agglomeration_handler.cc

/* -----------------------------------------------------------------------------
 *
 * SPDX-License-Identifier: LGPL-2.1-or-later
 * Copyright (C) 2025 by Marco Feder, Pasquale Claudio Africa, Xinping Gui,
 * Andrea Cangiani
 *
 * This file is part of the deal.II code gallery.
 *
 * -----------------------------------------------------------------------------
 */

#include <deal.II/base/quadrature_lib.h>
#include <deal.II/lac/sparsity_tools.h>

#include <agglomeration_handler.h>

template <int dim, int spacedim>
AgglomerationHandler<dim, spacedim>::AgglomerationHandler(
  const GridTools::Cache<dim, spacedim> &cache_tria)
  : cached_tria(std::make_unique<GridTools::Cache<dim, spacedim>>(
      cache_tria.get_triangulation(),
      cache_tria.get_mapping()))
  , communicator(cache_tria.get_triangulation().get_communicator())
{
  Assert(dim == spacedim, ExcNotImplemented("Not available with codim > 0"));
  Assert(dim == 2 || dim == 3, ExcImpossibleInDim(1));
  Assert((dynamic_cast<const parallel::shared::Triangulation<dim, spacedim> *>(
            &cached_tria->get_triangulation()) == nullptr),
         ExcNotImplemented());
  Assert(cached_tria->get_triangulation().n_active_cells() > 0,
         ExcMessage(
           "The triangulation must not be empty upon calling this function."));

  n_agglomerations = 0;
  hybrid_mesh      = false;
  initialize_agglomeration_data(cached_tria);
}



template <int dim, int spacedim>
typename AgglomerationHandler<dim, spacedim>::agglomeration_iterator
AgglomerationHandler<dim, spacedim>::define_agglomerate(
  const AgglomerationContainer &cells)
{
  Assert(cells.size() > 0, ExcMessage("No cells to be agglomerated."));

  if (cells.size() == 1)
    hybrid_mesh = true; // mesh is made also by classical cells

  // First index drives the selection of the master cell. After that, store the
  // master cell.
  const types::global_cell_index global_master_idx =
    cells[0]->global_active_cell_index();
  const types::global_cell_index master_idx = cells[0]->active_cell_index();
  master_cells_container.push_back(cells[0]);
  master_slave_relationships[global_master_idx] = -1;

  const typename DoFHandler<dim>::active_cell_iterator cell_dh =
    cells[0]->as_dof_handler_iterator(agglo_dh);
  cell_dh->set_active_fe_index(CellAgglomerationType::master);

  // Store slave cells and save the relationship with the parent
  std::vector<typename Triangulation<dim, spacedim>::active_cell_iterator>
    slaves;
  slaves.reserve(cells.size() - 1);
  // exclude first cell since it's the master cell
  for (auto it = ++cells.begin(); it != cells.end(); ++it)
    {
      slaves.push_back(*it);
      master_slave_relationships[(*it)->global_active_cell_index()] =
        global_master_idx; // mark each slave
      master_slave_relationships_iterators[(*it)->active_cell_index()] =
        cells[0];

      const typename DoFHandler<dim>::active_cell_iterator cell =
        (*it)->as_dof_handler_iterator(agglo_dh);
      cell->set_active_fe_index(CellAgglomerationType::slave); // slave cell

      // If we have a p::d::T, check that all cells are in the same subdomain.
      // If serial, just check that the subdomain_id is invalid.
      Assert(((*it)->subdomain_id() == tria->locally_owned_subdomain() ||
              tria->locally_owned_subdomain() == numbers::invalid_subdomain_id),
             ExcInternalError());
    }

  master_slave_relationships_iterators[master_idx] =
    cells[0]; // set iterator to master cell

  // Store the slaves of each master
  master2slaves[master_idx] = slaves;
  // Save to which polygon this agglomerate correspond
  master2polygon[master_idx] = n_agglomerations;

  ++n_agglomerations; // an agglomeration has been performed, record it

  create_bounding_box(cells); // fill the vector of bboxes

  // Finally, return a polygonal iterator to the polytope just constructed.
  return {cells[0], this};
}

template <int dim, int spacedim>
typename AgglomerationHandler<dim, spacedim>::agglomeration_iterator
AgglomerationHandler<dim, spacedim>::define_agglomerate(
  const AgglomerationContainer &cells,
  const unsigned int            fecollection_size)
{
  Assert(cells.size() > 0, ExcMessage("No cells to be agglomerated."));

  if (cells.size() == 1)
    hybrid_mesh = true; // mesh is made also by classical cells

  // First index drives the selection of the master cell. After that, store the
  // master cell.
  const types::global_cell_index global_master_idx =
    cells[0]->global_active_cell_index();
  const types::global_cell_index master_idx = cells[0]->active_cell_index();
  master_cells_container.push_back(cells[0]);
  master_slave_relationships[global_master_idx] = -1;

  const typename DoFHandler<dim>::active_cell_iterator cell_dh =
    cells[0]->as_dof_handler_iterator(agglo_dh);
  cell_dh->set_active_fe_index(CellAgglomerationType::master);

  // Store slave cells and save the relationship with the parent
  std::vector<typename Triangulation<dim, spacedim>::active_cell_iterator>
    slaves;
  slaves.reserve(cells.size() - 1);
  // exclude first cell since it's the master cell
  for (auto it = ++cells.begin(); it != cells.end(); ++it)
    {
      slaves.push_back(*it);
      master_slave_relationships[(*it)->global_active_cell_index()] =
        global_master_idx; // mark each slave
      master_slave_relationships_iterators[(*it)->active_cell_index()] =
        cells[0];

      const typename DoFHandler<dim>::active_cell_iterator cell =
        (*it)->as_dof_handler_iterator(agglo_dh);
      cell->set_active_fe_index(
        fecollection_size); // slave cell (the last index)

      // If we have a p::d::T, check that all cells are in the same subdomain.
      // If serial, just check that the subdomain_id is invalid.
      Assert(((*it)->subdomain_id() == tria->locally_owned_subdomain() ||
              tria->locally_owned_subdomain() == numbers::invalid_subdomain_id),
             ExcInternalError());
    }

  master_slave_relationships_iterators[master_idx] =
    cells[0]; // set iterator to master cell

  // Store the slaves of each master
  master2slaves[master_idx] = slaves;
  // Save to which polygon this agglomerate correspond
  master2polygon[master_idx] = n_agglomerations;

  ++n_agglomerations; // an agglomeration has been performed, record it

  create_bounding_box(cells); // fill the vector of bboxes

  // Finally, return a polygonal iterator to the polytope just constructed.
  return {cells[0], this};
}


template <int dim, int spacedim>
void
AgglomerationHandler<dim, spacedim>::initialize_fe_values(
  const Quadrature<dim>     &cell_quadrature,
  const UpdateFlags         &flags,
  const Quadrature<dim - 1> &face_quadrature,
  const UpdateFlags         &face_flags)
{
  agglomeration_quad       = cell_quadrature;
  agglomeration_flags      = flags;
  agglomeration_face_quad  = face_quadrature;
  agglomeration_face_flags = face_flags | internal_agglomeration_face_flags;


  no_values =
    std::make_unique<FEValues<dim>>(*mapping,
                                    dummy_fe,
                                    agglomeration_quad,
                                    update_quadrature_points |
                                      update_JxW_values); // only for quadrature
  no_face_values = std::make_unique<FEFaceValues<dim>>(
    *mapping,
    dummy_fe,
    agglomeration_face_quad,
    update_quadrature_points | update_JxW_values |
      update_normal_vectors); // only for quadrature
}

template <int dim, int spacedim>
void
AgglomerationHandler<dim, spacedim>::initialize_fe_values(
  const hp::QCollection<dim>     &cell_qcollection,
  const UpdateFlags              &flags,
  const hp::QCollection<dim - 1> &face_qcollection,
  const UpdateFlags              &face_flags)
{
  agglomeration_quad_collection      = cell_qcollection;
  agglomeration_flags                = flags;
  agglomeration_face_quad_collection = face_qcollection;
  agglomeration_face_flags = face_flags | internal_agglomeration_face_flags;

  mapping_collection  = hp::MappingCollection<dim>(*mapping);
  dummy_fe_collection = hp::FECollection<dim, spacedim>(dummy_fe);
  hp_no_values        = std::make_unique<hp::FEValues<dim>>(
    mapping_collection,
    dummy_fe_collection,
    agglomeration_quad_collection,
    update_quadrature_points | update_JxW_values); // only for quadrature

  hp_no_face_values = std::make_unique<hp::FEFaceValues<dim>>(
    mapping_collection,
    dummy_fe_collection,
    agglomeration_face_quad_collection,
    update_quadrature_points | update_JxW_values |
      update_normal_vectors); // only for quadrature
}



template <int dim, int spacedim>
unsigned int
AgglomerationHandler<dim, spacedim>::n_agglomerated_faces_per_cell(
  const typename Triangulation<dim, spacedim>::active_cell_iterator &cell) const
{
  unsigned int n_neighbors = 0;
  for (const auto &f : cell->face_indices())
    {
      const auto &neighboring_cell = cell->neighbor(f);
      if ((cell->face(f)->at_boundary()) ||
          (neighboring_cell->is_active() &&
           !are_cells_agglomerated(cell, neighboring_cell)))
        {
          ++n_neighbors;
        }
    }
  return n_neighbors;
}



template <int dim, int spacedim>
void
AgglomerationHandler<dim, spacedim>::initialize_agglomeration_data(
  const std::unique_ptr<GridTools::Cache<dim, spacedim>> &cache_tria)
{
  tria    = &(cache_tria->get_triangulation());
  mapping = &(cache_tria->get_mapping());

  agglo_dh.reinit(*tria);

  if (const auto parallel_tria = dynamic_cast<
        const dealii::parallel::TriangulationBase<dim, spacedim> *>(&*tria))
    {
      const std::weak_ptr<const Utilities::MPI::Partitioner> cells_partitioner =
        parallel_tria->global_active_cell_index_partitioner();
      master_slave_relationships.reinit(
        cells_partitioner.lock()->locally_owned_range(), communicator);
    }
  else
    {
      master_slave_relationships.reinit(tria->n_active_cells(), MPI_COMM_SELF);
    }

  polytope_cache.clear();
  bboxes.clear();

  // First, update the pointer
  cached_tria = std::make_unique<GridTools::Cache<dim, spacedim>>(
    cache_tria->get_triangulation(), cache_tria->get_mapping());

  connect_to_tria_signals();
  n_agglomerations = 0;
}



template <int dim, int spacedim>
void
AgglomerationHandler<dim, spacedim>::distribute_agglomerated_dofs(
  const FiniteElement<dim> &fe_space)
{
  if (dynamic_cast<const FE_DGQ<dim> *>(&fe_space))
    fe = std::make_unique<FE_DGQ<dim>>(fe_space.degree);
  else if (dynamic_cast<const FE_SimplexDGP<dim> *>(&fe_space))
    fe = std::make_unique<FE_SimplexDGP<dim>>(fe_space.degree);
  else
    AssertThrow(
      false,
      ExcNotImplemented(
        "Currently, this interface supports only DGQ and DGP bases."));

  box_mapping = std::make_unique<MappingBox<dim>>(
    bboxes,
    master2polygon); // construct bounding box mapping

  if (hybrid_mesh)
    {
      // the mesh is composed by standard and agglomerate cells. initialize
      // classes needed for standard cells in order to treat that finite
      // element space as defined on a standard shape and not on the
      // BoundingBox.
      standard_scratch =
        std::make_unique<ScratchData>(*mapping,
                                      *fe,
                                      QGauss<dim>(2 * fe_space.degree + 2),
                                      internal_agglomeration_flags);
    }


  fe_collection.push_back(*fe); // master
  fe_collection.push_back(
    FE_Nothing<dim, spacedim>(fe->reference_cell())); // slave

  initialize_hp_structure();

  // in case the tria is distributed, communicate ghost information with
  // neighboring ranks
  const bool needs_ghost_info =
    dynamic_cast<const parallel::TriangulationBase<dim, spacedim> *>(&*tria) !=
    nullptr;
  if (needs_ghost_info)
    setup_ghost_polytopes();

  setup_connectivity_of_agglomeration();

  if (needs_ghost_info)
    exchange_interface_values();
}

template <int dim, int spacedim>
void
AgglomerationHandler<dim, spacedim>::distribute_agglomerated_dofs(
  const hp::FECollection<dim, spacedim> &fe_collection_in)
{
  is_hp_collection = true;

  hp_fe_collection = std::make_unique<hp::FECollection<dim, spacedim>>(
    fe_collection_in); // copy the input collection

  box_mapping = std::make_unique<MappingBox<dim>>(
    bboxes,
    master2polygon); // construct bounding box mapping


  if (hybrid_mesh)
    {
      AssertThrow(false,
                  ExcNotImplemented(
                    "Hybrid mesh is not implemented for hp::FECollection."));
    }

  for (unsigned int i = 0; i < fe_collection_in.size(); ++i)
    {
      if (dynamic_cast<const FESystem<dim> *>(&fe_collection_in[i]))
        {
          // System case
          for (unsigned int b = 0; b < fe_collection_in[i].n_base_elements();
               ++b)
            {
              if (!(dynamic_cast<const FE_DGQ<dim> *>(
                      &fe_collection_in[i].base_element(b)) ||
                    dynamic_cast<const FE_SimplexDGP<dim> *>(
                      &fe_collection_in[i].base_element(b)) ||
                    dynamic_cast<const FE_Nothing<dim> *>(
                      &fe_collection_in[i].base_element(b))))
                AssertThrow(
                  false,
                  ExcNotImplemented(
                    "Currently, this interface supports only DGQ and DGP bases."));
            }
        }
      else
        {
          // Scalar case
          if (!(dynamic_cast<const FE_DGQ<dim> *>(&fe_collection_in[i]) ||
                dynamic_cast<const FE_SimplexDGP<dim> *>(&fe_collection_in[i])))
            AssertThrow(
              false,
              ExcNotImplemented(
                "Currently, this interface supports only DGQ and DGP bases."));
        }
      fe_collection.push_back(fe_collection_in[i]);
    }

  Assert(fe_collection[0].n_components() >= 1,
         ExcMessage("Invalid FE: must have at least one component."));
  if (fe_collection[0].n_components() == 1)
    {
      fe_collection.push_back(FE_Nothing<dim, spacedim>());
    }
  else if (fe_collection[0].n_components() > 1)
    {
      std::vector<const FiniteElement<dim, spacedim> *> base_elements;
      std::vector<unsigned int>                         multiplicities;
      for (unsigned int b = 0; b < fe_collection[0].n_base_elements(); ++b)
        {
          base_elements.push_back(new FE_Nothing<dim, spacedim>());
          multiplicities.push_back(fe_collection[0].element_multiplicity(b));
        }
      FESystem<dim, spacedim> fe_system_nothing(base_elements, multiplicities);
      for (const auto *ptr : base_elements)
        delete ptr;
      fe_collection.push_back(fe_system_nothing);
    }

  initialize_hp_structure();

  // in case the tria is distributed, communicate ghost information with
  // neighboring ranks
  const bool needs_ghost_info =
    dynamic_cast<const parallel::TriangulationBase<dim, spacedim> *>(&*tria) !=
    nullptr;
  if (needs_ghost_info)
    setup_ghost_polytopes();

  setup_connectivity_of_agglomeration();

  if (needs_ghost_info)
    exchange_interface_values();
}

template <int dim, int spacedim>
void
AgglomerationHandler<dim, spacedim>::create_bounding_box(
  const AgglomerationContainer &polytope)
{
  Assert(n_agglomerations > 0,
         ExcMessage("No agglomeration has been performed."));
  Assert(dim > 1, ExcNotImplemented());

  std::vector<Point<spacedim>> pts; // store all the vertices
  for (const auto &cell : polytope)
    for (const auto i : cell->vertex_indices())
      pts.push_back(cell->vertex(i));

  bboxes.emplace_back(pts);
}



template <int dim, int spacedim>
void
AgglomerationHandler<dim, spacedim>::setup_connectivity_of_agglomeration()
{
  Assert(master_cells_container.size() > 0,
         ExcMessage("No agglomeration has been performed."));
  Assert(
    agglo_dh.n_dofs() > 0,
    ExcMessage(
      "The DoFHandler associated to the agglomeration has not been initialized."
      "It's likely that you forgot to distribute the DoFs. You may want"
      "to check if a call to `initialize_hp_structure()` has been done."));

  number_of_agglomerated_faces.resize(master2polygon.size(), 0);
  for (const auto &cell : master_cells_container)
    {
      internal::AgglomerationHandlerImplementation<dim, spacedim>::
        setup_master_neighbor_connectivity(cell, *this);
    }

  if (Utilities::MPI::job_supports_mpi())
    {
      // communicate the number of faces
      recv_n_faces = Utilities::MPI::some_to_some(communicator, local_n_faces);

      // send information about boundaries and neighboring polytopes id
      recv_bdary_info =
        Utilities::MPI::some_to_some(communicator, local_bdary_info);

      recv_ghosted_master_id =
        Utilities::MPI::some_to_some(communicator, local_ghosted_master_id);
    }
}



template <int dim, int spacedim>
void
AgglomerationHandler<dim, spacedim>::exchange_interface_values()
{
  const unsigned int dofs_per_cell = fe->dofs_per_cell;
  for (const auto &polytope : polytope_iterators())
    {
      if (polytope->is_locally_owned())
        {
          const unsigned int n_faces = polytope->n_faces();
          for (unsigned int f = 0; f < n_faces; ++f)
            {
              if (!polytope->at_boundary(f))
                {
                  const auto &neigh_polytope = polytope->neighbor(f);
                  if (!neigh_polytope->is_locally_owned())
                    {
                      // Neighboring polytope is ghosted.

                      // Compute shape functions at the interface
                      const auto &current_fe = reinit(polytope, f);

                      std::vector<Point<spacedim>> qpoints_to_send =
                        current_fe.get_quadrature_points();

                      const std::vector<double> &jxws_to_send =
                        current_fe.get_JxW_values();

                      const std::vector<Tensor<1, spacedim>> &normals_to_send =
                        current_fe.get_normal_vectors();


                      const types::subdomain_id neigh_rank =
                        neigh_polytope->subdomain_id();

                      std::pair<CellId, unsigned int> cell_and_face{
                        polytope->id(), f};
                      // Prepare data to send
                      local_qpoints[neigh_rank].emplace(cell_and_face,
                                                        qpoints_to_send);

                      local_jxws[neigh_rank].emplace(cell_and_face,
                                                     jxws_to_send);

                      local_normals[neigh_rank].emplace(cell_and_face,
                                                        normals_to_send);


                      const unsigned int n_qpoints = qpoints_to_send.size();

                      // TODO: check `agglomeration_flags` before computing
                      // values and gradients.
                      std::vector<std::vector<double>> values_per_qpoints(
                        dofs_per_cell);

                      std::vector<std::vector<Tensor<1, spacedim>>>
                        gradients_per_qpoints(dofs_per_cell);

                      for (unsigned int i = 0; i < dofs_per_cell; ++i)
                        {
                          values_per_qpoints[i].resize(n_qpoints);
                          gradients_per_qpoints[i].resize(n_qpoints);
                          for (unsigned int q = 0; q < n_qpoints; ++q)
                            {
                              values_per_qpoints[i][q] =
                                current_fe.shape_value(i, q);
                              gradients_per_qpoints[i][q] =
                                current_fe.shape_grad(i, q);
                            }
                        }

                      local_values[neigh_rank].emplace(cell_and_face,
                                                       values_per_qpoints);
                      local_gradients[neigh_rank].emplace(
                        cell_and_face, gradients_per_qpoints);
                    }
                }
            }
        }
    }

  // Finally, exchange with neighboring ranks
  recv_qpoints   = Utilities::MPI::some_to_some(communicator, local_qpoints);
  recv_jxws      = Utilities::MPI::some_to_some(communicator, local_jxws);
  recv_normals   = Utilities::MPI::some_to_some(communicator, local_normals);
  recv_values    = Utilities::MPI::some_to_some(communicator, local_values);
  recv_gradients = Utilities::MPI::some_to_some(communicator, local_gradients);
}



template <int dim, int spacedim>
Quadrature<dim>
AgglomerationHandler<dim, spacedim>::agglomerated_quadrature(
  const typename AgglomerationHandler<dim, spacedim>::AgglomerationContainer
    &cells,
  const typename Triangulation<dim, spacedim>::active_cell_iterator
    &master_cell) const
{
  Assert(is_master_cell(master_cell),
         ExcMessage("This must be a master cell."));

  std::vector<Point<dim>> vec_pts;
  std::vector<double>     vec_JxWs;

  if (!is_hp_collection)
    {
      // Original version: handle case without hp::FECollection
      for (const auto &dummy_cell : cells)
        {
          no_values->reinit(dummy_cell);
          auto q_points    = no_values->get_quadrature_points(); // real qpoints
          const auto &JxWs = no_values->get_JxW_values();

          std::transform(q_points.begin(),
                         q_points.end(),
                         std::back_inserter(vec_pts),
                         [&](const Point<spacedim> &p) { return p; });
          std::transform(JxWs.begin(),
                         JxWs.end(),
                         std::back_inserter(vec_JxWs),
                         [&](const double w) { return w; });
        }
    }
  else
    {
      // Handle the hp::FECollection case
      const auto &master_cell_as_dh_iterator =
        master_cell->as_dof_handler_iterator(agglo_dh);
      for (const auto &dummy_cell : cells)
        {
          // The following verbose call is necessary to handle cases where
          // different slave cells on different polytopes use different
          // quadrature rules. If the hp::QCollection contains multiple
          // elements, calling hp_no_values->reinit(dummy_cell) won't work
          // because it cannot infer the correct quadrature rule. By explicitly
          // passing the active FE index as q_index, and setting mapping_index
          // and fe_index to 0, we ensure that the dummy cell uses the same
          // quadrature rule as its corresponding master cell. This assumes a
          // one-to-one correspondence between hp::QCollection and
          // hp::FECollection, which is the convention in deal.II. However, this
          // implementation does not support cases where hp::QCollection and
          // hp::FECollection have different sizes.
          // TODO: Refactor the architecture to better handle numerical
          // integration for hp::QCollection.
          hp_no_values->reinit(dummy_cell,
                               master_cell_as_dh_iterator->active_fe_index(),
                               0,
                               0);
          auto q_points = hp_no_values->get_present_fe_values()
                            .get_quadrature_points(); // real qpoints
          const auto &JxWs =
            hp_no_values->get_present_fe_values().get_JxW_values();

          std::transform(q_points.begin(),
                         q_points.end(),
                         std::back_inserter(vec_pts),
                         [&](const Point<spacedim> &p) { return p; });
          std::transform(JxWs.begin(),
                         JxWs.end(),
                         std::back_inserter(vec_JxWs),
                         [&](const double w) { return w; });
        }
    }

  // Map back each point in real space by using the map associated to the
  // bounding box.
  std::vector<Point<dim>> unit_points(vec_pts.size());
  const auto             &bbox =
    bboxes[master2polygon.at(master_cell->active_cell_index())];
  unit_points.reserve(vec_pts.size());

  for (unsigned int i = 0; i < vec_pts.size(); i++)
    unit_points[i] = bbox.real_to_unit(vec_pts[i]);

  return Quadrature<dim>(unit_points, vec_JxWs);
}



template <int dim, int spacedim>
void
AgglomerationHandler<dim, spacedim>::initialize_hp_structure()
{
  Assert(agglo_dh.get_triangulation().n_cells() > 0,
         ExcMessage(
           "Triangulation must not be empty upon calling this function."));
  Assert(n_agglomerations > 0,
         ExcMessage("No agglomeration has been performed."));

  agglo_dh.distribute_dofs(fe_collection);
  // euler_mapping = std::make_unique<
  //   MappingFEField<dim, spacedim,
  //   LinearAlgebra::distributed::Vector<double>>>( euler_dh, euler_vector);
}



template <int dim, int spacedim>
const FEValues<dim, spacedim> &
AgglomerationHandler<dim, spacedim>::reinit(
  const AgglomerationIterator<dim, spacedim> &polytope) const
{
  // Assert(euler_mapping,
  //        ExcMessage("The mapping describing the physical element stemming
  //        from "
  //                   "agglomeration has not been set up."));

  const auto &deal_cell = polytope->as_dof_handler_iterator(agglo_dh);

  // First check if the polytope is made just by a single cell. If so, use
  // classical FEValues
  // if (polytope->n_background_cells() == 1)
  //   return standard_scratch->reinit(deal_cell);

  const auto &agglo_cells = polytope->get_agglomerate();

  Quadrature<dim> agglo_quad = agglomerated_quadrature(agglo_cells, deal_cell);

  if (!is_hp_collection)
    {
      // Original version: handle case without hp::FECollection
      agglomerated_scratch = std::make_unique<ScratchData>(*box_mapping,
                                                           fe_collection[0],
                                                           agglo_quad,
                                                           agglomeration_flags);
    }
  else
    {
      // Handle the hp::FECollection case
      agglomerated_scratch = std::make_unique<ScratchData>(*box_mapping,
                                                           polytope->get_fe(),
                                                           agglo_quad,
                                                           agglomeration_flags);
    }
  return agglomerated_scratch->reinit(deal_cell);
}



template <int dim, int spacedim>
const FEValuesBase<dim, spacedim> &
AgglomerationHandler<dim, spacedim>::reinit_master(
  const typename DoFHandler<dim, spacedim>::active_cell_iterator &cell,
  const unsigned int                                              face_index,
  std::unique_ptr<NonMatching::FEImmersedSurfaceValues<spacedim>>
    &agglo_isv_ptr) const
{
  return internal::AgglomerationHandlerImplementation<dim, spacedim>::
    reinit_master(cell, face_index, agglo_isv_ptr, *this);
}



template <int dim, int spacedim>
const FEValuesBase<dim, spacedim> &
AgglomerationHandler<dim, spacedim>::reinit(
  const AgglomerationIterator<dim, spacedim> &polytope,
  const unsigned int                          face_index) const
{
  // Assert(euler_mapping,
  //        ExcMessage("The mapping describing the physical element stemming
  //        from "
  //                   "agglomeration has not been set up."));

  const auto &deal_cell = polytope->as_dof_handler_iterator(agglo_dh);
  Assert(is_master_cell(deal_cell), ExcMessage("This should be true."));

  return internal::AgglomerationHandlerImplementation<dim, spacedim>::
    reinit_master(deal_cell, face_index, agglomerated_isv_bdary, *this);
}



template <int dim, int spacedim>
std::pair<const FEValuesBase<dim, spacedim> &,
          const FEValuesBase<dim, spacedim> &>
AgglomerationHandler<dim, spacedim>::reinit_interface(
  const AgglomerationIterator<dim, spacedim> &polytope_in,
  const AgglomerationIterator<dim, spacedim> &neigh_polytope,
  const unsigned int                          local_in,
  const unsigned int                          local_neigh) const
{
  // If current and neighboring polytopes are both locally owned, then compute
  // the jump in the classical way without needing information about ghosted
  // entities.
  if (polytope_in->is_locally_owned() && neigh_polytope->is_locally_owned())
    {
      const auto &cell_in = polytope_in->as_dof_handler_iterator(agglo_dh);
      const auto &neigh_cell =
        neigh_polytope->as_dof_handler_iterator(agglo_dh);

      const auto &fe_in =
        internal::AgglomerationHandlerImplementation<dim, spacedim>::
          reinit_master(cell_in, local_in, agglomerated_isv, *this);
      const auto &fe_out =
        internal::AgglomerationHandlerImplementation<dim, spacedim>::
          reinit_master(neigh_cell, local_neigh, agglomerated_isv_neigh, *this);
      std::pair<const FEValuesBase<dim, spacedim> &,
                const FEValuesBase<dim, spacedim> &>
        my_p(fe_in, fe_out);

      return my_p;
    }
  else
    {
      Assert((polytope_in->is_locally_owned() &&
              !neigh_polytope->is_locally_owned()),
             ExcInternalError());

      const auto &cell = polytope_in->as_dof_handler_iterator(agglo_dh);
      const auto &bbox = bboxes[master2polygon.at(cell->active_cell_index())];
      // const double bbox_measure = bbox.volume();

      const unsigned int neigh_rank = neigh_polytope->subdomain_id();
      const CellId      &neigh_id   = neigh_polytope->id();

      // Retrieve qpoints,JxWs, normals sent previously from the neighboring
      // rank.
      std::vector<Point<spacedim>> &real_qpoints =
        recv_qpoints.at(neigh_rank).at({neigh_id, local_neigh});

      const auto &JxWs = recv_jxws.at(neigh_rank).at({neigh_id, local_neigh});

      std::vector<Tensor<1, spacedim>> &normals =
        recv_normals.at(neigh_rank).at({neigh_id, local_neigh});

      // Apply the necessary scalings due to the bbox.
      std::vector<Point<spacedim>> final_unit_q_points;
      std::transform(real_qpoints.begin(),
                     real_qpoints.end(),
                     std::back_inserter(final_unit_q_points),
                     [&](const Point<spacedim> &p) {
                       return bbox.real_to_unit(p);
                     });

      // std::vector<double>         scale_factors(final_unit_q_points.size());
      // std::vector<double>         scaled_weights(final_unit_q_points.size());
      // std::vector<Tensor<1, dim>> scaled_normals(final_unit_q_points.size());

      // Since we received normal vectors from a neighbor, we have to swap
      // the
      // // sign of the vector in order to have outward normals.
      // for (unsigned int q = 0; q < final_unit_q_points.size(); ++q)
      //   {
      //     for (unsigned int direction = 0; direction < spacedim; ++direction)
      //       scaled_normals[q][direction] =
      //         normals[q][direction] * (bbox.side_length(direction));

      //     scaled_normals[q] *= -1;

      //     scaled_weights[q] =
      //       (JxWs[q] * scaled_normals[q].norm()) / bbox_measure;
      //     scaled_normals[q] /= scaled_normals[q].norm();
      //   }
      for (unsigned int q = 0; q < final_unit_q_points.size(); ++q)
        normals[q] *= -1;


      NonMatching::ImmersedSurfaceQuadrature<dim, spacedim> surface_quad(
        final_unit_q_points, JxWs, normals);

      agglomerated_isv =
        std::make_unique<NonMatching::FEImmersedSurfaceValues<spacedim>>(
          *box_mapping, *fe, surface_quad, agglomeration_face_flags);


      agglomerated_isv->reinit(cell);

      std::pair<const FEValuesBase<dim, spacedim> &,
                const FEValuesBase<dim, spacedim> &>
        my_p(*agglomerated_isv, *agglomerated_isv);

      return my_p;
    }
}



template <int dim, int spacedim>
template <typename SparsityPatternType, typename Number>
void
AgglomerationHandler<dim, spacedim>::create_agglomeration_sparsity_pattern(
  SparsityPatternType             &dsp,
  const AffineConstraints<Number> &constraints,
  const bool                       keep_constrained_dofs,
  const types::subdomain_id        subdomain_id)
{
  Assert(n_agglomerations > 0,
         ExcMessage("The agglomeration has not been set up correctly."));
  Assert(dsp.empty(),
         ExcMessage(
           "The Sparsity pattern must be empty upon calling this function."));

  const IndexSet &locally_owned_dofs = agglo_dh.locally_owned_dofs();
  const IndexSet  locally_relevant_dofs =
    DoFTools::extract_locally_relevant_dofs(agglo_dh);

  if constexpr (std::is_same_v<SparsityPatternType, DynamicSparsityPattern>)
    dsp.reinit(locally_owned_dofs.size(),
               locally_owned_dofs.size(),
               locally_relevant_dofs);
  else if constexpr (std::is_same_v<SparsityPatternType,
                                    TrilinosWrappers::SparsityPattern>)
    dsp.reinit(locally_owned_dofs, communicator);
  else
    AssertThrow(false, ExcNotImplemented());

  // Create the sparsity pattern corresponding only to volumetric terms. The
  // fluxes needed by DG methods will be filled later.
  DoFTools::make_sparsity_pattern(
    agglo_dh, dsp, constraints, keep_constrained_dofs, subdomain_id);


  if (!is_hp_collection)
    {
      // Original version: handle case without hp::FECollection
      const unsigned int dofs_per_cell = agglo_dh.get_fe(0).n_dofs_per_cell();
      std::vector<types::global_dof_index> current_dof_indices(dofs_per_cell);
      std::vector<types::global_dof_index> neighbor_dof_indices(dofs_per_cell);

      // Loop over all locally owned polytopes, find the neighbor (also ghosted)
      // and add fluxes to the sparsity pattern.
      for (const auto &polytope : polytope_iterators())
        {
          if (polytope->is_locally_owned())
            {
              const unsigned int n_current_faces = polytope->n_faces();
              polytope->get_dof_indices(current_dof_indices);
              for (unsigned int f = 0; f < n_current_faces; ++f)
                {
                  const auto &neigh_polytope = polytope->neighbor(f);
                  if (neigh_polytope.state() == IteratorState::valid)
                    {
                      neigh_polytope->get_dof_indices(neighbor_dof_indices);
                      constraints.add_entries_local_to_global(
                        current_dof_indices,
                        neighbor_dof_indices,
                        dsp,
                        keep_constrained_dofs,
                        {});
                    }
                }
            }
        }
    }
  else
    {
      // Handle the hp::FECollection case

      // Loop over all locally owned polytopes, find the neighbor (also ghosted)
      // and add fluxes to the sparsity pattern.
      for (const auto &polytope : polytope_iterators())
        {
          if (polytope->is_locally_owned())
            {
              const unsigned int current_dofs_per_cell =
                polytope->get_fe().dofs_per_cell;
              std::vector<types::global_dof_index> current_dof_indices(
                current_dofs_per_cell);

              const unsigned int n_current_faces = polytope->n_faces();
              polytope->get_dof_indices(current_dof_indices);
              for (unsigned int f = 0; f < n_current_faces; ++f)
                {
                  const auto &neigh_polytope = polytope->neighbor(f);
                  if (neigh_polytope.state() == IteratorState::valid)
                    {
                      const unsigned int neighbor_dofs_per_cell =
                        neigh_polytope->get_fe().dofs_per_cell;
                      std::vector<types::global_dof_index> neighbor_dof_indices(
                        neighbor_dofs_per_cell);

                      neigh_polytope->get_dof_indices(neighbor_dof_indices);
                      constraints.add_entries_local_to_global(
                        current_dof_indices,
                        neighbor_dof_indices,
                        dsp,
                        keep_constrained_dofs,
                        {});
                    }
                }
            }
        }
    }



  if constexpr (std::is_same_v<SparsityPatternType,
                               TrilinosWrappers::SparsityPattern>)
    dsp.compress();
}



template <int dim, int spacedim>
void
AgglomerationHandler<dim, spacedim>::setup_ghost_polytopes()
{
  [[maybe_unused]] const auto parallel_triangulation =
    dynamic_cast<const parallel::TriangulationBase<dim, spacedim> *>(&*tria);
  Assert(parallel_triangulation != nullptr, ExcInternalError());

  const unsigned int                   n_dofs_per_cell = fe->dofs_per_cell;
  std::vector<types::global_dof_index> global_dof_indices(n_dofs_per_cell);
  for (const auto &polytope : polytope_iterators())
    if (polytope->is_locally_owned())
      {
        const CellId &master_cell_id = polytope->id();

        const auto polytope_dh = polytope->as_dof_handler_iterator(agglo_dh);
        polytope_dh->get_dof_indices(global_dof_indices);


        const auto &agglomerate = polytope->get_agglomerate();

        for (const auto &cell : agglomerate)
          {
            // interior, locally owned, cell
            for (const auto &f : cell->face_indices())
              {
                if (!cell->at_boundary(f))
                  {
                    const auto &neighbor = cell->neighbor(f);
                    if (neighbor->is_ghost())
                      {
                        // key of the map: the rank to which send the data
                        const types::subdomain_id neigh_rank =
                          neighbor->subdomain_id();

                        // inform the "standard" neighbor about the neighboring
                        // id and its master cell
                        local_cell_ids_neigh_cell[neigh_rank].emplace(
                          cell->id(), master_cell_id);

                        // inform the neighboring rank that this master cell
                        // (hence polytope) has the following DoF indices
                        local_ghost_dofs[neigh_rank].emplace(
                          master_cell_id, global_dof_indices);

                        // ...same for bounding boxes
                        const auto &bbox = bboxes[polytope->index()];
                        local_ghosted_bbox[neigh_rank].emplace(master_cell_id,
                                                               bbox);
                      }
                  }
              }
          }
      }

  recv_cell_ids_neigh_cell =
    Utilities::MPI::some_to_some(communicator, local_cell_ids_neigh_cell);

  // Exchange with neighboring ranks the neighboring bounding boxes
  recv_ghosted_bbox =
    Utilities::MPI::some_to_some(communicator, local_ghosted_bbox);

  // Exchange with neighboring ranks the neighboring ghosted DoFs
  recv_ghost_dofs =
    Utilities::MPI::some_to_some(communicator, local_ghost_dofs);
}



namespace dealii
{
  namespace internal
  {
    template <int dim, int spacedim>
    class AgglomerationHandlerImplementation
    {
    public:
      static const FEValuesBase<dim, spacedim> &
      reinit_master(
        const typename DoFHandler<dim, spacedim>::active_cell_iterator &cell,
        const unsigned int face_index,
        std::unique_ptr<NonMatching::FEImmersedSurfaceValues<spacedim>>
                                                  &agglo_isv_ptr,
        const AgglomerationHandler<dim, spacedim> &handler)
      {
        Assert(handler.is_master_cell(cell),
               ExcMessage("This cell must be a master one."));

        AgglomerationIterator<dim, spacedim> it{cell, &handler};
        const auto &neigh_polytope = it->neighbor(face_index);

        const CellId polytope_in_id = cell->id();

        // Retrieve the bounding box of the agglomeration
        const auto &bbox =
          handler.bboxes[handler.master2polygon.at(cell->active_cell_index())];

        CellId polytope_out_id;
        if (neigh_polytope.state() == IteratorState::valid)
          polytope_out_id = neigh_polytope->id();
        else
          polytope_out_id = polytope_in_id; // on the boundary. Same id

        const auto &common_face = handler.polytope_cache.interface.at(
          {polytope_in_id, polytope_out_id});

        std::vector<Point<spacedim>> final_unit_q_points;
        std::vector<double>          final_weights;
        std::vector<Tensor<1, dim>>  final_normals;

        if (!handler.is_hp_collection)
          {
            // Original version: handle case without hp::FECollection
            const unsigned int expected_qpoints =
              common_face.size() * handler.agglomeration_face_quad.size();
            final_unit_q_points.reserve(expected_qpoints);
            final_weights.reserve(expected_qpoints);
            final_normals.reserve(expected_qpoints);


            for (const auto &[deal_cell, local_face_idx] : common_face)
              {
                handler.no_face_values->reinit(deal_cell, local_face_idx);

                const auto &q_points =
                  handler.no_face_values->get_quadrature_points();
                const auto &JxWs = handler.no_face_values->get_JxW_values();
                const auto &normals =
                  handler.no_face_values->get_normal_vectors();

                const unsigned int n_qpoints_agglo = q_points.size();

                for (unsigned int q = 0; q < n_qpoints_agglo; ++q)
                  {
                    final_unit_q_points.push_back(
                      bbox.real_to_unit(q_points[q]));
                    final_weights.push_back(JxWs[q]);
                    final_normals.push_back(normals[q]);
                  }
              }
          }
        else
          {
            // Handle the hp::FECollection case
            unsigned int higher_order_quad_index = cell->active_fe_index();
            if (neigh_polytope.state() == IteratorState::valid)
              if (handler
                    .agglomeration_face_quad_collection[cell->active_fe_index()]
                    .size() <
                  handler
                    .agglomeration_face_quad_collection[neigh_polytope
                                                          ->active_fe_index()]
                    .size())
                higher_order_quad_index = neigh_polytope->active_fe_index();

            const unsigned int expected_qpoints =
              common_face.size() *
              handler
                .agglomeration_face_quad_collection[higher_order_quad_index]
                .size();
            final_unit_q_points.reserve(expected_qpoints);
            final_weights.reserve(expected_qpoints);
            final_normals.reserve(expected_qpoints);

            for (const auto &[deal_cell, local_face_idx] : common_face)
              {
                handler.hp_no_face_values->reinit(
                  deal_cell, local_face_idx, higher_order_quad_index, 0, 0);

                const auto &q_points =
                  handler.hp_no_face_values->get_present_fe_values()
                    .get_quadrature_points();
                const auto &JxWs =
                  handler.hp_no_face_values->get_present_fe_values()
                    .get_JxW_values();
                const auto &normals =
                  handler.hp_no_face_values->get_present_fe_values()
                    .get_normal_vectors();

                const unsigned int n_qpoints_agglo = q_points.size();

                for (unsigned int q = 0; q < n_qpoints_agglo; ++q)
                  {
                    final_unit_q_points.push_back(
                      bbox.real_to_unit(q_points[q]));
                    final_weights.push_back(JxWs[q]);
                    final_normals.push_back(normals[q]);
                  }
              }
          }


        NonMatching::ImmersedSurfaceQuadrature<dim, spacedim> surface_quad(
          final_unit_q_points, final_weights, final_normals);

        if (!handler.is_hp_collection)
          {
            agglo_isv_ptr =
              std::make_unique<NonMatching::FEImmersedSurfaceValues<spacedim>>(
                *(handler.box_mapping),
                *(handler.fe),
                surface_quad,
                handler.agglomeration_face_flags);
          }
        else
          {
            agglo_isv_ptr =
              std::make_unique<NonMatching::FEImmersedSurfaceValues<spacedim>>(
                *(handler.box_mapping),
                cell->get_fe(),
                surface_quad,
                handler.agglomeration_face_flags);
          }

        agglo_isv_ptr->reinit(cell);

        return *agglo_isv_ptr;
      }



      /**
       * Given an agglomeration described by the master cell `master_cell`,
       * this function:
       * - enumerates the faces of the agglomeration
       * - stores who is the neighbor, the local face indices from outside and
       * inside*/
      static void
      setup_master_neighbor_connectivity(
        const typename Triangulation<dim, spacedim>::active_cell_iterator
                                                  &master_cell,
        const AgglomerationHandler<dim, spacedim> &handler)
      {
        Assert(
          handler.master_slave_relationships[master_cell
                                               ->global_active_cell_index()] ==
            -1,
          ExcMessage("The present cell with index " +
                     std::to_string(master_cell->global_active_cell_index()) +
                     "is not a master one."));

        const auto &agglomeration = handler.get_agglomerate(master_cell);
        const types::global_cell_index current_polytope_index =
          handler.master2polygon.at(master_cell->active_cell_index());

        CellId current_polytope_id = master_cell->id();


        std::set<types::global_cell_index> visited_polygonal_neighbors;

        std::map<unsigned int, CellId> face_to_neigh_id;

        std::map<unsigned int, bool> is_face_at_boundary;

        // same as above, but with CellId
        std::set<CellId> visited_polygonal_neighbors_id;
        unsigned int     ghost_counter = 0;

        for (const auto &cell : agglomeration)
          {
            const types::global_cell_index cell_index =
              cell->active_cell_index();

            const CellId cell_id = cell->id();

            for (const auto f : cell->face_indices())
              {
                const auto &neighboring_cell = cell->neighbor(f);

                const bool valid_neighbor =
                  neighboring_cell.state() == IteratorState::valid;

                if (valid_neighbor)
                  {
                    if (neighboring_cell->is_locally_owned() &&
                        !handler.are_cells_agglomerated(cell, neighboring_cell))
                      {
                        // - cell is not on the boundary,
                        // - it's not agglomerated with the neighbor. If so,
                        // it's a neighbor of the present agglomeration
                        // std::cout << " (from rank) "
                        //           << Utilities::MPI::this_mpi_process(
                        //                handler.communicator)
                        //           << std::endl;

                        // std::cout
                        //   << "neighbor locally owned? " << std::boolalpha
                        //   << neighboring_cell->is_locally_owned() <<
                        //   std::endl;
                        // if (neighboring_cell->is_ghost())
                        //   handler.ghosted_indices.push_back(
                        //     neighboring_cell->active_cell_index());

                        // a new face of the agglomeration has been
                        // discovered.
                        handler.polygon_boundary[master_cell].push_back(
                          cell->face(f));

                        // global index of neighboring deal.II cell
                        const types::global_cell_index neighboring_cell_index =
                          neighboring_cell->active_cell_index();

                        // master cell for the neighboring polytope
                        const auto &master_of_neighbor =
                          handler.master_slave_relationships_iterators.at(
                            neighboring_cell_index);

                        const auto nof = cell->neighbor_of_neighbor(f);

                        if (handler.is_slave_cell(neighboring_cell))
                          {
                            // index of the neighboring polytope
                            const types::global_cell_index
                              neighbor_polytope_index =
                                handler.master2polygon.at(
                                  master_of_neighbor->active_cell_index());

                            CellId neighbor_polytope_id =
                              master_of_neighbor->id();

                            if (visited_polygonal_neighbors.find(
                                  neighbor_polytope_index) ==
                                std::end(visited_polygonal_neighbors))
                              {
                                // found a neighbor

                                const unsigned int n_face =
                                  handler.number_of_agglomerated_faces
                                    [current_polytope_index];

                                handler.polytope_cache.cell_face_at_boundary[{
                                  current_polytope_index, n_face}] = {
                                  false, master_of_neighbor};

                                is_face_at_boundary[n_face] = true;

                                ++handler.number_of_agglomerated_faces
                                    [current_polytope_index];

                                visited_polygonal_neighbors.insert(
                                  neighbor_polytope_index);
                              }


                            if (handler.polytope_cache.visited_cell_and_faces
                                  .find({cell_index, f}) ==
                                std::end(handler.polytope_cache
                                           .visited_cell_and_faces))
                              {
                                handler.polytope_cache
                                  .interface[{current_polytope_id,
                                              neighbor_polytope_id}]
                                  .emplace_back(cell, f);

                                handler.polytope_cache.visited_cell_and_faces
                                  .insert({cell_index, f});
                              }


                            if (handler.polytope_cache.visited_cell_and_faces
                                  .find({neighboring_cell_index, nof}) ==
                                std::end(handler.polytope_cache
                                           .visited_cell_and_faces))
                              {
                                handler.polytope_cache
                                  .interface[{neighbor_polytope_id,
                                              current_polytope_id}]
                                  .emplace_back(neighboring_cell, nof);

                                handler.polytope_cache.visited_cell_and_faces
                                  .insert({neighboring_cell_index, nof});
                              }
                          }
                        else
                          {
                            // neighboring cell is a master

                            // save the pair of neighboring cells
                            const types::global_cell_index
                              neighbor_polytope_index =
                                handler.master2polygon.at(
                                  neighboring_cell_index);

                            CellId neighbor_polytope_id =
                              neighboring_cell->id();

                            if (visited_polygonal_neighbors.find(
                                  neighbor_polytope_index) ==
                                std::end(visited_polygonal_neighbors))
                              {
                                // found a neighbor
                                const unsigned int n_face =
                                  handler.number_of_agglomerated_faces
                                    [current_polytope_index];


                                handler.polytope_cache.cell_face_at_boundary[{
                                  current_polytope_index, n_face}] = {
                                  false, neighboring_cell};

                                is_face_at_boundary[n_face] = true;

                                ++handler.number_of_agglomerated_faces
                                    [current_polytope_index];

                                visited_polygonal_neighbors.insert(
                                  neighbor_polytope_index);
                              }



                            if (handler.polytope_cache.visited_cell_and_faces
                                  .find({cell_index, f}) ==
                                std::end(handler.polytope_cache
                                           .visited_cell_and_faces))
                              {
                                handler.polytope_cache
                                  .interface[{current_polytope_id,
                                              neighbor_polytope_id}]
                                  .emplace_back(cell, f);

                                handler.polytope_cache.visited_cell_and_faces
                                  .insert({cell_index, f});
                              }

                            if (handler.polytope_cache.visited_cell_and_faces
                                  .find({neighboring_cell_index, nof}) ==
                                std::end(handler.polytope_cache
                                           .visited_cell_and_faces))
                              {
                                handler.polytope_cache
                                  .interface[{neighbor_polytope_id,
                                              current_polytope_id}]
                                  .emplace_back(neighboring_cell, nof);

                                handler.polytope_cache.visited_cell_and_faces
                                  .insert({neighboring_cell_index, nof});
                              }
                          }
                      }
                    else if (neighboring_cell->is_ghost())
                      {
                        const auto nof = cell->neighbor_of_neighbor(f);

                        // from neighboring rank,receive the association
                        // between standard cell ids and neighboring polytope.
                        // This tells to the current rank that the
                        // neighboring cell has the following CellId as master
                        // cell.
                        const auto &check_neigh_poly_ids =
                          handler.recv_cell_ids_neigh_cell.at(
                            neighboring_cell->subdomain_id());

                        const CellId neighboring_cell_id =
                          neighboring_cell->id();

                        const CellId &check_neigh_polytope_id =
                          check_neigh_poly_ids.at(neighboring_cell_id);

                        // const auto master_index =
                        // master_indices[ghost_counter];

                        if (visited_polygonal_neighbors_id.find(
                              check_neigh_polytope_id) ==
                            std::end(visited_polygonal_neighbors_id))
                          {
                            handler.polytope_cache.cell_face_at_boundary[{
                              current_polytope_index,
                              handler.number_of_agglomerated_faces
                                [current_polytope_index]}] = {false,
                                                              neighboring_cell};


                            // record the cell id of the neighboring polytope
                            handler.polytope_cache.ghosted_master_id[{
                              current_polytope_id,
                              handler.number_of_agglomerated_faces
                                [current_polytope_index]}] =
                              check_neigh_polytope_id;


                            const unsigned int n_face =
                              handler.number_of_agglomerated_faces
                                [current_polytope_index];

                            face_to_neigh_id[n_face] = check_neigh_polytope_id;

                            is_face_at_boundary[n_face] = false;


                            // increment number of faces
                            ++handler.number_of_agglomerated_faces
                                [current_polytope_index];

                            visited_polygonal_neighbors_id.insert(
                              check_neigh_polytope_id);

                            // ghosted polytope has been found, increment
                            // ghost counter
                            ++ghost_counter;
                          }



                        if (handler.polytope_cache.visited_cell_and_faces_id
                              .find({cell_id, f}) ==
                            std::end(
                              handler.polytope_cache.visited_cell_and_faces_id))
                          {
                            handler.polytope_cache
                              .interface[{current_polytope_id,
                                          check_neigh_polytope_id}]
                              .emplace_back(cell, f);

                            // std::cout << "ADDED ("
                            //           << cell->active_cell_index() << ")
                            //           BETWEEN "
                            //           << current_polytope_id << " e "
                            //           << check_neigh_polytope_id <<
                            //           std::endl;

                            handler.polytope_cache.visited_cell_and_faces_id
                              .insert({cell_id, f});
                          }


                        if (handler.polytope_cache.visited_cell_and_faces_id
                              .find({neighboring_cell_id, nof}) ==
                            std::end(
                              handler.polytope_cache.visited_cell_and_faces_id))
                          {
                            handler.polytope_cache
                              .interface[{check_neigh_polytope_id,
                                          current_polytope_id}]
                              .emplace_back(neighboring_cell, nof);

                            handler.polytope_cache.visited_cell_and_faces_id
                              .insert({neighboring_cell_id, nof});
                          }
                      }
                  }
                else if (cell->face(f)->at_boundary())
                  {
                    // Boundary face of a boundary cell.
                    // Note that the neighboring cell must be invalid.

                    handler.polygon_boundary[master_cell].push_back(
                      cell->face(f));

                    if (visited_polygonal_neighbors.find(
                          std::numeric_limits<unsigned int>::max()) ==
                        std::end(visited_polygonal_neighbors))
                      {
                        // boundary face. Notice that `neighboring_cell` is
                        // invalid here.
                        handler.polytope_cache.cell_face_at_boundary[{
                          current_polytope_index,
                          handler.number_of_agglomerated_faces
                            [current_polytope_index]}] = {true,
                                                          neighboring_cell};

                        const unsigned int n_face =
                          handler.number_of_agglomerated_faces
                            [current_polytope_index];

                        is_face_at_boundary[n_face] = true;

                        ++handler.number_of_agglomerated_faces
                            [current_polytope_index];

                        visited_polygonal_neighbors.insert(
                          std::numeric_limits<unsigned int>::max());
                      }



                    if (handler.polytope_cache.visited_cell_and_faces.find(
                          {cell_index, f}) ==
                        std::end(handler.polytope_cache.visited_cell_and_faces))
                      {
                        handler.polytope_cache
                          .interface[{current_polytope_id, current_polytope_id}]
                          .emplace_back(cell, f);

                        handler.polytope_cache.visited_cell_and_faces.insert(
                          {cell_index, f});
                      }
                  }
              } // loop over faces
          } // loop over all cells of agglomerate



        if (ghost_counter > 0)
          {
            const auto parallel_triangulation = dynamic_cast<
              const dealii::parallel::TriangulationBase<dim, spacedim> *>(
              &(*handler.tria));

            const unsigned int n_faces_current_poly =
              handler.number_of_agglomerated_faces[current_polytope_index];

            // Communicate to neighboring ranks that current_polytope_id has
            // a number of faces equal to n_faces_current_poly faces:
            // current_polytope_id -> n_faces_current_poly
            for (const unsigned int neigh_rank :
                 parallel_triangulation->ghost_owners())
              {
                handler.local_n_faces[neigh_rank].emplace(current_polytope_id,
                                                          n_faces_current_poly);

                handler.local_bdary_info[neigh_rank].emplace(
                  current_polytope_id, is_face_at_boundary);

                handler.local_ghosted_master_id[neigh_rank].emplace(
                  current_polytope_id, face_to_neigh_id);
              }
          }
      }
    };



  } // namespace internal
} // namespace dealii



template class AgglomerationHandler<1>;
template void
AgglomerationHandler<1>::create_agglomeration_sparsity_pattern(
  DynamicSparsityPattern          &sparsity_pattern,
  const AffineConstraints<double> &constraints,
  const bool                       keep_constrained_dofs,
  const types::subdomain_id        subdomain_id);

template void
AgglomerationHandler<1>::create_agglomeration_sparsity_pattern(
  TrilinosWrappers::SparsityPattern &sparsity_pattern,
  const AffineConstraints<double>   &constraints,
  const bool                         keep_constrained_dofs,
  const types::subdomain_id          subdomain_id);

template class AgglomerationHandler<2>;
template void
AgglomerationHandler<2>::create_agglomeration_sparsity_pattern(
  DynamicSparsityPattern          &sparsity_pattern,
  const AffineConstraints<double> &constraints,
  const bool                       keep_constrained_dofs,
  const types::subdomain_id        subdomain_id);

template void
AgglomerationHandler<2>::create_agglomeration_sparsity_pattern(
  TrilinosWrappers::SparsityPattern &sparsity_pattern,
  const AffineConstraints<double>   &constraints,
  const bool                         keep_constrained_dofs,
  const types::subdomain_id          subdomain_id);

template class AgglomerationHandler<3>;
template void
AgglomerationHandler<3>::create_agglomeration_sparsity_pattern(
  DynamicSparsityPattern          &sparsity_pattern,
  const AffineConstraints<double> &constraints,
  const bool                       keep_constrained_dofs,
  const types::subdomain_id        subdomain_id);

template void
AgglomerationHandler<3>::create_agglomeration_sparsity_pattern(
  TrilinosWrappers::SparsityPattern &sparsity_pattern,
  const AffineConstraints<double>   &constraints,
  const bool                         keep_constrained_dofs,
  const types::subdomain_id          subdomain_id);