dof_map.C

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// The libMesh Finite Element Library.
// Copyright (C) 2002-2018 Benjamin S. Kirk, John W. Peterson, Roy H. Stogner

// This library is free software; you can 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.

// This library is distributed in the hope that it will be useful,
// but WITHOUT ANY WARRANTY; without even the implied warranty of
// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
// Lesser General Public  License for more details.

// You should have received a copy of the GNU Lesser General Public
// License along with this library; if not, write to the Free Software
// Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA



// Local Includes
#include "libmesh/coupling_matrix.h"
#include "libmesh/default_coupling.h"
#include "libmesh/dense_matrix.h"
#include "libmesh/dense_vector_base.h"
#include "libmesh/dirichlet_boundaries.h"
#include "libmesh/dof_map.h"
#include "libmesh/elem.h"
#include "libmesh/fe_interface.h"
#include "libmesh/fe_type.h"
#include "libmesh/fe_base.h" // FEBase::build() for continuity test
#include "libmesh/ghosting_functor.h"
#include "libmesh/libmesh_logging.h"
#include "libmesh/mesh_base.h"
#include "libmesh/mesh_tools.h"
#include "libmesh/numeric_vector.h"
#include "libmesh/parallel.h"
#include "libmesh/parallel_sync.h"
#include "libmesh/periodic_boundaries.h"
#include "libmesh/sparse_matrix.h"
#include "libmesh/sparsity_pattern.h"
#include "libmesh/string_to_enum.h"
#include "libmesh/threads.h"
#include "libmesh/mesh_subdivision_support.h"
#include "libmesh/int_range.h"

// C++ Includes
#include <set>
#include <algorithm> // for std::fill, std::equal_range, std::max, std::lower_bound, etc.
#include <sstream>
#include <unordered_map>

namespace libMesh
{

// ------------------------------------------------------------
// DofMap member functions
std::unique_ptr<SparsityPattern::Build>
DofMap::build_sparsity (const MeshBase & mesh) const
{
  libmesh_assert (mesh.is_prepared());

  LOG_SCOPE("build_sparsity()", "DofMap");

  // Compute the sparsity structure of the global matrix.  This can be
  // fed into a PetscMatrix to allocate exactly the number of nonzeros
  // necessary to store the matrix.  This algorithm should be linear
  // in the (# of elements)*(# nodes per element)

  // We can be more efficient in the threaded sparsity pattern assembly
  // if we don't need the exact pattern.  For some sparse matrix formats
  // a good upper bound will suffice.

  // See if we need to include sparsity pattern entries for coupling
  // between neighbor dofs
  bool implicit_neighbor_dofs = this->use_coupled_neighbor_dofs(mesh);

  // We can compute the sparsity pattern in parallel on multiple
  // threads.  The goal is for each thread to compute the full sparsity
  // pattern for a subset of elements.  These sparsity patterns can
  // be efficiently merged in the SparsityPattern::Build::join()
  // method, especially if there is not too much overlap between them.
  // Even better, if the full sparsity pattern is not needed then
  // the number of nonzeros per row can be estimated from the
  // sparsity patterns created on each thread.
  std::unique_ptr<SparsityPattern::Build> sp
    (new SparsityPattern::Build (mesh,
                                 *this,
                                 this->_dof_coupling,
                                 this->_coupling_functors,
                                 implicit_neighbor_dofs,
                                 need_full_sparsity_pattern));

  Threads::parallel_reduce (ConstElemRange (mesh.active_local_elements_begin(),
                                            mesh.active_local_elements_end()), *sp);

  sp->parallel_sync();

#ifndef NDEBUG
  // Avoid declaring these variables unless asserts are enabled.
  const processor_id_type proc_id        = mesh.processor_id();
  const dof_id_type n_dofs_on_proc = this->n_dofs_on_processor(proc_id);
#endif
  libmesh_assert_equal_to (sp->sparsity_pattern.size(), n_dofs_on_proc);

  // Check to see if we have any extra stuff to add to the sparsity_pattern
  if (_extra_sparsity_function)
    {
      if (_augment_sparsity_pattern)
        {
          libmesh_here();
          libMesh::out << "WARNING:  You have specified both an extra sparsity function and object.\n"
                       << "          Are you sure this is what you meant to do??"
                       << std::endl;
        }

      _extra_sparsity_function
        (sp->sparsity_pattern, sp->n_nz,
         sp->n_oz, _extra_sparsity_context);
    }

  if (_augment_sparsity_pattern)
    _augment_sparsity_pattern->augment_sparsity_pattern
      (sp->sparsity_pattern, sp->n_nz, sp->n_oz);

  return std::unique_ptr<SparsityPattern::Build>(sp.release());
}



DofMap::DofMap(const unsigned int number,
               MeshBase & mesh) :
  ParallelObject (mesh.comm()),
  _dof_coupling(nullptr),
  _error_on_cyclic_constraint(false),
  _variables(),
  _variable_groups(),
  _variable_group_numbers(),
  _sys_number(number),
  _mesh(mesh),
  _matrices(),
  _first_df(),
  _end_df(),
  _first_scalar_df(),
  _send_list(),
  _augment_sparsity_pattern(nullptr),
  _extra_sparsity_function(nullptr),
  _extra_sparsity_context(nullptr),
  _augment_send_list(nullptr),
  _extra_send_list_function(nullptr),
  _extra_send_list_context(nullptr),
  _default_coupling(new DefaultCoupling()),
  _default_evaluating(new DefaultCoupling()),
  need_full_sparsity_pattern(false),
  _n_nz(nullptr),
  _n_oz(nullptr),
  _n_dfs(0),
  _n_SCALAR_dofs(0)
#ifdef LIBMESH_ENABLE_AMR
  , _n_old_dfs(0),
  _first_old_df(),
  _end_old_df(),
  _first_old_scalar_df()
#endif
#ifdef LIBMESH_ENABLE_CONSTRAINTS
  , _dof_constraints()
  , _stashed_dof_constraints()
  , _primal_constraint_values()
  , _adjoint_constraint_values()
#endif
#ifdef LIBMESH_ENABLE_NODE_CONSTRAINTS
  , _node_constraints()
#endif
#ifdef LIBMESH_ENABLE_PERIODIC
  , _periodic_boundaries(new PeriodicBoundaries)
#endif
#ifdef LIBMESH_ENABLE_DIRICHLET
  , _dirichlet_boundaries(new DirichletBoundaries)
  , _adjoint_dirichlet_boundaries()
#endif
  , _implicit_neighbor_dofs_initialized(false),
  _implicit_neighbor_dofs(false)
{
  _matrices.clear();

  _default_coupling->set_mesh(&_mesh);
  _default_evaluating->set_mesh(&_mesh);
  _default_evaluating->set_n_levels(1);

#ifdef LIBMESH_ENABLE_PERIODIC
  _default_coupling->set_periodic_boundaries(_periodic_boundaries.get());
  _default_evaluating->set_periodic_boundaries(_periodic_boundaries.get());
#endif

  this->add_coupling_functor(*_default_coupling);
  this->add_algebraic_ghosting_functor(*_default_evaluating);
}



// Destructor
DofMap::~DofMap()
{
  this->clear();

  // clear() resets all but the default DofMap-based functors.  We
  // need to remove those from the mesh too before we die.
  _mesh.remove_ghosting_functor(*_default_coupling);
  _mesh.remove_ghosting_functor(*_default_evaluating);

#ifdef LIBMESH_ENABLE_DIRICHLET
  for (auto & bnd : _adjoint_dirichlet_boundaries)
    delete bnd;
#endif
}


#ifdef LIBMESH_ENABLE_PERIODIC

bool DofMap::is_periodic_boundary (const boundary_id_type boundaryid) const
{
  if (_periodic_boundaries->count(boundaryid) != 0)
    return true;

  return false;
}

#endif



// void DofMap::add_variable (const Variable & var)
// {
//   libmesh_not_implemented();
//   _variables.push_back (var);
// }


void DofMap::set_error_on_cyclic_constraint(bool error_on_cyclic_constraint)
{
  _error_on_cyclic_constraint = error_on_cyclic_constraint;
}



void DofMap::add_variable_group (const VariableGroup & var_group)
{
  const unsigned int vg = cast_int<unsigned int>(_variable_groups.size());

  _variable_groups.push_back(var_group);

  VariableGroup & new_var_group = _variable_groups.back();

  for (unsigned int var=0; var<new_var_group.n_variables(); var++)
    {
      _variables.push_back (new_var_group(var));
      _variable_group_numbers.push_back (vg);
    }
}



void DofMap::attach_matrix (SparseMatrix<Number> & matrix)
{
  parallel_object_only();

  // We shouldn't be trying to re-attach the same matrices repeatedly
  libmesh_assert (std::find(_matrices.begin(), _matrices.end(),
                            &matrix) == _matrices.end());

  _matrices.push_back(&matrix);

  matrix.attach_dof_map (*this);

  // If we've already computed sparsity, then it's too late
  // to wait for "compute_sparsity" to help with sparse matrix
  // initialization, and we need to handle this matrix individually
  bool computed_sparsity_already =
    ((_n_nz && !_n_nz->empty()) ||
     (_n_oz && !_n_oz->empty()));
  this->comm().max(computed_sparsity_already);
  if (computed_sparsity_already &&
      matrix.need_full_sparsity_pattern())
    {
      // We'd better have already computed the full sparsity pattern
      // if we need it here
      libmesh_assert(need_full_sparsity_pattern);
      libmesh_assert(_sp.get());

      matrix.update_sparsity_pattern (_sp->sparsity_pattern);
    }

  if (matrix.need_full_sparsity_pattern())
    need_full_sparsity_pattern = true;
}



bool DofMap::is_attached (SparseMatrix<Number> & matrix)
{
  return (std::find(_matrices.begin(), _matrices.end(),
                    &matrix) != _matrices.end());
}



DofObject * DofMap::node_ptr(MeshBase & mesh, dof_id_type i) const
{
  return mesh.node_ptr(i);
}



DofObject * DofMap::elem_ptr(MeshBase & mesh, dof_id_type i) const
{
  return mesh.elem_ptr(i);
}



template <typename iterator_type>
void DofMap::set_nonlocal_dof_objects(iterator_type objects_begin,
                                      iterator_type objects_end,
                                      MeshBase & mesh,
                                      dofobject_accessor objects)
{
  // This function must be run on all processors at once
  parallel_object_only();

  // First, iterate over local objects to find out how many
  // are on each processor
  std::unordered_map<processor_id_type, dof_id_type> ghost_objects_from_proc;

  iterator_type it  = objects_begin;

  for (; it != objects_end; ++it)
    {
      DofObject * obj = *it;

      if (obj)
        {
          processor_id_type obj_procid = obj->processor_id();
          // We'd better be completely partitioned by now
          libmesh_assert_not_equal_to (obj_procid, DofObject::invalid_processor_id);
          ghost_objects_from_proc[obj_procid]++;
        }
    }

  // Request sets to send to each processor
  std::map<processor_id_type, std::vector<dof_id_type>>
    requested_ids;

  // We know how many of our objects live on each processor, so
  // reserve() space for requests from each.
  auto ghost_end = ghost_objects_from_proc.end();
  for (processor_id_type p=0, np = this->n_processors(); p != np; ++p)
    if (p != this->processor_id())
      {
        auto ghost_it = ghost_objects_from_proc.find(p);
        if (ghost_it != ghost_end)
          requested_ids[p].reserve(ghost_it->second);
      }

  for (it = objects_begin; it != objects_end; ++it)
    {
      DofObject * obj = *it;
      if (obj->processor_id() != DofObject::invalid_processor_id)
        requested_ids[obj->processor_id()].push_back(obj->id());
    }
#ifdef DEBUG
  for (processor_id_type p=0, np = this->n_processors(); p != np; ++p)
    {
      if (ghost_objects_from_proc.count(p))
        libmesh_assert_equal_to (requested_ids[p].size(), ghost_objects_from_proc[p]);
      else
        libmesh_assert(!requested_ids.count(p));
    }
#endif

  typedef std::vector<dof_id_type> datum;

  auto gather_functor =
    [this, &mesh, &objects]
    (processor_id_type,
     const std::vector<dof_id_type> & ids,
     std::vector<datum> & data)
    {
      // Fill those requests
      const unsigned int
        sys_num      = this->sys_number(),
        n_var_groups = this->n_variable_groups();

      const std::size_t query_size = ids.size();

      data.resize(query_size);
      for (auto & d : data)
        d.resize(2 * n_var_groups);

      for (std::size_t i=0; i != query_size; ++i)
        {
          DofObject * requested = (this->*objects)(mesh, ids[i]);
          libmesh_assert(requested);
          libmesh_assert_equal_to (requested->processor_id(), this->processor_id());
          libmesh_assert_equal_to (requested->n_var_groups(sys_num), n_var_groups);
          for (unsigned int vg=0; vg != n_var_groups; ++vg)
            {
              unsigned int n_comp_g =
                requested->n_comp_group(sys_num, vg);
              data[i][vg] = n_comp_g;
              dof_id_type my_first_dof = n_comp_g ?
                requested->vg_dof_base(sys_num, vg) : 0;
              libmesh_assert_not_equal_to (my_first_dof, DofObject::invalid_id);
              data[i][n_var_groups+vg] = my_first_dof;
            }
        }
    };

  auto action_functor =
    [this, &mesh, &objects]
    (processor_id_type libmesh_dbg_var(pid),
     const std::vector<dof_id_type> & ids,
     const std::vector<datum> & data)
    {
      const unsigned int
        sys_num      = this->sys_number(),
        n_var_groups = this->n_variable_groups();

      // Copy the id changes we've now been informed of
      for (auto i : index_range(ids))
        {
          DofObject * requested = (this->*objects)(mesh, ids[i]);
          libmesh_assert(requested);
          libmesh_assert_equal_to (requested->processor_id(), pid);
          for (unsigned int vg=0; vg != n_var_groups; ++vg)
            {
              unsigned int n_comp_g =
                cast_int<unsigned int>(data[i][vg]);
              requested->set_n_comp_group(sys_num, vg, n_comp_g);
              if (n_comp_g)
                {
                  dof_id_type my_first_dof = data[i][n_var_groups+vg];
                  libmesh_assert_not_equal_to (my_first_dof, DofObject::invalid_id);
                  requested->set_vg_dof_base
                    (sys_num, vg, my_first_dof);
                }
            }
        }
    };

  datum * ex = nullptr;
  Parallel::pull_parallel_vector_data
    (this->comm(), requested_ids, gather_functor, action_functor, ex);

#ifdef DEBUG
  // Double check for invalid dofs
  for (it = objects_begin; it != objects_end; ++it)
    {
      DofObject * obj = *it;
      libmesh_assert (obj);
      unsigned int num_variables = obj->n_vars(this->sys_number());
      for (unsigned int v=0; v != num_variables; ++v)
        {
          unsigned int n_comp =
            obj->n_comp(this->sys_number(), v);
          dof_id_type my_first_dof = n_comp ?
            obj->dof_number(this->sys_number(), v, 0) : 0;
          libmesh_assert_not_equal_to (my_first_dof, DofObject::invalid_id);
        }
    }
#endif
}



void DofMap::reinit(MeshBase & mesh)
{
  libmesh_assert (mesh.is_prepared());

  LOG_SCOPE("reinit()", "DofMap");

  // We ought to reconfigure our default coupling functor.
  //
  // The user might have removed it from our coupling functors set,
  // but if so, who cares, this reconfiguration is cheap.

  // Avoid calling set_dof_coupling() with an empty/non-nullptr
  // _dof_coupling matrix which may happen when there are actually no
  // variables on the system.
  if (this->_dof_coupling && this->_dof_coupling->empty() && !this->n_variables())
    this->_dof_coupling = nullptr;
  _default_coupling->set_dof_coupling(this->_dof_coupling);

  // By default we may want 0 or 1 levels of coupling
  unsigned int standard_n_levels =
    this->use_coupled_neighbor_dofs(mesh);
  _default_coupling->set_n_levels
    (std::max(_default_coupling->n_levels(), standard_n_levels));

  // But we *don't* want to restrict to a CouplingMatrix unless the
  // user does so manually; the original libMesh behavior was to put
  // ghost indices on the send_list regardless of variable.
  //_default_evaluating->set_dof_coupling(this->_dof_coupling);

  const unsigned int
    sys_num      = this->sys_number(),
    n_var_groups = this->n_variable_groups();

  // The DofObjects need to know how many variable groups we have, and
  // how many variables there are in each group.
  std::vector<unsigned int> n_vars_per_group;  n_vars_per_group.reserve (n_var_groups);

  for (unsigned int vg=0; vg<n_var_groups; vg++)
    n_vars_per_group.push_back (this->variable_group(vg).n_variables());

#ifdef LIBMESH_ENABLE_AMR

  //------------------------------------------------------------
  // Clear the old_dof_objects for all the nodes
  // and elements so that we can overwrite them
  for (auto & node : mesh.node_ptr_range())
    {
      node->clear_old_dof_object();
      libmesh_assert (!node->old_dof_object);
    }

  for (auto & elem : mesh.element_ptr_range())
    {
      elem->clear_old_dof_object();
      libmesh_assert (!elem->old_dof_object);
    }


  //------------------------------------------------------------
  // Set the old_dof_objects for the elements that
  // weren't just created, if these old dof objects
  // had variables
  for (auto & elem : mesh.element_ptr_range())
    {
      // Skip the elements that were just refined
      if (elem->refinement_flag() == Elem::JUST_REFINED)
        continue;

      for (unsigned int n=0; n<elem->n_nodes(); n++)
        {
          Node & node = elem->node_ref(n);

          if (node.old_dof_object == nullptr)
            if (node.has_dofs(sys_num))
              node.set_old_dof_object();
        }

      libmesh_assert (!elem->old_dof_object);

      if (elem->has_dofs(sys_num))
        elem->set_old_dof_object();
    }

#endif // #ifdef LIBMESH_ENABLE_AMR


  //------------------------------------------------------------
  // Then set the number of variables for each \p DofObject
  // equal to n_variables() for this system.  This will
  // handle new \p DofObjects that may have just been created

  // All the nodes
  for (auto & node : mesh.node_ptr_range())
    node->set_n_vars_per_group(sys_num, n_vars_per_group);

  // All the elements
  for (auto & elem : mesh.element_ptr_range())
    elem->set_n_vars_per_group(sys_num, n_vars_per_group);

  // Zero _n_SCALAR_dofs, it will be updated below.
  this->_n_SCALAR_dofs = 0;

  //------------------------------------------------------------
  // Next allocate space for the DOF indices
  for (unsigned int vg=0; vg<n_var_groups; vg++)
    {
      const VariableGroup & vg_description = this->variable_group(vg);

      const unsigned int n_var_in_group = vg_description.n_variables();
      const FEType & base_fe_type        = vg_description.type();

      // Don't need to loop over elements for a SCALAR variable
      // Just increment _n_SCALAR_dofs
      if (base_fe_type.family == SCALAR)
        {
          this->_n_SCALAR_dofs += base_fe_type.order.get_order()*n_var_in_group;
          continue;
        }

      // This should be constant even on p-refined elements
      const bool extra_hanging_dofs =
        FEInterface::extra_hanging_dofs(base_fe_type);

      // For all the active elements, count vertex degrees of freedom.
      for (auto & elem : mesh.active_element_ptr_range())
        {
          libmesh_assert(elem);

          // Skip the numbering if this variable is
          // not active on this element's subdomain
          if (!vg_description.active_on_subdomain(elem->subdomain_id()))
            continue;

          const ElemType type = elem->type();
          const unsigned int dim = elem->dim();

          FEType fe_type = base_fe_type;

#ifdef LIBMESH_ENABLE_AMR
          // Make sure we haven't done more p refinement than we can
          // handle
          if (elem->p_level() + base_fe_type.order >
              FEInterface::max_order(base_fe_type, type))
            {
              libmesh_assert_less_msg(static_cast<unsigned int>(base_fe_type.order.get_order()),
                                      FEInterface::max_order(base_fe_type,type),
                                      "ERROR: Finite element "
                                      << Utility::enum_to_string(base_fe_type.family)
                                      << " on geometric element "
                                      << Utility::enum_to_string(type)
                                      << "\nonly supports FEInterface::max_order = "
                                      << FEInterface::max_order(base_fe_type,type)
                                      << ", not fe_type.order = "
                                      << base_fe_type.order);

#  ifdef DEBUG
              libMesh::err << "WARNING: Finite element "
                           << Utility::enum_to_string(base_fe_type.family)
                           << " on geometric element "
                           << Utility::enum_to_string(type) << std::endl
                           << "could not be p refined past FEInterface::max_order = "
                           << FEInterface::max_order(base_fe_type,type)
                           << std::endl;
#  endif
              elem->set_p_level(FEInterface::max_order(base_fe_type,type)
                                - base_fe_type.order);
            }
#endif

          fe_type.order = static_cast<Order>(fe_type.order +
                                             elem->p_level());

          // Allocate the vertex DOFs
          for (unsigned int n=0; n<elem->n_nodes(); n++)
            {
              Node & node = elem->node_ref(n);

              if (elem->is_vertex(n))
                {
                  const unsigned int old_node_dofs =
                    node.n_comp_group(sys_num, vg);

                  const unsigned int vertex_dofs =
                    std::max(FEInterface::n_dofs_at_node(dim, fe_type,
                                                         type, n),
                             old_node_dofs);

                  // Some discontinuous FEs have no vertex dofs
                  if (vertex_dofs > old_node_dofs)
                    {
                      node.set_n_comp_group(sys_num, vg,
                                            vertex_dofs);

                      // Abusing dof_number to set a "this is a
                      // vertex" flag
                      node.set_vg_dof_base(sys_num, vg,
                                           vertex_dofs);

                      // libMesh::out << "sys_num,vg,old_node_dofs,vertex_dofs="
                      //       << sys_num << ","
                      //       << vg << ","
                      //       << old_node_dofs << ","
                      //       << vertex_dofs << '\n',
                      // node.debug_buffer();

                      // libmesh_assert_equal_to (vertex_dofs, node.n_comp(sys_num, vg));
                      // libmesh_assert_equal_to (vertex_dofs, node.vg_dof_base(sys_num, vg));
                    }
                }
            }
        } // done counting vertex dofs

      // count edge & face dofs next
      for (auto & elem : mesh.active_element_ptr_range())
        {
          libmesh_assert(elem);

          // Skip the numbering if this variable is
          // not active on this element's subdomain
          if (!vg_description.active_on_subdomain(elem->subdomain_id()))
            continue;

          const ElemType type = elem->type();
          const unsigned int dim = elem->dim();

          FEType fe_type = base_fe_type;
          fe_type.order = static_cast<Order>(fe_type.order +
                                             elem->p_level());

          // Allocate the edge and face DOFs
          for (unsigned int n=0; n<elem->n_nodes(); n++)
            {
              Node & node = elem->node_ref(n);

              const unsigned int old_node_dofs =
                node.n_comp_group(sys_num, vg);

              const unsigned int vertex_dofs = old_node_dofs?
                cast_int<unsigned int>(node.vg_dof_base (sys_num,vg)):0;

              const unsigned int new_node_dofs =
                FEInterface::n_dofs_at_node(dim, fe_type, type, n);

              // We've already allocated vertex DOFs
              if (elem->is_vertex(n))
                {
                  libmesh_assert_greater_equal (old_node_dofs, vertex_dofs);
                  // //if (vertex_dofs < new_node_dofs)
                  //   libMesh::out << "sys_num,vg,old_node_dofs,vertex_dofs,new_node_dofs="
                  //                << sys_num << ","
                  //                << vg << ","
                  //                << old_node_dofs << ","
                  //                << vertex_dofs << ","
                  //                << new_node_dofs << '\n',
                  //     node.debug_buffer();

                  libmesh_assert_greater_equal (vertex_dofs,   new_node_dofs);
                }
              // We need to allocate the rest
              else
                {
                  // If this has no dofs yet, it needs no vertex
                  // dofs, so we just give it edge or face dofs
                  if (!old_node_dofs)
                    {
                      node.set_n_comp_group(sys_num, vg,
                                            new_node_dofs);
                      // Abusing dof_number to set a "this has no
                      // vertex dofs" flag
                      if (new_node_dofs)
                        node.set_vg_dof_base(sys_num, vg, 0);
                    }

                  // If this has dofs, but has no vertex dofs,
                  // it may still need more edge or face dofs if
                  // we're p-refined.
                  else if (vertex_dofs == 0)
                    {
                      if (new_node_dofs > old_node_dofs)
                        {
                          node.set_n_comp_group(sys_num, vg,
                                                new_node_dofs);

                          node.set_vg_dof_base(sys_num, vg,
                                               vertex_dofs);
                        }
                    }
                  // If this is another element's vertex,
                  // add more (non-overlapping) edge/face dofs if
                  // necessary
                  else if (extra_hanging_dofs)
                    {
                      if (new_node_dofs > old_node_dofs - vertex_dofs)
                        {
                          node.set_n_comp_group(sys_num, vg,
                                                vertex_dofs + new_node_dofs);

                          node.set_vg_dof_base(sys_num, vg,
                                               vertex_dofs);
                        }
                    }
                  // If this is another element's vertex, add any
                  // (overlapping) edge/face dofs if necessary
                  else
                    {
                      libmesh_assert_greater_equal (old_node_dofs, vertex_dofs);
                      if (new_node_dofs > old_node_dofs)
                        {
                          node.set_n_comp_group(sys_num, vg,
                                                new_node_dofs);

                          node.set_vg_dof_base (sys_num, vg,
                                                vertex_dofs);
                        }
                    }
                }
            }
          // Allocate the element DOFs
          const unsigned int dofs_per_elem =
            FEInterface::n_dofs_per_elem(dim, fe_type,
                                         type);

          elem->set_n_comp_group(sys_num, vg, dofs_per_elem);

        }
    } // end loop over variable groups

  // Calling DofMap::reinit() by itself makes little sense,
  // so we won't bother with nonlocal DofObjects.
  // Those will be fixed by distribute_dofs

  //------------------------------------------------------------
  // Finally, clear all the current DOF indices
  // (distribute_dofs expects them cleared!)
  this->invalidate_dofs(mesh);
}



void DofMap::invalidate_dofs(MeshBase & mesh) const
{
  const unsigned int sys_num = this->sys_number();

  // All the nodes
  for (auto & node : mesh.node_ptr_range())
    node->invalidate_dofs(sys_num);

  // All the active elements.
  for (auto & elem : mesh.active_element_ptr_range())
    elem->invalidate_dofs(sys_num);
}



void DofMap::clear()
{
  // we don't want to clear
  // the coupling matrix!
  // It should not change...
  //_dof_coupling->clear();
  //
  // But it would be inconsistent to leave our coupling settings
  // through a clear()...
  _dof_coupling = nullptr;

  // Reset ghosting functor statuses
  {
    for (const auto & gf : _coupling_functors)
      {
        libmesh_assert(gf);
        _mesh.remove_ghosting_functor(*gf);
      }
    this->_coupling_functors.clear();

    // Go back to default coupling

    _default_coupling->set_dof_coupling(this->_dof_coupling);
    _default_coupling->set_n_levels(this->use_coupled_neighbor_dofs(this->_mesh));

    this->add_coupling_functor(*_default_coupling);
  }


  {
    for (const auto & gf : _algebraic_ghosting_functors)
      {
        libmesh_assert(gf);
        _mesh.remove_ghosting_functor(*gf);
      }
    this->_algebraic_ghosting_functors.clear();

    // Go back to default send_list generation

    // _default_evaluating->set_dof_coupling(this->_dof_coupling);
    _default_evaluating->set_n_levels(1);
    this->add_algebraic_ghosting_functor(*_default_evaluating);
  }

  _variables.clear();
  _variable_groups.clear();
  _variable_group_numbers.clear();
  _first_df.clear();
  _end_df.clear();
  _first_scalar_df.clear();
  _send_list.clear();
  this->clear_sparsity();
  need_full_sparsity_pattern = false;

#ifdef LIBMESH_ENABLE_AMR

  _dof_constraints.clear();
  _stashed_dof_constraints.clear();
  _primal_constraint_values.clear();
  _adjoint_constraint_values.clear();
  _n_old_dfs = 0;
  _first_old_df.clear();
  _end_old_df.clear();
  _first_old_scalar_df.clear();

#endif

  _matrices.clear();

  _n_dfs = 0;
}



void DofMap::distribute_dofs (MeshBase & mesh)
{
  // This function must be run on all processors at once
  parallel_object_only();

  // Log how long it takes to distribute the degrees of freedom
  LOG_SCOPE("distribute_dofs()", "DofMap");

  libmesh_assert (mesh.is_prepared());

  const processor_id_type proc_id = this->processor_id();
  const processor_id_type n_proc  = this->n_processors();

  //  libmesh_assert_greater (this->n_variables(), 0);
  libmesh_assert_less (proc_id, n_proc);

  // re-init in case the mesh has changed
  this->reinit(mesh);

  // By default distribute variables in a
  // var-major fashion, but allow run-time
  // specification
  bool node_major_dofs = libMesh::on_command_line ("--node-major-dofs");

  // The DOF counter, will be incremented as we encounter
  // new degrees of freedom
  dof_id_type next_free_dof = 0;

  // Clear the send list before we rebuild it
  _send_list.clear();

  // Set temporary DOF indices on this processor
  if (node_major_dofs)
    this->distribute_local_dofs_node_major (next_free_dof, mesh);
  else
    this->distribute_local_dofs_var_major (next_free_dof, mesh);

  // Get DOF counts on all processors
  std::vector<dof_id_type> dofs_on_proc(n_proc, 0);
  this->comm().allgather(next_free_dof, dofs_on_proc);

  // Resize and fill the _first_df and _end_df arrays
#ifdef LIBMESH_ENABLE_AMR
  _first_old_df = _first_df;
  _end_old_df = _end_df;
#endif

  _first_df.resize(n_proc);
  _end_df.resize (n_proc);

  // Get DOF offsets
  _first_df[0] = 0;
  for (processor_id_type i=1; i < n_proc; ++i)
    _first_df[i] = _end_df[i-1] = _first_df[i-1] + dofs_on_proc[i-1];
  _end_df[n_proc-1] = _first_df[n_proc-1] + dofs_on_proc[n_proc-1];

  // Clear all the current DOF indices
  // (distribute_dofs expects them cleared!)
  this->invalidate_dofs(mesh);

  next_free_dof = _first_df[proc_id];

  // Set permanent DOF indices on this processor
  if (node_major_dofs)
    this->distribute_local_dofs_node_major (next_free_dof, mesh);
  else
    this->distribute_local_dofs_var_major (next_free_dof, mesh);

  libmesh_assert_equal_to (next_free_dof, _end_df[proc_id]);

  //------------------------------------------------------------
  // At this point, all n_comp and dof_number values on local
  // DofObjects should be correct, but a DistributedMesh might have
  // incorrect values on non-local DofObjects.  Let's request the
  // correct values from each other processor.

  if (this->n_processors() > 1)
    {
      this->set_nonlocal_dof_objects(mesh.nodes_begin(),
                                     mesh.nodes_end(),
                                     mesh, &DofMap::node_ptr);

      this->set_nonlocal_dof_objects(mesh.elements_begin(),
                                     mesh.elements_end(),
                                     mesh, &DofMap::elem_ptr);
    }

#ifdef DEBUG
  {
    const unsigned int
      sys_num = this->sys_number();

    // Processors should all agree on DoF ids for the newly numbered
    // system.
    MeshTools::libmesh_assert_valid_dof_ids(mesh, sys_num);

    // DoF processor ids should match DofObject processor ids
    for (auto & node : mesh.node_ptr_range())
      {
        DofObject const * const dofobj = node;
        const processor_id_type obj_proc_id = dofobj->processor_id();

        for (unsigned int v=0; v != dofobj->n_vars(sys_num); ++v)
          for (unsigned int c=0; c != dofobj->n_comp(sys_num,v); ++c)
            {
              const dof_id_type dofid = dofobj->dof_number(sys_num,v,c);
              libmesh_assert_greater_equal (dofid, this->first_dof(obj_proc_id));
              libmesh_assert_less (dofid, this->end_dof(obj_proc_id));
            }
      }

    for (auto & elem : mesh.element_ptr_range())
      {
        DofObject const * const dofobj = elem;
        const processor_id_type obj_proc_id = dofobj->processor_id();

        for (unsigned int v=0; v != dofobj->n_vars(sys_num); ++v)
          for (unsigned int c=0; c != dofobj->n_comp(sys_num,v); ++c)
            {
              const dof_id_type dofid = dofobj->dof_number(sys_num,v,c);
              libmesh_assert_greater_equal (dofid, this->first_dof(obj_proc_id));
              libmesh_assert_less (dofid, this->end_dof(obj_proc_id));
            }
      }
  }
#endif

  // Set the total number of degrees of freedom, then start finding
  // SCALAR degrees of freedom
#ifdef LIBMESH_ENABLE_AMR
  _n_old_dfs = _n_dfs;
  _first_old_scalar_df = _first_scalar_df;
#endif
  _n_dfs = _end_df[n_proc-1];
  _first_scalar_df.clear();
  _first_scalar_df.resize(this->n_variables(), DofObject::invalid_id);
  dof_id_type current_SCALAR_dof_index = n_dofs() - n_SCALAR_dofs();

  // Calculate and cache the initial DoF indices for SCALAR variables.
  // This is an O(N_vars) calculation so we want to do it once per
  // renumbering rather than once per SCALAR_dof_indices() call

  for (unsigned int v=0; v<this->n_variables(); v++)
    if (this->variable(v).type().family == SCALAR)
      {
        _first_scalar_df[v] = current_SCALAR_dof_index;
        current_SCALAR_dof_index += this->variable(v).type().order.get_order();
      }

  // Allow our GhostingFunctor objects to reinit if necessary
  for (const auto & gf : _algebraic_ghosting_functors)
    {
      libmesh_assert(gf);
      gf->dofmap_reinit();
    }

  for (const auto & gf : _coupling_functors)
    {
      libmesh_assert(gf);
      gf->dofmap_reinit();
    }

  // Note that in the add_neighbors_to_send_list nodes on processor
  // boundaries that are shared by multiple elements are added for
  // each element.
  this->add_neighbors_to_send_list(mesh);

  // Here we used to clean up that data structure; now System and
  // EquationSystems call that for us, after we've added constraint
  // dependencies to the send_list too.
  // this->sort_send_list ();
}


void DofMap::local_variable_indices(std::vector<dof_id_type> & idx,
                                    const MeshBase & mesh,
                                    unsigned int var_num) const
{
  // Count dofs in the *exact* order that distribute_dofs numbered
  // them, so that we can assume ascending indices and use push_back
  // instead of find+insert.

  const unsigned int sys_num       = this->sys_number();

  // If this isn't a SCALAR variable, we need to find all its field
  // dofs on the mesh
  if (this->variable_type(var_num).family != SCALAR)
    {
      const Variable & var(this->variable(var_num));

      for (auto & elem : mesh.active_local_element_ptr_range())
        {
          if (!var.active_on_subdomain(elem->subdomain_id()))
            continue;

          // Only count dofs connected to active
          // elements on this processor.
          const unsigned int n_nodes = elem->n_nodes();

          // First get any new nodal DOFS
          for (unsigned int n=0; n<n_nodes; n++)
            {
              Node & node = elem->node_ref(n);

              if (node.processor_id() < this->processor_id())
                continue;

              const unsigned int n_comp = node.n_comp(sys_num, var_num);
              for(unsigned int i=0; i<n_comp; i++)
                {
                  const dof_id_type index = node.dof_number(sys_num,var_num,i);
                  libmesh_assert (this->local_index(index));

                  if (idx.empty() || index > idx.back())
                    idx.push_back(index);
                }
            }

          // Next get any new element DOFS
          const unsigned int n_comp = elem->n_comp(sys_num, var_num);
          for (unsigned int i=0; i<n_comp; i++)
            {
              const dof_id_type index = elem->dof_number(sys_num,var_num,i);
              if (idx.empty() || index > idx.back())
                idx.push_back(index);
            }
        } // done looping over elements


      // we may have missed assigning DOFs to nodes that we own
      // but to which we have no connected elements matching our
      // variable restriction criterion.  this will happen, for example,
      // if variable V is restricted to subdomain S.  We may not own
      // any elements which live in S, but we may own nodes which are
      // *connected* to elements which do.  in this scenario these nodes
      // will presently have unnumbered DOFs. we need to take care of
      // them here since we own them and no other processor will touch them.
      for (const auto & node : mesh.local_node_ptr_range())
        {
          libmesh_assert(node);

          const unsigned int n_comp = node->n_comp(sys_num, var_num);
          for (unsigned int i=0; i<n_comp; i++)
            {
              const dof_id_type index = node->dof_number(sys_num,var_num,i);
              if (idx.empty() || index > idx.back())
                idx.push_back(index);
            }
        }
    }
  // Otherwise, count up the SCALAR dofs, if we're on the processor
  // that holds this SCALAR variable
  else if (this->processor_id() == (this->n_processors()-1))
    {
      std::vector<dof_id_type> di_scalar;
      this->SCALAR_dof_indices(di_scalar,var_num);
      idx.insert( idx.end(), di_scalar.begin(), di_scalar.end());
    }
}


void DofMap::distribute_local_dofs_node_major(dof_id_type & next_free_dof,
                                              MeshBase & mesh)
{
  const unsigned int sys_num       = this->sys_number();
  const unsigned int n_var_groups  = this->n_variable_groups();

  // Our numbering here must be kept consistent with the numbering
  // scheme assumed by DofMap::local_variable_indices!

  //-------------------------------------------------------------------------
  // First count and assign temporary numbers to local dofs
  for (auto & elem : mesh.active_local_element_ptr_range())
    {
      // Only number dofs connected to active
      // elements on this processor.
      const unsigned int n_nodes = elem->n_nodes();

      // First number the nodal DOFS
      for (unsigned int n=0; n<n_nodes; n++)
        {
          Node & node = elem->node_ref(n);

          for (unsigned vg=0; vg<n_var_groups; vg++)
            {
              const VariableGroup & vg_description(this->variable_group(vg));

              if ((vg_description.type().family != SCALAR) &&
                  (vg_description.active_on_subdomain(elem->subdomain_id())))
                {
                  // assign dof numbers (all at once) if this is
                  // our node and if they aren't already there
                  if ((node.n_comp_group(sys_num,vg) > 0) &&
                      (node.processor_id() == this->processor_id()) &&
                      (node.vg_dof_base(sys_num,vg) ==
                       DofObject::invalid_id))
                    {
                      node.set_vg_dof_base(sys_num, vg,
                                           next_free_dof);
                      next_free_dof += (vg_description.n_variables()*
                                        node.n_comp_group(sys_num,vg));
                      //node.debug_buffer();
                    }
                }
            }
        }

      // Now number the element DOFS
      for (unsigned vg=0; vg<n_var_groups; vg++)
        {
          const VariableGroup & vg_description(this->variable_group(vg));

          if ((vg_description.type().family != SCALAR) &&
              (vg_description.active_on_subdomain(elem->subdomain_id())))
            if (elem->n_comp_group(sys_num,vg) > 0)
              {
                libmesh_assert_equal_to (elem->vg_dof_base(sys_num,vg),
                                         DofObject::invalid_id);

                elem->set_vg_dof_base(sys_num,
                                      vg,
                                      next_free_dof);

                next_free_dof += (vg_description.n_variables()*
                                  elem->n_comp(sys_num,vg));
              }
        }
    } // done looping over elements


  // we may have missed assigning DOFs to nodes that we own
  // but to which we have no connected elements matching our
  // variable restriction criterion.  this will happen, for example,
  // if variable V is restricted to subdomain S.  We may not own
  // any elements which live in S, but we may own nodes which are
  // *connected* to elements which do.  in this scenario these nodes
  // will presently have unnumbered DOFs. we need to take care of
  // them here since we own them and no other processor will touch them.
  for (auto & node : mesh.local_node_ptr_range())
    for (unsigned vg=0; vg<n_var_groups; vg++)
      {
        const VariableGroup & vg_description(this->variable_group(vg));

        if (node->n_comp_group(sys_num,vg))
          if (node->vg_dof_base(sys_num,vg) == DofObject::invalid_id)
            {
              node->set_vg_dof_base (sys_num,
                                     vg,
                                     next_free_dof);

              next_free_dof += (vg_description.n_variables()*
                                node->n_comp(sys_num,vg));
            }
      }

  // Finally, count up the SCALAR dofs
  this->_n_SCALAR_dofs = 0;
  for (unsigned vg=0; vg<n_var_groups; vg++)
    {
      const VariableGroup & vg_description(this->variable_group(vg));

      if (vg_description.type().family == SCALAR)
        {
          this->_n_SCALAR_dofs += (vg_description.n_variables()*
                                   vg_description.type().order.get_order());
          continue;
        }
    }

  // Only increment next_free_dof if we're on the processor
  // that holds this SCALAR variable
  if (this->processor_id() == (this->n_processors()-1))
    next_free_dof += _n_SCALAR_dofs;

#ifdef DEBUG
  {
    // libMesh::out << "next_free_dof=" << next_free_dof << std::endl
    //       << "_n_SCALAR_dofs=" << _n_SCALAR_dofs << std::endl;

    // Make sure we didn't miss any nodes
    MeshTools::libmesh_assert_valid_procids<Node>(mesh);

    for (auto & node : mesh.local_node_ptr_range())
      {
        unsigned int n_var_g = node->n_var_groups(this->sys_number());
        for (unsigned int vg=0; vg != n_var_g; ++vg)
          {
            unsigned int n_comp_g =
              node->n_comp_group(this->sys_number(), vg);
            dof_id_type my_first_dof = n_comp_g ?
              node->vg_dof_base(this->sys_number(), vg) : 0;
            libmesh_assert_not_equal_to (my_first_dof, DofObject::invalid_id);
          }
      }
  }
#endif // DEBUG
}



void DofMap::distribute_local_dofs_var_major(dof_id_type & next_free_dof,
                                             MeshBase & mesh)
{
  const unsigned int sys_num      = this->sys_number();
  const unsigned int n_var_groups = this->n_variable_groups();

  // Our numbering here must be kept consistent with the numbering
  // scheme assumed by DofMap::local_variable_indices!

  //-------------------------------------------------------------------------
  // First count and assign temporary numbers to local dofs
  for (unsigned vg=0; vg<n_var_groups; vg++)
    {
      const VariableGroup & vg_description(this->variable_group(vg));

      const unsigned int n_vars_in_group = vg_description.n_variables();

      // Skip the SCALAR dofs
      if (vg_description.type().family == SCALAR)
        continue;

      for (auto & elem : mesh.active_local_element_ptr_range())
        {
          // Only number dofs connected to active elements on this
          // processor and only variables which are active on on this
          // element's subdomain.
          if (!vg_description.active_on_subdomain(elem->subdomain_id()))
            continue;

          const unsigned int n_nodes = elem->n_nodes();

          // First number the nodal DOFS
          for (unsigned int n=0; n<n_nodes; n++)
            {
              Node & node = elem->node_ref(n);

              // assign dof numbers (all at once) if this is
              // our node and if they aren't already there
              if ((node.n_comp_group(sys_num,vg) > 0) &&
                  (node.processor_id() == this->processor_id()) &&
                  (node.vg_dof_base(sys_num,vg) ==
                   DofObject::invalid_id))
                {
                  node.set_vg_dof_base(sys_num, vg, next_free_dof);

                  next_free_dof += (n_vars_in_group*
                                    node.n_comp_group(sys_num,vg));
                }
            }

          // Now number the element DOFS
          if (elem->n_comp_group(sys_num,vg) > 0)
            {
              libmesh_assert_equal_to (elem->vg_dof_base(sys_num,vg),
                                       DofObject::invalid_id);

              elem->set_vg_dof_base(sys_num,
                                    vg,
                                    next_free_dof);

              next_free_dof += (n_vars_in_group*
                                elem->n_comp_group(sys_num,vg));
            }
        } // end loop on elements

      // we may have missed assigning DOFs to nodes that we own
      // but to which we have no connected elements matching our
      // variable restriction criterion.  this will happen, for example,
      // if variable V is restricted to subdomain S.  We may not own
      // any elements which live in S, but we may own nodes which are
      // *connected* to elements which do.  in this scenario these nodes
      // will presently have unnumbered DOFs. we need to take care of
      // them here since we own them and no other processor will touch them.
      for (auto & node : mesh.local_node_ptr_range())
        if (node->n_comp_group(sys_num,vg))
          if (node->vg_dof_base(sys_num,vg) == DofObject::invalid_id)
            {
              node->set_vg_dof_base (sys_num,
                                     vg,
                                     next_free_dof);

              next_free_dof += (n_vars_in_group*
                                node->n_comp_group(sys_num,vg));
            }
    } // end loop on variable groups

  // Finally, count up the SCALAR dofs
  this->_n_SCALAR_dofs = 0;
  for (unsigned vg=0; vg<n_var_groups; vg++)
    {
      const VariableGroup & vg_description(this->variable_group(vg));

      if (vg_description.type().family == SCALAR)
        {
          this->_n_SCALAR_dofs += (vg_description.n_variables()*
                                   vg_description.type().order.get_order());
          continue;
        }
    }

  // Only increment next_free_dof if we're on the processor
  // that holds this SCALAR variable
  if (this->processor_id() == (this->n_processors()-1))
    next_free_dof += _n_SCALAR_dofs;

#ifdef DEBUG
  {
    // Make sure we didn't miss any nodes
    MeshTools::libmesh_assert_valid_procids<Node>(mesh);

    for (auto & node : mesh.local_node_ptr_range())
      {
        unsigned int n_var_g = node->n_var_groups(this->sys_number());
        for (unsigned int vg=0; vg != n_var_g; ++vg)
          {
            unsigned int n_comp_g =
              node->n_comp_group(this->sys_number(), vg);
            dof_id_type my_first_dof = n_comp_g ?
              node->vg_dof_base(this->sys_number(), vg) : 0;
            libmesh_assert_not_equal_to (my_first_dof, DofObject::invalid_id);
          }
      }
  }
#endif // DEBUG
}



void
DofMap::
merge_ghost_functor_outputs(GhostingFunctor::map_type & elements_to_ghost,
                            std::set<CouplingMatrix *> & temporary_coupling_matrices,
                            const std::set<GhostingFunctor *>::iterator & gf_begin,
                            const std::set<GhostingFunctor *>::iterator & gf_end,
                            const MeshBase::const_element_iterator & elems_begin,
                            const MeshBase::const_element_iterator & elems_end,
                            processor_id_type p)
{
  for (const auto & gf : as_range(gf_begin, gf_end))
    {
      GhostingFunctor::map_type more_elements_to_ghost;

      libmesh_assert(gf);
      (*gf)(elems_begin, elems_end, p, more_elements_to_ghost);

      for (const auto & pr : more_elements_to_ghost)
        {
          GhostingFunctor::map_type::iterator existing_it =
            elements_to_ghost.find (pr.first);
          if (existing_it == elements_to_ghost.end())
            elements_to_ghost.insert(pr);
          else
            {
              if (existing_it->second)
                {
                  if (pr.second)
                    {
                      // If this isn't already a temporary
                      // then we need to make one so we'll
                      // have a non-const matrix to merge
                      if (temporary_coupling_matrices.empty() ||
                          temporary_coupling_matrices.find(const_cast<CouplingMatrix *>(existing_it->second)) == temporary_coupling_matrices.end())
                        {
                          CouplingMatrix * cm = new CouplingMatrix(*existing_it->second);
                          temporary_coupling_matrices.insert(cm);
                          existing_it->second = cm;
                        }
                      const_cast<CouplingMatrix &>(*existing_it->second) &= *pr.second;
                    }
                  else
                    {
                      // Any existing_it matrix merged with a full
                      // matrix (symbolized as nullptr) gives another
                      // full matrix (symbolizable as nullptr).

                      // So if existing_it->second is a temporary then
                      // we don't need it anymore; we might as well
                      // remove it to keep the set of temporaries
                      // small.
                      std::set<CouplingMatrix *>::iterator temp_it =
                        temporary_coupling_matrices.find(const_cast<CouplingMatrix *>(existing_it->second));
                      if (temp_it != temporary_coupling_matrices.end())
                        temporary_coupling_matrices.erase(temp_it);

                      existing_it->second = nullptr;
                    }
                }
              // else we have a nullptr already, then we have a full
              // coupling matrix, already, and merging with anything
              // else won't change that, so we're done.
            }
        }
    }
}



void DofMap::add_neighbors_to_send_list(MeshBase & mesh)
{
  LOG_SCOPE("add_neighbors_to_send_list()", "DofMap");

  const unsigned int n_var  = this->n_variables();

  MeshBase::const_element_iterator       local_elem_it
    = mesh.active_local_elements_begin();
  const MeshBase::const_element_iterator local_elem_end
    = mesh.active_local_elements_end();

  GhostingFunctor::map_type elements_to_send;

  // Man, I wish we had guaranteed unique_ptr availability...
  std::set<CouplingMatrix *> temporary_coupling_matrices;

  // We need to add dofs to the send list if they've been directly
  // requested by an algebraic ghosting functor or they've been
  // indirectly requested by a coupling functor.
  this->merge_ghost_functor_outputs(elements_to_send,
                                    temporary_coupling_matrices,
                                    this->algebraic_ghosting_functors_begin(),
                                    this->algebraic_ghosting_functors_end(),
                                    local_elem_it, local_elem_end, mesh.processor_id());

  this->merge_ghost_functor_outputs(elements_to_send,
                                    temporary_coupling_matrices,
                                    this->coupling_functors_begin(),
                                    this->coupling_functors_end(),
                                    local_elem_it, local_elem_end, mesh.processor_id());

  // Making a list of non-zero coupling matrix columns is an
  // O(N_var^2) operation.  We cache it so we only have to do it once
  // per CouplingMatrix and not once per element.
  std::map<const CouplingMatrix *, std::vector<unsigned int>>
    column_variable_lists;

  for (auto & pr : elements_to_send)
    {
      const Elem * const partner = pr.first;

      // We asked ghosting functors not to give us local elements
      libmesh_assert_not_equal_to
        (partner->processor_id(), this->processor_id());

      const CouplingMatrix * ghost_coupling = pr.second;

      // Loop over any present coupling matrix column variables if we
      // have a coupling matrix, or just add all variables to
      // send_list if not.
      if (ghost_coupling)
        {
          libmesh_assert_equal_to (ghost_coupling->size(), n_var);

          // Try to find a cached list of column variables.
          std::map<const CouplingMatrix *, std::vector<unsigned int>>::const_iterator
            column_variable_list = column_variable_lists.find(ghost_coupling);

          // If we didn't find it, then we need to create it.
          if (column_variable_list == column_variable_lists.end())
            {
              std::pair<std::map<const CouplingMatrix *, std::vector<unsigned int>>::iterator, bool>
                inserted_variable_list_pair = column_variable_lists.insert(std::make_pair(ghost_coupling,
                                                                                          std::vector<unsigned int>()));
              column_variable_list = inserted_variable_list_pair.first;

              std::vector<unsigned int> & new_variable_list =
                inserted_variable_list_pair.first->second;

              std::vector<unsigned char> has_variable(n_var, false);

              for (unsigned int vi = 0; vi != n_var; ++vi)
                {
                  ConstCouplingRow ccr(vi, *ghost_coupling);

                  for (const auto & vj : ccr)
                    has_variable[vj] = true;
                }
              for (unsigned int vj = 0; vj != n_var; ++vj)
                {
                  if (has_variable[vj])
                    new_variable_list.push_back(vj);
                }
            }

          const std::vector<unsigned int> & variable_list =
            column_variable_list->second;

          for (const auto & vj : variable_list)
            {
              std::vector<dof_id_type> di;
              this->dof_indices (partner, di, vj);

              // Insert the remote DOF indices into the send list
              for (auto d : di)
                if (!this->local_index(d))
                  _send_list.push_back(d);
            }
        }
      else
        {
          std::vector<dof_id_type> di;
          this->dof_indices (partner, di);

          // Insert the remote DOF indices into the send list
          for (const auto & dof : di)
            if (!this->local_index(dof))
              _send_list.push_back(dof);
        }

    }

  // We're now done with any merged coupling matrices we had to create.
  for (auto & mat : temporary_coupling_matrices)
    delete mat;

  //-------------------------------------------------------------------------
  // Our coupling functors added dofs from neighboring elements to the
  // send list, but we may still need to add non-local dofs from local
  // elements.
  //-------------------------------------------------------------------------

  // Loop over the active local elements, adding all active elements
  // that neighbor an active local element to the send list.
  for ( ; local_elem_it != local_elem_end; ++local_elem_it)
    {
      const Elem * elem = *local_elem_it;

      std::vector<dof_id_type> di;
      this->dof_indices (elem, di);

      // Insert the remote DOF indices into the send list
      for (const auto & dof : di)
        if (!this->local_index(dof))
          _send_list.push_back(dof);
    }
}



void DofMap::prepare_send_list ()
{
  LOG_SCOPE("prepare_send_list()", "DofMap");

  // Check to see if we have any extra stuff to add to the send_list
  if (_extra_send_list_function)
    {
      if (_augment_send_list)
        {
          libmesh_here();
          libMesh::out << "WARNING:  You have specified both an extra send list function and object.\n"
                       << "          Are you sure this is what you meant to do??"
                       << std::endl;
        }

      _extra_send_list_function(_send_list, _extra_send_list_context);
    }

  if (_augment_send_list)
    _augment_send_list->augment_send_list (_send_list);

  // First sort the send list.  After this
  // duplicated elements will be adjacent in the
  // vector
  std::sort(_send_list.begin(), _send_list.end());

  // Now use std::unique to remove duplicate entries
  std::vector<dof_id_type>::iterator new_end =
    std::unique (_send_list.begin(), _send_list.end());

  // Remove the end of the send_list.  Use the "swap trick"
  // from Effective STL
  std::vector<dof_id_type> (_send_list.begin(), new_end).swap (_send_list);
}


void DofMap::set_implicit_neighbor_dofs(bool implicit_neighbor_dofs)
{
  _implicit_neighbor_dofs_initialized = true;
  _implicit_neighbor_dofs = implicit_neighbor_dofs;
}


bool DofMap::use_coupled_neighbor_dofs(const MeshBase & mesh) const
{
  // If we were asked on the command line, then we need to
  // include sensitivities between neighbor degrees of freedom
  bool implicit_neighbor_dofs =
    libMesh::on_command_line ("--implicit-neighbor-dofs");

  // If the user specifies --implicit-neighbor-dofs 0, then
  // presumably he knows what he is doing and we won't try to
  // automatically turn it on even when all the variables are
  // discontinuous.
  if (implicit_neighbor_dofs)
    {
      // No flag provided defaults to 'true'
      int flag = 1;
      flag = libMesh::command_line_next ("--implicit-neighbor-dofs", flag);

      if (!flag)
        {
          // The user said --implicit-neighbor-dofs 0, so he knows
          // what he is doing and really doesn't want it.
          return false;
        }
    }

  // Possibly override the commandline option, if set_implicit_neighbor_dofs
  // has been called.
  if (_implicit_neighbor_dofs_initialized)
    {
      implicit_neighbor_dofs = _implicit_neighbor_dofs;

      // Again, if the user explicitly says implicit_neighbor_dofs = false,
      // then we return here.
      if (!implicit_neighbor_dofs)
        return false;
    }

  // Look at all the variables in this system.  If every one is
  // discontinuous then the user must be doing DG/FVM, so be nice
  // and force implicit_neighbor_dofs=true.
  {
    bool all_discontinuous_dofs = true;

    for (unsigned int var=0; var<this->n_variables(); var++)
      if (FEAbstract::build (mesh.mesh_dimension(),
                             this->variable_type(var))->get_continuity() !=  DISCONTINUOUS)
        all_discontinuous_dofs = false;

    if (all_discontinuous_dofs)
      implicit_neighbor_dofs = true;
  }

  return implicit_neighbor_dofs;
}



void DofMap::compute_sparsity(const MeshBase & mesh)
{
  _sp = this->build_sparsity(mesh);

  // It is possible that some \p SparseMatrix implementations want to
  // see the sparsity pattern before we throw it away.  If so, we
  // share a view of its arrays, and we pass it in to the matrices.
  if (need_full_sparsity_pattern)
    {
      _n_nz = &_sp->n_nz;
      _n_oz = &_sp->n_oz;

      for (const auto & mat : _matrices)
        mat->update_sparsity_pattern (_sp->sparsity_pattern);
    }
  // If we don't need the full sparsity pattern anymore, steal the
  // arrays we do need and free the rest of the memory
  else
    {
      if (!_n_nz)
        _n_nz = new std::vector<dof_id_type>();
      _n_nz->swap(_sp->n_nz);
      if (!_n_oz)
        _n_oz = new std::vector<dof_id_type>();
      _n_oz->swap(_sp->n_oz);

      _sp.reset();
    }
}



void DofMap::clear_sparsity()
{
  if (need_full_sparsity_pattern)
    {
      libmesh_assert(_sp.get());
      libmesh_assert(!_n_nz || _n_nz == &_sp->n_nz);
      libmesh_assert(!_n_oz || _n_oz == &_sp->n_oz);
      _sp.reset();
    }
  else
    {
      libmesh_assert(!_sp.get());
      delete _n_nz;
      delete _n_oz;
    }
  _n_nz = nullptr;
  _n_oz = nullptr;
}



void DofMap::remove_default_ghosting()
{
  this->remove_coupling_functor(this->default_coupling());
  this->remove_algebraic_ghosting_functor(this->default_algebraic_ghosting());
}



void DofMap::add_default_ghosting()
{
  this->add_coupling_functor(this->default_coupling());
  this->add_algebraic_ghosting_functor(this->default_algebraic_ghosting());
}



void
DofMap::add_coupling_functor(GhostingFunctor & coupling_functor,
                             bool to_mesh)
{
  _coupling_functors.insert(&coupling_functor);
  if (to_mesh)
    _mesh.add_ghosting_functor(coupling_functor);
}



void
DofMap::remove_coupling_functor(GhostingFunctor & coupling_functor)
{
  _coupling_functors.erase(&coupling_functor);
  _mesh.remove_ghosting_functor(coupling_functor);
}



void
DofMap::add_algebraic_ghosting_functor(GhostingFunctor & evaluable_functor,
                                       bool to_mesh)
{
  _algebraic_ghosting_functors.insert(&evaluable_functor);
  if (to_mesh)
    _mesh.add_ghosting_functor(evaluable_functor);
}



void
DofMap::remove_algebraic_ghosting_functor(GhostingFunctor & evaluable_functor)
{
  _algebraic_ghosting_functors.erase(&evaluable_functor);
  _mesh.remove_ghosting_functor(evaluable_functor);
}



void DofMap::extract_local_vector (const NumericVector<Number> & Ug,
                                   const std::vector<dof_id_type> & dof_indices_in,
                                   DenseVectorBase<Number> & Ue) const
{
  const unsigned int n_original_dofs = dof_indices_in.size();

#ifdef LIBMESH_ENABLE_AMR

  // Trivial mapping
  libmesh_assert_equal_to (dof_indices_in.size(), Ue.size());
  bool has_constrained_dofs = false;

  for (unsigned int il=0; il != n_original_dofs; ++il)
    {
      const dof_id_type ig = dof_indices_in[il];

      if (this->is_constrained_dof (ig)) has_constrained_dofs = true;

      libmesh_assert_less (ig, Ug.size());

      Ue.el(il) = Ug(ig);
    }

  // If the element has any constrained DOFs then we need
  // to account for them in the mapping.  This will handle
  // the case that the input vector is not constrained.
  if (has_constrained_dofs)
    {
      // Copy the input DOF indices.
      std::vector<dof_id_type> constrained_dof_indices(dof_indices_in);

      DenseMatrix<Number> C;
      DenseVector<Number> H;

      this->build_constraint_matrix_and_vector (C, H, constrained_dof_indices);

      libmesh_assert_equal_to (dof_indices_in.size(), C.m());
      libmesh_assert_equal_to (constrained_dof_indices.size(), C.n());

      // zero-out Ue
      Ue.zero();

      // compute Ue = C Ug, with proper mapping.
      for (unsigned int i=0; i != n_original_dofs; i++)
        {
          Ue.el(i) = H(i);

          const unsigned int n_constrained =
            cast_int<unsigned int>(constrained_dof_indices.size());
          for (unsigned int j=0; j<n_constrained; j++)
            {
              const dof_id_type jg = constrained_dof_indices[j];

              //          If Ug is a serial or ghosted vector, then this assert is
              //          overzealous.  If Ug is a parallel vector, then this assert
              //          is redundant.
              //    libmesh_assert ((jg >= Ug.first_local_index()) &&
              //    (jg <  Ug.last_local_index()));

              Ue.el(i) += C(i,j)*Ug(jg);
            }
        }
    }

#else

  // Trivial mapping

  libmesh_assert_equal_to (n_original_dofs, Ue.size());

  for (unsigned int il=0; il<n_original_dofs; il++)
    {
      const dof_id_type ig = dof_indices_in[il];

      libmesh_assert ((ig >= Ug.first_local_index()) && (ig <  Ug.last_local_index()));

      Ue.el(il) = Ug(ig);
    }

#endif
}

void DofMap::dof_indices (const Elem * const elem,
                          std::vector<dof_id_type> & di) const
{
  // We now allow elem==nullptr to request just SCALAR dofs
  // libmesh_assert(elem);

  // If we are asking for current indices on an element, it ought to
  // be an active element (or a Side proxy, which also thinks it's
  // active)
  libmesh_assert(!elem || elem->active());

  LOG_SCOPE("dof_indices()", "DofMap");

  // Clear the DOF indices vector
  di.clear();

  const unsigned int n_var_groups  = this->n_variable_groups();

#ifdef DEBUG
  // Check that sizes match in DEBUG mode
  std::size_t tot_size = 0;
#endif

  if (elem && elem->type() == TRI3SUBDIVISION)
    {
      // Subdivision surface FE require the 1-ring around elem
      const Tri3Subdivision * sd_elem = static_cast<const Tri3Subdivision *>(elem);

      // Ghost subdivision elements have no real dofs
      if (!sd_elem->is_ghost())
        {
          // Determine the nodes contributing to element elem
          std::vector<const Node *> elem_nodes;
          MeshTools::Subdivision::find_one_ring(sd_elem, elem_nodes);

          // Get the dof numbers
          for (unsigned int vg=0; vg<n_var_groups; vg++)
            {
              const VariableGroup & var = this->variable_group(vg);
              const unsigned int vars_in_group = var.n_variables();

              if (var.type().family == SCALAR &&
                  var.active_on_subdomain(elem->subdomain_id()))
                {
                  for (unsigned int vig=0; vig != vars_in_group; ++vig)
                    {
#ifdef DEBUG
                      tot_size += var.type().order;
#endif
                      std::vector<dof_id_type> di_new;
                      this->SCALAR_dof_indices(di_new,var.number(vig));
                      di.insert( di.end(), di_new.begin(), di_new.end());
                    }
                }
              else
                for (unsigned int vig=0; vig != vars_in_group; ++vig)
                  {
                    _dof_indices(*elem, elem->p_level(), di, vg, vig,
                                 elem_nodes.data(),
                                 cast_int<unsigned int>(elem_nodes.size())
#ifdef DEBUG
                                 , var.number(vig), tot_size
#endif
                                 );
                  }
            }
        }

      return;
    }

  // Get the dof numbers for each variable
  const unsigned int n_nodes = elem ? elem->n_nodes() : 0;
  for (unsigned int vg=0; vg<n_var_groups; vg++)
    {
      const VariableGroup & var = this->variable_group(vg);
      const unsigned int vars_in_group = var.n_variables();

      if (var.type().family == SCALAR &&
          (!elem ||
           var.active_on_subdomain(elem->subdomain_id())))
        {
          for (unsigned int vig=0; vig != vars_in_group; ++vig)
            {
#ifdef DEBUG
              tot_size += var.type().order;
#endif
              std::vector<dof_id_type> di_new;
              this->SCALAR_dof_indices(di_new,var.number(vig));
              di.insert( di.end(), di_new.begin(), di_new.end());
            }
        }
      else if (elem)
        for (unsigned int vig=0; vig != vars_in_group; ++vig)
          {
            _dof_indices(*elem, elem->p_level(), di, vg, vig,
                         elem->get_nodes(), n_nodes
#ifdef DEBUG
                         , var.number(vig), tot_size
#endif
                     );
          }
    }

#ifdef DEBUG
  libmesh_assert_equal_to (tot_size, di.size());
#endif
}


void DofMap::dof_indices (const Elem * const elem,
                          std::vector<dof_id_type> & di,
                          const unsigned int vn,
                          int p_level) const
{
  // We now allow elem==nullptr to request just SCALAR dofs
  // libmesh_assert(elem);

  LOG_SCOPE("dof_indices()", "DofMap");

  // Clear the DOF indices vector
  di.clear();

  // Use the default p refinement level?
  if (p_level == -12345)
    p_level = elem ? elem->p_level() : 0;

  const unsigned int vg = this->_variable_group_numbers[vn];
  const VariableGroup & var = this->variable_group(vg);
  const unsigned int vig = vn - var.number();

#ifdef DEBUG
  // Check that sizes match in DEBUG mode
  std::size_t tot_size = 0;
#endif

  if (elem && elem->type() == TRI3SUBDIVISION)
    {
      // Subdivision surface FE require the 1-ring around elem
      const Tri3Subdivision * sd_elem = static_cast<const Tri3Subdivision *>(elem);

      // Ghost subdivision elements have no real dofs
      if (!sd_elem->is_ghost())
        {
          // Determine the nodes contributing to element elem
          std::vector<const Node *> elem_nodes;
          MeshTools::Subdivision::find_one_ring(sd_elem, elem_nodes);

          _dof_indices(*elem, p_level, di, vg, vig, elem_nodes.data(),
                       cast_int<unsigned int>(elem_nodes.size())
#ifdef DEBUG
                       , vn, tot_size
#endif
                       );
        }

      return;
    }

  // Get the dof numbers
  if (var.type().family == SCALAR &&
      (!elem ||
       var.active_on_subdomain(elem->subdomain_id())))
    {
#ifdef DEBUG
      tot_size += var.type().order;
#endif
      std::vector<dof_id_type> di_new;
      this->SCALAR_dof_indices(di_new,vn);
      di.insert( di.end(), di_new.begin(), di_new.end());
    }
  else if (elem)
    _dof_indices(*elem, p_level, di, vg, vig, elem->get_nodes(),
                 elem->n_nodes()
#ifdef DEBUG
                 , vn, tot_size
#endif
                 );

#ifdef DEBUG
  libmesh_assert_equal_to (tot_size, di.size());
#endif
}


void DofMap::dof_indices (const Node * const node,
                          std::vector<dof_id_type> & di) const
{
  // We allow node==nullptr to request just SCALAR dofs
  // libmesh_assert(elem);

  LOG_SCOPE("dof_indices(Node)", "DofMap");

  // Clear the DOF indices vector
  di.clear();

  const unsigned int n_var_groups  = this->n_variable_groups();
  const unsigned int sys_num = this->sys_number();

  // Get the dof numbers
  for (unsigned int vg=0; vg<n_var_groups; vg++)
    {
      const VariableGroup & var = this->variable_group(vg);
      const unsigned int vars_in_group = var.n_variables();

      if (var.type().family == SCALAR)
        {
          for (unsigned int vig=0; vig != vars_in_group; ++vig)
            {
              std::vector<dof_id_type> di_new;
              this->SCALAR_dof_indices(di_new,var.number(vig));
              di.insert( di.end(), di_new.begin(), di_new.end());
            }
        }
      else
        {
          const int n_comp = node->n_comp_group(sys_num,vg);
          for (unsigned int vig=0; vig != vars_in_group; ++vig)
            {
              for (int i=0; i != n_comp; ++i)
                {
                  const dof_id_type d =
                    node->dof_number(sys_num, vg, vig, i, n_comp);
                  libmesh_assert_not_equal_to
                    (d, DofObject::invalid_id);
                  di.push_back(d);
                }
            }
        }
    }
}


void DofMap::dof_indices (const Node * const node,
                          std::vector<dof_id_type> & di,
                          const unsigned int vn) const
{
  if (vn == libMesh::invalid_uint)
    {
      this->dof_indices(node, di);
      return;
    }

  // We allow node==nullptr to request just SCALAR dofs
  // libmesh_assert(elem);

  LOG_SCOPE("dof_indices(Node)", "DofMap");

  // Clear the DOF indices vector
  di.clear();

  const unsigned int sys_num = this->sys_number();

  // Get the dof numbers
  const unsigned int vg = this->_variable_group_numbers[vn];
  const VariableGroup & var = this->variable_group(vg);

  if (var.type().family == SCALAR)
    {
      std::vector<dof_id_type> di_new;
      this->SCALAR_dof_indices(di_new,vn);
      di.insert( di.end(), di_new.begin(), di_new.end());
    }
  else
    {
      const unsigned int vig = vn - var.number();
      const int n_comp = node->n_comp_group(sys_num,vg);
      for (int i=0; i != n_comp; ++i)
        {
          const dof_id_type d =
            node->dof_number(sys_num, vg, vig, i, n_comp);
          libmesh_assert_not_equal_to
            (d, DofObject::invalid_id);
          di.push_back(d);
        }
    }
}


void DofMap::_dof_indices (const Elem & elem,
                           int p_level,
                           std::vector<dof_id_type> & di,
                           const unsigned int vg,
                           const unsigned int vig,
                           const Node * const * nodes,
                           unsigned int       n_nodes
#ifdef DEBUG
                           ,
                           const unsigned int v,
                           std::size_t & tot_size
#endif
                           ) const
{
  const VariableGroup & var = this->variable_group(vg);

  if (var.active_on_subdomain(elem.subdomain_id()))
    {
      const ElemType type        = elem.type();
      const unsigned int sys_num = this->sys_number();
      const unsigned int dim     = elem.dim();
#ifdef LIBMESH_ENABLE_INFINITE_ELEMENTS
      const bool is_inf          = elem.infinite();
#endif

      // Increase the polynomial order on p refined elements
      FEType fe_type = var.type();
      fe_type.order = static_cast<Order>(fe_type.order + p_level);

      const bool extra_hanging_dofs =
        FEInterface::extra_hanging_dofs(fe_type);

#ifdef DEBUG
      // The number of dofs per element is non-static for subdivision FE
      if (fe_type.family == SUBDIVISION)
        tot_size += n_nodes;
      else
        tot_size += FEInterface::n_dofs(dim,fe_type,type);
#endif

      const FEInterface::n_dofs_at_node_ptr ndan =
        FEInterface::n_dofs_at_node_function(dim, fe_type);

      // Get the node-based DOF numbers
      for (unsigned int n=0; n != n_nodes; n++)
        {
          const Node * node      = nodes[n];

          // Cache the intermediate lookups that are common to every
          // component
#ifdef DEBUG
          const std::pair<unsigned int, unsigned int>
            vg_and_offset = node->var_to_vg_and_offset(sys_num,v);
          libmesh_assert_equal_to (vg, vg_and_offset.first);
          libmesh_assert_equal_to (vig, vg_and_offset.second);
#endif
          const unsigned int n_comp = node->n_comp_group(sys_num,vg);

          // There is a potential problem with h refinement.  Imagine a
          // quad9 that has a linear FE on it.  Then, on the hanging side,
          // it can falsely identify a DOF at the mid-edge node. This is why
          // we go through FEInterface instead of node->n_comp() directly.
          const unsigned int nc =
#ifdef LIBMESH_ENABLE_INFINITE_ELEMENTS
            is_inf ?
            FEInterface::n_dofs_at_node(dim, fe_type, type, n) :
#endif
            ndan (type, fe_type.order, n);

          // If this is a non-vertex on a hanging node with extra
          // degrees of freedom, we use the non-vertex dofs (which
          // come in reverse order starting from the end, to
          // simplify p refinement)
          if (extra_hanging_dofs && !elem.is_vertex(n))
            {
              const int dof_offset = n_comp - nc;

              // We should never have fewer dofs than necessary on a
              // node unless we're getting indices on a parent element,
              // and we should never need the indices on such a node
              if (dof_offset < 0)
                {
                  libmesh_assert(!elem.active());
                  di.resize(di.size() + nc, DofObject::invalid_id);
                }
              else
                for (int i=n_comp-1; i>=dof_offset; i--)
                  {
                    const dof_id_type d =
                      node->dof_number(sys_num, vg, vig, i, n_comp);
                    libmesh_assert_not_equal_to (d, DofObject::invalid_id);
                    di.push_back(d);
                  }
            }
          // If this is a vertex or an element without extra hanging
          // dofs, our dofs come in forward order coming from the
          // beginning
          else
            for (unsigned int i=0; i<nc; i++)
              {
                const dof_id_type d =
                  node->dof_number(sys_num, vg, vig, i, n_comp);
                libmesh_assert_not_equal_to (d, DofObject::invalid_id);
                di.push_back(d);
              }
        }

      // If there are any element-based DOF numbers, get them
      const unsigned int nc = FEInterface::n_dofs_per_elem(dim,
                                                           fe_type,
                                                           type);
      // We should never have fewer dofs than necessary on an
      // element unless we're getting indices on a parent element,
      // and we should never need those indices
      if (nc != 0)
        {
          const unsigned int n_comp = elem.n_comp_group(sys_num,vg);
          if (elem.n_systems() > sys_num && nc <= n_comp)
            {
              for (unsigned int i=0; i<nc; i++)
                {
                  const dof_id_type d =
                    elem.dof_number(sys_num, vg, vig, i, n_comp);
                  libmesh_assert_not_equal_to (d, DofObject::invalid_id);

                  di.push_back(d);
                }
            }
          else
            {
              libmesh_assert(!elem.active() || fe_type.family == LAGRANGE || fe_type.family == SUBDIVISION);
              di.resize(di.size() + nc, DofObject::invalid_id);
            }
        }
    }
}



void DofMap::SCALAR_dof_indices (std::vector<dof_id_type> & di,
                                 const unsigned int vn,
#ifdef LIBMESH_ENABLE_AMR
                                 const bool old_dofs
#else
                                 const bool
#endif
                                 ) const
{
  LOG_SCOPE("SCALAR_dof_indices()", "DofMap");

  libmesh_assert(this->variable(vn).type().family == SCALAR);

#ifdef LIBMESH_ENABLE_AMR
  // If we're asking for old dofs then we'd better have some
  if (old_dofs)
    libmesh_assert_greater_equal(n_old_dofs(), n_SCALAR_dofs());

  dof_id_type my_idx = old_dofs ?
    this->_first_old_scalar_df[vn] : this->_first_scalar_df[vn];
#else
  dof_id_type my_idx = this->_first_scalar_df[vn];
#endif

  libmesh_assert_not_equal_to(my_idx, DofObject::invalid_id);

  // The number of SCALAR dofs comes from the variable order
  const int n_dofs_vn = this->variable(vn).type().order.get_order();

  di.resize(n_dofs_vn);
  for (int i = 0; i != n_dofs_vn; ++i)
    di[i] = my_idx++;
}



bool DofMap::semilocal_index (dof_id_type dof_index) const
{
  // If it's not in the local indices
  if (!this->local_index(dof_index))
    {
      // and if it's not in the ghost indices, then we're not
      // semilocal
      if (!std::binary_search(_send_list.begin(), _send_list.end(), dof_index))
        return false;
    }

  return true;
}



bool DofMap::all_semilocal_indices (const std::vector<dof_id_type> & dof_indices_in) const
{
  // We're all semilocal unless we find a counterexample
  for (const auto & di : dof_indices_in)
    if (!this->semilocal_index(di))
      return false;

  return true;
}



template <typename DofObjectSubclass>
bool DofMap::is_evaluable(const DofObjectSubclass & obj,
                          unsigned int var_num) const
{
  // Everything is evaluable on a local object
  if (obj.processor_id() == this->processor_id())
    return true;

  std::vector<dof_id_type> di;

  if (var_num == libMesh::invalid_uint)
    this->dof_indices(&obj, di);
  else
    this->dof_indices(&obj, di, var_num);

  return this->all_semilocal_indices(di);
}



#ifdef LIBMESH_ENABLE_AMR

void DofMap::old_dof_indices (const Elem * const elem,
                              std::vector<dof_id_type> & di,
                              const unsigned int vn) const
{
  LOG_SCOPE("old_dof_indices()", "DofMap");

  libmesh_assert(elem);

  const ElemType type              = elem->type();
  const unsigned int sys_num       = this->sys_number();
  const unsigned int n_var_groups  = this->n_variable_groups();
  const unsigned int dim           = elem->dim();
#ifdef LIBMESH_ENABLE_INFINITE_ELEMENTS
  const bool is_inf                = elem->infinite();
#endif

  // If we have dof indices stored on the elem, and there's no chance
  // that we only have those indices because we were just p refined,
  // then we should have old dof indices too.
  libmesh_assert(!elem->has_dofs(sys_num) ||
                 elem->p_refinement_flag() == Elem::JUST_REFINED ||
                 elem->old_dof_object);

  // Clear the DOF indices vector.
  di.clear();

  // Determine the nodes contributing to element elem
  std::vector<const Node *> elem_nodes;
  const Node * const * nodes_ptr;
  unsigned int n_nodes;
  if (elem->type() == TRI3SUBDIVISION)
    {
      // Subdivision surface FE require the 1-ring around elem
      const Tri3Subdivision * sd_elem = static_cast<const Tri3Subdivision *>(elem);
      MeshTools::Subdivision::find_one_ring(sd_elem, elem_nodes);
      nodes_ptr = elem_nodes.data();
      n_nodes = cast_int<unsigned int>(elem_nodes.size());
    }
  else
    {
      // All other FE use only the nodes of elem itself
      nodes_ptr = elem->get_nodes();
      n_nodes = elem->n_nodes();
    }

  // Get the dof numbers
  for (unsigned int vg=0; vg<n_var_groups; vg++)
    {
      const VariableGroup & var = this->variable_group(vg);
      const unsigned int vars_in_group = var.n_variables();

      for (unsigned int vig=0; vig<vars_in_group; vig++)
        {
          const unsigned int v = var.number(vig);
          if ((vn == v) || (vn == libMesh::invalid_uint))
            {
              if (var.type().family == SCALAR &&
                  (!elem ||
                   var.active_on_subdomain(elem->subdomain_id())))
                {
                  // We asked for this variable, so add it to the vector.
                  std::vector<dof_id_type> di_new;
                  this->SCALAR_dof_indices(di_new,v,true);
                  di.insert( di.end(), di_new.begin(), di_new.end());
                }
              else
                if (var.active_on_subdomain(elem->subdomain_id()))
                  { // Do this for all the variables if one was not specified
                    // or just for the specified variable

                    // Increase the polynomial order on p refined elements,
                    // but make sure you get the right polynomial order for
                    // the OLD degrees of freedom
                    int p_adjustment = 0;
                    if (elem->p_refinement_flag() == Elem::JUST_REFINED)
                      {
                        libmesh_assert_greater (elem->p_level(), 0);
                        p_adjustment = -1;
                      }
                    else if (elem->p_refinement_flag() == Elem::JUST_COARSENED)
                      {
                        p_adjustment = 1;
                      }
                    FEType fe_type = var.type();
                    fe_type.order = static_cast<Order>(fe_type.order +
                                                       elem->p_level() +
                                                       p_adjustment);

                    const bool extra_hanging_dofs =
                      FEInterface::extra_hanging_dofs(fe_type);

                    const FEInterface::n_dofs_at_node_ptr ndan =
                      FEInterface::n_dofs_at_node_function(dim, fe_type);

                    // Get the node-based DOF numbers
                    for (unsigned int n=0; n<n_nodes; n++)
                      {
                        const Node * node = nodes_ptr[n];
                        const DofObject * old_dof_obj = node->old_dof_object;
                        libmesh_assert(old_dof_obj);

                        // There is a potential problem with h refinement.  Imagine a
                        // quad9 that has a linear FE on it.  Then, on the hanging side,
                        // it can falsely identify a DOF at the mid-edge node. This is why
                        // we call FEInterface instead of node->n_comp() directly.
                        const unsigned int nc =
#ifdef LIBMESH_ENABLE_INFINITE_ELEMENTS
                          is_inf ?
                          FEInterface::n_dofs_at_node(dim, fe_type, type, n) :
#endif
                          ndan (type, fe_type.order, n);

                        const int n_comp = old_dof_obj->n_comp_group(sys_num,vg);

                        // If this is a non-vertex on a hanging node with extra
                        // degrees of freedom, we use the non-vertex dofs (which
                        // come in reverse order starting from the end, to
                        // simplify p refinement)
                        if (extra_hanging_dofs && !elem->is_vertex(n))
                          {
                            const int dof_offset = n_comp - nc;

                            // We should never have fewer dofs than necessary on a
                            // node unless we're getting indices on a parent element
                            // or a just-coarsened element
                            if (dof_offset < 0)
                              {
                                libmesh_assert(!elem->active() || elem->refinement_flag() ==
                                               Elem::JUST_COARSENED);
                                di.resize(di.size() + nc, DofObject::invalid_id);
                              }
                            else
                              for (int i=n_comp-1; i>=dof_offset; i--)
                                {
                                  const dof_id_type d =
                                    old_dof_obj->dof_number(sys_num, vg, vig, i, n_comp);
                                  libmesh_assert_not_equal_to (d, DofObject::invalid_id);
                                  di.push_back(d);
                                }
                          }
                        // If this is a vertex or an element without extra hanging
                        // dofs, our dofs come in forward order coming from the
                        // beginning
                        else
                          for (unsigned int i=0; i<nc; i++)
                            {
                              const dof_id_type d =
                                old_dof_obj->dof_number(sys_num, vg, vig, i, n_comp);
                              libmesh_assert_not_equal_to (d, DofObject::invalid_id);
                              di.push_back(d);
                            }
                      }

                    // If there are any element-based DOF numbers, get them
                    const unsigned int nc = FEInterface::n_dofs_per_elem(dim,
                                                                         fe_type,
                                                                         type);

                    // We should never have fewer dofs than necessary on an
                    // element unless we're getting indices on a parent element
                    // or a just-coarsened element
                    if (nc != 0)
                      {
                        const DofObject * old_dof_obj = elem->old_dof_object;
                        libmesh_assert(old_dof_obj);

                        const unsigned int n_comp =
                          old_dof_obj->n_comp_group(sys_num,vg);

                        if (old_dof_obj->n_systems() > sys_num &&
                            nc <= n_comp)
                          {

                            for (unsigned int i=0; i<nc; i++)
                              {
                                const dof_id_type d =
                                  old_dof_obj->dof_number(sys_num, vg, vig, i, n_comp);

                                di.push_back(d);
                              }
                          }
                        else
                          {
                            libmesh_assert(!elem->active() || fe_type.family == LAGRANGE ||
                                           elem->refinement_flag() == Elem::JUST_COARSENED);
                            di.resize(di.size() + nc, DofObject::invalid_id);
                          }
                      }
                  }
            }
        } // end loop over variables within group
    } // end loop over variable groups
}

#endif // LIBMESH_ENABLE_AMR


#ifdef LIBMESH_ENABLE_CONSTRAINTS

void DofMap::find_connected_dofs (std::vector<dof_id_type> & elem_dofs) const
{
  typedef std::set<dof_id_type> RCSet;

  // First insert the DOFS we already depend on into the set.
  RCSet dof_set (elem_dofs.begin(), elem_dofs.end());

  bool done = true;

  // Next insert any dofs those might be constrained in terms
  // of.  Note that in this case we may not be done:  Those may
  // in turn depend on others.  So, we need to repeat this process
  // in that case until the system depends only on unconstrained
  // degrees of freedom.
  for (const auto & dof : elem_dofs)
    if (this->is_constrained_dof(dof))
      {
        // If the DOF is constrained
        DofConstraints::const_iterator
          pos = _dof_constraints.find(dof);

        libmesh_assert (pos != _dof_constraints.end());

        const DofConstraintRow & constraint_row = pos->second;

        // adaptive p refinement currently gives us lots of empty constraint
        // rows - we should optimize those DoFs away in the future.  [RHS]
        //libmesh_assert (!constraint_row.empty());

        // Add the DOFs this dof is constrained in terms of.
        // note that these dofs might also be constrained, so
        // we will need to call this function recursively.
        for (const auto & pr : constraint_row)
          if (!dof_set.count (pr.first))
            {
              dof_set.insert (pr.first);
              done = false;
            }
      }


  // If not done then we need to do more work
  // (obviously :-) )!
  if (!done)
    {
      // Fill the vector with the contents of the set
      elem_dofs.clear();
      elem_dofs.insert (elem_dofs.end(),
                        dof_set.begin(), dof_set.end());


      // May need to do this recursively.  It is possible
      // that we just replaced a constrained DOF with another
      // constrained DOF.
      this->find_connected_dofs (elem_dofs);

    } // end if (!done)
}

#endif // LIBMESH_ENABLE_CONSTRAINTS



void DofMap::print_info(std::ostream & os) const
{
  os << this->get_info();
}



std::string DofMap::get_info() const
{
  std::ostringstream os;

  // If we didn't calculate the exact sparsity pattern, the threaded
  // sparsity pattern assembly may have just given us an upper bound
  // on sparsity.
  const char * may_equal = " <= ";

  // If we calculated the exact sparsity pattern, then we can report
  // exact bandwidth figures:
  for (const auto & mat : _matrices)
    if (mat->need_full_sparsity_pattern())
      may_equal = " = ";

  dof_id_type max_n_nz = 0, max_n_oz = 0;
  long double avg_n_nz = 0, avg_n_oz = 0;

  if (_n_nz)
    {
      for (const auto & val : *_n_nz)
        {
          max_n_nz = std::max(max_n_nz, val);
          avg_n_nz += val;
        }

      std::size_t n_nz_size = _n_nz->size();

      this->comm().max(max_n_nz);
      this->comm().sum(avg_n_nz);
      this->comm().sum(n_nz_size);

      avg_n_nz /= std::max(n_nz_size,std::size_t(1));

      libmesh_assert(_n_oz);

      for (const auto & val : *_n_oz)
        {
          max_n_oz = std::max(max_n_oz, val);
          avg_n_oz += val;
        }

      std::size_t n_oz_size = _n_oz->size();

      this->comm().max(max_n_oz);
      this->comm().sum(avg_n_oz);
      this->comm().sum(n_oz_size);

      avg_n_oz /= std::max(n_oz_size,std::size_t(1));
    }

  os << "    DofMap Sparsity\n      Average  On-Processor Bandwidth"
     << may_equal << avg_n_nz << '\n';

  os << "      Average Off-Processor Bandwidth"
     << may_equal << avg_n_oz << '\n';

  os << "      Maximum  On-Processor Bandwidth"
     << may_equal << max_n_nz << '\n';

  os << "      Maximum Off-Processor Bandwidth"
     << may_equal << max_n_oz << std::endl;

#ifdef LIBMESH_ENABLE_CONSTRAINTS

  std::size_t n_constraints = 0, max_constraint_length = 0,
    n_rhss = 0;
  long double avg_constraint_length = 0.;

  for (const auto & pr : _dof_constraints)
    {
      // Only count local constraints, then sum later
      const dof_id_type constrained_dof = pr.first;
      if (!this->local_index(constrained_dof))
        continue;

      const DofConstraintRow & row = pr.second;
      std::size_t rowsize = row.size();

      max_constraint_length = std::max(max_constraint_length,
                                       rowsize);
      avg_constraint_length += rowsize;
      n_constraints++;

      if (_primal_constraint_values.count(constrained_dof))
        n_rhss++;
    }

  this->comm().sum(n_constraints);
  this->comm().sum(n_rhss);
  this->comm().sum(avg_constraint_length);
  this->comm().max(max_constraint_length);

  os << "    DofMap Constraints\n      Number of DoF Constraints = "
     << n_constraints;
  if (n_rhss)
    os << '\n'
       << "      Number of Heterogenous Constraints= " << n_rhss;
  if (n_constraints)
    {
      avg_constraint_length /= n_constraints;

      os << '\n'
         << "      Average DoF Constraint Length= " << avg_constraint_length;
    }

#ifdef LIBMESH_ENABLE_NODE_CONSTRAINTS
  std::size_t n_node_constraints = 0, max_node_constraint_length = 0,
    n_node_rhss = 0;
  long double avg_node_constraint_length = 0.;

  for (const auto & pr : _node_constraints)
    {
      // Only count local constraints, then sum later
      const Node * node = pr.first;
      if (node->processor_id() != this->processor_id())
        continue;

      const NodeConstraintRow & row = pr.second.first;
      std::size_t rowsize = row.size();

      max_node_constraint_length = std::max(max_node_constraint_length,
                                            rowsize);
      avg_node_constraint_length += rowsize;
      n_node_constraints++;

      if (pr.second.second != Point(0))
        n_node_rhss++;
    }

  this->comm().sum(n_node_constraints);
  this->comm().sum(n_node_rhss);
  this->comm().sum(avg_node_constraint_length);
  this->comm().max(max_node_constraint_length);

  os << "\n      Number of Node Constraints = " << n_node_constraints;
  if (n_node_rhss)
    os << '\n'
       << "      Number of Heterogenous Node Constraints= " << n_node_rhss;
  if (n_node_constraints)
    {
      avg_node_constraint_length /= n_node_constraints;
      os << "\n      Maximum Node Constraint Length= " << max_node_constraint_length
         << '\n'
         << "      Average Node Constraint Length= " << avg_node_constraint_length;
    }
#endif // LIBMESH_ENABLE_NODE_CONSTRAINTS

  os << std::endl;

#endif // LIBMESH_ENABLE_CONSTRAINTS

  return os.str();
}


template bool DofMap::is_evaluable<Elem>(const Elem &, unsigned int) const;
template bool DofMap::is_evaluable<Node>(const Node &, unsigned int) const;

} // namespace libMesh