dgnavierstokes.hh

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// -*- tab-width: 2; indent-tabs-mode: nil -*-
// vi: set et ts=2 sw=2 sts=2:
 
#ifndef DUNE_PDELAB_LOCALOPERATOR_DGNAVIERSTOKES_HH
#define DUNE_PDELAB_LOCALOPERATOR_DGNAVIERSTOKES_HH
 
#include <dune/common/exceptions.hh>
#include <dune/common/fvector.hh>
#include <dune/common/parametertreeparser.hh>
 
#include <dune/geometry/referenceelements.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>
 
namespace Dune {
  namespace PDELab {
 
    template<typename PRM>
    class DGNavierStokes :
      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 };
      enum { doLambdaBoundary = true };
 
      DGNavierStokes (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
      {
        // dimensions
        const unsigned int dim = EG::Geometry::mydimension;
 
        // subspaces
        using namespace Indices;
        using LFSV_PFS_V = TypeTree::Child<LFSV,_0>;
        const auto& lfsv_pfs_v = child(lfsv,_0);
 
        // ... we assume all velocity components are the same
        using LFSV_V = TypeTree::Child<LFSV_PFS_V,_0>;
        const auto& lfsv_v = child(lfsv_pfs_v,_0);
        const unsigned int vsize = lfsv_v.size();
        using LFSV_P = TypeTree::Child<LFSV,_1>;
        const auto& lfsv_p = child(lfsv,_1);
        const unsigned int psize = lfsv_p.size();
 
        // domain and range field type
        using FESwitch_V = FiniteElementInterfaceSwitch<typename LFSV_V::Traits::FiniteElementType >;
        using BasisSwitch_V = BasisInterfaceSwitch<typename FESwitch_V::Basis >;
        using RT = typename BasisSwitch_V::Range;
        using RF = typename BasisSwitch_V::RangeField;
        using FESwitch_P = FiniteElementInterfaceSwitch<typename LFSV_P::Traits::FiniteElementType >;
        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);
 
        // Initialize vectors outside for loop
        std::vector<RT> phi_v(vsize);
        Dune::FieldVector<RF,dim> vu(0.0);
        std::vector<RT> phi_p(psize);
        std::vector<Dune::FieldMatrix<RF,1,dim> > grad_phi_v(vsize);
        Dune::FieldMatrix<RF,dim,dim> jacu(0.0);
 
        // 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)
            if(navier) {
              FESwitch_V::basis(lfsv_v.finiteElement()).evaluateFunction(local,phi_v);
 
              for(unsigned int d=0; d<dim; ++d) {
                vu[d] = 0.0;
                const auto& lfsu_v = lfsv_pfs_v.child(d);
                for(size_type i=0; i<vsize; i++)
                  vu[d] += x(lfsu_v,i) * phi_v[i];
              }
            } // end navier
 
            // and value of pressure shape functions
            FESwitch_P::basis(lfsv_p.finiteElement()).evaluateFunction(local,phi_p);
 
            // compute pressure
            RF p(0.0);
            for(size_type i=0; i<psize; i++)
              p += x(lfsv_p,i) * phi_p[i];
 
            // compute gradients
            BasisSwitch_V::gradient(FESwitch_V::basis(lfsv_v.finiteElement()),
                                    geo, local, grad_phi_v);
 
            // compute velocity jacobian
            for(unsigned int d = 0; d<dim; ++d) {
              jacu[d] = 0.0;
              const auto& lfsv_v = lfsv_pfs_v.child(d);
              for(size_type i=0; i<vsize; i++)
                jacu[d].axpy(x(lfsv_v,i), grad_phi_v[i][0]);
            }
 
            auto detj = geo.integrationElement(ip.position());
            auto weight = ip.weight() * detj;
 
            for(unsigned int d = 0; d<dim; ++d) {
              const auto& lfsv_v = lfsv_pfs_v.child(d);
 
              for(size_type i=0; i<vsize; i++) {
                //================================================//
                // \int (mu*grad_u*grad_v)
                //================================================//
                r.accumulate(lfsv_v,i, mu * (jacu[d]*grad_phi_v[i][0]) * weight);
 
                // Assemble symmetric part for (grad u)^T
                if(full_tensor)
                  for(unsigned int dd = 0; dd<dim; ++dd)
                    r.accumulate(lfsv_v,i, mu * jacu[dd][d] * grad_phi_v[i][0][dd] * weight);
 
                //================================================//
                // \int -p \nabla\cdot v
                //================================================//
                r.accumulate(lfsv_v,i, -p * grad_phi_v[i][0][d] * weight);
 
                //================================================//
                // \int \rho ((u\cdot\nabla ) u )\cdot v
                //================================================//
                if(navier) {
                  // compute u * grad u_d
                  auto u_nabla_u = vu * jacu[d];
 
                  r.accumulate(lfsv_v,i, rho * u_nabla_u * phi_v[i] * weight);
                } // end navier
 
              } // end i
            } // end d
 
            //================================================//
            // \int -q \nabla\cdot u
            //================================================//
            for(size_type i=0; i<psize; i++)
              for(unsigned int d = 0; d < dim; ++d)
                // divergence of u is the trace of the velocity jacobian
                r.accumulate(lfsv_p,i, -jacu[d][d] * 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
      {
        // dimensions
        const unsigned int dim = EG::Geometry::mydimension;
 
        // subspaces
        using namespace Indices;
        using LFSV_PFS_V = TypeTree::Child<LFSV,_0>;
        const auto& lfsv_pfs_v = child(lfsv,_0);
 
        // ... we assume all velocity components are the same
        using LFSV_V = TypeTree::Child<LFSV_PFS_V,_0>;
        const auto& lfsv_v = child(lfsv_pfs_v,_0);
        const unsigned int vsize = lfsv_v.size();
        using LFSV_P = TypeTree::Child<LFSV,_1>;
        const auto& lfsv_p = child(lfsv,_1);
        const unsigned int psize = lfsv_p.size();
 
        // domain and range field type
        using FESwitch_V = FiniteElementInterfaceSwitch<typename LFSV_V::Traits::FiniteElementType >;
        using BasisSwitch_V = BasisInterfaceSwitch<typename FESwitch_V::Basis >;
        using RT = typename BasisSwitch_V::Range;
        using RF = typename BasisSwitch_V::RangeField;
        using FESwitch_P = FiniteElementInterfaceSwitch<typename LFSV_P::Traits::FiniteElementType >;
        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);
 
        // Initialize vectors outside for loop
        std::vector<RT> phi_v(vsize);
        Dune::FieldVector<RF,dim> vu(0.0);
        std::vector<RT> phi_p(psize);
        std::vector<Dune::FieldMatrix<RF,1,dim> > grad_phi_v(vsize);
        Dune::FieldVector<RF,dim> gradu_dv(0.0);
 
        // 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)
            if(navier) {
              FESwitch_V::basis(lfsv_v.finiteElement()).evaluateFunction(local,phi_v);
 
              for(unsigned int d=0; d<dim; ++d) {
                vu[d] = 0.0;
                const auto& lfsu_v = lfsv_pfs_v.child(d);
                for(size_type i=0; i<vsize; i++)
                  vu[d] += x(lfsu_v,i) * phi_v[i];
              }
            } // end navier
 
            // and value of pressure shape functions
            FESwitch_P::basis(lfsv_p.finiteElement()).evaluateFunction(local,phi_p);
 
            // compute gradients
            BasisSwitch_V::gradient(FESwitch_V::basis(lfsv_v.finiteElement()),
                                    geo, local, grad_phi_v);
 
            auto detj = geo.integrationElement(ip.position());
            auto weight = ip.weight() * detj;
 
            for(unsigned int dv = 0; dv<dim; ++dv) {
              const LFSV_V& lfsv_v = lfsv_pfs_v.child(dv);
 
              // gradient of dv-th velocity component
              gradu_dv = 0.0;
              if(navier)
                for(size_type l=0; l<vsize; ++l)
                  gradu_dv.axpy(x(lfsv_v,l), grad_phi_v[l][0]);
 
              for(size_type i=0; i<vsize; i++) {
 
                for(size_type j=0; j<vsize; j++) {
                  //================================================//
                  // \int (mu*grad_u*grad_v)
                  //================================================//
                  mat.accumulate(lfsv_v,i,lfsv_v,j, mu * (grad_phi_v[j][0]*grad_phi_v[i][0]) * weight);
 
                  // Assemble symmetric part for (grad u)^T
                  if(full_tensor)
                    for(unsigned int du = 0; du<dim; ++du) {
                      const auto& lfsu_v = lfsv_pfs_v.child(du);
                      mat.accumulate(lfsv_v,i,lfsu_v,j, mu * (grad_phi_v[j][0][dv]*grad_phi_v[i][0][du]) * weight);
                    }
                }
 
                //================================================//
                // - q * div u
                // - p * div v
                //================================================//
                for(size_type j=0; j<psize; j++) {
                  mat.accumulate(lfsv_p,j,lfsv_v,i, -phi_p[j] * grad_phi_v[i][0][dv] * incomp_scaling * weight);
                  mat.accumulate(lfsv_v,i,lfsv_p,j, -phi_p[j] * grad_phi_v[i][0][dv] * weight);
                }
 
                //================================================//
                // \int \rho ((u\cdot\nabla ) u )\cdot v
                //================================================//
                if(navier) {
 
                  // block diagonal contribution
                  for(size_type j=0; j<vsize; j++)
                    mat.accumulate(lfsv_v,i,lfsv_v,j, rho * (vu * grad_phi_v[j][0]) * phi_v[i] * weight);
 
                  // remaining contribution
                  for(unsigned int du = 0; du < dim; ++du) {
                    const auto& lfsu_v = lfsv_pfs_v.child(du);
                    for(size_type j=0; j<vsize; j++)
                      mat.accumulate(lfsv_v,i,lfsu_v,j, rho * phi_v[j] * gradu_dv[du] * phi_v[i] * weight);
                  }
 
                } // end navier
 
              } // end i
            } // end dv
 
          } // end loop quadrature points
      } // end jacobian_volume
 
      // skeleton integral depending on test and ansatz functions
      // 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;
 
        // subspaces
        using namespace Indices;
        using LFSV_PFS_V = TypeTree::Child<LFSV,_0>;
        const auto& lfsv_s_pfs_v = child(lfsv_s,_0);
        const auto& lfsv_n_pfs_v = child(lfsv_n,_0);
 
        // ... we assume all velocity components are the same
        using LFSV_V = TypeTree::Child<LFSV_PFS_V,_0>;
        const auto& lfsv_s_v = child(lfsv_s_pfs_v,_0);
        const auto& lfsv_n_v = child(lfsv_n_pfs_v,_0);
        const unsigned int vsize_s = lfsv_s_v.size();
        const unsigned int vsize_n = lfsv_n_v.size();
        using LFSV_P = TypeTree::Child<LFSV,_1>;
        const auto& lfsv_s_p = child(lfsv_s,_1);
        const auto& lfsv_n_p = child(lfsv_n,_1);
        const unsigned int psize_s = lfsv_s_p.size();
        const unsigned int psize_n = lfsv_n_p.size();
 
        // domain and range field type
        using FESwitch_V = FiniteElementInterfaceSwitch<typename LFSV_V::Traits::FiniteElementType >;
        using BasisSwitch_V = BasisInterfaceSwitch<typename FESwitch_V::Basis >;
        using RT = typename BasisSwitch_V::Range;
        using RF = typename BasisSwitch_V::RangeField;
        using FESwitch_P = FiniteElementInterfaceSwitch<typename LFSV_P::Traits::FiniteElementType >;
 
        // 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_s_v.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);
 
        // Initialize vectors outside for loop
        std::vector<RT> phi_v_s(vsize_s);
        std::vector<RT> phi_v_n(vsize_n);
        Dune::FieldVector<RF,dim> u_s(0.0), u_n(0.0);
        std::vector<RT> phi_p_s(psize_s);
        std::vector<RT> phi_p_n(psize_n);
        std::vector<Dune::FieldMatrix<RF,1,dim> > grad_phi_v_s(vsize_s);
        std::vector<Dune::FieldMatrix<RF,1,dim> > grad_phi_v_n(vsize_n);
        Dune::FieldMatrix<RF,dim,dim> jacu_s(0.0), jacu_n(0.0);
 
        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());
 
            // value of velocity shape functions
            FESwitch_V::basis(lfsv_s_v.finiteElement()).evaluateFunction(local_s,phi_v_s);
            FESwitch_V::basis(lfsv_n_v.finiteElement()).evaluateFunction(local_n,phi_v_n);
 
            // evaluate u
            assert(vsize_s == vsize_n);
            for(unsigned int d=0; d<dim; ++d) {
              u_s[d] = 0.0;
              u_n[d] = 0.0;
              const auto& lfsv_s_v = lfsv_s_pfs_v.child(d);
              const auto& lfsv_n_v = lfsv_n_pfs_v.child(d);
              for(unsigned int i=0; i<vsize_s; i++) {
                u_s[d] += x_s(lfsv_s_v,i) * phi_v_s[i];
                u_n[d] += x_n(lfsv_n_v,i) * phi_v_n[i];
              }
            }
 
            // value of pressure shape functions
            FESwitch_P::basis(lfsv_s_p.finiteElement()).evaluateFunction(local_s,phi_p_s);
            FESwitch_P::basis(lfsv_n_p.finiteElement()).evaluateFunction(local_n,phi_p_n);
 
            // evaluate pressure
            assert(psize_s == psize_n);
            RF p_s(0.0), p_n(0.0);
            for(unsigned int i=0; i<psize_s; i++) {
              p_s += x_s(lfsv_s_p,i) * phi_p_s[i];
              p_n += x_n(lfsv_n_p,i) * phi_p_n[i];
            }
 
            // compute gradients
            BasisSwitch_V::gradient(FESwitch_V::basis(lfsv_s_v.finiteElement()),
                                    geo_inside, local_s, grad_phi_v_s);
 
            BasisSwitch_V::gradient(FESwitch_V::basis(lfsv_n_v.finiteElement()),
                                    geo_outside, local_n, grad_phi_v_n);
 
            // evaluate velocity jacobian
            for(unsigned int d=0; d<dim; ++d) {
              jacu_s[d] = 0.0;
              jacu_n[d] = 0.0;
              const auto& lfsv_s_v = lfsv_s_pfs_v.child(d);
              const auto& lfsv_n_v = lfsv_n_pfs_v.child(d);
              for(unsigned int i=0; i<vsize_s; i++) {
                jacu_s[d].axpy(x_s(lfsv_s_v,i), grad_phi_v_s[i][0]);
                jacu_n[d].axpy(x_n(lfsv_n_v,i), grad_phi_v_n[i][0]);
              }
            }
 
            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;
 
            for(unsigned int d=0; d<dim; ++d) {
              const auto& lfsv_s_v = lfsv_s_pfs_v.child(d);
              const auto& lfsv_n_v = lfsv_n_pfs_v.child(d);
 
              //================================================//
              // diffusion term
              //================================================//
              auto val = 0.5 * ((jacu_s[d] * normal) + (jacu_n[d] * normal)) * factor;
              for(unsigned int i=0; i<vsize_s; i++) {
                r_s.accumulate(lfsv_s_v,i, -val * phi_v_s[i]);
                r_n.accumulate(lfsv_n_v,i, val * phi_v_n[i]);
 
                if(full_tensor) {
                  for(unsigned int dd=0; dd<dim; ++dd) {
                    auto Tval = 0.5 * (jacu_s[dd][d] + jacu_n[dd][d]) * normal[dd] * factor;
                    r_s.accumulate(lfsv_s_v,i, -Tval * phi_v_s[i]);
                    r_n.accumulate(lfsv_n_v,i, Tval * phi_v_n[i]);
                  }
                }
              } // end i
 
              //================================================//
              // (non-)symmetric IP term
              //================================================//
              auto jumpu_d  = u_s[d] - u_n[d];
              for(unsigned int i=0; i<vsize_s; i++) {
                r_s.accumulate(lfsv_s_v,i, epsilon * 0.5 * (grad_phi_v_s[i][0] * normal) * jumpu_d * factor);
                r_n.accumulate(lfsv_n_v,i, epsilon * 0.5 * (grad_phi_v_n[i][0] * normal) * jumpu_d * factor);
 
                if(full_tensor) {
                  for(unsigned int dd=0; dd<dim; ++dd) {
                    r_s.accumulate(lfsv_s_v,i, epsilon * 0.5 * grad_phi_v_s[i][0][dd] * (u_s[dd] - u_n[dd]) * normal[d] * factor);
                    r_n.accumulate(lfsv_n_v,i, epsilon * 0.5 * grad_phi_v_n[i][0][dd] * (u_s[dd] - u_n[dd]) * normal[d] * factor);
                  }
                }
              } // end i
 
              //================================================//
              // standard IP term integral
              //================================================//
              for(unsigned int i=0; i<vsize_s; i++) {
                r_s.accumulate(lfsv_s_v,i, penalty_factor * jumpu_d * phi_v_s[i] * factor);
                r_n.accumulate(lfsv_n_v,i, -penalty_factor * jumpu_d * phi_v_n[i] * factor);
              } // end i
 
              //================================================//
              // pressure-velocity-coupling in momentum equation
              //================================================//
              auto mean_p = 0.5*(p_s + p_n);
              for(unsigned int i=0; i<vsize_s; i++) {
                r_s.accumulate(lfsv_s_v,i, mean_p * phi_v_s[i] * normal[d] * weight);
                r_n.accumulate(lfsv_n_v,i, -mean_p * phi_v_n[i] * normal[d] * weight);
              } // end i
            } // end d
 
            //================================================//
            // incompressibility constraint
            //================================================//
            auto jumpu_n = (u_s*normal) - (u_n*normal);
            for(unsigned int i=0; i<psize_s; i++) {
              r_s.accumulate(lfsv_s_p,i, 0.5 * phi_p_s[i] * jumpu_n * incomp_scaling * weight);
              r_n.accumulate(lfsv_n_p,i, 0.5 * phi_p_n[i] * jumpu_n * incomp_scaling * weight);
            } // end i
 
          } // end loop quadrature points
      } // end alpha_skeleton
 
      // jacobian of skeleton term
      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;
 
        // subspaces
        using namespace Indices;
        using LFSV_PFS_V = TypeTree::Child<LFSV,_0>;
        const auto& lfsv_s_pfs_v = child(lfsv_s,_0);
        const auto& lfsv_n_pfs_v = child(lfsv_n,_0);
 
        // ... we assume all velocity components are the same
        using LFSV_V = TypeTree::Child<LFSV_PFS_V,_0>;
        const auto& lfsv_s_v = child(lfsv_s_pfs_v,_0);
        const auto& lfsv_n_v = child(lfsv_n_pfs_v,_0);
        const unsigned int vsize_s = lfsv_s_v.size();
        const unsigned int vsize_n = lfsv_n_v.size();
        using LFSV_P = TypeTree::Child<LFSV,_1>;
        const auto& lfsv_s_p = child(lfsv_s,_1);
        const auto& lfsv_n_p = child(lfsv_n,_1);
        const unsigned int psize_s = lfsv_s_p.size();
        const unsigned int psize_n = lfsv_n_p.size();
 
        // domain and range field type
        using FESwitch_V = FiniteElementInterfaceSwitch<typename LFSV_V::Traits::FiniteElementType >;
        using BasisSwitch_V = BasisInterfaceSwitch<typename FESwitch_V::Basis >;
        using RT = typename BasisSwitch_V::Range;
        using RF = typename BasisSwitch_V::RangeField;
        using FESwitch_P = FiniteElementInterfaceSwitch<typename LFSV_P::Traits::FiniteElementType >;
 
        // 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_s_v.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);
 
        // Initialize vectors outside for loop
        std::vector<RT> phi_v_s(vsize_s);
        std::vector<RT> phi_v_n(vsize_n);
        std::vector<RT> phi_p_s(psize_s);
        std::vector<RT> phi_p_n(psize_n);
        std::vector<Dune::FieldMatrix<RF,1,dim> > grad_phi_v_s(vsize_s);
        std::vector<Dune::FieldMatrix<RF,1,dim> > grad_phi_v_n(vsize_n);
 
        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());
 
            // value of velocity shape functions
            FESwitch_V::basis(lfsv_s_v.finiteElement()).evaluateFunction(local_s,phi_v_s);
            FESwitch_V::basis(lfsv_n_v.finiteElement()).evaluateFunction(local_n,phi_v_n);
            // and value of pressure shape functions
            FESwitch_P::basis(lfsv_s_p.finiteElement()).evaluateFunction(local_s,phi_p_s);
            FESwitch_P::basis(lfsv_n_p.finiteElement()).evaluateFunction(local_n,phi_p_n);
 
            // compute gradients
            BasisSwitch_V::gradient(FESwitch_V::basis(lfsv_s_v.finiteElement()),
                                    geo_inside, local_s, grad_phi_v_s);
 
            BasisSwitch_V::gradient(FESwitch_V::basis(lfsv_n_v.finiteElement()),
                                    geo_outside, local_n, grad_phi_v_n);
 
            auto normal = ig.unitOuterNormal(ip.position());
            auto weight = ip.weight()*geo.integrationElement(ip.position());
            auto mu = prm.mu(ig,ip.position());
 
            assert(vsize_s == vsize_n);
            auto factor = mu * weight;
 
            for(unsigned int d = 0; d < dim; ++d) {
              const auto& lfsv_s_v = lfsv_s_pfs_v.child(d);
              const auto& lfsv_n_v = lfsv_n_pfs_v.child(d);
 
              //================================================//
              // - (\mu \int < \nabla u > . normal . [v])
              // \mu \int \frac{\sigma}{|\gamma|^\beta} [u] \cdot [v]
              //================================================//
              for(unsigned int i=0; i<vsize_s; ++i) {
 
                for(unsigned int j=0; j<vsize_s; ++j) {
                  auto val = (0.5*(grad_phi_v_s[i][0]*normal)*phi_v_s[j]) * factor;
                  mat_ss.accumulate(lfsv_s_v,j,lfsv_s_v,i, -val);
                  mat_ss.accumulate(lfsv_s_v,i,lfsv_s_v,j, epsilon * val);
                  mat_ss.accumulate(lfsv_s_v,i,lfsv_s_v,j, phi_v_s[i] * phi_v_s[j] * penalty_factor * factor);
 
                  // Assemble symmetric part for (grad u)^T
                  if(full_tensor) {
                    for(unsigned int dd = 0; dd < dim; ++dd) {
                      auto Tval = (0.5*(grad_phi_v_s[i][0][d]*normal[dd])*phi_v_s[j]) * factor;
                      const auto& lfsv_s_v_dd = lfsv_s_pfs_v.child(dd);
                      mat_ss.accumulate(lfsv_s_v,j,lfsv_s_v_dd,i, - Tval);
                      mat_ss.accumulate(lfsv_s_v_dd,i,lfsv_s_v,j, epsilon*Tval );
                    }
                  }
                }
 
                for(unsigned int j=0; j<vsize_n; ++j) {
                  // the normal vector flipped, thus the sign flips
                  auto val = (-0.5*(grad_phi_v_s[i][0]*normal)*phi_v_n[j]) * factor;
                  mat_ns.accumulate(lfsv_n_v,j,lfsv_s_v,i,- val);
                  mat_sn.accumulate(lfsv_s_v,i,lfsv_n_v,j, epsilon*val);
                  mat_ns.accumulate(lfsv_n_v,j,lfsv_s_v,i, -phi_v_s[i] * phi_v_n[j] * penalty_factor * factor);
 
                  // Assemble symmetric part for (grad u)^T
                  if(full_tensor) {
                    for (unsigned int dd=0;dd<dim;++dd) {
                      auto Tval = (-0.5*(grad_phi_v_s[i][0][d]*normal[dd])*phi_v_n[j]) * factor;
                      const auto& lfsv_s_v_dd = lfsv_s_pfs_v.child(dd);
                      mat_ns.accumulate(lfsv_n_v,j,lfsv_s_v_dd,i,- Tval);
                      mat_sn.accumulate(lfsv_s_v_dd,i,lfsv_n_v,j, epsilon*Tval);
                    }
                  }
                }
 
                for(unsigned int j=0; j<vsize_s; ++j) {
                  auto val = (0.5*(grad_phi_v_n[i][0]*normal)*phi_v_s[j]) * factor;
                  mat_sn.accumulate(lfsv_s_v,j,lfsv_n_v,i, - val);
                  mat_ns.accumulate(lfsv_n_v,i,lfsv_s_v,j, epsilon*val );
                  mat_sn.accumulate(lfsv_s_v,j,lfsv_n_v,i, -phi_v_n[i] * phi_v_s[j] * penalty_factor * factor);
 
                  // Assemble symmetric part for (grad u)^T
                  if(full_tensor) {
                    for (unsigned int dd=0;dd<dim;++dd) {
                      auto Tval = (0.5*(grad_phi_v_n[i][0][d]*normal[dd])*phi_v_s[j]) * factor;
                      const auto& lfsv_n_v_dd = lfsv_n_pfs_v.child(dd);
                      mat_sn.accumulate(lfsv_s_v,j,lfsv_n_v_dd,i, - Tval);
                      mat_ns.accumulate(lfsv_n_v_dd,i,lfsv_s_v,j, epsilon*Tval );
                    }
                  }
                }
 
                for(unsigned int j=0; j<vsize_n; ++j) {
                  // the normal vector flipped, thus the sign flips
                  auto val = (-0.5*(grad_phi_v_n[i][0]*normal)*phi_v_n[j]) * factor;
                  mat_nn.accumulate(lfsv_n_v,j,lfsv_n_v,i, - val);
                  mat_nn.accumulate(lfsv_n_v,i,lfsv_n_v,j, epsilon*val);
                  mat_nn.accumulate(lfsv_n_v,j,lfsv_n_v,i, phi_v_n[i] * phi_v_n[j] * penalty_factor * factor);
 
                  // Assemble symmetric part for (grad u)^T
                  if(full_tensor) {
                    for (unsigned int dd=0;dd<dim;++dd) {
                      auto Tval = (-0.5*(grad_phi_v_n[i][0][d]*normal[dd])*phi_v_n[j]) * factor;
                      const auto& lfsv_n_v_dd = lfsv_n_pfs_v.child(dd);
                      mat_nn.accumulate(lfsv_n_v,j,lfsv_n_v_dd,i,- Tval);
                      mat_nn.accumulate(lfsv_n_v_dd,i,lfsv_n_v,j, epsilon*Tval);
                    }
                  }
                }
 
                //================================================//
                // \int <q> [u] n
                // \int <p> [v] n
                //================================================//
                for(unsigned int j=0; j<psize_s; ++j) {
                  auto val = 0.5*(phi_p_s[j]*normal[d]*phi_v_s[i]) * weight;
                  mat_ss.accumulate(lfsv_s_v,i,lfsv_s_p,j, val);
                  mat_ss.accumulate(lfsv_s_p,j,lfsv_s_v,i, val * incomp_scaling);
                }
 
                for(unsigned int j=0; j<psize_n; ++j) {
                  auto val = 0.5*(phi_p_n[j]*normal[d]*phi_v_s[i]) * weight;
                  mat_sn.accumulate(lfsv_s_v,i,lfsv_n_p,j, val);
                  mat_ns.accumulate(lfsv_n_p,j,lfsv_s_v,i, val * incomp_scaling);
                }
 
                for (unsigned int j=0; j<psize_s;++j) {
                  // the normal vector flipped, thus the sign flips
                  auto val = -0.5*(phi_p_s[j]*normal[d]*phi_v_n[i]) * weight;
                  mat_ns.accumulate(lfsv_n_v,i,lfsv_s_p,j, val);
                  mat_sn.accumulate(lfsv_s_p,j,lfsv_n_v,i, val * incomp_scaling);
                }
 
                for (unsigned int j=0; j<psize_n;++j) {
                  // the normal vector flipped, thus the sign flips
                  auto val = -0.5*(phi_p_n[j]*normal[d]*phi_v_n[i]) * weight;
                  mat_nn.accumulate(lfsv_n_v,i,lfsv_n_p,j, val);
                  mat_nn.accumulate(lfsv_n_p,j,lfsv_n_v,i, val * incomp_scaling);
                }
              } // end i
            } // end d
 
          } // 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;
 
        // subspaces
        using namespace Indices;
        using LFSV_PFS_V = TypeTree::Child<LFSV,_0>;
        const auto& lfsv_pfs_v = child(lfsv,_0);
 
        // ... we assume all velocity components are the same
        using LFSV_V = TypeTree::Child<LFSV_PFS_V,_0>;
        const auto& lfsv_v = child(lfsv_pfs_v,_0);
        const unsigned int vsize = lfsv_v.size();
        using LFSV_P = TypeTree::Child<LFSV,_1>;
        const auto& lfsv_p = child(lfsv,_1);
        const unsigned int psize = lfsv_p.size();
 
        // domain and range field type
        using FESwitch_V = FiniteElementInterfaceSwitch<typename LFSV_V::Traits::FiniteElementType >;
        using BasisSwitch_V = BasisInterfaceSwitch<typename FESwitch_V::Basis >;
        using RT = typename BasisSwitch_V::Range;
        using RF = typename BasisSwitch_V::RangeField;
        using FESwitch_P = FiniteElementInterfaceSwitch<typename LFSV_P::Traits::FiniteElementType >;
 
        // 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);
 
        // Initialize vectors outside for loop
        std::vector<RT> phi_v(vsize);
        Dune::FieldVector<RF,dim> u(0.0);
        std::vector<Dune::FieldMatrix<RF,1,dim> > grad_phi_v(vsize);
        Dune::FieldMatrix<RF,dim,dim> jacu(0.0);
 
        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());
 
            // value of velocity shape functions
            FESwitch_V::basis(lfsv_v.finiteElement()).evaluateFunction(local,phi_v);
 
            // evaluate u
            for(unsigned int d=0; d<dim; ++d) {
              u[d] = 0.0;
              const LFSV_V& lfsv_v = lfsv_pfs_v.child(d);
              for(unsigned int i=0; i<vsize; i++)
                u[d] += x(lfsv_v,i) * phi_v[i];
            }
 
            // value of pressure shape functions
            std::vector<RT> phi_p(psize);
            FESwitch_P::basis(lfsv_p.finiteElement()).evaluateFunction(local,phi_p);
 
            // evaluate pressure
            RF p(0.0);
            for(unsigned int i=0; i<psize; i++)
              p += x(lfsv_p,i) * phi_p[i];
 
            // compute gradients
            BasisSwitch_V::gradient(FESwitch_V::basis(lfsv_v.finiteElement()),
                                    geo_inside, local, grad_phi_v);
 
            // evaluate velocity jacobian
            for(unsigned int d=0; d<dim; ++d) {
              jacu[d] = 0.0;
              const LFSV_V& lfsv_v = lfsv_pfs_v.child(d);
              for(unsigned int i=0; i<vsize; i++)
                jacu[d].axpy(x(lfsv_v,i), grad_phi_v[i][0]);
            }
 
            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()));
 
            // Slip factor smoothly switching between slip and no slip conditions.
            RF slip_factor = 0.0;
            using BoundarySlipSwitch = NavierStokesDGImp::VariableBoundarySlipSwitch<PRM>;
            if (bctype == BC::SlipVelocity)
              // Calls boundarySlip(..) function of parameter
              // class if available, i.e. if
              // enable_variable_slip is defined. Otherwise
              // returns 1.0;
              slip_factor = BoundarySlipSwitch::boundarySlip(prm,ig,ip.position());
 
            // velocity boundary condition
            if (bctype == BC::VelocityDirichlet || bctype == BC::SlipVelocity)
              {
                // on BC::VelocityDirichlet: 1.0 - slip_factor = 1.0
                auto factor = weight * (1.0 - slip_factor);
 
                for(unsigned int d = 0; d < dim; ++d) {
                  const auto& lfsv_v = lfsv_pfs_v.child(d);
 
                  auto val = (jacu[d] * normal) * factor * mu;
                  for(unsigned int i=0; i<vsize; i++) {
                    //================================================//
                    // - (\mu \int \nabla u. normal . v)
                    //================================================//
                    r.accumulate(lfsv_v,i, -val * phi_v[i]);
                    r.accumulate(lfsv_v,i, epsilon * mu * (grad_phi_v[i][0] * normal) * u[d] * factor);
 
                    if(full_tensor) {
                      for(unsigned int dd=0; dd<dim; ++dd) {
                        r.accumulate(lfsv_v,i, -mu * jacu[dd][d] * normal[dd] * phi_v[i] * factor);
                        r.accumulate(lfsv_v,i, epsilon * mu * grad_phi_v[i][0][dd] * u[dd] * normal[d] * factor);
                      }
                    }
 
                    //================================================//
                    // \mu \int \sigma / |\gamma|^\beta v u
                    //================================================//
                    r.accumulate(lfsv_v,i, u[d] * phi_v[i] * mu * penalty_factor * factor);
 
                    //================================================//
                    // \int p v n
                    //================================================//
                    r.accumulate(lfsv_v,i, p * phi_v[i] * normal[d] * weight);
                  } // end i
                } // end d
 
                //================================================//
                // \int q u n
                //================================================//
                for(unsigned int i=0; i<psize; i++) {
                  r.accumulate(lfsv_p,i, phi_p[i] * (u * normal) * incomp_scaling * weight);
                }
              } // Velocity Dirichlet
 
            if(bctype == BC::SlipVelocity)
              {
                auto factor = weight * (slip_factor);
 
                RF ten_sum = 1.0;
                if(full_tensor)
                  ten_sum = 2.0;
 
                for(unsigned int d = 0; d < dim; ++d) {
                  const auto& lfsv_v = lfsv_pfs_v.child(d);
 
                  for(unsigned int i=0; i<vsize; i++) {
                    //================================================//
                    // - (\mu \int \nabla u. normal . v)
                    //================================================//
                    for(unsigned int dd = 0; dd < dim; ++dd)
                      r.accumulate(lfsv_v,i, -ten_sum * mu * (jacu[dd] * normal) * normal[dd] *phi_v[i] * normal[d] * factor);
                    r.accumulate(lfsv_v,i, epsilon * ten_sum * mu * (u * normal) * (grad_phi_v[i][0] * normal) * normal[d] * factor);
 
                    //================================================//
                    // \mu \int \sigma / |\gamma|^\beta v u
                    //================================================//
                    r.accumulate(lfsv_v,i, mu * penalty_factor * (u * normal) * phi_v[i] * normal[d] * factor);
                  } // end i
                } // end d
 
              } // Slip Velocity
          } // 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;
 
        // subspaces
        using namespace Indices;
        using LFSV_PFS_V = TypeTree::Child<LFSV,_0>;
        const auto& lfsv_pfs_v = child(lfsv,_0);
 
        // ... we assume all velocity components are the same
        using LFSV_V = TypeTree::Child<LFSV_PFS_V,_0>;
        const auto& lfsv_v = child(lfsv_pfs_v,_0);
        const unsigned int vsize = lfsv_v.size();
        using LFSV_P = TypeTree::Child<LFSV,_1>;
        const auto& lfsv_p = child(lfsv,_1);
        const unsigned int psize = lfsv_p.size();
 
        // domain and range field type
        using FESwitch_V = FiniteElementInterfaceSwitch<typename LFSV_V::Traits::FiniteElementType >;
        using BasisSwitch_V = BasisInterfaceSwitch<typename FESwitch_V::Basis >;
        using RT = typename BasisSwitch_V::Range;
        using RF = typename BasisSwitch_V::RangeField;
        using FESwitch_P = FiniteElementInterfaceSwitch<typename LFSV_P::Traits::FiniteElementType >;
 
        // 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);
 
        // Initialize vectors outside for loop
        std::vector<RT> phi_v(vsize);
        std::vector<RT> phi_p(psize);
        std::vector<Dune::FieldMatrix<RF,1,dim> > grad_phi_v(vsize);
 
        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());
 
            // value of velocity shape functions
            FESwitch_V::basis(lfsv_v.finiteElement()).evaluateFunction(local,phi_v);
            // and value of pressure shape functions
            FESwitch_P::basis(lfsv_p.finiteElement()).evaluateFunction(local,phi_p);
 
            BasisSwitch_V::gradient(FESwitch_V::basis(lfsv_v.finiteElement()),
                                    geo_inside, local, grad_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()));
 
            // Slip factor smoothly switching between slip and no slip conditions.
            RF slip_factor = 0.0;
            using BoundarySlipSwitch = NavierStokesDGImp::VariableBoundarySlipSwitch<PRM>;
            if (bctype == BC::SlipVelocity)
              // Calls boundarySlip(..) function of parameter
              // class if available, i.e. if
              // enable_variable_slip is defined. Otherwise
              // returns 1.0;
              slip_factor = BoundarySlipSwitch::boundarySlip(prm,ig,ip.position());
 
            // velocity boundary condition
            if (bctype == BC::VelocityDirichlet || bctype == BC::SlipVelocity)
              {
                // on BC::VelocityDirichlet: 1.0 - slip_factor = 1.0
                auto factor = weight * (1.0 - slip_factor);
 
                for(unsigned int d = 0; d < dim; ++d) {
                  const auto& lfsv_v = lfsv_pfs_v.child(d);
 
                  for(unsigned int i=0; i<vsize; i++) {
 
                    for(unsigned int j=0; j<vsize; j++) {
                      //================================================//
                      // - (\mu \int \nabla u. normal . v)
                      //================================================//
                      auto val = ((grad_phi_v[j][0]*normal)*phi_v[i]) * factor * mu;
                      mat.accumulate(lfsv_v,i,lfsv_v,j, - val);
                      mat.accumulate(lfsv_v,j,lfsv_v,i, epsilon*val);
 
                      // Assemble symmetric part for (grad u)^T
                      if(full_tensor) {
                        for(unsigned int dd = 0; dd < dim; ++dd) {
                          const auto& lfsv_v_dd = lfsv_pfs_v.child(dd);
                          auto Tval = ((grad_phi_v[j][0][d]*normal[dd])*phi_v[i]) * factor * mu;
                          mat.accumulate(lfsv_v,i,lfsv_v_dd,j, - Tval);
                          mat.accumulate(lfsv_v_dd,j,lfsv_v,i, epsilon*Tval);
                        }
                      }
                      //================================================//
                      // \mu \int \sigma / |\gamma|^\beta v u
                      //================================================//
                      mat.accumulate(lfsv_v,j,lfsv_v,i, phi_v[i] * phi_v[j] * mu * penalty_factor * factor);
                    }
 
                    //================================================//
                    // \int q u n
                    // \int p v n
                    //================================================//
                    for(unsigned int j=0; j<psize; j++) {
                      mat.accumulate(lfsv_p,j,lfsv_v,i, (phi_p[j]*normal[d]*phi_v[i]) * weight * incomp_scaling);
                      mat.accumulate(lfsv_v,i,lfsv_p,j, (phi_p[j]*normal[d]*phi_v[i]) * weight);
                    }
                  } // end i
                } // end d
 
              } // Velocity Dirichlet
 
            if (bctype == BC::SlipVelocity)
              {
                auto factor = weight * (slip_factor);
 
                //================================================//
                // - (\mu \int \nabla u. normal . v)
                //================================================//
 
                for (unsigned int i=0;i<vsize;++i) // ansatz
                  {
                    for (unsigned int j=0;j<vsize;++j) // test
                      {
                        RF ten_sum = 1.0;
 
                        // Assemble symmetric part for (grad u)^T
                        if(full_tensor)
                          ten_sum = 2.0;
 
                        auto val = ten_sum * ((grad_phi_v[j][0]*normal)*phi_v[i]) * factor * mu;
                        for (unsigned int d=0;d<dim;++d)
                          {
                            const auto& lfsv_v_d = lfsv_pfs_v.child(d);
 
                            for (unsigned int dd=0;dd<dim;++dd)
                              {
                                const auto& lfsv_v_dd = lfsv_pfs_v.child(dd);
 
                                mat.accumulate(lfsv_v_dd,i,lfsv_v_d,j, -val*normal[d]*normal[dd]);
                                mat.accumulate(lfsv_v_d,j,lfsv_v_dd,i, epsilon*val*normal[d]*normal[dd]);
                              }
                          }
                      }
                  }
 
                //================================================//
                // \mu \int \sigma / |\gamma|^\beta v u
                //================================================//
                auto p_factor = mu * penalty_factor * factor;
                for (unsigned int i=0;i<vsize;++i)
                  {
                    for (unsigned int j=0;j<vsize;++j)
                      {
                        auto val = phi_v[i]*phi_v[j] * p_factor;
                        for (unsigned int d=0;d<dim;++d)
                          {
                            const auto& lfsv_v_d = lfsv_pfs_v.child(d);
                            for (unsigned int dd=0;dd<dim;++dd)
                              {
                                const auto& lfsv_v_dd = lfsv_pfs_v.child(dd);
                                mat.accumulate(lfsv_v_d,j,lfsv_v_dd,i, val*normal[d]*normal[dd]);
                              }
                          }
                      }
                  }
 
              } // Slip Velocity
          } // 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
      {
        // dimensions
        static const unsigned int dim = EG::Geometry::mydimension;
 
        // subspaces
        using namespace Indices;
        using LFSV_PFS_V = TypeTree::Child<LFSV,_0>;
        const auto& lfsv_pfs_v = child(lfsv,_0);
 
        // we assume all velocity components are the same type
        using LFSV_V = TypeTree::Child<LFSV_PFS_V,_0>;
        const auto& lfsv_v = child(lfsv_pfs_v,_0);
        const unsigned int vsize = lfsv_v.size();
        using LFSV_P = TypeTree::Child<LFSV,_1>;
        const auto& lfsv_p = child(lfsv,_1);
        const unsigned int psize = lfsv_p.size();
 
        // domain and range field type
        using FESwitch_V = FiniteElementInterfaceSwitch<typename LFSV_V::Traits::FiniteElementType >;
        using BasisSwitch_V = BasisInterfaceSwitch<typename FESwitch_V::Basis >;
        using RT = typename BasisSwitch_V::Range;
        using FESwitch_P = FiniteElementInterfaceSwitch<typename LFSV_P::Traits::FiniteElementType >;
        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;
 
        // Initialize vectors outside for loop
        std::vector<RT> phi_v(vsize);
        std::vector<RT> phi_p(psize);
 
        // 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
            FESwitch_V::basis(lfsv_v.finiteElement()).evaluateFunction(local,phi_v);
 
            // values of pressure shape functions
            FESwitch_P::basis(lfsv_p.finiteElement()).evaluateFunction(local,phi_p);
 
            auto weight = ip.weight() * geo.integrationElement(ip.position());
 
            // evaluate source term
            auto fval(prm.f(cell,local));
 
            //================================================//
            // \int (f*v)
            //================================================//
            auto factor = weight;
            for (unsigned int d=0; d<dim; d++) {
              const auto& lfsv_v = lfsv_pfs_v.child(d);
 
              // and store for each velocity component
              for (size_type i=0; i<vsize; i++) {
                auto val = phi_v[i]*factor;
                r.accumulate(lfsv_v,i, -fval[d] * val);
              }
            }
 
            auto g2 = prm.g2(eg,ip.position());
 
            // integrate div u * psi_i
            for (size_t i=0; i<lfsv_p.size(); i++) {
              r.accumulate(lfsv_p,i, g2 * phi_p[i] * factor);
            }
 
          } // end loop quadrature points
      } // end lambda_volume
 
      // boundary integral independent of ansatz functions,
      // Neumann and Dirichlet boundary conditions, DG penalty term's right hand side
      template<typename IG, typename LFSV, typename R>
      void lambda_boundary (const IG& ig, const LFSV& lfsv, R& r) const
      {
        // dimensions
        static const unsigned int dim = IG::Entity::dimension;
 
        // subspaces
        using namespace Indices;
        using LFSV_PFS_V = TypeTree::Child<LFSV,_0>;
        const auto& lfsv_pfs_v = child(lfsv,_0);
 
        // ... we assume all velocity components are the same
        using LFSV_V = TypeTree::Child<LFSV_PFS_V,_0>;
        const auto& lfsv_v = child(lfsv_pfs_v,_0);
        const unsigned int vsize = lfsv_v.size();
        using LFSV_P = TypeTree::Child<LFSV,_1>;
        const auto& lfsv_p = child(lfsv,_1);
        const unsigned int psize = lfsv_p.size();
 
        // domain and range field type
        using FESwitch_V = FiniteElementInterfaceSwitch<typename LFSV_V::Traits::FiniteElementType >;
        using BasisSwitch_V = BasisInterfaceSwitch<typename FESwitch_V::Basis >;
        using DF = typename BasisSwitch_V::DomainField;
        using RT = typename BasisSwitch_V::Range;
        using RF = typename BasisSwitch_V::RangeField;
        using FESwitch_P = FiniteElementInterfaceSwitch<typename LFSV_P::Traits::FiniteElementType >;
 
        // 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-2);
        const int jac_order = geo.type().isSimplex() ? 0 : 1;
        const int qorder = 2*v_order + det_jac_order + jac_order + superintegration_order;
 
        auto epsilon = prm.epsilonIPSymmetryFactor();
        auto incomp_scaling = prm.incompressibilityScaling(current_dt);
 
        // Initialize vectors outside for loop
        std::vector<RT> phi_v(vsize);
        std::vector<RT> phi_p(psize);
        std::vector<Dune::FieldMatrix<RF,1,dim> > grad_phi_v(vsize);
 
        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
            Dune::FieldVector<DF,dim-1> flocal = ip.position();
            Dune::FieldVector<DF,dim> local = geo_in_inside.global(flocal);
            //Dune::FieldVector<DF,dimw> global = ig.geometry().global(flocal);
 
            // value of velocity shape functions
            FESwitch_V::basis(lfsv_v.finiteElement()).evaluateFunction(local,phi_v);
            // and value of pressure shape functions
            FESwitch_P::basis(lfsv_p.finiteElement()).evaluateFunction(local,phi_p);
 
            // compute gradients
            BasisSwitch_V::gradient(FESwitch_V::basis(lfsv_v.finiteElement()),
                                    geo_inside, local, grad_phi_v);
 
            auto normal = ig.unitOuterNormal(ip.position());
            auto weight = ip.weight()*geo.integrationElement(ip.position());
            auto mu = prm.mu(ig,flocal);
 
            // evaluate boundary condition type
            auto bctype(prm.bctype(ig,flocal));
 
            if (bctype == BC::VelocityDirichlet)
              {
                auto u0(prm.g(cell_inside,local));
 
                auto factor = mu * weight;
                for(unsigned int d = 0; d < dim; ++d) {
                  const auto& lfsv_v = lfsv_pfs_v.child(d);
 
                  for(unsigned int i=0; i<vsize; i++) {
                    //================================================//
                    // \mu \int \nabla v \cdot u_0 \cdot n
                    //================================================//
                    r.accumulate(lfsv_v,i, -epsilon * (grad_phi_v[i][0] * normal) * factor * u0[d]);
 
                    // Assemble symmetric part for (grad u)^T
                    if(full_tensor) {
                      for(unsigned int dd = 0; dd < dim; ++dd) {
                        const auto& lfsv_v_dd = lfsv_pfs_v.child(dd);
                        auto Tval = (grad_phi_v[i][0][d]*normal[dd]) * factor;
                        r.accumulate(lfsv_v_dd,i, -epsilon * Tval * u0[d]);
                      }
                    }
                    //================================================//
                    // \int \sigma / |\gamma|^\beta v u_0
                    //================================================//
                    r.accumulate(lfsv_v,i, -phi_v[i] * penalty_factor * u0[d] * factor);
 
                  } // end i
                } // end d
 
                //================================================//
                // \int q u_0 n
                //================================================//
                for (unsigned int i=0;i<psize;++i) // test
                  {
                    auto val = phi_p[i]*(u0 * normal) * weight;
                    r.accumulate(lfsv_p,i, - val * incomp_scaling);
                  }
              } // end BC velocity
            if (bctype == BC::StressNeumann)
              {
                auto stress(prm.j(ig,flocal,normal));
 
                //std::cout << "Pdirichlet\n";
                //================================================//
                // \int p u n
                //================================================//
                for(unsigned int d = 0; d < dim; ++d) {
                  const auto& lfsv_v = lfsv_pfs_v.child(d);
 
                  for(unsigned int i=0; i<vsize; i++)
                    r.accumulate(lfsv_v,i, stress[d] * phi_v[i] * weight);
                }
              }
          } // end loop quadrature points
      } // end lambda_boundary
 
    private :
      PRM& prm;
      const int superintegration_order;
      Real current_dt;
    }; // end class DGNavierStokes
 
  } // end namespace PDELab
} // end namespace Dune
#endif // DUNE_PDELAB_LOCALOPERATOR_DGNAVIERSTOKES_HH