry of intersection in local coordinates of cell_inside
        auto geo_in_inside = ig.geometryInInside();
 
        // Determine quadrature order
        const int v_order = FESwitch_V::basis(lfsv_v.finiteElement()).order();
        const int det_jac_order = geo.type().isSimplex() ? 0 : (dim-1);
        const int qorder = 2*v_order + det_jac_order + superintegration_order;
 
        auto epsilon = prm.epsilonIPSymmetryFactor();
        auto incomp_scaling = prm.incompressibilityScaling(current_dt);
 
        auto penalty_factor = prm.getFaceIP(geo,geo_inside);
 
        // loop over quadrature points and integrate normal flux
        for (const auto& ip : quadratureRule(geo,qorder))
          {
            // position of quadrature point in local coordinates of element
            auto local = geo_in_inside.global(ip.position());
 
            // values of velocity shape functions
            std::vector<Range_V> phi_v(lfsv_v.size());
            FESwitch_V::basis(lfsv_v.finiteElement()).evaluateFunction(local,phi_v);
 
            // values of pressure shape functions
            std::vector<Range_P> phi_p(lfsv_p.size());
            FESwitch_P::basis(lfsv_p.finiteElement()).evaluateFunction(local,phi_p);
 
            // evaluate jacobian of basis functions on reference element
            std::vector<Dune::FieldMatrix<RF,dim,dim> > jac_phi_v(lfsu_v.size());
            VectorBasisSwitch_V::jacobian
              (FESwitch_V::basis(lfsv_v.finiteElement()), geo_inside, local, jac_phi_v);
 
            // compute pressure value
            Range_P val_p(0.0);
            for (size_type i=0; i<lfsu_p.size(); i++)
              val_p.axpy(x(lfsu_p,i),phi_p[i]);
 
            // compute u and velocity jacobian
            Range_V val_u(0.0);
            Dune::FieldMatrix<RF,dim,dim> jac_u(0.0);
            for (size_type i=0; i<lfsu_v.size(); i++){
              val_u.axpy(x(lfsu_v,i),phi_v[i]);
              jac_u.axpy(x(lfsu_v,i),jac_phi_v[i]);
            }
 
            auto normal = ig.unitOuterNormal(ip.position());
            auto weight = ip.weight()*geo.integrationElement(ip.position());
            auto mu = prm.mu(ig,ip.position());
 
            // evaluate boundary condition type
            auto bctype(prm.bctype(ig,ip.position()));
 
            if (bctype == BC::VelocityDirichlet) {
              // compute jump relative to Dirichlet value
              auto u0(prm.g(cell_inside,local));
              auto jump = val_u - u0;
 
              // compute flux of velocity jacobian
              Dune::FieldVector<DF,dimw> flux_jac_u(0.0);
              add_compute_flux(jac_u,normal,flux_jac_u);
 
              for (size_t i=0; i<lfsv_v.size(); i++) {
                //================================================//
                // diffusion term
                //================================================//
                r.accumulate(lfsv_v,i, -mu * (flux_jac_u * phi_v[i]) * weight);
 
                //================================================//
                // (non-)symmetric IP term
                //================================================//
                Dune::FieldVector<DF,dimw> flux_jac_phi(0.0);
                add_compute_flux(jac_phi_v[i],normal,flux_jac_phi);
                r.accumulate(lfsv_v,i, mu * epsilon * (flux_jac_phi*jump) * weight);
 
                //================================================//
                // standard IP term integral
                //================================================//
                r.accumulate(lfsv_v,i, mu * (jump*phi_v[i]) * penalty_factor * weight);
 
                //================================================//
                // pressure-velocity-coupling in momentum equation
                //================================================//
                r.accumulate(lfsv_v,i, val_p * (phi_v[i]*normal) * weight);
              } // end i
 
              //================================================//
              // incompressibility constraint
              //================================================//
              for(size_type i=0; i<lfsv_p.size(); i++) {
                r.accumulate(lfsv_p,i, phi_p[i] * (jump*normal) * incomp_scaling * weight);
              }
            } // Velocity Dirichlet
 
            if (bctype == BC::StressNeumann) {
              auto stress(prm.j(ig,ip.position(),normal));
 
              for(size_type i=0; i<lfsv_v.size(); i++) {
                r.accumulate(lfsv_v,i, (stress*phi_v[i]) * weight);
              }
            } // Pressure Dirichlet
 
          } // end loop quadrature points
      } // end alpha_boundary
 
      // jacobian of boundary term
      template<typename IG, typename LFSU, typename X, typename LFSV,
               typename LocalMatrix>
      void jacobian_boundary (const IG& ig,
                              const LFSU& lfsu, const X& x, const LFSV& lfsv,
                              LocalMatrix& mat) const
      {
        // dimensions
        const unsigned int dim = IG::Entity::dimension;
        const unsigned int dimw = IG::coorddimension;
 
        // subspaces
        using namespace Indices;
        using LFSV_V = TypeTree::Child<LFSV,_0>;
        const auto& lfsv_v = child(lfsv,_0);
        const auto& lfsu_v = child(lfsu,_0);
 
        using LFSV_P = TypeTree::Child<LFSV,_1>;
        const auto& lfsv_p = child(lfsv,_1);
        const auto& lfsu_p = child(lfsu,_1);
 
        // domain and range field type
        using FESwitch_V = FiniteElementInterfaceSwitch<typename LFSV_V::Traits::FiniteElementType >;
        using BasisSwitch_V = BasisInterfaceSwitch<typename FESwitch_V::Basis >;
        using VectorBasisSwitch_V = VectorBasisInterfaceSwitch<typename FESwitch_V::Basis >;
        using FESwitch_P = FiniteElementInterfaceSwitch<typename LFSV_P::Traits::FiniteElementType >;
        using BasisSwitch_P = BasisInterfaceSwitch<typename FESwitch_P::Basis >;
        using DF = typename BasisSwitch_V::DomainField;
        using RF = typename BasisSwitch_V::RangeField;
        using Range_V = typename BasisSwitch_V::Range;
        using Range_P = typename BasisSwitch_P::Range;
        using size_type = typename LFSV::Traits::SizeType;
 
        // References to inside cell
        const auto& cell_inside = ig.inside();
 
        // Get geometries
        auto geo = ig.geometry();
        auto geo_inside = cell_inside.geometry();
 
        // Get geometry of intersection in local coordinates of cell_inside
        auto geo_in_inside = ig.geometryInInside();
 
        // Determine quadrature order
        const int v_order = FESwitch_V::basis(lfsv_v.finiteElement()).order();
        const int det_jac_order = geo.type().isSimplex() ? 0 : (dim-1);
        const int qorder = 2*v_order + det_jac_order + superintegration_order;
 
        auto epsilon = prm.epsilonIPSymmetryFactor();
        auto incomp_scaling = prm.incompressibilityScaling(current_dt);
 
        auto penalty_factor = prm.getFaceIP(geo,geo_inside);
 
        // loop over quadrature points and integrate normal flux
        for (const auto& ip : quadratureRule(geo,qorder))
          {
            // position of quadrature point in local coordinates of element
            auto local = geo_in_inside.global(ip.position());
 
            // values of velocity shape functions
            std::vector<Range_V> phi_v(lfsv_v.size());
            FESwitch_V::basis(lfsv_v.finiteElement()).evaluateFunction(local,phi_v);
 
            // values of pressure shape functions
            std::vector<Range_P> phi_p(lfsv_p.size());
            FESwitch_P::basis(lfsv_p.finiteElement()).evaluateFunction(local,phi_p);
 
            // evaluate jacobian of basis functions on reference element
            std::vector<Dune::FieldMatrix<RF,dim,dim> > jac_phi_v(lfsu_v.size());
            VectorBasisSwitch_V::jacobian
              (FESwitch_V::basis(lfsv_v.finiteElement()), geo_inside, local, jac_phi_v);
 
            auto normal = ig.unitOuterNormal(ip.position());
            auto weight = ip.weight()*geo.integrationElement(ip.position());
            auto mu = prm.mu(ig,ip.position());
 
            // evaluate boundary condition type
            auto bctype(prm.bctype(ig,ip.position()));
 
            if (bctype == BC::VelocityDirichlet) {
 
              for(size_type i=0; i<lfsv_v.size(); i++) {
                // compute flux
                Dune::FieldVector<DF,dimw> flux_jac_phi_i(0.0);
                add_compute_flux(jac_phi_v[i],normal,flux_jac_phi_i);
 
                for(size_type j=0; j<lfsu_v.size(); j++) {
                  //================================================//
                  // diffusion term
                  // (non-)symmetric IP term
                  //================================================//
                  Dune::FieldVector<DF,dimw> flux_jac_phi_j(0.0);
                  add_compute_flux(jac_phi_v[j],normal,flux_jac_phi_j);
 
                  mat.accumulate(lfsv_v,i,lfsu_v,j, -mu * (flux_jac_phi_j*phi_v[i]) * weight);
                  mat.accumulate(lfsv_v,i,lfsu_v,j, mu * epsilon * (flux_jac_phi_i*phi_v[j])  *weight);
 
                  //================================================//
                  // standard IP term integral
                  //================================================//
                  mat.accumulate(lfsv_v,i,lfsu_v,j, mu * (phi_v[j]*phi_v[i]) * penalty_factor * weight);
                }
 
                //================================================//
                // pressure-velocity-coupling in momentum equation
                //================================================//
                for(size_type j=0; j<lfsu_p.size(); j++) {
                  mat.accumulate(lfsv_v,i,lfsu_p,j, phi_p[j] * (phi_v[i]*normal) * weight);
                }
              } // end i
 
              //================================================//
              // incompressibility constraint
              //================================================//
              for(size_type i=0; i<lfsv_p.size(); i++) {
                for(size_type j=0; j<lfsu_v.size(); j++) {
                  mat.accumulate(lfsv_p,i,lfsu_v,j, phi_p[i] * (phi_v[j]*normal) * incomp_scaling * weight);
                }
              }
 
            } // Velocity Dirichlet
 
          } // end loop quadrature points
      } // end jacobian_boundary
 
      // volume integral depending only on test functions,
      // contains f on the right hand side
      template<typename EG, typename LFSV, typename R>
      void lambda_volume (const EG& eg, const LFSV& lfsv, R& r) const
      {
        const unsigned int dim = EG::Geometry::mydimension;
 
        // subspaces
        using namespace Indices;
        using LFSV_V = TypeTree::Child<LFSV,_0>;
        const auto& lfsv_v = child(lfsv,_0);
 
        // domain and range field type
        using FESwitch_V = FiniteElementInterfaceSwitch<typename LFSV_V::Traits::FiniteElementType >;
        using BasisSwitch_V = BasisInterfaceSwitch<typename FESwitch_V::Basis >;
        using Range_V = typename BasisSwitch_V::Range;
        using size_type = typename LFSV::Traits::SizeType;
 
        // Get cell
        const auto& cell = eg.entity();
 
        // Get geometries
        auto geo = eg.geometry();
 
        // Determine quadrature order
        const int v_order = FESwitch_V::basis(lfsv_v.finiteElement()).order();
        const int det_jac_order = geo.type().isSimplex() ? 0 : (dim-1);
        const int qorder = 2*v_order + det_jac_order + superintegration_order;
 
        // loop over quadrature points
        for (const auto& ip : quadratureRule(geo,qorder))
          {
            auto local = ip.position();
            //const Dune::FieldVector<DF,dimw> global = eg.geometry().global(local);
 
            // values of velocity shape functions
            std::vector<Range_V> phi_v(lfsv_v.size());
            FESwitch_V::basis(lfsv_v.finiteElement()).evaluateFunction(local,phi_v);
 
            auto weight = ip.weight() * geo.integrationElement(ip.position());
 
            // evaluate source term
            auto fval(prm.f(cell,local));
 
            //================================================//
            // \int (f*v)
            //================================================//
            for(size_type i=0; i<lfsv_v.size(); i++)
              r.accumulate(lfsv_v,i, -(fval*phi_v[i]) * weight);
 
          } // end loop quadrature points
      } // end lambda_volume
 
    private :
 
      template<class M, class RF>
      static void contraction(const M & a, const M & b, RF & v)
      {
        v = 0;
        const int an = a.N();
        const int am = a.M();
        for(int r=0; r<an; ++r)
          for(int c=0; c<am; ++c)
            v += a[r][c] * b[r][c];
      }
 
      template<class DU, class R>
      static void add_compute_flux(const DU & du, const R & n, R & result)
      {
        const int N = du.N();
        const int M = du.M();
        for(int r=0; r<N; ++r)
          for(int c=0; c<M; ++c)
            result[r] += du[r][c] * n[c];
      }
 
      PRM& prm;
      const int superintegration_order;
      Real current_dt;
    }; // end class DGNavierStokesVelVecFEM
 
  } // end namespace PDELab
} // end namespace Dune
#endif // DUNE_PDELAB_LOCALOPERATOR_DGNAVIERSTOKESVELVECFEM_HH