ovlp_amg_dg_backend.hh

#ifndef DUNE_PDELAB_BACKEND_ISTL_OVLP_AMG_DG_BACKEND_HH
#define DUNE_PDELAB_BACKEND_ISTL_OVLP_AMG_DG_BACKEND_HH
 
#include <dune/common/parametertree.hh>
#include <dune/common/power.hh>
 
#include <dune/istl/matrixmatrix.hh>
 
#include <dune/grid/common/datahandleif.hh>
#include <dune/pdelab/backend/istl/vector.hh>
#include <dune/pdelab/backend/istl/bcrsmatrix.hh>
#include <dune/pdelab/backend/istl/bcrsmatrixbackend.hh>
#include <dune/pdelab/backend/istl/ovlpistlsolverbackend.hh>
#include <dune/pdelab/gridoperator/gridoperator.hh>
#include <dune/pdelab/localoperator/flags.hh>
#include <dune/pdelab/localoperator/idefault.hh>
#include <dune/pdelab/localoperator/pattern.hh>
#include <dune/pdelab/localoperator/defaultimp.hh>
 
namespace Dune {
  namespace PDELab {
    //***********************************************************
    // A data handle / function to communicate matrix entries
    // in the overlap. It turned out that it is actually not
    // required to do that, but anyway it is there now.
    //***********************************************************
 
    template<class GFS>
    class LocalGlobalMapDataHandle
      : public Dune::CommDataHandleIF<LocalGlobalMapDataHandle<GFS>,int>
    {
    public:
      // some types from the grid view
      typedef typename GFS::Traits::GridView GV;
      typedef typename GV::IndexSet IndexSet;
      typedef typename IndexSet::IndexType IndexType;
      typedef typename GV::Grid Grid;
      typedef typename Grid::Traits::GlobalIdSet GlobalIdSet;
      typedef typename GlobalIdSet::IdType IdType;
 
      typedef int DataType;
 
      // map types
      typedef std::map<IndexType,IdType> LocalToGlobalMap;
      typedef std::map<IdType,IndexType> GlobalToLocalMap;
 
      bool contains (int dim, int codim) const
      {
        return (codim==0);
      }
 
      bool fixedSize (int dim, int codim) const
      {
        return true;
      }
 
      template<class EntityType>
      size_t size (EntityType& e) const
      {
        return 1;
      }
 
      template<class MessageBuffer, class EntityType>
      void gather (MessageBuffer& buff, const EntityType& e) const
      {
        // fill map
        IndexType myindex = indexSet.index(e);
        IdType myid = globalIdSet.id(e);
        l2g[myindex] = myid;
        g2l[myid] = myindex;
        //std::cout << gv.comm().rank() << ": gather local=" << myindex << " global=" << myid << std::endl;
 
        buff.write(0); // this is a dummy, we are not interested in the data
      }
 
      template<class MessageBuffer, class EntityType>
      void scatter (MessageBuffer& buff, const EntityType& e, size_t n)
      {
        DataType x;
        buff.read(x); // this is a dummy, we are not interested in the data
 
        IndexType myindex = indexSet.index(e);
        IdType myid = globalIdSet.id(e);
        l2g[myindex] = myid;
        g2l[myid] = myindex;
        //std::cout << gv.comm().rank() << ": scatter local=" << myindex << " global=" << myid << std::endl;
      }
 
      LocalGlobalMapDataHandle (const GFS& gfs_, LocalToGlobalMap& l2g_, GlobalToLocalMap& g2l_)
        : gfs(gfs_), gv(gfs.gridView()), indexSet(gv.indexSet()),
          grid(gv.grid()), globalIdSet(grid.globalIdSet()),
          l2g(l2g_), g2l(g2l_)
      {
      }
 
    private:
      const GFS& gfs;
      GV gv;
      const IndexSet& indexSet;
      const Grid& grid;
      const GlobalIdSet& globalIdSet;
      LocalToGlobalMap& l2g;
      GlobalToLocalMap& g2l;
    };
 
 
    // A DataHandle class to exchange rows of a sparse matrix
    template<class GFS, class M> // mapper type and vector type
    class MatrixExchangeDataHandle
      : public Dune::CommDataHandleIF<MatrixExchangeDataHandle<GFS,M>,
                                      std::pair<typename GFS::Traits::GridView::Grid::Traits::GlobalIdSet::IdType,
                                                typename M::block_type> >
    {
    public:
      // some types from the grid view
      typedef typename GFS::Traits::GridView GV;
      typedef typename GV::IndexSet IndexSet;
      typedef typename IndexSet::IndexType IndexType;
      typedef typename GV::Grid Grid;
      typedef typename Grid::Traits::GlobalIdSet GlobalIdSet;
      typedef typename GlobalIdSet::IdType IdType;
 
      // some types from the matrix
      typedef typename M::block_type B;
 
      typedef std::pair<IdType,B> DataType;
 
      // map types
      typedef std::map<IndexType,IdType> LocalToGlobalMap;
      typedef std::map<IdType,IndexType> GlobalToLocalMap;
 
      bool contains (int dim, int codim) const
      {
        return (codim==0);
      }
 
      bool fixedSize (int dim, int codim) const
      {
        return false;
      }
 
      template<class EntityType>
      size_t size (EntityType& e) const
      {
        IndexType myindex = indexSet.index(e);
        typename M::row_type myrow = m[myindex];
        typename M::row_type::iterator endit=myrow.end();
        size_t count=0;
        for (typename M::row_type::iterator it=myrow.begin(); it!=endit; ++it)
          {
            typename LocalToGlobalMap::const_iterator find=l2g.find(it.index());
            if (find!=l2g.end())
              {
                count++;
              }
          }
        //std::cout << gv.comm().rank() << ": row=" << myindex << " size=" << count << std::endl;
        return count;
      }
 
      template<class MessageBuffer, class EntityType>
      void gather (MessageBuffer& buff, const EntityType& e) const
      {
        IndexType myindex = indexSet.index(e);
        //std::cout << gv.comm().rank() << ": begin gather local=" << myindex << std::endl;
        typename M::row_type myrow = m[myindex];
        typename M::row_type::iterator endit=myrow.end();
        size_t count = 0;
        for (typename M::row_type::iterator it=myrow.begin(); it!=endit; ++it)
          {
            typename LocalToGlobalMap::const_iterator find=l2g.find(it.index());
            if (find!=l2g.end())
              {
                buff.write(std::make_pair(find->second,*it));
                //std::cout << gv.comm().rank() << ":   ==> local=" << find->first << " global=" << find->second << std::endl;
                count++;
              }
          }
        //std::cout << gv.comm().rank() << ": end gather row=" << myindex << " size=" << count << std::endl;
      }
 
      template<class MessageBuffer, class EntityType>
      void scatter (MessageBuffer& buff, const EntityType& e, size_t n)
      {
        IndexType myindex = indexSet.index(e);
        std::cout << gv.comm().rank() << ": begin scatter local=" << myindex << " size=" << n << std::endl;
        DataType x;
        size_t count = 0;
        for (size_t i=0; i<n; ++i)
          {
            buff.read(x);
            std::cout << gv.comm().rank() << ":   --> received global=" << x.first << std::endl;
            typename GlobalToLocalMap::const_iterator find=g2l.find(x.first);
            if (find!=g2l.end())
              {
                IndexType nbindex=find->second;
                if (m.exists(myindex,nbindex))
                  {
                    m[myindex][nbindex] = x.second;
                    B block(x.second);
                    block -= m2[myindex][nbindex];
                    std::cout << gv.comm().rank() << ":   compare i=" << myindex << " j=" << nbindex
                              << " norm=" << block.infinity_norm() << std::endl;
 
                    count++;
                    //std::cout << gv.comm().rank() << ":   --> local=" << find->first << " global=" << find->second << std::endl;
                  }
              }
          }
        //std::cout << gv.comm().rank() << ": end scatter row=" << myindex << " size=" << count << std::endl;
      }
 
      MatrixExchangeDataHandle (const GFS& gfs_, M& m_, const LocalToGlobalMap& l2g_, const GlobalToLocalMap& g2l_,M& m2_)
        : gfs(gfs_), m(m_), gv(gfs.gridView()), indexSet(gv.indexSet()),
          grid(gv.grid()), globalIdSet(grid.globalIdSet()),
          l2g(l2g_), g2l(g2l_), m2(m2_)
      {
      }
 
    private:
      const GFS& gfs;
      M& m;
      GV gv;
      const IndexSet& indexSet;
      const Grid& grid;
      const GlobalIdSet& globalIdSet;
      const LocalToGlobalMap& l2g;
      const GlobalToLocalMap& g2l;
      M& m2;
    };
 
 
  template<class GFS, class T, class A, int n, int m>
  void restore_overlap_entries (const GFS& gfs, Dune::BCRSMatrix<Dune::FieldMatrix<T,n,m>,A>& matrix,
                                Dune::BCRSMatrix<Dune::FieldMatrix<T,n,m>,A>& matrix2)
  {
    typedef Dune::FieldMatrix<T,n,m> B;
    typedef Dune::BCRSMatrix<B,A> M; // m is of type M
    typedef typename LocalGlobalMapDataHandle<GFS>::LocalToGlobalMap LocalToGlobalMap;
    typedef typename LocalGlobalMapDataHandle<GFS>::GlobalToLocalMap GlobalToLocalMap;
 
    // build up two maps to associate local indices and global ids
    LocalToGlobalMap l2g;
    GlobalToLocalMap g2l;
    LocalGlobalMapDataHandle<GFS> lgdh(gfs,l2g,g2l);
    if (gfs.gridView().comm().size()>1)
      gfs.gridView().communicate(lgdh,Dune::InteriorBorder_All_Interface,Dune::ForwardCommunication);
 
    // exchange matrix entries
    MatrixExchangeDataHandle<GFS,M> mexdh(gfs,matrix,l2g,g2l,matrix2);
    if (gfs.gridView().comm().size()>1)
      gfs.gridView().communicate(mexdh,Dune::InteriorBorder_All_Interface,Dune::ForwardCommunication);
  }
 
  //***********************************************************
  // The DG/AMG preconditioner in the overlapping case
  //***********************************************************
 
  template<class DGGFS, class DGMatrix, class DGPrec, class DGCC,
           class CGGFS, class CGPrec, class CGCC,
           class P, class DGHelper, class Comm>
  class OvlpDGAMGPrec
    : public Dune::Preconditioner<Dune::PDELab::Backend::Vector<DGGFS,typename DGPrec::domain_type::field_type>,
                                  Dune::PDELab::Backend::Vector<DGGFS,typename DGPrec::range_type::field_type>>
  {
    const DGGFS& dggfs;
    DGMatrix& dgmatrix;
    DGPrec& dgprec;
    const DGCC& dgcc;
    const CGGFS& cggfs;
    CGPrec& cgprec;
    const CGCC& cgcc;
    P& p;
    const DGHelper& dghelper;
    const Comm& comm;
    int n1,n2;
 
  public:
    using V = Dune::PDELab::Backend::Vector<DGGFS,typename DGPrec::domain_type::field_type>;
    using W = Dune::PDELab::Backend::Vector<DGGFS,typename DGPrec::range_type::field_type>;
    using X = Backend::Native<V>;
    using Y = Backend::Native<W>;
    using CGV = Dune::PDELab::Backend::Vector<CGGFS,typename CGPrec::domain_type::field_type>;
    using CGW = Dune::PDELab::Backend::Vector<CGGFS,typename CGPrec::range_type::field_type>;
 
    // define the category
    SolverCategory::Category category() const override
    {
      return SolverCategory::overlapping;
    }
 
    OvlpDGAMGPrec (const DGGFS& dggfs_, DGMatrix& dgmatrix_, DGPrec& dgprec_, const DGCC& dgcc_,
                   const CGGFS& cggfs_, CGPrec& cgprec_, const CGCC& cgcc_, P& p_,
                   const DGHelper& dghelper_, const Comm& comm_, int n1_, int n2_)
      : dggfs(dggfs_), dgmatrix(dgmatrix_), dgprec(dgprec_), dgcc(dgcc_),
        cggfs(cggfs_), cgprec(cgprec_), cgcc(cgcc_), p(p_), dghelper(dghelper_),
        comm(comm_), n1(n1_), n2(n2_)
    {
    }
 
    virtual void pre (V& x, W& b) override
    {
      using Backend::native;
      dgprec.pre(native(x),native(b));
      CGW cgd(cggfs,0.0);
      CGV cgv(cggfs,0.0);
      cgprec.pre(native(cgv),native(cgd));
    }
 
    virtual void apply (V& x, const W& b) override
    {
      using Backend::native;
      // need local copies to store defect and solution
      W d(b);
      Dune::PDELab::set_constrained_dofs(dgcc,0.0,d);
      V v(x); // only to get correct size
 
      // pre-smoothing on DG matrix
      for (int i=0; i<n1; i++)
        {
          using Backend::native;
          v = 0.0;
          dgprec.apply(native(v),native(d));
          Dune::PDELab::AddDataHandle<DGGFS,V> adddh(dggfs,v);
          if (dggfs.gridView().comm().size()>1)
            dggfs.gridView().communicate(adddh,Dune::All_All_Interface,Dune::ForwardCommunication);
          dgmatrix.mmv(native(v),native(d));
          Dune::PDELab::set_constrained_dofs(dgcc,0.0,d);
          x += v;
        }
 
      // restrict defect to CG subspace
      dghelper.maskForeignDOFs(d); // DG defect is additive for overlap 1, but in case we use more
      CGW cgd(cggfs,0.0);
      p.mtv(native(d),native(cgd));
      Dune::PDELab::AddDataHandle<CGGFS,CGW> adddh(cggfs,cgd);
      if (cggfs.gridView().comm().size()>1)
        cggfs.gridView().communicate(adddh,Dune::All_All_Interface,Dune::ForwardCommunication); // now we have consistent defect on coarse grid
      Dune::PDELab::set_constrained_dofs(cgcc,0.0,cgd);
      comm.project(native(cgd));
      CGV cgv(cggfs,0.0);
 
 
      // call preconditioner
      cgprec.apply(native(cgv),native(cgd));
 
      // prolongate correction
      p.mv(native(cgv),native(v));
      dgmatrix.mmv(native(v),native(d));
      Dune::PDELab::set_constrained_dofs(dgcc,0.0,d);
      x += v;
 
      // post-smoothing on DG matrix
      for (int i=0; i<n2; i++)
        {
          v = 0.0;
          dgprec.apply(native(v),native(d));
          Dune::PDELab::AddDataHandle<DGGFS,V> adddh(dggfs,v);
          if (dggfs.gridView().comm().size()>1)
            dggfs.gridView().communicate(adddh,Dune::All_All_Interface,Dune::ForwardCommunication);
          dgmatrix.mmv(native(v),native(d));
          Dune::PDELab::set_constrained_dofs(dgcc,0.0,d);
          x += v;
        }
    }
 
    virtual void post (V& x) override
    {
      dgprec.post(Backend::native(x));
      CGV cgv(cggfs,0.0);
      cgprec.post(Backend::native(cgv));
    }
  };
 
//***********************************************************
// The DG/AMG linear solver backend in the overlapping case
//***********************************************************
 
template<class DGGO, class DGCC, class CGGFS, class CGCC, class TransferLOP,
         template<class,class,class,int> class DGPrec, template<class> class Solver, int s=96>
class ISTLBackend_OVLP_AMG_4_DG :
  public Dune::PDELab::OVLPScalarProductImplementation<typename DGGO::Traits::TrialGridFunctionSpace>,
  public Dune::PDELab::LinearResultStorage
{
public:
  // DG grid function space
  typedef typename DGGO::Traits::TrialGridFunctionSpace GFS;
 
  // vectors and matrices on DG level
  typedef typename DGGO::Traits::Jacobian M; // wrapped istl DG matrix
  typedef typename DGGO::Traits::Domain V;   // wrapped istl DG vector
  typedef Backend::Native<M> Matrix;         // istl DG matrix
  typedef Backend::Native<V> Vector;         // istl DG vector
  typedef typename Vector::field_type field_type;
 
  // vectors and matrices on CG level
  using CGV = Dune::PDELab::Backend::Vector<CGGFS,field_type>; // wrapped istl CG vector
  typedef Backend::Native<CGV> CGVector;                       // istl CG vector
 
  // prolongation matrix
  typedef Dune::PDELab::ISTL::BCRSMatrixBackend<> MBE;
  typedef Dune::PDELab::EmptyTransformation CC;
  typedef TransferLOP CGTODGLOP; // local operator
  typedef Dune::PDELab::GridOperator<CGGFS,GFS,CGTODGLOP,MBE,field_type,field_type,field_type,CC,CC> PGO;
  typedef typename PGO::Jacobian PMatrix; // wrapped ISTL prolongation matrix
  typedef Backend::Native<PMatrix> P;     // ISTL prolongation matrix
 
  // CG subspace matrix
  typedef typename Dune::TransposedMatMultMatResult<P,Matrix>::type PTADG;
  typedef typename Dune::MatMultMatResult<PTADG,P>::type CGMatrix; // istl coarse space matrix
 
  // AMG in CG-subspace
  typedef typename Dune::PDELab::ISTL::CommSelector<s,Dune::MPIHelper::isFake>::type Comm;
  typedef Dune::OverlappingSchwarzOperator<CGMatrix,CGVector,CGVector,Comm> ParCGOperator;
  typedef Dune::SeqSSOR<CGMatrix,CGVector,CGVector,1> Smoother;
  typedef Dune::BlockPreconditioner<CGVector,CGVector,Comm,Smoother> ParSmoother;
  typedef Dune::Amg::AMG<ParCGOperator,CGVector,ParSmoother,Comm> AMG;
  typedef Dune::Amg::Parameters Parameters;
 
private:
 
  const GFS& gfs;
  DGGO& dggo;
  const DGCC& dgcc;
  CGGFS& cggfs;
  const CGCC& cgcc;
  std::shared_ptr<AMG> amg;
  Parameters amg_parameters;
  unsigned maxiter;
  int verbose;
  bool reuse;
  bool firstapply;
  bool usesuperlu;
  std::size_t low_order_space_entries_per_row;
 
  // Parameters for DG smoother
  double smoother_relaxation = 0.92; // Relaxation parameter of DG smoother
  int n1=3; // Number of DG pre-smoothing steps
  int n2=3; // Number of DG post-smoothing steps
 
  CGTODGLOP cgtodglop;  // local operator to assemble prolongation matrix
  PGO pgo;              // grid operator to assemble prolongation matrix
  PMatrix pmatrix;      // wrapped prolongation matrix
 
  class EmptyLop : public Dune::PDELab::NumericalJacobianApplyVolume<EmptyLop>,
                   public Dune::PDELab::FullVolumePattern,
                   public Dune::PDELab::LocalOperatorDefaultFlags,
                   public Dune::PDELab::InstationaryLocalOperatorDefaultMethods<double>
  {
  };
 
  // grid operator with empty local operator for constraints assembly in parallel
  typedef Dune::PDELab::GridOperator<CGGFS,CGGFS,EmptyLop,MBE,field_type,field_type,field_type,CGCC,CGCC> CGGO;
  // CG-subspace matrix
  typedef typename CGGO::Jacobian CGM;
  CGM acg;
 
public:
 
  // access to prolongation matrix
  PMatrix& prolongation_matrix ()
  {
    return pmatrix;
  }
 
  void setParameters(const Parameters& amg_parameters_)
  {
    amg_parameters = amg_parameters_;
  }
 
  const Parameters& parameters() const
  {
    return amg_parameters;
  }
 
  void setReuse(bool reuse_)
  {
    reuse = reuse_;
  }
 
  bool getReuse() const
  {
    return reuse;
  }
 
  ISTLBackend_OVLP_AMG_4_DG(DGGO& dggo_, const DGCC& dgcc_, CGGFS& cggfs_, const CGCC& cgcc_,
                            unsigned maxiter_=5000, int verbose_=1, bool reuse_=false,
                            bool usesuperlu_=true)
    : Dune::PDELab::OVLPScalarProductImplementation<typename DGGO::Traits::TrialGridFunctionSpace>(dggo_.trialGridFunctionSpace())
    , gfs(dggo_.trialGridFunctionSpace())
    , dggo(dggo_)
    , dgcc(dgcc_)
    , cggfs(cggfs_)
    , cgcc(cgcc_)
    , amg_parameters(15,2000)
    , maxiter(maxiter_)
    , verbose(verbose_)
    , reuse(reuse_)
    , firstapply(true)
    , usesuperlu(usesuperlu_)
    , low_order_space_entries_per_row(StaticPower<3,GFS::Traits::GridView::dimension>::power)
    , cgtodglop()
    , pgo(cggfs,dggo.trialGridFunctionSpace(),cgtodglop,MBE(low_order_space_entries_per_row))
    , pmatrix(pgo)
    , acg(Backend::attached_container())
  {
    amg_parameters.setDefaultValuesIsotropic(GFS::Traits::GridViewType::Traits::Grid::dimension);
    amg_parameters.setDebugLevel(verbose_);
#if !HAVE_SUPERLU
    if (usesuperlu == true)
      {
        if (gfs.gridView().comm().rank()==0)
          std::cout << "WARNING: You are using AMG without SuperLU!"
                    << " Please consider installing SuperLU,"
                    << " or set the usesuperlu flag to false"
                    << " to suppress this warning." << std::endl;
      }
#endif
 
    // assemble prolongation matrix; this will not change from one apply to the next
    pmatrix = 0.0;
    if (verbose>0 && gfs.gridView().comm().rank()==0) std::cout << "allocated prolongation matrix of size " << pmatrix.N() << " x " << pmatrix.M() << std::endl;
    CGV cgx(cggfs,0.0);         // need vector to call jacobian
    pgo.jacobian(cgx,pmatrix);
  }
 
 
  ISTLBackend_OVLP_AMG_4_DG(DGGO& dggo_, const DGCC& dgcc_, CGGFS& cggfs_, const CGCC& cgcc_,
                            const ParameterTree& params)
    : Dune::PDELab::OVLPScalarProductImplementation<typename DGGO::Traits::TrialGridFunctionSpace>(dggo_.trialGridFunctionSpace())
    , gfs(dggo_.trialGridFunctionSpace())
    , dggo(dggo_)
    , dgcc(dgcc_)
    , cggfs(cggfs_)
    , cgcc(cgcc_)
    , maxiter(params.get<int>("max_iterations",5000))
    , amg_parameters(15,2000)
    , verbose(params.get<int>("verbose",1))
    , reuse(params.get<bool>("reuse",false))
    , firstapply(true)
    , usesuperlu(params.get<bool>("use_superlu",true))
    , low_order_space_entries_per_row(params.get<std::size_t>("low_order_space.entries_per_row",StaticPower<3,GFS::Traits::GridView::dimension>::power))
    , cgtodglop()
    , pgo(cggfs,dggo.trialGridFunctionSpace(),cgtodglop,MBE(low_order_space_entries_per_row))
    , pmatrix(pgo)
    , acg(Backend::attached_container())
  {
    amg_parameters.setDefaultValuesIsotropic(GFS::Traits::GridViewType::Traits::Grid::dimension);
    amg_parameters.setDebugLevel(params.get<int>("verbose",1));
#if !HAVE_SUPERLU
    if (usesuperlu == true)
      {
        if (gfs.gridView().comm().rank()==0)
          std::cout << "WARNING: You are using AMG without SuperLU!"
                    << " Please consider installing SuperLU,"
                    << " or set the usesuperlu flag to false"
                    << " to suppress this warning." << std::endl;
      }
#endif
 
    // assemble prolongation matrix; this will not change from one apply to the next
    pmatrix = 0.0;
    if (verbose>0 && gfs.gridView().comm().rank()==0) std::cout << "allocated prolongation matrix of size " << pmatrix.N() << " x " << pmatrix.M() << std::endl;
    CGV cgx(cggfs,0.0);         // need vector to call jacobian
    pgo.jacobian(cgx,pmatrix);
  }
 
 
  void setDGSmootherRelaxation(double relaxation_)
  {
    smoother_relaxation = relaxation_;
  }
 
  void setNoDGPreSmoothSteps(int n1_)
  {
    n1 = n1_;
  }
 
  void setNoDGPostSmoothSteps(int n2_)
  {
    n2 = n2_;
  }
 
 
  void apply (M& A, V& z, V& r, typename Dune::template FieldTraits<typename V::ElementType >::real_type reduction)
  {
    using Backend::native;
    // make operator and scalar product for overlapping solver
    typedef Dune::PDELab::OverlappingOperator<DGCC,M,V,V> POP;
    POP pop(dgcc,A);
    typedef Dune::PDELab::OVLPScalarProduct<GFS,V> PSP;
    PSP psp(*this);
 
    // compute CG matrix
    // make grid operator with empty local operator => matrix data type and constraints assembly
    EmptyLop emptylop;
    CGGO cggo(cggfs,cgcc,cggfs,cgcc,emptylop,MBE(low_order_space_entries_per_row));
 
    // do triple matrix product ACG = P^T ADG P; this is purely local
    Dune::Timer watch;
    watch.reset();
    // only do triple matrix product if the matrix changes
    double triple_product_time = 0.0;
    // no need to set acg here back to zero, this is done in matMultmat
    if(reuse == false || firstapply == true) {
      PTADG ptadg;
      Dune::transposeMatMultMat(ptadg,native(pmatrix),native(A)); // 1a
      //Dune::transposeMatMultMat(ptadg,native(pmatrix),native(A2));   // 1b
      Dune::matMultMat(native(acg),ptadg,native(pmatrix));
      triple_product_time = watch.elapsed();
      if (verbose>0 && gfs.gridView().comm().rank()==0)
        std::cout << "=== triple matrix product " << triple_product_time << " s" << std::endl;
      //Dune::printmatrix(std::cout,native(acg),"triple product matrix","row",10,2);
      CGV cgx(cggfs,0.0);     // need vector to call jacobian
      cggo.jacobian(cgx,acg); // insert trivial rows at processor boundaries
      //std::cout << "CG constraints: " << cgcc.size() << " out of " << cggfs.globalSize() << std::endl;
    }
    else if(verbose>0 && gfs.gridView().comm().rank()==0)
      std::cout << "=== reuse CG matrix, SKIPPING triple matrix product " << std::endl;
 
    // NOW we need to insert the processor boundary conditions in DG matrix
    typedef Dune::PDELab::GridOperator<GFS,GFS,EmptyLop,MBE,field_type,field_type,field_type,DGCC,DGCC> DGGOEmpty;
    DGGOEmpty dggoempty(gfs,dgcc,gfs,dgcc,emptylop,MBE(1 << GFS::Traits::GridView::dimension));
    dggoempty.jacobian(z,A);
 
    // and in the residual
    Dune::PDELab::set_constrained_dofs(dgcc,0.0,r);
 
    // now set up parallel AMG solver for the CG subspace
    Comm oocc(gfs.gridView().comm());
    typedef Dune::PDELab::ISTL::ParallelHelper<CGGFS> CGHELPER;
    CGHELPER cghelper(cggfs,2);
    cghelper.createIndexSetAndProjectForAMG(acg,oocc);
    ParCGOperator paroop(native(acg),oocc);
    Dune::OverlappingSchwarzScalarProduct<CGVector,Comm> sp(oocc);
 
    typedef typename Dune::Amg::SmootherTraits<ParSmoother>::Arguments SmootherArgs;
    SmootherArgs smootherArgs;
    smootherArgs.iterations = 1;
    smootherArgs.relaxationFactor = 1.0;
    typedef Dune::Amg::CoarsenCriterion<Dune::Amg::SymmetricCriterion<CGMatrix,Dune::Amg::FirstDiagonal> > Criterion;
    Criterion criterion(amg_parameters);
    watch.reset();
 
    // only construct a new AMG for the CG-subspace if the matrix changes
    double amg_setup_time = 0.0;
    if(reuse == false || firstapply == true) {
      amg.reset(new AMG(paroop,criterion,smootherArgs,oocc));
      firstapply = false;
      amg_setup_time = watch.elapsed();
      if (verbose>0 && gfs.gridView().comm().rank()==0)
        std::cout << "=== AMG setup " <<amg_setup_time << " s" << std::endl;
    }
    else if (verbose>0 && gfs.gridView().comm().rank()==0)
      std::cout << "=== reuse CG matrix, SKIPPING AMG setup " << std::endl;
 
    // set up hybrid DG/CG preconditioner
    typedef DGPrec<Matrix,Vector,Vector,1> DGPrecType;
    DGPrecType dgprec(native(A),1,smoother_relaxation);
    typedef Dune::PDELab::ISTL::ParallelHelper<GFS> DGHELPER;
    typedef OvlpDGAMGPrec<GFS,Matrix,DGPrecType,DGCC,CGGFS,AMG,CGCC,P,DGHELPER,Comm> HybridPrec;
    HybridPrec hybridprec(gfs,native(A),dgprec,dgcc,cggfs,*amg,cgcc,native(pmatrix),
                          this->parallelHelper(),oocc,n1,n2);
 
    // /********/
    // /* Test */
    // /********/
    // Solver<CGVector> testsolver(paroop,sp,amg,1e-8,100,2);
    // CGV cgxx(cggfs,0.0);
    // CGV cgdd(cggfs,1.0);
    // Dune::InverseOperatorResult statstat;
    // testsolver.apply(native(cgxx),native(cgdd),statstat);
    // /********/
 
    // set up solver
    int verb=verbose;
    if (gfs.gridView().comm().rank()>0) verb=0;
    Solver<V> solver(pop,psp,hybridprec,reduction,maxiter,verb);
 
    // solve
    Dune::InverseOperatorResult stat;
    watch.reset();
    solver.apply(z,r,stat);
    double amg_solve_time = watch.elapsed();
    if (verbose>0 && gfs.gridView().comm().rank()==0) std::cout << "=== Hybrid total solve time " << amg_solve_time+amg_setup_time+triple_product_time << " s" << std::endl;
    res.converged  = stat.converged;
    res.iterations = stat.iterations;
    res.elapsed    = amg_solve_time+amg_setup_time+triple_product_time;
    res.reduction  = stat.reduction;
    res.conv_rate  = stat.conv_rate;
  }
 
};
}
}
#endif // DUNE_PDELAB_BACKEND_ISTL_OVLP_AMG_DG_BACKEND_HH