fe_q_bubbles.cc

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//
// Copyright (C) 2012 - 2018 by the deal.II authors
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#include <deal.II/base/quadrature.h>
#include <deal.II/base/qprojector.h>
#include <deal.II/base/template_constraints.h>
#include <deal.II/fe/fe_q_bubbles.h>
#include <deal.II/fe/fe_nothing.h>
#include <deal.II/fe/fe_tools.h>
#include <deal.II/fe/fe_values.h>
#include <deal.II/fe/mapping_q1.h>
#include <deal.II/base/quadrature_lib.h>
#include <deal.II/dofs/dof_accessor.h>
#include <deal.II/dofs/dof_handler.h>
#include <deal.II/grid/tria.h>

#include <deal.II/grid/grid_generator.h>


#include <vector>
#include <sstream>
#include <memory>
#include <deal.II/base/std_cxx14/memory.h>

DEAL_II_NAMESPACE_OPEN


namespace internal
{
  namespace FE_Q_Bubbles
  {
    namespace
    {
      template <int dim, int spacedim>
      inline
      void
      compute_embedding_matrices(const ::FE_Q_Bubbles<dim, spacedim> &fe,
                                 std::vector<std::vector<FullMatrix<double> > > &matrices,
                                 const bool                                 isotropic_only)
      {
        const unsigned int dpc    = fe.dofs_per_cell;
        const unsigned int degree = fe.degree;

        // Initialize quadrature formula on fine cells
        std::unique_ptr<Quadrature<dim> > q_fine;
        Quadrature<1> q_dummy(std::vector<Point<1> >(1), std::vector<double> (1,1.));
        switch (dim)
          {
          case 1:
            if (spacedim==1)
              q_fine = std_cxx14::make_unique<QGauss<dim>>(degree+1);
            else if (spacedim==2)
              q_fine = std_cxx14::make_unique<QAnisotropic<dim>>(QGauss<1>(degree+1), q_dummy);
            else
              q_fine = std_cxx14::make_unique<QAnisotropic<dim>>(QGauss<1>(degree+1), q_dummy, q_dummy);
            break;
          case 2:
            if (spacedim==2)
              q_fine = std_cxx14::make_unique<QGauss<dim>>(degree+1);
            else
              q_fine = std_cxx14::make_unique<QAnisotropic<dim>>(QGauss<1>(degree+1), QGauss<1>(degree+1), q_dummy);
            break;
          case 3:
            q_fine = std_cxx14::make_unique<QGauss<dim>>(degree+1);
            break;
          default:
            Assert(false, ExcInternalError());
          }

        Assert(q_fine.get() != nullptr, ExcInternalError());
        const unsigned int nq = q_fine->size();

        // loop over all possible refinement cases
        unsigned int ref_case = (isotropic_only)
                                ? RefinementCase<dim>::isotropic_refinement
                                : RefinementCase<dim>::cut_x;
        for (; ref_case <= RefinementCase<dim>::isotropic_refinement; ++ref_case)
          {
            const unsigned int nc
              = GeometryInfo<dim>::n_children(RefinementCase<dim>(ref_case));

            for (unsigned int i=0; i<nc; ++i)
              {
                Assert(matrices[ref_case-1][i].n() == dpc,
                       ExcDimensionMismatch(matrices[ref_case-1][i].n(),dpc));
                Assert(matrices[ref_case-1][i].m() == dpc,
                       ExcDimensionMismatch(matrices[ref_case-1][i].m(),dpc));
              }

            // create a respective refinement on the triangulation
            ::Triangulation<dim, spacedim> tr;
            GridGenerator::hyper_cube (tr, 0, 1);
            tr.begin_active()->set_refine_flag(RefinementCase<dim>(ref_case));
            tr.execute_coarsening_and_refinement();

            ::DoFHandler<dim, spacedim> dh(tr);
            dh.distribute_dofs(fe);

            ::FEValues<dim, spacedim> fine
            (StaticMappingQ1<dim,spacedim>::mapping, fe, *q_fine,
             update_quadrature_points
             | update_JxW_values | update_values);

            const unsigned int n_dofs = dh.n_dofs();

            FullMatrix<double> fine_mass(n_dofs);
            FullMatrix<double> coarse_rhs_matrix(n_dofs, dpc);

            std::vector<std::vector<types::global_dof_index> > child_ldi
            (nc, std::vector<types::global_dof_index>(fe.dofs_per_cell));

            //now create the mass matrix and all the right_hand sides
            unsigned int child_no = 0;
            typename ::DoFHandler<dim>::active_cell_iterator cell
              = dh.begin_active();
            for (; cell!=dh.end(); ++cell, ++child_no)
              {
                fine.reinit(cell);
                cell->get_dof_indices(child_ldi[child_no]);

                for (unsigned int q=0; q<nq; ++q)
                  for (unsigned int i=0; i<dpc; ++i)
                    for (unsigned int j=0; j<dpc; ++j)
                      {
                        const unsigned int gdi=child_ldi[child_no][i];
                        const unsigned int gdj=child_ldi[child_no][j];
                        fine_mass(gdi, gdj)+=fine.shape_value(i,q)
                                             *fine.shape_value(j,q)
                                             *fine.JxW(q);
                        Point<dim> quad_tmp;
                        for (unsigned int k=0; k<dim; ++k)
                          quad_tmp(k) = fine.quadrature_point(q)(k);
                        coarse_rhs_matrix(gdi, j)
                        +=fine.shape_value(i,q)
                          *fe.shape_value(j, quad_tmp)
                          *fine.JxW(q);
                      }
              }

            //now solve for all right-hand sides simultaneously
            ::FullMatrix<double> solution (n_dofs, dpc);
            fine_mass.gauss_jordan();
            fine_mass.mmult(solution, coarse_rhs_matrix);

            //and distribute to the fine cell matrices
            for (unsigned int child_no=0; child_no<nc; ++child_no)
              for (unsigned int i=0; i<dpc; ++i)
                for (unsigned int j=0; j<dpc; ++j)
                  {
                    const unsigned int gdi=child_ldi[child_no][i];
                    //remove small entries
                    if (std::fabs(solution(gdi, j)) > 1.e-12)
                      matrices[ref_case-1][child_no](i,j)=solution(gdi, j);
                  }
          }
      }
    }
  }
}


template <int dim, int spacedim>
FE_Q_Bubbles<dim,spacedim>::FE_Q_Bubbles (const unsigned int q_degree)
  :
  FE_Q_Base<TensorProductPolynomialsBubbles<dim>, dim, spacedim> (
    TensorProductPolynomialsBubbles<dim>(Polynomials::generate_complete_Lagrange_basis(QGaussLobatto<1>(q_degree+1).get_points())),
    FiniteElementData<dim>(get_dpo_vector(q_degree),
                           1, q_degree+1,
                           FiniteElementData<dim>::H1),
    get_riaf_vector(q_degree)),
  n_bubbles((q_degree<=1)?1:dim)
{
  Assert (q_degree > 0,
          ExcMessage ("This element can only be used for polynomial degrees "
                      "greater than zero"));

  this->initialize(QGaussLobatto<1>(q_degree+1).get_points());

  // adjust unit support point for discontinuous node
  Point<dim> point;
  for (unsigned int d=0; d<dim; ++d)
    point[d] = 0.5;
  for (unsigned int i=0; i<n_bubbles; ++i)
    this->unit_support_points.push_back(point);
  AssertDimension(this->dofs_per_cell, this->unit_support_points.size());

  this->reinit_restriction_and_prolongation_matrices();
  if (dim == spacedim)
    {
      internal::FE_Q_Bubbles::compute_embedding_matrices
      (*this, this->prolongation, false);
      // Fill restriction matrices with L2-projection
      FETools::compute_projection_matrices (*this, this->restriction);
    }

}



template <int dim, int spacedim>
FE_Q_Bubbles<dim,spacedim>::FE_Q_Bubbles (const Quadrature<1> &points)
  :
  FE_Q_Base<TensorProductPolynomialsBubbles<dim>, dim, spacedim> (
    TensorProductPolynomialsBubbles<dim>(Polynomials::generate_complete_Lagrange_basis(points.get_points())),
    FiniteElementData<dim>(get_dpo_vector(points.size()-1),
                           1, points.size(),
                           FiniteElementData<dim>::H1),
    get_riaf_vector(points.size()-1)),
  n_bubbles((points.size()-1<=1)?1:dim)
{
  Assert (points.size() > 1,
          ExcMessage ("This element can only be used for polynomial degrees "
                      "at least one"));

  this->initialize(points.get_points());

  // adjust unit support point for discontinuous node
  Point<dim> point;
  for (unsigned int d=0; d<dim; ++d)
    point[d] = 0.5;
  for (unsigned int i=0; i< n_bubbles; ++i)
    this->unit_support_points.push_back(point);
  AssertDimension(this->dofs_per_cell, this->unit_support_points.size());

  this->reinit_restriction_and_prolongation_matrices();
  if (dim == spacedim)
    {
      internal::FE_Q_Bubbles::compute_embedding_matrices
      (*this, this->prolongation, false);
      // Fill restriction matrices with L2-projection
      FETools::compute_projection_matrices (*this, this->restriction);
    }
}



template <int dim, int spacedim>
std::string
FE_Q_Bubbles<dim,spacedim>::get_name () const
{
  // note that the FETools::get_fe_by_name function depends on the
  // particular format of the string this function returns, so they have to be
  // kept in synch

  std::ostringstream namebuf;
  bool type = true;
  const unsigned int n_points = this->degree;
  std::vector<double> points(n_points);
  const unsigned int dofs_per_cell = this->dofs_per_cell;
  const std::vector<Point<dim> > &unit_support_points = this->unit_support_points;
  unsigned int index = 0;

  // Decode the support points in one coordinate direction.
  for (unsigned int j=0; j<dofs_per_cell; j++)
    {
      if ((dim>1) ? (unit_support_points[j](1)==0 &&
                     ((dim>2) ? unit_support_points[j](2)==0: true)) : true)
        {
          if (index == 0)
            points[index] = unit_support_points[j](0);
          else if (index == 1)
            points[n_points-1] = unit_support_points[j](0);
          else
            points[index-1] = unit_support_points[j](0);

          index++;
        }
    }
  // Do not consider the discontinuous node for dimension 1
  Assert (index == n_points || (dim==1 && index == n_points+n_bubbles),
          ExcMessage ("Could not decode support points in one coordinate direction."));

  // Check whether the support points are equidistant.
  for (unsigned int j=0; j<n_points; j++)
    if (std::fabs(points[j] - (double)j/(this->degree-1)) > 1e-15)
      {
        type = false;
        break;
      }

  if (type == true)
    {
      if (this->degree > 3)
        namebuf << "FE_Q_Bubbles<"
                << Utilities::dim_string(dim,spacedim)
                << ">(QIterated(QTrapez()," << this->degree-1 << "))";
      else
        namebuf << "FE_Q_Bubbles<"
                << Utilities::dim_string(dim,spacedim)
                << ">(" << this->degree-1 << ")";
    }
  else
    {

      // Check whether the support points come from QGaussLobatto.
      const QGaussLobatto<1> points_gl(n_points);
      type = true;
      for (unsigned int j=0; j<n_points; j++)
        if (points[j] != points_gl.point(j)(0))
          {
            type = false;
            break;
          }
      if (type == true)
        namebuf << "FE_Q_Bubbles<" << dim << ">(" << this->degree-1 << ")";
      else
        namebuf << "FE_Q_Bubbles<" << dim << ">(QUnknownNodes(" << this->degree << "))";
    }
  return namebuf.str();
}



template <int dim, int spacedim>
std::unique_ptr<FiniteElement<dim,spacedim> >
FE_Q_Bubbles<dim,spacedim>::clone() const
{
  return std_cxx14::make_unique<FE_Q_Bubbles<dim,spacedim>>(*this);
}



template <int dim, int spacedim>
void
FE_Q_Bubbles<dim,spacedim>::
convert_generalized_support_point_values_to_dof_values(const std::vector<Vector<double> > &support_point_values,
                                                       std::vector<double>    &nodal_values) const
{
  Assert (support_point_values.size() == this->unit_support_points.size(),
          ExcDimensionMismatch(support_point_values.size(),
                               this->unit_support_points.size()));
  Assert (nodal_values.size() == this->dofs_per_cell,
          ExcDimensionMismatch(nodal_values.size(),this->dofs_per_cell));
  Assert (support_point_values[0].size() == this->n_components(),
          ExcDimensionMismatch(support_point_values[0].size(),this->n_components()));

  for (unsigned int i=0; i<this->dofs_per_cell-1; ++i)
    {
      const std::pair<unsigned int, unsigned int> index
        = this->system_to_component_index(i);
      nodal_values[i] = support_point_values[i](index.first);
    }

  // We don't use the bubble functions for local interpolation
  for (unsigned int i = 0; i<n_bubbles; ++i)
    nodal_values[nodal_values.size()-i-1] = 0.;
}



template <int dim, int spacedim>
void
FE_Q_Bubbles<dim,spacedim>::
get_interpolation_matrix (const FiniteElement<dim,spacedim> &x_source_fe,
                          FullMatrix<double>       &interpolation_matrix) const
{
  // We don't know how to do this properly, yet.
  // However, for SolutionTransfer to work we need to provide an implementation
  // for the case that the x_source_fe is identical to this FE
  typedef FE_Q_Bubbles<dim,spacedim> FEQBUBBLES;

  AssertThrow ((x_source_fe.get_name().find ("FE_Q_Bubbles<") == 0)
               ||
               (dynamic_cast<const FEQBUBBLES *>(&x_source_fe) != nullptr)
               ,
               (typename FiniteElement<dim,spacedim>::
                ExcInterpolationNotImplemented()));
  Assert (interpolation_matrix.m() == this->dofs_per_cell,
          ExcDimensionMismatch (interpolation_matrix.m(),
                                this->dofs_per_cell));
  Assert (interpolation_matrix.n() == x_source_fe.dofs_per_cell,
          ExcDimensionMismatch (interpolation_matrix.m(),
                                x_source_fe.dofs_per_cell));

  //Provide a short cut in case we are just inquiring the identity
  auto casted_fe = dynamic_cast<const FEQBUBBLES *>(&x_source_fe);
  if (casted_fe != nullptr && casted_fe->degree == this->degree)
    for (unsigned int i=0; i<interpolation_matrix.m(); ++i)
      interpolation_matrix.set(i,i,1.);
  //else we need to do more...
  else
    Assert(false, (typename FiniteElement<dim,spacedim>::
                   ExcInterpolationNotImplemented()));
}



template <int dim, int spacedim>
std::vector<bool>
FE_Q_Bubbles<dim,spacedim>::get_riaf_vector(const unsigned int q_deg)
{
  unsigned int n_cont_dofs = Utilities::fixed_power<dim>(q_deg+1);
  const unsigned int n_bubbles = (q_deg<=1?1:dim);
  return std::vector<bool> (n_cont_dofs+n_bubbles,true);
}



template <int dim, int spacedim>
std::vector<unsigned int>
FE_Q_Bubbles<dim,spacedim>::get_dpo_vector(const unsigned int q_deg)
{
  std::vector<unsigned int> dpo(dim+1, 1U);
  for (unsigned int i=1; i<dpo.size(); ++i)
    dpo[i]=dpo[i-1]*(q_deg-1);

  dpo[dim]+=(q_deg<=1?1:dim);//all the bubble functions are discontinuous
  return dpo;
}



template <int dim, int spacedim>
bool
FE_Q_Bubbles<dim,spacedim>::has_support_on_face
(const unsigned int shape_index,
 const unsigned int face_index) const
{
  // discontinuous functions have no support on faces
  if (shape_index >= this->n_dofs_per_cell()-n_bubbles)
    return false;
  else
    return FE_Q_Base<TensorProductPolynomialsBubbles<dim>,dim,spacedim>::has_support_on_face(shape_index, face_index);
}



template <int dim, int spacedim>
const FullMatrix<double> &
FE_Q_Bubbles<dim,spacedim>::get_prolongation_matrix
(const unsigned int child,
 const RefinementCase<dim> &refinement_case) const
{
  Assert (refinement_case<RefinementCase<dim>::isotropic_refinement+1,
          ExcIndexRange(refinement_case,0,RefinementCase<dim>::isotropic_refinement+1));
  Assert (refinement_case!=RefinementCase<dim>::no_refinement,
          ExcMessage("Prolongation matrices are only available for refined cells!"));
  Assert (child<GeometryInfo<dim>::n_children(refinement_case),
          ExcIndexRange(child,0,GeometryInfo<dim>::n_children(refinement_case)));

  Assert (this->prolongation[refinement_case-1][child].n() != 0,
          ExcMessage("This prolongation matrix has not been computed yet!"));
  // finally return the matrix
  return this->prolongation[refinement_case-1][child];
}



template <int dim, int spacedim>
const FullMatrix<double> &
FE_Q_Bubbles<dim,spacedim>::get_restriction_matrix
(const unsigned int child,
 const RefinementCase<dim> &refinement_case) const
{
  Assert (refinement_case<RefinementCase<dim>::isotropic_refinement+1,
          ExcIndexRange(refinement_case,0,RefinementCase<dim>::isotropic_refinement+1));
  Assert (refinement_case!=RefinementCase<dim>::no_refinement,
          ExcMessage("Restriction matrices are only available for refined cells!"));
  Assert (child<GeometryInfo<dim>::n_children(refinement_case),
          ExcIndexRange(child,0,GeometryInfo<dim>::n_children(refinement_case)));

  Assert(this->restriction[refinement_case-1][child].n() != 0,
         ExcMessage("This restriction matrix has not been computed yet!"));

  //finally return the matrix
  return this->restriction[refinement_case-1][child];
}

// explicit instantiations
#include "fe_q_bubbles.inst"

DEAL_II_NAMESPACE_CLOSE