mg_transfer_internal.cc

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// ---------------------------------------------------------------------
//
// Copyright (C) 2016 - 2023 by the deal.II authors
//
// This file is part of the deal.II library.
//
// The deal.II library is free software; you can use it, redistribute
// it, and/or modify it under the terms of the GNU Lesser General
// Public License as published by the Free Software Foundation; either
// version 2.1 of the License, or (at your option) any later version.
// The full text of the license can be found in the file LICENSE.md at
// the top level directory of deal.II.
//
// ---------------------------------------------------------------------
 
 
#include <deal.II/distributed/tria.h>
 
#include <deal.II/dofs/dof_tools.h>
 
#include <deal.II/fe/fe_tools.h>
 
#include <deal.II/matrix_free/shape_info.h>
 
#include <deal.II/multigrid/mg_transfer_internal.h>
 
#include <memory>
 
DEAL_II_NAMESPACE_OPEN
 
namespace internal
{
  namespace MGTransfer
  {
    // Internal data structure that is used in the MPI communication in
    // fill_copy_indices().  It represents an entry in the copy_indices* map,
    // that associates a level dof index with a global dof index.
    struct DoFPair
    {
      unsigned int            level;
      types::global_dof_index global_dof_index;
      types::global_dof_index level_dof_index;
 
      DoFPair(const unsigned int            level,
              const types::global_dof_index global_dof_index,
              const types::global_dof_index level_dof_index)
        : level(level)
        , global_dof_index(global_dof_index)
        , level_dof_index(level_dof_index)
      {}
 
      DoFPair()
        : level(numbers::invalid_unsigned_int)
        , global_dof_index(numbers::invalid_dof_index)
        , level_dof_index(numbers::invalid_dof_index)
      {}
    };
 
    template <int dim, int spacedim>
    void
    fill_copy_indices(
      const DoFHandler<dim, spacedim> &dof_handler,
      const MGConstrainedDoFs *        mg_constrained_dofs,
      std::vector<std::vector<
        std::pair<types::global_dof_index, types::global_dof_index>>>
        &copy_indices,
      std::vector<std::vector<
        std::pair<types::global_dof_index, types::global_dof_index>>>
        &copy_indices_global_mine,
      std::vector<std::vector<
        std::pair<types::global_dof_index, types::global_dof_index>>>
        &        copy_indices_level_mine,
      const bool skip_interface_dofs)
    {
      // Now we are filling the variables copy_indices*, which are essentially
      // maps from global to mgdof for each level stored as a std::vector of
      // pairs. We need to split this map on each level depending on the
      // ownership of the global and mgdof, so that we later do not access
      // non-local elements in copy_to/from_mg.
      // We keep track in the bitfield dof_touched which global dof has been
      // processed already (on the current level). This is the same as the
      // multigrid running in serial.
 
      // map cpu_index -> vector of data
      // that will be copied into copy_indices_level_mine
      std::vector<DoFPair> send_data_temp;
 
      const unsigned int n_levels =
        dof_handler.get_triangulation().n_global_levels();
      copy_indices.resize(n_levels);
      copy_indices_global_mine.resize(n_levels);
      copy_indices_level_mine.resize(n_levels);
      const IndexSet &owned_dofs = dof_handler.locally_owned_dofs();
 
      const unsigned int dofs_per_cell = dof_handler.get_fe().n_dofs_per_cell();
      std::vector<types::global_dof_index> global_dof_indices(dofs_per_cell);
      std::vector<types::global_dof_index> level_dof_indices(dofs_per_cell);
 
      for (unsigned int level = 0; level < n_levels; ++level)
        {
          std::vector<bool> dof_touched(owned_dofs.n_elements(), false);
          const IndexSet &  owned_level_dofs =
            dof_handler.locally_owned_mg_dofs(level);
 
          // for the most common case where copy_indices are locally owned
          // both globally and on the level, we want to skip collecting pairs
          // and later sorting them. instead, we insert these indices into a
          // plain vector
          std::vector<types::global_dof_index> unrolled_copy_indices;
 
          copy_indices_level_mine[level].clear();
          copy_indices_global_mine[level].clear();
 
          for (const auto &level_cell :
               dof_handler.active_cell_iterators_on_level(level))
            {
              if (dof_handler.get_triangulation().locally_owned_subdomain() !=
                    numbers::invalid_subdomain_id &&
                  (level_cell->level_subdomain_id() ==
                     numbers::artificial_subdomain_id ||
                   level_cell->subdomain_id() ==
                     numbers::artificial_subdomain_id))
                continue;
 
              unrolled_copy_indices.resize(owned_dofs.n_elements(),
                                           numbers::invalid_dof_index);
 
              // get the dof numbers of this cell for the global and the
              // level-wise numbering
              level_cell->get_dof_indices(global_dof_indices);
              level_cell->get_mg_dof_indices(level_dof_indices);
 
              for (unsigned int i = 0; i < dofs_per_cell; ++i)
                {
                  // we need to ignore if the DoF is on a refinement edge
                  // (hanging node)
                  if (skip_interface_dofs && mg_constrained_dofs != nullptr &&
                      mg_constrained_dofs->at_refinement_edge(
                        level, level_dof_indices[i]))
                    continue;
 
                  // First check whether we own any of the active dof index
                  // and the level one. This check involves locally owned
                  // indices which often consist only of a single range, so
                  // they are cheap to look up.
                  const types::global_dof_index global_index_in_set =
                    owned_dofs.index_within_set(global_dof_indices[i]);
                  bool global_mine =
                    global_index_in_set != numbers::invalid_dof_index;
                  bool level_mine =
                    owned_level_dofs.is_element(level_dof_indices[i]);
 
                  if (global_mine && level_mine)
                    {
                      // we own both the active dof index and the level one ->
                      // set them into the vector, indexed by the local index
                      // range of the active dof
                      unrolled_copy_indices[global_index_in_set] =
                        level_dof_indices[i];
                    }
                  else if (global_mine &&
                           dof_touched[global_index_in_set] == false)
                    {
                      copy_indices_global_mine[level].emplace_back(
                        global_dof_indices[i], level_dof_indices[i]);
 
                      // send this to the owner of the level_dof:
                      send_data_temp.emplace_back(level,
                                                  global_dof_indices[i],
                                                  level_dof_indices[i]);
                      dof_touched[global_index_in_set] = true;
                    }
                  else
                    {
                      // somebody will send those to me
                    }
                }
            }
 
          // we now translate the plain vector for the copy_indices field into
          // the expected format of a pair of indices
          if (!unrolled_copy_indices.empty())
            {
              copy_indices[level].clear();
 
              // reserve the full length in case we did not hit global-mine
              // indices, so we expect all indices to come into copy_indices
              if (copy_indices_global_mine[level].empty())
                copy_indices[level].reserve(unrolled_copy_indices.size());
 
              // owned_dofs.nth_index_in_set(i) in this query is
              // usually cheap to look up as there are few ranges in
              // the locally owned part
              for (unsigned int i = 0; i < unrolled_copy_indices.size(); ++i)
                if (unrolled_copy_indices[i] != numbers::invalid_dof_index)
                  copy_indices[level].emplace_back(
                    owned_dofs.nth_index_in_set(i), unrolled_copy_indices[i]);
            }
        }
 
      const ::parallel::TriangulationBase<dim, spacedim> *tria =
        (dynamic_cast<const ::parallel::TriangulationBase<dim, spacedim>
                        *>(&dof_handler.get_triangulation()));
      AssertThrow(
        send_data_temp.size() == 0 || tria != nullptr,
        ExcMessage(
          "We should only be sending information with a parallel Triangulation!"));
 
#ifdef DEAL_II_WITH_MPI
      if (tria && Utilities::MPI::sum(send_data_temp.size(),
                                      tria->get_communicator()) > 0)
        {
          const std::set<types::subdomain_id> &neighbors =
            tria->level_ghost_owners();
          std::map<int, std::vector<DoFPair>> send_data;
 
          std::sort(send_data_temp.begin(),
                    send_data_temp.end(),
                    [](const DoFPair &lhs, const DoFPair &rhs) {
                      if (lhs.level < rhs.level)
                        return true;
                      if (lhs.level > rhs.level)
                        return false;
 
                      if (lhs.level_dof_index < rhs.level_dof_index)
                        return true;
                      if (lhs.level_dof_index > rhs.level_dof_index)
                        return false;
 
                      if (lhs.global_dof_index < rhs.global_dof_index)
                        return true;
                      else
                        return false;
                    });
          send_data_temp.erase(
            std::unique(send_data_temp.begin(),
                        send_data_temp.end(),
                        [](const DoFPair &lhs, const DoFPair &rhs) {
                          return (lhs.level == rhs.level) &&
                                 (lhs.level_dof_index == rhs.level_dof_index) &&
                                 (lhs.global_dof_index == rhs.global_dof_index);
                        }),
            send_data_temp.end());
 
          for (unsigned int level = 0; level < n_levels; ++level)
            {
              const IndexSet &owned_level_dofs =
                dof_handler.locally_owned_mg_dofs(level);
 
              std::vector<types::global_dof_index> level_dof_indices;
              std::vector<types::global_dof_index> global_dof_indices;
              for (const auto &dofpair : send_data_temp)
                if (dofpair.level == level)
                  {
                    level_dof_indices.push_back(dofpair.level_dof_index);
                    global_dof_indices.push_back(dofpair.global_dof_index);
                  }
 
              IndexSet is_ghost(owned_level_dofs.size());
              is_ghost.add_indices(level_dof_indices.begin(),
                                   level_dof_indices.end());
 
              AssertThrow(level_dof_indices.size() == is_ghost.n_elements(),
                          ExcMessage("Size does not match!"));
 
              const auto index_owner =
                Utilities::MPI::compute_index_owner(owned_level_dofs,
                                                    is_ghost,
                                                    tria->get_communicator());
 
              AssertThrow(level_dof_indices.size() == index_owner.size(),
                          ExcMessage("Size does not match!"));
 
              for (unsigned int i = 0; i < index_owner.size(); ++i)
                send_data[index_owner[i]].emplace_back(level,
                                                       global_dof_indices[i],
                                                       level_dof_indices[i]);
            }
 
 
          // Protect the send/recv logic with a mutex:
          static Utilities::MPI::CollectiveMutex      mutex;
          Utilities::MPI::CollectiveMutex::ScopedLock lock(
            mutex, tria->get_communicator());
 
          const int mpi_tag =
            Utilities::MPI::internal::Tags::mg_transfer_fill_copy_indices;
 
          // * send
          std::vector<MPI_Request> requests;
          {
            for (const auto dest : neighbors)
              {
                requests.push_back(MPI_Request());
                std::vector<DoFPair> &data = send_data[dest];
 
                const int ierr =
                  MPI_Isend(data.data(),
                            data.size() * sizeof(decltype(*data.data())),
                            MPI_BYTE,
                            dest,
                            mpi_tag,
                            tria->get_communicator(),
                            &*requests.rbegin());
                AssertThrowMPI(ierr);
              }
          }
 
          // * receive
          {
            // We should get one message from each of our neighbors
            std::vector<DoFPair> receive_buffer;
            for (unsigned int counter = 0; counter < neighbors.size();
                 ++counter)
              {
                MPI_Status status;
                int        ierr = MPI_Probe(MPI_ANY_SOURCE,
                                     mpi_tag,
                                     tria->get_communicator(),
                                     &status);
                AssertThrowMPI(ierr);
                int len;
                ierr = MPI_Get_count(&status, MPI_BYTE, &len);
                AssertThrowMPI(ierr);
 
                if (len == 0)
                  {
                    ierr = MPI_Recv(nullptr,
                                    0,
                                    MPI_BYTE,
                                    status.MPI_SOURCE,
                                    status.MPI_TAG,
                                    tria->get_communicator(),
                                    &status);
                    AssertThrowMPI(ierr);
                    continue;
                  }
 
                int count = len / sizeof(DoFPair);
                Assert(static_cast<int>(count * sizeof(DoFPair)) == len,
                       ExcInternalError());
                receive_buffer.resize(count);
 
                void *ptr = receive_buffer.data();
                ierr      = MPI_Recv(ptr,
                                len,
                                MPI_BYTE,
                                status.MPI_SOURCE,
                                status.MPI_TAG,
                                tria->get_communicator(),
                                &status);
                AssertThrowMPI(ierr);
 
                for (const auto &dof_pair : receive_buffer)
                  {
                    copy_indices_level_mine[dof_pair.level].emplace_back(
                      dof_pair.global_dof_index, dof_pair.level_dof_index);
                  }
              }
          }
 
          // * wait for all MPI_Isend to complete
          if (requests.size() > 0)
            {
              const int ierr = MPI_Waitall(requests.size(),
                                           requests.data(),
                                           MPI_STATUSES_IGNORE);
              AssertThrowMPI(ierr);
              requests.clear();
            }
#  ifdef DEBUG
          // Make sure in debug mode, that everybody sent/received all packages
          // on this level. If a deadlock occurs here, the list of expected
          // senders is not computed correctly.
          const int ierr = MPI_Barrier(tria->get_communicator());
          AssertThrowMPI(ierr);
#  endif
        }
#endif
 
      // Sort the indices, except the copy_indices which already are
      // sorted. This will produce more reliable debug output for regression
      // tests and won't hurt performance even in release mode because the
      // non-owned indices are a small subset of all unknowns.
      std::less<std::pair<types::global_dof_index, types::global_dof_index>>
        compare;
      for (auto &level_indices : copy_indices_level_mine)
        std::sort(level_indices.begin(), level_indices.end(), compare);
      for (auto &level_indices : copy_indices_global_mine)
        std::sort(level_indices.begin(), level_indices.end(), compare);
    }
 
 
 
    // initialize the vectors needed for the transfer (and merge with the
    // content in copy_indices_global_mine)
    void
    reinit_level_partitioner(
      const IndexSet &                      locally_owned,
      std::vector<types::global_dof_index> &ghosted_level_dofs,
      const std::shared_ptr<const Utilities::MPI::Partitioner>
        &                                                 external_partitioner,
      const MPI_Comm                                      communicator,
      std::shared_ptr<const Utilities::MPI::Partitioner> &target_partitioner,
      Table<2, unsigned int> &copy_indices_global_mine)
    {
      std::sort(ghosted_level_dofs.begin(), ghosted_level_dofs.end());
      IndexSet ghosted_dofs(locally_owned.size());
      ghosted_dofs.add_indices(ghosted_level_dofs.begin(),
                               std::unique(ghosted_level_dofs.begin(),
                                           ghosted_level_dofs.end()));
      ghosted_dofs.compress();
 
      // Add possible ghosts from the previous content in the vector
      if (target_partitioner.get() != nullptr &&
          target_partitioner->size() == locally_owned.size())
        {
          ghosted_dofs.add_indices(target_partitioner->ghost_indices());
        }
 
      // check if the given partitioner's ghosts represent a superset of the
      // ghosts we require in this function
      const int ghosts_locally_contained =
        (external_partitioner.get() != nullptr &&
         (external_partitioner->ghost_indices() & ghosted_dofs) ==
           ghosted_dofs) ?
          1 :
          0;
      if (external_partitioner.get() != nullptr &&
          Utilities::MPI::min(ghosts_locally_contained, communicator) == 1)
        {
          // shift the local number of the copy indices according to the new
          // partitioner that we are going to use during the access to the
          // entries
          if (target_partitioner.get() != nullptr &&
              target_partitioner->size() == locally_owned.size())
            for (unsigned int i = 0; i < copy_indices_global_mine.n_cols(); ++i)
              copy_indices_global_mine(1, i) =
                external_partitioner->global_to_local(
                  target_partitioner->local_to_global(
                    copy_indices_global_mine(1, i)));
          target_partitioner = external_partitioner;
        }
      else
        {
          if (target_partitioner.get() != nullptr &&
              target_partitioner->size() == locally_owned.size())
            for (unsigned int i = 0; i < copy_indices_global_mine.n_cols(); ++i)
              copy_indices_global_mine(1, i) =
                locally_owned.n_elements() +
                ghosted_dofs.index_within_set(
                  target_partitioner->local_to_global(
                    copy_indices_global_mine(1, i)));
          target_partitioner =
            std::make_shared<Utilities::MPI::Partitioner>(locally_owned,
                                                          ghosted_dofs,
                                                          communicator);
        }
    }
 
 
 
    // Transform the ghost indices to local index space for the vector
    inline void
    copy_indices_to_mpi_local_numbers(
      const Utilities::MPI::Partitioner &         part,
      const std::vector<types::global_dof_index> &mine,
      const std::vector<types::global_dof_index> &remote,
      std::vector<unsigned int> &                 localized_indices)
    {
      localized_indices.resize(mine.size() + remote.size(),
                               numbers::invalid_unsigned_int);
      for (unsigned int i = 0; i < mine.size(); ++i)
        if (mine[i] != numbers::invalid_dof_index)
          localized_indices[i] = part.global_to_local(mine[i]);
 
      for (unsigned int i = 0; i < remote.size(); ++i)
        if (remote[i] != numbers::invalid_dof_index)
          localized_indices[i + mine.size()] = part.global_to_local(remote[i]);
    }
 
 
 
    // given the collection of child cells in lexicographic ordering as seen
    // from the parent, compute the first index of the given child
    template <int dim>
    unsigned int
    compute_shift_within_children(const unsigned int child,
                                  const unsigned int fe_shift_1d,
                                  const unsigned int fe_degree)
    {
      // we put the degrees of freedom of all child cells in lexicographic
      // ordering
      unsigned int c_tensor_index[dim];
      unsigned int tmp = child;
      for (unsigned int d = 0; d < dim; ++d)
        {
          c_tensor_index[d] = tmp % 2;
          tmp /= 2;
        }
      const unsigned int n_child_dofs_1d = fe_degree + 1 + fe_shift_1d;
      unsigned int       factor          = 1;
      unsigned int       shift           = fe_shift_1d * c_tensor_index[0];
      for (unsigned int d = 1; d < dim; ++d)
        {
          factor *= n_child_dofs_1d;
          shift = shift + factor * fe_shift_1d * c_tensor_index[d];
        }
      return shift;
    }
 
 
 
    // puts the indices on the given child cell in lexicographic ordering with
    // respect to the collection of all child cells as seen from the parent
    template <int dim>
    void
    add_child_indices(
      const unsigned int                          child,
      const unsigned int                          fe_shift_1d,
      const unsigned int                          fe_degree,
      const std::vector<unsigned int> &           lexicographic_numbering,
      const std::vector<types::global_dof_index> &local_dof_indices,
      types::global_dof_index *                   target_indices)
    {
      const unsigned int n_child_dofs_1d = fe_degree + 1 + fe_shift_1d;
      const unsigned int shift =
        compute_shift_within_children<dim>(child, fe_shift_1d, fe_degree);
      const unsigned int n_components =
        local_dof_indices.size() / Utilities::fixed_power<dim>(fe_degree + 1);
      types::global_dof_index *indices = target_indices + shift;
      const unsigned int       n_scalar_cell_dofs =
        Utilities::fixed_power<dim>(n_child_dofs_1d);
      for (unsigned int c = 0, m = 0; c < n_components; ++c)
        for (unsigned int k = 0; k < (dim > 2 ? (fe_degree + 1) : 1); ++k)
          for (unsigned int j = 0; j < (dim > 1 ? (fe_degree + 1) : 1); ++j)
            for (unsigned int i = 0; i < (fe_degree + 1); ++i, ++m)
              {
                const unsigned int index =
                  c * n_scalar_cell_dofs +
                  k * n_child_dofs_1d * n_child_dofs_1d + j * n_child_dofs_1d +
                  i;
                Assert(indices[index] == numbers::invalid_dof_index ||
                         indices[index] ==
                           local_dof_indices[lexicographic_numbering[m]],
                       ExcInternalError());
                indices[index] = local_dof_indices[lexicographic_numbering[m]];
              }
    }
 
 
 
    template <int dim, typename Number>
    void
    setup_element_info(ElementInfo<Number> &   elem_info,
                       const FiniteElement<1> &fe,
                       const DoFHandler<dim> & dof_handler)
    {
      // currently, we have only FE_Q and FE_DGQ type elements implemented
      elem_info.n_components = dof_handler.get_fe().element_multiplicity(0);
      AssertDimension(Utilities::fixed_power<dim>(fe.n_dofs_per_cell()) *
                        elem_info.n_components,
                      dof_handler.get_fe().n_dofs_per_cell());
      AssertDimension(fe.degree, dof_handler.get_fe().degree);
      elem_info.fe_degree             = fe.degree;
      elem_info.element_is_continuous = fe.n_dofs_per_vertex() > 0;
      Assert(fe.n_dofs_per_vertex() < 2, ExcNotImplemented());
 
      // step 1.2: get renumbering of 1d basis functions to lexicographic
      // numbers. The distinction according to fe.n_dofs_per_vertex() is to
      // support both continuous and discontinuous bases.
      std::vector<unsigned int> renumbering(fe.n_dofs_per_cell());
      {
        AssertIndexRange(fe.n_dofs_per_vertex(), 2);
        renumbering[0] = 0;
        for (unsigned int i = 0; i < fe.n_dofs_per_line(); ++i)
          renumbering[i + fe.n_dofs_per_vertex()] =
            GeometryInfo<1>::vertices_per_cell * fe.n_dofs_per_vertex() + i;
        if (fe.n_dofs_per_vertex() > 0)
          renumbering[fe.n_dofs_per_cell() - fe.n_dofs_per_vertex()] =
            fe.n_dofs_per_vertex();
      }
 
      // step 1.3: create a dummy 1d quadrature formula to extract the
      // lexicographic numbering for the elements
      Assert(fe.n_dofs_per_vertex() == 0 || fe.n_dofs_per_vertex() == 1,
             ExcNotImplemented());
      const unsigned int shift = fe.n_dofs_per_cell() - fe.n_dofs_per_vertex();
      const unsigned int n_child_dofs_1d =
        (fe.n_dofs_per_vertex() > 0 ? (2 * fe.n_dofs_per_cell() - 1) :
                                      (2 * fe.n_dofs_per_cell()));
 
      elem_info.n_child_cell_dofs =
        elem_info.n_components * Utilities::fixed_power<dim>(n_child_dofs_1d);
      const Quadrature<1> dummy_quadrature(
        std::vector<Point<1>>(1, Point<1>()));
      internal::MatrixFreeFunctions::ShapeInfo<Number> shape_info;
      shape_info.reinit(dummy_quadrature, dof_handler.get_fe(), 0);
      elem_info.lexicographic_numbering = shape_info.lexicographic_numbering;
 
      // step 1.4: get the 1d prolongation matrix and combine from both children
      elem_info.prolongation_matrix_1d.resize(fe.n_dofs_per_cell() *
                                              n_child_dofs_1d);
 
      for (unsigned int c = 0; c < GeometryInfo<1>::max_children_per_cell; ++c)
        for (unsigned int i = 0; i < fe.n_dofs_per_cell(); ++i)
          for (unsigned int j = 0; j < fe.n_dofs_per_cell(); ++j)
            elem_info
              .prolongation_matrix_1d[i * n_child_dofs_1d + j + c * shift] =
              fe.get_prolongation_matrix(c)(renumbering[j], renumbering[i]);
    }
 
 
 
    // Sets up most of the internal data structures of the MGTransferMatrixFree
    // class
    template <int dim, typename Number>
    void
    setup_transfer(
      const DoFHandler<dim> &  dof_handler,
      const MGConstrainedDoFs *mg_constrained_dofs,
      const std::vector<std::shared_ptr<const Utilities::MPI::Partitioner>>
        &                                     external_partitioners,
      ElementInfo<Number> &                   elem_info,
      std::vector<std::vector<unsigned int>> &level_dof_indices,
      std::vector<std::vector<std::pair<unsigned int, unsigned int>>>
        &                        parent_child_connect,
      std::vector<unsigned int> &n_owned_level_cells,
      std::vector<std::vector<std::vector<unsigned short>>> &dirichlet_indices,
      std::vector<std::vector<Number>> &                     weights_on_refined,
      std::vector<Table<2, unsigned int>> &copy_indices_global_mine,
      MGLevelObject<std::shared_ptr<const Utilities::MPI::Partitioner>>
        &target_partitioners)
    {
      level_dof_indices.clear();
      parent_child_connect.clear();
      n_owned_level_cells.clear();
      dirichlet_indices.clear();
      weights_on_refined.clear();
 
      // we collect all child DoFs of a mother cell together. For faster
      // tensorized operations, we align the degrees of freedom
      // lexicographically. We distinguish FE_Q elements and FE_DGQ elements
 
      const ::Triangulation<dim> &tria = dof_handler.get_triangulation();
 
      // ---------------------------- 1. Extract 1d info about the finite
      // element step 1.1: create a 1d copy of the finite element from FETools
      // where we substitute the template argument
      AssertDimension(dof_handler.get_fe().n_base_elements(), 1);
      std::string fe_name = dof_handler.get_fe().base_element(0).get_name();
      {
        const std::size_t template_starts = fe_name.find_first_of('<');
        Assert(fe_name[template_starts + 1] ==
                 (dim == 1 ? '1' : (dim == 2 ? '2' : '3')),
               ExcInternalError());
        fe_name[template_starts + 1] = '1';
      }
      const std::unique_ptr<FiniteElement<1>> fe(
        FETools::get_fe_by_name<1, 1>(fe_name));
 
      setup_element_info(elem_info, *fe, dof_handler);
 
 
      // ---------- 2. Extract and match dof indices between child and parent
      const unsigned int n_levels = tria.n_global_levels();
      level_dof_indices.resize(n_levels);
      parent_child_connect.resize(n_levels - 1);
      n_owned_level_cells.resize(n_levels - 1);
      std::vector<std::vector<unsigned int>> coarse_level_indices(n_levels - 1);
      for (unsigned int level = 0;
           level < std::min(tria.n_levels(), n_levels - 1);
           ++level)
        coarse_level_indices[level].resize(tria.n_raw_cells(level),
                                           numbers::invalid_unsigned_int);
      std::vector<types::global_dof_index> local_dof_indices(
        dof_handler.get_fe().n_dofs_per_cell());
      dirichlet_indices.resize(n_levels - 1);
 
      AssertDimension(target_partitioners.max_level(), n_levels - 1);
      Assert(external_partitioners.empty() ||
               external_partitioners.size() == n_levels,
             ExcDimensionMismatch(external_partitioners.size(), n_levels));
 
      for (unsigned int level = n_levels - 1; level > 0; --level)
        {
          unsigned int                         counter = 0;
          std::vector<types::global_dof_index> global_level_dof_indices;
          std::vector<types::global_dof_index> global_level_dof_indices_remote;
          std::vector<types::global_dof_index> ghosted_level_dofs;
          std::vector<types::global_dof_index> global_level_dof_indices_l0;
          std::vector<types::global_dof_index> ghosted_level_dofs_l0;
 
          // step 2.1: loop over the cells on the coarse side
          typename DoFHandler<dim>::cell_iterator cell,
            endc = dof_handler.end(level - 1);
          for (cell = dof_handler.begin(level - 1); cell != endc; ++cell)
            {
              // need to look into a cell if it has children and it is locally
              // owned
              if (!cell->has_children())
                continue;
 
              bool consider_cell =
                (tria.locally_owned_subdomain() ==
                   numbers::invalid_subdomain_id ||
                 cell->level_subdomain_id() == tria.locally_owned_subdomain());
 
              // due to the particular way we store DoF indices (via children),
              // we also need to add the DoF indices for coarse cells where we
              // own at least one child
              const bool cell_is_remote = !consider_cell;
              for (unsigned int c = 0;
                   c < GeometryInfo<dim>::max_children_per_cell;
                   ++c)
                if (cell->child(c)->level_subdomain_id() ==
                    tria.locally_owned_subdomain())
                  {
                    consider_cell = true;
                    break;
                  }
 
              if (!consider_cell)
                continue;
 
              // step 2.2: loop through children and append the dof indices to
              // the appropriate list. We need separate lists for the owned
              // coarse cell case (which will be part of
              // restriction/prolongation between level-1 and level) and the
              // remote case (which needs to store DoF indices for the
              // operations between level and level+1).
              AssertDimension(cell->n_children(),
                              GeometryInfo<dim>::max_children_per_cell);
              std::vector<types::global_dof_index> &next_indices =
                cell_is_remote ? global_level_dof_indices_remote :
                                 global_level_dof_indices;
              const std::size_t start_index = next_indices.size();
              next_indices.resize(start_index + elem_info.n_child_cell_dofs,
                                  numbers::invalid_dof_index);
              for (unsigned int c = 0;
                   c < GeometryInfo<dim>::max_children_per_cell;
                   ++c)
                {
                  if (cell_is_remote && cell->child(c)->level_subdomain_id() !=
                                          tria.locally_owned_subdomain())
                    continue;
                  cell->child(c)->get_mg_dof_indices(local_dof_indices);
 
                  resolve_identity_constraints(mg_constrained_dofs,
                                               level,
                                               local_dof_indices);
 
                  const IndexSet &owned_level_dofs =
                    dof_handler.locally_owned_mg_dofs(level);
                  for (const auto local_dof_index : local_dof_indices)
                    if (!owned_level_dofs.is_element(local_dof_index))
                      ghosted_level_dofs.push_back(local_dof_index);
 
                  add_child_indices<dim>(c,
                                         fe->n_dofs_per_cell() -
                                           fe->n_dofs_per_vertex(),
                                         fe->degree,
                                         elem_info.lexicographic_numbering,
                                         local_dof_indices,
                                         &next_indices[start_index]);
 
                  // step 2.3 store the connectivity to the parent
                  if (cell->child(c)->has_children() &&
                      (tria.locally_owned_subdomain() ==
                         numbers::invalid_subdomain_id ||
                       cell->child(c)->level_subdomain_id() ==
                         tria.locally_owned_subdomain()))
                    {
                      const unsigned int child_index =
                        coarse_level_indices[level][cell->child(c)->index()];
                      AssertIndexRange(child_index,
                                       parent_child_connect[level].size());
                      unsigned int parent_index = counter;
                      // remote cells, i.e., cells where we work on a further
                      // level but are not treated on the current level, need to
                      // be placed at the end of the list; however, we do not
                      // yet know the exact position in the array, so shift
                      // their parent index by the number of cells so we can set
                      // the correct number after the end of this loop
                      if (cell_is_remote)
                        parent_index =
                          start_index / elem_info.n_child_cell_dofs +
                          tria.n_cells(level);
                      parent_child_connect[level][child_index] =
                        std::make_pair(parent_index, c);
                      AssertIndexRange(dof_handler.get_fe().n_dofs_per_cell(),
                                       static_cast<unsigned short>(-1));
 
                      // set Dirichlet boundary conditions (as a list of
                      // constrained DoFs) for the child
                      if (mg_constrained_dofs != nullptr)
                        for (unsigned int i = 0;
                             i < dof_handler.get_fe().n_dofs_per_cell();
                             ++i)
                          if (mg_constrained_dofs->is_boundary_index(
                                level,
                                local_dof_indices
                                  [elem_info.lexicographic_numbering[i]]))
                            dirichlet_indices[level][child_index].push_back(i);
                    }
                }
              if (!cell_is_remote)
                {
                  AssertIndexRange(static_cast<unsigned int>(cell->index()),
                                   coarse_level_indices[level - 1].size());
                  coarse_level_indices[level - 1][cell->index()] = counter++;
                }
 
              // step 2.4: include indices for the coarsest cells. we still
              // insert the indices as if they were from a child in order to use
              // the same code (the coarsest level does not matter much in terms
              // of memory, so we gain in code simplicity)
              if (level == 1 && !cell_is_remote)
                {
                  cell->get_mg_dof_indices(local_dof_indices);
 
                  resolve_identity_constraints(mg_constrained_dofs,
                                               level - 1,
                                               local_dof_indices);
 
                  const IndexSet &owned_level_dofs_l0 =
                    dof_handler.locally_owned_mg_dofs(0);
                  for (const auto local_dof_index : local_dof_indices)
                    if (!owned_level_dofs_l0.is_element(local_dof_index))
                      ghosted_level_dofs_l0.push_back(local_dof_index);
 
                  const std::size_t start_index =
                    global_level_dof_indices_l0.size();
                  global_level_dof_indices_l0.resize(
                    start_index + elem_info.n_child_cell_dofs,
                    numbers::invalid_dof_index);
                  add_child_indices<dim>(
                    0,
                    fe->n_dofs_per_cell() - fe->n_dofs_per_vertex(),
                    fe->degree,
                    elem_info.lexicographic_numbering,
                    local_dof_indices,
                    &global_level_dof_indices_l0[start_index]);
 
                  dirichlet_indices[0].emplace_back();
                  if (mg_constrained_dofs != nullptr)
                    for (unsigned int i = 0;
                         i < dof_handler.get_fe().n_dofs_per_cell();
                         ++i)
                      if (mg_constrained_dofs->is_boundary_index(
                            0,
                            local_dof_indices[elem_info
                                                .lexicographic_numbering[i]]))
                        dirichlet_indices[0].back().push_back(i);
                }
            }
 
          // step 2.5: store information about the current level and prepare the
          // Dirichlet indices and parent-child relationship for the next
          // coarser level
          AssertDimension(counter * elem_info.n_child_cell_dofs,
                          global_level_dof_indices.size());
          n_owned_level_cells[level - 1] = counter;
          dirichlet_indices[level - 1].resize(counter);
          parent_child_connect[level - 1].resize(
            counter,
            std::make_pair(numbers::invalid_unsigned_int,
                           numbers::invalid_unsigned_int));
 
          // step 2.6: put the cells with remotely owned parent to the end of
          // the list (these are needed for the transfer from level to level+1
          // but not for the transfer from level-1 to level).
          if (level < n_levels - 1)
            for (std::vector<std::pair<unsigned int, unsigned int>>::iterator
                   i = parent_child_connect[level].begin();
                 i != parent_child_connect[level].end();
                 ++i)
              if (i->first >= tria.n_cells(level))
                {
                  i->first -= tria.n_cells(level);
                  i->first += counter;
                }
 
          // step 2.7: Initialize the partitioner for the ghosted vector
          //
          // We use a vector based on the target partitioner handed in also in
          // the base class for keeping ghosted transfer indices. To avoid
          // keeping two very similar vectors, we keep one single ghosted
          // vector that is augmented/filled here.
          reinit_level_partitioner(dof_handler.locally_owned_mg_dofs(level),
                                   ghosted_level_dofs,
                                   external_partitioners.empty() ?
                                     nullptr :
                                     external_partitioners[level],
                                   tria.get_communicator(),
                                   target_partitioners[level],
                                   copy_indices_global_mine[level]);
 
          copy_indices_to_mpi_local_numbers(*target_partitioners[level],
                                            global_level_dof_indices,
                                            global_level_dof_indices_remote,
                                            level_dof_indices[level]);
          // step 2.8: Initialize the ghosted vector for level 0
          if (level == 1)
            {
              for (unsigned int i = 0; i < parent_child_connect[0].size(); ++i)
                parent_child_connect[0][i] = std::make_pair(i, 0U);
 
              reinit_level_partitioner(dof_handler.locally_owned_mg_dofs(0),
                                       ghosted_level_dofs_l0,
                                       external_partitioners.empty() ?
                                         nullptr :
                                         external_partitioners[0],
                                       tria.get_communicator(),
                                       target_partitioners[0],
                                       copy_indices_global_mine[0]);
 
              copy_indices_to_mpi_local_numbers(
                *target_partitioners[0],
                global_level_dof_indices_l0,
                std::vector<types::global_dof_index>(),
                level_dof_indices[0]);
            }
        }
 
      // ---------------------- 3. compute weights to make restriction additive
 
      const unsigned int n_child_dofs_1d =
        fe->degree + 1 + fe->n_dofs_per_cell() - fe->n_dofs_per_vertex();
 
      // get the valence of the individual components and compute the weights as
      // the inverse of the valence
      weights_on_refined.resize(n_levels - 1);
      for (unsigned int level = 1; level < n_levels; ++level)
        {
          LinearAlgebra::distributed::Vector<Number> touch_count(
            target_partitioners[level]);
          for (unsigned int c = 0; c < n_owned_level_cells[level - 1]; ++c)
            for (unsigned int j = 0; j < elem_info.n_child_cell_dofs; ++j)
              touch_count.local_element(
                level_dof_indices[level][elem_info.n_child_cell_dofs * c +
                                         j]) += Number(1.);
          touch_count.compress(VectorOperation::add);
          touch_count.update_ghost_values();
 
          std::vector<unsigned int> degree_to_3(n_child_dofs_1d);
          degree_to_3[0] = 0;
          for (unsigned int i = 1; i < n_child_dofs_1d - 1; ++i)
            degree_to_3[i] = 1;
          degree_to_3.back() = 2;
 
          // we only store 3^dim weights because all dofs on a line have the
          // same valence, and all dofs on a quad have the same valence.
          weights_on_refined[level - 1].resize(n_owned_level_cells[level - 1] *
                                               Utilities::fixed_power<dim>(3));
          for (unsigned int c = 0; c < n_owned_level_cells[level - 1]; ++c)
            for (unsigned int k = 0, m = 0; k < (dim > 2 ? n_child_dofs_1d : 1);
                 ++k)
              for (unsigned int j = 0; j < (dim > 1 ? n_child_dofs_1d : 1); ++j)
                {
                  unsigned int shift = 9 * degree_to_3[k] + 3 * degree_to_3[j];
                  for (unsigned int i = 0; i < n_child_dofs_1d; ++i, ++m)
                    weights_on_refined[level -
                                       1][c * Utilities::fixed_power<dim>(3) +
                                          shift + degree_to_3[i]] =
                      Number(1.) /
                      touch_count.local_element(
                        level_dof_indices[level]
                                         [elem_info.n_child_cell_dofs * c + m]);
                }
        }
    }
 
 
 
    void
    resolve_identity_constraints(
      const MGConstrainedDoFs *             mg_constrained_dofs,
      const unsigned int                    level,
      std::vector<types::global_dof_index> &dof_indices)
    {
      if (mg_constrained_dofs != nullptr &&
          mg_constrained_dofs->get_level_constraints(level).n_constraints() > 0)
        for (auto &ind : dof_indices)
          if (mg_constrained_dofs->get_level_constraints(level)
                .is_identity_constrained(ind))
            {
              Assert(mg_constrained_dofs->get_level_constraints(level)
                         .get_constraint_entries(ind)
                         ->size() == 1,
                     ExcInternalError());
              ind = mg_constrained_dofs->get_level_constraints(level)
                      .get_constraint_entries(ind)
                      ->front()
                      .first;
            }
    }
 
  } // namespace MGTransfer
} // namespace internal
 
// Explicit instantiations
 
#include "mg_transfer_internal.inst"
 
DEAL_II_NAMESPACE_CLOSE