fespace.cpp

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// Copyright (c) 2010-2026, Lawrence Livermore National Security, LLC. Produced
// at the Lawrence Livermore National Laboratory. All Rights reserved. See files
// LICENSE and NOTICE for details. LLNL-CODE-806117.
//
// This file is part of the MFEM library. For more information and source code
// availability visit https://mfem.org.
//
// MFEM is free software; you can redistribute it and/or modify it under the
// terms of the BSD-3 license. We welcome feedback and contributions, see file
// CONTRIBUTING.md for details.
 
// Implementation of FiniteElementSpace
 
#include "../general/text.hpp"
#include "../general/forall.hpp"
#include "../mesh/mesh_headers.hpp"
#include "fem.hpp"
#include "ceed/interface/util.hpp"
 
#include "derefmat_op.hpp"
 
#include <algorithm>
#include <cmath>
#include <cstdarg>
#include <unordered_map>
#include <unordered_set>
 
using namespace std;
 
namespace mfem
{
 
FiniteElementSpace::FiniteElementSpace()
   : mesh(NULL), fec(NULL), vdim(0), ordering(Ordering::byNODES),
     ndofs(0), nvdofs(0), nedofs(0), nfdofs(0), nbdofs(0),
     bdofs(NULL),
     elem_dof(NULL), elem_fos(NULL), bdr_elem_dof(NULL), bdr_elem_fos(NULL),
     face_dof(NULL),
     NURBSext(NULL), own_ext(false),
     cP_is_set(false),
     Th(Operator::ANY_TYPE),
     sequence(0), mesh_sequence(0), orders_changed(false), relaxed_hp(false)
{ }
 
FiniteElementSpace::FiniteElementSpace(const FiniteElementSpace &orig,
                                       Mesh *mesh_,
                                       const FiniteElementCollection *fec_)
{
   mesh_ = mesh_ ? mesh_ : orig.mesh;
   fec_ = fec_ ? fec_ : orig.fec;
 
   NURBSExtension *nurbs_ext = NULL;
   if (orig.NURBSext && orig.NURBSext != orig.mesh->NURBSext)
   {
#ifdef MFEM_USE_MPI
      ParNURBSExtension *pNURBSext =
         dynamic_cast<ParNURBSExtension *>(orig.NURBSext);
      if (pNURBSext)
      {
         nurbs_ext = new ParNURBSExtension(*pNURBSext);
      }
      else
#endif
      {
         nurbs_ext = new NURBSExtension(*orig.NURBSext);
      }
   }
 
   Constructor(mesh_, nurbs_ext, fec_, orig.vdim, orig.ordering);
}
 
FiniteElementSpace::FiniteElementSpace(Mesh *mesh,
                                       const FiniteElementCollection *fec,
                                       int vdim, int ordering)
{ Constructor(mesh, NULL, fec, vdim, ordering); }
 
FiniteElementSpace::FiniteElementSpace(Mesh *mesh, NURBSExtension *ext,
                                       const FiniteElementCollection *fec,
                                       int vdim, int ordering)
{ Constructor(mesh, ext, fec, vdim, ordering); }
 
void FiniteElementSpace::CopyProlongationAndRestriction(
   const FiniteElementSpace &fes, const Array<int> *perm)
{
   MFEM_VERIFY(cP == NULL, "");
   MFEM_VERIFY(cR == NULL, "");
 
   SparseMatrix *perm_mat = NULL, *perm_mat_tr = NULL;
   if (perm)
   {
      // Note: although n and fes.GetVSize() are typically equal, in
      // variable-order spaces they may differ, since nonconforming edges/faces
      // my have fictitious DOFs.
      int n = perm->Size();
      perm_mat = new SparseMatrix(n, fes.GetVSize());
      for (int i=0; i<n; ++i)
      {
         real_t s;
         int j = DecodeDof((*perm)[i], s);
         perm_mat->Set(i, j, s);
      }
      perm_mat->Finalize();
      perm_mat_tr = Transpose(*perm_mat);
   }
 
   if (fes.GetConformingProlongation() != NULL)
   {
      if (perm) { cP.reset(Mult(*perm_mat, *fes.GetConformingProlongation())); }
      else { cP.reset(new SparseMatrix(*fes.GetConformingProlongation())); }
      cP_is_set = true;
   }
   else if (perm != NULL)
   {
      cP.reset(perm_mat);
      cP_is_set = true;
      perm_mat = NULL;
   }
   if (fes.GetConformingRestriction() != NULL)
   {
      if (perm) { cR.reset(Mult(*fes.GetConformingRestriction(), *perm_mat_tr)); }
      else { cR.reset(new SparseMatrix(*fes.GetConformingRestriction())); }
   }
   else if (perm != NULL)
   {
      cR.reset(perm_mat_tr);
      perm_mat_tr = NULL;
   }
 
   delete perm_mat;
   delete perm_mat_tr;
}
 
void FiniteElementSpace::SetProlongation(const SparseMatrix& p)
{
#ifdef MFEM_USE_MPI
   MFEM_VERIFY(dynamic_cast<const ParFiniteElementSpace*>(this) == NULL,
               "Attempting to set serial prolongation operator for "
               "parallel finite element space.");
#endif
 
   if (!cP)
   {
      cP = std::unique_ptr<SparseMatrix>(new SparseMatrix(p));
   }
   else
   {
      *cP = p;
   }
   cP_is_set = true;
}
 
void FiniteElementSpace::SetRestriction(const SparseMatrix& r)
{
#ifdef MFEM_USE_MPI
   MFEM_VERIFY(dynamic_cast<const ParFiniteElementSpace*>(this) == NULL,
               "Attempting to set serial restriction operator for "
               "parallel finite element space.");
#endif
 
   if (!cR)
   {
      cR = std::unique_ptr<SparseMatrix>(new SparseMatrix(r));
   }
   else
   {
      *cR = r;
   }
}
 
void FiniteElementSpace::SetElementOrder(int i, int p)
{
   MFEM_VERIFY(mesh_sequence == mesh->GetSequence(),
               "Space has not been Updated() after a Mesh change.");
   MFEM_VERIFY(i >= 0 && i < GetNE(), "Invalid element index");
   MFEM_VERIFY(p >= 0 && p <= MaxVarOrder, "Order out of range");
   MFEM_ASSERT(!elem_order.Size() || elem_order.Size() == GetNE(),
               "Internal error");
 
   const bool change = elem_order.Size() == 0 || elem_order[i] != p;
   if (elem_order.Size() == 0)  // convert space to variable-order space
   {
      elem_order.SetSize(GetNE());
      elem_order = fec->GetOrder();
   }
 
   if (change)
   {
      elem_order[i] = p;
      orders_changed = true;
   }
 
   variableOrder = true;
}
 
int FiniteElementSpace::GetElementOrder(int i) const
{
   MFEM_VERIFY(mesh_sequence == mesh->GetSequence(),
               "Space has not been Updated() after a Mesh change.");
   MFEM_VERIFY(i >= 0 && i < GetNE(), "Invalid element index");
   MFEM_ASSERT(!elem_order.Size() || elem_order.Size() == GetNE(),
               "Internal error");
 
   return GetElementOrderImpl(i);
}
 
int FiniteElementSpace::GetElementOrderImpl(int i) const
{
   // (this is an internal version of GetElementOrder without asserts and checks)
   return elem_order.Size() ? elem_order[i] : fec->GetOrder();
}
 
void FiniteElementSpace::GetVDofs(int vd, Array<int>& dofs, int ndofs_) const
{
   if (ndofs_ < 0) { ndofs_ = this->ndofs; }
 
   if (ordering == Ordering::byNODES)
   {
      for (int i = 0; i < dofs.Size(); i++)
      {
         dofs[i] = Ordering::Map<Ordering::byNODES>(ndofs_, vdim, i, vd);
      }
   }
   else
   {
      for (int i = 0; i < dofs.Size(); i++)
      {
         dofs[i] = Ordering::Map<Ordering::byVDIM>(ndofs_, vdim, i, vd);
      }
   }
}
 
void FiniteElementSpace::DofsToVDofs(Array<int> &dofs, int ndofs_) const
{
   if (vdim == 1) { return; }
   if (ndofs_ < 0) { ndofs_ = this->ndofs; }
 
   if (ordering == Ordering::byNODES)
   {
      Ordering::DofsToVDofs<Ordering::byNODES>(ndofs_, vdim, dofs);
   }
   else
   {
      Ordering::DofsToVDofs<Ordering::byVDIM>(ndofs_, vdim, dofs);
   }
}
 
void FiniteElementSpace::DofsToVDofs(int vd, Array<int> &dofs, int ndofs_) const
{
   if (vdim == 1) { return; }
   if (ndofs_ < 0) { ndofs_ = this->ndofs; }
 
   if (ordering == Ordering::byNODES)
   {
      for (int i = 0; i < dofs.Size(); i++)
      {
         dofs[i] = Ordering::Map<Ordering::byNODES>(ndofs_, vdim, dofs[i], vd);
      }
   }
   else
   {
      for (int i = 0; i < dofs.Size(); i++)
      {
         dofs[i] = Ordering::Map<Ordering::byVDIM>(ndofs_, vdim, dofs[i], vd);
      }
   }
}
 
int FiniteElementSpace::DofToVDof(int dof, int vd, int ndofs_) const
{
   if (vdim == 1) { return dof; }
   if (ndofs_ < 0) { ndofs_ = this->ndofs; }
 
   if (ordering == Ordering::byNODES)
   {
      return Ordering::Map<Ordering::byNODES>(ndofs_, vdim, dof, vd);
   }
   else
   {
      return Ordering::Map<Ordering::byVDIM>(ndofs_, vdim, dof, vd);
   }
}
 
// static function
void FiniteElementSpace::AdjustVDofs(Array<int> &vdofs)
{
   int n = vdofs.Size(), *vdof = vdofs;
   for (int i = 0; i < n; i++) { vdof[i] = UnsignIndex(vdof[i]); }
}
 
void FiniteElementSpace::GetElementVDofs(int i, Array<int> &vdofs,
                                         DofTransformation &doftrans) const
{
   GetElementDofs(i, vdofs, doftrans);
   DofsToVDofs(vdofs);
   doftrans.SetVDim(vdim, ordering);
}
 
DofTransformation *
FiniteElementSpace::GetElementVDofs(int i, Array<int> &vdofs) const
{
   GetElementVDofs(i, vdofs, DoFTrans);
   return DoFTrans.GetDofTransformation() ? &DoFTrans : NULL;
}
 
void FiniteElementSpace::GetBdrElementVDofs(int i, Array<int> &vdofs,
                                            DofTransformation &doftrans) const
{
   GetBdrElementDofs(i, vdofs, doftrans);
   DofsToVDofs(vdofs);
   doftrans.SetVDim(vdim, ordering);
}
 
DofTransformation *
FiniteElementSpace::GetBdrElementVDofs(int i, Array<int> &vdofs) const
{
   GetBdrElementVDofs(i, vdofs, DoFTrans);
   return DoFTrans.GetDofTransformation() ? &DoFTrans : NULL;
}
 
void FiniteElementSpace::GetPatchVDofs(int i, Array<int> &vdofs) const
{
   GetPatchDofs(i, vdofs);
   DofsToVDofs(vdofs);
}
 
void FiniteElementSpace::GetFaceVDofs(int i, Array<int> &vdofs) const
{
   GetFaceDofs(i, vdofs);
   DofsToVDofs(vdofs);
}
 
void FiniteElementSpace::GetEdgeVDofs(int i, Array<int> &vdofs) const
{
   GetEdgeDofs(i, vdofs);
   DofsToVDofs(vdofs);
}
 
void FiniteElementSpace::GetVertexVDofs(int i, Array<int> &vdofs) const
{
   GetVertexDofs(i, vdofs);
   DofsToVDofs(vdofs);
}
 
void FiniteElementSpace::GetElementInteriorVDofs(int i, Array<int> &vdofs) const
{
   GetElementInteriorDofs(i, vdofs);
   DofsToVDofs(vdofs);
}
 
void FiniteElementSpace::GetEdgeInteriorVDofs(int i, Array<int> &vdofs) const
{
   GetEdgeInteriorDofs(i, vdofs);
   DofsToVDofs(vdofs);
}
 
void FiniteElementSpace::BuildElementToDofTable() const
{
   if (elem_dof) { return; }
 
   // TODO: can we call GetElementDofs only once per element?
   Table *el_dof = new Table;
   Table *el_fos = (mesh->Dimension() > 2) ? (new Table) : NULL;
   Array<int> dofs;
   Array<int> F, Fo;
   el_dof->MakeI(mesh->GetNE());
   if (el_fos) { el_fos->MakeI(mesh->GetNE()); }
   for (int i = 0; i < mesh->GetNE(); i++)
   {
      GetElementDofs(i, dofs);
      el_dof->AddColumnsInRow(i, dofs.Size());
 
      if (el_fos)
      {
         mesh->GetElementFaces(i, F, Fo);
         el_fos->AddColumnsInRow(i, Fo.Size());
      }
   }
   el_dof->MakeJ();
   if (el_fos) { el_fos->MakeJ(); }
   for (int i = 0; i < mesh->GetNE(); i++)
   {
      GetElementDofs(i, dofs);
      el_dof->AddConnections(i, (int *)dofs, dofs.Size());
 
      if (el_fos)
      {
         mesh->GetElementFaces(i, F, Fo);
         el_fos->AddConnections(i, (int *)Fo, Fo.Size());
      }
   }
   el_dof->ShiftUpI();
   if (el_fos) { el_fos->ShiftUpI(); }
   elem_dof = el_dof;
   elem_fos = el_fos;
}
 
void FiniteElementSpace::BuildBdrElementToDofTable() const
{
   if (bdr_elem_dof) { return; }
 
   Table *bel_dof = new Table;
   Table *bel_fos = (mesh->Dimension() == 3) ? (new Table) : NULL;
   Array<int> dofs;
   int F, Fo;
   bel_dof->MakeI(mesh->GetNBE());
   if (bel_fos) { bel_fos->MakeI(mesh->GetNBE()); }
   for (int i = 0; i < mesh->GetNBE(); i++)
   {
      GetBdrElementDofs(i, dofs);
      bel_dof->AddColumnsInRow(i, dofs.Size());
 
      if (bel_fos)
      {
         bel_fos->AddAColumnInRow(i);
      }
   }
   bel_dof->MakeJ();
   if (bel_fos) { bel_fos->MakeJ(); }
   for (int i = 0; i < mesh->GetNBE(); i++)
   {
      GetBdrElementDofs(i, dofs);
      bel_dof->AddConnections(i, (int *)dofs, dofs.Size());
 
      if (bel_fos)
      {
         mesh->GetBdrElementFace(i, &F, &Fo);
         bel_fos->AddConnection(i, Fo);
      }
   }
   bel_dof->ShiftUpI();
   if (bel_fos) { bel_fos->ShiftUpI(); }
   bdr_elem_dof = bel_dof;
   bdr_elem_fos = bel_fos;
}
 
void FiniteElementSpace::BuildFaceToDofTable() const
{
   // Here, "face" == (dim-1)-dimensional mesh entity.
 
   if (face_dof) { return; }
 
   if (NURBSext) { BuildNURBSFaceToDofTable(); return; }
 
   Table *fc_dof = new Table;
   Array<int> dofs;
   fc_dof->MakeI(mesh->GetNumFaces());
   for (int i = 0; i < fc_dof->Size(); i++)
   {
      GetFaceDofs(i, dofs, 0);
      fc_dof->AddColumnsInRow(i, dofs.Size());
   }
   fc_dof->MakeJ();
   for (int i = 0; i < fc_dof->Size(); i++)
   {
      GetFaceDofs(i, dofs, 0);
      fc_dof->AddConnections(i, (int *)dofs, dofs.Size());
   }
   fc_dof->ShiftUpI();
   face_dof = fc_dof;
}
 
void FiniteElementSpace::RebuildElementToDofTable()
{
   delete elem_dof;
   delete elem_fos;
   elem_dof = NULL;
   elem_fos = NULL;
   BuildElementToDofTable();
}
 
void FiniteElementSpace::ReorderElementToDofTable()
{
   Array<int> dof_marker(ndofs);
 
   dof_marker = -1;
 
   int *J = elem_dof->GetJ(), nnz = elem_dof->Size_of_connections();
   for (int k = 0, dof_counter = 0; k < nnz; k++)
   {
      const int sdof = J[k]; // signed dof
      const int dof = UnsignIndex(sdof);
      int new_dof = dof_marker[dof];
      if (new_dof < 0)
      {
         dof_marker[dof] = new_dof = dof_counter++;
      }
      // Preserve the sign of sdof
      J[k] = (sdof < 0) ? FlipIndexSign(new_dof) : new_dof;
   }
}
 
void FiniteElementSpace::BuildDofToArrays_() const
{
   if (dof_elem_array.Size()) { return; }
 
   BuildElementToDofTable();
 
   dof_elem_array.SetSize (ndofs);
   dof_ldof_array.SetSize (ndofs);
   dof_elem_array = -1;
   for (int i = 0; i < mesh -> GetNE(); i++)
   {
      const int *dofs = elem_dof -> GetRow(i);
      const int n = elem_dof -> RowSize(i);
      for (int j = 0; j < n; j++)
      {
         int dof = DecodeDof(dofs[j]);
         if (dof_elem_array[dof] < 0)
         {
            dof_elem_array[dof] = i;
            dof_ldof_array[dof] = j;
         }
      }
   }
}
 
void FiniteElementSpace::BuildDofToBdrArrays() const
{
   if (dof_bdr_elem_array.Size()) { return; }
 
   BuildBdrElementToDofTable();
 
   dof_bdr_elem_array.SetSize (ndofs);
   dof_bdr_ldof_array.SetSize (ndofs);
   dof_bdr_elem_array = -1;
   for (int i = 0; i < mesh -> GetNBE(); i++)
   {
      const int *dofs = bdr_elem_dof -> GetRow(i);
      const int n = bdr_elem_dof -> RowSize(i);
      for (int j = 0; j < n; j++)
      {
         int dof = DecodeDof(dofs[j]);
         if (dof_bdr_elem_array[dof] < 0)
         {
            dof_bdr_elem_array[dof] = i;
            dof_bdr_ldof_array[dof] = j;
         }
      }
   }
}
 
void MarkDofs(const Array<int> &dofs, Array<int> &mark_array)
{
   for (auto d : dofs)
   {
      mark_array[UnsignIndex(d)] = -1;
   }
}
 
void FiniteElementSpace::GetEssentialVDofs(const Array<int> &bdr_attr_is_ess,
                                           Array<int> &ess_vdofs,
                                           int component) const
{
   Array<int> dofs;
   ess_vdofs.SetSize(GetVSize());
   ess_vdofs = 0;
   for (int i = 0; i < GetNBE(); i++)
   {
      if (bdr_attr_is_ess[GetBdrAttribute(i)-1])
      {
         if (component < 0)
         {
            // Mark all components.
            GetBdrElementVDofs(i, dofs);
         }
         else
         {
            GetBdrElementDofs(i, dofs);
            for (auto &d : dofs) { d = DofToVDof(d, component); }
         }
         MarkDofs(dofs, ess_vdofs);
      }
   }
 
   // mark possible hidden boundary edges in a non-conforming mesh, also
   // local DOFs affected by boundary elements on other processors
   if (Nonconforming())
   {
      Array<int> bdr_verts, bdr_edges, bdr_faces;
      mesh->ncmesh->GetBoundaryClosure(bdr_attr_is_ess, bdr_verts, bdr_edges,
                                       bdr_faces);
      for (auto v : bdr_verts)
      {
         if (component < 0)
         {
            GetVertexVDofs(v, dofs);
         }
         else
         {
            GetVertexDofs(v, dofs);
            for (auto &d : dofs) { d = DofToVDof(d, component); }
         }
         MarkDofs(dofs, ess_vdofs);
      }
      for (auto e : bdr_edges)
      {
         if (component < 0)
         {
            GetEdgeVDofs(e, dofs);
         }
         else
         {
            GetEdgeDofs(e, dofs);
            for (auto &d : dofs) { d = DofToVDof(d, component); }
         }
         MarkDofs(dofs, ess_vdofs);
      }
      for (auto f : bdr_faces)
      {
         if (component < 0)
         {
            GetEntityVDofs(2, f, dofs);
         }
         else
         {
            GetEntityDofs(2, f, dofs);
            for (auto &d : dofs) { d = DofToVDof(d, component); }
         }
         MarkDofs(dofs, ess_vdofs);
      }
   }
}
 
void FiniteElementSpace::GetEssentialTrueDofs(const Array<int> &bdr_attr_is_ess,
                                              Array<int> &ess_tdof_list,
                                              int component) const
{
   Array<int> ess_vdofs, ess_tdofs;
   GetEssentialVDofs(bdr_attr_is_ess, ess_vdofs, component);
   const SparseMatrix *R = GetConformingRestriction();
   if (!R)
   {
      ess_tdofs.MakeRef(ess_vdofs);
   }
   else
   {
      R->BooleanMult(ess_vdofs, ess_tdofs);
#ifdef MFEM_DEBUG
      // Verify that in boolean arithmetic: P^T ess_dofs = R ess_dofs
      Array<int> ess_tdofs2(ess_tdofs.Size());
      GetConformingProlongation()->BooleanMultTranspose(ess_vdofs, ess_tdofs2);
 
      int counter = 0;
      std::string error_msg = "failed dof: ";
      auto ess_tdofs_ = ess_tdofs.HostRead();
      auto ess_tdofs2_ = ess_tdofs2.HostRead();
      for (int i = 0; i < ess_tdofs2.Size(); ++i)
      {
         if (bool(ess_tdofs_[i]) != bool(ess_tdofs2_[i]))
         {
            error_msg += std::to_string(i) += "(R ";
            error_msg += std::to_string(bool(ess_tdofs_[i])) += " P^T ";
            error_msg += std::to_string(bool(ess_tdofs2_[i])) += ") ";
            counter++;
         }
      }
 
      MFEM_ASSERT(R->Height() == GetConformingProlongation()->Width(), "!");
      MFEM_ASSERT(R->Width() == GetConformingProlongation()->Height(), "!");
      MFEM_ASSERT(R->Width() == ess_vdofs.Size(), "!");
      MFEM_VERIFY(counter == 0, "internal MFEM error: counter = " << counter
                  << ' ' << error_msg);
#endif
   }
   MarkerToList(ess_tdofs, ess_tdof_list);
}
 
void FiniteElementSpace::GetBoundaryTrueDofs(Array<int> &boundary_dofs,
                                             int component)
{
   if (mesh->bdr_attributes.Size())
   {
      Array<int> ess_bdr(mesh->bdr_attributes.Max());
      ess_bdr = 1;
      GetEssentialTrueDofs(ess_bdr, boundary_dofs, component);
   }
   else
   {
      boundary_dofs.DeleteAll();
   }
}
 
void FiniteElementSpace::GetExteriorVDofs(Array<int> &ext_vdofs,
                                          int component) const
{
   Array<int> dofs;
   ext_vdofs.SetSize(GetVSize());
   ext_vdofs = 0;
 
   Array<int> ext_face_marker;
   mesh->GetExteriorFaceMarker(ext_face_marker);
   for (int i = 0; i < ext_face_marker.Size(); i++)
   {
      if (ext_face_marker[i])
      {
         if (component < 0)
         {
            // Mark all components.
            GetFaceDofs(i, dofs);
            DofsToVDofs(dofs);
         }
         else
         {
            GetFaceDofs(i, dofs);
            for (auto &d : dofs) { d = DofToVDof(d, component); }
         }
         MarkDofs(dofs, ext_vdofs);
      }
   }
}
 
void FiniteElementSpace::GetExteriorTrueDofs(Array<int> &ext_tdof_list,
                                             int component) const
{
   Array<int> ext_vdofs, ext_tdofs;
   GetExteriorVDofs(ext_vdofs, component);
   const SparseMatrix *R = GetConformingRestriction();
   if (!R)
   {
      ext_tdofs.MakeRef(ext_vdofs);
   }
   else
   {
      R->BooleanMult(ext_vdofs, ext_tdofs);
#ifdef MFEM_DEBUG
      // Verify that in boolean arithmetic: P^T ext_dofs = R ext_dofs
      Array<int> ext_tdofs2(ext_tdofs.Size());
      GetConformingProlongation()->BooleanMultTranspose(ext_vdofs, ext_tdofs2);
 
      int counter = 0;
      std::string error_msg = "failed dof: ";
      auto ext_tdofs_ = ext_tdofs.HostRead();
      auto ext_tdofs2_ = ext_tdofs2.HostRead();
      for (int i = 0; i < ext_tdofs2.Size(); ++i)
      {
         if (bool(ext_tdofs_[i]) != bool(ext_tdofs2_[i]))
         {
            error_msg += std::to_string(i) += "(R ";
            error_msg += std::to_string(bool(ext_tdofs_[i])) += " P^T ";
            error_msg += std::to_string(bool(ext_tdofs2_[i])) += ") ";
            counter++;
         }
      }
 
      MFEM_ASSERT(R->Height() == GetConformingProlongation()->Width(), "!");
      MFEM_ASSERT(R->Width() == GetConformingProlongation()->Height(), "!");
      MFEM_ASSERT(R->Width() == ext_vdofs.Size(), "!");
      MFEM_VERIFY(counter == 0, "internal MFEM error: counter = " << counter
                  << ' ' << error_msg);
#endif
   }
   MarkerToList(ext_tdofs, ext_tdof_list);
}
 
// static method
void FiniteElementSpace::MarkerToList(const Array<int> &marker,
                                      Array<int> &list)
{
   int num_marked = 0;
   marker.HostRead(); // make sure we can read the array on host
   for (int i = 0; i < marker.Size(); i++)
   {
      if (marker[i]) { num_marked++; }
   }
   list.SetSize(0);
   list.HostWrite();
   list.Reserve(num_marked);
   for (int i = 0; i < marker.Size(); i++)
   {
      if (marker[i]) { list.Append(i); }
   }
}
 
// static method
void FiniteElementSpace::ListToMarker(const Array<int> &list, int marker_size,
                                      Array<int> &marker, int mark_val)
{
   list.HostRead(); // make sure we can read the array on host
   marker.SetSize(marker_size);
   marker.HostWrite();
   marker = 0;
   for (int i = 0; i < list.Size(); i++)
   {
      marker[list[i]] = mark_val;
   }
}
 
void FiniteElementSpace::ConvertToConformingVDofs(const Array<int> &dofs,
                                                  Array<int> &cdofs)
{
   GetConformingProlongation();
   if (cP) { cP->BooleanMultTranspose(dofs, cdofs); }
   else { dofs.Copy(cdofs); }
}
 
void FiniteElementSpace::ConvertFromConformingVDofs(const Array<int> &cdofs,
                                                    Array<int> &dofs)
{
   GetConformingRestriction();
   if (cR) { cR->BooleanMultTranspose(cdofs, dofs); }
   else { cdofs.Copy(dofs); }
}
 
SparseMatrix *
FiniteElementSpace::D2C_GlobalRestrictionMatrix (FiniteElementSpace *cfes)
{
   int i, j;
   Array<int> d_vdofs, c_vdofs;
   SparseMatrix *R;
 
   R = new SparseMatrix (cfes -> GetVSize(), GetVSize());
 
   for (i = 0; i < mesh -> GetNE(); i++)
   {
      this -> GetElementVDofs (i, d_vdofs);
      cfes -> GetElementVDofs (i, c_vdofs);
 
#ifdef MFEM_DEBUG
      if (d_vdofs.Size() != c_vdofs.Size())
      {
         mfem_error ("FiniteElementSpace::D2C_GlobalRestrictionMatrix (...)");
      }
#endif
 
      for (j = 0; j < d_vdofs.Size(); j++)
      {
         R -> Set (c_vdofs[j], d_vdofs[j], 1.0);
      }
   }
 
   R -> Finalize();
 
   return R;
}
 
SparseMatrix *
FiniteElementSpace::D2Const_GlobalRestrictionMatrix(FiniteElementSpace *cfes)
{
   int i, j;
   Array<int> d_dofs, c_dofs;
   SparseMatrix *R;
 
   R = new SparseMatrix (cfes -> GetNDofs(), ndofs);
 
   for (i = 0; i < mesh -> GetNE(); i++)
   {
      this -> GetElementDofs (i, d_dofs);
      cfes -> GetElementDofs (i, c_dofs);
 
#ifdef MFEM_DEBUG
      if (c_dofs.Size() != 1)
         mfem_error ("FiniteElementSpace::"
                     "D2Const_GlobalRestrictionMatrix (...)");
#endif
 
      for (j = 0; j < d_dofs.Size(); j++)
      {
         R -> Set (c_dofs[0], d_dofs[j], 1.0);
      }
   }
 
   R -> Finalize();
 
   return R;
}
 
SparseMatrix *
FiniteElementSpace::H2L_GlobalRestrictionMatrix (FiniteElementSpace *lfes)
{
   SparseMatrix *R;
   DenseMatrix loc_restr;
   Array<int> l_dofs, h_dofs, l_vdofs, h_vdofs;
 
   int lvdim = lfes->GetVDim();
   R = new SparseMatrix (lvdim * lfes -> GetNDofs(), lvdim * ndofs);
 
   Geometry::Type cached_geom = Geometry::INVALID;
   const FiniteElement *h_fe = NULL;
   const FiniteElement *l_fe = NULL;
   IsoparametricTransformation T;
 
   for (int i = 0; i < mesh -> GetNE(); i++)
   {
      this -> GetElementDofs (i, h_dofs);
      lfes -> GetElementDofs (i, l_dofs);
 
      // Assuming 'loc_restr' depends only on the Geometry::Type.
      const Geometry::Type geom = mesh->GetElementBaseGeometry(i);
      if (geom != cached_geom)
      {
         h_fe = this -> GetFE (i);
         l_fe = lfes -> GetFE (i);
         T.SetIdentityTransformation(h_fe->GetGeomType());
         h_fe->Project(*l_fe, T, loc_restr);
         cached_geom = geom;
      }
 
      for (int vd = 0; vd < lvdim; vd++)
      {
         l_dofs.Copy(l_vdofs);
         lfes->DofsToVDofs(vd, l_vdofs);
 
         h_dofs.Copy(h_vdofs);
         this->DofsToVDofs(vd, h_vdofs);
 
         R -> SetSubMatrix (l_vdofs, h_vdofs, loc_restr, 1);
      }
   }
 
   R -> Finalize();
 
   return R;
}
 
void FiniteElementSpace::AddDependencies(
   SparseMatrix& deps, Array<int>& master_dofs, Array<int>& slave_dofs,
   DenseMatrix& I, int skipfirst)
{
   for (int i = skipfirst; i < slave_dofs.Size(); i++)
   {
      const int sdof = slave_dofs[i];
      if (!deps.RowSize(sdof)) // not processed yet
      {
         for (int j = 0; j < master_dofs.Size(); j++)
         {
            const real_t coef = I(i, j);
            if (std::abs(coef) > 1e-12)
            {
               const int mdof = master_dofs[j];
               if (mdof != sdof && mdof != FlipIndexSign(sdof))
               {
                  deps.Add(sdof, mdof, coef);
               }
            }
         }
      }
   }
}
 
void FiniteElementSpace::AddEdgeFaceDependencies(
   SparseMatrix &deps, Array<int> &master_dofs, const FiniteElement *master_fe,
   Array<int> &slave_dofs, int slave_face, const DenseMatrix *pm) const
{
   // In variable-order spaces in 3D, we need to only constrain interior face
   // DOFs (this is done one level up), since edge dependencies can be more
   // complex and are primarily handled by edge-edge dependencies. The one
   // exception is edges of slave faces that lie in the interior of the master
   // face, which are not covered by edge-edge relations. This function finds
   // such edges and makes them constrained by the master face.
   // See also https://github.com/mfem/mfem/pull/1423#issuecomment-633916643
 
   Array<int> V, E, Eo; // TODO: LocalArray
   mesh->GetFaceVertices(slave_face, V);
   mesh->GetFaceEdges(slave_face, E, Eo);
   MFEM_ASSERT(V.Size() == E.Size(), "");
 
   DenseMatrix I;
   IsoparametricTransformation edge_T;
   edge_T.SetFE(&SegmentFE);
 
   // constrain each edge of the slave face
   for (int i = 0; i < E.Size(); i++)
   {
      int a = i, b = (i+1) % V.Size();
      if (V[a] > V[b]) { std::swap(a, b); }
 
      DenseMatrix &edge_pm = edge_T.GetPointMat();
      edge_pm.SetSize(2, 2);
 
      // copy two points from the face point matrix
      real_t mid[2];
      for (int j = 0; j < 2; j++)
      {
         edge_pm(j, 0) = (*pm)(j, a);
         edge_pm(j, 1) = (*pm)(j, b);
         mid[j] = 0.5*((*pm)(j, a) + (*pm)(j, b));
      }
 
      // check that the edge does not coincide with the master face's edge
      const real_t eps = 1e-14;
      if (mid[0] > eps && mid[0] < 1-eps &&
          mid[1] > eps && mid[1] < 1-eps)
      {
         int order = GetEdgeDofs(E[i], slave_dofs, 0);
 
         const auto *edge_fe = fec->GetFE(Geometry::SEGMENT, order);
         edge_fe->GetTransferMatrix(*master_fe, edge_T, I);
 
         AddDependencies(deps, master_dofs, slave_dofs, I, 0);
      }
   }
}
 
bool FiniteElementSpace::DofFinalizable(int dof, const Array<bool>& finalized,
                                        const SparseMatrix& deps)
{
   const int* dep = deps.GetRowColumns(dof);
   int ndep = deps.RowSize(dof);
 
   // are all constraining DOFs finalized?
   for (int i = 0; i < ndep; i++)
   {
      if (!finalized[dep[i]]) { return false; }
   }
   return true;
}
 
int FiniteElementSpace::GetDegenerateFaceDofs(int index, Array<int> &dofs,
                                              Geometry::Type master_geom,
                                              int variant) const
{
   // In NC meshes with prisms/tets, a special constraint occurs where a
   // prism/tet edge is slave to another element's face (see illustration
   // here: https://github.com/mfem/mfem/pull/713#issuecomment-495786362)
   // Rather than introduce a new edge-face constraint type, we handle such
   // cases as degenerate face-face constraints, where the point-matrix
   // rectangle has zero height. This method returns DOFs for the first edge
   // of the rectangle, duplicated in the orthogonal direction, to resemble
   // DOFs for a quadrilateral face. The extra DOFs are ignored by
   // FiniteElementSpace::AddDependencies.
 
   Array<int> edof;
   int order = GetEdgeDofs(FlipIndexSign(index), edof, variant);
 
   int nv = fec->DofForGeometry(Geometry::POINT);
   int ne = fec->DofForGeometry(Geometry::SEGMENT);
   int nn = 2*nv + ne;
 
   dofs.SetSize(nn*nn);
   if (!dofs.Size()) { return 0; }
 
   dofs = edof[0];
 
   // copy first two vertex DOFs
   for (int i = 0; i < nv; i++)
   {
      dofs[i] = edof[i];
      dofs[nv+i] = edof[nv+i];
   }
   // copy first edge DOFs
   int face_vert = Geometry::NumVerts[master_geom];
   for (int i = 0; i < ne; i++)
   {
      dofs[face_vert*nv + i] = edof[2*nv + i];
   }
 
   return order;
}
 
int FiniteElementSpace::GetNumBorderDofs(Geometry::Type geom, int order) const
{
   // return the number of vertex and edge DOFs that precede inner DOFs
   const int nv = fec->GetNumDof(Geometry::POINT, order);
   const int ne = fec->GetNumDof(Geometry::SEGMENT, order);
 
   return Geometry::NumVerts[geom] * (geom == Geometry::SEGMENT ? nv : (nv + ne));
}
 
int FiniteElementSpace::GetEntityDofs(int entity, int index, Array<int> &dofs,
                                      Geometry::Type master_geom,
                                      int variant) const
{
   switch (entity)
   {
      case 0:
         GetVertexDofs(index, dofs);
         return 0;
 
      case 1:
         return GetEdgeDofs(index, dofs, variant);
 
      default:
         if (index >= 0)
         {
            return GetFaceDofs(index, dofs, variant);
         }
         else
         {
            return GetDegenerateFaceDofs(index, dofs, master_geom, variant);
         }
   }
}
 
int FiniteElementSpace::GetEntityVDofs(int entity, int index, Array<int> &dofs,
                                       Geometry::Type master_geom,
                                       int variant) const
{
   const int n = GetEntityDofs(entity, index, dofs, master_geom, variant);
   DofsToVDofs(dofs);
   return n;
}
 
// Variable-order spaces: enforce minimum rule on conforming edges/faces
void FiniteElementSpace::VariableOrderMinimumRule(SparseMatrix & deps) const
{
   if (!IsVariableOrder()) { return; }
 
   Array<int> master_dofs, slave_dofs;
 
   IsoparametricTransformation T;
   DenseMatrix I;
 
   for (int entity = 1; entity < mesh->Dimension(); entity++)
   {
      const Table &ent_dofs = (entity == 1) ? var_edge_dofs : var_face_dofs;
      const int num_ent = (entity == 1) ? mesh->GetNEdges() : mesh->GetNFaces();
      MFEM_ASSERT(ent_dofs.Size() >= num_ent+1, "");
 
      // add constraints within edges/faces holding multiple DOF sets
      Geometry::Type last_geom = Geometry::INVALID;
      for (int i = 0; i < num_ent; i++)
      {
         if (ent_dofs.RowSize(i) <= 1) { continue; }
 
         Geometry::Type geom =
            (entity == 1) ? Geometry::SEGMENT : mesh->GetFaceGeometry(i);
 
         if (geom != last_geom)
         {
            T.SetIdentityTransformation(geom);
            last_geom = geom;
         }
 
         // get lowest order variant DOFs and FE
         const int p = GetEntityDofs(entity, i, master_dofs, geom, 0);
         const auto *master_fe = fec->GetFE(geom, p);
         if (!master_fe) { break; }
 
         // constrain all higher order DOFs: interpolate lowest order function
         for (int variant = 1; ; variant++)
         {
            const int q = GetEntityDofs(entity, i, slave_dofs, geom, variant);
            if (q < 0) { break; }
 
            const auto *slave_fe = fec->GetFE(geom, q);
            slave_fe->GetTransferMatrix(*master_fe, T, I);
 
            AddDependencies(deps, master_dofs, slave_dofs, I);
         }
      }
   }
}
 
void FiniteElementSpace::BuildConformingInterpolation() const
{
#ifdef MFEM_USE_MPI
   MFEM_VERIFY(dynamic_cast<const ParFiniteElementSpace*>(this) == NULL,
               "This method should not be used with a ParFiniteElementSpace!");
#endif
 
   if (cP_is_set) { return; }
   cP_is_set = true;
 
   if (FEColl()->GetContType() == FiniteElementCollection::DISCONTINUOUS)
   {
      cP.reset();
      cR.reset();
      cR_hp.reset();
      R_transpose.reset();
      return;
   }
 
   Array<int> master_dofs, slave_dofs, highest_dofs;
 
   IsoparametricTransformation T;
   DenseMatrix I;
 
   // For each slave DOF, the dependency matrix will contain a row that
   // expresses the slave DOF as a linear combination of its immediate master
   // DOFs. Rows of independent DOFs will remain empty.
   SparseMatrix deps(ndofs);
 
   // Inverse dependencies for the cR_hp matrix in variable-order spaces:
   // For each master edge/face with more DOF sets, the inverse dependency
   // matrix contains a row that expresses the master true DOF (lowest order)
   // as a linear combination of the highest order set of DOFs.
   SparseMatrix inv_deps(ndofs);
 
   VariableOrderMinimumRule(deps);
 
   // Collect local face/edge dependencies, starting with faces
   for (int entity = 2; entity >= 1; entity--)
   {
      const NCMesh::NCList &list = mesh->ncmesh->GetNCList(entity);
      if (!list.masters.Size()) { continue; }
 
      // loop through all master edges/faces, constrain their slave edges/faces
      for (const NCMesh::Master &master : list.masters)
      {
         Geometry::Type master_geom = master.Geom();
 
         const int p = GetEntityDofs(entity, master.index, master_dofs,
                                     master_geom);
         if (!master_dofs.Size()) { continue; }
 
         const FiniteElement *master_fe = fec->GetFE(master_geom, p);
         if (!master_fe) { continue; }
 
         switch (master_geom)
         {
            case Geometry::SQUARE:   T.SetFE(&QuadrilateralFE); break;
            case Geometry::TRIANGLE: T.SetFE(&TriangleFE); break;
            case Geometry::SEGMENT:  T.SetFE(&SegmentFE); break;
            default: MFEM_ABORT("unsupported geometry");
         }
 
         for (int si = master.slaves_begin; si < master.slaves_end; si++)
         {
            const NCMesh::Slave &slave = list.slaves[si];
 
            int q = GetEntityDofs(entity, slave.index, slave_dofs, master_geom);
            if (!slave_dofs.Size()) { break; }
 
            const FiniteElement *slave_fe = fec->GetFE(slave.Geom(), q);
            list.OrientedPointMatrix(slave, T.GetPointMat());
            slave_fe->GetTransferMatrix(*master_fe, T, I);
 
            // variable-order spaces: face edges need to be handled separately
            int skipfirst = 0;
            if (IsVariableOrder() && entity == 2 && slave.index >= 0)
            {
               skipfirst = GetNumBorderDofs(master_geom, q);
            }
 
            // make each slave DOF dependent on all master DOFs
            AddDependencies(deps, master_dofs, slave_dofs, I, skipfirst);
 
            if (skipfirst)
            {
               // constrain internal edge DOFs if they were skipped
               const auto *pm = list.point_matrices[master_geom][slave.matrix];
               AddEdgeFaceDependencies(deps, master_dofs, master_fe,
                                       slave_dofs, slave.index, pm);
            }
         }
 
         // Add inverse dependencies for the cR_hp matrix; if a master has
         // more DOF sets, the lowest order set interpolates the highest one.
         if (IsVariableOrder())
         {
            int nvar = GetNVariants(entity, master.index);
            if (nvar > 1)
            {
               const int q = GetEntityDofs(entity, master.index, highest_dofs,
                                           master_geom, nvar-1);
               const auto *highest_fe = fec->GetFE(master_geom, q);
 
               T.SetIdentityTransformation(master_geom);
               master_fe->GetTransferMatrix(*highest_fe, T, I);
 
               // add dependencies only for the inner dofs
               const int skip = GetNumBorderDofs(master_geom, p);
               AddDependencies(inv_deps, highest_dofs, master_dofs, I, skip);
            }
         }
      }
   }
 
   deps.Finalize();
   inv_deps.Finalize();
 
   // DOFs that stayed independent are true DOFs
   int n_true_dofs = 0;
   for (int i = 0; i < ndofs; i++)
   {
      if (!deps.RowSize(i)) { n_true_dofs++; }
   }
 
   // if all dofs are true dofs leave cP and cR NULL
   if (n_true_dofs == ndofs)
   {
      cP.reset();
      cR.reset();
      cR_hp.reset();
      R_transpose.reset();
      return;
   }
 
   // create the conforming prolongation matrix cP
   cP.reset(new SparseMatrix(ndofs, n_true_dofs));
 
   // create the conforming restriction matrix cR
   int *cR_J;
   {
      int *cR_I = Memory<int>(n_true_dofs+1);
      real_t *cR_A = Memory<real_t>(n_true_dofs);
      cR_J = Memory<int>(n_true_dofs);
      for (int i = 0; i < n_true_dofs; i++)
      {
         cR_I[i] = i;
         cR_A[i] = 1.0;
      }
      cR_I[n_true_dofs] = n_true_dofs;
      cR.reset(new SparseMatrix(cR_I, cR_J, cR_A, n_true_dofs, ndofs));
   }
 
   // In variable-order spaces, create the restriction matrix cR_hp, which is
   // similar to cR but has interpolation of the master edge/face DOFs of
   // maximum order per edge/face, since the maximum order is on an adjacent
   // element (e.g. where projection would be computed).
   if (IsVariableOrder())
   {
      cR_hp.reset(new SparseMatrix(n_true_dofs, ndofs));
   }
   else
   {
      cR_hp.reset();
   }
 
   Array<bool> finalized(ndofs);
   finalized = false;
 
   Array<int> cols;
   Vector srow;
 
   // Put identity in the prolongation matrix for true DOFs, and set cR_hp
   for (int i = 0, true_dof = 0; i < ndofs; i++)
   {
      if (!deps.RowSize(i)) // true dof
      {
         cP->Add(i, true_dof, 1.0);
         cR_J[true_dof] = i;
         finalized[i] = true;
 
         if (cR_hp)
         {
            if (inv_deps.RowSize(i))
            {
               inv_deps.GetRow(i, cols, srow);
               cR_hp->AddRow(true_dof, cols, srow);
            }
            else
            {
               cR_hp->Add(true_dof, i, 1.0);
            }
         }
 
         true_dof++;
      }
   }
 
   // Now calculate cP rows of slave DOFs as combinations of cP rows of their
   // master DOFs. It is possible that some slave DOFs depend on DOFs that are
   // themselves slaves. Here we resolve such indirect constraints by first
   // calculating rows of the cP matrix for DOFs whose master DOF cP rows are
   // already known (in the first iteration these are the true DOFs). In the
   // second iteration, slaves of slaves can be 'finalized' (given a row in the
   // cP matrix), in the third iteration slaves of slaves of slaves, etc.
   bool finished;
   int n_finalized = n_true_dofs;
   do
   {
      finished = true;
      for (int dof = 0; dof < ndofs; dof++)
      {
         if (!finalized[dof] && DofFinalizable(dof, finalized, deps))
         {
            const int* dep_col = deps.GetRowColumns(dof);
            const real_t* dep_coef = deps.GetRowEntries(dof);
            int n_dep = deps.RowSize(dof);
 
            for (int j = 0; j < n_dep; j++)
            {
               cP->GetRow(dep_col[j], cols, srow);
               srow *= dep_coef[j];
               cP->AddRow(dof, cols, srow);
            }
 
            finalized[dof] = true;
            n_finalized++;
            finished = false;
         }
      }
   }
   while (!finished);
 
   // If everything is consistent (mesh, face orientations, etc.), we should
   // be able to finalize all slave DOFs, otherwise it's a serious error.
   MFEM_VERIFY(n_finalized == ndofs,
               "Error creating cP matrix: n_finalized = "
               << n_finalized << ", ndofs = " << ndofs);
 
   cP->Finalize();
   if (cR_hp) { cR_hp->Finalize(); }
 
   if (vdim > 1)
   {
      MakeVDimMatrix(*cP);
      MakeVDimMatrix(*cR);
      if (cR_hp) { MakeVDimMatrix(*cR_hp); }
   }
}
 
void FiniteElementSpace::MakeVDimMatrix(SparseMatrix &mat) const
{
   if (vdim == 1) { return; }
 
   int height = mat.Height();
   int width = mat.Width();
 
   SparseMatrix *vmat = new SparseMatrix(vdim*height, vdim*width);
 
   Array<int> dofs, vdofs;
   Vector srow;
   for (int i = 0; i < height; i++)
   {
      mat.GetRow(i, dofs, srow);
      for (int vd = 0; vd < vdim; vd++)
      {
         dofs.Copy(vdofs);
         DofsToVDofs(vd, vdofs, width);
         vmat->SetRow(DofToVDof(i, vd, height), vdofs, srow);
      }
   }
   vmat->Finalize();
 
   mat.Swap(*vmat);
   delete vmat;
}
 
 
const SparseMatrix* FiniteElementSpace::GetConformingProlongation() const
{
   if (Conforming()) { return NULL; }
   if (!cP_is_set) { BuildConformingInterpolation(); }
   return cP.get();
}
 
const SparseMatrix* FiniteElementSpace::GetConformingRestriction() const
{
   if (Conforming()) { return NULL; }
   if (!cP_is_set) { BuildConformingInterpolation(); }
   if (cR && !R_transpose) { R_transpose.reset(new TransposeOperator(*cR)); }
   return cR.get();
}
 
const SparseMatrix* FiniteElementSpace::GetHpConformingRestriction() const
{
   if (Conforming()) { return NULL; }
   if (!cP_is_set) { BuildConformingInterpolation(); }
   return IsVariableOrder() ? cR_hp.get() : cR.get();
}
 
const Operator *FiniteElementSpace::GetRestrictionTransposeOperator() const
{
   GetRestrictionOperator(); // Ensure that R_transpose is built
   return R_transpose.get();
}
 
int FiniteElementSpace::GetNConformingDofs() const
{
   const SparseMatrix* P = GetConformingProlongation();
   return P ? (P->Width() / vdim) : ndofs;
}
 
int FiniteElementSpace::GetVectorDim() const
{
   const FiniteElement *fe = GetTypicalFE();
   if (fe->GetRangeType() == FiniteElement::SCALAR)
   {
      return GetVDim();
   }
   return GetVDim()*std::max(GetMesh()->SpaceDimension(), fe->GetRangeDim());
}
 
int FiniteElementSpace::GetCurlDim() const
{
   const FiniteElement *fe = GetTypicalFE();
   if (fe->GetRangeType() == FiniteElement::SCALAR)
   {
      return 2 * GetMesh()->SpaceDimension() - 3;
   }
   return GetVDim()*fe->GetCurlDim();
}
 
const ElementRestrictionOperator *FiniteElementSpace::GetElementRestriction(
   ElementDofOrdering e_ordering) const
{
   // Check if we have a discontinuous space using the FE collection:
   if (IsDGSpace())
   {
      // TODO: when VDIM is 1, we can return IdentityOperator.
      if (L2E_nat.Ptr() == NULL)
      {
         // The input L-vector layout is:
         // * ND x NE x VDIM, for Ordering::byNODES, or
         // * VDIM x ND x NE, for Ordering::byVDIM.
         // The output E-vector layout is: ND x VDIM x NE.
         L2E_nat.Reset(new L2ElementRestriction(*this));
      }
      return L2E_nat.Is<ElementRestrictionOperator>();
   }
   if (e_ordering == ElementDofOrdering::LEXICOGRAPHIC)
   {
      if (L2E_lex.Ptr() == NULL)
      {
         L2E_lex.Reset(new ElementRestriction(*this, e_ordering));
      }
      return L2E_lex.Is<ElementRestrictionOperator>();
   }
   // e_ordering == ElementDofOrdering::NATIVE
   if (L2E_nat.Ptr() == NULL)
   {
      L2E_nat.Reset(new ElementRestriction(*this, e_ordering));
   }
   return L2E_nat.Is<ElementRestrictionOperator>();
}
 
const FaceRestriction *FiniteElementSpace::GetFaceRestriction(
   ElementDofOrdering f_ordering, FaceType type, L2FaceValues mul) const
{
   const bool is_dg_space = IsDGSpace();
   const L2FaceValues m = (is_dg_space && mul==L2FaceValues::DoubleValued) ?
                          L2FaceValues::DoubleValued : L2FaceValues::SingleValued;
   auto key = std::make_tuple(is_dg_space, f_ordering, type, m);
   auto itr = L2F.find(key);
   if (itr != L2F.end())
   {
      return itr->second.get();
   }
   else
   {
      std::unique_ptr<FaceRestriction> res;
      if (is_dg_space)
      {
         if (Conforming())
         {
            res.reset(new L2FaceRestriction(*this, f_ordering, type, m));
         }
         else
         {
            res.reset(new NCL2FaceRestriction(*this, f_ordering, type, m));
         }
      }
      else if (dynamic_cast<const DG_Interface_FECollection*>(fec))
      {
         res.reset(new L2InterfaceFaceRestriction(*this, f_ordering, type));
      }
      else
      {
         res.reset(new ConformingFaceRestriction(*this, f_ordering, type));
      }
      return L2F.emplace(key, std::move(res)).first->second.get();
   }
}
 
const InterpolationManager &FiniteElementSpace::GetInterpolationManager(
   ElementDofOrdering f_ordering, FaceType type) const
{
   const auto key = make_tuple(f_ordering, type);
 
   auto it = interpolations.find(key);
   if (it != interpolations.end())
   {
      return *it->second;
   }
   else
   {
      auto interp = make_unique<InterpolationManager>(*this, f_ordering, type);
 
      int face_idx = 0;
      for (int f = 0; f < mesh->GetNumFacesWithGhost(); ++f)
      {
         Mesh::FaceInformation face = mesh->GetFaceInformation(f);
         if (!face.IsOfFaceType(type) || face.IsNonconformingCoarse())
         {
            continue;
         }
         if (face.IsConforming() || face.IsBoundary())
         {
            interp->RegisterFaceConformingInterpolation(face, face_idx);
         }
         else
         {
            interp->RegisterFaceCoarseToFineInterpolation(face, face_idx);
         }
         ++face_idx;
      }
 
      // Transform the interpolation matrix map into contiguous memory.
      interp->LinearizeInterpolatorMapIntoVector();
      interp->InitializeNCInterpConfig();
 
      return *interpolations.emplace(key, std::move(interp)).first->second;
   }
}
 
const QuadratureInterpolator *FiniteElementSpace::GetQuadratureInterpolator(
   const IntegrationRule &ir) const
{
   if (!QuadratureInterpolator::SupportsFESpace(*this))
   {
      return nullptr;
   }
 
   for (int i = 0; i < E2Q_array.Size(); i++)
   {
      const QuadratureInterpolator *qi = E2Q_array[i];
      if (qi->IntRule == &ir) { return qi; }
   }
 
   QuadratureInterpolator *qi = new QuadratureInterpolator(*this, ir);
   E2Q_array.Append(qi);
   return qi;
}
 
const QuadratureInterpolator *FiniteElementSpace::GetQuadratureInterpolator(
   const QuadratureSpace &qs) const
{
   if (!QuadratureInterpolator::SupportsFESpace(*this))
   {
      return nullptr;
   }
 
   for (int i = 0; i < E2Q_array.Size(); i++)
   {
      const QuadratureInterpolator *qi = E2Q_array[i];
      if (qi->qspace == &qs) { return qi; }
   }
 
   QuadratureInterpolator *qi = new QuadratureInterpolator(*this, qs);
   E2Q_array.Append(qi);
   return qi;
}
 
const FaceQuadratureInterpolator
*FiniteElementSpace::GetFaceQuadratureInterpolator(
   const IntegrationRule &ir, FaceType type) const
{
   if (!FaceQuadratureInterpolator::SupportsFESpace(*this))
   {
      return nullptr;
   }
 
   if (type==FaceType::Interior)
   {
      for (int i = 0; i < E2IFQ_array.Size(); i++)
      {
         const FaceQuadratureInterpolator *qi = E2IFQ_array[i];
         if (qi->IntRule == &ir) { return qi; }
      }
 
      FaceQuadratureInterpolator *qi = new FaceQuadratureInterpolator(*this, ir,
                                                                      type);
      E2IFQ_array.Append(qi);
      return qi;
   }
   else //Boundary
   {
      for (int i = 0; i < E2BFQ_array.Size(); i++)
      {
         const FaceQuadratureInterpolator *qi = E2BFQ_array[i];
         if (qi->IntRule == &ir) { return qi; }
      }
 
      FaceQuadratureInterpolator *qi = new FaceQuadratureInterpolator(*this, ir,
                                                                      type);
      E2BFQ_array.Append(qi);
      return qi;
   }
}
 
SparseMatrix *FiniteElementSpace::RefinementMatrix_main(
   const int coarse_ndofs, const Table &coarse_elem_dof,
   const Table *coarse_elem_fos, const DenseTensor localP[]) const
{
   /// TODO: Implement DofTransformation support
 
   MFEM_VERIFY(mesh->GetLastOperation() == Mesh::REFINE, "");
 
   Array<int> dofs, coarse_dofs, coarse_vdofs;
   Vector row;
 
   Mesh::GeometryList elem_geoms(*mesh);
 
   SparseMatrix *P;
   if (elem_geoms.Size() == 1)
   {
      const int coarse_ldof = localP[elem_geoms[0]].SizeJ();
      P = new SparseMatrix(GetVSize(), coarse_ndofs*vdim, coarse_ldof);
   }
   else
   {
      P = new SparseMatrix(GetVSize(), coarse_ndofs*vdim);
   }
 
   Array<int> mark(P->Height());
   mark = 0;
 
   const CoarseFineTransformations &rtrans = mesh->GetRefinementTransforms();
 
   for (int k = 0; k < mesh->GetNE(); k++)
   {
      const Embedding &emb = rtrans.embeddings[k];
      const Geometry::Type geom = mesh->GetElementBaseGeometry(k);
      const DenseMatrix &lP = localP[geom](emb.matrix);
      const int fine_ldof = localP[geom].SizeI();
 
      elem_dof->GetRow(k, dofs);
      coarse_elem_dof.GetRow(emb.parent, coarse_dofs);
 
      for (int vd = 0; vd < vdim; vd++)
      {
         coarse_dofs.Copy(coarse_vdofs);
         DofsToVDofs(vd, coarse_vdofs, coarse_ndofs);
 
         for (int i = 0; i < fine_ldof; i++)
         {
            const int r = DofToVDof(dofs[i], vd);
            const int m = UnsignIndex(r);
 
            if (!mark[m])
            {
               lP.GetRow(i, row);
               P->SetRow(r, coarse_vdofs, row);
               mark[m] = 1;
            }
         }
      }
   }
 
   MFEM_ASSERT(mark.Sum() == P->Height(), "Not all rows of P set.");
   if (elem_geoms.Size() != 1) { P->Finalize(); }
   return P;
}
 
SparseMatrix *FiniteElementSpace::VariableOrderRefinementMatrix(
   const int coarse_ndofs, const Table &coarse_elem_dof) const
{
   MFEM_VERIFY(mesh->GetLastOperation() == Mesh::REFINE, "");
 
   Array<int> dofs, coarse_dofs, coarse_vdofs;
   Vector row;
 
   Mesh::GeometryList elem_geoms(*mesh);
 
   SparseMatrix *P = new SparseMatrix(GetVSize(), coarse_ndofs*vdim);
 
   Array<int> mark(P->Height());
   mark = 0;
 
   const CoarseFineTransformations &rtrans = mesh->GetRefinementTransforms();
   DenseMatrix lP;
   IsoparametricTransformation isotr;
   for (int k = 0; k < mesh->GetNE(); k++)
   {
      const Embedding &emb = rtrans.embeddings[k];
      const Geometry::Type geom = mesh->GetElementBaseGeometry(k);
 
      const FiniteElement *fe = GetFE(k);
      isotr.SetIdentityTransformation(geom);
      const int ldof = fe->GetDof();
      lP.SetSize(ldof, ldof);
      const DenseTensor &pmats = rtrans.point_matrices[geom];
      isotr.SetPointMat(pmats(emb.matrix));
      fe->GetLocalInterpolation(isotr, lP);
 
      const int fine_ldof = lP.Height();
 
      elem_dof->GetRow(k, dofs);
      coarse_elem_dof.GetRow(emb.parent, coarse_dofs);
 
      for (int vd = 0; vd < vdim; vd++)
      {
         coarse_dofs.Copy(coarse_vdofs);
         DofsToVDofs(vd, coarse_vdofs, coarse_ndofs);
 
         for (int i = 0; i < fine_ldof; i++)
         {
            const int r = DofToVDof(dofs[i], vd);
            const int m = UnsignIndex(r);
 
            if (!mark[m])
            {
               lP.GetRow(i, row);
               P->SetRow(r, coarse_vdofs, row);
               mark[m] = 1;
            }
         }
      }
   }
 
   MFEM_VERIFY(mark.Sum() == P->Height(), "Not all rows of P set.");
   P->Finalize();
   return P;
}
 
void FiniteElementSpace::GetLocalRefinementMatrices(
   Geometry::Type geom, DenseTensor &localP) const
{
   const FiniteElement *fe = fec->FiniteElementForGeometry(geom);
 
   const CoarseFineTransformations &rtrans = mesh->GetRefinementTransforms();
   const DenseTensor &pmats = rtrans.point_matrices[geom];
 
   int nmat = pmats.SizeK();
   int ldof = fe->GetDof();
 
   IsoparametricTransformation isotr;
   isotr.SetIdentityTransformation(geom);
 
   // calculate local interpolation matrices for all refinement types
   localP.SetSize(ldof, ldof, nmat);
   for (int i = 0; i < nmat; i++)
   {
      isotr.SetPointMat(pmats(i));
      fe->GetLocalInterpolation(isotr, localP(i));
   }
}
 
SparseMatrix* FiniteElementSpace::RefinementMatrix(int old_ndofs,
                                                   const Table* old_elem_dof,
                                                   const Table* old_elem_fos)
{
   MFEM_VERIFY(GetNE() >= old_elem_dof->Size(),
               "Previous mesh is not coarser.");
 
   Mesh::GeometryList elem_geoms(*mesh);
   if (!IsVariableOrder())
   {
      DenseTensor localP[Geometry::NumGeom];
      for (int i = 0; i < elem_geoms.Size(); i++)
      {
         GetLocalRefinementMatrices(elem_geoms[i], localP[elem_geoms[i]]);
      }
      return RefinementMatrix_main(old_ndofs, *old_elem_dof, old_elem_fos,
                                   localP);
   }
   else
   {
      return VariableOrderRefinementMatrix(old_ndofs, *old_elem_dof);
   }
}
 
FiniteElementSpace::RefinementOperator::RefinementOperator(
   const FiniteElementSpace* fespace, Table* old_elem_dof, Table* old_elem_fos,
   int old_ndofs)
   : fespace(fespace),
     old_elem_dof(old_elem_dof),
     old_elem_fos(old_elem_fos)
{
   MFEM_VERIFY(fespace->GetNE() >= old_elem_dof->Size(),
               "Previous mesh is not coarser.");
 
   width = old_ndofs * fespace->GetVDim();
   height = fespace->GetVSize();
 
   Mesh::GeometryList elem_geoms(*fespace->GetMesh());
 
   if (!fespace->IsVariableOrder())
   {
      for (int i = 0; i < elem_geoms.Size(); i++)
      {
         fespace->GetLocalRefinementMatrices(elem_geoms[i], localP[elem_geoms[i]]);
      }
   }
 
   ConstructDoFTransArray();
}
 
FiniteElementSpace::RefinementOperator::RefinementOperator(
   const FiniteElementSpace *fespace, const FiniteElementSpace *coarse_fes)
   : Operator(fespace->GetVSize(), coarse_fes->GetVSize()),
     fespace(fespace), old_elem_dof(NULL), old_elem_fos(NULL)
{
   Mesh::GeometryList elem_geoms(*fespace->GetMesh());
 
   if (!fespace->IsVariableOrder())
   {
      for (int i = 0; i < elem_geoms.Size(); i++)
      {
         fespace->GetLocalRefinementMatrices(*coarse_fes, elem_geoms[i],
                                             localP[elem_geoms[i]]);
      }
   }
 
   // Make a copy of the coarse elem_dof Table.
   old_elem_dof = new Table(coarse_fes->GetElementToDofTable());
 
   // Make a copy of the coarse elem_fos Table if it exists.
   if (coarse_fes->GetElementToFaceOrientationTable())
   {
      old_elem_fos = new Table(*coarse_fes->GetElementToFaceOrientationTable());
   }
 
   ConstructDoFTransArray();
}
 
FiniteElementSpace::RefinementOperator::~RefinementOperator()
{
   delete old_elem_dof;
   delete old_elem_fos;
   for (int i=0; i<old_DoFTransArray.Size(); i++)
   {
      delete old_DoFTransArray[i];
   }
}
 
void FiniteElementSpace::RefinementOperator::ConstructDoFTransArray()
{
   old_DoFTransArray.SetSize(Geometry::NUM_GEOMETRIES);
   for (int i=0; i<old_DoFTransArray.Size(); i++)
   {
      old_DoFTransArray[i] = NULL;
   }
 
   const FiniteElementCollection *fec_ref = fespace->FEColl();
   if (dynamic_cast<const ND_FECollection*>(fec_ref))
   {
      const FiniteElement *nd_tri =
         fec_ref->FiniteElementForGeometry(Geometry::TRIANGLE);
      if (nd_tri)
      {
         old_DoFTransArray[Geometry::TRIANGLE] =
            new ND_TriDofTransformation(nd_tri->GetOrder());
      }
 
      const FiniteElement *nd_tet =
         fec_ref->FiniteElementForGeometry(Geometry::TETRAHEDRON);
      if (nd_tet)
      {
         old_DoFTransArray[Geometry::TETRAHEDRON] =
            new ND_TetDofTransformation(nd_tet->GetOrder());
      }
 
      const FiniteElement *nd_pri =
         fec_ref->FiniteElementForGeometry(Geometry::PRISM);
      if (nd_pri)
      {
         old_DoFTransArray[Geometry::PRISM] =
            new ND_WedgeDofTransformation(nd_pri->GetOrder());
      }
 
      const FiniteElement *nd_pyr =
         fec_ref->FiniteElementForGeometry(Geometry::PYRAMID);
      if (nd_pyr)
      {
         old_DoFTransArray[Geometry::PYRAMID] =
            new ND_PyramidDofTransformation(nd_pyr->GetOrder());
      }
   }
}
 
void FiniteElementSpace::RefinementOperator::Mult(const Vector &x,
                                                  Vector &y) const
{
   Mesh* mesh_ref = fespace->GetMesh();
   const CoarseFineTransformations &trans_ref =
      mesh_ref->GetRefinementTransforms();
 
   Array<int> dofs, vdofs, old_dofs, old_vdofs, old_Fo;
 
   int rvdim = fespace->GetVDim();
   int old_ndofs = width / rvdim;
 
   Vector subY, subX;
 
   DenseMatrix eP;
   IsoparametricTransformation isotr;
   DofTransformation doftrans;
 
   for (int k = 0; k < mesh_ref->GetNE(); k++)
   {
      const Embedding &emb = trans_ref.embeddings[k];
      const Geometry::Type geom = mesh_ref->GetElementBaseGeometry(k);
      if (fespace->IsVariableOrder())
      {
         const FiniteElement *fe = fespace->GetFE(k);
         isotr.SetIdentityTransformation(geom);
         const int ldof = fe->GetDof();
         eP.SetSize(ldof, ldof);
         const DenseTensor &pmats = trans_ref.point_matrices[geom];
         isotr.SetPointMat(pmats(emb.matrix));
         fe->GetLocalInterpolation(isotr, eP);
      }
      const DenseMatrix &lP = (fespace->IsVariableOrder()) ? eP : localP[geom](
                                 emb.matrix);
 
      subY.SetSize(lP.Height());
 
      fespace->GetElementDofs(k, dofs, doftrans);
      old_elem_dof->GetRow(emb.parent, old_dofs);
 
      if (doftrans.IsIdentity())
      {
         for (int vd = 0; vd < rvdim; vd++)
         {
            dofs.Copy(vdofs);
            fespace->DofsToVDofs(vd, vdofs);
            old_dofs.Copy(old_vdofs);
            fespace->DofsToVDofs(vd, old_vdofs, old_ndofs);
 
            x.GetSubVector(old_vdofs, subX);
            lP.Mult(subX, subY);
            y.SetSubVector(vdofs, subY);
         }
      }
      else
      {
         old_elem_fos->GetRow(emb.parent, old_Fo);
         old_DoFTrans.SetDofTransformation(*old_DoFTransArray[geom]);
         old_DoFTrans.SetFaceOrientations(old_Fo);
 
         doftrans.SetVDim();
         for (int vd = 0; vd < rvdim; vd++)
         {
            dofs.Copy(vdofs);
            fespace->DofsToVDofs(vd, vdofs);
            old_dofs.Copy(old_vdofs);
            fespace->DofsToVDofs(vd, old_vdofs, old_ndofs);
 
            x.GetSubVector(old_vdofs, subX);
            old_DoFTrans.InvTransformPrimal(subX);
            lP.Mult(subX, subY);
            doftrans.TransformPrimal(subY);
            y.SetSubVector(vdofs, subY);
         }
         doftrans.SetVDim(rvdim, fespace->GetOrdering());
      }
   }
}
 
void FiniteElementSpace::RefinementOperator::MultTranspose(const Vector &x,
                                                           Vector &y) const
{
   y = 0.0;
 
   Mesh* mesh_ref = fespace->GetMesh();
   const CoarseFineTransformations &trans_ref =
      mesh_ref->GetRefinementTransforms();
 
   Array<char> processed(fespace->GetVSize());
   processed = 0;
 
   Array<int> f_dofs, c_dofs, f_vdofs, c_vdofs, old_Fo;
 
   int rvdim = fespace->GetVDim();
   int old_ndofs = width / rvdim;
 
   Vector subY, subX, subYt;
 
   DenseMatrix eP;
   IsoparametricTransformation isotr;
   const FiniteElement *fe = nullptr;
   DofTransformation doftrans;
 
   for (int k = 0; k < mesh_ref->GetNE(); k++)
   {
      const Embedding &emb = trans_ref.embeddings[k];
      const Geometry::Type geom = mesh_ref->GetElementBaseGeometry(k);
 
      if (fespace->IsVariableOrder())
      {
         fe = fespace->GetFE(k);
         isotr.SetIdentityTransformation(geom);
         const int ldof = fe->GetDof();
         eP.SetSize(ldof);
         const DenseTensor &pmats = trans_ref.point_matrices[geom];
         isotr.SetPointMat(pmats(emb.matrix));
         fe->GetLocalInterpolation(isotr, eP);
      }
 
      const DenseMatrix &lP = (fespace->IsVariableOrder()) ? eP : localP[geom](
                                 emb.matrix);
 
      fespace->GetElementDofs(k, f_dofs, doftrans);
      old_elem_dof->GetRow(emb.parent, c_dofs);
 
      if (doftrans.IsIdentity())
      {
         subY.SetSize(lP.Width());
 
         for (int vd = 0; vd < rvdim; vd++)
         {
            f_dofs.Copy(f_vdofs);
            fespace->DofsToVDofs(vd, f_vdofs);
            c_dofs.Copy(c_vdofs);
            fespace->DofsToVDofs(vd, c_vdofs, old_ndofs);
 
            x.GetSubVector(f_vdofs, subX);
            for (int p = 0; p < f_dofs.Size(); ++p)
            {
               if (processed[DecodeDof(f_dofs[p])])
               {
                  subX[p] = 0.0;
               }
            }
            lP.MultTranspose(subX, subY);
            y.AddElementVector(c_vdofs, subY);
         }
      }
      else
      {
         subYt.SetSize(lP.Width());
 
         old_elem_fos->GetRow(emb.parent, old_Fo);
         old_DoFTrans.SetDofTransformation(*old_DoFTransArray[geom]);
         old_DoFTrans.SetFaceOrientations(old_Fo);
 
         doftrans.SetVDim();
         for (int vd = 0; vd < rvdim; vd++)
         {
            f_dofs.Copy(f_vdofs);
            fespace->DofsToVDofs(vd, f_vdofs);
            c_dofs.Copy(c_vdofs);
            fespace->DofsToVDofs(vd, c_vdofs, old_ndofs);
 
            x.GetSubVector(f_vdofs, subX);
            doftrans.InvTransformDual(subX);
            for (int p = 0; p < f_dofs.Size(); ++p)
            {
               if (processed[DecodeDof(f_dofs[p])])
               {
                  subX[p] = 0.0;
               }
            }
            lP.MultTranspose(subX, subYt);
            old_DoFTrans.TransformDual(subYt);
            y.AddElementVector(c_vdofs, subYt);
         }
         doftrans.SetVDim(rvdim, fespace->GetOrdering());
      }
 
      for (int p = 0; p < f_dofs.Size(); ++p)
      {
         processed[DecodeDof(f_dofs[p])] = 1;
      }
   }
}
 
namespace internal
{
 
// Used in GetCoarseToFineMap() below.
struct RefType
{
   Geometry::Type geom;
   int num_children;
   const Pair<int,int> *children;
 
   RefType(Geometry::Type g, int n, const Pair<int,int> *c)
      : geom(g), num_children(n), children(c) { }
 
   bool operator<(const RefType &other) const
   {
      if (geom < other.geom) { return true; }
      if (geom > other.geom) { return false; }
      if (num_children < other.num_children) { return true; }
      if (num_children > other.num_children) { return false; }
      for (int i = 0; i < num_children; i++)
      {
         if (children[i].one < other.children[i].one) { return true; }
         if (children[i].one > other.children[i].one) { return false; }
      }
      return false; // everything is equal
   }
};
 
void GetCoarseToFineMap(const CoarseFineTransformations &cft,
                        const mfem::Mesh &fine_mesh,
                        Table &coarse_to_fine,
                        Array<int> &coarse_to_ref_type,
                        Table &ref_type_to_matrix,
                        Array<Geometry::Type> &ref_type_to_geom)
{
   const int fine_ne = cft.embeddings.Size();
   int coarse_ne = -1;
   for (int i = 0; i < fine_ne; i++)
   {
      coarse_ne = std::max(coarse_ne, cft.embeddings[i].parent);
   }
   coarse_ne++;
 
   coarse_to_ref_type.SetSize(coarse_ne);
   coarse_to_fine.SetDims(coarse_ne, fine_ne);
 
   Array<int> cf_i(coarse_to_fine.GetI(), coarse_ne+1);
   Array<Pair<int,int> > cf_j(fine_ne);
   cf_i = 0;
   for (int i = 0; i < fine_ne; i++)
   {
      cf_i[cft.embeddings[i].parent+1]++;
   }
   cf_i.PartialSum();
   MFEM_ASSERT(cf_i.Last() == cf_j.Size(), "internal error");
   for (int i = 0; i < fine_ne; i++)
   {
      const Embedding &e = cft.embeddings[i];
      cf_j[cf_i[e.parent]].one = e.matrix; // used as sort key below
      cf_j[cf_i[e.parent]].two = i;
      cf_i[e.parent]++;
   }
   std::copy_backward(cf_i.begin(), cf_i.end()-1, cf_i.end());
   cf_i[0] = 0;
   for (int i = 0; i < coarse_ne; i++)
   {
      std::sort(&cf_j[cf_i[i]], cf_j.GetData() + cf_i[i+1]);
   }
   for (int i = 0; i < fine_ne; i++)
   {
      coarse_to_fine.GetJ()[i] = cf_j[i].two;
   }
 
   using std::map;
   using std::pair;
 
   map<RefType,int> ref_type_map;
   for (int i = 0; i < coarse_ne; i++)
   {
      const int num_children = cf_i[i+1]-cf_i[i];
      MFEM_ASSERT(num_children > 0, "");
      const int fine_el = cf_j[cf_i[i]].two;
      // Assuming the coarse and the fine elements have the same geometry:
      const Geometry::Type geom = fine_mesh.GetElementBaseGeometry(fine_el);
      const RefType ref_type(geom, num_children, &cf_j[cf_i[i]]);
      pair<map<RefType,int>::iterator,bool> res =
         ref_type_map.insert(
            pair<const RefType,int>(ref_type, (int)ref_type_map.size()));
      coarse_to_ref_type[i] = res.first->second;
   }
 
   ref_type_to_matrix.MakeI((int)ref_type_map.size());
   ref_type_to_geom.SetSize((int)ref_type_map.size());
   for (map<RefType,int>::iterator it = ref_type_map.begin();
        it != ref_type_map.end(); ++it)
   {
      ref_type_to_matrix.AddColumnsInRow(it->second, it->first.num_children);
      ref_type_to_geom[it->second] = it->first.geom;
   }
 
   ref_type_to_matrix.MakeJ();
   for (map<RefType,int>::iterator it = ref_type_map.begin();
        it != ref_type_map.end(); ++it)
   {
      const RefType &rt = it->first;
      for (int j = 0; j < rt.num_children; j++)
      {
         ref_type_to_matrix.AddConnection(it->second, rt.children[j].one);
      }
   }
   ref_type_to_matrix.ShiftUpI();
}
 
} // namespace internal
 
 
/// TODO: Implement DofTransformation support
FiniteElementSpace::DerefinementOperator::DerefinementOperator(
   const FiniteElementSpace *f_fes, const FiniteElementSpace *c_fes,
   BilinearFormIntegrator *mass_integ)
   : Operator(c_fes->GetVSize(), f_fes->GetVSize()),
     fine_fes(f_fes)
{
   MFEM_VERIFY(c_fes->GetOrdering() == f_fes->GetOrdering() &&
               c_fes->GetVDim() == f_fes->GetVDim(),
               "incompatible coarse and fine FE spaces");
 
   IsoparametricTransformation emb_tr;
   Mesh *f_mesh = f_fes->GetMesh();
   const CoarseFineTransformations &rtrans = f_mesh->GetRefinementTransforms();
 
   Mesh::GeometryList elem_geoms(*f_mesh);
   DenseTensor localP[Geometry::NumGeom], localM[Geometry::NumGeom];
   for (int gi = 0; gi < elem_geoms.Size(); gi++)
   {
      const Geometry::Type geom = elem_geoms[gi];
      DenseTensor &lP = localP[geom], &lM = localM[geom];
      const FiniteElement *fine_fe =
         f_fes->fec->FiniteElementForGeometry(geom);
      const FiniteElement *coarse_fe =
         c_fes->fec->FiniteElementForGeometry(geom);
      const DenseTensor &pmats = rtrans.point_matrices[geom];
 
      lP.SetSize(fine_fe->GetDof(), coarse_fe->GetDof(), pmats.SizeK());
      lM.SetSize(fine_fe->GetDof(),   fine_fe->GetDof(), pmats.SizeK());
      emb_tr.SetIdentityTransformation(geom);
      for (int i = 0; i < pmats.SizeK(); i++)
      {
         emb_tr.SetPointMat(pmats(i));
         // Get the local interpolation matrix for this refinement type
         fine_fe->GetTransferMatrix(*coarse_fe, emb_tr, lP(i));
         // Get the local mass matrix for this refinement type
         mass_integ->AssembleElementMatrix(*fine_fe, emb_tr, lM(i));
      }
   }
 
   Table ref_type_to_matrix;
   internal::GetCoarseToFineMap(rtrans, *f_mesh, coarse_to_fine,
                                coarse_to_ref_type, ref_type_to_matrix,
                                ref_type_to_geom);
   MFEM_ASSERT(coarse_to_fine.Size() == c_fes->GetNE(), "");
 
   const int total_ref_types = ref_type_to_geom.Size();
   int num_ref_types[Geometry::NumGeom], num_fine_elems[Geometry::NumGeom];
   Array<int> ref_type_to_coarse_elem_offset(total_ref_types);
   ref_type_to_fine_elem_offset.SetSize(total_ref_types);
   std::fill(num_ref_types, num_ref_types+Geometry::NumGeom, 0);
   std::fill(num_fine_elems, num_fine_elems+Geometry::NumGeom, 0);
   for (int i = 0; i < total_ref_types; i++)
   {
      Geometry::Type g = ref_type_to_geom[i];
      ref_type_to_coarse_elem_offset[i] = num_ref_types[g];
      ref_type_to_fine_elem_offset[i] = num_fine_elems[g];
      num_ref_types[g]++;
      num_fine_elems[g] += ref_type_to_matrix.RowSize(i);
   }
   DenseTensor localPtMP[Geometry::NumGeom];
   for (int g = 0; g < Geometry::NumGeom; g++)
   {
      if (num_ref_types[g] == 0) { continue; }
      const int fine_dofs = localP[g].SizeI();
      const int coarse_dofs = localP[g].SizeJ();
      localPtMP[g].SetSize(coarse_dofs, coarse_dofs, num_ref_types[g]);
      localR[g].SetSize(coarse_dofs, fine_dofs, num_fine_elems[g]);
   }
   for (int i = 0; i < total_ref_types; i++)
   {
      Geometry::Type g = ref_type_to_geom[i];
      DenseMatrix &lPtMP = localPtMP[g](ref_type_to_coarse_elem_offset[i]);
      int lR_offset = ref_type_to_fine_elem_offset[i]; // offset in localR[g]
      const int *mi = ref_type_to_matrix.GetRow(i);
      const int nm = ref_type_to_matrix.RowSize(i);
      lPtMP = 0.0;
      for (int s = 0; s < nm; s++)
      {
         DenseMatrix &lP = localP[g](mi[s]);
         DenseMatrix &lM = localM[g](mi[s]);
         DenseMatrix &lR = localR[g](lR_offset+s);
         MultAtB(lP, lM, lR); // lR = lP^T lM
         mfem::AddMult(lR, lP, lPtMP); // lPtMP += lP^T lM lP
      }
      DenseMatrixInverse lPtMP_inv(lPtMP);
      for (int s = 0; s < nm; s++)
      {
         DenseMatrix &lR = localR[g](lR_offset+s);
         lPtMP_inv.Mult(lR); // lR <- (P^T M P)^{-1} P^T M
      }
   }
 
   // Make a copy of the coarse element-to-dof Table.
   coarse_elem_dof = new Table(c_fes->GetElementToDofTable());
}
 
FiniteElementSpace::DerefinementOperator::~DerefinementOperator()
{
   delete coarse_elem_dof;
}
 
void FiniteElementSpace::DerefinementOperator::Mult(const Vector &x,
                                                    Vector &y) const
{
   Array<int> c_vdofs, f_vdofs;
   Vector loc_x, loc_y;
   DenseMatrix loc_x_mat, loc_y_mat;
   const int fine_vdim = fine_fes->GetVDim();
   const int coarse_ndofs = height/fine_vdim;
   for (int coarse_el = 0; coarse_el < coarse_to_fine.Size(); coarse_el++)
   {
      coarse_elem_dof->GetRow(coarse_el, c_vdofs);
      fine_fes->DofsToVDofs(c_vdofs, coarse_ndofs);
      loc_y.SetSize(c_vdofs.Size());
      loc_y = 0.0;
      loc_y_mat.UseExternalData(loc_y.GetData(), c_vdofs.Size()/fine_vdim,
                                fine_vdim);
      const int ref_type = coarse_to_ref_type[coarse_el];
      const Geometry::Type geom = ref_type_to_geom[ref_type];
      const int *fine_elems = coarse_to_fine.GetRow(coarse_el);
      const int num_fine_elems = coarse_to_fine.RowSize(coarse_el);
      const int lR_offset = ref_type_to_fine_elem_offset[ref_type];
      for (int s = 0; s < num_fine_elems; s++)
      {
         const DenseMatrix &lR = localR[geom](lR_offset+s);
         fine_fes->GetElementVDofs(fine_elems[s], f_vdofs);
         x.GetSubVector(f_vdofs, loc_x);
         loc_x_mat.UseExternalData(loc_x.GetData(), f_vdofs.Size()/fine_vdim,
                                   fine_vdim);
         mfem::AddMult(lR, loc_x_mat, loc_y_mat);
      }
      y.SetSubVector(c_vdofs, loc_y);
   }
}
 
void FiniteElementSpace::GetLocalDerefinementMatrices(Geometry::Type geom,
                                                      DenseTensor &localR) const
{
   const FiniteElement *fe = fec->FiniteElementForGeometry(geom);
 
   const CoarseFineTransformations &dtrans =
      mesh->ncmesh->GetDerefinementTransforms();
   const DenseTensor &pmats = dtrans.point_matrices[geom];
 
   const int nmat = pmats.SizeK();
   const int ldof = fe->GetDof();
 
   IsoparametricTransformation isotr;
   isotr.SetIdentityTransformation(geom);
 
   // calculate local restriction matrices for all refinement types
   localR.SetSize(ldof, ldof, nmat);
   for (int i = 0; i < nmat; i++)
   {
      isotr.SetPointMat(pmats(i));
      fe->GetLocalRestriction(isotr, localR(i));
   }
}
 
SparseMatrix* FiniteElementSpace::DerefinementMatrix(int old_ndofs,
                                                     const Table* old_elem_dof,
                                                     const Table* old_elem_fos)
{
   /// TODO: Implement DofTransformation support
 
   MFEM_VERIFY(Nonconforming(), "Not implemented for conforming meshes.");
   MFEM_VERIFY(old_ndofs, "Missing previous (finer) space.");
   MFEM_VERIFY(ndofs <= old_ndofs, "Previous space is not finer.");
 
   Array<int> dofs, old_dofs, old_vdofs;
   Vector row;
 
   Mesh::GeometryList elem_geoms(*mesh);
 
   DenseTensor localR[Geometry::NumGeom];
   if (!IsVariableOrder())
   {
      for (int i = 0; i < elem_geoms.Size(); i++)
      {
         GetLocalDerefinementMatrices(elem_geoms[i], localR[elem_geoms[i]]);
      }
   }
 
   SparseMatrix *R = new SparseMatrix(ndofs*vdim, old_ndofs*vdim);
 
   Array<int> mark(R->Height());
   mark = 0;
 
   const CoarseFineTransformations &dtrans =
      mesh->ncmesh->GetDerefinementTransforms();
 
   MFEM_ASSERT(dtrans.embeddings.Size() == old_elem_dof->Size(), "");
 
   bool is_dg = FEColl()->GetContType() == FiniteElementCollection::DISCONTINUOUS;
   int num_marked = 0;
   const FiniteElement *fe = nullptr;
   DenseMatrix localRVO; //for variable-order only
   for (int k = 0; k < dtrans.embeddings.Size(); k++)
   {
      const Embedding &emb = dtrans.embeddings[k];
      Geometry::Type geom = mesh->GetElementBaseGeometry(emb.parent);
 
      if (IsVariableOrder())
      {
         fe = GetFE(emb.parent);
         const DenseTensor &pmats = dtrans.point_matrices[geom];
         const int ldof = fe->GetDof();
 
         IsoparametricTransformation isotr;
         isotr.SetIdentityTransformation(geom);
 
         localRVO.SetSize(ldof, ldof);
         isotr.SetPointMat(pmats(emb.matrix));
         // Local restriction is size ldofxldof assuming that the parent and
         // child are of same polynomial order.
         fe->GetLocalRestriction(isotr, localRVO);
      }
      DenseMatrix &lR = IsVariableOrder() ? localRVO : localR[geom](emb.matrix);
 
      elem_dof->GetRow(emb.parent, dofs);
      old_elem_dof->GetRow(k, old_dofs);
      MFEM_VERIFY(old_dofs.Size() == dofs.Size(),
                  "Parent and child must have same #dofs.");
 
      for (int vd = 0; vd < vdim; vd++)
      {
         old_dofs.Copy(old_vdofs);
         DofsToVDofs(vd, old_vdofs, old_ndofs);
 
         for (int i = 0; i < lR.Height(); i++)
         {
            if (!std::isfinite(lR(i, 0))) { continue; }
 
            const int r = DofToVDof(dofs[i], vd);
            const int m = UnsignIndex(r);
 
            if (is_dg || !mark[m])
            {
               lR.GetRow(i, row);
               R->SetRow(r, old_vdofs, row);
 
               mark[m] = 1;
               num_marked++;
            }
         }
      }
   }
 
   if (!is_dg && !IsVariableOrder())
   {
      MFEM_VERIFY(num_marked == R->Height(),
                  "internal error: not all rows of R were set.");
   }
 
   R->Finalize(); // no-op if fixed width
   return R;
}
 
void FiniteElementSpace::GetLocalRefinementMatrices(
   const FiniteElementSpace &coarse_fes, Geometry::Type geom,
   DenseTensor &localP) const
{
   // Assumptions: see the declaration of the method.
 
   const FiniteElement *fine_fe = fec->FiniteElementForGeometry(geom);
   const FiniteElement *coarse_fe =
      coarse_fes.fec->FiniteElementForGeometry(geom);
 
   const CoarseFineTransformations &rtrans = mesh->GetRefinementTransforms();
   const DenseTensor &pmats = rtrans.point_matrices[geom];
 
   int nmat = pmats.SizeK();
 
   IsoparametricTransformation isotr;
   isotr.SetIdentityTransformation(geom);
 
   // Calculate the local interpolation matrices for all refinement types
   localP.SetSize(fine_fe->GetDof(), coarse_fe->GetDof(), nmat);
   for (int i = 0; i < nmat; i++)
   {
      isotr.SetPointMat(pmats(i));
      fine_fe->GetTransferMatrix(*coarse_fe, isotr, localP(i));
   }
}
 
void FiniteElementSpace::Constructor(Mesh *mesh_, NURBSExtension *NURBSext_,
                                     const FiniteElementCollection *fec_,
                                     int vdim_, int ordering_)
{
   mesh = mesh_;
   fec = fec_;
   vdim = vdim_;
   ordering = (Ordering::Type) ordering_;
 
   elem_dof = NULL;
   elem_fos = NULL;
   face_dof = NULL;
 
   sequence = 0;
   orders_changed = false;
   relaxed_hp = false;
 
   Th.SetType(Operator::ANY_TYPE);
 
   const NURBSFECollection *nurbs_fec =
      dynamic_cast<const NURBSFECollection *>(fec_);
 
   if (nurbs_fec)
   {
      MFEM_VERIFY(mesh_->NURBSext, "NURBS FE space requires a NURBS mesh.");
 
      if (NURBSext_ == NULL)
      {
         NURBSext = mesh_->NURBSext;
         own_ext = 0;
      }
      else
      {
         NURBSext = NURBSext_;
         own_ext = 1;
      }
      UpdateNURBS();
      cP.reset();
      cR.reset();
      cR_hp.reset();
      R_transpose.reset();
      cP_is_set = false;
 
      ConstructDoFTransArray();
   }
   else
   {
      NURBSext = NULL;
      own_ext = 0;
      Construct();
   }
 
   BuildElementToDofTable();
}
 
void FiniteElementSpace::ConstructDoFTransArray()
{
   DestroyDoFTransArray();
 
   DoFTransArray.SetSize(Geometry::NUM_GEOMETRIES);
   for (int i=0; i<DoFTransArray.Size(); i++)
   {
      DoFTransArray[i] = NULL;
   }
   if (mesh->Dimension() < 3) { return; }
   if (dynamic_cast<const ND_FECollection*>(fec))
   {
      const FiniteElement *nd_tri =
         fec->FiniteElementForGeometry(Geometry::TRIANGLE);
      if (nd_tri)
      {
         DoFTransArray[Geometry::TRIANGLE] =
            new ND_TriDofTransformation(nd_tri->GetOrder());
      }
 
      const FiniteElement *nd_tet =
         fec->FiniteElementForGeometry(Geometry::TETRAHEDRON);
      if (nd_tet)
      {
         DoFTransArray[Geometry::TETRAHEDRON] =
            new ND_TetDofTransformation(nd_tet->GetOrder());
      }
 
      const FiniteElement *nd_pri =
         fec->FiniteElementForGeometry(Geometry::PRISM);
      if (nd_pri)
      {
         DoFTransArray[Geometry::PRISM] =
            new ND_WedgeDofTransformation(nd_pri->GetOrder());
      }
 
      const FiniteElement *nd_pyr =
         fec->FiniteElementForGeometry(Geometry::PYRAMID);
      if (nd_pyr)
      {
         DoFTransArray[Geometry::PYRAMID] =
            new ND_PyramidDofTransformation(nd_pyr->GetOrder());
      }
   }
}
 
NURBSExtension *FiniteElementSpace::StealNURBSext()
{
   if (NURBSext && !own_ext)
   {
      mfem_error("FiniteElementSpace::StealNURBSext");
   }
   own_ext = 0;
 
   return NURBSext;
}
 
void FiniteElementSpace::UpdateNURBS()
{
   MFEM_VERIFY(NURBSext, "NURBSExt not defined.");
 
   nvdofs = 0;
   nedofs = 0;
   nfdofs = 0;
   nbdofs = 0;
   bdofs = NULL;
 
   delete face_dof;
   face_dof = NULL;
   face_to_be.DeleteAll();
 
   // Depending on the element type create the appropriate extensions
   // for the individual components.
   dynamic_cast<const NURBSFECollection *>(fec)->Reset();
 
   if (dynamic_cast<const NURBS_HDivFECollection *>(fec))
   {
      VNURBSext.SetSize(mesh->Dimension());
      for (int d = 0; d < mesh->Dimension(); d++)
      {
         VNURBSext[d] = NURBSext->GetDivExtension(d);
      }
   }
 
   if (dynamic_cast<const NURBS_HCurlFECollection *>(fec))
   {
      VNURBSext.SetSize(mesh->Dimension());
      for (int d = 0; d < mesh->Dimension(); d++)
      {
         VNURBSext[d] = NURBSext->GetCurlExtension(d);
      }
   }
 
   // If required: concatenate the dof tables of the individual components into
   // one dof table for the vector fespace.
   if (VNURBSext.Size() == 2)
   {
      int offset1 = VNURBSext[0]->GetNDof();
      ndofs = VNURBSext[0]->GetNDof() + VNURBSext[1]->GetNDof();
 
      // Merge Tables
      elem_dof = new Table(*VNURBSext[0]->GetElementDofTable(),
                           *VNURBSext[1]->GetElementDofTable(),offset1 );
 
      bdr_elem_dof = new Table(*VNURBSext[0]->GetBdrElementDofTable(),
                               *VNURBSext[1]->GetBdrElementDofTable(),offset1);
   }
   else if (VNURBSext.Size() == 3)
   {
      int offset1 = VNURBSext[0]->GetNDof();
      int offset2 = offset1 + VNURBSext[1]->GetNDof();
      ndofs = offset2 + VNURBSext[2]->GetNDof();
 
      // Merge Tables
      elem_dof = new Table(*VNURBSext[0]->GetElementDofTable(),
                           *VNURBSext[1]->GetElementDofTable(),offset1,
                           *VNURBSext[2]->GetElementDofTable(),offset2);
 
      bdr_elem_dof = new Table(*VNURBSext[0]->GetBdrElementDofTable(),
                               *VNURBSext[1]->GetBdrElementDofTable(),offset1,
                               *VNURBSext[2]->GetBdrElementDofTable(),offset2);
   }
   else
   {
      ndofs = NURBSext->GetNDof();
      elem_dof = NURBSext->GetElementDofTable();
      bdr_elem_dof = NURBSext->GetBdrElementDofTable();
   }
   mesh_sequence = mesh->GetSequence();
   sequence++;
}
 
void FiniteElementSpace::BuildNURBSFaceToDofTable() const
{
   if (face_dof) { return; }
 
   const int dim = mesh->Dimension();
 
   // Find bdr to face mapping
   face_to_be.SetSize(GetNF());
   face_to_be = -1;
   for (int b = 0; b < GetNBE(); b++)
   {
      int f = mesh->GetBdrElementFaceIndex(b);
      face_to_be[f] = b;
   }
 
   // Loop over faces in correct order, to prevent a sort
   // Sort will destroy orientation info in ordering of dofs
   Array<Connection> face_dof_list;
   Array<int> row;
   for (int f = 0; f < GetNF(); f++)
   {
      int b = face_to_be[f];
      if (b == -1) { continue; }
      // FIXME: this assumes that the boundary element and the face element have
      //        the same orientation.
      if (dim > 1)
      {
         const Element *fe = mesh->GetFace(f);
         const Element *be = mesh->GetBdrElement(b);
         const int nv = be->GetNVertices();
         const int *fv = fe->GetVertices();
         const int *bv = be->GetVertices();
         for (int i = 0; i < nv; i++)
         {
            MFEM_VERIFY(fv[i] == bv[i],
                        "non-matching face and boundary elements detected!");
         }
      }
      GetBdrElementDofs(b, row);
      Connection conn(f,0);
      for (int i = 0; i < row.Size(); i++)
      {
         conn.to = row[i];
         face_dof_list.Append(conn);
      }
   }
   face_dof = new Table(GetNF(), face_dof_list);
}
 
void FiniteElementSpace::Construct()
{
   // This method should be used only for non-NURBS spaces.
   MFEM_VERIFY(!NURBSext, "internal error");
 
   // Variable-order space needs a nontrivial P matrix + also ghost elements
   // in parallel, we thus require the mesh to be NC.
   MFEM_VERIFY(!IsVariableOrder() || Nonconforming(),
               "Variable-order space requires a nonconforming mesh.");
 
   elem_dof = NULL;
   elem_fos = NULL;
   bdr_elem_dof = NULL;
   bdr_elem_fos = NULL;
   face_dof = NULL;
 
   ndofs = 0;
   nvdofs = nedofs = nfdofs = nbdofs = 0;
   bdofs = NULL;
 
   cP.reset();
   cR.reset();
   cR_hp.reset();
   cP_is_set = false;
   R_transpose.reset();
   // 'Th' is initialized/destroyed before this method is called.
 
   int dim = mesh->Dimension();
   int order = fec->GetOrder();
 
   MFEM_VERIFY((mesh->GetNumGeometries(dim) > 0) || (mesh->GetNE() == 0),
               "Mesh was not correctly finalized.");
 
   bool mixed_elements = (mesh->GetNumGeometries(dim) > 1);
   bool mixed_faces = (dim > 2 && mesh->GetNumGeometries(2) > 1);
 
   Array<VarOrderBits> edge_orders, face_orders, edge_elem_orders,
         face_elem_orders;
 
   if (IsVariableOrder())
   {
      // for variable-order spaces, calculate orders of edges and faces
      CalcEdgeFaceVarOrders(edge_orders, face_orders, edge_elem_orders,
                            face_elem_orders, skip_edge, skip_face);
   }
   else if (mixed_faces)
   {
      // for mixed faces we also create the var_face_dofs table, see below
      face_orders.SetSize(mesh->GetNFaces());
      face_orders = (VarOrderBits(1) << order);
   }
 
   // assign vertex DOFs
   if (mesh->GetNV())
   {
      nvdofs = mesh->GetNV() * fec->GetNumDof(Geometry::POINT, order);
   }
 
   // assign edge DOFs
   if (mesh->GetNEdges())
   {
      if (IsVariableOrder())
      {
         nedofs = MakeDofTable(1, edge_orders, var_edge_dofs, &var_edge_orders);
         MakeDofTable(1, edge_elem_orders, loc_var_edge_dofs,
                      &loc_var_edge_orders);
         // Set lnedofs from the last row of loc_var_edge_dofs
         Array<int> lastRow;
         loc_var_edge_dofs.GetRow(loc_var_edge_dofs.Size() - 1, lastRow);
         MFEM_ASSERT(lastRow.Size() == 1, "");
         lnedofs = lastRow[0];
      }
      else
      {
         // the simple case: all edges are of the same order
         nedofs = mesh->GetNEdges() * fec->GetNumDof(Geometry::SEGMENT, order);
         var_edge_dofs.Clear(); // ensure any old var_edge_dof table is dumped.
      }
   }
 
   // assign face DOFs
   if (mesh->GetNFaces())
   {
      if (IsVariableOrder() || mixed_faces)
      {
         // NOTE: for simplicity, we also use Table var_face_dofs for mixed faces
         nfdofs = MakeDofTable(2, face_orders, var_face_dofs,
                               IsVariableOrder() ? &var_face_orders : NULL);
         uni_fdof = -1;
 
         if (IsVariableOrder())
         {
            MakeDofTable(2, face_elem_orders, loc_var_face_dofs,
                         &loc_var_face_orders);
            // Set lnfdofs from the last row of loc_var_face_dofs
            Array<int> lastRow;
            loc_var_face_dofs.GetRow(loc_var_face_dofs.Size() - 1, lastRow);
            MFEM_ASSERT(lastRow.Size() == 1, "");
            lnfdofs = lastRow[0];
         }
      }
      else
      {
         // the simple case: all faces are of the same geometry and order
         uni_fdof = fec->GetNumDof(mesh->GetTypicalFaceGeometry(), order);
         nfdofs = mesh->GetNFaces() * uni_fdof;
         var_face_dofs.Clear(); // ensure any old var_face_dof table is dumped.
      }
   }
 
   // assign internal ("bubble") DOFs
   if (mesh->GetNE() && dim > 0)
   {
      if (IsVariableOrder() || mixed_elements)
      {
         bdofs = new int[mesh->GetNE()+1];
         bdofs[0] = 0;
         for (int i = 0; i < mesh->GetNE(); i++)
         {
            int p = GetElementOrderImpl(i);
            nbdofs += fec->GetNumDof(mesh->GetElementGeometry(i), p);
            bdofs[i+1] = nbdofs;
         }
      }
      else
      {
         // the simple case: all elements are the same
         bdofs = NULL;
         Geometry::Type geom = mesh->GetElementGeometry(0);
         nbdofs = mesh->GetNE() * fec->GetNumDof(geom, order);
      }
   }
 
   ndofs = nvdofs + nedofs + nfdofs + nbdofs;
 
   ConstructDoFTransArray();
 
   // record the current mesh sequence number to detect refinement etc.
   mesh_sequence = mesh->GetSequence();
 
   // increment our sequence number to let GridFunctions know they need updating
   sequence++;
 
   // DOFs are now assigned according to current element orders
   orders_changed = false;
 
   // Do not build elem_dof Table here: in parallel it has to be constructed
   // later.
}
 
void DofMapHelper(int entity, const Table & var_ent_dofs,
                  const Table & loc_var_ent_dofs,
                  const Array<char> & var_ent_orders,
                  const Array<char> & loc_var_ent_orders,
                  Array<int> & all2local, int & ndof_all, int & ndof_loc)
{
   const int osall0 = var_ent_dofs.GetI()[entity];
   const int osall1 = var_ent_dofs.GetI()[entity + 1];
 
   const int osloc0 = loc_var_ent_dofs.GetI()[entity];
   const int osloc1 = loc_var_ent_dofs.GetI()[entity + 1];
 
   // loc_var_ent_orders must be a subset of var_ent_orders
   int j = osall0;
   for (int i=osloc0; i<osloc1; ++i)  // Loop over local variants
   {
      const int order = loc_var_ent_orders[i];
      // Find the variant in var_ent_orders with the same order
      int na = var_ent_dofs.GetJ()[j + 1] - var_ent_dofs.GetJ()[j];
      while (var_ent_orders[j] != order && j < osall1 - 1)
      {
         j++;
         ndof_all += na;
         na = var_ent_dofs.GetJ()[j + 1] - var_ent_dofs.GetJ()[j];
      }
 
      MFEM_ASSERT(var_ent_orders[j] == order, "");
 
      const int n = loc_var_ent_dofs.GetJ()[i + 1] - loc_var_ent_dofs.GetJ()[i];
 
      MFEM_ASSERT(n == na &&
                  n == var_ent_dofs.GetJ()[j + 1] - var_ent_dofs.GetJ()[j], "");
 
      for (int k=0; k<n; ++k) { all2local[ndof_all + k] = ndof_loc + k; }
 
      ndof_loc += n;
      ndof_all += na;
      j++;
   }
 
   // Reach the end of all variants for ndof_all
   while (j < osall1)
   {
      const int na = var_ent_dofs.GetJ()[j + 1] - var_ent_dofs.GetJ()[j];
      ndof_all += na;
      j++;
   }
}
 
void FiniteElementSpace::SetVarOrderLocalDofs()
{
   if (!IsVariableOrder()) { return; }
 
   // Set a map from all DOFs to local DOFs
   all2local.SetSize(ndofs);
   all2local = -1;
 
   // Vertex DOFs simply have the identity mapping
   for (int i=0; i<nvdofs; ++i)
   {
      all2local[i] = i;
   }
 
   // Redefine local edge DOFs
   int ndof_all = nvdofs;
   int ndof_loc = nvdofs;
   if (mesh->GetNEdges())
   {
      for (int edge=0; edge<mesh->GetNEdges(); ++edge)
      {
         DofMapHelper(edge, var_edge_dofs, loc_var_edge_dofs, var_edge_orders,
                      loc_var_edge_orders, all2local, ndof_all, ndof_loc);
      }
 
      MFEM_ASSERT(ndof_loc - nvdofs == lnedofs, "");
      nedofs = lnedofs;
   }
 
   // Redefine local face DOFs
   if (mesh->GetNFaces())
   {
      for (int face=0; face<mesh->GetNFaces(); ++face)
      {
         DofMapHelper(face, var_face_dofs, loc_var_face_dofs, var_face_orders,
                      loc_var_face_orders, all2local, ndof_all, ndof_loc);
      }
 
      MFEM_ASSERT(ndof_loc - nvdofs - lnedofs == lnfdofs, "");
      nfdofs = lnfdofs;
   }
 
   // The remaining DOFs simply have the identity mapping
   for (int i=ndof_all; i<ndofs; ++i)
   {
      all2local[i] = ndof_loc + i - ndof_all;
   }
 
   ndofs = nvdofs + nedofs + nfdofs + nbdofs;
}
 
int FiniteElementSpace::MinOrder(VarOrderBits bits)
{
   MFEM_ASSERT(bits != 0, "invalid bit mask");
   for (int order = 0; bits != 0; order++, bits >>= 1)
   {
      if (bits & 1) { return order; }
   }
   return 0;
}
 
// For the serial FiniteElementSpace, there are no ghost elements, and this
// function just sets the sizes of edge_orders and face_orders, initializing to
// 0.
void FiniteElementSpace::ApplyGhostElementOrdersToEdgesAndFaces(
   Array<VarOrderBits> &edge_orders,
   Array<VarOrderBits> &face_orders) const
{
   edge_orders.SetSize(mesh->GetNEdges());
   face_orders.SetSize(mesh->GetNFaces());
 
   edge_orders = 0;
   face_orders = 0;
}
 
void FiniteElementSpace::CalcEdgeFaceVarOrders(
   Array<VarOrderBits> &edge_orders, Array<VarOrderBits> &face_orders,
   Array<VarOrderBits> &edge_elem_orders, Array<VarOrderBits> &face_elem_orders,
   Array<bool> &skip_edges, Array<bool> &skip_faces) const
{
   MFEM_ASSERT(Nonconforming(), "");
 
   const bool localVar = elem_order.Size() == mesh->GetNE();
   const int baseOrder = fec->GetOrder();
 
   ApplyGhostElementOrdersToEdgesAndFaces(edge_orders, face_orders);
 
   edge_elem_orders.SetSize(mesh->GetNEdges());
   face_elem_orders.SetSize(mesh->GetNFaces());
 
   edge_elem_orders = 0;
   face_elem_orders = 0;
 
   edge_min_nghb_order.SetSize(mesh->ncmesh->GetNEdges());
   face_min_nghb_order.SetSize(mesh->ncmesh->GetNFaces());
 
   edge_min_nghb_order = MaxVarOrder + 1;
   face_min_nghb_order = MaxVarOrder + 1;
 
   // Calculate initial edge/face orders, as required by incident elements.
   // For each edge/face we accumulate in a bit-mask the orders of elements
   // sharing the edge/face.
   Array<int> E, F, ori;
   for (int i = 0; i < mesh->GetNE(); i++)
   {
      const int order = localVar ? elem_order[i] : baseOrder;
      MFEM_ASSERT(order <= MaxVarOrder, "");
      const VarOrderBits mask = (VarOrderBits(1) << order);
 
      mesh->GetElementEdges(i, E, ori);
      for (int j = 0; j < E.Size(); j++)
      {
         edge_orders[E[j]] |= mask;
         edge_elem_orders[E[j]] |= mask;
 
         if (order < edge_min_nghb_order[E[j]])
         {
            edge_min_nghb_order[E[j]] = order;
         }
      }
 
      if (mesh->Dimension() > 2)
      {
         mesh->GetElementFaces(i, F, ori);
         for (int j = 0; j < F.Size(); j++)
         {
            face_orders[F[j]] |= mask;
            face_elem_orders[F[j]] |= mask;
 
            if (order < face_min_nghb_order[F[j]])
            {
               face_min_nghb_order[F[j]] = order;
            }
         }
      }
   }
 
   if (relaxed_hp)
   {
      // for relaxed conformity we don't need the masters to match the minimum
      // orders of the slaves, we can stop now
      return;
   }
 
   // Iterate while minimum orders propagate by master/slave relations
   // (and new orders also propagate from faces to incident edges).
   // See https://github.com/mfem/mfem/pull/1423#issuecomment-638930559
   // for an illustration of why this is necessary in hp meshes.
   bool done;
   do
   {
      std::set<int> changedEdges;
      std::set<int> changedFaces;
 
      const int numEdges = mesh->GetNEdges();
 
      // Propagate from slave edges to master edges
      const NCMesh::NCList &edge_list = mesh->ncmesh->GetEdgeList();
      for (const NCMesh::Master &master : edge_list.masters)
      {
         VarOrderBits slave_orders = 0;
         for (int i = master.slaves_begin; i < master.slaves_end; i++)
         {
            slave_orders |= edge_orders[edge_list.slaves[i].index];
         }
 
         if (slave_orders == 0)
         {
            continue;
         }
 
         const int min_order_slaves = MinOrder(slave_orders);
         if (edge_orders[master.index] == 0 ||
             min_order_slaves < MinOrder(edge_orders[master.index]))
         {
            edge_orders[master.index] |= VarOrderBits(1) << min_order_slaves;
            changedEdges.insert(master.index);
         }
 
         // Also apply the minimum order to all the slave edges, since they must
         // interpolate the master edge, which has the minimum order.
         const VarOrderBits min_mask = VarOrderBits(1) << MinOrder(
                                          edge_orders[master.index]);
         for (int i = master.slaves_begin; i < master.slaves_end; i++)
         {
            if (edge_list.slaves[i].index >= numEdges)
            {
               continue;   // Skip ghost edges
            }
 
            const VarOrderBits eo0 = edge_orders[edge_list.slaves[i].index];
            edge_orders[edge_list.slaves[i].index] |= min_mask;
            if (eo0 != edge_orders[edge_list.slaves[i].index])
            {
               changedEdges.insert(edge_list.slaves[i].index);
            }
         }
      }
 
      // Propagate from slave faces(+edges) to master faces.
      const int numFaces = mesh->GetNumFaces();
 
      const NCMesh::NCList &face_list = mesh->ncmesh->GetFaceList();
 
      for (const NCMesh::Master &master : face_list.masters)
      {
         VarOrderBits slave_orders = 0;
 
         for (int i = master.slaves_begin; i < master.slaves_end; i++)
         {
            const NCMesh::Slave &slave = face_list.slaves[i];
 
            if (slave.index >= 0)
            {
               // Note that master.index >= numFaces occurs for ghost master faces.
 
               slave_orders |= face_orders[slave.index];
 
               if (slave.index >= numFaces)
               {
                  continue;  // Skip ghost faces
               }
 
               mesh->GetFaceEdges(slave.index, E, ori);
               for (int j = 0; j < E.Size(); j++)
               {
                  slave_orders |= edge_orders[E[j]];
               }
            }
            else
            {
               // degenerate face (i.e., edge-face constraint)
               slave_orders |= edge_orders[FlipIndexSign(slave.index)];
            }
         }
 
         if (slave_orders == 0)
         {
            continue;
         }
 
         const int min_order_slaves = MinOrder(slave_orders);
         if (face_orders[master.index] == 0 ||
             min_order_slaves < MinOrder(face_orders[master.index]))
         {
            face_orders[master.index] |= VarOrderBits(1) << min_order_slaves;
            changedFaces.insert(master.index);
         }
 
         // Also apply the minimum order to all the slave faces, since they must
         // interpolate the master face, which has the minimum order.
         const VarOrderBits min_mask =
            VarOrderBits(1) << MinOrder(face_orders[master.index]);
         for (int i = master.slaves_begin; i < master.slaves_end; i++)
         {
            const NCMesh::Slave &slave = face_list.slaves[i];
 
            if (slave.index >= 0 && slave.index < numFaces)  // Skip ghost faces
            {
               const VarOrderBits fo0 = face_orders[slave.index];
               face_orders[slave.index] |= min_mask;
               if (fo0 != face_orders[slave.index])
               {
                  changedFaces.insert(slave.index);
               }
            }
         }
      }
 
      // Make sure edges support (new) orders required by incident faces.
      for (int i = 0; i < mesh->GetNFaces(); i++)
      {
         mesh->GetFaceEdges(i, E, ori);
         for (int j = 0; j < E.Size(); j++)
         {
            const VarOrderBits eo0 = edge_orders[E[j]];
            edge_orders[E[j]] |= face_orders[i];
            if (eo0 != edge_orders[E[j]])
            {
               changedEdges.insert(E[j]);
            }
         }
      }
 
      // In the parallel case, OrderPropagation communicates orders on updated
      // edges and faces.
      done = OrderPropagation(changedEdges, changedFaces,
                              edge_orders, face_orders);
   }
   while (!done);
 
   GhostFaceOrderToEdges(face_orders, edge_orders);
 
   // Some ghost edges and faces (3D) may not have any orders applied, since we
   // only communicate orders of neighboring ghost elements. Such ghost entities
   // are marked here, to be skipped by BuildParallelConformingInterpolation as
   // master entities constraining slave entity DOFs.
 
   skip_edges.SetSize(edge_orders.Size());
   skip_edges = false;
 
   skip_faces.SetSize(face_orders.Size());
   skip_faces = false;
 
   for (int i=0; i<edge_orders.Size(); ++i)
   {
      if (edge_orders[i] == 0)
      {
         skip_edges[i] = true;
      }
   }
 
   for (int i=0; i<face_orders.Size(); ++i)
   {
      if (face_orders[i] == 0)
      {
         skip_faces[i] = true;
      }
   }
}
 
int FiniteElementSpace::MakeDofTable(int ent_dim,
                                     const Array<VarOrderBits> &entity_orders,
                                     Table &entity_dofs,
                                     Array<char> *var_ent_order)
{
   // The tables var_edge_dofs and var_face_dofs hold DOF assignments for edges
   // and faces of a variable-order space, in which each edge/face may host
   // several DOF sets, called DOF set variants. Example: an edge 'i' shared by
   // 4 hexes of orders 2, 3, 4, 5 will hold four DOF sets, each starting at
   // indices e.g. 100, 101, 103, 106, respectively. These numbers are stored
   // in row 'i' of var_edge_dofs. Variant zero is always the lowest order DOF
   // set, followed by consecutive ranges of higher order DOFs. Variable-order
   // faces are handled similarly by var_face_dofs. The tables are empty for
   // constant-order spaces.
 
   int num_ent = entity_orders.Size();
   int total_dofs = 0;
   int total_dofs_nonghost = 0;
 
   Array<Connection> list;
   list.Reserve(2*num_ent);
 
   if (var_ent_order)
   {
      var_ent_order->SetSize(0);
      var_ent_order->Reserve(num_ent);
   }
 
   int nonGhost = num_ent;
   if (IsVariableOrder())
   {
      nonGhost -= (ent_dim == 1) ? NumGhostEdges() : NumGhostFaces();
   }
 
   // assign DOFs according to order bit masks
   for (int i = 0; i < num_ent; i++)
   {
      auto geom = Geometry::SEGMENT;  // ent_dim == 1 case
      if (ent_dim != 1)
      {
         // TODO: put this logic in mesh->GetFaceGeometry?
         if (i >= nonGhost)  // if ghost
         {
            geom = mesh->ncmesh->GetFaceGeometry(i);
         }
         else
         {
            geom = mesh->GetFaceGeometry(i);
         }
      }
 
      VarOrderBits bits = entity_orders[i];
      for (int order = 0; bits != 0; order++, bits >>= 1)
      {
         if (bits & 1)
         {
            const int dofs = fec->GetNumDof(geom, order);
            list.Append(Connection(i, total_dofs));
            total_dofs += dofs;
            if (i < nonGhost) { total_dofs_nonghost += dofs; }
            if (var_ent_order) { var_ent_order->Append(order); }
         }
      }
   }
 
   // append a dummy row as terminator
   list.Append(Connection(num_ent, total_dofs));
 
   // build the table
   entity_dofs.MakeFromList(num_ent+1, list);
   return total_dofs_nonghost;
}
 
int FiniteElementSpace::FindDofs(const Table &var_dof_table,
                                 int row, int ndof) const
{
   const int *beg = var_dof_table.GetRow(row);
   const int *end = var_dof_table.GetRow(row + 1); // terminator, see above
 
   while (beg < end)
   {
      // return the appropriate range of DOFs
      if ((beg[1] - beg[0]) == ndof) { return beg[0]; }
      beg++;
   }
 
   MFEM_ABORT("DOFs not found for ndof = " << ndof);
   return 0;
}
 
int FiniteElementSpace::GetEdgeOrder(int edge, int variant) const
{
   if (!IsVariableOrder()) { return fec->GetOrder(); }
 
   if (edge >= var_edge_dofs.Size())
   {
      return ghost_edge_orders[edge - var_edge_dofs.Size()];
   }
 
   const int* beg = var_edge_dofs.GetRow(edge);
   const int* end = var_edge_dofs.GetRow(edge + 1);
   if (variant >= end - beg) { return -1; } // past last variant
 
   return var_edge_orders[var_edge_dofs.GetI()[edge] + variant];
}
 
int FiniteElementSpace::GetFaceOrder(int face, int variant) const
{
   if (!IsVariableOrder())
   {
      // face order can be different from fec->GetOrder()
      Geometry::Type geom = mesh->GetFaceGeometry(face);
      return fec->FiniteElementForGeometry(geom)->GetOrder();
   }
 
   if (face >= var_face_dofs.Size())
   {
      return ghost_face_orders[face - var_face_dofs.Size()];
   }
 
   const int* beg = var_face_dofs.GetRow(face);
   const int* end = var_face_dofs.GetRow(face + 1);
   if (variant >= end - beg) { return -1; } // past last variant
 
   return var_face_orders[var_face_dofs.GetI()[face] + variant];
}
 
int FiniteElementSpace::GetNVariants(int entity, int index) const
{
   MFEM_ASSERT(IsVariableOrder(), "");
   const Table &dof_table = (entity == 1) ? var_edge_dofs : var_face_dofs;
 
   MFEM_ASSERT(index >= 0 && index < dof_table.Size(), "");
   return dof_table.GetRow(index + 1) - dof_table.GetRow(index);
}
 
static const char* msg_orders_changed =
   "Element orders changed, you need to Update() the space first.";
 
void FiniteElementSpace::GetElementDofs(int elem, Array<int> &dofs,
                                        DofTransformation &doftrans) const
{
   MFEM_VERIFY(!orders_changed, msg_orders_changed);
 
   doftrans.SetDofTransformation(nullptr);
 
   if (elem_dof)
   {
      elem_dof->GetRow(elem, dofs);
 
      if (DoFTransArray[mesh->GetElementBaseGeometry(elem)])
      {
         Array<int> Fo;
         elem_fos -> GetRow (elem, Fo);
         doftrans.SetDofTransformation(
            *DoFTransArray[mesh->GetElementBaseGeometry(elem)]);
         doftrans.SetFaceOrientations(Fo);
         doftrans.SetVDim();
      }
      return;
   }
 
   Array<int> V, E, Eo, F, Fo; // TODO: LocalArray
 
   const int dim = mesh->Dimension();
   const auto geom = mesh->GetElementGeometry(elem);
   const int order = GetElementOrderImpl(elem);
 
   const int nv = fec->GetNumDof(Geometry::POINT, order);
   const int ne = (dim > 1) ? fec->GetNumDof(Geometry::SEGMENT, order) : 0;
   const int nb = (dim > 0) ? fec->GetNumDof(geom, order) : 0;
 
   if (nv) { mesh->GetElementVertices(elem, V); }
   if (ne) { mesh->GetElementEdges(elem, E, Eo); }
 
   int nfd = 0;
   if (dim > 2 && fec->HasFaceDofs(geom, order))
   {
      mesh->GetElementFaces(elem, F, Fo);
      for (int i = 0; i < F.Size(); i++)
      {
         nfd += fec->GetNumDof(mesh->GetFaceGeometry(F[i]), order);
      }
      if (DoFTransArray[mesh->GetElementBaseGeometry(elem)])
      {
         doftrans.SetDofTransformation(
            *DoFTransArray[mesh->GetElementBaseGeometry(elem)]);
         doftrans.SetFaceOrientations(Fo);
         doftrans.SetVDim();
      }
   }
 
   dofs.SetSize(0);
   dofs.Reserve(nv*V.Size() + ne*E.Size() + nfd + nb);
 
   if (nv) // vertex DOFs
   {
      for (int i = 0; i < V.Size(); i++)
      {
         for (int j = 0; j < nv; j++)
         {
            dofs.Append(V[i]*nv + j);
         }
      }
   }
 
   if (ne) // edge DOFs
   {
      for (int i = 0; i < E.Size(); i++)
      {
         int ebase = IsVariableOrder() ? FindEdgeDof(E[i], ne) : E[i]*ne;
         const int *ind = fec->GetDofOrdering(Geometry::SEGMENT, order, Eo[i]);
 
         for (int j = 0; j < ne; j++)
         {
            dofs.Append(EncodeDof(nvdofs + ebase, ind[j]));
         }
      }
   }
 
   if (nfd) // face DOFs
   {
      for (int i = 0; i < F.Size(); i++)
      {
         auto fgeom = mesh->GetFaceGeometry(F[i]);
         int nf = fec->GetNumDof(fgeom, order);
 
         int fbase = (var_face_dofs.Size() > 0) ? FindFaceDof(F[i], nf) : F[i]*nf;
         const int *ind = fec->GetDofOrdering(fgeom, order, Fo[i]);
 
         for (int j = 0; j < nf; j++)
         {
            dofs.Append(EncodeDof(nvdofs + nedofs + fbase, ind[j]));
         }
      }
   }
 
   if (nb) // interior ("bubble") DOFs
   {
      int bbase = bdofs ? bdofs[elem] : elem*nb;
      bbase += nvdofs + nedofs + nfdofs;
 
      for (int j = 0; j < nb; j++)
      {
         dofs.Append(bbase + j);
      }
   }
}
 
DofTransformation *FiniteElementSpace::GetElementDofs(int elem,
                                                      Array<int> &dofs) const
{
   GetElementDofs(elem, dofs, DoFTrans);
   return DoFTrans.GetDofTransformation() ? &DoFTrans : NULL;
}
 
void FiniteElementSpace::GetBdrElementDofs(int bel, Array<int> &dofs,
                                           DofTransformation &doftrans) const
{
   MFEM_VERIFY(!orders_changed, msg_orders_changed);
 
   doftrans.SetDofTransformation(nullptr);
 
   if (bdr_elem_dof)
   {
      bdr_elem_dof->GetRow(bel, dofs);
 
      if (DoFTransArray[mesh->GetBdrElementGeometry(bel)])
      {
         Array<int> Fo;
         bdr_elem_fos -> GetRow (bel, Fo);
         doftrans.SetDofTransformation(
            *DoFTransArray[mesh->GetBdrElementGeometry(bel)]);
         doftrans.SetFaceOrientations(Fo);
         doftrans.SetVDim();
      }
      return;
   }
 
   Array<int> V, E, Eo; // TODO: LocalArray
   int F, oF;
 
   int dim = mesh->Dimension();
   auto geom = mesh->GetBdrElementGeometry(bel);
   int order = fec->GetOrder();
 
   if (elem_order.Size()) // determine order from adjacent element
   {
      int elem, info;
      mesh->GetBdrElementAdjacentElement(bel, elem, info);
      order = elem_order[elem];
   }
 
   int nv = fec->GetNumDof(Geometry::POINT, order);
   int ne = (dim > 1) ? fec->GetNumDof(Geometry::SEGMENT, order) : 0;
   int nf = (dim > 2) ? fec->GetNumDof(geom, order) : 0;
 
   if (nv) { mesh->GetBdrElementVertices(bel, V); }
   if (ne) { mesh->GetBdrElementEdges(bel, E, Eo); }
   if (nf)
   {
      mesh->GetBdrElementFace(bel, &F, &oF);
 
      if (DoFTransArray[mesh->GetBdrElementGeometry(bel)])
      {
         mfem::Array<int> Fo(1);
         Fo[0] = oF;
         doftrans.SetDofTransformation(
            *DoFTransArray[mesh->GetBdrElementGeometry(bel)]);
         doftrans.SetFaceOrientations(Fo);
         doftrans.SetVDim();
      }
   }
 
   dofs.SetSize(0);
   dofs.Reserve(nv*V.Size() + ne*E.Size() + nf);
 
   if (nv) // vertex DOFs
   {
      for (int i = 0; i < V.Size(); i++)
      {
         for (int j = 0; j < nv; j++)
         {
            dofs.Append(V[i]*nv + j);
         }
      }
   }
 
   if (ne) // edge DOFs
   {
      for (int i = 0; i < E.Size(); i++)
      {
         int ebase = IsVariableOrder() ? FindEdgeDof(E[i], ne) : E[i]*ne;
         const int *ind = fec->GetDofOrdering(Geometry::SEGMENT, order, Eo[i]);
 
         for (int j = 0; j < ne; j++)
         {
            dofs.Append(EncodeDof(nvdofs + ebase, ind[j]));
         }
      }
   }
 
   if (nf) // face DOFs
   {
      int fbase = (var_face_dofs.Size() > 0) ? FindFaceDof(F, nf) : F*nf;
      const int *ind = fec->GetDofOrdering(geom, order, oF);
 
      for (int j = 0; j < nf; j++)
      {
         dofs.Append(EncodeDof(nvdofs + nedofs + fbase, ind[j]));
      }
   }
}
 
DofTransformation *FiniteElementSpace::GetBdrElementDofs(int bel,
                                                         Array<int> &dofs) const
{
   GetBdrElementDofs(bel, dofs, DoFTrans);
   return DoFTrans.GetDofTransformation() ? &DoFTrans : NULL;
}
 
int FiniteElementSpace::GetFaceDofs(int face, Array<int> &dofs,
                                    int variant) const
{
   MFEM_VERIFY(!orders_changed, msg_orders_changed);
 
   // If face_dof is already built, use it.
   // If it is not and we have a NURBS space, build the face_dof and use it.
   if ((face_dof && variant == 0) ||
       (NURBSext && (BuildNURBSFaceToDofTable(), true)))
   {
      face_dof->GetRow(face, dofs);
      return fec->GetOrder();
   }
 
   int order, nf, fbase;
   int dim = mesh->Dimension();
   auto fgeom = (dim > 2) ? mesh->GetFaceGeometry(face) : Geometry::INVALID;
 
   if (var_face_dofs.Size() > 0) // variable orders or *mixed* faces
   {
      const int* beg = var_face_dofs.GetRow(face);
      const int* end = var_face_dofs.GetRow(face + 1);
      if (variant >= end - beg) { return -1; } // past last face DOFs
 
      fbase = beg[variant];
      nf = beg[variant+1] - fbase;
 
      order = !IsVariableOrder() ? fec->GetOrder() :
              var_face_orders[var_face_dofs.GetI()[face] + variant];
      MFEM_ASSERT(fec->GetNumDof(fgeom, order) == nf, [&]()
      {
         std::stringstream msg;
         msg << "fec->GetNumDof(" << (fgeom == Geometry::SQUARE ? "square" : "triangle")
             << ", " << order << ") = " << fec->GetNumDof(fgeom, order) << " nf " << nf;
         msg << " face " << face << " variant " << variant << std::endl;
         return msg.str();
      }());
   }
   else
   {
      if (variant > 0) { return -1; }
      order = fec->GetOrder();
      nf = (dim > 2) ? fec->GetNumDof(fgeom, order) : 0;
      fbase = face*nf;
   }
 
   // for 1D, 2D and 3D faces
   int nv = fec->GetNumDof(Geometry::POINT, order);
   int ne = (dim > 1) ? fec->GetNumDof(Geometry::SEGMENT, order) : 0;
 
   Array<int> V, E, Eo;
   if (nv) { mesh->GetFaceVertices(face, V); }
   if (ne) { mesh->GetFaceEdges(face, E, Eo); }
 
   dofs.SetSize(0);
   dofs.Reserve(V.Size() * nv + E.Size() * ne + nf);
 
   if (nv) // vertex DOFs
   {
      for (int i = 0; i < V.Size(); i++)
      {
         for (int j = 0; j < nv; j++)
         {
            dofs.Append(V[i]*nv + j);
         }
      }
   }
   if (ne) // edge DOFs
   {
      for (int i = 0; i < E.Size(); i++)
      {
         int ebase = IsVariableOrder() ? FindEdgeDof(E[i], ne) : E[i]*ne;
         const int *ind = fec->GetDofOrdering(Geometry::SEGMENT, order, Eo[i]);
 
         for (int j = 0; j < ne; j++)
         {
            dofs.Append(EncodeDof(nvdofs + ebase, ind[j]));
         }
      }
   }
   for (int j = 0; j < nf; j++)
   {
      dofs.Append(nvdofs + nedofs + fbase + j);
   }
 
   return order;
}
 
int FiniteElementSpace::GetEdgeDofs(int edge, Array<int> &dofs,
                                    int variant) const
{
   MFEM_VERIFY(!orders_changed, msg_orders_changed);
 
   int order, ne, base;
   if (IsVariableOrder())
   {
      const int* beg = var_edge_dofs.GetRow(edge);
      const int* end = var_edge_dofs.GetRow(edge + 1);
      if (variant >= end - beg) { return -1; } // past last edge DOFs
 
      base = beg[variant];
      ne = beg[variant+1] - base;
 
      order = var_edge_orders[var_edge_dofs.GetI()[edge] + variant];
      MFEM_ASSERT(fec->GetNumDof(Geometry::SEGMENT, order) == ne, "");
   }
   else
   {
      if (variant > 0) { return -1; }
      order = fec->GetOrder();
      ne = fec->GetNumDof(Geometry::SEGMENT, order);
      base = edge*ne;
   }
 
   Array<int> V; // TODO: LocalArray
   int nv = fec->GetNumDof(Geometry::POINT, order);
   if (nv) { mesh->GetEdgeVertices(edge, V); }
 
   dofs.SetSize(0);
   dofs.Reserve(2*nv + ne);
 
   for (int i = 0; i < 2; i++)
   {
      for (int j = 0; j < nv; j++)
      {
         dofs.Append(V[i]*nv + j);
      }
   }
   for (int j = 0; j < ne; j++)
   {
      dofs.Append(nvdofs + base + j);
   }
 
   return order;
}
 
void FiniteElementSpace::GetVertexDofs(int i, Array<int> &dofs) const
{
   int nv = fec->DofForGeometry(Geometry::POINT);
   dofs.SetSize(nv);
   for (int j = 0; j < nv; j++)
   {
      dofs[j] = i*nv+j;
   }
}
 
void FiniteElementSpace::GetElementInteriorDofs(int i, Array<int> &dofs) const
{
   MFEM_VERIFY(!orders_changed, msg_orders_changed);
 
   int nb = fec->GetNumDof(mesh->GetElementGeometry(i), GetElementOrderImpl(i));
   int base = bdofs ? bdofs[i] : i*nb;
 
   dofs.SetSize(nb);
   base += nvdofs + nedofs + nfdofs;
   for (int j = 0; j < nb; j++)
   {
      dofs[j] = base + j;
   }
}
 
int FiniteElementSpace::GetNumElementInteriorDofs(int i) const
{
   return fec->GetNumDof(mesh->GetElementGeometry(i),
                         GetElementOrderImpl(i));
}
 
void FiniteElementSpace::GetFaceInteriorDofs(int i, Array<int> &dofs) const
{
   MFEM_VERIFY(!IsVariableOrder(), "not implemented");
 
   int nf, base;
   if (var_face_dofs.Size() > 0) // mixed faces
   {
      base = var_face_dofs.GetRow(i)[0];
      nf = var_face_dofs.GetRow(i)[1] - base;
   }
   else
   {
      auto geom = mesh->GetTypicalFaceGeometry();
      nf = fec->GetNumDof(geom, fec->GetOrder());
      base = i*nf;
   }
 
   dofs.SetSize(nf);
   for (int j = 0; j < nf; j++)
   {
      dofs[j] = nvdofs + nedofs + base + j;
   }
}
 
void FiniteElementSpace::GetEdgeInteriorDofs(int i, Array<int> &dofs) const
{
   MFEM_VERIFY(!IsVariableOrder(), "not implemented");
 
   int ne = fec->DofForGeometry(Geometry::SEGMENT);
   dofs.SetSize (ne);
   for (int j = 0, k = nvdofs+i*ne; j < ne; j++, k++)
   {
      dofs[j] = k;
   }
}
 
void FiniteElementSpace::GetPatchDofs(int patch, Array<int> &dofs) const
{
   MFEM_ASSERT(NURBSext,
               "FiniteElementSpace::GetPatchDofs needs a NURBSExtension");
   NURBSext->GetPatchDofs(patch, dofs);
}
 
const FiniteElement *FiniteElementSpace::GetFE(int i) const
{
   if (i < 0 || i >= mesh->GetNE())
   {
      if (mesh->GetNE() == 0)
      {
         MFEM_ABORT("Empty MPI partitions are not permitted!");
      }
      MFEM_ABORT("Invalid element id:" << i << "; minimum allowed:" << 0 <<
                 ", maximum allowed:" << mesh->GetNE()-1);
   }
 
   const FiniteElement *FE =
      fec->GetFE(mesh->GetElementGeometry(i), GetElementOrderImpl(i));
 
   if (NURBSext)
   {
      NURBSext->LoadFE(i, FE);
   }
   else
   {
#ifdef MFEM_DEBUG
      // consistency check: fec->GetOrder() and FE->GetOrder() should return
      // the same value (for standard, constant-order spaces)
      if (!IsVariableOrder() && FE->GetDim() > 0)
      {
         MFEM_ASSERT(FE->GetOrder() == fec->GetOrder(),
                     "internal error: " <<
                     FE->GetOrder() << " != " << fec->GetOrder());
      }
#endif
   }
 
   return FE;
}
 
const FiniteElement *FiniteElementSpace::GetTypicalFE() const
{
   if (mesh->GetNE() > 0) { return GetFE(0); }
 
   Geometry::Type geom = mesh->GetTypicalElementGeometry();
   const FiniteElement *fe = fec->FiniteElementForGeometry(geom);
   MFEM_VERIFY(fe != nullptr, "Could not determine a typical FE!");
   return fe;
}
 
const FiniteElement *FiniteElementSpace::GetBE(int i) const
{
   int order = fec->GetOrder();
 
   if (IsVariableOrder()) // determine order from adjacent element
   {
      int elem, info;
      mesh->GetBdrElementAdjacentElement(i, elem, info);
      order = GetElementOrderImpl(elem);
   }
 
   const FiniteElement *BE;
   switch (mesh->Dimension())
   {
      case 1:
         BE = fec->GetFE(Geometry::POINT, order);
         break;
      case 2:
         BE = fec->GetFE(Geometry::SEGMENT, order);
         break;
      case 3:
      default:
         BE = fec->GetFE(mesh->GetBdrElementGeometry(i), order);
   }
 
   if (NURBSext)
   {
      NURBSext->LoadBE(i, BE);
   }
 
   return BE;
}
 
const FiniteElement *FiniteElementSpace::GetTypicalBE() const
{
   if (mesh->GetNBE() > 0) { return GetBE(0); }
 
   Geometry::Type geom = mesh->GetTypicalFaceGeometry();
   const FiniteElement *be = fec->FiniteElementForGeometry(geom);
   MFEM_VERIFY(be != nullptr, "Could not determine a typical BE!");
   return be;
}
 
const FiniteElement *FiniteElementSpace::GetFaceElement(int i) const
{
   MFEM_VERIFY(!IsVariableOrder(), "not implemented");
 
   const FiniteElement *fe;
   switch (mesh->Dimension())
   {
      case 1:
         fe = fec->FiniteElementForGeometry(Geometry::POINT);
         break;
      case 2:
         fe = fec->FiniteElementForGeometry(Geometry::SEGMENT);
         break;
      case 3:
      default:
         fe = fec->FiniteElementForGeometry(mesh->GetFaceGeometry(i));
   }
 
   if (NURBSext)
   {
      // Ensure 'face_to_be' is built:
      if (!face_dof) { BuildNURBSFaceToDofTable(); }
      MFEM_ASSERT(face_to_be[i] >= 0,
                  "NURBS mesh: only boundary faces are supported!");
      NURBSext->LoadBE(face_to_be[i], fe);
   }
 
   return fe;
}
 
const FiniteElement *FiniteElementSpace::GetTypicalFaceElement() const
{
   return fec->FiniteElementForGeometry(mesh->GetTypicalFaceGeometry());
}
 
const FiniteElement *FiniteElementSpace::GetEdgeElement(int i,
                                                        int variant) const
{
   MFEM_ASSERT(mesh->Dimension() > 1, "No edges with mesh dimension < 2");
 
   int eo = IsVariableOrder() ? GetEdgeOrder(i, variant) : fec->GetOrder();
   return fec->GetFE(Geometry::SEGMENT, eo);
}
 
const FiniteElement *FiniteElementSpace::GetTraceElement(
   int i, Geometry::Type geom_type) const
{
   return fec->GetTraceFE(geom_type, GetElementOrder(i));
}
 
const FiniteElement *FiniteElementSpace::GetTypicalTraceElement() const
{
   return fec->TraceFiniteElementForGeometry(mesh->GetTypicalFaceGeometry());
}
 
FiniteElementSpace::~FiniteElementSpace()
{
   Destroy();
}
 
void FiniteElementSpace::Destroy()
{
   R_transpose.reset();
   cR.reset();
   cR_hp.reset();
   cP.reset();
   Th.Clear();
   L2E_nat.Clear();
   L2E_lex.Clear();
   for (int i = 0; i < E2Q_array.Size(); i++)
   {
      delete E2Q_array[i];
   }
   E2Q_array.SetSize(0);
   L2F.clear();
   interpolations.clear();
   for (int i = 0; i < E2IFQ_array.Size(); i++)
   {
      delete E2IFQ_array[i];
   }
   E2IFQ_array.SetSize(0);
   for (int i = 0; i < E2BFQ_array.Size(); i++)
   {
      delete E2BFQ_array[i];
   }
   E2BFQ_array.SetSize(0);
 
   DestroyDoFTransArray();
 
   dof_elem_array.DeleteAll();
   dof_ldof_array.DeleteAll();
   dof_bdr_elem_array.DeleteAll();
   dof_bdr_ldof_array.DeleteAll();
 
   for (int i = 0; i < VNURBSext.Size(); i++)
   {
      delete VNURBSext[i];
   }
 
   if (NURBSext)
   {
      if (own_ext) { delete NURBSext; }
      delete face_dof;
      face_to_be.DeleteAll();
      if (VNURBSext.Size() > 0 )
      {
         delete elem_dof;
         delete bdr_elem_dof;
      }
   }
   else
   {
      delete elem_dof;
      delete elem_fos;
      delete bdr_elem_dof;
      delete bdr_elem_fos;
      delete face_dof;
      delete [] bdofs;
   }
   ceed::RemoveBasisAndRestriction(this);
}
 
void FiniteElementSpace::DestroyDoFTransArray()
{
   for (int i = 0; i < DoFTransArray.Size(); i++)
   {
      delete DoFTransArray[i];
   }
   DoFTransArray.SetSize(0);
}
 
void FiniteElementSpace::GetTransferOperator(
   const FiniteElementSpace &coarse_fes, OperatorHandle &T) const
{
   // Assumptions: see the declaration of the method.
 
   if (T.Type() == Operator::MFEM_SPARSEMAT)
   {
      if (!IsVariableOrder())
      {
         Mesh::GeometryList elem_geoms(*mesh);
 
         DenseTensor localP[Geometry::NumGeom];
         for (int i = 0; i < elem_geoms.Size(); i++)
         {
            GetLocalRefinementMatrices(coarse_fes, elem_geoms[i],
                                       localP[elem_geoms[i]]);
         }
         T.Reset(RefinementMatrix_main(coarse_fes.GetNDofs(),
                                       coarse_fes.GetElementToDofTable(),
                                       coarse_fes.
                                       GetElementToFaceOrientationTable(),
                                       localP));
      }
      else
      {
         T.Reset(VariableOrderRefinementMatrix(coarse_fes.GetNDofs(),
                                               coarse_fes.GetElementToDofTable()));
      }
   }
   else
   {
      T.Reset(new RefinementOperator(this, &coarse_fes));
   }
}
 
void FiniteElementSpace::GetTrueTransferOperator(
   const FiniteElementSpace &coarse_fes, OperatorHandle &T) const
{
   const SparseMatrix *coarse_P = coarse_fes.GetConformingProlongation();
 
   Operator::Type req_type = T.Type();
   GetTransferOperator(coarse_fes, T);
 
   if (req_type == Operator::MFEM_SPARSEMAT)
   {
      if (GetConformingRestriction())
      {
         T.Reset(mfem::Mult(*cR, *T.As<SparseMatrix>()));
      }
      if (coarse_P)
      {
         T.Reset(mfem::Mult(*T.As<SparseMatrix>(), *coarse_P));
      }
   }
   else
   {
      const int RP_case = bool(GetConformingRestriction()) + 2*bool(coarse_P);
      if (RP_case == 0) { return; }
      const bool owner = T.OwnsOperator();
      T.SetOperatorOwner(false);
      switch (RP_case)
      {
         case 1:
            T.Reset(new ProductOperator(cR.get(), T.Ptr(), false, owner));
            break;
         case 2:
            T.Reset(new ProductOperator(T.Ptr(), coarse_P, owner, false));
            break;
         case 3:
            T.Reset(new TripleProductOperator(
                       cR.get(), T.Ptr(), coarse_P, false, owner, false));
            break;
      }
   }
}
 
void FiniteElementSpace::UpdateElementOrders()
{
   Array<char> new_order(mesh->GetNE());
   switch (mesh->GetLastOperation())
   {
      case Mesh::REFINE:
      {
         const CoarseFineTransformations &cf_tr = mesh->GetRefinementTransforms();
         for (int i = 0; i < mesh->GetNE(); i++)
         {
            new_order[i] = elem_order[cf_tr.embeddings[i].parent];
         }
         break;
      }
      case Mesh::DEREFINE:
      {
         const CoarseFineTransformations &cf_tr =
            mesh->ncmesh->GetDerefinementTransforms();
         Table coarse_to_fine;
         cf_tr.MakeCoarseToFineTable(coarse_to_fine);
         Array<int> tabrow;
         for (int i = 0; i < coarse_to_fine.Size(); i++)
         {
            coarse_to_fine.GetRow(i, tabrow);
            // For now we require all children to be of same polynomial order.
            new_order[i] = elem_order[tabrow[0]];
         }
         break;
      }
      default:
         MFEM_ABORT("not implemented yet");
   }
 
   mfem::Swap(elem_order, new_order);
}
 
void FiniteElementSpace::Update(bool want_transform)
{
   lastUpdatePRef = false;
 
   if (!orders_changed)
   {
      if (mesh->GetSequence() == mesh_sequence)
      {
         return; // mesh and space are in sync, no-op
      }
      if (want_transform && mesh->GetSequence() != mesh_sequence + 1)
      {
         MFEM_ABORT("Error in update sequence. Space needs to be updated after "
                    "each mesh modification.");
      }
   }
   else
   {
      if (mesh->GetSequence() != mesh_sequence)
      {
         MFEM_ABORT("Updating space after both mesh change and element order "
                    "change is not supported. Please update separately after "
                    "each change.");
      }
   }
 
   if (NURBSext)
   {
      UpdateNURBS();
      return;
   }
 
   Table* old_elem_dof = NULL;
   Table* old_elem_fos = NULL;
   int old_ndofs;
   bool old_orders_changed = orders_changed;
 
   // save old DOF table
   if (want_transform)
   {
      old_elem_dof = elem_dof;
      old_elem_fos = elem_fos;
      elem_dof = NULL;
      elem_fos = NULL;
      old_ndofs = ndofs;
   }
 
   // update the 'elem_order' array if the mesh has changed
   if (IsVariableOrder() && mesh->GetSequence() != mesh_sequence)
   {
      UpdateElementOrders();
   }
 
   Destroy(); // calls Th.Clear()
   Construct();
   BuildElementToDofTable();
 
   if (want_transform)
   {
      MFEM_VERIFY(!old_orders_changed, "Interpolation for element order change "
                  "is not implemented yet, sorry.");
 
      // calculate appropriate GridFunction transformation
      switch (mesh->GetLastOperation())
      {
         case Mesh::REFINE:
         {
            if (Th.Type() != Operator::MFEM_SPARSEMAT)
            {
               Th.Reset(new RefinementOperator(this, old_elem_dof,
                                               old_elem_fos, old_ndofs));
               // The RefinementOperator takes ownership of 'old_elem_dof', so
               // we no longer own it:
               old_elem_dof = NULL;
               old_elem_fos = NULL;
            }
            else
            {
               // calculate fully assembled matrix
               Th.Reset(RefinementMatrix(old_ndofs, old_elem_dof,
                                         old_elem_fos));
            }
            break;
         }
 
         case Mesh::DEREFINE:
         {
            BuildConformingInterpolation();
#if 0
            Th.Reset(DerefinementMatrix(old_ndofs, old_elem_dof, old_elem_fos));
#else
            Th.Reset(new DerefineMatrixOp(*this, old_ndofs, old_elem_dof, old_elem_fos));
#endif
            if (IsVariableOrder())
            {
               if (cP && cR_hp)
               {
                  Th.SetOperatorOwner(false);
                  Th.Reset(new TripleProductOperator(cP.get(), cR_hp.get(), Th.Ptr(),
                                                     false, false, true));
               }
            }
            else
            {
               if (cP && cR)
               {
                  Th.SetOperatorOwner(false);
                  Th.Reset(new TripleProductOperator(cP.get(), cR.get(), Th.Ptr(),
                                                     false, false, true));
               }
            }
            break;
         }
 
         default:
            break;
      }
 
      delete old_elem_dof;
      delete old_elem_fos;
   }
}
 
void FiniteElementSpace::PRefineAndUpdate(const Array<pRefinement> & refs,
                                          bool want_transfer)
{
   if (want_transfer)
   {
      fesPrev.reset(new FiniteElementSpace(mesh, fec, vdim, ordering));
      for (int i = 0; i<mesh->GetNE(); i++)
      {
         fesPrev->SetElementOrder(i, GetElementOrder(i));
      }
      fesPrev->Update(false);
   }
 
   for (auto ref : refs)
   {
      SetElementOrder(ref.index, GetElementOrder(ref.index) + ref.delta);
   }
 
   Update(false);
 
   if (want_transfer)
   {
      PTh.reset(new PRefinementTransferOperator(*fesPrev, *this));
   }
 
   lastUpdatePRef = true;
}
 
bool FiniteElementSpace::PRefinementSupported()
{
   // Check whether the space type is L2 or H1
   if (!dynamic_cast<const L2_FECollection*>(fec) &&
       !dynamic_cast<const H1_FECollection*>(fec))
   {
      return false;
   }
 
   // Check whether the mesh is purely quadrilateral or hexahedral.
   const int dim = mesh->Dimension();
   Array<Geometry::Type> geoms;
   mesh->GetGeometries(dim, geoms);
   if (geoms.Size() != 1) { return false; }
   if (dim == 2 && geoms[0] != Geometry::Type::SQUARE) { return false; }
   else if (dim == 3 && geoms[0] != Geometry::Type::CUBE) { return false; }
 
   return true;
}
 
void FiniteElementSpace::UpdateMeshPointer(Mesh *new_mesh)
{
   mesh = new_mesh;
}
 
void FiniteElementSpace::GetNodePositions(const Vector &mesh_nodes,
                                          Vector &fes_node_pos,
                                          int fes_nodes_ordering) const
{
   Mesh *m = GetMesh();
   const int NE = m->GetNE();
 
   if (NE == 0) { fes_node_pos.SetSize(0); return; }
 
   const int dim = m->Dimension();
   Array<int> dofs;
   Vector e_xyz;
   fes_node_pos.SetSize(GetNDofs() * dim);
   const FiniteElementSpace *mesh_fes = m->GetNodalFESpace();
   FiniteElementSpace vector_fes(m, FEColl(), dim, fes_nodes_ordering);
 
   for (int e = 0; e < NE; e++)
   {
      mesh_fes->GetElementVDofs(e, dofs);
      const int mdof_cnt = dofs.Size() / dim;
      mesh_nodes.GetSubVector(dofs, e_xyz); //e_xyz is ordered by nodes here
 
      auto ir = GetFE(e)->GetNodes();
      const int fdof_cnt = ir.GetNPoints();
      Vector mesh_shape(mdof_cnt), gf_xyz(fdof_cnt * dim);
      for (int q = 0; q < fdof_cnt; q++)
      {
         mesh_fes->GetFE(e)->CalcShape(ir.IntPoint(q), mesh_shape);
         for (int d = 0; d < dim; d++)
         {
            Vector x(e_xyz.GetData() + d*mdof_cnt, mdof_cnt);
            gf_xyz(d*fdof_cnt + q) = x * mesh_shape; // order by nodes
         }
      }
 
      // reuse/resize dofs.
      vector_fes.GetElementVDofs(e, dofs);
      fes_node_pos.SetSubVector(dofs, gf_xyz);
   }
}
 
void FiniteElementSpace::Save(std::ostream &os) const
{
   int fes_format = 90; // the original format, v0.9
   bool nurbs_unit_weights = false;
 
   // Determine the format that should be used.
   if (!NURBSext)
   {
      // TODO: if this is a variable-order FE space, use fes_format = 100.
   }
   else
   {
      const NURBSFECollection *nurbs_fec =
         dynamic_cast<const NURBSFECollection *>(fec);
      MFEM_VERIFY(nurbs_fec, "invalid FE collection");
      nurbs_fec->SetOrder(NURBSext->GetOrder());
      const real_t eps = 5e-14;
      nurbs_unit_weights = (NURBSext->GetWeights().Min() >= 1.0-eps &&
                            NURBSext->GetWeights().Max() <= 1.0+eps);
      if ((NURBSext->GetOrder() == NURBSFECollection::VariableOrder) ||
          (NURBSext != mesh->NURBSext && !nurbs_unit_weights) ||
          (NURBSext->GetMaster().Size() != 0 ))
      {
         fes_format = 100; // v1.0 format
      }
   }
 
   os << (fes_format == 90 ?
          "FiniteElementSpace\n" : "MFEM FiniteElementSpace v1.0\n")
      << "FiniteElementCollection: " << fec->Name() << '\n'
      << "VDim: " << vdim << '\n'
      << "Ordering: " << ordering << '\n';
 
   if (fes_format == 100) // v1.0
   {
      if (!NURBSext)
      {
         // TODO: this is a variable-order FE space --> write 'element_orders'.
      }
      else if (NURBSext != mesh->NURBSext)
      {
         if (NURBSext->GetOrder() != NURBSFECollection::VariableOrder)
         {
            os << "NURBS_order\n" << NURBSext->GetOrder() << '\n';
         }
         else
         {
            os << "NURBS_orders\n";
            // 1 = do not write the size, just the entries:
            NURBSext->GetOrders().Save(os, 1);
         }
         // If periodic BCs are given, write connectivity
         if (NURBSext->GetMaster().Size() != 0 )
         {
            os <<"NURBS_periodic\n";
            NURBSext->GetMaster().Save(os);
            NURBSext->GetSlave().Save(os);
         }
         // If the weights are not unit, write them to the output:
         if (!nurbs_unit_weights)
         {
            os << "NURBS_weights\n";
            NURBSext->GetWeights().Print(os, 1);
         }
      }
      os << "End: MFEM FiniteElementSpace v1.0\n";
   }
}
 
std::shared_ptr<const PRefinementTransferOperator>
FiniteElementSpace::GetPrefUpdateOperator() { return PTh; }
 
void FiniteElementSpace
::GetEssentialBdrEdgesFaces(const Array<int> &bdr_attr_is_ess,
                            std::set<int> & edges, std::set<int> & faces) const
{
   const int dim = mesh->Dimension();
   MFEM_VERIFY(dim == 2 || dim == 3, "");
 
   for (int i = 0; i < GetNBE(); i++)
   {
      if (bdr_attr_is_ess[GetBdrAttribute(i)-1])
      {
         int f, o;
         mesh->GetBdrElementFace(i, &f, &o);
 
         if (dim == 3)
         {
            faces.insert(f);
            Array<int> edges_i, cor;
            mesh->GetBdrElementEdges(i, edges_i, cor);
            for (auto edge : edges_i)
            {
               edges.insert(edge);
            }
         }
         else
         {
            edges.insert(f);
         }
      }
   }
 
   if (Nonconforming())
   {
      Array<int> bdr_verts, bdr_edges, bdr_faces;
      mesh->ncmesh->GetBoundaryClosure(bdr_attr_is_ess, bdr_verts, bdr_edges,
                                       bdr_faces);
 
      for (auto e : bdr_edges)
      {
         edges.insert(e);
      }
 
      for (auto f : bdr_faces)
      {
         faces.insert(f);
      }
   }
}
 
void FiniteElementSpace::GetBoundaryLoopEdgeDofs(
   const Array<int> &boundary_element_indices,
   Array<int> &boundary_edge_dofs,
   Array<int> *dof_edges,
   Array<int> *dof_boundary_elements) const
{
   MFEM_VERIFY(mesh->Dimension() >= 2,
               "GetBoundaryLoopEdgeDofs requires 2D or 3D meshes to find edge objects");
 
   boundary_edge_dofs.SetSize(0);
   if (dof_edges) { dof_edges->SetSize(0); }
   if (dof_boundary_elements) { dof_boundary_elements->SetSize(0); }
 
   // A DOF that appears in exactly one selected boundary element lies on the
   // bounding loop; one appearing in two or more is interior to the boundary
   // region and is dropped. Count occurrences of each DOF (using scratch maps,
   // exposed only as parallel-indexed Array<int> below) and record, on first
   // sight, the local edge and boundary element carrying it.
   //
   // The count is over GetEdgeDofs, which returns endpoint vertex DOFs as well
   // as edge-interior DOFs (relevant for collections such as ND_R2D that carry
   // vertex DOFs). Edge-interior DOFs occur once per edge, so the count mainly
   // resolves vertex DOFs: a vertex shared by several elements is interior and
   // dropped, while a genuine loop-corner (open-curve endpoint) vertex is kept.
   // This is why we count GetEdgeDofs rather than collecting GetEdgeInteriorDofs,
   // which would omit the endpoint vertex DOFs the method is documented to keep.
   // The 3D removal criterion (any edge in two or more faces) matches the
   // parallel version rather than a parity toggle.
   std::unordered_map<int, int> dof_count, dof_edge, dof_belem;
   Array<int> edge_dofs, edges, edge_orientations;
 
   const int dim = mesh->Dimension();
   for (int i = 0; i < boundary_element_indices.Size(); ++i)
   {
      const int boundary_element_idx = boundary_element_indices[i];
      std::unordered_set<int> boundary_element_dofs;
 
      if (dim == 3)
      {
         // Boundary elements are 2D faces; extract their 1D edges.
         int face_index, face_orientation;
         mesh->GetBdrElementFace(boundary_element_idx, &face_index,
                                 &face_orientation);
         mesh->GetFaceEdges(face_index, edges, edge_orientations);
      }
      else
      {
         // Boundary elements are 1D segments, each being a single edge.
         mesh->GetBdrElementEdges(boundary_element_idx, edges, edge_orientations);
         MFEM_VERIFY(edges.Size() == 1,
                     "2D boundary element should have exactly one edge");
      }
 
      for (int j = 0; j < edges.Size(); ++j)
      {
         GetEdgeDofs(edges[j], edge_dofs);
         for (int k = 0; k < edge_dofs.Size(); ++k)
         {
            const int dof = edge_dofs[k];
            // Count each DOF once per boundary element and record metadata the
            // first time it is seen, so H1 DOFs shared by multiple edges of the
            // same element are not double counted.
            if (boundary_element_dofs.insert(dof).second &&
                dof_count[dof]++ == 0)
            {
               dof_edge[dof] = edges[j];
               dof_belem[dof] = boundary_element_idx;
            }
         }
      }
   }
 
   // Emit the DOFs seen in exactly one selected boundary element, in a
   // deterministic (increasing DOF index) order shared by all output arrays.
   std::vector<int> kept;
   kept.reserve(dof_count.size());
   for (const auto &[dof, count] : dof_count)
   {
      if (count == 1) { kept.push_back(dof); }
   }
   std::sort(kept.begin(), kept.end());
 
   boundary_edge_dofs.Reserve(static_cast<int>(kept.size()));
   if (dof_edges) { dof_edges->Reserve(static_cast<int>(kept.size())); }
   if (dof_boundary_elements)
   {
      dof_boundary_elements->Reserve(static_cast<int>(kept.size()));
   }
   for (int dof : kept)
   {
      boundary_edge_dofs.Append(dof);
      if (dof_edges) { dof_edges->Append(dof_edge[dof]); }
      if (dof_boundary_elements) { dof_boundary_elements->Append(dof_belem[dof]); }
   }
}
 
void FiniteElementSpace::GetBoundaryElementsByAttribute(
   const Array<int> &bdr_attrs,
   std::vector<Array<int>> &attr_to_elements) const
{
   // One (initially empty) list of boundary elements per requested attribute,
   // indexed to match bdr_attrs.
   attr_to_elements.assign(bdr_attrs.Size(), Array<int>());
 
   // Map attribute value -> position in bdr_attrs for quick lookup.
   std::unordered_map<int, int> attr_to_index;
   for (int i = 0; i < bdr_attrs.Size(); ++i)
   {
      attr_to_index[bdr_attrs[i]] = i;
   }
 
   // Bucket boundary elements by their attribute.
   for (int i = 0; i < mesh->GetNBE(); ++i)
   {
      int attr = mesh->GetBdrElement(i)->GetAttribute();
      auto it = attr_to_index.find(attr);
      if (it != attr_to_index.end())
      {
         attr_to_elements[it->second].Append(i);
      }
   }
}
 
void FiniteElementSpace::GetBoundaryElementsByAttribute(
   int bdr_attr, Array<int> &boundary_elements) const
{
   boundary_elements.SetSize(0);
 
   for (int i = 0; i < mesh->GetNBE(); ++i)
   {
      if (mesh->GetBdrElement(i)->GetAttribute() == bdr_attr)
      {
         boundary_elements.Append(i);
      }
   }
}
 
void FiniteElementSpace::ComputeLoopEdgeOrientations(
   const Array<int> &dof_edges,
   const Array<int> &dof_boundary_elements,
   const Vector &loop_normal,
   Array<int> &dof_orientations) const
{
   MFEM_VERIFY(dof_edges.Size() == dof_boundary_elements.Size(),
               "dof_edges and dof_boundary_elements must be parallel-indexed");
 
   const int ndof = dof_edges.Size();
   dof_orientations.SetSize(ndof);
 
   Array<int> edge_verts, bdr_elem_verts;
   Vector edge_vec(3), to_edge_vec(3), cross_product(3);
   for (int i = 0; i < ndof; i++)
   {
      const int edge_id = dof_edges[i];
      const int bdr_elem_idx = dof_boundary_elements[i];
 
      // Get edge vertices
      mesh->GetEdgeVertices(edge_id, edge_verts);
 
      const real_t *v0 = mesh->GetVertex(edge_verts[0]);
      const real_t *v1 = mesh->GetVertex(edge_verts[1]);
 
      // Get boundary element vertices
      mesh->GetBdrElement(bdr_elem_idx)->GetVertices(bdr_elem_verts);
 
      // Find the third vertex (not part of the edge)
      int third_vertex = -1;
      for (int j = 0; j < bdr_elem_verts.Size(); j++)
      {
         int v = bdr_elem_verts[j];
         if (v != edge_verts[0] && v != edge_verts[1])
         {
            third_vertex = v;
            break;
         }
      }
 
      if (third_vertex == -1)
      {
         MFEM_ABORT("Boundary element " << bdr_elem_idx << " has only 2 vertices, "
                    "but 3D boundary elements must have at least 3 vertices");
      }
 
      const real_t *v2 = mesh->GetVertex(third_vertex);
 
      // Edge vector
      for (int j = 0; j < 3; j++) { edge_vec[j] = v1[j] - v0[j]; }
 
      // Vector from third vertex to edge (use edge midpoint)
      for (int j = 0; j < 3; j++)
      {
         real_t edge_midpoint = (v0[j] + v1[j]) * 0.5;
         to_edge_vec[j] = edge_midpoint - v2[j];
      }
 
      // Cross product: to_edge × edge
      to_edge_vec.cross3D(edge_vec, cross_product);
 
      // Check alignment with loop normal
      real_t dot_product = cross_product * loop_normal;
      dof_orientations[i] = (dot_product > 0) ? 1 : -1;
   }
}
 
FiniteElementCollection *FiniteElementSpace::Load(Mesh *m, std::istream &input)
{
   string buff;
   int fes_format = 0, ord;
   FiniteElementCollection *r_fec;
 
   Destroy();
 
   input >> std::ws;
   getline(input, buff);  // 'FiniteElementSpace'
   filter_dos(buff);
   if (buff == "FiniteElementSpace") { fes_format = 90; /* v0.9 */ }
   else if (buff == "MFEM FiniteElementSpace v1.0") { fes_format = 100; }
   else { MFEM_ABORT("input stream is not a FiniteElementSpace!"); }
   getline(input, buff, ' '); // 'FiniteElementCollection:'
   input >> std::ws;
   getline(input, buff);
   filter_dos(buff);
   r_fec = FiniteElementCollection::New(buff.c_str());
   getline(input, buff, ' '); // 'VDim:'
   input >> vdim;
   getline(input, buff, ' '); // 'Ordering:'
   input >> ord;
 
   NURBSFECollection *nurbs_fec = dynamic_cast<NURBSFECollection*>(r_fec);
   if (nurbs_fec) { nurbs_fec->SetDim(m->Dimension()); }
   NURBSExtension *nurbs_ext = NULL;
   if (fes_format == 90) // original format, v0.9
   {
      if (nurbs_fec)
      {
         MFEM_VERIFY(m->NURBSext, "NURBS FE collection requires a NURBS mesh!");
         const int order = nurbs_fec->GetOrder();
         if (order != m->NURBSext->GetOrder() &&
             order != NURBSFECollection::VariableOrder)
         {
            nurbs_ext = new NURBSExtension(m->NURBSext, order);
         }
      }
   }
   else if (fes_format == 100) // v1.0
   {
      while (1)
      {
         skip_comment_lines(input, '#');
         MFEM_VERIFY(input.good(), "error reading FiniteElementSpace v1.0");
         getline(input, buff);
         filter_dos(buff);
         if (buff == "NURBS_order" || buff == "NURBS_orders")
         {
            MFEM_VERIFY(nurbs_fec,
                        buff << ": NURBS FE collection is required!");
            MFEM_VERIFY(m->NURBSext, buff << ": NURBS mesh is required!");
            MFEM_VERIFY(!nurbs_ext, buff << ": order redefinition!");
            if (buff == "NURBS_order")
            {
               int order;
               input >> order;
               nurbs_ext = new NURBSExtension(m->NURBSext, order);
            }
            else
            {
               Array<int> orders;
               orders.Load(m->NURBSext->GetNKV(), input);
               nurbs_ext = new NURBSExtension(m->NURBSext, orders);
            }
         }
         else if (buff == "NURBS_periodic")
         {
            Array<int> master, slave;
            master.Load(input);
            slave.Load(input);
            nurbs_ext->ConnectBoundaries(master,slave);
         }
         else if (buff == "NURBS_weights")
         {
            MFEM_VERIFY(nurbs_ext, "NURBS_weights: NURBS_orders have to be "
                        "specified before NURBS_weights!");
            nurbs_ext->GetWeights().Load(input, nurbs_ext->GetNDof());
         }
         else if (buff == "element_orders")
         {
            MFEM_VERIFY(!nurbs_fec, "section element_orders cannot be used "
                        "with a NURBS FE collection");
            MFEM_ABORT("element_orders: not implemented yet!");
         }
         else if (buff == "End: MFEM FiniteElementSpace v1.0")
         {
            break;
         }
         else
         {
            MFEM_ABORT("unknown section: " << buff);
         }
      }
   }
 
   Constructor(m, nurbs_ext, r_fec, vdim, ord);
 
   return r_fec;
}
 
ElementDofOrdering GetEVectorOrdering(const FiniteElementSpace& fes)
{
   return (UsesTensorBasis(fes) || fes.UsesRaggedTensorBasis()) ?
          ElementDofOrdering::LEXICOGRAPHIC:
          ElementDofOrdering::NATIVE;
}
} // namespace mfem