fe_pyramid_p.cc

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// -----------------------------------------------------------------------------
//
// SPDX-License-Identifier: Apache-2.0 WITH LLVM-exception OR LGPL-2.1-or-later
// Copyright (C) 2021 - 2026 by the deal.II authors
//
// This file is part of the deal.II library.
//
// Detailed license information governing the source code and contributions
// can be found in LICENSE.md and CONTRIBUTING.md at the top level directory.
//
// -----------------------------------------------------------------------------
 
#include <deal.II/base/config.h>
 
#include <deal.II/base/polynomials_barycentric.h>
#include <deal.II/base/qprojector.h>
#include <deal.II/base/quadrature_lib.h>
 
#include <deal.II/fe/fe_dgq.h>
#include <deal.II/fe/fe_nothing.h>
#include <deal.II/fe/fe_pyramid_p.h>
#include <deal.II/fe/fe_q.h>
#include <deal.II/fe/fe_simplex_p.h>
#include <deal.II/fe/fe_simplex_p_bubbles.h>
#include <deal.II/fe/fe_tools.h>
#include <deal.II/fe/fe_wedge_p.h>
 
DEAL_II_NAMESPACE_OPEN
 
namespace
{
  unsigned int
  compute_n_dofs(const unsigned int dim, const unsigned int degree)
  {
    AssertDimension(dim, 3);
    return (degree + 1) * (degree + 2) * (2 * degree + 3) / 6;
  }

  template <int dim>
  std::vector<Point<dim>>
  get_support_points(const unsigned int degree)
  {
    AssertDimension(dim, 3);
    Assert(degree > 0, ExcInternalError("Degree must be larger than 0."));
 
 
    std::vector<Point<dim>> support_points;
    support_points.resize(compute_n_dofs(dim, degree));
 
    const double z_equidistance = 1.0 / degree;
 
    // the support points on the 8 lines excluding the vertices
    const unsigned int n_dofs_per_line = degree - 1;
 
    // support points on the bottom quad face and on the 4 triangular faces,
    // on the triangular faces the number of points is the sum from 1 to
    // (degree - 2) so 4*0.5*(degree - 2)*(degree - 1)
    const unsigned int n_dofs_per_quad = n_dofs_per_line * n_dofs_per_line;
    const unsigned int total_dofs_faces =
      n_dofs_per_quad + 2 * (degree - 2) * (degree - 1);
 
    // starting indices for lines 4 - 7
    std::vector<unsigned int> start_lines(4);
    // line 4 starts after the DoFs at the vertices and the DoFs on the lines of
    // the bottom quad
    start_lines[0] = 5 + 4 * n_dofs_per_line;
    // the rest increments with the number of DoFs on the edges 4 - 7
    for (unsigned int i = 1; i < 4; ++i)
      start_lines[i] = start_lines[i - 1] + n_dofs_per_line;
 
    // same applies to the triangular faces 1 - 4
    std::vector<unsigned int> start_faces(4);
    start_faces[0] = 5 + 8 * n_dofs_per_line + n_dofs_per_quad;
 
    for (unsigned int i = 1; i < 4; ++i)
      start_faces[i] = start_faces[i - 1] + (degree - 2) * (degree - 1) / 2;
 
    unsigned int start_hex = 5 + 8 * n_dofs_per_line + total_dofs_faces;
 
    auto lift_point =
      [](const Point<2> &p2d, const double scale, const double z) {
        return Point<dim>(scale * (2.0 * p2d[0] - 1.0),
                          scale * (2.0 * p2d[1] - 1.0),
                          z);
      };
    {
      // this gives all info on the vertices, the first 4 edges and the
      // first face
      // switch to FE_Q when simplex supports electrostatic points
      // FE_Q<2> fe_q(degree);
      FE_Q<2> fe_q(QIterated<1>(QTrapezoid<1>(), degree));
 
      // vertices
      for (unsigned int v = 0; v < fe_q.reference_cell().n_vertices(); ++v)
        {
          support_points[v] =
            lift_point(fe_q.get_unit_support_points()[v], 1.0, 0.0);
        }
      // lines
      for (unsigned int l = 0;
           l < fe_q.reference_cell().n_lines() * fe_q.n_dofs_per_line();
           ++l)
        {
          support_points[5 + l] = lift_point(
            fe_q.get_unit_support_points()[fe_q.reference_cell().n_vertices() +
                                           l],
            1.0,
            0.0);
        }
      // quad
      for (unsigned int q = 0; q < fe_q.n_dofs_per_quad(); ++q)
        {
          support_points[5 + 8 * n_dofs_per_line + q] = lift_point(
            fe_q.get_unit_support_points()[fe_q.reference_cell().n_vertices() +
                                           fe_q.reference_cell().n_lines() *
                                             fe_q.n_dofs_per_line() +
                                           q],
            1.0,
            0.0);
        }
    }
    // now add the other layers
    for (unsigned int current_degree = degree - 1; current_degree > 0;
         --current_degree)
      {
        // switch to FE_Q when simplex supports electrostatic points
        // FE_Q<2> fe_q(current_degree);
        FE_Q<2> fe_q(QIterated<1>(QTrapezoid<1>(), current_degree));
 
 
        const auto  &points = fe_q.get_unit_support_points();
        unsigned int p      = 0;
 
        const double z     = (degree - current_degree) * z_equidistance;
        const double scale = current_degree * z_equidistance;
 
        // vertices are on lines
        for (unsigned int line = 0; line < fe_q.reference_cell().n_vertices();
             ++line)
          {
            support_points[start_lines[line]++] =
              lift_point(points[p++], scale, z);
          }
        // lines are on face
        for (unsigned int face = 0; face < fe_q.reference_cell().n_lines();
             ++face)
          {
            for (unsigned int n_dof = 0; n_dof < fe_q.n_dofs_per_line();
                 ++n_dof)
              support_points[start_faces[face]++] =
                lift_point(points[p++], scale, z);
          }
        // faces are on hex
        for (unsigned int hex = 0; hex < fe_q.n_dofs_per_quad(); ++hex)
          {
            support_points[start_hex++] = lift_point(points[p++], scale, z);
          }
      }
    Point<dim> tip;
    for (unsigned int d = 0; d < dim; ++d)
      {
        if (d == 2)
          tip[d] = 1.0;
        else
          tip[d] = 0.0;
      }
    support_points[4] = tip;
 
    return support_points;
  }
 

  template <int dim>
  internal::GenericDoFsPerObject
  get_dpo(const unsigned int                                degree,
          const typename FiniteElementData<dim>::Conformity conformity)
  {
    AssertDimension(dim, 3);
    internal::GenericDoFsPerObject dpo;
 
    if (conformity == FiniteElementData<dim>::L2)
      {
        dpo = internal::expand<3>({{0, 0, 0, compute_n_dofs(dim, degree)}},
                                  ReferenceCells::Pyramid);
      }
    else if (conformity == FiniteElementData<dim>::H1)
      {
        // the support points on the 8 lines excluding the vertices
        const unsigned int n_dofs_per_line = degree - 1;
 
        // support points on the bottom quad face and on the 4 triangular faces,
        // on the triangular faces the number of points is the sum from 1 to
        // (degree - 2) so 4*0.5*(degree - 2)*(degree - 1)
        const unsigned int n_dofs_per_quad = n_dofs_per_line * n_dofs_per_line;
        const unsigned int n_dofs_per_tri  = (degree - 2) * (degree - 1) / 2;
        const unsigned int total_dofs_faces =
          n_dofs_per_quad + 4 * n_dofs_per_tri;
        // total number of DoFs on a tri
        const unsigned int n_dofs_per_tri_inclusive =
          (degree + 1) * (degree + 2) / 2;
 
        const unsigned int n_dofs_total = compute_n_dofs(dim, degree);
 
        dpo.dofs_per_object_exclusive = {
          {1, 1, 1, 1, 1},
          {n_dofs_per_line,
           n_dofs_per_line,
           n_dofs_per_line,
           n_dofs_per_line,
           n_dofs_per_line,
           n_dofs_per_line,
           n_dofs_per_line,
           n_dofs_per_line},
          {n_dofs_per_quad,
           n_dofs_per_tri,
           n_dofs_per_tri,
           n_dofs_per_tri,
           n_dofs_per_tri},
          {n_dofs_total - 5 - 8 * n_dofs_per_line - total_dofs_faces}};
 
        dpo.dofs_per_object_inclusive = {{1, 1, 1, 1, 1},
                                         {
                                           degree + 1,
                                           degree + 1,
                                           degree + 1,
                                           degree + 1,
                                           degree + 1,
                                           degree + 1,
                                           degree + 1,
                                           degree + 1,
                                         },
                                         {(degree + 1) * (degree + 1),
                                          n_dofs_per_tri_inclusive,
                                          n_dofs_per_tri_inclusive,
                                          n_dofs_per_tri_inclusive,
                                          n_dofs_per_tri_inclusive},
                                         {n_dofs_total}};
 
        dpo.object_index = {
          {0, 1, 2, 3, 4},
          {5,
           5 + 1 * n_dofs_per_line,
           5 + 2 * n_dofs_per_line,
           5 + 3 * n_dofs_per_line,
           5 + 4 * n_dofs_per_line,
           5 + 5 * n_dofs_per_line,
           5 + 6 * n_dofs_per_line,
           5 + 7 * n_dofs_per_line},
          {5 + 8 * n_dofs_per_line,
           5 + 8 * n_dofs_per_line + n_dofs_per_quad,
           5 + 8 * n_dofs_per_line + n_dofs_per_quad + n_dofs_per_tri,
           5 + 8 * n_dofs_per_line + n_dofs_per_quad + 2 * n_dofs_per_tri,
           5 + 8 * n_dofs_per_line + n_dofs_per_quad + 3 * n_dofs_per_tri},
          {5 + 8 * n_dofs_per_line + total_dofs_faces}};
 
        dpo.first_object_index_on_face = {{0, 0, 0, 0, 0},
                                          {4, 3, 3, 3, 3},
                                          {4 + 4 * n_dofs_per_line,
                                           3 + 3 * n_dofs_per_line,
                                           3 + 3 * n_dofs_per_line,
                                           3 + 3 * n_dofs_per_line,
                                           3 + 3 * n_dofs_per_line}};
      }
    return dpo;
  }
} // namespace
 
 
template <int dim, int spacedim>
FE_PyramidPoly<dim, spacedim>::FE_PyramidPoly(
  const unsigned int                                degree,
  const internal::GenericDoFsPerObject              dpos,
  const std::vector<Point<dim>>                     support_points,
  const bool                                        prolongation_is_additive,
  const typename FiniteElementData<dim>::Conformity conformity)
  : ::FE_Poly<dim, spacedim>(
      ScalarLagrangePolynomialPyramid<dim>(degree,
                                           compute_n_dofs(dim, degree),
                                           support_points),
      FiniteElementData<dim>(dpos,
                             reinterpret_cast<const ReferenceCell<dim> &>(
                               ReferenceCells::Pyramid),
                             1,
                             degree,
                             conformity),
      std::vector<bool>(
        FiniteElementData<dim>(dpos,
                               reinterpret_cast<const ReferenceCell<dim> &>(
                                 ReferenceCells::Pyramid),
                               1,
                               degree)
          .dofs_per_cell,
        prolongation_is_additive),
      std::vector<ComponentMask>(
        FiniteElementData<dim>(dpos,
                               reinterpret_cast<const ReferenceCell<dim> &>(
                                 ReferenceCells::Pyramid),
                               1,
                               degree)
          .dofs_per_cell,
        ComponentMask(std::vector<bool>(1, true))))
{
  AssertDimension(dim, 3);
 
  for (auto &support_point : support_points)
    this->unit_support_points.emplace_back(support_point);
 
  if (conformity == FiniteElementData<dim>::H1)
    {
      // face support points
      this->unit_face_support_points.resize(this->reference_cell().n_faces());
 
      for (const auto f : this->reference_cell().face_indices())
        {
          const auto face_reference_cell =
            this->reference_cell().face_reference_cell(f);
 
          if (face_reference_cell == ReferenceCells::Quadrilateral)
            {
              // switch to FE_Q when simplex supports electrostatic points
              // FE_Q<2> fe_face(degree);
              FE_Q<2> fe_face(QIterated<1>(QTrapezoid<1>(), degree));
 
              for (const auto &face_support_point :
                   fe_face.get_unit_support_points())
                {
                  Point<dim - 1> p;
                  for (unsigned int d = 0; d < dim - 1; ++d)
                    p[d] = face_support_point[d];
                  this->unit_face_support_points[f].emplace_back(p);
                }
            }
          else if (face_reference_cell == ReferenceCells::Triangle)
            {
              FE_SimplexP<2> fe_face(degree);
              for (const auto &face_support_point :
                   fe_face.get_unit_support_points())
                {
                  Point<dim - 1> p;
                  for (unsigned int d = 0; d < dim - 1; ++d)
                    p[d] = face_support_point[d];
                  this->unit_face_support_points[f].emplace_back(p);
                }
            }
        }
    }
}
 
 
 
template <int dim, int spacedim>
void
FE_PyramidPoly<dim, spacedim>::
  convert_generalized_support_point_values_to_dof_values(
    const std::vector<Vector<double>> &support_point_values,
    std::vector<double>               &nodal_values) const
{
  AssertDimension(support_point_values.size(),
                  this->get_unit_support_points().size());
  AssertDimension(support_point_values.size(), nodal_values.size());
  AssertDimension(this->dofs_per_cell, nodal_values.size());
 
  for (unsigned int i = 0; i < this->dofs_per_cell; ++i)
    {
      AssertDimension(support_point_values[i].size(), 1);
 
      nodal_values[i] = support_point_values[i](0);
    }
}
 
 
 
template <int dim, int spacedim>
FE_PyramidP<dim, spacedim>::FE_PyramidP(const unsigned int degree)
  : FE_PyramidPoly<dim, spacedim>(degree,
                                  get_dpo<dim>(degree,
                                               FiniteElementData<dim>::H1),
                                  get_support_points<dim>(degree),
                                  false,
                                  FiniteElementData<dim>::H1)
{
  if (degree > 2)
    {
      // adjust DoFs on lines
      for (unsigned int i = 0; i < this->n_dofs_per_line(); ++i)
        this->adjust_line_dof_index_for_line_orientation_table[i] =
          this->n_dofs_per_line() - 1 - i - i;
 
      // do the quad face first
      const unsigned int n            = degree - 1;
      const unsigned int face_no_quad = 0;
      Assert(n * n == this->n_dofs_per_quad(face_no_quad), ExcInternalError());
      // see fe_q_base.cc
      for (unsigned int local = 0; local < this->n_dofs_per_quad(face_no_quad);
           ++local)
        {
          unsigned int i = local % n, j = local / n;
 
          // face_orientation=false, face_flip=false, face_rotation=false
          this->adjust_quad_dof_index_for_face_orientation_table[face_no_quad](
            local, internal::combined_face_orientation(false, false, false)) =
            j + i * n - local;
          // face_orientation=false, face_flip=false, face_rotation=true
          this->adjust_quad_dof_index_for_face_orientation_table[face_no_quad](
            local, internal::combined_face_orientation(false, true, false)) =
            i + (n - 1 - j) * n - local;
          // face_orientation=false, face_flip=true,  face_rotation=false
          this->adjust_quad_dof_index_for_face_orientation_table[face_no_quad](
            local, internal::combined_face_orientation(false, false, true)) =
            (n - 1 - j) + (n - 1 - i) * n - local;
          // face_orientation=false, face_flip=true,  face_rotation=true
          this->adjust_quad_dof_index_for_face_orientation_table[face_no_quad](
            local, internal::combined_face_orientation(false, true, true)) =
            (n - 1 - i) + j * n - local;
          // face_orientation=true,  face_flip=false, face_rotation=false
          this->adjust_quad_dof_index_for_face_orientation_table[face_no_quad](
            local, internal::combined_face_orientation(true, false, false)) = 0;
          // face_orientation=true,  face_flip=false, face_rotation=true
          this->adjust_quad_dof_index_for_face_orientation_table[face_no_quad](
            local, internal::combined_face_orientation(true, true, false)) =
            j + (n - 1 - i) * n - local;
          // face_orientation=true,  face_flip=true,  face_rotation=false
          this->adjust_quad_dof_index_for_face_orientation_table[face_no_quad](
            local, internal::combined_face_orientation(true, false, true)) =
            (n - 1 - i) + (n - 1 - j) * n - local;
          // face_orientation=true,  face_flip=true,  face_rotation=true
          this->adjust_quad_dof_index_for_face_orientation_table[face_no_quad](
            local, internal::combined_face_orientation(true, true, true)) =
            (n - 1 - j) + i * n - local;
        }
 
      // Now do it for the triangular faces
      // for degree 3 there is only 1 DoF on the face
      Assert(degree <= 3, ExcNotImplemented());
    }
}
 
 
 
template <int dim, int spacedim>
std::unique_ptr<FiniteElement<dim, spacedim>>
FE_PyramidP<dim, spacedim>::clone() const
{
  return std::make_unique<FE_PyramidP<dim, spacedim>>(*this);
}
 
 
 
template <int dim, int spacedim>
std::string
FE_PyramidP<dim, spacedim>::get_name() const
{
  std::ostringstream namebuf;
  namebuf << "FE_PyramidP<" << Utilities::dim_string(dim, spacedim) << ">("
          << this->degree << ")";
 
  return namebuf.str();
}
 
 
 
template <int dim, int spacedim>
FiniteElementDomination::Domination
FE_PyramidP<dim, spacedim>::compare_for_domination(
  const FiniteElement<dim, spacedim> &fe_other,
  const unsigned int                  codim) const
{
  Assert(codim <= dim, ExcImpossibleInDim(dim));
 
  // vertex/line/face domination
  // (if fe_other is derived from FE_SimplexDGP)
  // ------------------------------------
  if (codim > 0)
    if (dynamic_cast<const FE_SimplexDGP<dim, spacedim> *>(&fe_other) !=
        nullptr)
      // there are no requirements between continuous and discontinuous
      // elements
      return FiniteElementDomination::no_requirements;
 
  // vertex/line/face domination
  // (if fe_other is not derived from FE_SimplexDGP)
  // & cell domination
  // ----------------------------------------
  if (const FE_PyramidP<dim, spacedim> *fe_pp_other =
        dynamic_cast<const FE_PyramidP<dim, spacedim> *>(&fe_other))
    {
      if (this->degree < fe_pp_other->degree)
        return FiniteElementDomination::this_element_dominates;
      else if (this->degree == fe_pp_other->degree)
        return FiniteElementDomination::either_element_can_dominate;
      else
        return FiniteElementDomination::other_element_dominates;
    }
  else if (const FE_SimplexP<dim, spacedim> *fe_p_other =
             dynamic_cast<const FE_SimplexP<dim, spacedim> *>(&fe_other))
    {
      if (this->degree < fe_p_other->degree)
        return FiniteElementDomination::this_element_dominates;
      else if (this->degree == fe_p_other->degree)
        return FiniteElementDomination::either_element_can_dominate;
      else
        return FiniteElementDomination::other_element_dominates;
    }
  else if (const FE_Q<dim, spacedim> *fe_q_other =
             dynamic_cast<const FE_Q<dim, spacedim> *>(&fe_other))
    {
      if (this->degree < fe_q_other->degree)
        return FiniteElementDomination::this_element_dominates;
      else if (this->degree == fe_q_other->degree)
        return FiniteElementDomination::either_element_can_dominate;
      else
        return FiniteElementDomination::other_element_dominates;
    }
  else if (const FE_WedgeP<dim, spacedim> *fe_pp_other =
             dynamic_cast<const FE_WedgeP<dim, spacedim> *>(&fe_other))
    {
      if (this->degree < fe_pp_other->degree)
        return FiniteElementDomination::this_element_dominates;
      else if (this->degree == fe_pp_other->degree)
        return FiniteElementDomination::either_element_can_dominate;
      else
        return FiniteElementDomination::other_element_dominates;
    }
  else if (const FE_Nothing<dim, spacedim> *fe_nothing =
             dynamic_cast<const FE_Nothing<dim, spacedim> *>(&fe_other))
    {
      if (fe_nothing->is_dominating())
        return FiniteElementDomination::other_element_dominates;
      else
        // the FE_Nothing has no degrees of freedom and it is typically used
        // in a context where we don't require any continuity along the
        // interface
        return FiniteElementDomination::no_requirements;
    }
 
  DEAL_II_NOT_IMPLEMENTED();
  return FiniteElementDomination::neither_element_dominates;
}
 
 
 
template <int dim, int spacedim>
std::vector<std::pair<unsigned int, unsigned int>>
FE_PyramidP<dim, spacedim>::hp_vertex_dof_identities(
  const FiniteElement<dim, spacedim> &fe_other) const
{
  (void)fe_other;
 
  Assert((dynamic_cast<const FE_SimplexP<dim, spacedim> *>(&fe_other)) ||
           (dynamic_cast<const FE_Q<dim, spacedim> *>(&fe_other)) ||
           (dynamic_cast<const FE_PyramidP<dim, spacedim> *>(&fe_other)) ||
           (dynamic_cast<const FE_WedgeP<dim, spacedim> *>(&fe_other)),
         ExcNotImplemented());
 
  return {{0, 0}};
}
 
 
 
template <int dim, int spacedim>
std::vector<std::pair<unsigned int, unsigned int>>
FE_PyramidP<dim, spacedim>::hp_line_dof_identities(
  const FiniteElement<dim, spacedim> &fe_other) const
{
  Assert((dynamic_cast<const FE_SimplexP<dim, spacedim> *>(&fe_other)) ||
           (dynamic_cast<const FE_Q<dim, spacedim> *>(&fe_other)) ||
           (dynamic_cast<const FE_PyramidP<dim, spacedim> *>(&fe_other)) ||
           (dynamic_cast<const FE_WedgeP<dim, spacedim> *>(&fe_other)),
         ExcNotImplemented());
 
  std::vector<std::pair<unsigned int, unsigned int>> identities;
  // check if the support points are the same location on the line
  // the pyramid base is defined by the vertices [-1,-1,0], [1,-1,0],
  // [-1,1,0], [1,1,0], to avoid rescaling use the support points on the faces
  const auto &face_support_points = this->get_unit_face_support_points(0);
  const auto &face_support_points_other =
    fe_other.get_unit_face_support_points(0);
 
  // now just compare the DoFs on the line going from [0,0] to [1,0]
  // for a triangular face that is the first line
  // for a quad face that is the third line
  // adjust the offsets accordingly
  // face number 0 of the pyramid is a quad
  const unsigned int offset =
    this->reference_cell().face_reference_cell(0).n_vertices() +
    2 * this->n_dofs_per_line();
 
  const unsigned int offset_other =
    fe_other.reference_cell().face_reference_cell(0).is_hyper_cube() ?
      fe_other.reference_cell().face_reference_cell(0).n_vertices() +
        2 * fe_other.n_dofs_per_line() :
      fe_other.reference_cell().face_reference_cell(0).n_vertices();
 
  // now get the identities
  for (unsigned int i = 0; i < this->n_dofs_per_line(); ++i)
    for (unsigned int j = 0; j < fe_other.n_dofs_per_line(); ++j)
      if (face_support_points[i + offset].distance(
            face_support_points_other[j + offset_other]) < 1e-14)
        identities.emplace_back(i, j);
 
  return identities;
}
 
 
 
template <int dim, int spacedim>
std::vector<std::pair<unsigned int, unsigned int>>
FE_PyramidP<dim, spacedim>::hp_quad_dof_identities(
  const FiniteElement<dim, spacedim> &fe_other,
  const unsigned int                  face_no) const
{
  AssertIndexRange(face_no, 5);
  std::vector<std::pair<unsigned int, unsigned int>> identities;
 
  unsigned int face_no_neighbor;
 
  if (face_no == 0)
    {
      // on a quad, neighbor can be a hex, a pyramid or a wedge
      Assert((dynamic_cast<const FE_Q<dim, spacedim> *>(&fe_other)) ||
               (dynamic_cast<const FE_PyramidP<dim, spacedim> *>(&fe_other)) ||
               (dynamic_cast<const FE_WedgeP<dim, spacedim> *>(&fe_other)),
             ExcNotImplemented());
      // for the wedge the first quad face is face no. 2
      if (dynamic_cast<const FE_WedgeP<dim, spacedim> *>(&fe_other))
        face_no_neighbor = 2;
      else
        face_no_neighbor = 0;
    }
  else
    {
      Assert((dynamic_cast<const FE_SimplexP<dim, spacedim> *>(&fe_other)) ||
               (dynamic_cast<const FE_PyramidP<dim, spacedim> *>(&fe_other)) ||
               (dynamic_cast<const FE_WedgeP<dim, spacedim> *>(&fe_other)),
             ExcNotImplemented());
      // on tri, neighbor can be a tet, a pyramid or a wedge
      if (dynamic_cast<const FE_PyramidP<dim, spacedim> *>(&fe_other))
        face_no_neighbor = 1;
      else
        face_no_neighbor = 0;
    }
 
  // compare the face support points
  const auto &face_support_points = this->get_unit_face_support_points(face_no);
  const auto &face_support_points_other =
    fe_other.get_unit_face_support_points(face_no_neighbor);
 
  // get the offsets to only compare the DoFs within the face as the vertices
  // and lines were done before
  const auto face_reference_cell =
    this->reference_cell().face_reference_cell(face_no);
 
  Assert(face_reference_cell ==
           fe_other.reference_cell().face_reference_cell(face_no_neighbor),
         ExcInternalError());
 
  const unsigned int offset =
    face_reference_cell.n_vertices() +
    face_reference_cell.n_lines() * this->n_dofs_per_line();
 
  const unsigned int offset_other =
    face_reference_cell.n_vertices() +
    face_reference_cell.n_lines() * fe_other.n_dofs_per_line();
 
  // do the comparison
  for (unsigned int i = 0; i < this->n_dofs_per_quad(face_no); ++i)
    for (unsigned int j = 0; j < fe_other.n_dofs_per_quad(face_no_neighbor);
         ++j)
      if (face_support_points[i + offset].distance(
            face_support_points_other[j + offset_other]) < 1e-14)
        identities.emplace_back(i, j);
 
  return identities;
}
 
 
 
template <int dim, int spacedim>
FE_PyramidDGP<dim, spacedim>::FE_PyramidDGP(const unsigned int degree)
  : FE_PyramidPoly<dim, spacedim>(degree,
                                  get_dpo<dim>(degree,
                                               FiniteElementData<dim>::L2),
                                  get_support_points<dim>(degree),
                                  true,
                                  FiniteElementData<dim>::L2)
{}
 
 
 
template <int dim, int spacedim>
std::unique_ptr<FiniteElement<dim, spacedim>>
FE_PyramidDGP<dim, spacedim>::clone() const
{
  return std::make_unique<FE_PyramidDGP<dim, spacedim>>(*this);
}
 
 
 
template <int dim, int spacedim>
std::string
FE_PyramidDGP<dim, spacedim>::get_name() const
{
  std::ostringstream namebuf;
  namebuf << "FE_PyramidDGP<" << Utilities::dim_string(dim, spacedim) << ">("
          << this->degree << ")";
 
  return namebuf.str();
}
 
// explicit instantiations
#include "fe/fe_pyramid_p.inst"
 
DEAL_II_NAMESPACE_CLOSE