fe_coll.cpp

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// Copyright (c) 2010, Lawrence Livermore National Security, LLC. Produced at
// the Lawrence Livermore National Laboratory. LLNL-CODE-443211. All Rights
// reserved. See file COPYRIGHT for details.
//
// This file is part of the MFEM library. For more information and source code
// availability see http://mfem.org.
//
// MFEM is free software; you can redistribute it and/or modify it under the
// terms of the GNU Lesser General Public License (as published by the Free
// Software Foundation) version 2.1 dated February 1999.

#include "fem.hpp"
#include <cstdlib>
#include <cstring>
#include <cstdio>
#ifdef _WIN32
#define snprintf _snprintf_s
#endif

namespace mfem
{

using namespace std;

int FiniteElementCollection::HasFaceDofs(int GeomType) const
{
   switch (GeomType)
   {
      case Geometry::TETRAHEDRON: return DofForGeometry (Geometry::TRIANGLE);
      case Geometry::CUBE:        return DofForGeometry (Geometry::SQUARE);
      default:
         mfem_error ("FiniteElementCollection::HasFaceDofs:"
                     " unknown geometry type.");
   }
   return 0;
}

FiniteElementCollection *FiniteElementCollection::GetTraceCollection() const
{
   MFEM_ABORT("this method is not implemented in this derived class!");
   return NULL;
}

FiniteElementCollection *FiniteElementCollection::New(const char *name)
{
   FiniteElementCollection *fec = NULL;

   if (!strcmp(name, "Linear"))
   {
      fec = new LinearFECollection;
   }
   else if (!strcmp(name, "Quadratic"))
   {
      fec = new QuadraticFECollection;
   }
   else if (!strcmp(name, "QuadraticPos"))
   {
      fec = new QuadraticPosFECollection;
   }
   else if (!strcmp(name, "Cubic"))
   {
      fec = new CubicFECollection;
   }
   else if (!strcmp(name, "Const3D"))
   {
      fec = new Const3DFECollection;
   }
   else if (!strcmp(name, "Const2D"))
   {
      fec = new Const2DFECollection;
   }
   else if (!strcmp(name, "LinearDiscont2D"))
   {
      fec = new LinearDiscont2DFECollection;
   }
   else if (!strcmp(name, "GaussLinearDiscont2D"))
   {
      fec = new GaussLinearDiscont2DFECollection;
   }
   else if (!strcmp(name, "P1OnQuad"))
   {
      fec = new P1OnQuadFECollection;
   }
   else if (!strcmp(name, "QuadraticDiscont2D"))
   {
      fec = new QuadraticDiscont2DFECollection;
   }
   else if (!strcmp(name, "QuadraticPosDiscont2D"))
   {
      fec = new QuadraticPosDiscont2DFECollection;
   }
   else if (!strcmp(name, "GaussQuadraticDiscont2D"))
   {
      fec = new GaussQuadraticDiscont2DFECollection;
   }
   else if (!strcmp(name, "CubicDiscont2D"))
   {
      fec = new CubicDiscont2DFECollection;
   }
   else if (!strcmp(name, "LinearDiscont3D"))
   {
      fec = new LinearDiscont3DFECollection;
   }
   else if (!strcmp(name, "QuadraticDiscont3D"))
   {
      fec = new QuadraticDiscont3DFECollection;
   }
   else if (!strcmp(name, "LinearNonConf3D"))
   {
      fec = new LinearNonConf3DFECollection;
   }
   else if (!strcmp(name, "CrouzeixRaviart"))
   {
      fec = new CrouzeixRaviartFECollection;
   }
   else if (!strcmp(name, "ND1_3D"))
   {
      fec = new ND1_3DFECollection;
   }
   else if (!strcmp(name, "RT0_2D"))
   {
      fec = new RT0_2DFECollection;
   }
   else if (!strcmp(name, "RT1_2D"))
   {
      fec = new RT1_2DFECollection;
   }
   else if (!strcmp(name, "RT2_2D"))
   {
      fec = new RT2_2DFECollection;
   }
   else if (!strcmp(name, "RT0_3D"))
   {
      fec = new RT0_3DFECollection;
   }
   else if (!strcmp(name, "RT1_3D"))
   {
      fec = new RT1_3DFECollection;
   }
   else if (!strncmp(name, "H1_Trace_", 9))
   {
      fec = new H1_Trace_FECollection(atoi(name + 13), atoi(name + 9));
   }
   else if (!strncmp(name, "H1_", 3))
   {
      fec = new H1_FECollection(atoi(name + 7), atoi(name + 3));
   }
   else if (!strncmp(name, "H1Pos_Trace_", 12))
   {
      fec = new H1_Trace_FECollection(atoi(name + 16), atoi(name + 12),
                                      BasisType::Positive);
   }
   else if (!strncmp(name, "H1Pos_", 6))
   {
      fec = new H1Pos_FECollection(atoi(name + 10), atoi(name + 6));
   }
   else if (!strncmp(name, "H1_Trace@", 9))
   {
      fec = new H1_Trace_FECollection(atoi(name + 15), atoi(name + 11),
                                      BasisType::GetType(name[9]));
   }
   else if (!strncmp(name, "H1@", 3))
   {
      fec = new H1_FECollection(atoi(name + 9), atoi(name + 5),
                                BasisType::GetType(name[3]));
   }
   else if (!strncmp(name, "L2_T", 4))
      fec = new L2_FECollection(atoi(name + 10), atoi(name + 6),
                                atoi(name + 4));
   else if (!strncmp(name, "L2_", 3))
   {
      fec = new L2_FECollection(atoi(name + 7), atoi(name + 3));
   }
   else if (!strncmp(name, "L2Int_T", 7))
   {
      fec = new L2_FECollection(atoi(name + 13), atoi(name + 9),
                                atoi(name + 7), FiniteElement::INTEGRAL);
   }
   else if (!strncmp(name, "L2Int_", 6))
   {
      fec = new L2_FECollection(atoi(name + 10), atoi(name + 6),
                                BasisType::GaussLegendre,
                                FiniteElement::INTEGRAL);
   }
   else if (!strncmp(name, "RT_Trace_", 9))
   {
      fec = new RT_Trace_FECollection(atoi(name + 13), atoi(name + 9));
   }
   else if (!strncmp(name, "RT_ValTrace_", 12))
   {
      fec = new RT_Trace_FECollection(atoi(name + 16), atoi(name + 12),
                                      FiniteElement::VALUE);
   }
   else if (!strncmp(name, "RT_Trace@", 9))
   {
      fec = new RT_Trace_FECollection(atoi(name + 15), atoi(name + 11),
                                      FiniteElement::INTEGRAL,
                                      BasisType::GetType(name[9]));
   }
   else if (!strncmp(name, "RT_ValTrace@", 12))
   {
      fec = new RT_Trace_FECollection(atoi(name + 18), atoi(name + 14),
                                      FiniteElement::VALUE,
                                      BasisType::GetType(name[12]));
   }
   else if (!strncmp(name, "DG_Iface_", 9))
   {
      fec = new DG_Interface_FECollection(atoi(name + 13), atoi(name + 9));
   }
   else if (!strncmp(name, "DG_Iface@", 9))
   {
      fec = new DG_Interface_FECollection(atoi(name + 15), atoi(name + 11),
                                          FiniteElement::VALUE,
                                          BasisType::GetType(name[9]));
   }
   else if (!strncmp(name, "DG_IntIface_", 12))
   {
      fec = new DG_Interface_FECollection(atoi(name + 16), atoi(name + 12),
                                          FiniteElement::INTEGRAL);
   }
   else if (!strncmp(name, "DG_IntIface@", 12))
   {
      fec = new DG_Interface_FECollection(atoi(name + 18), atoi(name + 14),
                                          FiniteElement::INTEGRAL,
                                          BasisType::GetType(name[12]));
   }
   else if (!strncmp(name, "RT_", 3))
   {
      fec = new RT_FECollection(atoi(name + 7), atoi(name + 3));
   }
   else if (!strncmp(name, "RT@", 3))
   {
      fec = new RT_FECollection(atoi(name + 10), atoi(name + 6),
                                BasisType::GetType(name[3]),
                                BasisType::GetType(name[4]));
   }
   else if (!strncmp(name, "ND_Trace_", 9))
   {
      fec = new ND_Trace_FECollection(atoi(name + 13), atoi(name + 9));
   }
   else if (!strncmp(name, "ND_", 3))
   {
      fec = new ND_FECollection(atoi(name + 7), atoi(name + 3));
   }
   else if (!strncmp(name, "ND_Trace@", 9))
   {
      fec = new ND_Trace_FECollection(atoi(name + 16), atoi(name + 12),
                                      BasisType::GetType(name[9]),
                                      BasisType::GetType(name[10]));
   }
   else if (!strncmp(name, "ND@", 3))
   {
      fec = new ND_FECollection(atoi(name + 10), atoi(name + 6),
                                BasisType::GetType(name[3]),
                                BasisType::GetType(name[4]));
   }
   else if (!strncmp(name, "Local_", 6))
   {
      fec = new Local_FECollection(name + 6);
   }
   else if (!strncmp(name, "NURBS", 5))
   {
      fec = new NURBSFECollection(atoi(name + 5));
   }
   else
   {
      mfem_error("FiniteElementCollection::New : "
                 "Unknown FiniteElementCollection!");
   }
   MFEM_VERIFY(!strcmp(fec->Name(), name), "input name: \"" << name
               << "\" does not match the created collection name: \""
               << fec->Name() << '"');

   return fec;
}

template <Geometry::Type geom>
inline void FiniteElementCollection::GetNVE(int &nv, int &ne)
{
   typedef typename Geometry::Constants<geom> g_consts;

   nv = g_consts::NumVert;
   ne = g_consts::NumEdges;
}

template <Geometry::Type geom, typename v_t>
inline void FiniteElementCollection::
GetEdge(int &nv, v_t &v, int &ne, int &e, int &eo, const int edge_info)
{
   typedef typename Geometry::Constants<Geometry::SEGMENT> e_consts;
   typedef typename Geometry::Constants<geom> g_consts;

   nv = e_consts::NumVert;
   ne = 1;
   e = edge_info/64;
   eo = edge_info%64;
   MFEM_ASSERT(0 <= e && e < g_consts::NumEdges, "");
   MFEM_ASSERT(0 <= eo && eo < e_consts::NumOrient, "");
   v[0] = g_consts::Edges[e][0];
   v[1] = g_consts::Edges[e][1];
   v[0] = e_consts::Orient[eo][v[0]];
   v[1] = e_consts::Orient[eo][v[1]];
}

template <Geometry::Type geom, Geometry::Type f_geom,
          typename v_t, typename e_t, typename eo_t>
inline void FiniteElementCollection::
GetFace(int &nv, v_t &v, int &ne, e_t &e, eo_t &eo,
        int &nf, int &f, int &fg, int &fo, const int face_info)
{
   typedef typename Geometry::Constants<  geom> g_consts;
   typedef typename Geometry::Constants<f_geom> f_consts;

   nv = f_consts::NumVert;
   nf = 1;
   f = face_info/64;
   fg = f_geom;
   fo = face_info%64;
   MFEM_ASSERT(0 <= f && f < g_consts::NumFaces, "");
   MFEM_ASSERT(0 <= fo && fo < f_consts::NumOrient, "");
   for (int i = 0; i < f_consts::NumVert; i++)
   {
      v[i] = f_consts::Orient[fo][i];
      v[i] = g_consts::FaceVert[f][v[i]];
   }
   ne = f_consts::NumEdges;
   for (int i = 0; i < f_consts::NumEdges; i++)
   {
      int v0 = v[f_consts::Edges[i][0]];
      int v1 = v[f_consts::Edges[i][1]];
      int eor = 0;
      if (v0 > v1) { swap(v0, v1); eor = 1; }
      for (int j = g_consts::VertToVert::I[v0]; true; j++)
      {
         MFEM_ASSERT(j < g_consts::VertToVert::I[v0+1],
                     "internal error, edge not found");
         if (v1 == g_consts::VertToVert::J[j][0])
         {
            int en = g_consts::VertToVert::J[j][1];
            if (en < 0)
            {
               en = -1-en;
               eor = 1-eor;
            }
            e[i] = en;
            eo[i] = eor;
            break;
         }
      }
   }
}

void FiniteElementCollection::SubDofOrder(int Geom, int SDim, int Info,
                                          Array<int> &dofs) const
{
   // Info = 64 * SubIndex + SubOrientation
   MFEM_ASSERT(0 <= Geom && Geom < Geometry::NumGeom,
               "invalid Geom = " << Geom);
   MFEM_ASSERT(0 <= SDim && SDim <= Geometry::Dimension[Geom],
               "invalid SDim = " << SDim <<
               " for Geom = " << Geometry::Name[Geom]);

   const int nvd = DofForGeometry(Geometry::POINT);
   if (SDim == 0) // vertex
   {
      const int off = nvd*(Info/64);
      dofs.SetSize(nvd);
      for (int i = 0; i < nvd; i++)
      {
         dofs[i] = off + i;
      }
   }
   else
   {
      int v[4], e[4], eo[4], f[1], fg[1], fo[1];
      int av = 0, nv = 0, ae = 0, ne = 0, nf = 0;

      switch (Geom)
      {
         case Geometry::SEGMENT:
         {
            GetNVE<Geometry::SEGMENT>(av, ae);
            GetEdge<Geometry::SEGMENT>(nv, v, ne, e[0], eo[0], Info);
            break;
         }

         case Geometry::TRIANGLE:
         {
            GetNVE<Geometry::TRIANGLE>(av, ae);
            switch (SDim)
            {
               case 1:
                  GetEdge<Geometry::TRIANGLE>(nv, v, ne, e[0], eo[0], Info);
                  break;
               case 2:
                  GetFace<Geometry::TRIANGLE,Geometry::TRIANGLE>(
                     nv, v, ne, e, eo, nf, f[0], fg[0], fo[0], Info);
                  break;
               default:
                  goto not_supp;
            }
            break;
         }

         case Geometry::SQUARE:
         {
            GetNVE<Geometry::SQUARE>(av, ae);
            switch (SDim)
            {
               case 1:
                  GetEdge<Geometry::SQUARE>(nv, v, ne, e[0], eo[0], Info);
                  break;
               case 2:
                  GetFace<Geometry::SQUARE,Geometry::SQUARE>(
                     nv, v, ne, e, eo, nf, f[0], fg[0], fo[0], Info);
                  break;
               default:
                  goto not_supp;
            }
            break;
         }

         case Geometry::TETRAHEDRON:
         {
            GetNVE<Geometry::TETRAHEDRON>(av, ae);
            switch (SDim)
            {
               case 1:
                  GetEdge<Geometry::TETRAHEDRON>(nv, v, ne, e[0], eo[0], Info);
                  break;
               case 2:
                  GetFace<Geometry::TETRAHEDRON,Geometry::TRIANGLE>(
                     nv, v, ne, e, eo, nf, f[0], fg[0], fo[0], Info);
                  break;
               default:
                  goto not_supp;
            }
            break;
         }

         case Geometry::CUBE:
         {
            GetNVE<Geometry::CUBE>(av, ae);
            switch (SDim)
            {
               case 1:
                  GetEdge<Geometry::CUBE>(nv, v, ne, e[0], eo[0], Info);
                  break;
               case 2:
                  GetFace<Geometry::CUBE,Geometry::SQUARE>(
                     nv, v, ne, e, eo, nf, f[0], fg[0], fo[0], Info);
                  break;
               default:
                  goto not_supp;
            }
            break;
         }

         default:
            MFEM_ABORT("invalid Geom = " << Geom);
      }

      int ned = (ne > 0) ? DofForGeometry(Geometry::SEGMENT) : 0;

      // add vertex dofs
      dofs.SetSize(nv*nvd+ne*ned);
      for (int i = 0; i < nv; i++)
      {
         for (int j = 0; j < nvd; j++)
         {
            dofs[i*nvd+j] = v[i]*nvd+j;
         }
      }
      int l_off = nv*nvd, g_off = av*nvd;

      // add edge dofs
      if (ned > 0)
      {
         for (int i = 0; i < ne; i++)
         {
            const int *ed = DofOrderForOrientation(Geometry::SEGMENT,
                                                   eo[i] ? -1 : 1);
            for (int j = 0; j < ned; j++)
            {
               dofs[l_off+i*ned+j] =
                  ed[j] >= 0 ?
                  g_off+e[i]*ned+ed[j] :
                  -1-(g_off+e[i]*ned+(-1-ed[j]));
            }
         }
         l_off += ne*ned;
         g_off += ae*ned;
      }

      // add face dofs
      if (nf > 0)
      {
         const int nfd = DofForGeometry(fg[0]); // assume same face geometry
         dofs.SetSize(dofs.Size()+nf*nfd);
         for (int i = 0; i < nf; i++)
         {
            const int *fd = DofOrderForOrientation(fg[i], fo[i]);
            for (int j = 0; j < nfd; j++)
            {
               dofs[l_off+i*nfd+j] =
                  fd[j] >= 0 ?
                  g_off+f[i]*nfd+fd[j] :
                  -1-(g_off+f[i]*nfd+(-1-fd[j]));
            }
         }
      }

      // add volume dofs ...
   }
   return;

not_supp:
   MFEM_ABORT("Geom = " << Geometry::Name[Geom] <<
              ", SDim = " << SDim << " is not supported");
}

const FiniteElement *
LinearFECollection::FiniteElementForGeometry(int GeomType) const
{
   switch (GeomType)
   {
      case Geometry::POINT:       return &PointFE;
      case Geometry::SEGMENT:     return &SegmentFE;
      case Geometry::TRIANGLE:    return &TriangleFE;
      case Geometry::SQUARE:      return &QuadrilateralFE;
      case Geometry::TETRAHEDRON: return &TetrahedronFE;
      case Geometry::CUBE:        return &ParallelepipedFE;
      default:
         mfem_error ("LinearFECollection: unknown geometry type.");
   }
   return &SegmentFE; // Make some compilers happy
}

int LinearFECollection::DofForGeometry(int GeomType) const
{
   switch (GeomType)
   {
      case Geometry::POINT:       return 1;
      case Geometry::SEGMENT:     return 0;
      case Geometry::TRIANGLE:    return 0;
      case Geometry::SQUARE:      return 0;
      case Geometry::TETRAHEDRON: return 0;
      case Geometry::CUBE:        return 0;
      default:
         mfem_error ("LinearFECollection: unknown geometry type.");
   }
   return 0; // Make some compilers happy
}

int * LinearFECollection::DofOrderForOrientation(int GeomType, int Or) const
{
   return NULL;
}


const FiniteElement *
QuadraticFECollection::FiniteElementForGeometry(int GeomType) const
{
   switch (GeomType)
   {
      case Geometry::POINT:       return &PointFE;
      case Geometry::SEGMENT:     return &SegmentFE;
      case Geometry::TRIANGLE:    return &TriangleFE;
      case Geometry::SQUARE:      return &QuadrilateralFE;
      case Geometry::TETRAHEDRON: return &TetrahedronFE;
      case Geometry::CUBE:        return &ParallelepipedFE;
      default:
         mfem_error ("QuadraticFECollection: unknown geometry type.");
   }
   return &SegmentFE; // Make some compilers happy
}

int QuadraticFECollection::DofForGeometry(int GeomType) const
{
   switch (GeomType)
   {
      case Geometry::POINT:       return 1;
      case Geometry::SEGMENT:     return 1;
      case Geometry::TRIANGLE:    return 0;
      case Geometry::SQUARE:      return 1;
      case Geometry::TETRAHEDRON: return 0;
      case Geometry::CUBE:        return 1;
      default:
         mfem_error ("QuadraticFECollection: unknown geometry type.");
   }
   return 0; // Make some compilers happy
}

int * QuadraticFECollection::DofOrderForOrientation(int GeomType, int Or) const
{
   static int indexes[] = { 0 };

   return indexes;
}


const FiniteElement *
QuadraticPosFECollection::FiniteElementForGeometry(int GeomType) const
{
   switch (GeomType)
   {
      case Geometry::SEGMENT:     return &SegmentFE;
      case Geometry::SQUARE:      return &QuadrilateralFE;
      default:
         mfem_error ("QuadraticPosFECollection: unknown geometry type.");
   }
   return NULL; // Make some compilers happy
}

int QuadraticPosFECollection::DofForGeometry(int GeomType) const
{
   switch (GeomType)
   {
      case Geometry::POINT:       return 1;
      case Geometry::SEGMENT:     return 1;
      case Geometry::SQUARE:      return 1;
      default:
         mfem_error ("QuadraticPosFECollection: unknown geometry type.");
   }
   return 0; // Make some compilers happy
}

int * QuadraticPosFECollection::DofOrderForOrientation(int GeomType, int Or)
const
{
   static int indexes[] = { 0 };

   return indexes;
}


const FiniteElement *
CubicFECollection::FiniteElementForGeometry(int GeomType) const
{
   switch (GeomType)
   {
      case Geometry::POINT:       return &PointFE;
      case Geometry::SEGMENT:     return &SegmentFE;
      case Geometry::TRIANGLE:    return &TriangleFE;
      case Geometry::SQUARE:      return &QuadrilateralFE;
      case Geometry::TETRAHEDRON: return &TetrahedronFE;
      case Geometry::CUBE:        return &ParallelepipedFE;
      default:
         mfem_error ("CubicFECollection: unknown geometry type.");
   }
   return &SegmentFE; // Make some compilers happy
}

int CubicFECollection::DofForGeometry(int GeomType) const
{
   switch (GeomType)
   {
      case Geometry::POINT:       return 1;
      case Geometry::SEGMENT:     return 2;
      case Geometry::TRIANGLE:    return 1;
      case Geometry::SQUARE:      return 4;
      case Geometry::TETRAHEDRON: return 0;
      case Geometry::CUBE:        return 8;
      default:
         mfem_error ("CubicFECollection: unknown geometry type.");
   }
   return 0; // Make some compilers happy
}

int * CubicFECollection::DofOrderForOrientation(int GeomType, int Or) const
{
   if (GeomType == Geometry::SEGMENT)
   {
      static int ind_pos[] = { 0, 1 };
      static int ind_neg[] = { 1, 0 };

      if (Or < 0)
      {
         return ind_neg;
      }
      return ind_pos;
   }
   else if (GeomType == Geometry::TRIANGLE)
   {
      static int indexes[] = { 0 };

      return indexes;
   }
   else if (GeomType == Geometry::SQUARE)
   {
      static int sq_ind[8][4] = {{0, 1, 2, 3}, {0, 2, 1, 3},
         {2, 0, 3, 1}, {1, 0, 3, 2},
         {3, 2, 1, 0}, {3, 1, 2, 0},
         {1, 3, 0, 2}, {2, 3, 0, 1}
      };
      return sq_ind[Or];
   }

   return NULL;
}


const FiniteElement *
CrouzeixRaviartFECollection::FiniteElementForGeometry(int GeomType) const
{
   switch (GeomType)
   {
      case Geometry::SEGMENT:     return &SegmentFE;
      case Geometry::TRIANGLE:    return &TriangleFE;
      case Geometry::SQUARE:      return &QuadrilateralFE;
      default:
         mfem_error ("CrouzeixRaviartFECollection: unknown geometry type.");
   }
   return &SegmentFE; // Make some compilers happy
}

int CrouzeixRaviartFECollection::DofForGeometry(int GeomType) const
{
   switch (GeomType)
   {
      case Geometry::POINT:       return 0;
      case Geometry::SEGMENT:     return 1;
      case Geometry::TRIANGLE:    return 0;
      case Geometry::SQUARE:      return 0;
      default:
         mfem_error ("CrouzeixRaviartFECollection: unknown geometry type.");
   }
   return 0; // Make some compilers happy
}

int * CrouzeixRaviartFECollection::DofOrderForOrientation(int GeomType, int Or)
const
{
   static int indexes[] = { 0 };

   return indexes;
}


const FiniteElement *
RT0_2DFECollection::FiniteElementForGeometry(int GeomType) const
{
   switch (GeomType)
   {
      case Geometry::SEGMENT:     return &SegmentFE;
      case Geometry::TRIANGLE:    return &TriangleFE;
      case Geometry::SQUARE:      return &QuadrilateralFE;
      default:
         mfem_error ("RT0_2DFECollection: unknown geometry type.");
   }
   return &SegmentFE; // Make some compilers happy
}

int RT0_2DFECollection::DofForGeometry(int GeomType) const
{
   switch (GeomType)
   {
      case Geometry::POINT:       return 0;
      case Geometry::SEGMENT:     return 1;
      case Geometry::TRIANGLE:    return 0;
      case Geometry::SQUARE:      return 0;
      default:
         mfem_error ("RT0_2DFECollection: unknown geometry type.");
   }
   return 0; // Make some compilers happy
}

int * RT0_2DFECollection::DofOrderForOrientation(int GeomType, int Or)
const
{
   static int ind_pos[] = { 0 };
   static int ind_neg[] = { -1 };

   if (Or > 0)
   {
      return ind_pos;
   }
   return ind_neg;
}


const FiniteElement *
RT1_2DFECollection::FiniteElementForGeometry(int GeomType) const
{
   switch (GeomType)
   {
      case Geometry::SEGMENT:     return &SegmentFE;
      case Geometry::TRIANGLE:    return &TriangleFE;
      case Geometry::SQUARE:      return &QuadrilateralFE;
      default:
         mfem_error ("RT1_2DFECollection: unknown geometry type.");
   }
   return &SegmentFE; // Make some compilers happy
}

int RT1_2DFECollection::DofForGeometry(int GeomType) const
{
   switch (GeomType)
   {
      case Geometry::POINT:       return 0;
      case Geometry::SEGMENT:     return 2;
      case Geometry::TRIANGLE:    return 2;
      case Geometry::SQUARE:      return 4;
      default:
         mfem_error ("RT1_2DFECollection: unknown geometry type.");
   }
   return 0; // Make some compilers happy
}

int * RT1_2DFECollection::DofOrderForOrientation(int GeomType, int Or)
const
{
   static int ind_pos[] = {  0,  1 };
   static int ind_neg[] = { -2, -1 };

   if (Or > 0)
   {
      return ind_pos;
   }
   return ind_neg;
}

const FiniteElement *
RT2_2DFECollection::FiniteElementForGeometry(int GeomType) const
{
   switch (GeomType)
   {
      case Geometry::SEGMENT:     return &SegmentFE;
      case Geometry::TRIANGLE:    return &TriangleFE;
      case Geometry::SQUARE:      return &QuadrilateralFE;
      default:
         mfem_error ("RT2_2DFECollection: unknown geometry type.");
   }
   return &SegmentFE; // Make some compilers happy
}

int RT2_2DFECollection::DofForGeometry(int GeomType) const
{
   switch (GeomType)
   {
      case Geometry::POINT:       return 0;
      case Geometry::SEGMENT:     return 3;
      case Geometry::TRIANGLE:    return 6;
      case Geometry::SQUARE:      return 12;
      default:
         mfem_error ("RT2_2DFECollection: unknown geometry type.");
   }
   return 0; // Make some compilers happy
}

int * RT2_2DFECollection::DofOrderForOrientation(int GeomType, int Or)
const
{
   static int ind_pos[] = { 0, 1, 2 };
   static int ind_neg[] = { -3, -2, -1 };

   if (Or > 0)
   {
      return ind_pos;
   }
   return ind_neg;
}


const FiniteElement *
Const2DFECollection::FiniteElementForGeometry(int GeomType) const
{
   switch (GeomType)
   {
      case Geometry::TRIANGLE:    return &TriangleFE;
      case Geometry::SQUARE:      return &QuadrilateralFE;
      default:
         mfem_error ("Const2DFECollection: unknown geometry type.");
   }
   return &TriangleFE; // Make some compilers happy
}

int Const2DFECollection::DofForGeometry(int GeomType) const
{
   switch (GeomType)
   {
      case Geometry::POINT:       return 0;
      case Geometry::SEGMENT:     return 0;
      case Geometry::TRIANGLE:    return 1;
      case Geometry::SQUARE:      return 1;
      default:
         mfem_error ("Const2DFECollection: unknown geometry type.");
   }
   return 0; // Make some compilers happy
}

int * Const2DFECollection::DofOrderForOrientation(int GeomType, int Or)
const
{
   return NULL;
}


const FiniteElement *
LinearDiscont2DFECollection::FiniteElementForGeometry(int GeomType) const
{
   switch (GeomType)
   {
      case Geometry::TRIANGLE:    return &TriangleFE;
      case Geometry::SQUARE:      return &QuadrilateralFE;
      default:
         mfem_error ("LinearDiscont2DFECollection: unknown geometry type.");
   }
   return &TriangleFE; // Make some compilers happy
}

int LinearDiscont2DFECollection::DofForGeometry(int GeomType) const
{
   switch (GeomType)
   {
      case Geometry::POINT:       return 0;
      case Geometry::SEGMENT:     return 0;
      case Geometry::TRIANGLE:    return 3;
      case Geometry::SQUARE:      return 4;
      default:
         mfem_error ("LinearDiscont2DFECollection: unknown geometry type.");
   }
   return 0; // Make some compilers happy
}

int * LinearDiscont2DFECollection::DofOrderForOrientation(int GeomType, int Or)
const
{
   return NULL;
}


const FiniteElement *
GaussLinearDiscont2DFECollection::FiniteElementForGeometry(int GeomType) const
{
   switch (GeomType)
   {
      case Geometry::TRIANGLE:    return &TriangleFE;
      case Geometry::SQUARE:      return &QuadrilateralFE;
      default:
         mfem_error ("GaussLinearDiscont2DFECollection:"
                     " unknown geometry type.");
   }
   return &TriangleFE; // Make some compilers happy
}

int GaussLinearDiscont2DFECollection::DofForGeometry(int GeomType) const
{
   switch (GeomType)
   {
      case Geometry::POINT:       return 0;
      case Geometry::SEGMENT:     return 0;
      case Geometry::TRIANGLE:    return 3;
      case Geometry::SQUARE:      return 4;
      default:
         mfem_error ("GaussLinearDiscont2DFECollection:"
                     " unknown geometry type.");
   }
   return 0; // Make some compilers happy
}

int * GaussLinearDiscont2DFECollection::DofOrderForOrientation(
   int GeomType, int Or) const
{
   return NULL;
}


const FiniteElement *
P1OnQuadFECollection::FiniteElementForGeometry(int GeomType) const
{
   if (GeomType != Geometry::SQUARE)
   {
      mfem_error ("P1OnQuadFECollection: unknown geometry type.");
   }
   return &QuadrilateralFE;
}

int P1OnQuadFECollection::DofForGeometry(int GeomType) const
{
   switch (GeomType)
   {
      case Geometry::POINT:       return 0;
      case Geometry::SEGMENT:     return 0;
      case Geometry::SQUARE:      return 3;
      default:
         mfem_error ("P1OnQuadFECollection: unknown geometry type.");
   }
   return 0; // Make some compilers happy
}

int * P1OnQuadFECollection::DofOrderForOrientation(
   int GeomType, int Or) const
{
   return NULL;
}


const FiniteElement *
QuadraticDiscont2DFECollection::FiniteElementForGeometry(int GeomType) const
{
   switch (GeomType)
   {
      case Geometry::TRIANGLE:    return &TriangleFE;
      case Geometry::SQUARE:      return &QuadrilateralFE;
      default:
         mfem_error ("QuadraticDiscont2DFECollection: unknown geometry type.");
   }
   return &TriangleFE; // Make some compilers happy
}

int QuadraticDiscont2DFECollection::DofForGeometry(int GeomType) const
{
   switch (GeomType)
   {
      case Geometry::POINT:       return 0;
      case Geometry::SEGMENT:     return 0;
      case Geometry::TRIANGLE:    return 6;
      case Geometry::SQUARE:      return 9;
      default:
         mfem_error ("QuadraticDiscont2DFECollection: unknown geometry type.");
   }
   return 0; // Make some compilers happy
}

int * QuadraticDiscont2DFECollection::DofOrderForOrientation(
   int GeomType, int Or) const
{
   return NULL;
}


const FiniteElement *
QuadraticPosDiscont2DFECollection::FiniteElementForGeometry(int GeomType) const
{
   switch (GeomType)
   {
      case Geometry::SQUARE:  return &QuadrilateralFE;
      default:
         mfem_error ("QuadraticPosDiscont2DFECollection: unknown geometry type.");
   }
   return NULL; // Make some compilers happy
}

int QuadraticPosDiscont2DFECollection::DofForGeometry(int GeomType) const
{
   switch (GeomType)
   {
      case Geometry::POINT:       return 0;
      case Geometry::SEGMENT:     return 0;
      case Geometry::SQUARE:      return 9;
      default:
         mfem_error ("QuadraticPosDiscont2DFECollection: unknown geometry type.");
   }
   return 0; // Make some compilers happy
}


const FiniteElement *
GaussQuadraticDiscont2DFECollection::FiniteElementForGeometry(int GeomType)
const
{
   switch (GeomType)
   {
      case Geometry::TRIANGLE:    return &TriangleFE;
      case Geometry::SQUARE:      return &QuadrilateralFE;
      default:
         mfem_error ("GaussQuadraticDiscont2DFECollection:"
                     " unknown geometry type.");
   }
   return &QuadrilateralFE; // Make some compilers happy
}

int GaussQuadraticDiscont2DFECollection::DofForGeometry(int GeomType) const
{
   switch (GeomType)
   {
      case Geometry::POINT:       return 0;
      case Geometry::SEGMENT:     return 0;
      case Geometry::TRIANGLE:    return 6;
      case Geometry::SQUARE:      return 9;
      default:
         mfem_error ("GaussQuadraticDiscont2DFECollection:"
                     " unknown geometry type.");
   }
   return 0; // Make some compilers happy
}

int * GaussQuadraticDiscont2DFECollection::DofOrderForOrientation(
   int GeomType, int Or) const
{
   return NULL;
}


const FiniteElement *
CubicDiscont2DFECollection::FiniteElementForGeometry(int GeomType) const
{
   switch (GeomType)
   {
      case Geometry::TRIANGLE:    return &TriangleFE;
      case Geometry::SQUARE:      return &QuadrilateralFE;
      default:
         mfem_error ("CubicDiscont2DFECollection: unknown geometry type.");
   }
   return &TriangleFE; // Make some compilers happy
}

int CubicDiscont2DFECollection::DofForGeometry(int GeomType) const
{
   switch (GeomType)
   {
      case Geometry::POINT:       return 0;
      case Geometry::SEGMENT:     return 0;
      case Geometry::TRIANGLE:    return 10;
      case Geometry::SQUARE:      return 16;
      default:
         mfem_error ("CubicDiscont2DFECollection: unknown geometry type.");
   }
   return 0; // Make some compilers happy
}

int * CubicDiscont2DFECollection::DofOrderForOrientation(int GeomType, int Or)
const
{
   return NULL;
}


const FiniteElement *
LinearNonConf3DFECollection::FiniteElementForGeometry(int GeomType) const
{
   switch (GeomType)
   {
      case Geometry::TRIANGLE:    return &TriangleFE;
      case Geometry::SQUARE:      return &QuadrilateralFE;
      case Geometry::TETRAHEDRON: return &TetrahedronFE;
      case Geometry::CUBE:        return &ParallelepipedFE;
      default:
         mfem_error ("LinearNonConf3DFECollection: unknown geometry type.");
   }
   return &TriangleFE; // Make some compilers happy
}

int LinearNonConf3DFECollection::DofForGeometry(int GeomType) const
{
   switch (GeomType)
   {
      case Geometry::POINT:       return 0;
      case Geometry::SEGMENT:     return 0;
      case Geometry::TRIANGLE:    return 1;
      case Geometry::SQUARE:      return 1;
      case Geometry::TETRAHEDRON: return 0;
      case Geometry::CUBE:        return 0;
      default:
         mfem_error ("LinearNonConf3DFECollection: unknown geometry type.");
   }
   return 0; // Make some compilers happy
}

int * LinearNonConf3DFECollection::DofOrderForOrientation(int GeomType, int Or)
const
{
   static int indexes[] = { 0 };

   return indexes;
}


const FiniteElement *
Const3DFECollection::FiniteElementForGeometry(int GeomType) const
{
   switch (GeomType)
   {
      case Geometry::TETRAHEDRON: return &TetrahedronFE;
      case Geometry::CUBE:        return &ParallelepipedFE;
      default:
         mfem_error ("Const3DFECollection: unknown geometry type.");
   }
   return &TetrahedronFE; // Make some compilers happy
}

int Const3DFECollection::DofForGeometry(int GeomType) const
{
   switch (GeomType)
   {
      case Geometry::POINT:       return 0;
      case Geometry::SEGMENT:     return 0;
      case Geometry::TRIANGLE:    return 0;
      case Geometry::TETRAHEDRON: return 1;
      case Geometry::SQUARE:      return 0;
      case Geometry::CUBE:        return 1;
      default:
         mfem_error ("Const3DFECollection: unknown geometry type.");
   }
   return 0; // Make some compilers happy
}

int * Const3DFECollection::DofOrderForOrientation(int GeomType, int Or)
const
{
   return NULL;
}


const FiniteElement *
LinearDiscont3DFECollection::FiniteElementForGeometry(int GeomType) const
{
   switch (GeomType)
   {
      case Geometry::TETRAHEDRON: return &TetrahedronFE;
      case Geometry::CUBE:        return &ParallelepipedFE;
      default:
         mfem_error ("LinearDiscont3DFECollection: unknown geometry type.");
   }
   return &TetrahedronFE; // Make some compilers happy
}

int LinearDiscont3DFECollection::DofForGeometry(int GeomType) const
{
   switch (GeomType)
   {
      case Geometry::POINT:       return 0;
      case Geometry::SEGMENT:     return 0;
      case Geometry::TRIANGLE:    return 0;
      case Geometry::SQUARE:      return 0;
      case Geometry::TETRAHEDRON: return 4;
      case Geometry::CUBE:        return 8;
      default:
         mfem_error ("LinearDiscont3DFECollection: unknown geometry type.");
   }
   return 0; // Make some compilers happy
}

int * LinearDiscont3DFECollection::DofOrderForOrientation(int GeomType, int Or)
const
{
   return NULL;
}


const FiniteElement *
QuadraticDiscont3DFECollection::FiniteElementForGeometry(int GeomType) const
{
   switch (GeomType)
   {
      case Geometry::TETRAHEDRON: return &TetrahedronFE;
      case Geometry::CUBE:        return &ParallelepipedFE;
      default:
         mfem_error ("QuadraticDiscont3DFECollection: unknown geometry type.");
   }
   return &TetrahedronFE; // Make some compilers happy
}

int QuadraticDiscont3DFECollection::DofForGeometry(int GeomType) const
{
   switch (GeomType)
   {
      case Geometry::POINT:       return 0;
      case Geometry::SEGMENT:     return 0;
      case Geometry::TRIANGLE:    return 0;
      case Geometry::SQUARE:      return 0;
      case Geometry::TETRAHEDRON: return 10;
      case Geometry::CUBE:        return 27;
      default:
         mfem_error ("QuadraticDiscont3DFECollection: unknown geometry type.");
   }
   return 0; // Make some compilers happy
}

int * QuadraticDiscont3DFECollection::DofOrderForOrientation(
   int GeomType, int Or) const
{
   return NULL;
}

const FiniteElement *
RefinedLinearFECollection::FiniteElementForGeometry(int GeomType) const
{
   switch (GeomType)
   {
      case Geometry::POINT:       return &PointFE;
      case Geometry::SEGMENT:     return &SegmentFE;
      case Geometry::TRIANGLE:    return &TriangleFE;
      case Geometry::SQUARE:      return &QuadrilateralFE;
      case Geometry::TETRAHEDRON: return &TetrahedronFE;
      case Geometry::CUBE:        return &ParallelepipedFE;
      default:
         mfem_error ("RefinedLinearFECollection: unknown geometry type.");
   }
   return &SegmentFE; // Make some compilers happy
}

int RefinedLinearFECollection::DofForGeometry(int GeomType) const
{
   switch (GeomType)
   {
      case Geometry::POINT:       return 1;
      case Geometry::SEGMENT:     return 1;
      case Geometry::TRIANGLE:    return 0;
      case Geometry::SQUARE:      return 1;
      case Geometry::TETRAHEDRON: return 0;
      case Geometry::CUBE:        return 1;
      default:
         mfem_error ("RefinedLinearFECollection: unknown geometry type.");
   }
   return 0; // Make some compilers happy
}

int * RefinedLinearFECollection::DofOrderForOrientation(int GeomType,
                                                        int Or) const
{
   static int indexes[] = { 0 };

   return indexes;
}


const FiniteElement *
ND1_3DFECollection::FiniteElementForGeometry(int GeomType) const
{
   switch (GeomType)
   {
      case Geometry::CUBE:        return &HexahedronFE;
      case Geometry::TETRAHEDRON: return &TetrahedronFE;
      default:
         mfem_error ("ND1_3DFECollection: unknown geometry type.");
   }
   return &HexahedronFE; // Make some compilers happy
}

int ND1_3DFECollection::DofForGeometry(int GeomType) const
{
   switch (GeomType)
   {
      case Geometry::POINT:       return 0;
      case Geometry::SEGMENT:     return 1;
      case Geometry::TRIANGLE:    return 0;
      case Geometry::SQUARE:      return 0;
      case Geometry::TETRAHEDRON: return 0;
      case Geometry::CUBE:        return 0;
      default:
         mfem_error ("ND1_3DFECollection: unknown geometry type.");
   }
   return 0; // Make some compilers happy
}

int * ND1_3DFECollection::DofOrderForOrientation(int GeomType, int Or)
const
{
   static int ind_pos[] = { 0 };
   static int ind_neg[] = { -1 };

   if (Or > 0)
   {
      return ind_pos;
   }
   return ind_neg;
}


const FiniteElement *
RT0_3DFECollection::FiniteElementForGeometry(int GeomType) const
{
   switch (GeomType)
   {
      case Geometry::TRIANGLE:    return &TriangleFE;
      case Geometry::SQUARE:      return &QuadrilateralFE;
      case Geometry::CUBE:        return &HexahedronFE;
      case Geometry::TETRAHEDRON: return &TetrahedronFE;
      default:
         mfem_error ("RT0_3DFECollection: unknown geometry type.");
   }
   return &HexahedronFE; // Make some compilers happy
}

int RT0_3DFECollection::DofForGeometry(int GeomType) const
{
   switch (GeomType)
   {
      case Geometry::POINT:       return 0;
      case Geometry::SEGMENT:     return 0;
      case Geometry::TRIANGLE:    return 1;
      case Geometry::SQUARE:      return 1;
      case Geometry::TETRAHEDRON: return 0;
      case Geometry::CUBE:        return 0;
      default:
         mfem_error ("RT0_3DFECollection: unknown geometry type.");
   }
   return 0; // Make some compilers happy
}

int * RT0_3DFECollection::DofOrderForOrientation(int GeomType, int Or)
const
{
   static int ind_pos[] = { 0 };
   static int ind_neg[] = { -1 };

   if ((GeomType == Geometry::TRIANGLE) || (GeomType == Geometry::SQUARE))
   {
      if (Or % 2 == 0)
      {
         return ind_pos;
      }
      return ind_neg;
   }
   return NULL;
}

const FiniteElement *
RT1_3DFECollection::FiniteElementForGeometry(int GeomType) const
{
   switch (GeomType)
   {
      case Geometry::TRIANGLE:    return &TriangleFE;
      case Geometry::SQUARE:      return &QuadrilateralFE;
      case Geometry::CUBE:        return &HexahedronFE;
      default:
         mfem_error ("RT1_3DFECollection: unknown geometry type.");
   }
   return &HexahedronFE; // Make some compilers happy
}

int RT1_3DFECollection::DofForGeometry(int GeomType) const
{
   switch (GeomType)
   {
      case Geometry::POINT:       return 0;
      case Geometry::SEGMENT:     return 0;
      case Geometry::TRIANGLE:    return 2;
      case Geometry::SQUARE:      return 4;
      case Geometry::CUBE:        return 12;
      default:
         mfem_error ("RT1_3DFECollection: unknown geometry type.");
   }
   return 0; // Make some compilers happy
}

int * RT1_3DFECollection::DofOrderForOrientation(int GeomType, int Or)
const
{
   if (GeomType == Geometry::SQUARE)
   {
      static int sq_ind[8][4] =
      {
         {0, 1, 2, 3}, {-1, -3, -2, -4},
         {2, 0, 3, 1}, {-2, -1, -4, -3},
         {3, 2, 1, 0}, {-4, -2, -3, -1},
         {1, 3, 0, 2}, {-3, -4, -1, -2}
      };

      return sq_ind[Or];
   }
   else
   {
      return NULL;
   }
}


H1_FECollection::H1_FECollection(const int p, const int dim, const int type)
{
   const int pm1 = p - 1, pm2 = pm1 - 1, pm3 = pm2 - 1;

   int pt_type = BasisType::GetQuadrature1D(type);
   m_type = BasisType::Check(type);
   switch (type)
   {
      case BasisType::GaussLobatto:
      {
         snprintf(h1_name, 32, "H1_%dD_P%d", dim, p);
         break;
      }
      case BasisType::Positive:
      {
         snprintf(h1_name, 32, "H1Pos_%dD_P%d", dim, p);
         break;
      }
      default:
      {
         MFEM_VERIFY(Quadrature1D::CheckClosed(pt_type) !=
                     Quadrature1D::Invalid,
                     "unsupported BasisType: " << BasisType::Name(type));

         snprintf(h1_name, 32, "H1@%c_%dD_P%d",
                  (int)BasisType::GetChar(type), dim, p);
      }
   }

   for (int g = 0; g < Geometry::NumGeom; g++)
   {
      H1_dof[g] = 0;
      H1_Elements[g] = NULL;
   }
   for (int i = 0; i < 2; i++)
   {
      SegDofOrd[i] = NULL;
   }
   for (int i = 0; i < 6; i++)
   {
      TriDofOrd[i] = NULL;
   }
   for (int i = 0; i < 8; i++)
   {
      QuadDofOrd[i] = NULL;
   }

   H1_dof[Geometry::POINT] = 1;
   H1_Elements[Geometry::POINT] = new PointFiniteElement;

   if (dim >= 1)
   {
      H1_dof[Geometry::SEGMENT] = pm1;
      if (m_type == BasisType::Positive)
      {
         H1_Elements[Geometry::SEGMENT] = new H1Pos_SegmentElement(p);
      }
      else
      {
         H1_Elements[Geometry::SEGMENT] = new H1_SegmentElement(p, pt_type);
      }

      SegDofOrd[0] = new int[2*pm1];
      SegDofOrd[1] = SegDofOrd[0] + pm1;
      for (int i = 0; i < pm1; i++)
      {
         SegDofOrd[0][i] = i;
         SegDofOrd[1][i] = pm2 - i;
      }
   }

   if (dim >= 2)
   {
      H1_dof[Geometry::TRIANGLE] = (pm1*pm2)/2;
      H1_dof[Geometry::SQUARE] = pm1*pm1;
      if (m_type == BasisType::Positive)
      {
         H1_Elements[Geometry::TRIANGLE] = new H1Pos_TriangleElement(p);
         H1_Elements[Geometry::SQUARE] = new H1Pos_QuadrilateralElement(p);
      }
      else
      {
         H1_Elements[Geometry::TRIANGLE] = new H1_TriangleElement(p, pt_type);
         H1_Elements[Geometry::SQUARE] = new H1_QuadrilateralElement(p,pt_type);
      }

      const int &TriDof = H1_dof[Geometry::TRIANGLE];
      const int &QuadDof = H1_dof[Geometry::SQUARE];
      TriDofOrd[0] = new int[6*TriDof];
      for (int i = 1; i < 6; i++)
      {
         TriDofOrd[i] = TriDofOrd[i-1] + TriDof;
      }
      // see Mesh::GetTriOrientation in mesh/mesh.cpp
      for (int j = 0; j < pm2; j++)
      {
         for (int i = 0; i + j < pm2; i++)
         {
            int o = TriDof - ((pm1 - j)*(pm2 - j))/2 + i;
            int k = pm3 - j - i;
            TriDofOrd[0][o] = o;  // (0,1,2)
            TriDofOrd[1][o] = TriDof - ((pm1-j)*(pm2-j))/2 + k;  // (1,0,2)
            TriDofOrd[2][o] = TriDof - ((pm1-i)*(pm2-i))/2 + k;  // (2,0,1)
            TriDofOrd[3][o] = TriDof - ((pm1-k)*(pm2-k))/2 + i;  // (2,1,0)
            TriDofOrd[4][o] = TriDof - ((pm1-k)*(pm2-k))/2 + j;  // (1,2,0)
            TriDofOrd[5][o] = TriDof - ((pm1-i)*(pm2-i))/2 + j;  // (0,2,1)
         }
      }

      QuadDofOrd[0] = new int[8*QuadDof];
      for (int i = 1; i < 8; i++)
      {
         QuadDofOrd[i] = QuadDofOrd[i-1] + QuadDof;
      }
      // see Mesh::GetQuadOrientation in mesh/mesh.cpp
      for (int j = 0; j < pm1; j++)
      {
         for (int i = 0; i < pm1; i++)
         {
            int o = i + j*pm1;
            QuadDofOrd[0][o] = i + j*pm1;  // (0,1,2,3)
            QuadDofOrd[1][o] = j + i*pm1;  // (0,3,2,1)
            QuadDofOrd[2][o] = j + (pm2 - i)*pm1;  // (1,2,3,0)
            QuadDofOrd[3][o] = (pm2 - i) + j*pm1;  // (1,0,3,2)
            QuadDofOrd[4][o] = (pm2 - i) + (pm2 - j)*pm1;  // (2,3,0,1)
            QuadDofOrd[5][o] = (pm2 - j) + (pm2 - i)*pm1;  // (2,1,0,3)
            QuadDofOrd[6][o] = (pm2 - j) + i*pm1;  // (3,0,1,2)
            QuadDofOrd[7][o] = i + (pm2 - j)*pm1;  // (3,2,1,0)
         }
      }

      if (dim >= 3)
      {
         H1_dof[Geometry::TETRAHEDRON] = (TriDof*pm3)/3;
         H1_dof[Geometry::CUBE] = QuadDof*pm1;
         if (m_type == BasisType::Positive)
         {
            H1_Elements[Geometry::TETRAHEDRON] = new H1Pos_TetrahedronElement(p);
            H1_Elements[Geometry::CUBE] = new H1Pos_HexahedronElement(p);
         }
         else
         {
            H1_Elements[Geometry::TETRAHEDRON] =
               new H1_TetrahedronElement(p, pt_type);
            H1_Elements[Geometry::CUBE] = new H1_HexahedronElement(p, pt_type);
         }
      }
   }
}

int *H1_FECollection::DofOrderForOrientation(int GeomType, int Or) const
{
   if (GeomType == Geometry::SEGMENT)
   {
      return (Or > 0) ? SegDofOrd[0] : SegDofOrd[1];
   }
   else if (GeomType == Geometry::TRIANGLE)
   {
      return TriDofOrd[Or%6];
   }
   else if (GeomType == Geometry::SQUARE)
   {
      return QuadDofOrd[Or%8];
   }
   return NULL;
}

FiniteElementCollection *H1_FECollection::GetTraceCollection() const
{
   int p = H1_dof[Geometry::SEGMENT] + 1;
   int dim = -1;
   if (!strncmp(h1_name, "H1_", 3))
   {
      dim = atoi(h1_name + 3);
   }
   else if (!strncmp(h1_name, "H1Pos_", 6))
   {
      dim = atoi(h1_name + 6);
   }
   else if (!strncmp(h1_name, "H1@", 3))
   {
      dim = atoi(h1_name + 5);
   }
   return (dim < 0) ? NULL : new H1_Trace_FECollection(p, dim, m_type);
}

const int *H1_FECollection::GetDofMap(int GeomType) const
{
   MFEM_ASSERT(m_type != BasisType::Positive, "");
   const int *dof_map = NULL;
   const FiniteElement *fe = H1_Elements[GeomType];
   switch (GeomType)
   {
      case Geometry::SEGMENT:
         dof_map = dynamic_cast<const H1_SegmentElement *>(fe)
                   ->GetDofMap().GetData();
         break;
      case Geometry::SQUARE:
         dof_map = dynamic_cast<const H1_QuadrilateralElement *>(fe)
                   ->GetDofMap().GetData();
         break;
      case Geometry::CUBE:
         dof_map = dynamic_cast<const H1_HexahedronElement *>(fe)
                   ->GetDofMap().GetData();
         break;
      default:
         MFEM_ABORT("Geometry type " << Geometry::Name[GeomType] << " is not "
                    "implemented");
   }
   return dof_map;
}

H1_FECollection::~H1_FECollection()
{
   delete [] SegDofOrd[0];
   delete [] TriDofOrd[0];
   delete [] QuadDofOrd[0];
   for (int g = 0; g < Geometry::NumGeom; g++)
   {
      delete H1_Elements[g];
   }
}


H1_Trace_FECollection::H1_Trace_FECollection(const int p, const int dim,
                                             const int type)
   : H1_FECollection(p, dim-1, type)
{
   if (type == BasisType::GaussLobatto)
   {
      snprintf(h1_name, 32, "H1_Trace_%dD_P%d", dim, p);
   }
   else if (type == BasisType::Positive)
   {
      snprintf(h1_name, 32, "H1Pos_Trace_%dD_P%d", dim, p);
   }
   else // base class checks that type is closed
   {
      snprintf(h1_name, 32, "H1_Trace@%c_%dD_P%d",
               (int)BasisType::GetChar(type), dim, p);
   }
}


L2_FECollection::L2_FECollection(const int p, const int dim, const int type,
                                 const int map_type)
{
   int pt_type = BasisType::GetQuadrature1D(type);
   m_type = BasisType::Check(type);
   MFEM_VERIFY(pt_type != Quadrature1D::Invalid ||
               m_type == BasisType::Positive,
               "unsupported L2 basis type = " << BasisType::Name(type));
   const char *prefix = NULL;
   switch (map_type)
   {
      case FiniteElement::VALUE:    prefix = "L2";    break;
      case FiniteElement::INTEGRAL: prefix = "L2Int"; break;
      default:
         MFEM_ABORT("invalid map_type: " << map_type);
   }
   switch (type)
   {
      case BasisType::GaussLegendre:
         snprintf(d_name, 32, "%s_%dD_P%d", prefix, dim, p);
         break;
      default:
         snprintf(d_name, 32, "%s_T%d_%dD_P%d", prefix, type, dim, p);
   }

   for (int g = 0; g < Geometry::NumGeom; g++)
   {
      L2_Elements[g] = NULL;
      Tr_Elements[g] = NULL;
   }
   for (int i = 0; i < 2; i++)
   {
      SegDofOrd[i] = NULL;
   }
   for (int i = 0; i < 6; i++)
   {
      TriDofOrd[i] = NULL;
   }
   OtherDofOrd = NULL;

   if (dim == 1)
   {
      if (m_type == BasisType::Positive)
      {
         L2_Elements[Geometry::SEGMENT] = new L2Pos_SegmentElement(p);
      }
      else
      {
         L2_Elements[Geometry::SEGMENT] = new L2_SegmentElement(p, pt_type);
      }
      L2_Elements[Geometry::SEGMENT]->SetMapType(map_type);

      Tr_Elements[Geometry::POINT] = new PointFiniteElement;
      // No need to set the map_type for Tr_Elements.

      const int pp1 = p + 1;
      SegDofOrd[0] = new int[2*pp1];
      SegDofOrd[1] = SegDofOrd[0] + pp1;
      for (int i = 0; i <= p; i++)
      {
         SegDofOrd[0][i] = i;
         SegDofOrd[1][i] = p - i;
      }
   }
   else if (dim == 2)
   {
      if (m_type == BasisType::Positive)
      {
         L2_Elements[Geometry::TRIANGLE] = new L2Pos_TriangleElement(p);
         L2_Elements[Geometry::SQUARE] = new L2Pos_QuadrilateralElement(p);
      }
      else
      {
         L2_Elements[Geometry::TRIANGLE] = new L2_TriangleElement(p, pt_type);
         L2_Elements[Geometry::SQUARE] = new L2_QuadrilateralElement(p, pt_type);
      }
      L2_Elements[Geometry::TRIANGLE]->SetMapType(map_type);
      L2_Elements[Geometry::SQUARE]->SetMapType(map_type);
      // All trace elements use the default Gauss-Legendre points
      Tr_Elements[Geometry::SEGMENT] = new L2_SegmentElement(p);

      const int TriDof = L2_Elements[Geometry::TRIANGLE]->GetDof();
      TriDofOrd[0] = new int[6*TriDof];
      for (int i = 1; i < 6; i++)
      {
         TriDofOrd[i] = TriDofOrd[i-1] + TriDof;
      }
      const int pp1 = p + 1, pp2 = pp1 + 1;
      for (int j = 0; j <= p; j++)
      {
         for (int i = 0; i + j <= p; i++)
         {
            int o = TriDof - ((pp2 - j)*(pp1 - j))/2 + i;
            int k = p - j - i;
            TriDofOrd[0][o] = o;  // (0,1,2)
            TriDofOrd[1][o] = TriDof - ((pp2-j)*(pp1-j))/2 + k;  // (1,0,2)
            TriDofOrd[2][o] = TriDof - ((pp2-i)*(pp1-i))/2 + k;  // (2,0,1)
            TriDofOrd[3][o] = TriDof - ((pp2-k)*(pp1-k))/2 + i;  // (2,1,0)
            TriDofOrd[4][o] = TriDof - ((pp2-k)*(pp1-k))/2 + j;  // (1,2,0)
            TriDofOrd[5][o] = TriDof - ((pp2-i)*(pp1-i))/2 + j;  // (0,2,1)
         }
      }
      const int QuadDof = L2_Elements[Geometry::SQUARE]->GetDof();
      OtherDofOrd = new int[QuadDof];
      for (int j = 0; j < QuadDof; j++)
      {
         OtherDofOrd[j] = j; // for Or == 0
      }
   }
   else if (dim == 3)
   {
      if (m_type == BasisType::Positive)
      {
         L2_Elements[Geometry::TETRAHEDRON] = new L2Pos_TetrahedronElement(p);
         L2_Elements[Geometry::CUBE] = new L2Pos_HexahedronElement(p);
      }
      else
      {
         L2_Elements[Geometry::TETRAHEDRON] =
            new L2_TetrahedronElement(p, pt_type);
         L2_Elements[Geometry::CUBE] = new L2_HexahedronElement(p, pt_type);
      }
      L2_Elements[Geometry::TETRAHEDRON]->SetMapType(map_type);
      L2_Elements[Geometry::CUBE]->SetMapType(map_type);
      // All trace element use the default Gauss-Legendre nodal points
      Tr_Elements[Geometry::TRIANGLE] = new L2_TriangleElement(p);
      Tr_Elements[Geometry::SQUARE] = new L2_QuadrilateralElement(p);

      const int TetDof = L2_Elements[Geometry::TETRAHEDRON]->GetDof();
      const int HexDof = L2_Elements[Geometry::CUBE]->GetDof();
      const int MaxDof = std::max(TetDof, HexDof);
      OtherDofOrd = new int[MaxDof];
      for (int j = 0; j < MaxDof; j++)
      {
         OtherDofOrd[j] = j; // for Or == 0
      }
   }
   else
   {
      mfem::err << "L2_FECollection::L2_FECollection : dim = "
                << dim << endl;
      mfem_error();
   }
}

int *L2_FECollection::DofOrderForOrientation(int GeomType, int Or) const
{
   switch (GeomType)
   {
      case Geometry::SEGMENT:
         return (Or > 0) ? SegDofOrd[0] : SegDofOrd[1];

      case Geometry::TRIANGLE:
         return TriDofOrd[Or%6];

      default:
         return (Or == 0) ? OtherDofOrd : NULL;
   }
}

L2_FECollection::~L2_FECollection()
{
   delete [] OtherDofOrd;
   delete [] SegDofOrd[0];
   delete [] TriDofOrd[0];
   for (int i = 0; i < Geometry::NumGeom; i++)
   {
      delete L2_Elements[i];
      delete Tr_Elements[i];
   }
}


RT_FECollection::RT_FECollection(const int p, const int dim,
                                 const int cb_type, const int ob_type)
   : ob_type(ob_type)
{
   int cp_type = BasisType::GetQuadrature1D(cb_type);
   int op_type = BasisType::GetQuadrature1D(ob_type);

   if (Quadrature1D::CheckClosed(cp_type) == Quadrature1D::Invalid)
   {
      const char *cb_name = BasisType::Name(cb_type); // this may abort
      MFEM_ABORT("unknown closed BasisType: " << cb_name);
   }
   if (Quadrature1D::CheckOpen(op_type) == Quadrature1D::Invalid)
   {
      const char *ob_name = BasisType::Name(ob_type); // this may abort
      MFEM_ABORT("unknown open BasisType: " << ob_name);
   }

   InitFaces(p, dim, FiniteElement::INTEGRAL, true);

   if (cb_type == BasisType::GaussLobatto &&
       ob_type == BasisType::GaussLegendre)
   {
      snprintf(rt_name, 32, "RT_%dD_P%d", dim, p);
   }
   else
   {
      snprintf(rt_name, 32, "RT@%c%c_%dD_P%d", (int)BasisType::GetChar(cb_type),
               (int)BasisType::GetChar(ob_type), dim, p);
   }

   const int pp1 = p + 1;
   if (dim == 2)
   {
      // TODO: cp_type, op_type for triangles
      RT_Elements[Geometry::TRIANGLE] = new RT_TriangleElement(p);
      RT_dof[Geometry::TRIANGLE] = p*pp1;

      RT_Elements[Geometry::SQUARE] = new RT_QuadrilateralElement(p, cp_type,
                                                                  op_type);
      // two vector components * n_unk_face *
      RT_dof[Geometry::SQUARE] = 2*p*pp1;
   }
   else if (dim == 3)
   {
      // TODO: cp_type, op_type for tets
      RT_Elements[Geometry::TETRAHEDRON] = new RT_TetrahedronElement(p);
      RT_dof[Geometry::TETRAHEDRON] = p*pp1*(p + 2)/2;

      RT_Elements[Geometry::CUBE] = new RT_HexahedronElement(p, cp_type, op_type);
      RT_dof[Geometry::CUBE] = 3*p*pp1*pp1;
   }
   else
   {
      MFEM_ABORT("invalid dim = " << dim);
   }
}

// This is a special protected constructor only used by RT_Trace_FECollection
// and DG_Interface_FECollection
RT_FECollection::RT_FECollection(const int p, const int dim, const int map_type,
                                 const bool signs, const int ob_type)
   : ob_type(ob_type)
{
   if (Quadrature1D::CheckOpen(BasisType::GetQuadrature1D(ob_type)) ==
       Quadrature1D::Invalid)
   {
      const char *ob_name = BasisType::Name(ob_type); // this may abort
      MFEM_ABORT("Invalid open basis type: " << ob_name);
   }
   InitFaces(p, dim, map_type, signs);
}

void RT_FECollection::InitFaces(const int p, const int dim, const int map_type,
                                const bool signs)
{
   int op_type = BasisType::GetQuadrature1D(ob_type);

   MFEM_VERIFY(Quadrature1D::CheckOpen(op_type) != Quadrature1D::Invalid,
               "invalid open point type");

   const int pp1 = p + 1, pp2 = p + 2;

   for (int g = 0; g < Geometry::NumGeom; g++)
   {
      RT_Elements[g] = NULL;
      RT_dof[g] = 0;
   }
   // Degree of Freedom orderings
   for (int i = 0; i < 2; i++)
   {
      SegDofOrd[i] = NULL;
   }
   for (int i = 0; i < 6; i++)
   {
      TriDofOrd[i] = NULL;
   }
   for (int i = 0; i < 8; i++)
   {
      QuadDofOrd[i] = NULL;
   }

   if (dim == 2)
   {
      L2_SegmentElement *l2_seg = new L2_SegmentElement(p, op_type);
      l2_seg->SetMapType(map_type);
      RT_Elements[Geometry::SEGMENT] = l2_seg;
      RT_dof[Geometry::SEGMENT] = pp1;

      SegDofOrd[0] = new int[2*pp1];
      SegDofOrd[1] = SegDofOrd[0] + pp1;
      for (int i = 0; i <= p; i++)
      {
         SegDofOrd[0][i] = i;
         SegDofOrd[1][i] = signs ? (-1 - (p - i)) : (p - i);
      }
   }
   else if (dim == 3)
   {
      L2_TriangleElement *l2_tri = new L2_TriangleElement(p, op_type);
      l2_tri->SetMapType(map_type);
      RT_Elements[Geometry::TRIANGLE] = l2_tri;
      RT_dof[Geometry::TRIANGLE] = pp1*pp2/2;

      L2_QuadrilateralElement *l2_quad = new L2_QuadrilateralElement(p, op_type);
      l2_quad->SetMapType(map_type);
      RT_Elements[Geometry::SQUARE] = l2_quad;
      RT_dof[Geometry::SQUARE] = pp1*pp1;

      int TriDof = RT_dof[Geometry::TRIANGLE];
      TriDofOrd[0] = new int[6*TriDof];
      for (int i = 1; i < 6; i++)
      {
         TriDofOrd[i] = TriDofOrd[i-1] + TriDof;
      }
      // see Mesh::GetTriOrientation in mesh/mesh.cpp,
      // the constructor of H1_FECollection
      for (int j = 0; j <= p; j++)
      {
         for (int i = 0; i + j <= p; i++)
         {
            int o = TriDof - ((pp2 - j)*(pp1 - j))/2 + i;
            int k = p - j - i;
            TriDofOrd[0][o] = o;  // (0,1,2)
            TriDofOrd[1][o] = -1-(TriDof-((pp2-j)*(pp1-j))/2+k);  // (1,0,2)
            TriDofOrd[2][o] =     TriDof-((pp2-i)*(pp1-i))/2+k;   // (2,0,1)
            TriDofOrd[3][o] = -1-(TriDof-((pp2-k)*(pp1-k))/2+i);  // (2,1,0)
            TriDofOrd[4][o] =     TriDof-((pp2-k)*(pp1-k))/2+j;   // (1,2,0)
            TriDofOrd[5][o] = -1-(TriDof-((pp2-i)*(pp1-i))/2+j);  // (0,2,1)
            if (!signs)
            {
               for (int k = 1; k < 6; k += 2)
               {
                  TriDofOrd[k][o] = -1 - TriDofOrd[k][o];
               }
            }
         }
      }

      int QuadDof = RT_dof[Geometry::SQUARE];
      QuadDofOrd[0] = new int[8*QuadDof];
      for (int i = 1; i < 8; i++)
      {
         QuadDofOrd[i] = QuadDofOrd[i-1] + QuadDof;
      }
      // see Mesh::GetQuadOrientation in mesh/mesh.cpp
      for (int j = 0; j <= p; j++)
      {
         for (int i = 0; i <= p; i++)
         {
            int o = i + j*pp1;
            QuadDofOrd[0][o] = i + j*pp1;                    // (0,1,2,3)
            QuadDofOrd[1][o] = -1 - (j + i*pp1);             // (0,3,2,1)
            QuadDofOrd[2][o] = j + (p - i)*pp1;              // (1,2,3,0)
            QuadDofOrd[3][o] = -1 - ((p - i) + j*pp1);       // (1,0,3,2)
            QuadDofOrd[4][o] = (p - i) + (p - j)*pp1;        // (2,3,0,1)
            QuadDofOrd[5][o] = -1 - ((p - j) + (p - i)*pp1); // (2,1,0,3)
            QuadDofOrd[6][o] = (p - j) + i*pp1;              // (3,0,1,2)
            QuadDofOrd[7][o] = -1 - (i + (p - j)*pp1);       // (3,2,1,0)
            if (!signs)
            {
               for (int k = 1; k < 8; k += 2)
               {
                  QuadDofOrd[k][o] = -1 - QuadDofOrd[k][o];
               }
            }
         }
      }
   }
}

int *RT_FECollection::DofOrderForOrientation(int GeomType, int Or) const
{
   if (GeomType == Geometry::SEGMENT)
   {
      return (Or > 0) ? SegDofOrd[0] : SegDofOrd[1];
   }
   else if (GeomType == Geometry::TRIANGLE)
   {
      return TriDofOrd[Or%6];
   }
   else if (GeomType == Geometry::SQUARE)
   {
      return QuadDofOrd[Or%8];
   }
   return NULL;
}

FiniteElementCollection *RT_FECollection::GetTraceCollection() const
{
   int dim, p;
   if (!strncmp(rt_name, "RT_", 3))
   {
      dim = atoi(rt_name + 3);
      p = atoi(rt_name + 7);
   }
   else // rt_name = RT@.._.D_P*
   {
      dim = atoi(rt_name + 6);
      p = atoi(rt_name + 10);
   }
   return new RT_Trace_FECollection(p, dim, FiniteElement::INTEGRAL, ob_type);
}

RT_FECollection::~RT_FECollection()
{
   delete [] SegDofOrd[0];
   delete [] TriDofOrd[0];
   delete [] QuadDofOrd[0];
   for (int g = 0; g < Geometry::NumGeom; g++)
   {
      delete RT_Elements[g];
   }
}


RT_Trace_FECollection::RT_Trace_FECollection(const int p, const int dim,
                                             const int map_type,
                                             const int ob_type)
   : RT_FECollection(p, dim, map_type, true, ob_type)
{
   const char *prefix =
      (map_type == FiniteElement::INTEGRAL) ? "RT_Trace" : "RT_ValTrace";
   char ob_str[3] = { '\0', '\0', '\0' };

   if (ob_type != BasisType::GaussLegendre)
   {
      ob_str[0] = '@';
      ob_str[1] = BasisType::GetChar(ob_type);
   }
   snprintf(rt_name, 32, "%s%s_%dD_P%d", prefix, ob_str, dim, p);

   MFEM_VERIFY(dim == 2 || dim == 3, "Wrong dimension, dim = " << dim);
}


DG_Interface_FECollection::DG_Interface_FECollection(const int p, const int dim,
                                                     const int map_type,
                                                     const int ob_type)
   : RT_FECollection(p, dim, map_type, false, ob_type)
{
   MFEM_VERIFY(dim == 2 || dim == 3, "Wrong dimension, dim = " << dim);

   const char *prefix =
      (map_type == FiniteElement::VALUE) ? "DG_Iface" : "DG_IntIface";
   if (ob_type == BasisType::GaussLegendre)
   {
      snprintf(rt_name, 32, "%s_%dD_P%d", prefix, dim, p);
   }
   else
   {
      snprintf(rt_name, 32, "%s@%c_%dD_P%d", prefix,
               (int)BasisType::GetChar(ob_type), dim, p);
   }
}

ND_FECollection::ND_FECollection(const int p, const int dim,
                                 const int cb_type, const int ob_type)
{
   const int pm1 = p - 1, pm2 = p - 2;

   if (cb_type == BasisType::GaussLobatto &&
       ob_type == BasisType::GaussLegendre)
   {
      snprintf(nd_name, 32, "ND_%dD_P%d", dim, p);
   }
   else
   {
      snprintf(nd_name, 32, "ND@%c%c_%dD_P%d", (int)BasisType::GetChar(cb_type),
               (int)BasisType::GetChar(ob_type), dim, p);
   }

   for (int g = 0; g < Geometry::NumGeom; g++)
   {
      ND_Elements[g] = NULL;
      ND_dof[g] = 0;
   }
   for (int i = 0; i < 2; i++)
   {
      SegDofOrd[i] = NULL;
   }
   for (int i = 0; i < 6; i++)
   {
      TriDofOrd[i] = NULL;
   }
   for (int i = 0; i < 8; i++)
   {
      QuadDofOrd[i] = NULL;
   }

   int op_type = BasisType::GetQuadrature1D(ob_type);
   int cp_type = BasisType::GetQuadrature1D(cb_type);

   // Error checking
   if (Quadrature1D::CheckOpen(op_type) == Quadrature1D::Invalid)
   {
      const char *ob_name = BasisType::Name(ob_type);
      MFEM_ABORT("Invalid open basis point type: " << ob_name);
   }
   if (Quadrature1D::CheckClosed(cp_type) == Quadrature1D::Invalid)
   {
      const char *cb_name = BasisType::Name(cb_type);
      MFEM_ABORT("Invalid closed basis point type: " << cb_name);
   }

   if (dim >= 1)
   {
      ND_Elements[Geometry::SEGMENT] = new ND_SegmentElement(p, op_type);
      ND_dof[Geometry::SEGMENT] = p;

      SegDofOrd[0] = new int[2*p];
      SegDofOrd[1] = SegDofOrd[0] + p;
      for (int i = 0; i < p; i++)
      {
         SegDofOrd[0][i] = i;
         SegDofOrd[1][i] = -1 - (pm1 - i);
      }
   }

   if (dim >= 2)
   {
      ND_Elements[Geometry::SQUARE] = new ND_QuadrilateralElement(p, cp_type,
                                                                  op_type);
      ND_dof[Geometry::SQUARE] = 2*p*pm1;

      // TODO: cp_type and op_type for triangles
      ND_Elements[Geometry::TRIANGLE] = new ND_TriangleElement(p);
      ND_dof[Geometry::TRIANGLE] = p*pm1;

      int QuadDof = ND_dof[Geometry::SQUARE];
      QuadDofOrd[0] = new int[8*QuadDof];
      for (int i = 1; i < 8; i++)
      {
         QuadDofOrd[i] = QuadDofOrd[i-1] + QuadDof;
      }
      // see Mesh::GetQuadOrientation in mesh/mesh.cpp
      for (int j = 0; j < pm1; j++)
      {
         for (int i = 0; i < p; i++)
         {
            int d1 = i + j*p;            // x-component
            int d2 = p*pm1 + j + i*pm1;  // y-component
            // (0,1,2,3)
            QuadDofOrd[0][d1] = d1;
            QuadDofOrd[0][d2] = d2;
            // (0,3,2,1)
            QuadDofOrd[1][d1] = d2;
            QuadDofOrd[1][d2] = d1;
            // (1,2,3,0)
            // QuadDofOrd[2][d1] = p*pm1 + (pm2 - j) + i*pm1;
            // QuadDofOrd[2][d2] = -1 - ((pm1 - i) + j*p);
            QuadDofOrd[2][d1] = -1 - (p*pm1 + j + (pm1 - i)*pm1);
            QuadDofOrd[2][d2] = i + (pm2 - j)*p;
            // (1,0,3,2)
            QuadDofOrd[3][d1] = -1 - ((pm1 - i) + j*p);
            QuadDofOrd[3][d2] = p*pm1 + (pm2 - j) + i*pm1;
            // (2,3,0,1)
            QuadDofOrd[4][d1] = -1 - ((pm1 - i) + (pm2 - j)*p);
            QuadDofOrd[4][d2] = -1 - (p*pm1 + (pm2 - j) + (pm1 - i)*pm1);
            // (2,1,0,3)
            QuadDofOrd[5][d1] = -1 - (p*pm1 + (pm2 - j) + (pm1 - i)*pm1);
            QuadDofOrd[5][d2] = -1 - ((pm1 - i) + (pm2 - j)*p);
            // (3,0,1,2)
            // QuadDofOrd[6][d1] = -1 - (p*pm1 + j + (pm1 - i)*pm1);
            // QuadDofOrd[6][d2] = i + (pm2 - j)*p;
            QuadDofOrd[6][d1] = p*pm1 + (pm2 - j) + i*pm1;
            QuadDofOrd[6][d2] = -1 - ((pm1 - i) + j*p);
            // (3,2,1,0)
            QuadDofOrd[7][d1] = i + (pm2 - j)*p;
            QuadDofOrd[7][d2] = -1 - (p*pm1 + j + (pm1 - i)*pm1);
         }
      }

      int TriDof = ND_dof[Geometry::TRIANGLE];
      TriDofOrd[0] = new int[6*TriDof];
      for (int i = 1; i < 6; i++)
      {
         TriDofOrd[i] = TriDofOrd[i-1] + TriDof;
      }
      // see Mesh::GetTriOrientation in mesh/mesh.cpp,
      // the constructor of H1_FECollection
      for (int j = 0; j <= pm2; j++)
      {
         for (int i = 0; i + j <= pm2; i++)
         {
            int k1 = p*pm1 - (p - j)*(pm1 - j) + 2*i;
            int k2 = p*pm1 - (p - i)*(pm1 - i) + 2*j;
            // (0,1,2)
            TriDofOrd[0][k1  ] = k1;
            TriDofOrd[0][k1+1] = k1 + 1;
            // (0,2,1)
            TriDofOrd[5][k1  ] = k2 + 1;
            TriDofOrd[5][k1+1] = k2;

            // The other orientations can not be supported with the current
            // interface. The method Mesh::ReorientTetMesh will ensure that
            // only orientations 0 and 5 are generated.
         }
      }
   }

   if (dim >= 3)
   {
      ND_Elements[Geometry::CUBE] = new ND_HexahedronElement(p, cp_type, op_type);
      ND_dof[Geometry::CUBE] = 3*p*pm1*pm1;

      // TODO: cp_type and op_type for tets
      ND_Elements[Geometry::TETRAHEDRON] = new ND_TetrahedronElement(p);
      ND_dof[Geometry::TETRAHEDRON] = p*pm1*pm2/2;
   }
}

int *ND_FECollection::DofOrderForOrientation(int GeomType, int Or) const
{
   if (GeomType == Geometry::SEGMENT)
   {
      return (Or > 0) ? SegDofOrd[0] : SegDofOrd[1];
   }
   else if (GeomType == Geometry::TRIANGLE)
   {
      if (Or != 0 && Or != 5)
      {
         MFEM_ABORT("ND_FECollection::DofOrderForOrientation: "
                    "triangle face orientation " << Or << " is not supported! "
                    "Use Mesh::ReorientTetMesh to fix it.");
      }
      return TriDofOrd[Or%6];
   }
   else if (GeomType == Geometry::SQUARE)
   {
      return QuadDofOrd[Or%8];
   }
   return NULL;
}

FiniteElementCollection *ND_FECollection::GetTraceCollection() const
{
   int p, dim, cb_type, ob_type;

   p = ND_dof[Geometry::SEGMENT];
   if (nd_name[2] == '_') // ND_
   {
      dim = atoi(nd_name + 3);
      cb_type = BasisType::GaussLobatto;
      ob_type = BasisType::GaussLegendre;
   }
   else // ND@
   {
      dim = atoi(nd_name + 6);
      cb_type = BasisType::GetType(nd_name[3]);
      ob_type = BasisType::GetType(nd_name[4]);
   }
   return new ND_Trace_FECollection(p, dim, cb_type, ob_type);
}

ND_FECollection::~ND_FECollection()
{
   delete [] SegDofOrd[0];
   delete [] TriDofOrd[0];
   delete [] QuadDofOrd[0];
   for (int g = 0; g < Geometry::NumGeom; g++)
   {
      delete ND_Elements[g];
   }
}


ND_Trace_FECollection::ND_Trace_FECollection(const int p, const int dim,
                                             const int cb_type,
                                             const int ob_type)
   : ND_FECollection(p, dim-1, cb_type, ob_type)
{
   if (cb_type == BasisType::GaussLobatto &&
       ob_type == BasisType::GaussLegendre)
   {
      snprintf(nd_name, 32, "ND_Trace_%dD_P%d", dim, p);
   }
   else
   {
      snprintf(nd_name, 32, "ND_Trace@%c%c_%dD_P%d",
               (int)BasisType::GetChar(cb_type),
               (int)BasisType::GetChar(ob_type), dim, p);
   }
}


Local_FECollection::Local_FECollection(const char *fe_name)
{
   snprintf(d_name, 32, "Local_%s", fe_name);

   Local_Element = NULL;

   if (!strcmp(fe_name, "BiCubic2DFiniteElement") ||
       !strcmp(fe_name, "Quad_Q3"))
   {
      GeomType = Geometry::SQUARE;
      Local_Element = new BiCubic2DFiniteElement;
   }
   else if (!strcmp(fe_name, "Nedelec1HexFiniteElement") ||
            !strcmp(fe_name, "Hex_ND1"))
   {
      GeomType = Geometry::CUBE;
      Local_Element = new Nedelec1HexFiniteElement;
   }
   else if (!strncmp(fe_name, "H1_", 3))
   {
      GeomType = Geometry::SQUARE;
      Local_Element = new H1_QuadrilateralElement(atoi(fe_name + 7));
   }
   else if (!strncmp(fe_name, "H1Pos_", 6))
   {
      GeomType = Geometry::SQUARE;
      Local_Element = new H1Pos_QuadrilateralElement(atoi(fe_name + 10));
   }
   else if (!strncmp(fe_name, "L2_", 3))
   {
      GeomType = Geometry::SQUARE;
      Local_Element = new L2_QuadrilateralElement(atoi(fe_name + 7));
   }
   else
   {
      mfem::err << "Local_FECollection::Local_FECollection : fe_name = "
                << fe_name << endl;
      mfem_error();
   }
}


void NURBSFECollection::Allocate(int Order)
{
   SegmentFE        = new NURBS1DFiniteElement(Order);
   QuadrilateralFE  = new NURBS2DFiniteElement(Order);
   ParallelepipedFE = new NURBS3DFiniteElement(Order);

   snprintf(name, 16, "NURBS%i", Order);
}

void NURBSFECollection::Deallocate()
{
   delete ParallelepipedFE;
   delete QuadrilateralFE;
   delete SegmentFE;
}

const FiniteElement *
NURBSFECollection::FiniteElementForGeometry(int GeomType) const
{
   switch (GeomType)
   {
      case Geometry::SEGMENT:     return SegmentFE;
      case Geometry::SQUARE:      return QuadrilateralFE;
      case Geometry::CUBE:        return ParallelepipedFE;
      default:
         mfem_error ("NURBSFECollection: unknown geometry type.");
   }
   return SegmentFE; // Make some compilers happy
}

int NURBSFECollection::DofForGeometry(int GeomType) const
{
   mfem_error("NURBSFECollection::DofForGeometry");
   return 0; // Make some compilers happy
}

int *NURBSFECollection::DofOrderForOrientation(int GeomType, int Or) const
{
   mfem_error("NURBSFECollection::DofOrderForOrientation");
   return NULL;
}

FiniteElementCollection *NURBSFECollection::GetTraceCollection() const
{
   MFEM_ABORT("NURBS finite elements can not be statically condensed!");
   return NULL;
}

}