fe.cpp

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// Copyright (c) 2010-2020, 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.

// Finite Element classes

#include "fe.hpp"
#include "fe_coll.hpp"
#include "../mesh/nurbs.hpp"
#include "bilininteg.hpp"
#include <cmath>

namespace mfem
{

using namespace std;

FiniteElement::FiniteElement(int D, Geometry::Type G, int Do, int O, int F)
   : Nodes(Do)
{
   Dim = D ; GeomType = G ; Dof = Do ; Order = O ; FuncSpace = F;
   RangeType = SCALAR;
   MapType = VALUE;
   DerivType = NONE;
   DerivRangeType = SCALAR;
   DerivMapType = VALUE;
   for (int i = 0; i < Geometry::MaxDim; i++) { Orders[i] = -1; }
#ifndef MFEM_THREAD_SAFE
   vshape.SetSize(Dof, Dim);
#endif
}

void FiniteElement::CalcVShape (
   const IntegrationPoint &ip, DenseMatrix &shape) const
{
   mfem_error ("FiniteElement::CalcVShape (ip, ...)\n"
               "   is not implemented for this class!");
}

void FiniteElement::CalcVShape (
   ElementTransformation &Trans, DenseMatrix &shape) const
{
   mfem_error ("FiniteElement::CalcVShape (trans, ...)\n"
               "   is not implemented for this class!");
}

void FiniteElement::CalcDivShape (
   const IntegrationPoint &ip, Vector &divshape) const
{
   mfem_error ("FiniteElement::CalcDivShape (ip, ...)\n"
               "   is not implemented for this class!");
}

void FiniteElement::CalcPhysDivShape(
   ElementTransformation &Trans, Vector &div_shape) const
{
   CalcDivShape(Trans.GetIntPoint(), div_shape);
   div_shape *= (1.0 / Trans.Weight());
}

void FiniteElement::CalcCurlShape(const IntegrationPoint &ip,
                                  DenseMatrix &curl_shape) const
{
   mfem_error ("FiniteElement::CalcCurlShape (ip, ...)\n"
               "   is not implemented for this class!");
}

void FiniteElement::CalcPhysCurlShape(ElementTransformation &Trans,
                                      DenseMatrix &curl_shape) const
{
   switch (Dim)
   {
      case 3:
      {
#ifdef MFEM_THREAD_SAFE
         DenseMatrix vshape(Dof, Dim);
#endif
         CalcCurlShape(Trans.GetIntPoint(), vshape);
         MultABt(vshape, Trans.Jacobian(), curl_shape);
         curl_shape *= (1.0 / Trans.Weight());
         break;
      }
      case 2:
         // This is valid for both 2x2 and 3x2 Jacobians
         CalcCurlShape(Trans.GetIntPoint(), curl_shape);
         curl_shape *= (1.0 / Trans.Weight());
         break;
      default:
         MFEM_ABORT("Invalid dimension, Dim = " << Dim);
   }
}

void FiniteElement::GetFaceDofs(int face, int **dofs, int *ndofs) const
{
   mfem_error ("FiniteElement::GetFaceDofs (...)");
}

void FiniteElement::CalcHessian (const IntegrationPoint &ip,
                                 DenseMatrix &h) const
{
   mfem_error ("FiniteElement::CalcHessian (...) is not overloaded !");
}

void FiniteElement::GetLocalInterpolation (ElementTransformation &Trans,
                                           DenseMatrix &I) const
{
   mfem_error ("GetLocalInterpolation (...) is not overloaded !");
}

void FiniteElement::GetLocalRestriction(ElementTransformation &,
                                        DenseMatrix &) const
{
   mfem_error("FiniteElement::GetLocalRestriction() is not overloaded !");
}

void FiniteElement::GetTransferMatrix(const FiniteElement &fe,
                                      ElementTransformation &Trans,
                                      DenseMatrix &I) const
{
   MFEM_ABORT("method is not overloaded !");
}

void FiniteElement::Project (
   Coefficient &coeff, ElementTransformation &Trans, Vector &dofs) const
{
   mfem_error ("FiniteElement::Project (...) is not overloaded !");
}

void FiniteElement::Project (
   VectorCoefficient &vc, ElementTransformation &Trans, Vector &dofs) const
{
   mfem_error ("FiniteElement::Project (...) (vector) is not overloaded !");
}

void FiniteElement::ProjectMatrixCoefficient(
   MatrixCoefficient &mc, ElementTransformation &T, Vector &dofs) const
{
   mfem_error("FiniteElement::ProjectMatrixCoefficient() is not overloaded !");
}

void FiniteElement::ProjectDelta(int vertex, Vector &dofs) const
{
   mfem_error("FiniteElement::ProjectDelta(...) is not implemented for "
              "this element!");
}

void FiniteElement::Project(
   const FiniteElement &fe, ElementTransformation &Trans, DenseMatrix &I) const
{
   mfem_error("FiniteElement::Project(...) (fe version) is not implemented "
              "for this element!");
}

void FiniteElement::ProjectGrad(
   const FiniteElement &fe, ElementTransformation &Trans,
   DenseMatrix &grad) const
{
   mfem_error("FiniteElement::ProjectGrad(...) is not implemented for "
              "this element!");
}

void FiniteElement::ProjectCurl(
   const FiniteElement &fe, ElementTransformation &Trans,
   DenseMatrix &curl) const
{
   mfem_error("FiniteElement::ProjectCurl(...) is not implemented for "
              "this element!");
}

void FiniteElement::ProjectDiv(
   const FiniteElement &fe, ElementTransformation &Trans,
   DenseMatrix &div) const
{
   mfem_error("FiniteElement::ProjectDiv(...) is not implemented for "
              "this element!");
}

void FiniteElement::CalcPhysShape(ElementTransformation &Trans,
                                  Vector &shape) const
{
   CalcShape(Trans.GetIntPoint(), shape);
   if (MapType == INTEGRAL)
   {
      shape /= Trans.Weight();
   }
}

void FiniteElement::CalcPhysDShape(ElementTransformation &Trans,
                                   DenseMatrix &dshape) const
{
   MFEM_ASSERT(MapType == VALUE, "");
#ifdef MFEM_THREAD_SAFE
   DenseMatrix vshape(Dof, Dim);
#endif
   CalcDShape(Trans.GetIntPoint(), vshape);
   Mult(vshape, Trans.InverseJacobian(), dshape);
}

void FiniteElement::CalcPhysLaplacian(ElementTransformation &Trans,
                                      Vector &Laplacian) const
{
   MFEM_ASSERT(MapType == VALUE, "");

   // Simpler routine if mapping is affine
   if (Trans.Hessian().FNorm2() < 1e-20)
   {
      CalcPhysLinLaplacian(Trans, Laplacian);
      return;
   }

   // Compute full Hessian first if non-affine
   int size = (Dim*(Dim+1))/2;
   DenseMatrix hess(Dof, size);
   CalcPhysHessian(Trans,hess);

   if (Dim == 3)
   {
      for (int nd = 0; nd < Dof; nd++)
      {
         Laplacian[nd] = hess(nd,0) + hess(nd,4) + hess(nd,5);
      }
   }
   else if (Dim == 2)
   {
      for (int nd = 0; nd < Dof; nd++)
      {
         Laplacian[nd] = hess(nd,0) + hess(nd,2);
      }
   }
   else
   {
      for (int nd = 0; nd < Dof; nd++)
      {
         Laplacian[nd] = hess(nd,0);
      }
   }
}


// Assume a linear mapping
void FiniteElement::CalcPhysLinLaplacian(ElementTransformation &Trans,
                                         Vector &Laplacian) const
{
   MFEM_ASSERT(MapType == VALUE, "");
   int size = (Dim*(Dim+1))/2;
   DenseMatrix hess(Dof, size);
   DenseMatrix Gij(Dim,Dim);
   Vector scale(size);

   CalcHessian (Trans.GetIntPoint(), hess);
   MultAAt(Trans.InverseJacobian(), Gij);

   if (Dim == 3)
   {
      scale[0] =   Gij(0,0);
      scale[1] = 2*Gij(0,1);
      scale[2] = 2*Gij(0,2);

      scale[3] = 2*Gij(1,2);
      scale[4] =   Gij(2,2);

      scale[5] =   Gij(1,1);
   }
   else if (Dim == 2)
   {
      scale[0] =   Gij(0,0);
      scale[1] = 2*Gij(0,1);
      scale[2] =   Gij(1,1);
   }
   else
   {
      scale[0] =   Gij(0,0);
   }

   for (int nd = 0; nd < Dof; nd++)
   {
      Laplacian[nd] = 0.0;
      for (int ii = 0; ii < size; ii++)
      {
         Laplacian[nd] += hess(nd,ii)*scale[ii];
      }
   }

}

void  FiniteElement::CalcPhysHessian(ElementTransformation &Trans,
                                     DenseMatrix& Hessian) const
{
   MFEM_ASSERT(MapType == VALUE, "");

   // Roll 2-Tensors in vectors and 4-Tensor in Matrix, exploiting symmetry
   Array<int> map(Dim*Dim);
   if (Dim == 3)
   {
      map[0] = 0;
      map[1] = 1;
      map[2] = 2;

      map[3] = 1;
      map[4] = 5;
      map[5] = 3;

      map[6] = 2;
      map[7] = 3;
      map[8] = 4;
   }
   else if (Dim == 2)
   {
      map[0] = 0;
      map[1] = 1;

      map[2] = 1;
      map[3] = 2;
   }
   else
   {
      map[0] = 0;
   }

   // Hessian in ref coords
   int size = (Dim*(Dim+1))/2;
   DenseMatrix hess(Dof, size);
   CalcHessian(Trans.GetIntPoint(), hess);

   // Gradient in physical coords
   if (Trans.Hessian().FNorm2() > 1e-10)
   {
      DenseMatrix grad(Dof, Dim);
      CalcPhysDShape(Trans, grad);
      DenseMatrix gmap(Dof, size);
      Mult(grad,Trans.Hessian(),gmap);
      hess -= gmap;
   }

   // LHM
   DenseMatrix lhm(size,size);
   DenseMatrix invJ = Trans.Jacobian();
   lhm = 0.0;
   for (int i = 0; i < Dim; i++)
   {
      for (int j = 0; j < Dim; j++)
      {
         for (int k = 0; k < Dim; k++)
         {
            for (int l = 0; l < Dim; l++)
            {
               lhm(map[i*Dim+j],map[k*Dim+l]) += invJ(i,k)*invJ(j,l);
            }
         }
      }
   }
   // Correct multiplicity
   Vector mult(size);
   mult = 0.0;
   for (int i = 0; i < Dim*Dim; i++) { mult[map[i]]++; }
   lhm.InvRightScaling(mult);

   // Hessian in physical coords
   lhm.Invert();
   Mult( hess, lhm, Hessian);
}

const DofToQuad &FiniteElement::GetDofToQuad(const IntegrationRule &,
                                             DofToQuad::Mode) const
{
   mfem_error("FiniteElement::GetDofToQuad(...) is not implemented for "
              "this element!");
   return *dof2quad_array[0]; // suppress a warning
}

FiniteElement::~FiniteElement()
{
   for (int i = 0; i < dof2quad_array.Size(); i++)
   {
      delete dof2quad_array[i];
   }
}


void ScalarFiniteElement::NodalLocalInterpolation (
   ElementTransformation &Trans, DenseMatrix &I,
   const ScalarFiniteElement &fine_fe) const
{
   double v[Geometry::MaxDim];
   Vector vv (v, Dim);
   IntegrationPoint f_ip;

#ifdef MFEM_THREAD_SAFE
   Vector c_shape(Dof);
#endif

   MFEM_ASSERT(MapType == fine_fe.GetMapType(), "");

   I.SetSize(fine_fe.Dof, Dof);
   for (int i = 0; i < fine_fe.Dof; i++)
   {
      Trans.Transform(fine_fe.Nodes.IntPoint(i), vv);
      f_ip.Set(v, Dim);
      CalcShape(f_ip, c_shape);
      for (int j = 0; j < Dof; j++)
         if (fabs(I(i,j) = c_shape(j)) < 1.0e-12)
         {
            I(i,j) = 0.0;
         }
   }
   if (MapType == INTEGRAL)
   {
      // assuming Trans is linear; this should be ok for all refinement types
      Trans.SetIntPoint(&Geometries.GetCenter(GeomType));
      I *= Trans.Weight();
   }
}

void ScalarFiniteElement::ScalarLocalInterpolation(
   ElementTransformation &Trans, DenseMatrix &I,
   const ScalarFiniteElement &fine_fe) const
{
   // General "interpolation", defined by L2 projection

   double v[Geometry::MaxDim];
   Vector vv (v, Dim);
   IntegrationPoint f_ip;

   const int fs = fine_fe.GetDof(), cs = this->GetDof();
   I.SetSize(fs, cs);
   Vector fine_shape(fs), coarse_shape(cs);
   DenseMatrix fine_mass(fs), fine_coarse_mass(fs, cs); // initialized with 0
   const int ir_order = GetOrder() + fine_fe.GetOrder();
   const IntegrationRule &ir = IntRules.Get(fine_fe.GetGeomType(), ir_order);

   for (int i = 0; i < ir.GetNPoints(); i++)
   {
      const IntegrationPoint &ip = ir.IntPoint(i);
      fine_fe.CalcShape(ip, fine_shape);
      Trans.Transform(ip, vv);
      f_ip.Set(v, Dim);
      this->CalcShape(f_ip, coarse_shape);

      AddMult_a_VVt(ip.weight, fine_shape, fine_mass);
      AddMult_a_VWt(ip.weight, fine_shape, coarse_shape, fine_coarse_mass);
   }

   DenseMatrixInverse fine_mass_inv(fine_mass);
   fine_mass_inv.Mult(fine_coarse_mass, I);

   if (MapType == INTEGRAL)
   {
      // assuming Trans is linear; this should be ok for all refinement types
      Trans.SetIntPoint(&Geometries.GetCenter(GeomType));
      I *= Trans.Weight();
   }
}

const DofToQuad &ScalarFiniteElement::GetDofToQuad(const IntegrationRule &ir,
                                                   DofToQuad::Mode mode) const
{
   MFEM_VERIFY(mode == DofToQuad::FULL, "invalid mode requested");

   for (int i = 0; i < dof2quad_array.Size(); i++)
   {
      const DofToQuad &d2q = *dof2quad_array[i];
      if (d2q.IntRule == &ir && d2q.mode == mode) { return d2q; }
   }

   DofToQuad *d2q = new DofToQuad;
   const int nqpt = ir.GetNPoints();
   d2q->FE = this;
   d2q->IntRule = &ir;
   d2q->mode = mode;
   d2q->ndof = Dof;
   d2q->nqpt = nqpt;
   d2q->B.SetSize(nqpt*Dof);
   d2q->Bt.SetSize(Dof*nqpt);
   d2q->G.SetSize(nqpt*Dim*Dof);
   d2q->Gt.SetSize(Dof*nqpt*Dim);
#ifdef MFEM_THREAD_SAFE
   Vector c_shape(Dof);
   DenseMatrix vshape(Dof, Dim);
#endif
   for (int i = 0; i < nqpt; i++)
   {
      const IntegrationPoint &ip = ir.IntPoint(i);
      CalcShape(ip, c_shape);
      for (int j = 0; j < Dof; j++)
      {
         d2q->B[i+nqpt*j] = d2q->Bt[j+Dof*i] = c_shape(j);
      }
      CalcDShape(ip, vshape);
      for (int d = 0; d < Dim; d++)
      {
         for (int j = 0; j < Dof; j++)
         {
            d2q->G[i+nqpt*(d+Dim*j)] = d2q->Gt[j+Dof*(i+nqpt*d)] = vshape(j,d);
         }
      }
   }
   dof2quad_array.Append(d2q);
   return *d2q;
}

// protected method
const DofToQuad &ScalarFiniteElement::GetTensorDofToQuad(
   const TensorBasisElement &tb,
   const IntegrationRule &ir, DofToQuad::Mode mode) const
{
   MFEM_VERIFY(mode == DofToQuad::TENSOR, "invalid mode requested");

   for (int i = 0; i < dof2quad_array.Size(); i++)
   {
      const DofToQuad &d2q = *dof2quad_array[i];
      if (d2q.IntRule == &ir && d2q.mode == mode) { return d2q; }
   }

   DofToQuad *d2q = new DofToQuad;
   const Poly_1D::Basis &basis_1d = tb.GetBasis1D();
   const int ndof = Order + 1;
   const int nqpt = (int)floor(pow(ir.GetNPoints(), 1.0/Dim) + 0.5);
   d2q->FE = this;
   d2q->IntRule = &ir;
   d2q->mode = mode;
   d2q->ndof = ndof;
   d2q->nqpt = nqpt;
   d2q->B.SetSize(nqpt*ndof);
   d2q->Bt.SetSize(ndof*nqpt);
   d2q->G.SetSize(nqpt*ndof);
   d2q->Gt.SetSize(ndof*nqpt);
   Vector val(ndof), grad(ndof);
   for (int i = 0; i < nqpt; i++)
   {
      // The first 'nqpt' points in 'ir' have the same x-coordinates as those
      // of the 1D rule.
      basis_1d.Eval(ir.IntPoint(i).x, val, grad);
      for (int j = 0; j < ndof; j++)
      {
         d2q->B[i+nqpt*j] = d2q->Bt[j+ndof*i] = val(j);
         d2q->G[i+nqpt*j] = d2q->Gt[j+ndof*i] = grad(j);
      }
   }
   dof2quad_array.Append(d2q);
   return *d2q;
}


void NodalFiniteElement::ProjectCurl_2D(
   const FiniteElement &fe, ElementTransformation &Trans,
   DenseMatrix &curl) const
{
   MFEM_ASSERT(GetMapType() == FiniteElement::INTEGRAL, "");

   DenseMatrix curl_shape(fe.GetDof(), 1);

   curl.SetSize(Dof, fe.GetDof());
   for (int i = 0; i < Dof; i++)
   {
      fe.CalcCurlShape(Nodes.IntPoint(i), curl_shape);
      for (int j = 0; j < fe.GetDof(); j++)
      {
         curl(i,j) = curl_shape(j,0);
      }
   }
}

void InvertLinearTrans(ElementTransformation &trans,
                       const IntegrationPoint &pt, Vector &x)
{
   // invert a linear transform with one Newton step
   IntegrationPoint p0;
   p0.Set3(0, 0, 0);
   trans.Transform(p0, x);

   double store[3];
   Vector v(store, x.Size());
   pt.Get(v, x.Size());
   v -= x;

   trans.InverseJacobian().Mult(v, x);
}

void NodalFiniteElement::GetLocalRestriction(ElementTransformation &Trans,
                                             DenseMatrix &R) const
{
   IntegrationPoint ipt;
   Vector pt(&ipt.x, Dim);

#ifdef MFEM_THREAD_SAFE
   Vector c_shape(Dof);
#endif

   Trans.SetIntPoint(&Nodes[0]);

   for (int j = 0; j < Dof; j++)
   {
      InvertLinearTrans(Trans, Nodes[j], pt);
      if (Geometries.CheckPoint(GeomType, ipt)) // do we need an epsilon here?
      {
         CalcShape(ipt, c_shape);
         R.SetRow(j, c_shape);
      }
      else
      {
         // Set the whole row to avoid valgrind warnings in R.Threshold().
         R.SetRow(j, infinity());
      }
   }
   R.Threshold(1e-12);
}

void NodalFiniteElement::Project (
   Coefficient &coeff, ElementTransformation &Trans, Vector &dofs) const
{
   for (int i = 0; i < Dof; i++)
   {
      const IntegrationPoint &ip = Nodes.IntPoint(i);
      // some coefficients expect that Trans.IntPoint is the same
      // as the second argument of Eval
      Trans.SetIntPoint(&ip);
      dofs(i) = coeff.Eval (Trans, ip);
      if (MapType == INTEGRAL)
      {
         dofs(i) *= Trans.Weight();
      }
   }
}

void NodalFiniteElement::Project (
   VectorCoefficient &vc, ElementTransformation &Trans, Vector &dofs) const
{
   MFEM_ASSERT(dofs.Size() == vc.GetVDim()*Dof, "");
   Vector x(vc.GetVDim());

   for (int i = 0; i < Dof; i++)
   {
      const IntegrationPoint &ip = Nodes.IntPoint(i);
      Trans.SetIntPoint(&ip);
      vc.Eval (x, Trans, ip);
      if (MapType == INTEGRAL)
      {
         x *= Trans.Weight();
      }
      for (int j = 0; j < x.Size(); j++)
      {
         dofs(Dof*j+i) = x(j);
      }
   }
}

void NodalFiniteElement::ProjectMatrixCoefficient(
   MatrixCoefficient &mc, ElementTransformation &T, Vector &dofs) const
{
   // (mc.height x mc.width) @ DOFs -> (Dof x mc.width x mc.height) in dofs
   MFEM_ASSERT(dofs.Size() == mc.GetHeight()*mc.GetWidth()*Dof, "");
   DenseMatrix MQ(mc.GetHeight(), mc.GetWidth());

   for (int k = 0; k < Dof; k++)
   {
      T.SetIntPoint(&Nodes.IntPoint(k));
      mc.Eval(MQ, T, Nodes.IntPoint(k));
      if (MapType == INTEGRAL) { MQ *= T.Weight(); }
      for (int r = 0; r < MQ.Height(); r++)
      {
         for (int d = 0; d < MQ.Width(); d++)
         {
            dofs(k+Dof*(d+MQ.Width()*r)) = MQ(r,d);
         }
      }
   }
}

void NodalFiniteElement::Project(
   const FiniteElement &fe, ElementTransformation &Trans, DenseMatrix &I) const
{
   if (fe.GetRangeType() == SCALAR)
   {
      MFEM_ASSERT(MapType == fe.GetMapType(), "");

      Vector shape(fe.GetDof());

      I.SetSize(Dof, fe.GetDof());
      for (int k = 0; k < Dof; k++)
      {
         fe.CalcShape(Nodes.IntPoint(k), shape);
         for (int j = 0; j < shape.Size(); j++)
         {
            I(k,j) = (fabs(shape(j)) < 1e-12) ? 0.0 : shape(j);
         }
      }
   }
   else
   {
      DenseMatrix vshape(fe.GetDof(), Trans.GetSpaceDim());

      I.SetSize(vshape.Width()*Dof, fe.GetDof());
      for (int k = 0; k < Dof; k++)
      {
         Trans.SetIntPoint(&Nodes.IntPoint(k));
         fe.CalcVShape(Trans, vshape);
         if (MapType == INTEGRAL)
         {
            vshape *= Trans.Weight();
         }
         for (int j = 0; j < vshape.Height(); j++)
            for (int d = 0; d < vshape.Width(); d++)
            {
               I(k+d*Dof,j) = vshape(j,d);
            }
      }
   }
}

void NodalFiniteElement::ProjectGrad(
   const FiniteElement &fe, ElementTransformation &Trans,
   DenseMatrix &grad) const
{
   MFEM_ASSERT(fe.GetMapType() == VALUE, "");
   MFEM_ASSERT(Trans.GetSpaceDim() == Dim, "")

   DenseMatrix dshape(fe.GetDof(), Dim), grad_k(fe.GetDof(), Dim), Jinv(Dim);

   grad.SetSize(Dim*Dof, fe.GetDof());
   for (int k = 0; k < Dof; k++)
   {
      const IntegrationPoint &ip = Nodes.IntPoint(k);
      fe.CalcDShape(ip, dshape);
      Trans.SetIntPoint(&ip);
      CalcInverse(Trans.Jacobian(), Jinv);
      Mult(dshape, Jinv, grad_k);
      if (MapType == INTEGRAL)
      {
         grad_k *= Trans.Weight();
      }
      for (int j = 0; j < grad_k.Height(); j++)
         for (int d = 0; d < Dim; d++)
         {
            grad(k+d*Dof,j) = grad_k(j,d);
         }
   }
}

void NodalFiniteElement::ProjectDiv(
   const FiniteElement &fe, ElementTransformation &Trans,
   DenseMatrix &div) const
{
   double detJ;
   Vector div_shape(fe.GetDof());

   div.SetSize(Dof, fe.GetDof());
   for (int k = 0; k < Dof; k++)
   {
      const IntegrationPoint &ip = Nodes.IntPoint(k);
      fe.CalcDivShape(ip, div_shape);
      if (MapType == VALUE)
      {
         Trans.SetIntPoint(&ip);
         detJ = Trans.Weight();
         for (int j = 0; j < div_shape.Size(); j++)
         {
            div(k,j) = (fabs(div_shape(j)) < 1e-12) ? 0.0 : div_shape(j)/detJ;
         }
      }
      else
      {
         for (int j = 0; j < div_shape.Size(); j++)
         {
            div(k,j) = (fabs(div_shape(j)) < 1e-12) ? 0.0 : div_shape(j);
         }
      }
   }
}


void PositiveFiniteElement::Project(
   Coefficient &coeff, ElementTransformation &Trans, Vector &dofs) const
{
   for (int i = 0; i < Dof; i++)
   {
      const IntegrationPoint &ip = Nodes.IntPoint(i);
      Trans.SetIntPoint(&ip);
      dofs(i) = coeff.Eval(Trans, ip);
   }
}

void PositiveFiniteElement::Project(
   VectorCoefficient &vc, ElementTransformation &Trans, Vector &dofs) const
{
   MFEM_ASSERT(dofs.Size() == vc.GetVDim()*Dof, "");
   Vector x(vc.GetVDim());

   for (int i = 0; i < Dof; i++)
   {
      const IntegrationPoint &ip = Nodes.IntPoint(i);
      Trans.SetIntPoint(&ip);
      vc.Eval (x, Trans, ip);
      for (int j = 0; j < x.Size(); j++)
      {
         dofs(Dof*j+i) = x(j);
      }
   }
}

void PositiveFiniteElement::Project(
   const FiniteElement &fe, ElementTransformation &Trans, DenseMatrix &I) const
{
   const NodalFiniteElement *nfe =
      dynamic_cast<const NodalFiniteElement *>(&fe);

   if (nfe && Dof == nfe->GetDof())
   {
      nfe->Project(*this, Trans, I);
      I.Invert();
   }
   else
   {
      // local L2 projection
      DenseMatrix pos_mass, mixed_mass;
      MassIntegrator mass_integ;

      mass_integ.AssembleElementMatrix(*this, Trans, pos_mass);
      mass_integ.AssembleElementMatrix2(fe, *this, Trans, mixed_mass);

      DenseMatrixInverse pos_mass_inv(pos_mass);
      I.SetSize(Dof, fe.GetDof());
      pos_mass_inv.Mult(mixed_mass, I);
   }
}


void VectorFiniteElement::CalcShape (
   const IntegrationPoint &ip, Vector &shape ) const
{
   mfem_error ("Error: Cannot use scalar CalcShape(...) function with\n"
               "   VectorFiniteElements!");
}

void VectorFiniteElement::CalcDShape (
   const IntegrationPoint &ip, DenseMatrix &dshape ) const
{
   mfem_error ("Error: Cannot use scalar CalcDShape(...) function with\n"
               "   VectorFiniteElements!");
}

void VectorFiniteElement::SetDerivMembers()
{
   switch (MapType)
   {
      case H_DIV:
         DerivType = DIV;
         DerivRangeType = SCALAR;
         DerivMapType = INTEGRAL;
         break;
      case H_CURL:
         switch (Dim)
         {
            case 3: // curl: 3D H_CURL -> 3D H_DIV
               DerivType = CURL;
               DerivRangeType = VECTOR;
               DerivMapType = H_DIV;
               break;
            case 2:
               // curl: 2D H_CURL -> INTEGRAL
               DerivType = CURL;
               DerivRangeType = SCALAR;
               DerivMapType = INTEGRAL;
               break;
            case 1:
               DerivType = NONE;
               DerivRangeType = SCALAR;
               DerivMapType = INTEGRAL;
               break;
            default:
               MFEM_ABORT("Invalid dimension, Dim = " << Dim);
         }
         break;
      default:
         MFEM_ABORT("Invalid MapType = " << MapType);
   }
}

void VectorFiniteElement::CalcVShape_RT (
   ElementTransformation &Trans, DenseMatrix &shape) const
{
   MFEM_ASSERT(MapType == H_DIV, "");
#ifdef MFEM_THREAD_SAFE
   DenseMatrix vshape(Dof, Dim);
#endif
   CalcVShape(Trans.GetIntPoint(), vshape);
   MultABt(vshape, Trans.Jacobian(), shape);
   shape *= (1.0 / Trans.Weight());
}

void VectorFiniteElement::CalcVShape_ND (
   ElementTransformation &Trans, DenseMatrix &shape) const
{
   MFEM_ASSERT(MapType == H_CURL, "");
#ifdef MFEM_THREAD_SAFE
   DenseMatrix vshape(Dof, Dim);
#endif
   CalcVShape(Trans.GetIntPoint(), vshape);
   Mult(vshape, Trans.InverseJacobian(), shape);
}

void VectorFiniteElement::Project_RT(
   const double *nk, const Array<int> &d2n,
   VectorCoefficient &vc, ElementTransformation &Trans, Vector &dofs) const
{
   double vk[Geometry::MaxDim];
   const int sdim = Trans.GetSpaceDim();
   MFEM_ASSERT(vc.GetVDim() == sdim, "");
   Vector xk(vk, sdim);
   const bool square_J = (Dim == sdim);

   for (int k = 0; k < Dof; k++)
   {
      Trans.SetIntPoint(&Nodes.IntPoint(k));
      vc.Eval(xk, Trans, Nodes.IntPoint(k));
      // dof_k = nk^t adj(J) xk
      dofs(k) = Trans.AdjugateJacobian().InnerProduct(vk, nk + d2n[k]*Dim);
      if (!square_J) { dofs(k) /= Trans.Weight(); }
   }
}

void VectorFiniteElement::ProjectMatrixCoefficient_RT(
   const double *nk, const Array<int> &d2n,
   MatrixCoefficient &mc, ElementTransformation &T, Vector &dofs) const
{
   // project the rows of the matrix coefficient in an RT space

   const int sdim = T.GetSpaceDim();
   MFEM_ASSERT(mc.GetWidth() == sdim, "");
   const bool square_J = (Dim == sdim);
   DenseMatrix MQ(mc.GetHeight(), mc.GetWidth());
   Vector nk_phys(sdim), dofs_k(MQ.Height());
   MFEM_ASSERT(dofs.Size() == Dof*MQ.Height(), "");

   for (int k = 0; k < Dof; k++)
   {
      T.SetIntPoint(&Nodes.IntPoint(k));
      mc.Eval(MQ, T, Nodes.IntPoint(k));
      // nk_phys = adj(J)^t nk
      T.AdjugateJacobian().MultTranspose(nk + d2n[k]*Dim, nk_phys);
      if (!square_J) { nk_phys /= T.Weight(); }
      MQ.Mult(nk_phys, dofs_k);
      for (int r = 0; r < MQ.Height(); r++)
      {
         dofs(k+Dof*r) = dofs_k(r);
      }
   }
}

void VectorFiniteElement::Project_RT(
   const double *nk, const Array<int> &d2n, const FiniteElement &fe,
   ElementTransformation &Trans, DenseMatrix &I) const
{
   if (fe.GetRangeType() == SCALAR)
   {
      double vk[Geometry::MaxDim];
      Vector shape(fe.GetDof());
      int sdim = Trans.GetSpaceDim();

      I.SetSize(Dof, sdim*fe.GetDof());
      for (int k = 0; k < Dof; k++)
      {
         const IntegrationPoint &ip = Nodes.IntPoint(k);

         fe.CalcShape(ip, shape);
         Trans.SetIntPoint(&ip);
         Trans.AdjugateJacobian().MultTranspose(nk + d2n[k]*Dim, vk);
         if (fe.GetMapType() == INTEGRAL)
         {
            double w = 1.0/Trans.Weight();
            for (int d = 0; d < Dim; d++)
            {
               vk[d] *= w;
            }
         }

         for (int j = 0; j < shape.Size(); j++)
         {
            double s = shape(j);
            if (fabs(s) < 1e-12)
            {
               s = 0.0;
            }
            for (int d = 0; d < sdim; d++)
            {
               I(k,j+d*shape.Size()) = s*vk[d];
            }
         }
      }
   }
   else
   {
      mfem_error("VectorFiniteElement::Project_RT (fe version)");
   }
}

void VectorFiniteElement::ProjectGrad_RT(
   const double *nk, const Array<int> &d2n, const FiniteElement &fe,
   ElementTransformation &Trans, DenseMatrix &grad) const
{
   if (Dim != 2)
   {
      mfem_error("VectorFiniteElement::ProjectGrad_RT works only in 2D!");
   }

   DenseMatrix dshape(fe.GetDof(), fe.GetDim());
   Vector grad_k(fe.GetDof());
   double tk[2];

   grad.SetSize(Dof, fe.GetDof());
   for (int k = 0; k < Dof; k++)
   {
      fe.CalcDShape(Nodes.IntPoint(k), dshape);
      tk[0] = nk[d2n[k]*Dim+1];
      tk[1] = -nk[d2n[k]*Dim];
      dshape.Mult(tk, grad_k);
      for (int j = 0; j < grad_k.Size(); j++)
      {
         grad(k,j) = (fabs(grad_k(j)) < 1e-12) ? 0.0 : grad_k(j);
      }
   }
}

void VectorFiniteElement::ProjectCurl_ND(
   const double *tk, const Array<int> &d2t, const FiniteElement &fe,
   ElementTransformation &Trans, DenseMatrix &curl) const
{
#ifdef MFEM_THREAD_SAFE
   DenseMatrix curlshape(fe.GetDof(), Dim);
   DenseMatrix curlshape_J(fe.GetDof(), Dim);
   DenseMatrix J(Dim, Dim);
#else
   curlshape.SetSize(fe.GetDof(), Dim);
   curlshape_J.SetSize(fe.GetDof(), Dim);
   J.SetSize(Dim, Dim);
#endif

   Vector curl_k(fe.GetDof());

   curl.SetSize(Dof, fe.GetDof());
   for (int k = 0; k < Dof; k++)
   {
      const IntegrationPoint &ip = Nodes.IntPoint(k);

      // calculate J^t * J / |J|
      Trans.SetIntPoint(&ip);
      MultAtB(Trans.Jacobian(), Trans.Jacobian(), J);
      J *= 1.0 / Trans.Weight();

      // transform curl of shapes (rows) by J^t * J / |J|
      fe.CalcCurlShape(ip, curlshape);
      Mult(curlshape, J, curlshape_J);

      curlshape_J.Mult(tk + d2t[k]*Dim, curl_k);
      for (int j = 0; j < curl_k.Size(); j++)
      {
         curl(k,j) = (fabs(curl_k(j)) < 1e-12) ? 0.0 : curl_k(j);
      }
   }
}

void VectorFiniteElement::ProjectCurl_RT(
   const double *nk, const Array<int> &d2n, const FiniteElement &fe,
   ElementTransformation &Trans, DenseMatrix &curl) const
{
   DenseMatrix curl_shape(fe.GetDof(), Dim);
   Vector curl_k(fe.GetDof());

   curl.SetSize(Dof, fe.GetDof());
   for (int k = 0; k < Dof; k++)
   {
      fe.CalcCurlShape(Nodes.IntPoint(k), curl_shape);
      curl_shape.Mult(nk + d2n[k]*Dim, curl_k);
      for (int j = 0; j < curl_k.Size(); j++)
      {
         curl(k,j) = (fabs(curl_k(j)) < 1e-12) ? 0.0 : curl_k(j);
      }
   }
}

void VectorFiniteElement::Project_ND(
   const double *tk, const Array<int> &d2t,
   VectorCoefficient &vc, ElementTransformation &Trans, Vector &dofs) const
{
   double vk[Geometry::MaxDim];
   Vector xk(vk, vc.GetVDim());

   for (int k = 0; k < Dof; k++)
   {
      Trans.SetIntPoint(&Nodes.IntPoint(k));

      vc.Eval(xk, Trans, Nodes.IntPoint(k));
      // dof_k = xk^t J tk
      dofs(k) = Trans.Jacobian().InnerProduct(tk + d2t[k]*Dim, vk);
   }
}

void VectorFiniteElement::ProjectMatrixCoefficient_ND(
   const double *tk, const Array<int> &d2t,
   MatrixCoefficient &mc, ElementTransformation &T, Vector &dofs) const
{
   // project the rows of the matrix coefficient in an ND space

   const int sdim = T.GetSpaceDim();
   MFEM_ASSERT(mc.GetWidth() == sdim, "");
   DenseMatrix MQ(mc.GetHeight(), mc.GetWidth());
   Vector tk_phys(sdim), dofs_k(MQ.Height());
   MFEM_ASSERT(dofs.Size() == Dof*MQ.Height(), "");

   for (int k = 0; k < Dof; k++)
   {
      T.SetIntPoint(&Nodes.IntPoint(k));
      mc.Eval(MQ, T, Nodes.IntPoint(k));
      // tk_phys = J tk
      T.Jacobian().Mult(tk + d2t[k]*Dim, tk_phys);
      MQ.Mult(tk_phys, dofs_k);
      for (int r = 0; r < MQ.Height(); r++)
      {
         dofs(k+Dof*r) = dofs_k(r);
      }
   }
}

void VectorFiniteElement::Project_ND(
   const double *tk, const Array<int> &d2t, const FiniteElement &fe,
   ElementTransformation &Trans, DenseMatrix &I) const
{
   if (fe.GetRangeType() == SCALAR)
   {
      int sdim = Trans.GetSpaceDim();
      double vk[Geometry::MaxDim];
      Vector shape(fe.GetDof());

      I.SetSize(Dof, sdim*fe.GetDof());
      for (int k = 0; k < Dof; k++)
      {
         const IntegrationPoint &ip = Nodes.IntPoint(k);

         fe.CalcShape(ip, shape);
         Trans.SetIntPoint(&ip);
         Trans.Jacobian().Mult(tk + d2t[k]*Dim, vk);
         if (fe.GetMapType() == INTEGRAL)
         {
            double w = 1.0/Trans.Weight();
            for (int d = 0; d < sdim; d++)
            {
               vk[d] *= w;
            }
         }

         for (int j = 0; j < shape.Size(); j++)
         {
            double s = shape(j);
            if (fabs(s) < 1e-12)
            {
               s = 0.0;
            }
            for (int d = 0; d < sdim; d++)
            {
               I(k, j + d*shape.Size()) = s*vk[d];
            }
         }
      }
   }
   else
   {
      mfem_error("VectorFiniteElement::Project_ND (fe version)");
   }
}

void VectorFiniteElement::ProjectGrad_ND(
   const double *tk, const Array<int> &d2t, const FiniteElement &fe,
   ElementTransformation &Trans, DenseMatrix &grad) const
{
   MFEM_ASSERT(fe.GetMapType() == VALUE, "");

   DenseMatrix dshape(fe.GetDof(), fe.GetDim());
   Vector grad_k(fe.GetDof());

   grad.SetSize(Dof, fe.GetDof());
   for (int k = 0; k < Dof; k++)
   {
      fe.CalcDShape(Nodes.IntPoint(k), dshape);
      dshape.Mult(tk + d2t[k]*Dim, grad_k);
      for (int j = 0; j < grad_k.Size(); j++)
      {
         grad(k,j) = (fabs(grad_k(j)) < 1e-12) ? 0.0 : grad_k(j);
      }
   }
}

void VectorFiniteElement::LocalInterpolation_RT(
   const VectorFiniteElement &cfe, const double *nk, const Array<int> &d2n,
   ElementTransformation &Trans, DenseMatrix &I) const
{
   MFEM_ASSERT(MapType == cfe.GetMapType(), "");

   double vk[Geometry::MaxDim];
   Vector xk(vk, Dim);
   IntegrationPoint ip;
#ifdef MFEM_THREAD_SAFE
   DenseMatrix vshape(cfe.GetDof(), cfe.GetDim());
#else
   DenseMatrix vshape(cfe.vshape.Data(), cfe.GetDof(), cfe.GetDim());
#endif
   I.SetSize(Dof, vshape.Height());

   // assuming Trans is linear; this should be ok for all refinement types
   Trans.SetIntPoint(&Geometries.GetCenter(GeomType));
   const DenseMatrix &adjJ = Trans.AdjugateJacobian();
   for (int k = 0; k < Dof; k++)
   {
      Trans.Transform(Nodes.IntPoint(k), xk);
      ip.Set3(vk);
      cfe.CalcVShape(ip, vshape);
      // xk = |J| J^{-t} n_k
      adjJ.MultTranspose(nk + d2n[k]*Dim, vk);
      // I_k = vshape_k.adj(J)^t.n_k, k=1,...,Dof
      for (int j = 0; j < vshape.Height(); j++)
      {
         double Ikj = 0.;
         for (int i = 0; i < Dim; i++)
         {
            Ikj += vshape(j, i) * vk[i];
         }
         I(k, j) = (fabs(Ikj) < 1e-12) ? 0.0 : Ikj;
      }
   }
}

void VectorFiniteElement::LocalInterpolation_ND(
   const VectorFiniteElement &cfe, const double *tk, const Array<int> &d2t,
   ElementTransformation &Trans, DenseMatrix &I) const
{
   double vk[Geometry::MaxDim];
   Vector xk(vk, Dim);
   IntegrationPoint ip;
#ifdef MFEM_THREAD_SAFE
   DenseMatrix vshape(cfe.GetDof(), cfe.GetDim());
#else
   DenseMatrix vshape(cfe.vshape.Data(), cfe.GetDof(), cfe.GetDim());
#endif
   I.SetSize(Dof, vshape.Height());

   // assuming Trans is linear; this should be ok for all refinement types
   Trans.SetIntPoint(&Geometries.GetCenter(GeomType));
   const DenseMatrix &J = Trans.Jacobian();
   for (int k = 0; k < Dof; k++)
   {
      Trans.Transform(Nodes.IntPoint(k), xk);
      ip.Set3(vk);
      cfe.CalcVShape(ip, vshape);
      // xk = J t_k
      J.Mult(tk + d2t[k]*Dim, vk);
      // I_k = vshape_k.J.t_k, k=1,...,Dof
      for (int j = 0; j < vshape.Height(); j++)
      {
         double Ikj = 0.;
         for (int i = 0; i < Dim; i++)
         {
            Ikj += vshape(j, i) * vk[i];
         }
         I(k, j) = (fabs(Ikj) < 1e-12) ? 0.0 : Ikj;
      }
   }
}

void VectorFiniteElement::LocalRestriction_RT(
   const double *nk, const Array<int> &d2n, ElementTransformation &Trans,
   DenseMatrix &R) const
{
   double pt_data[Geometry::MaxDim];
   IntegrationPoint ip;
   Vector pt(pt_data, Dim);

#ifdef MFEM_THREAD_SAFE
   DenseMatrix vshape(Dof, Dim);
#endif

   Trans.SetIntPoint(&Geometries.GetCenter(GeomType));
   const DenseMatrix &J = Trans.Jacobian();
   const double weight = Trans.Weight();
   for (int j = 0; j < Dof; j++)
   {
      InvertLinearTrans(Trans, Nodes.IntPoint(j), pt);
      ip.Set(pt_data, Dim);
      if (Geometries.CheckPoint(GeomType, ip)) // do we need an epsilon here?
      {
         CalcVShape(ip, vshape);
         J.MultTranspose(nk+Dim*d2n[j], pt_data);
         pt /= weight;
         for (int k = 0; k < Dof; k++)
         {
            double R_jk = 0.0;
            for (int d = 0; d < Dim; d++)
            {
               R_jk += vshape(k,d)*pt_data[d];
            }
            R(j,k) = R_jk;
         }
      }
      else
      {
         // Set the whole row to avoid valgrind warnings in R.Threshold().
         R.SetRow(j, infinity());
      }
   }
   R.Threshold(1e-12);
}

void VectorFiniteElement::LocalRestriction_ND(
   const double *tk, const Array<int> &d2t, ElementTransformation &Trans,
   DenseMatrix &R) const
{
   double pt_data[Geometry::MaxDim];
   IntegrationPoint ip;
   Vector pt(pt_data, Dim);

#ifdef MFEM_THREAD_SAFE
   DenseMatrix vshape(Dof, Dim);
#endif

   Trans.SetIntPoint(&Geometries.GetCenter(GeomType));
   const DenseMatrix &Jinv = Trans.InverseJacobian();
   for (int j = 0; j < Dof; j++)
   {
      InvertLinearTrans(Trans, Nodes.IntPoint(j), pt);
      ip.Set(pt_data, Dim);
      if (Geometries.CheckPoint(GeomType, ip)) // do we need an epsilon here?
      {
         CalcVShape(ip, vshape);
         Jinv.Mult(tk+Dim*d2t[j], pt_data);
         for (int k = 0; k < Dof; k++)
         {
            double R_jk = 0.0;
            for (int d = 0; d < Dim; d++)
            {
               R_jk += vshape(k,d)*pt_data[d];
            }
            R(j,k) = R_jk;
         }
      }
      else
      {
         // Set the whole row to avoid valgrind warnings in R.Threshold().
         R.SetRow(j, infinity());
      }
   }
   R.Threshold(1e-12);
}


PointFiniteElement::PointFiniteElement()
   : NodalFiniteElement(0, Geometry::POINT, 1, 0)
{
   Nodes.IntPoint(0).x = 0.0;
}

void PointFiniteElement::CalcShape(const IntegrationPoint &ip,
                                   Vector &shape) const
{
   shape(0) = 1.;
}

void PointFiniteElement::CalcDShape(const IntegrationPoint &ip,
                                    DenseMatrix &dshape) const
{
   // dshape is (1 x 0) - nothing to compute
}

Linear1DFiniteElement::Linear1DFiniteElement()
   : NodalFiniteElement(1, Geometry::SEGMENT, 2, 1)
{
   Nodes.IntPoint(0).x = 0.0;
   Nodes.IntPoint(1).x = 1.0;
}

void Linear1DFiniteElement::CalcShape(const IntegrationPoint &ip,
                                      Vector &shape) const
{
   shape(0) = 1. - ip.x;
   shape(1) = ip.x;
}

void Linear1DFiniteElement::CalcDShape(const IntegrationPoint &ip,
                                       DenseMatrix &dshape) const
{
   dshape(0,0) = -1.;
   dshape(1,0) =  1.;
}

Linear2DFiniteElement::Linear2DFiniteElement()
   : NodalFiniteElement(2, Geometry::TRIANGLE, 3, 1)
{
   Nodes.IntPoint(0).x = 0.0;
   Nodes.IntPoint(0).y = 0.0;
   Nodes.IntPoint(1).x = 1.0;
   Nodes.IntPoint(1).y = 0.0;
   Nodes.IntPoint(2).x = 0.0;
   Nodes.IntPoint(2).y = 1.0;
}

void Linear2DFiniteElement::CalcShape(const IntegrationPoint &ip,
                                      Vector &shape) const
{
   shape(0) = 1. - ip.x - ip.y;
   shape(1) = ip.x;
   shape(2) = ip.y;
}

void Linear2DFiniteElement::CalcDShape(const IntegrationPoint &ip,
                                       DenseMatrix &dshape) const
{
   dshape(0,0) = -1.; dshape(0,1) = -1.;
   dshape(1,0) =  1.; dshape(1,1) =  0.;
   dshape(2,0) =  0.; dshape(2,1) =  1.;
}

BiLinear2DFiniteElement::BiLinear2DFiniteElement()
   : NodalFiniteElement(2, Geometry::SQUARE, 4, 1, FunctionSpace::Qk)
{
   Nodes.IntPoint(0).x = 0.0;
   Nodes.IntPoint(0).y = 0.0;
   Nodes.IntPoint(1).x = 1.0;
   Nodes.IntPoint(1).y = 0.0;
   Nodes.IntPoint(2).x = 1.0;
   Nodes.IntPoint(2).y = 1.0;
   Nodes.IntPoint(3).x = 0.0;
   Nodes.IntPoint(3).y = 1.0;
}

void BiLinear2DFiniteElement::CalcShape(const IntegrationPoint &ip,
                                        Vector &shape) const
{
   shape(0) = (1. - ip.x) * (1. - ip.y) ;
   shape(1) = ip.x * (1. - ip.y) ;
   shape(2) = ip.x * ip.y ;
   shape(3) = (1. - ip.x) * ip.y ;
}

void BiLinear2DFiniteElement::CalcDShape(const IntegrationPoint &ip,
                                         DenseMatrix &dshape) const
{
   dshape(0,0) = -1. + ip.y; dshape(0,1) = -1. + ip.x ;
   dshape(1,0) =  1. - ip.y; dshape(1,1) = -ip.x ;
   dshape(2,0) =  ip.y ;     dshape(2,1) = ip.x ;
   dshape(3,0) = -ip.y ;     dshape(3,1) = 1. - ip.x ;
}

void BiLinear2DFiniteElement::CalcHessian(
   const IntegrationPoint &ip, DenseMatrix &h) const
{
   h(0,0) = 0.;   h(0,1) =  1.;   h(0,2) = 0.;
   h(1,0) = 0.;   h(1,1) = -1.;   h(1,2) = 0.;
   h(2,0) = 0.;   h(2,1) =  1.;   h(2,2) = 0.;
   h(3,0) = 0.;   h(3,1) = -1.;   h(3,2) = 0.;
}


GaussLinear2DFiniteElement::GaussLinear2DFiniteElement()
   : NodalFiniteElement(2, Geometry::TRIANGLE, 3, 1, FunctionSpace::Pk)
{
   Nodes.IntPoint(0).x = 1./6.;
   Nodes.IntPoint(0).y = 1./6.;
   Nodes.IntPoint(1).x = 2./3.;
   Nodes.IntPoint(1).y = 1./6.;
   Nodes.IntPoint(2).x = 1./6.;
   Nodes.IntPoint(2).y = 2./3.;
}

void GaussLinear2DFiniteElement::CalcShape(const IntegrationPoint &ip,
                                           Vector &shape) const
{
   const double x = ip.x, y = ip.y;

   shape(0) = 5./3. - 2. * (x + y);
   shape(1) = 2. * (x - 1./6.);
   shape(2) = 2. * (y - 1./6.);
}

void GaussLinear2DFiniteElement::CalcDShape(const IntegrationPoint &ip,
                                            DenseMatrix &dshape) const
{
   dshape(0,0) = -2.;  dshape(0,1) = -2.;
   dshape(1,0) =  2.;  dshape(1,1) =  0.;
   dshape(2,0) =  0.;  dshape(2,1) =  2.;
}

void GaussLinear2DFiniteElement::ProjectDelta(int vertex, Vector &dofs) const
{
   dofs(vertex)       = 2./3.;
   dofs((vertex+1)%3) = 1./6.;
   dofs((vertex+2)%3) = 1./6.;
}


// 0.5-0.5/sqrt(3) and 0.5+0.5/sqrt(3)
const double GaussBiLinear2DFiniteElement::p[] =
{ 0.2113248654051871177454256, 0.7886751345948128822545744 };

GaussBiLinear2DFiniteElement::GaussBiLinear2DFiniteElement()
   : NodalFiniteElement(2, Geometry::SQUARE, 4, 1, FunctionSpace::Qk)
{
   Nodes.IntPoint(0).x = p[0];
   Nodes.IntPoint(0).y = p[0];
   Nodes.IntPoint(1).x = p[1];
   Nodes.IntPoint(1).y = p[0];
   Nodes.IntPoint(2).x = p[1];
   Nodes.IntPoint(2).y = p[1];
   Nodes.IntPoint(3).x = p[0];
   Nodes.IntPoint(3).y = p[1];
}

void GaussBiLinear2DFiniteElement::CalcShape(const IntegrationPoint &ip,
                                             Vector &shape) const
{
   const double x = ip.x, y = ip.y;

   shape(0) = 3. * (p[1] - x) * (p[1] - y);
   shape(1) = 3. * (x - p[0]) * (p[1] - y);
   shape(2) = 3. * (x - p[0]) * (y - p[0]);
   shape(3) = 3. * (p[1] - x) * (y - p[0]);
}

void GaussBiLinear2DFiniteElement::CalcDShape(const IntegrationPoint &ip,
                                              DenseMatrix &dshape) const
{
   const double x = ip.x, y = ip.y;

   dshape(0,0) = 3. * (y - p[1]);  dshape(0,1) = 3. * (x - p[1]);
   dshape(1,0) = 3. * (p[1] - y);  dshape(1,1) = 3. * (p[0] - x);
   dshape(2,0) = 3. * (y - p[0]);  dshape(2,1) = 3. * (x - p[0]);
   dshape(3,0) = 3. * (p[0] - y);  dshape(3,1) = 3. * (p[1] - x);
}

void GaussBiLinear2DFiniteElement::ProjectDelta(int vertex, Vector &dofs) const
{
#if 1
   dofs(vertex)       = p[1]*p[1];
   dofs((vertex+1)%4) = p[0]*p[1];
   dofs((vertex+2)%4) = p[0]*p[0];
   dofs((vertex+3)%4) = p[0]*p[1];
#else
   dofs = 1.0;
#endif
}


P1OnQuadFiniteElement::P1OnQuadFiniteElement()
   : NodalFiniteElement(2, Geometry::SQUARE, 3, 1, FunctionSpace::Qk)
{
   Nodes.IntPoint(0).x = 0.0;
   Nodes.IntPoint(0).y = 0.0;
   Nodes.IntPoint(1).x = 1.0;
   Nodes.IntPoint(1).y = 0.0;
   Nodes.IntPoint(2).x = 0.0;
   Nodes.IntPoint(2).y = 1.0;
}

void P1OnQuadFiniteElement::CalcShape(const IntegrationPoint &ip,
                                      Vector &shape) const
{
   shape(0) = 1. - ip.x - ip.y;
   shape(1) = ip.x;
   shape(2) = ip.y;
}

void P1OnQuadFiniteElement::CalcDShape(const IntegrationPoint &ip,
                                       DenseMatrix &dshape) const
{
   dshape(0,0) = -1.; dshape(0,1) = -1.;
   dshape(1,0) =  1.; dshape(1,1) =  0.;
   dshape(2,0) =  0.; dshape(2,1) =  1.;
}


Quad1DFiniteElement::Quad1DFiniteElement()
   : NodalFiniteElement(1, Geometry::SEGMENT, 3, 2)
{
   Nodes.IntPoint(0).x = 0.0;
   Nodes.IntPoint(1).x = 1.0;
   Nodes.IntPoint(2).x = 0.5;
}

void Quad1DFiniteElement::CalcShape(const IntegrationPoint &ip,
                                    Vector &shape) const
{
   double x = ip.x;
   double l1 = 1.0 - x, l2 = x, l3 = 2. * x - 1.;

   shape(0) = l1 * (-l3);
   shape(1) = l2 * l3;
   shape(2) = 4. * l1 * l2;
}

void Quad1DFiniteElement::CalcDShape(const IntegrationPoint &ip,
                                     DenseMatrix &dshape) const
{
   double x = ip.x;

   dshape(0,0) = 4. * x - 3.;
   dshape(1,0) = 4. * x - 1.;
   dshape(2,0) = 4. - 8. * x;
}


QuadPos1DFiniteElement::QuadPos1DFiniteElement()
   : PositiveFiniteElement(1, Geometry::SEGMENT, 3, 2)
{
   Nodes.IntPoint(0).x = 0.0;
   Nodes.IntPoint(1).x = 1.0;
   Nodes.IntPoint(2).x = 0.5;
}

void QuadPos1DFiniteElement::CalcShape(const IntegrationPoint &ip,
                                       Vector &shape) const
{
   const double x = ip.x, x1 = 1. - x;

   shape(0) = x1 * x1;
   shape(1) = x * x;
   shape(2) = 2. * x * x1;
}

void QuadPos1DFiniteElement::CalcDShape(const IntegrationPoint &ip,
                                        DenseMatrix &dshape) const
{
   const double x = ip.x;

   dshape(0,0) = 2. * x - 2.;
   dshape(1,0) = 2. * x;
   dshape(2,0) = 2. - 4. * x;
}

Quad2DFiniteElement::Quad2DFiniteElement()
   : NodalFiniteElement(2, Geometry::TRIANGLE, 6, 2)
{
   Nodes.IntPoint(0).x = 0.0;
   Nodes.IntPoint(0).y = 0.0;
   Nodes.IntPoint(1).x = 1.0;
   Nodes.IntPoint(1).y = 0.0;
   Nodes.IntPoint(2).x = 0.0;
   Nodes.IntPoint(2).y = 1.0;
   Nodes.IntPoint(3).x = 0.5;
   Nodes.IntPoint(3).y = 0.0;
   Nodes.IntPoint(4).x = 0.5;
   Nodes.IntPoint(4).y = 0.5;
   Nodes.IntPoint(5).x = 0.0;
   Nodes.IntPoint(5).y = 0.5;
}

void Quad2DFiniteElement::CalcShape(const IntegrationPoint &ip,
                                    Vector &shape) const
{
   double x = ip.x, y = ip.y;
   double l1 = 1.-x-y, l2 = x, l3 = y;

   shape(0) = l1 * (2. * l1 - 1.);
   shape(1) = l2 * (2. * l2 - 1.);
   shape(2) = l3 * (2. * l3 - 1.);
   shape(3) = 4. * l1 * l2;
   shape(4) = 4. * l2 * l3;
   shape(5) = 4. * l3 * l1;
}

void Quad2DFiniteElement::CalcDShape(const IntegrationPoint &ip,
                                     DenseMatrix &dshape) const
{
   double x = ip.x, y = ip.y;

   dshape(0,0) =
      dshape(0,1) = 4. * (x + y) - 3.;

   dshape(1,0) = 4. * x - 1.;
   dshape(1,1) = 0.;

   dshape(2,0) = 0.;
   dshape(2,1) = 4. * y - 1.;

   dshape(3,0) = -4. * (2. * x + y - 1.);
   dshape(3,1) = -4. * x;

   dshape(4,0) = 4. * y;
   dshape(4,1) = 4. * x;

   dshape(5,0) = -4. * y;
   dshape(5,1) = -4. * (x + 2. * y - 1.);
}

void Quad2DFiniteElement::CalcHessian (const IntegrationPoint &ip,
                                       DenseMatrix &h) const
{
   h(0,0) = 4.;
   h(0,1) = 4.;
   h(0,2) = 4.;

   h(1,0) = 4.;
   h(1,1) = 0.;
   h(1,2) = 0.;

   h(2,0) = 0.;
   h(2,1) = 0.;
   h(2,2) = 4.;

   h(3,0) = -8.;
   h(3,1) = -4.;
   h(3,2) =  0.;

   h(4,0) = 0.;
   h(4,1) = 4.;
   h(4,2) = 0.;

   h(5,0) =  0.;
   h(5,1) = -4.;
   h(5,2) = -8.;
}

void Quad2DFiniteElement::ProjectDelta(int vertex, Vector &dofs) const
{
#if 0
   dofs = 1.;
#else
   dofs = 0.;
   dofs(vertex) = 1.;
   switch (vertex)
   {
      case 0: dofs(3) = 0.25; dofs(5) = 0.25; break;
      case 1: dofs(3) = 0.25; dofs(4) = 0.25; break;
      case 2: dofs(4) = 0.25; dofs(5) = 0.25; break;
   }
#endif
}


const double GaussQuad2DFiniteElement::p[] =
{ 0.0915762135097707434595714634022015, 0.445948490915964886318329253883051 };

GaussQuad2DFiniteElement::GaussQuad2DFiniteElement()
   : NodalFiniteElement(2, Geometry::TRIANGLE, 6, 2), A(6), D(6,2), pol(6)
{
   Nodes.IntPoint(0).x = p[0];
   Nodes.IntPoint(0).y = p[0];
   Nodes.IntPoint(1).x = 1. - 2. * p[0];
   Nodes.IntPoint(1).y = p[0];
   Nodes.IntPoint(2).x = p[0];
   Nodes.IntPoint(2).y = 1. - 2. * p[0];
   Nodes.IntPoint(3).x = p[1];
   Nodes.IntPoint(3).y = p[1];
   Nodes.IntPoint(4).x = 1. - 2. * p[1];
   Nodes.IntPoint(4).y = p[1];
   Nodes.IntPoint(5).x = p[1];
   Nodes.IntPoint(5).y = 1. - 2. * p[1];

   for (int i = 0; i < 6; i++)
   {
      const double x = Nodes.IntPoint(i).x, y = Nodes.IntPoint(i).y;
      A(0,i) = 1.;
      A(1,i) = x;
      A(2,i) = y;
      A(3,i) = x * x;
      A(4,i) = x * y;
      A(5,i) = y * y;
   }

   A.Invert();
}

void GaussQuad2DFiniteElement::CalcShape(const IntegrationPoint &ip,
                                         Vector &shape) const
{
   const double x = ip.x, y = ip.y;
   pol(0) = 1.;
   pol(1) = x;
   pol(2) = y;
   pol(3) = x * x;
   pol(4) = x * y;
   pol(5) = y * y;

   A.Mult(pol, shape);
}

void GaussQuad2DFiniteElement::CalcDShape(const IntegrationPoint &ip,
                                          DenseMatrix &dshape) const
{
   const double x = ip.x, y = ip.y;
   D(0,0) = 0.;      D(0,1) = 0.;
   D(1,0) = 1.;      D(1,1) = 0.;
   D(2,0) = 0.;      D(2,1) = 1.;
   D(3,0) = 2. *  x; D(3,1) = 0.;
   D(4,0) = y;       D(4,1) = x;
   D(5,0) = 0.;      D(5,1) = 2. * y;

   Mult(A, D, dshape);
}


BiQuad2DFiniteElement::BiQuad2DFiniteElement()
   : NodalFiniteElement(2, Geometry::SQUARE, 9, 2, FunctionSpace::Qk)
{
   Nodes.IntPoint(0).x = 0.0;
   Nodes.IntPoint(0).y = 0.0;
   Nodes.IntPoint(1).x = 1.0;
   Nodes.IntPoint(1).y = 0.0;
   Nodes.IntPoint(2).x = 1.0;
   Nodes.IntPoint(2).y = 1.0;
   Nodes.IntPoint(3).x = 0.0;
   Nodes.IntPoint(3).y = 1.0;
   Nodes.IntPoint(4).x = 0.5;
   Nodes.IntPoint(4).y = 0.0;
   Nodes.IntPoint(5).x = 1.0;
   Nodes.IntPoint(5).y = 0.5;
   Nodes.IntPoint(6).x = 0.5;
   Nodes.IntPoint(6).y = 1.0;
   Nodes.IntPoint(7).x = 0.0;
   Nodes.IntPoint(7).y = 0.5;
   Nodes.IntPoint(8).x = 0.5;
   Nodes.IntPoint(8).y = 0.5;
}

void BiQuad2DFiniteElement::CalcShape(const IntegrationPoint &ip,
                                      Vector &shape) const
{
   double x = ip.x, y = ip.y;
   double l1x, l2x, l3x, l1y, l2y, l3y;

   l1x = (x - 1.) * (2. * x - 1);
   l2x = 4. * x * (1. - x);
   l3x = x * (2. * x - 1.);
   l1y = (y - 1.) * (2. * y - 1);
   l2y = 4. * y * (1. - y);
   l3y = y * (2. * y - 1.);

   shape(0) = l1x * l1y;
   shape(4) = l2x * l1y;
   shape(1) = l3x * l1y;
   shape(7) = l1x * l2y;
   shape(8) = l2x * l2y;
   shape(5) = l3x * l2y;
   shape(3) = l1x * l3y;
   shape(6) = l2x * l3y;
   shape(2) = l3x * l3y;
}

void BiQuad2DFiniteElement::CalcDShape(const IntegrationPoint &ip,
                                       DenseMatrix &dshape) const
{
   double x = ip.x, y = ip.y;
   double l1x, l2x, l3x, l1y, l2y, l3y;
   double d1x, d2x, d3x, d1y, d2y, d3y;

   l1x = (x - 1.) * (2. * x - 1);
   l2x = 4. * x * (1. - x);
   l3x = x * (2. * x - 1.);
   l1y = (y - 1.) * (2. * y - 1);
   l2y = 4. * y * (1. - y);
   l3y = y * (2. * y - 1.);

   d1x = 4. * x - 3.;
   d2x = 4. - 8. * x;
   d3x = 4. * x - 1.;
   d1y = 4. * y - 3.;
   d2y = 4. - 8. * y;
   d3y = 4. * y - 1.;

   dshape(0,0) = d1x * l1y;
   dshape(0,1) = l1x * d1y;

   dshape(4,0) = d2x * l1y;
   dshape(4,1) = l2x * d1y;

   dshape(1,0) = d3x * l1y;
   dshape(1,1) = l3x * d1y;

   dshape(7,0) = d1x * l2y;
   dshape(7,1) = l1x * d2y;

   dshape(8,0) = d2x * l2y;
   dshape(8,1) = l2x * d2y;

   dshape(5,0) = d3x * l2y;
   dshape(5,1) = l3x * d2y;

   dshape(3,0) = d1x * l3y;
   dshape(3,1) = l1x * d3y;

   dshape(6,0) = d2x * l3y;
   dshape(6,1) = l2x * d3y;

   dshape(2,0) = d3x * l3y;
   dshape(2,1) = l3x * d3y;
}

void BiQuad2DFiniteElement::ProjectDelta(int vertex, Vector &dofs) const
{
#if 0
   dofs = 1.;
#else
   dofs = 0.;
   dofs(vertex) = 1.;
   switch (vertex)
   {
      case 0: dofs(4) = 0.25; dofs(7) = 0.25; break;
      case 1: dofs(4) = 0.25; dofs(5) = 0.25; break;
      case 2: dofs(5) = 0.25; dofs(6) = 0.25; break;
      case 3: dofs(6) = 0.25; dofs(7) = 0.25; break;
   }
   dofs(8) = 1./16.;
#endif
}


H1Ser_QuadrilateralElement::H1Ser_QuadrilateralElement(const int p)
   : ScalarFiniteElement(2, Geometry::SQUARE, (p*p + 3*p +6) / 2, p,
                         FunctionSpace::Qk)
{
   // Store the dof_map of the associated TensorBasisElement, which will be used
   // to create the serendipity dof map.  Its size is larger than the size of
   // the serendipity element.
   TensorBasisElement tbeTemp =
      TensorBasisElement(2, p, BasisType::GaussLobatto,
                         TensorBasisElement::DofMapType::Sr_DOF_MAP);
   const Array<int> tp_dof_map = tbeTemp.GetDofMap();

   const double *cp = poly1d.ClosedPoints(p, BasisType::GaussLobatto);

   // Fixing the Nodes is exactly the same as the H1_QuadrilateralElement
   // constructor except we only use those values of the associated tensor
   // product dof_map that are <= the number of serendipity Dofs e.g. only DoFs
   // 0-7 out of the 9 tensor product dofs (at quadratic order)
   int o = 0;

   for (int j = 0; j <= p; j++)
   {
      for (int i = 0; i <= p; i++)
      {
         if (tp_dof_map[o] < Nodes.Size())
         {
            Nodes.IntPoint(tp_dof_map[o]).x = cp[i];
            Nodes.IntPoint(tp_dof_map[o]).y = cp[j];
         }
         o++;
      }
   }
}

void H1Ser_QuadrilateralElement::CalcShape(const IntegrationPoint &ip,
                                           Vector &shape) const
{
   int p = (this)->GetOrder();
   double x = ip.x, y = ip.y;

   Poly_1D::Basis edgeNodalBasis(poly1d.GetBasis(p, BasisType::GaussLobatto));
   Vector nodalX(p+1);
   Vector nodalY(p+1);

   edgeNodalBasis.Eval(x, nodalX);
   edgeNodalBasis.Eval(y, nodalY);

   // First, fix edge-based shape functions. Use a nodal interpolant for edge
   // points, weighted by the linear function that vanishes on opposite edge.
   for (int i = 0; i < p-1; i++)
   {
      shape(4 + 0*(p-1) + i) = (nodalX(i+1))*(1.-y);         // south edge 0->1
      shape(4 + 1*(p-1) + i) = (nodalY(i+1))*x;              // east edge  1->2
      shape(4 + 3*(p-1) - i - 1) = (nodalX(i+1)) * y;        // north edge 3->2
      shape(4 + 4*(p-1) - i - 1) = (nodalY(i+1)) * (1. - x); // west edge  0->3
   }

   BiLinear2DFiniteElement bilinear = BiLinear2DFiniteElement();
   Vector bilinearsAtIP(4);
   bilinear.CalcShape(ip, bilinearsAtIP);

   const double *edgePts(poly1d.ClosedPoints(p, BasisType::GaussLobatto));

   // Next, set the shape function associated with vertex V, evaluated at (x,y)
   // to be: bilinear function associated to V, evaluated at (x,y) - sum (shape
   // function at edge point P, weighted by bilinear function for V evaluated at
   // P) where the sum is taken only for points P on edges incident to V.

   double vtx0fix =0;
   double vtx1fix =0;
   double vtx2fix =0;
   double vtx3fix =0;
   for (int i = 0; i<p-1; i++)
   {
      vtx0fix += (1-edgePts[i+1])*(shape(4 + i) +
                                   shape(4 + 4*(p-1) - i - 1)); // bot+left edge
      vtx1fix += (1-edgePts[i+1])*(shape(4 + 1*(p-1) + i) +
                                   shape(4 + (p-2)-i));        // right+bot edge
      vtx2fix += (1-edgePts[i+1])*(shape(4 + 2*(p-1) + i) +
                                   shape(1 + 2*p-i));          // top+right edge
      vtx3fix += (1-edgePts[i+1])*(shape(4 + 3*(p-1) + i) +
                                   shape(3*p - i));            // left+top edge
   }
   shape(0) = bilinearsAtIP(0) - vtx0fix;
   shape(1) = bilinearsAtIP(1) - vtx1fix;
   shape(2) = bilinearsAtIP(2) - vtx2fix;
   shape(3) = bilinearsAtIP(3) - vtx3fix;

   // Interior basis functions appear starting at order p=4. These are non-nodal
   // bubble functions.
   if (p > 3)
   {
      double *legX = new double[p-1];
      double *legY = new double[p-1];
      Poly_1D *storeLegendre = new Poly_1D();

      storeLegendre->CalcLegendre(p-2, x, legX);
      storeLegendre->CalcLegendre(p-2, y, legY);

      int interior_total = 0;
      for (int j = 4; j < p + 1; j++)
      {
         for (int k = 0; k < j-3; k++)
         {
            shape(4 + 4*(p-1) + interior_total)
               = legX[k] * legY[j-4-k] * x * (1. - x) * y * (1. - y);
            interior_total++;
         }
      }

      delete[] legX;
      delete[] legY;
      delete storeLegendre;
   }
}

void H1Ser_QuadrilateralElement::CalcDShape(const IntegrationPoint &ip,
                                            DenseMatrix &dshape) const
{
   int p = (this)->GetOrder();
   double x = ip.x, y = ip.y;

   Poly_1D::Basis edgeNodalBasis(poly1d.GetBasis(p, BasisType::GaussLobatto));
   Vector nodalX(p+1);
   Vector DnodalX(p+1);
   Vector nodalY(p+1);
   Vector DnodalY(p+1);

   edgeNodalBasis.Eval(x, nodalX, DnodalX);
   edgeNodalBasis.Eval(y, nodalY, DnodalY);

   for (int i = 0; i < p-1; i++)
   {
      dshape(4 + 0*(p-1) + i,0) =  DnodalX(i+1) * (1.-y);
      dshape(4 + 0*(p-1) + i,1) = -nodalX(i+1);
      dshape(4 + 1*(p-1) + i,0) =  nodalY(i+1);
      dshape(4 + 1*(p-1) + i,1) =  DnodalY(i+1)*x;
      dshape(4 + 3*(p-1) - i - 1,0) =  DnodalX(i+1)*y;
      dshape(4 + 3*(p-1) - i - 1,1) =  nodalX(i+1);
      dshape(4 + 4*(p-1) - i - 1,0) = -nodalY(i+1);
      dshape(4 + 4*(p-1) - i - 1,1) =  DnodalY(i+1) * (1.-x);
   }

   BiLinear2DFiniteElement bilinear = BiLinear2DFiniteElement();
   DenseMatrix DbilinearsAtIP(4);
   bilinear.CalcDShape(ip, DbilinearsAtIP);

   const double *edgePts(poly1d.ClosedPoints(p, BasisType::GaussLobatto));

   dshape(0,0) = DbilinearsAtIP(0,0);
   dshape(0,1) = DbilinearsAtIP(0,1);
   dshape(1,0) = DbilinearsAtIP(1,0);
   dshape(1,1) = DbilinearsAtIP(1,1);
   dshape(2,0) = DbilinearsAtIP(2,0);
   dshape(2,1) = DbilinearsAtIP(2,1);
   dshape(3,0) = DbilinearsAtIP(3,0);
   dshape(3,1) = DbilinearsAtIP(3,1);

   for (int i = 0; i<p-1; i++)
   {
      dshape(0,0) -= (1-edgePts[i+1])*(dshape(4 + 0*(p-1) + i, 0) +
                                       dshape(4 + 4*(p-1) - i - 1,0));
      dshape(0,1) -= (1-edgePts[i+1])*(dshape(4 + 0*(p-1) + i, 1) +
                                       dshape(4 + 4*(p-1) - i - 1,1));
      dshape(1,0) -= (1-edgePts[i+1])*(dshape(4 + 1*(p-1) + i, 0) +
                                       dshape(4 + (p-2)-i, 0));
      dshape(1,1) -= (1-edgePts[i+1])*(dshape(4 + 1*(p-1) + i, 1) +
                                       dshape(4 + (p-2)-i, 1));
      dshape(2,0) -= (1-edgePts[i+1])*(dshape(4 + 2*(p-1) + i, 0) +
                                       dshape(1 + 2*p-i, 0));
      dshape(2,1) -= (1-edgePts[i+1])*(dshape(4 + 2*(p-1) + i, 1) +
                                       dshape(1 + 2*p-i, 1));
      dshape(3,0) -= (1-edgePts[i+1])*(dshape(4 + 3*(p-1) + i, 0) +
                                       dshape(3*p - i, 0));
      dshape(3,1) -= (1-edgePts[i+1])*(dshape(4 + 3*(p-1) + i, 1) +
                                       dshape(3*p - i, 1));
   }

   if (p > 3)
   {
      double *legX = new double[p-1];
      double *legY = new double[p-1];
      double *DlegX = new double[p-1];
      double *DlegY = new double[p-1];
      Poly_1D *storeLegendre = new Poly_1D();

      storeLegendre->CalcLegendre(p-2, x, legX, DlegX);
      storeLegendre->CalcLegendre(p-2, y, legY, DlegY);

      int interior_total = 0;
      for (int j = 4; j < p + 1; j++)
      {
         for (int k = 0; k < j-3; k++)
         {
            dshape(4 + 4*(p-1) + interior_total, 0) =
               legY[j-4-k]*y*(1-y) * (DlegX[k]*x*(1-x) + legX[k]*(1-2*x));
            dshape(4 + 4*(p-1) + interior_total, 1) =
               legX[k]*x*(1-x) * (DlegY[j-4-k]*y*(1-y) + legY[j-4-k]*(1-2*y));
            interior_total++;
         }
      }
      delete[] legX;
      delete[] legY;
      delete[] DlegX;
      delete[] DlegY;
      delete storeLegendre;
   }
}

void H1Ser_QuadrilateralElement::GetLocalInterpolation(ElementTransformation
                                                       &Trans,
                                                       DenseMatrix &I) const
{
   // For p<=4, the basis is nodal; for p>4, the quad-interior functions are
   // non-nodal.
   if (Order <= 4)
   {
      NodalLocalInterpolation(Trans, I, *this);
   }
   else
   {
      ScalarLocalInterpolation(Trans, I, *this);
   }
}


BiQuadPos2DFiniteElement::BiQuadPos2DFiniteElement()
   : PositiveFiniteElement(2, Geometry::SQUARE, 9, 2, FunctionSpace::Qk)
{
   Nodes.IntPoint(0).x = 0.0;
   Nodes.IntPoint(0).y = 0.0;
   Nodes.IntPoint(1).x = 1.0;
   Nodes.IntPoint(1).y = 0.0;
   Nodes.IntPoint(2).x = 1.0;
   Nodes.IntPoint(2).y = 1.0;
   Nodes.IntPoint(3).x = 0.0;
   Nodes.IntPoint(3).y = 1.0;
   Nodes.IntPoint(4).x = 0.5;
   Nodes.IntPoint(4).y = 0.0;
   Nodes.IntPoint(5).x = 1.0;
   Nodes.IntPoint(5).y = 0.5;
   Nodes.IntPoint(6).x = 0.5;
   Nodes.IntPoint(6).y = 1.0;
   Nodes.IntPoint(7).x = 0.0;
   Nodes.IntPoint(7).y = 0.5;
   Nodes.IntPoint(8).x = 0.5;
   Nodes.IntPoint(8).y = 0.5;
}

void BiQuadPos2DFiniteElement::CalcShape(const IntegrationPoint &ip,
                                         Vector &shape) const
{
   double x = ip.x, y = ip.y;
   double l1x, l2x, l3x, l1y, l2y, l3y;

   l1x = (1. - x) * (1. - x);
   l2x = 2. * x * (1. - x);
   l3x = x * x;
   l1y = (1. - y) * (1. - y);
   l2y = 2. * y * (1. - y);
   l3y = y * y;

   shape(0) = l1x * l1y;
   shape(4) = l2x * l1y;
   shape(1) = l3x * l1y;
   shape(7) = l1x * l2y;
   shape(8) = l2x * l2y;
   shape(5) = l3x * l2y;
   shape(3) = l1x * l3y;
   shape(6) = l2x * l3y;
   shape(2) = l3x * l3y;
}

void BiQuadPos2DFiniteElement::CalcDShape(const IntegrationPoint &ip,
                                          DenseMatrix &dshape) const
{
   double x = ip.x, y = ip.y;
   double l1x, l2x, l3x, l1y, l2y, l3y;
   double d1x, d2x, d3x, d1y, d2y, d3y;

   l1x = (1. - x) * (1. - x);
   l2x = 2. * x * (1. - x);
   l3x = x * x;
   l1y = (1. - y) * (1. - y);
   l2y = 2. * y * (1. - y);
   l3y = y * y;

   d1x = 2. * x - 2.;
   d2x = 2. - 4. * x;
   d3x = 2. * x;
   d1y = 2. * y - 2.;
   d2y = 2. - 4. * y;
   d3y = 2. * y;

   dshape(0,0) = d1x * l1y;
   dshape(0,1) = l1x * d1y;

   dshape(4,0) = d2x * l1y;
   dshape(4,1) = l2x * d1y;

   dshape(1,0) = d3x * l1y;
   dshape(1,1) = l3x * d1y;

   dshape(7,0) = d1x * l2y;
   dshape(7,1) = l1x * d2y;

   dshape(8,0) = d2x * l2y;
   dshape(8,1) = l2x * d2y;

   dshape(5,0) = d3x * l2y;
   dshape(5,1) = l3x * d2y;

   dshape(3,0) = d1x * l3y;
   dshape(3,1) = l1x * d3y;

   dshape(6,0) = d2x * l3y;
   dshape(6,1) = l2x * d3y;

   dshape(2,0) = d3x * l3y;
   dshape(2,1) = l3x * d3y;
}

void BiQuadPos2DFiniteElement::GetLocalInterpolation(
   ElementTransformation &Trans, DenseMatrix &I) const
{
   double s[9];
   IntegrationPoint tr_ip;
   Vector xx(&tr_ip.x, 2), shape(s, 9);

   for (int i = 0; i < 9; i++)
   {
      Trans.Transform(Nodes.IntPoint(i), xx);
      CalcShape(tr_ip, shape);
      for (int j = 0; j < 9; j++)
         if (fabs(I(i,j) = s[j]) < 1.0e-12)
         {
            I(i,j) = 0.0;
         }
   }
   for (int i = 0; i < 9; i++)
   {
      double *d = &I(0,i);
      d[4] = 2. * d[4] - 0.5 * (d[0] + d[1]);
      d[5] = 2. * d[5] - 0.5 * (d[1] + d[2]);
      d[6] = 2. * d[6] - 0.5 * (d[2] + d[3]);
      d[7] = 2. * d[7] - 0.5 * (d[3] + d[0]);
      d[8] = 4. * d[8] - 0.5 * (d[4] + d[5] + d[6] + d[7]) -
             0.25 * (d[0] + d[1] + d[2] + d[3]);
   }
}

void BiQuadPos2DFiniteElement::Project(
   Coefficient &coeff, ElementTransformation &Trans, Vector &dofs) const
{
   double *d = dofs;

   for (int i = 0; i < 9; i++)
   {
      const IntegrationPoint &ip = Nodes.IntPoint(i);
      Trans.SetIntPoint(&ip);
      d[i] = coeff.Eval(Trans, ip);
   }
   d[4] = 2. * d[4] - 0.5 * (d[0] + d[1]);
   d[5] = 2. * d[5] - 0.5 * (d[1] + d[2]);
   d[6] = 2. * d[6] - 0.5 * (d[2] + d[3]);
   d[7] = 2. * d[7] - 0.5 * (d[3] + d[0]);
   d[8] = 4. * d[8] - 0.5 * (d[4] + d[5] + d[6] + d[7]) -
          0.25 * (d[0] + d[1] + d[2] + d[3]);
}

void BiQuadPos2DFiniteElement::Project (
   VectorCoefficient &vc, ElementTransformation &Trans,
   Vector &dofs) const
{
   double v[3];
   Vector x (v, vc.GetVDim());

   for (int i = 0; i < 9; i++)
   {
      const IntegrationPoint &ip = Nodes.IntPoint(i);
      Trans.SetIntPoint(&ip);
      vc.Eval (x, Trans, ip);
      for (int j = 0; j < x.Size(); j++)
      {
         dofs(9*j+i) = v[j];
      }
   }
   for (int j = 0; j < x.Size(); j++)
   {
      double *d = &dofs(9*j);

      d[4] = 2. * d[4] - 0.5 * (d[0] + d[1]);
      d[5] = 2. * d[5] - 0.5 * (d[1] + d[2]);
      d[6] = 2. * d[6] - 0.5 * (d[2] + d[3]);
      d[7] = 2. * d[7] - 0.5 * (d[3] + d[0]);
      d[8] = 4. * d[8] - 0.5 * (d[4] + d[5] + d[6] + d[7]) -
             0.25 * (d[0] + d[1] + d[2] + d[3]);
   }
}


GaussBiQuad2DFiniteElement::GaussBiQuad2DFiniteElement()
   : NodalFiniteElement(2, Geometry::SQUARE, 9, 2, FunctionSpace::Qk)
{
   const double p1 = 0.5*(1.-sqrt(3./5.));

   Nodes.IntPoint(0).x = p1;
   Nodes.IntPoint(0).y = p1;
   Nodes.IntPoint(4).x = 0.5;
   Nodes.IntPoint(4).y = p1;
   Nodes.IntPoint(1).x = 1.-p1;
   Nodes.IntPoint(1).y = p1;
   Nodes.IntPoint(7).x = p1;
   Nodes.IntPoint(7).y = 0.5;
   Nodes.IntPoint(8).x = 0.5;
   Nodes.IntPoint(8).y = 0.5;
   Nodes.IntPoint(5).x = 1.-p1;
   Nodes.IntPoint(5).y = 0.5;
   Nodes.IntPoint(3).x = p1;
   Nodes.IntPoint(3).y = 1.-p1;
   Nodes.IntPoint(6).x = 0.5;
   Nodes.IntPoint(6).y = 1.-p1;
   Nodes.IntPoint(2).x = 1.-p1;
   Nodes.IntPoint(2).y = 1.-p1;
}

void GaussBiQuad2DFiniteElement::CalcShape(const IntegrationPoint &ip,
                                           Vector &shape) const
{
   const double a = sqrt(5./3.);
   const double p1 = 0.5*(1.-sqrt(3./5.));

   double x = a*(ip.x-p1), y = a*(ip.y-p1);
   double l1x, l2x, l3x, l1y, l2y, l3y;

   l1x = (x - 1.) * (2. * x - 1);
   l2x = 4. * x * (1. - x);
   l3x = x * (2. * x - 1.);
   l1y = (y - 1.) * (2. * y - 1);
   l2y = 4. * y * (1. - y);
   l3y = y * (2. * y - 1.);

   shape(0) = l1x * l1y;
   shape(4) = l2x * l1y;
   shape(1) = l3x * l1y;
   shape(7) = l1x * l2y;
   shape(8) = l2x * l2y;
   shape(5) = l3x * l2y;
   shape(3) = l1x * l3y;
   shape(6) = l2x * l3y;
   shape(2) = l3x * l3y;
}

void GaussBiQuad2DFiniteElement::CalcDShape(const IntegrationPoint &ip,
                                            DenseMatrix &dshape) const
{
   const double a = sqrt(5./3.);
   const double p1 = 0.5*(1.-sqrt(3./5.));

   double x = a*(ip.x-p1), y = a*(ip.y-p1);
   double l1x, l2x, l3x, l1y, l2y, l3y;
   double d1x, d2x, d3x, d1y, d2y, d3y;

   l1x = (x - 1.) * (2. * x - 1);
   l2x = 4. * x * (1. - x);
   l3x = x * (2. * x - 1.);
   l1y = (y - 1.) * (2. * y - 1);
   l2y = 4. * y * (1. - y);
   l3y = y * (2. * y - 1.);

   d1x = a * (4. * x - 3.);
   d2x = a * (4. - 8. * x);
   d3x = a * (4. * x - 1.);
   d1y = a * (4. * y - 3.);
   d2y = a * (4. - 8. * y);
   d3y = a * (4. * y - 1.);

   dshape(0,0) = d1x * l1y;
   dshape(0,1) = l1x * d1y;

   dshape(4,0) = d2x * l1y;
   dshape(4,1) = l2x * d1y;

   dshape(1,0) = d3x * l1y;
   dshape(1,1) = l3x * d1y;

   dshape(7,0) = d1x * l2y;
   dshape(7,1) = l1x * d2y;

   dshape(8,0) = d2x * l2y;
   dshape(8,1) = l2x * d2y;

   dshape(5,0) = d3x * l2y;
   dshape(5,1) = l3x * d2y;

   dshape(3,0) = d1x * l3y;
   dshape(3,1) = l1x * d3y;

   dshape(6,0) = d2x * l3y;
   dshape(6,1) = l2x * d3y;

   dshape(2,0) = d3x * l3y;
   dshape(2,1) = l3x * d3y;
}

BiCubic2DFiniteElement::BiCubic2DFiniteElement()
   : NodalFiniteElement (2, Geometry::SQUARE, 16, 3, FunctionSpace::Qk)
{
   Nodes.IntPoint(0).x = 0.;
   Nodes.IntPoint(0).y = 0.;
   Nodes.IntPoint(1).x = 1.;
   Nodes.IntPoint(1).y = 0.;
   Nodes.IntPoint(2).x = 1.;
   Nodes.IntPoint(2).y = 1.;
   Nodes.IntPoint(3).x = 0.;
   Nodes.IntPoint(3).y = 1.;
   Nodes.IntPoint(4).x = 1./3.;
   Nodes.IntPoint(4).y = 0.;
   Nodes.IntPoint(5).x = 2./3.;
   Nodes.IntPoint(5).y = 0.;
   Nodes.IntPoint(6).x = 1.;
   Nodes.IntPoint(6).y = 1./3.;
   Nodes.IntPoint(7).x = 1.;
   Nodes.IntPoint(7).y = 2./3.;
   Nodes.IntPoint(8).x = 2./3.;
   Nodes.IntPoint(8).y = 1.;
   Nodes.IntPoint(9).x = 1./3.;
   Nodes.IntPoint(9).y = 1.;
   Nodes.IntPoint(10).x = 0.;
   Nodes.IntPoint(10).y = 2./3.;
   Nodes.IntPoint(11).x = 0.;
   Nodes.IntPoint(11).y = 1./3.;
   Nodes.IntPoint(12).x = 1./3.;
   Nodes.IntPoint(12).y = 1./3.;
   Nodes.IntPoint(13).x = 2./3.;
   Nodes.IntPoint(13).y = 1./3.;
   Nodes.IntPoint(14).x = 1./3.;
   Nodes.IntPoint(14).y = 2./3.;
   Nodes.IntPoint(15).x = 2./3.;
   Nodes.IntPoint(15).y = 2./3.;
}

void BiCubic2DFiniteElement::CalcShape(
   const IntegrationPoint &ip, Vector &shape) const
{
   double x = ip.x, y = ip.y;

   double w1x, w2x, w3x, w1y, w2y, w3y;
   double l0x, l1x, l2x, l3x, l0y, l1y, l2y, l3y;

   w1x = x - 1./3.; w2x = x - 2./3.; w3x = x - 1.;
   w1y = y - 1./3.; w2y = y - 2./3.; w3y = y - 1.;

   l0x = (- 4.5) * w1x * w2x * w3x;
   l1x = ( 13.5) *   x * w2x * w3x;
   l2x = (-13.5) *   x * w1x * w3x;
   l3x = (  4.5) *   x * w1x * w2x;

   l0y = (- 4.5) * w1y * w2y * w3y;
   l1y = ( 13.5) *   y * w2y * w3y;
   l2y = (-13.5) *   y * w1y * w3y;
   l3y = (  4.5) *   y * w1y * w2y;

   shape(0)  = l0x * l0y;
   shape(1)  = l3x * l0y;
   shape(2)  = l3x * l3y;
   shape(3)  = l0x * l3y;
   shape(4)  = l1x * l0y;
   shape(5)  = l2x * l0y;
   shape(6)  = l3x * l1y;
   shape(7)  = l3x * l2y;
   shape(8)  = l2x * l3y;
   shape(9)  = l1x * l3y;
   shape(10) = l0x * l2y;
   shape(11) = l0x * l1y;
   shape(12) = l1x * l1y;
   shape(13) = l2x * l1y;
   shape(14) = l1x * l2y;
   shape(15) = l2x * l2y;
}

void BiCubic2DFiniteElement::CalcDShape(
   const IntegrationPoint &ip, DenseMatrix &dshape) const
{
   double x = ip.x, y = ip.y;

   double w1x, w2x, w3x, w1y, w2y, w3y;
   double l0x, l1x, l2x, l3x, l0y, l1y, l2y, l3y;
   double d0x, d1x, d2x, d3x, d0y, d1y, d2y, d3y;

   w1x = x - 1./3.; w2x = x - 2./3.; w3x = x - 1.;
   w1y = y - 1./3.; w2y = y - 2./3.; w3y = y - 1.;

   l0x = (- 4.5) * w1x * w2x * w3x;
   l1x = ( 13.5) *   x * w2x * w3x;
   l2x = (-13.5) *   x * w1x * w3x;
   l3x = (  4.5) *   x * w1x * w2x;

   l0y = (- 4.5) * w1y * w2y * w3y;
   l1y = ( 13.5) *   y * w2y * w3y;
   l2y = (-13.5) *   y * w1y * w3y;
   l3y = (  4.5) *   y * w1y * w2y;

   d0x = -5.5 + ( 18. - 13.5 * x) * x;
   d1x =  9.  + (-45. + 40.5 * x) * x;
   d2x = -4.5 + ( 36. - 40.5 * x) * x;
   d3x =  1.  + (- 9. + 13.5 * x) * x;

   d0y = -5.5 + ( 18. - 13.5 * y) * y;
   d1y =  9.  + (-45. + 40.5 * y) * y;
   d2y = -4.5 + ( 36. - 40.5 * y) * y;
   d3y =  1.  + (- 9. + 13.5 * y) * y;

   dshape( 0,0) = d0x * l0y;   dshape( 0,1) = l0x * d0y;
   dshape( 1,0) = d3x * l0y;   dshape( 1,1) = l3x * d0y;
   dshape( 2,0) = d3x * l3y;   dshape( 2,1) = l3x * d3y;
   dshape( 3,0) = d0x * l3y;   dshape( 3,1) = l0x * d3y;
   dshape( 4,0) = d1x * l0y;   dshape( 4,1) = l1x * d0y;
   dshape( 5,0) = d2x * l0y;   dshape( 5,1) = l2x * d0y;
   dshape( 6,0) = d3x * l1y;   dshape( 6,1) = l3x * d1y;
   dshape( 7,0) = d3x * l2y;   dshape( 7,1) = l3x * d2y;
   dshape( 8,0) = d2x * l3y;   dshape( 8,1) = l2x * d3y;
   dshape( 9,0) = d1x * l3y;   dshape( 9,1) = l1x * d3y;
   dshape(10,0) = d0x * l2y;   dshape(10,1) = l0x * d2y;
   dshape(11,0) = d0x * l1y;   dshape(11,1) = l0x * d1y;
   dshape(12,0) = d1x * l1y;   dshape(12,1) = l1x * d1y;
   dshape(13,0) = d2x * l1y;   dshape(13,1) = l2x * d1y;
   dshape(14,0) = d1x * l2y;   dshape(14,1) = l1x * d2y;
   dshape(15,0) = d2x * l2y;   dshape(15,1) = l2x * d2y;
}

void BiCubic2DFiniteElement::CalcHessian(
   const IntegrationPoint &ip, DenseMatrix &h) const
{
   double x = ip.x, y = ip.y;

   double w1x, w2x, w3x, w1y, w2y, w3y;
   double l0x, l1x, l2x, l3x, l0y, l1y, l2y, l3y;
   double d0x, d1x, d2x, d3x, d0y, d1y, d2y, d3y;
   double h0x, h1x, h2x, h3x, h0y, h1y, h2y, h3y;

   w1x = x - 1./3.; w2x = x - 2./3.; w3x = x - 1.;
   w1y = y - 1./3.; w2y = y - 2./3.; w3y = y - 1.;

   l0x = (- 4.5) * w1x * w2x * w3x;
   l1x = ( 13.5) *   x * w2x * w3x;
   l2x = (-13.5) *   x * w1x * w3x;
   l3x = (  4.5) *   x * w1x * w2x;

   l0y = (- 4.5) * w1y * w2y * w3y;
   l1y = ( 13.5) *   y * w2y * w3y;
   l2y = (-13.5) *   y * w1y * w3y;
   l3y = (  4.5) *   y * w1y * w2y;

   d0x = -5.5 + ( 18. - 13.5 * x) * x;
   d1x =  9.  + (-45. + 40.5 * x) * x;
   d2x = -4.5 + ( 36. - 40.5 * x) * x;
   d3x =  1.  + (- 9. + 13.5 * x) * x;

   d0y = -5.5 + ( 18. - 13.5 * y) * y;
   d1y =  9.  + (-45. + 40.5 * y) * y;
   d2y = -4.5 + ( 36. - 40.5 * y) * y;
   d3y =  1.  + (- 9. + 13.5 * y) * y;

   h0x = -27. * x + 18.;
   h1x =  81. * x - 45.;
   h2x = -81. * x + 36.;
   h3x =  27. * x -  9.;

   h0y = -27. * y + 18.;
   h1y =  81. * y - 45.;
   h2y = -81. * y + 36.;
   h3y =  27. * y -  9.;

   h( 0,0) = h0x * l0y;   h( 0,1) = d0x * d0y;   h( 0,2) = l0x * h0y;
   h( 1,0) = h3x * l0y;   h( 1,1) = d3x * d0y;   h( 1,2) = l3x * h0y;
   h( 2,0) = h3x * l3y;   h( 2,1) = d3x * d3y;   h( 2,2) = l3x * h3y;
   h( 3,0) = h0x * l3y;   h( 3,1) = d0x * d3y;   h( 3,2) = l0x * h3y;
   h( 4,0) = h1x * l0y;   h( 4,1) = d1x * d0y;   h( 4,2) = l1x * h0y;
   h( 5,0) = h2x * l0y;   h( 5,1) = d2x * d0y;   h( 5,2) = l2x * h0y;
   h( 6,0) = h3x * l1y;   h( 6,1) = d3x * d1y;   h( 6,2) = l3x * h1y;
   h( 7,0) = h3x * l2y;   h( 7,1) = d3x * d2y;   h( 7,2) = l3x * h2y;
   h( 8,0) = h2x * l3y;   h( 8,1) = d2x * d3y;   h( 8,2) = l2x * h3y;
   h( 9,0) = h1x * l3y;   h( 9,1) = d1x * d3y;   h( 9,2) = l1x * h3y;
   h(10,0) = h0x * l2y;   h(10,1) = d0x * d2y;   h(10,2) = l0x * h2y;
   h(11,0) = h0x * l1y;   h(11,1) = d0x * d1y;   h(11,2) = l0x * h1y;
   h(12,0) = h1x * l1y;   h(12,1) = d1x * d1y;   h(12,2) = l1x * h1y;
   h(13,0) = h2x * l1y;   h(13,1) = d2x * d1y;   h(13,2) = l2x * h1y;
   h(14,0) = h1x * l2y;   h(14,1) = d1x * d2y;   h(14,2) = l1x * h2y;
   h(15,0) = h2x * l2y;   h(15,1) = d2x * d2y;   h(15,2) = l2x * h2y;
}


Cubic1DFiniteElement::Cubic1DFiniteElement()
   : NodalFiniteElement(1, Geometry::SEGMENT, 4, 3)
{
   Nodes.IntPoint(0).x = 0.0;
   Nodes.IntPoint(1).x = 1.0;
   Nodes.IntPoint(2).x = 0.33333333333333333333;
   Nodes.IntPoint(3).x = 0.66666666666666666667;
}

void Cubic1DFiniteElement::CalcShape(const IntegrationPoint &ip,
                                     Vector &shape) const
{
   double x = ip.x;
   double l1 = x,
          l2 = (1.0-x),
          l3 = (0.33333333333333333333-x),
          l4 = (0.66666666666666666667-x);

   shape(0) =   4.5 * l2 * l3 * l4;
   shape(1) =   4.5 * l1 * l3 * l4;
   shape(2) =  13.5 * l1 * l2 * l4;
   shape(3) = -13.5 * l1 * l2 * l3;
}

void Cubic1DFiniteElement::CalcDShape(const IntegrationPoint &ip,
                                      DenseMatrix &dshape) const
{
   double x = ip.x;

   dshape(0,0) = -5.5 + x * (18. - 13.5 * x);
   dshape(1,0) = 1. - x * (9. - 13.5 * x);
   dshape(2,0) = 9. - x * (45. - 40.5 * x);
   dshape(3,0) = -4.5 + x * (36. - 40.5 * x);
}


Cubic2DFiniteElement::Cubic2DFiniteElement()
   : NodalFiniteElement(2, Geometry::TRIANGLE, 10, 3)
{
   Nodes.IntPoint(0).x = 0.0;
   Nodes.IntPoint(0).y = 0.0;
   Nodes.IntPoint(1).x = 1.0;
   Nodes.IntPoint(1).y = 0.0;
   Nodes.IntPoint(2).x = 0.0;
   Nodes.IntPoint(2).y = 1.0;
   Nodes.IntPoint(3).x = 0.33333333333333333333;
   Nodes.IntPoint(3).y = 0.0;
   Nodes.IntPoint(4).x = 0.66666666666666666667;
   Nodes.IntPoint(4).y = 0.0;
   Nodes.IntPoint(5).x = 0.66666666666666666667;
   Nodes.IntPoint(5).y = 0.33333333333333333333;
   Nodes.IntPoint(6).x = 0.33333333333333333333;
   Nodes.IntPoint(6).y = 0.66666666666666666667;
   Nodes.IntPoint(7).x = 0.0;
   Nodes.IntPoint(7).y = 0.66666666666666666667;
   Nodes.IntPoint(8).x = 0.0;
   Nodes.IntPoint(8).y = 0.33333333333333333333;
   Nodes.IntPoint(9).x = 0.33333333333333333333;
   Nodes.IntPoint(9).y = 0.33333333333333333333;
}

void Cubic2DFiniteElement::CalcShape(const IntegrationPoint &ip,
                                     Vector &shape) const
{
   double x = ip.x, y = ip.y;
   double l1 = (-1. + x + y),
          lx = (-1. + 3.*x),
          ly = (-1. + 3.*y);

   shape(0) = -0.5*l1*(3.*l1 + 1.)*(3.*l1 + 2.);
   shape(1) =  0.5*x*(lx - 1.)*lx;
   shape(2) =  0.5*y*(-1. + ly)*ly;
   shape(3) =  4.5*x*l1*(3.*l1 + 1.);
   shape(4) = -4.5*x*lx*l1;
   shape(5) =  4.5*x*lx*y;
   shape(6) =  4.5*x*y*ly;
   shape(7) = -4.5*y*l1*ly;
   shape(8) =  4.5*y*l1*(1. + 3.*l1);
   shape(9) = -27.*x*y*l1;
}

void Cubic2DFiniteElement::CalcDShape(const IntegrationPoint &ip,
                                      DenseMatrix &dshape) const
{
   double x = ip.x, y = ip.y;

   dshape(0,0) =  0.5*(-11. + 36.*y - 9.*(x*(-4. + 3.*x) + 6.*x*y + 3.*y*y));
   dshape(1,0) =  1. + 4.5*x*(-2. + 3.*x);
   dshape(2,0) =  0.;
   dshape(3,0) =  4.5*(2. + 9.*x*x - 5.*y + 3.*y*y + 2.*x*(-5. + 6.*y));
   dshape(4,0) = -4.5*(1. - 1.*y + x*(-8. + 9.*x + 6.*y));
   dshape(5,0) =  4.5*(-1. + 6.*x)*y;
   dshape(6,0) =  4.5*y*(-1. + 3.*y);
   dshape(7,0) =  4.5*(1. - 3.*y)*y;
   dshape(8,0) =  4.5*y*(-5. + 6.*x + 6.*y);
   dshape(9,0) =  -27.*y*(-1. + 2.*x + y);

   dshape(0,1) =  0.5*(-11. + 36.*y - 9.*(x*(-4. + 3.*x) + 6.*x*y + 3.*y*y));
   dshape(1,1) =  0.;
   dshape(2,1) =  1. + 4.5*y*(-2. + 3.*y);
   dshape(3,1) =  4.5*x*(-5. + 6.*x + 6.*y);
   dshape(4,1) =  4.5*(1. - 3.*x)*x;
   dshape(5,1) =  4.5*x*(-1. + 3.*x);
   dshape(6,1) =  4.5*x*(-1. + 6.*y);
   dshape(7,1) = -4.5*(1. + x*(-1. + 6.*y) + y*(-8. + 9.*y));
   dshape(8,1) =  4.5*(2. + 3.*x*x + y*(-10. + 9.*y) + x*(-5. + 12.*y));
   dshape(9,1) = -27.*x*(-1. + x + 2.*y);
}

void Cubic2DFiniteElement::CalcHessian (const IntegrationPoint &ip,
                                        DenseMatrix &h) const
{
   double x = ip.x, y = ip.y;

   h(0,0) = 18.-27.*(x+y);
   h(0,1) = 18.-27.*(x+y);
   h(0,2) = 18.-27.*(x+y);

   h(1,0) = -9.+27.*x;
   h(1,1) = 0.;
   h(1,2) = 0.;

   h(2,0) = 0.;
   h(2,1) = 0.;
   h(2,2) = -9.+27.*y;

   h(3,0) = -45.+81.*x+54.*y;
   h(3,1) = -22.5+54.*x+27.*y;
   h(3,2) = 27.*x;

   h(4,0) = 36.-81.*x-27.*y;
   h(4,1) = 4.5-27.*x;
   h(4,2) = 0.;

   h(5,0) = 27.*y;
   h(5,1) = -4.5+27.*x;
   h(5,2) = 0.;

   h(6,0) = 0.;
   h(6,1) = -4.5+27.*y;
   h(6,2) = 27.*x;

   h(7,0) = 0.;
   h(7,1) = 4.5-27.*y;
   h(7,2) = 36.-27.*x-81.*y;

   h(8,0) = 27.*y;
   h(8,1) = -22.5+27.*x+54.*y;
   h(8,2) = -45.+54.*x+81.*y;

   h(9,0) = -54.*y;
   h(9,1) = 27.-54.*(x+y);
   h(9,2) = -54.*x;
}


Cubic3DFiniteElement::Cubic3DFiniteElement()
   : NodalFiniteElement(3, Geometry::TETRAHEDRON, 20, 3)
{
   Nodes.IntPoint(0).x = 0;
   Nodes.IntPoint(0).y = 0;
   Nodes.IntPoint(0).z = 0;
   Nodes.IntPoint(1).x = 1.;
   Nodes.IntPoint(1).y = 0;
   Nodes.IntPoint(1).z = 0;
   Nodes.IntPoint(2).x = 0;
   Nodes.IntPoint(2).y = 1.;
   Nodes.IntPoint(2).z = 0;
   Nodes.IntPoint(3).x = 0;
   Nodes.IntPoint(3).y = 0;
   Nodes.IntPoint(3).z = 1.;
   Nodes.IntPoint(4).x = 0.3333333333333333333333333333;
   Nodes.IntPoint(4).y = 0;
   Nodes.IntPoint(4).z = 0;
   Nodes.IntPoint(5).x = 0.6666666666666666666666666667;
   Nodes.IntPoint(5).y = 0;
   Nodes.IntPoint(5).z = 0;
   Nodes.IntPoint(6).x = 0;
   Nodes.IntPoint(6).y = 0.3333333333333333333333333333;
   Nodes.IntPoint(6).z = 0;
   Nodes.IntPoint(7).x = 0;
   Nodes.IntPoint(7).y = 0.6666666666666666666666666667;
   Nodes.IntPoint(7).z = 0;
   Nodes.IntPoint(8).x = 0;
   Nodes.IntPoint(8).y = 0;
   Nodes.IntPoint(8).z = 0.3333333333333333333333333333;
   Nodes.IntPoint(9).x = 0;
   Nodes.IntPoint(9).y = 0;
   Nodes.IntPoint(9).z = 0.6666666666666666666666666667;
   Nodes.IntPoint(10).x = 0.6666666666666666666666666667;
   Nodes.IntPoint(10).y = 0.3333333333333333333333333333;
   Nodes.IntPoint(10).z = 0;
   Nodes.IntPoint(11).x = 0.3333333333333333333333333333;
   Nodes.IntPoint(11).y = 0.6666666666666666666666666667;
   Nodes.IntPoint(11).z = 0;
   Nodes.IntPoint(12).x = 0.6666666666666666666666666667;
   Nodes.IntPoint(12).y = 0;
   Nodes.IntPoint(12).z = 0.3333333333333333333333333333;
   Nodes.IntPoint(13).x = 0.3333333333333333333333333333;
   Nodes.IntPoint(13).y = 0;
   Nodes.IntPoint(13).z = 0.6666666666666666666666666667;
   Nodes.IntPoint(14).x = 0;
   Nodes.IntPoint(14).y = 0.6666666666666666666666666667;
   Nodes.IntPoint(14).z = 0.3333333333333333333333333333;
   Nodes.IntPoint(15).x = 0;
   Nodes.IntPoint(15).y = 0.3333333333333333333333333333;
   Nodes.IntPoint(15).z = 0.6666666666666666666666666667;
   Nodes.IntPoint(16).x = 0.3333333333333333333333333333;
   Nodes.IntPoint(16).y = 0.3333333333333333333333333333;
   Nodes.IntPoint(16).z = 0.3333333333333333333333333333;
   Nodes.IntPoint(17).x = 0;
   Nodes.IntPoint(17).y = 0.3333333333333333333333333333;
   Nodes.IntPoint(17).z = 0.3333333333333333333333333333;
   Nodes.IntPoint(18).x = 0.3333333333333333333333333333;
   Nodes.IntPoint(18).y = 0;
   Nodes.IntPoint(18).z = 0.3333333333333333333333333333;
   Nodes.IntPoint(19).x = 0.3333333333333333333333333333;
   Nodes.IntPoint(19).y = 0.3333333333333333333333333333;
   Nodes.IntPoint(19).z = 0;
}

void Cubic3DFiniteElement::CalcShape(const IntegrationPoint &ip,
                                     Vector &shape) const
{
   double x = ip.x, y = ip.y, z = ip.z;

   shape(0) = -((-1 + x + y + z)*(-2 + 3*x + 3*y + 3*z)*
                (-1 + 3*x + 3*y + 3*z))/2.;
   shape(4) = (9*x*(-1 + x + y + z)*(-2 + 3*x + 3*y + 3*z))/2.;
   shape(5) = (-9*x*(-1 + 3*x)*(-1 + x + y + z))/2.;
   shape(1) = (x*(2 + 9*(-1 + x)*x))/2.;
   shape(6) = (9*y*(-1 + x + y + z)*(-2 + 3*x + 3*y + 3*z))/2.;
   shape(19) = -27*x*y*(-1 + x + y + z);
   shape(10) = (9*x*(-1 + 3*x)*y)/2.;
   shape(7) = (-9*y*(-1 + 3*y)*(-1 + x + y + z))/2.;
   shape(11) = (9*x*y*(-1 + 3*y))/2.;
   shape(2) = (y*(2 + 9*(-1 + y)*y))/2.;
   shape(8) = (9*z*(-1 + x + y + z)*(-2 + 3*x + 3*y + 3*z))/2.;
   shape(18) = -27*x*z*(-1 + x + y + z);
   shape(12) = (9*x*(-1 + 3*x)*z)/2.;
   shape(17) = -27*y*z*(-1 + x + y + z);
   shape(16) = 27*x*y*z;
   shape(14) = (9*y*(-1 + 3*y)*z)/2.;
   shape(9) = (-9*z*(-1 + x + y + z)*(-1 + 3*z))/2.;
   shape(13) = (9*x*z*(-1 + 3*z))/2.;
   shape(15) = (9*y*z*(-1 + 3*z))/2.;
   shape(3) = (z*(2 + 9*(-1 + z)*z))/2.;
}

void Cubic3DFiniteElement::CalcDShape(const IntegrationPoint &ip,
                                      DenseMatrix &dshape) const
{
   double x = ip.x, y = ip.y, z = ip.z;

   dshape(0,0) = (-11 + 36*y + 36*z - 9*(3*pow(x,2) + 3*pow(y + z,2) +
                                         x*(-4 + 6*y + 6*z)))/2.;
   dshape(0,1) = (-11 + 36*y + 36*z - 9*(3*pow(x,2) + 3*pow(y + z,2) +
                                         x*(-4 + 6*y + 6*z)))/2.;
   dshape(0,2) = (-11 + 36*y + 36*z - 9*(3*pow(x,2) + 3*pow(y + z,2) +
                                         x*(-4 + 6*y + 6*z)))/2.;
   dshape(4,0) = (9*(9*pow(x,2) + (-1 + y + z)*(-2 + 3*y + 3*z) +
                     2*x*(-5 + 6*y + 6*z)))/2.;
   dshape(4,1) = (9*x*(-5 + 6*x + 6*y + 6*z))/2.;
   dshape(4,2) = (9*x*(-5 + 6*x + 6*y + 6*z))/2.;
   dshape(5,0) = (-9*(1 - y - z + x*(-8 + 9*x + 6*y + 6*z)))/2.;
   dshape(5,1) = (9*(1 - 3*x)*x)/2.;
   dshape(5,2) = (9*(1 - 3*x)*x)/2.;
   dshape(1,0) = 1 + (9*x*(-2 + 3*x))/2.;
   dshape(1,1) = 0;
   dshape(1,2) = 0;
   dshape(6,0) = (9*y*(-5 + 6*x + 6*y + 6*z))/2.;
   dshape(6,1) = (9*(2 + 3*pow(x,2) - 10*y - 5*z + 3*(y + z)*(3*y + z) +
                     x*(-5 + 12*y + 6*z)))/2.;
   dshape(6,2) = (9*y*(-5 + 6*x + 6*y + 6*z))/2.;
   dshape(19,0) = -27*y*(-1 + 2*x + y + z);
   dshape(19,1) = -27*x*(-1 + x + 2*y + z);
   dshape(19,2) = -27*x*y;
   dshape(10,0) = (9*(-1 + 6*x)*y)/2.;
   dshape(10,1) = (9*x*(-1 + 3*x))/2.;
   dshape(10,2) = 0;
   dshape(7,0) = (9*(1 - 3*y)*y)/2.;
   dshape(7,1) = (-9*(1 + x*(-1 + 6*y) - z + y*(-8 + 9*y + 6*z)))/2.;
   dshape(7,2) = (9*(1 - 3*y)*y)/2.;
   dshape(11,0) = (9*y*(-1 + 3*y))/2.;
   dshape(11,1) = (9*x*(-1 + 6*y))/2.;
   dshape(11,2) = 0;
   dshape(2,0) = 0;
   dshape(2,1) = 1 + (9*y*(-2 + 3*y))/2.;
   dshape(2,2) = 0;
   dshape(8,0) = (9*z*(-5 + 6*x + 6*y + 6*z))/2.;
   dshape(8,1) = (9*z*(-5 + 6*x + 6*y + 6*z))/2.;
   dshape(8,2) = (9*(2 + 3*pow(x,2) - 5*y - 10*z + 3*(y + z)*(y + 3*z) +
                     x*(-5 + 6*y + 12*z)))/2.;
   dshape(18,0) = -27*z*(-1 + 2*x + y + z);
   dshape(18,1) = -27*x*z;
   dshape(18,2) = -27*x*(-1 + x + y + 2*z);
   dshape(12,0) = (9*(-1 + 6*x)*z)/2.;
   dshape(12,1) = 0;
   dshape(12,2) = (9*x*(-1 + 3*x))/2.;
   dshape(17,0) = -27*y*z;
   dshape(17,1) = -27*z*(-1 + x + 2*y + z);
   dshape(17,2) = -27*y*(-1 + x + y + 2*z);
   dshape(16,0) = 27*y*z;
   dshape(16,1) = 27*x*z;
   dshape(16,2) = 27*x*y;
   dshape(14,0) = 0;
   dshape(14,1) = (9*(-1 + 6*y)*z)/2.;
   dshape(14,2) = (9*y*(-1 + 3*y))/2.;
   dshape(9,0) = (9*(1 - 3*z)*z)/2.;
   dshape(9,1) = (9*(1 - 3*z)*z)/2.;
   dshape(9,2) = (9*(-1 + x + y + 8*z - 6*(x + y)*z - 9*pow(z,2)))/2.;
   dshape(13,0) = (9*z*(-1 + 3*z))/2.;
   dshape(13,1) = 0;
   dshape(13,2) = (9*x*(-1 + 6*z))/2.;
   dshape(15,0) = 0;
   dshape(15,1) = (9*z*(-1 + 3*z))/2.;
   dshape(15,2) = (9*y*(-1 + 6*z))/2.;
   dshape(3,0) = 0;
   dshape(3,1) = 0;
   dshape(3,2) = 1 + (9*z*(-2 + 3*z))/2.;
}


P0TriangleFiniteElement::P0TriangleFiniteElement()
   : NodalFiniteElement(2, Geometry::TRIANGLE, 1, 0)
{
   Nodes.IntPoint(0).x = 0.333333333333333333;
   Nodes.IntPoint(0).y = 0.333333333333333333;
}

void P0TriangleFiniteElement::CalcShape(const IntegrationPoint &ip,
                                        Vector &shape) const
{
   shape(0) = 1.0;
}

void P0TriangleFiniteElement::CalcDShape(const IntegrationPoint &ip,
                                         DenseMatrix &dshape) const
{
   dshape(0,0) = 0.0;
   dshape(0,1) = 0.0;
}


P0QuadFiniteElement::P0QuadFiniteElement()
   : NodalFiniteElement(2, Geometry::SQUARE, 1, 0, FunctionSpace::Qk)
{
   Nodes.IntPoint(0).x = 0.5;
   Nodes.IntPoint(0).y = 0.5;
}

void P0QuadFiniteElement::CalcShape(const IntegrationPoint &ip,
                                    Vector &shape) const
{
   shape(0) = 1.0;
}

void P0QuadFiniteElement::CalcDShape(const IntegrationPoint &ip,
                                     DenseMatrix &dshape) const
{
   dshape(0,0) = 0.0;
   dshape(0,1) = 0.0;
}


Linear3DFiniteElement::Linear3DFiniteElement()
   : NodalFiniteElement(3, Geometry::TETRAHEDRON, 4, 1)
{
   Nodes.IntPoint(0).x = 0.0;
   Nodes.IntPoint(0).y = 0.0;
   Nodes.IntPoint(0).z = 0.0;
   Nodes.IntPoint(1).x = 1.0;
   Nodes.IntPoint(1).y = 0.0;
   Nodes.IntPoint(1).z = 0.0;
   Nodes.IntPoint(2).x = 0.0;
   Nodes.IntPoint(2).y = 1.0;
   Nodes.IntPoint(2).z = 0.0;
   Nodes.IntPoint(3).x = 0.0;
   Nodes.IntPoint(3).y = 0.0;
   Nodes.IntPoint(3).z = 1.0;
}

void Linear3DFiniteElement::CalcShape(const IntegrationPoint &ip,
                                      Vector &shape) const
{
   shape(0) = 1. - ip.x - ip.y - ip.z;
   shape(1) = ip.x;
   shape(2) = ip.y;
   shape(3) = ip.z;
}

void Linear3DFiniteElement::CalcDShape(const IntegrationPoint &ip,
                                       DenseMatrix &dshape) const
{
   if (dshape.Height() == 4)
   {
      double *A = &dshape(0,0);
      A[0] = -1.; A[4] = -1.; A[8]  = -1.;
      A[1] =  1.; A[5] =  0.; A[9]  =  0.;
      A[2] =  0.; A[6] =  1.; A[10] =  0.;
      A[3] =  0.; A[7] =  0.; A[11] =  1.;
   }
   else
   {
      dshape(0,0) = -1.; dshape(0,1) = -1.; dshape(0,2) = -1.;
      dshape(1,0) =  1.; dshape(1,1) =  0.; dshape(1,2) =  0.;
      dshape(2,0) =  0.; dshape(2,1) =  1.; dshape(2,2) =  0.;
      dshape(3,0) =  0.; dshape(3,1) =  0.; dshape(3,2) =  1.;
   }
}

void Linear3DFiniteElement::GetFaceDofs (int face, int **dofs, int *ndofs)
const
{
   static int face_dofs[4][3] = {{1, 2, 3}, {0, 2, 3}, {0, 1, 3}, {0, 1, 2}};

   *ndofs = 3;
   *dofs  = face_dofs[face];
}


Quadratic3DFiniteElement::Quadratic3DFiniteElement()
   : NodalFiniteElement(3, Geometry::TETRAHEDRON, 10, 2)
{
   Nodes.IntPoint(0).x = 0.0;
   Nodes.IntPoint(0).y = 0.0;
   Nodes.IntPoint(0).z = 0.0;
   Nodes.IntPoint(1).x = 1.0;
   Nodes.IntPoint(1).y = 0.0;
   Nodes.IntPoint(1).z = 0.0;
   Nodes.IntPoint(2).x = 0.0;
   Nodes.IntPoint(2).y = 1.0;
   Nodes.IntPoint(2).z = 0.0;
   Nodes.IntPoint(3).x = 0.0;
   Nodes.IntPoint(3).y = 0.0;
   Nodes.IntPoint(3).z = 1.0;
   Nodes.IntPoint(4).x = 0.5;
   Nodes.IntPoint(4).y = 0.0;
   Nodes.IntPoint(4).z = 0.0;
   Nodes.IntPoint(5).x = 0.0;
   Nodes.IntPoint(5).y = 0.5;
   Nodes.IntPoint(5).z = 0.0;
   Nodes.IntPoint(6).x = 0.0;
   Nodes.IntPoint(6).y = 0.0;
   Nodes.IntPoint(6).z = 0.5;
   Nodes.IntPoint(7).x = 0.5;
   Nodes.IntPoint(7).y = 0.5;
   Nodes.IntPoint(7).z = 0.0;
   Nodes.IntPoint(8).x = 0.5;
   Nodes.IntPoint(8).y = 0.0;
   Nodes.IntPoint(8).z = 0.5;
   Nodes.IntPoint(9).x = 0.0;
   Nodes.IntPoint(9).y = 0.5;
   Nodes.IntPoint(9).z = 0.5;
}

void Quadratic3DFiniteElement::CalcShape(const IntegrationPoint &ip,
                                         Vector &shape) const
{
   double L0, L1, L2, L3;

   L0 = 1. - ip.x - ip.y - ip.z;
   L1 = ip.x;
   L2 = ip.y;
   L3 = ip.z;

   shape(0) = L0 * ( 2.0 * L0 - 1.0 );
   shape(1) = L1 * ( 2.0 * L1 - 1.0 );
   shape(2) = L2 * ( 2.0 * L2 - 1.0 );
   shape(3) = L3 * ( 2.0 * L3 - 1.0 );
   shape(4) = 4.0 * L0 * L1;
   shape(5) = 4.0 * L0 * L2;
   shape(6) = 4.0 * L0 * L3;
   shape(7) = 4.0 * L1 * L2;
   shape(8) = 4.0 * L1 * L3;
   shape(9) = 4.0 * L2 * L3;
}

void Quadratic3DFiniteElement::CalcDShape(const IntegrationPoint &ip,
                                          DenseMatrix &dshape) const
{
   double x, y, z, L0;

   x = ip.x;
   y = ip.y;
   z = ip.z;
   L0 = 1.0 - x - y - z;

   dshape(0,0) = dshape(0,1) = dshape(0,2) = 1.0 - 4.0 * L0;
   dshape(1,0) = -1.0 + 4.0 * x; dshape(1,1) = 0.0; dshape(1,2) = 0.0;
   dshape(2,0) = 0.0; dshape(2,1) = -1.0 + 4.0 * y; dshape(2,2) = 0.0;
   dshape(3,0) = dshape(3,1) = 0.0; dshape(3,2) = -1.0 + 4.0 * z;
   dshape(4,0) = 4.0 * (L0 - x); dshape(4,1) = dshape(4,2) = -4.0 * x;
   dshape(5,0) = dshape(5,2) = -4.0 * y; dshape(5,1) = 4.0 * (L0 - y);
   dshape(6,0) = dshape(6,1) = -4.0 * z; dshape(6,2) = 4.0 * (L0 - z);
   dshape(7,0) = 4.0 * y; dshape(7,1) = 4.0 * x; dshape(7,2) = 0.0;
   dshape(8,0) = 4.0 * z; dshape(8,1) = 0.0; dshape(8,2) = 4.0 * x;
   dshape(9,0) = 0.0; dshape(9,1) = 4.0 * z; dshape(9,2) = 4.0 * y;
}

TriLinear3DFiniteElement::TriLinear3DFiniteElement()
   : NodalFiniteElement(3, Geometry::CUBE, 8, 1, FunctionSpace::Qk)
{
   Nodes.IntPoint(0).x = 0.0;
   Nodes.IntPoint(0).y = 0.0;
   Nodes.IntPoint(0).z = 0.0;

   Nodes.IntPoint(1).x = 1.0;
   Nodes.IntPoint(1).y = 0.0;
   Nodes.IntPoint(1).z = 0.0;

   Nodes.IntPoint(2).x = 1.0;
   Nodes.IntPoint(2).y = 1.0;
   Nodes.IntPoint(2).z = 0.0;

   Nodes.IntPoint(3).x = 0.0;
   Nodes.IntPoint(3).y = 1.0;
   Nodes.IntPoint(3).z = 0.0;

   Nodes.IntPoint(4).x = 0.0;
   Nodes.IntPoint(4).y = 0.0;
   Nodes.IntPoint(4).z = 1.0;

   Nodes.IntPoint(5).x = 1.0;
   Nodes.IntPoint(5).y = 0.0;
   Nodes.IntPoint(5).z = 1.0;

   Nodes.IntPoint(6).x = 1.0;
   Nodes.IntPoint(6).y = 1.0;
   Nodes.IntPoint(6).z = 1.0;

   Nodes.IntPoint(7).x = 0.0;
   Nodes.IntPoint(7).y = 1.0;
   Nodes.IntPoint(7).z = 1.0;
}

void TriLinear3DFiniteElement::CalcShape(const IntegrationPoint &ip,
                                         Vector &shape) const
{
   double x = ip.x, y = ip.y, z = ip.z;
   double ox = 1.-x, oy = 1.-y, oz = 1.-z;

   shape(0) = ox * oy * oz;
   shape(1) =  x * oy * oz;
   shape(2) =  x *  y * oz;
   shape(3) = ox *  y * oz;
   shape(4) = ox * oy *  z;
   shape(5) =  x * oy *  z;
   shape(6) =  x *  y *  z;
   shape(7) = ox *  y *  z;
}

void TriLinear3DFiniteElement::CalcDShape(const IntegrationPoint &ip,
                                          DenseMatrix &dshape) const
{
   double x = ip.x, y = ip.y, z = ip.z;
   double ox = 1.-x, oy = 1.-y, oz = 1.-z;

   dshape(0,0) = - oy * oz;
   dshape(0,1) = - ox * oz;
   dshape(0,2) = - ox * oy;

   dshape(1,0) =   oy * oz;
   dshape(1,1) = -  x * oz;
   dshape(1,2) = -  x * oy;

   dshape(2,0) =    y * oz;
   dshape(2,1) =    x * oz;
   dshape(2,2) = -  x *  y;

   dshape(3,0) = -  y * oz;
   dshape(3,1) =   ox * oz;
   dshape(3,2) = - ox *  y;

   dshape(4,0) = - oy *  z;
   dshape(4,1) = - ox *  z;
   dshape(4,2) =   ox * oy;

   dshape(5,0) =   oy *  z;
   dshape(5,1) = -  x *  z;
   dshape(5,2) =    x * oy;

   dshape(6,0) =    y *  z;
   dshape(6,1) =    x *  z;
   dshape(6,2) =    x *  y;

   dshape(7,0) = -  y *  z;
   dshape(7,1) =   ox *  z;
   dshape(7,2) =   ox *  y;
}


P0SegmentFiniteElement::P0SegmentFiniteElement(int Ord)
   : NodalFiniteElement(1, Geometry::SEGMENT, 1, Ord)   // defaul Ord = 0
{
   Nodes.IntPoint(0).x = 0.5;
}

void P0SegmentFiniteElement::CalcShape(const IntegrationPoint &ip,
                                       Vector &shape) const
{
   shape(0) = 1.0;
}

void P0SegmentFiniteElement::CalcDShape(const IntegrationPoint &ip,
                                        DenseMatrix &dshape) const
{
   dshape(0,0) = 0.0;
}

CrouzeixRaviartFiniteElement::CrouzeixRaviartFiniteElement()
   : NodalFiniteElement(2, Geometry::TRIANGLE, 3, 1)
{
   Nodes.IntPoint(0).x = 0.5;
   Nodes.IntPoint(0).y = 0.0;
   Nodes.IntPoint(1).x = 0.5;
   Nodes.IntPoint(1).y = 0.5;
   Nodes.IntPoint(2).x = 0.0;
   Nodes.IntPoint(2).y = 0.5;
}

void CrouzeixRaviartFiniteElement::CalcShape(const IntegrationPoint &ip,
                                             Vector &shape) const
{
   shape(0) =  1.0 - 2.0 * ip.y;
   shape(1) = -1.0 + 2.0 * ( ip.x + ip.y );
   shape(2) =  1.0 - 2.0 * ip.x;
}

void CrouzeixRaviartFiniteElement::CalcDShape(const IntegrationPoint &ip,
                                              DenseMatrix &dshape) const
{
   dshape(0,0) =  0.0; dshape(0,1) = -2.0;
   dshape(1,0) =  2.0; dshape(1,1) =  2.0;
   dshape(2,0) = -2.0; dshape(2,1) =  0.0;
}

CrouzeixRaviartQuadFiniteElement::CrouzeixRaviartQuadFiniteElement()
// the FunctionSpace should be rotated (45 degrees) Q_1
// i.e. the span of { 1, x, y, x^2 - y^2 }
   : NodalFiniteElement(2, Geometry::SQUARE, 4, 2, FunctionSpace::Qk)
{
   Nodes.IntPoint(0).x = 0.5;
   Nodes.IntPoint(0).y = 0.0;
   Nodes.IntPoint(1).x = 1.0;
   Nodes.IntPoint(1).y = 0.5;
   Nodes.IntPoint(2).x = 0.5;
   Nodes.IntPoint(2).y = 1.0;
   Nodes.IntPoint(3).x = 0.0;
   Nodes.IntPoint(3).y = 0.5;
}

void CrouzeixRaviartQuadFiniteElement::CalcShape(const IntegrationPoint &ip,
                                                 Vector &shape) const
{
   const double l1 = ip.x+ip.y-0.5, l2 = 1.-l1, l3 = ip.x-ip.y+0.5, l4 = 1.-l3;

   shape(0) = l2 * l3;
   shape(1) = l1 * l3;
   shape(2) = l1 * l4;
   shape(3) = l2 * l4;
}

void CrouzeixRaviartQuadFiniteElement::CalcDShape(const IntegrationPoint &ip,
                                                  DenseMatrix &dshape) const
{
   const double x2 = 2.*ip.x, y2 = 2.*ip.y;

   dshape(0,0) =  1. - x2; dshape(0,1) = -2. + y2;
   dshape(1,0) =       x2; dshape(1,1) =  1. - y2;
   dshape(2,0) =  1. - x2; dshape(2,1) =       y2;
   dshape(3,0) = -2. + x2; dshape(3,1) =  1. - y2;
}


RT0TriangleFiniteElement::RT0TriangleFiniteElement()
   : VectorFiniteElement(2, Geometry::TRIANGLE, 3, 1, H_DIV)
{
   Nodes.IntPoint(0).x = 0.5;
   Nodes.IntPoint(0).y = 0.0;
   Nodes.IntPoint(1).x = 0.5;
   Nodes.IntPoint(1).y = 0.5;
   Nodes.IntPoint(2).x = 0.0;
   Nodes.IntPoint(2).y = 0.5;
}

void RT0TriangleFiniteElement::CalcVShape(const IntegrationPoint &ip,
                                          DenseMatrix &shape) const
{
   double x = ip.x, y = ip.y;

   shape(0,0) = x;
   shape(0,1) = y - 1.;
   shape(1,0) = x;
   shape(1,1) = y;
   shape(2,0) = x - 1.;
   shape(2,1) = y;
}

void RT0TriangleFiniteElement::CalcDivShape(const IntegrationPoint &ip,
                                            Vector &divshape) const
{
   divshape(0) = 2.;
   divshape(1) = 2.;
   divshape(2) = 2.;
}

const double RT0TriangleFiniteElement::nk[3][2] =
{ {0, -1}, {1, 1}, {-1, 0} };

void RT0TriangleFiniteElement::GetLocalInterpolation (
   ElementTransformation &Trans, DenseMatrix &I) const
{
   int k, j;
#ifdef MFEM_THREAD_SAFE
   DenseMatrix vshape(Dof, Dim);
   DenseMatrix Jinv(Dim);
#endif

#ifdef MFEM_DEBUG
   for (k = 0; k < 3; k++)
   {
      CalcVShape (Nodes.IntPoint(k), vshape);
      for (j = 0; j < 3; j++)
      {
         double d = vshape(j,0)*nk[k][0]+vshape(j,1)*nk[k][1];
         if (j == k) { d -= 1.0; }
         if (fabs(d) > 1.0e-12)
         {
            mfem::err << "RT0TriangleFiniteElement::GetLocalInterpolation (...)\n"
                      " k = " << k << ", j = " << j << ", d = " << d << endl;
            mfem_error();
         }
      }
   }
#endif

   IntegrationPoint ip;
   ip.x = ip.y = 0.0;
   Trans.SetIntPoint (&ip);
   // Trans must be linear
   // set Jinv = |J| J^{-t} = adj(J)^t
   CalcAdjugateTranspose (Trans.Jacobian(), Jinv);
   double vk[2];
   Vector xk (vk, 2);

   for (k = 0; k < 3; k++)
   {
      Trans.Transform (Nodes.IntPoint (k), xk);
      ip.x = vk[0]; ip.y = vk[1];
      CalcVShape (ip, vshape);
      //  vk = |J| J^{-t} nk
      vk[0] = Jinv(0,0)*nk[k][0]+Jinv(0,1)*nk[k][1];
      vk[1] = Jinv(1,0)*nk[k][0]+Jinv(1,1)*nk[k][1];
      for (j = 0; j < 3; j++)
         if (fabs (I(k,j) = vshape(j,0)*vk[0]+vshape(j,1)*vk[1]) < 1.0e-12)
         {
            I(k,j) = 0.0;
         }
   }
}

void RT0TriangleFiniteElement::Project (
   VectorCoefficient &vc, ElementTransformation &Trans,
   Vector &dofs) const
{
   double vk[2];
   Vector xk (vk, 2);
#ifdef MFEM_THREAD_SAFE
   DenseMatrix Jinv(Dim);
#endif

   for (int k = 0; k < 3; k++)
   {
      Trans.SetIntPoint (&Nodes.IntPoint (k));
      // set Jinv = |J| J^{-t} = adj(J)^t
      CalcAdjugateTranspose (Trans.Jacobian(), Jinv);

      vc.Eval (xk, Trans, Nodes.IntPoint (k));
      //  xk^t |J| J^{-t} nk
      dofs(k) = (vk[0] * ( Jinv(0,0)*nk[k][0]+Jinv(0,1)*nk[k][1] ) +
                 vk[1] * ( Jinv(1,0)*nk[k][0]+Jinv(1,1)*nk[k][1] ));
   }
}

RT0QuadFiniteElement::RT0QuadFiniteElement()
   : VectorFiniteElement(2, Geometry::SQUARE, 4, 1, H_DIV, FunctionSpace::Qk)
{
   Nodes.IntPoint(0).x = 0.5;
   Nodes.IntPoint(0).y = 0.0;
   Nodes.IntPoint(1).x = 1.0;
   Nodes.IntPoint(1).y = 0.5;
   Nodes.IntPoint(2).x = 0.5;
   Nodes.IntPoint(2).y = 1.0;
   Nodes.IntPoint(3).x = 0.0;
   Nodes.IntPoint(3).y = 0.5;
}

void RT0QuadFiniteElement::CalcVShape(const IntegrationPoint &ip,
                                      DenseMatrix &shape) const
{
   double x = ip.x, y = ip.y;

   shape(0,0) = 0;
   shape(0,1) = y - 1.;
   shape(1,0) = x;
   shape(1,1) = 0;
   shape(2,0) = 0;
   shape(2,1) = y;
   shape(3,0) = x - 1.;
   shape(3,1) = 0;
}

void RT0QuadFiniteElement::CalcDivShape(const IntegrationPoint &ip,
                                        Vector &divshape) const
{
   divshape(0) = 1.;
   divshape(1) = 1.;
   divshape(2) = 1.;
   divshape(3) = 1.;
}

const double RT0QuadFiniteElement::nk[4][2] =
{ {0, -1}, {1, 0}, {0, 1}, {-1, 0} };

void RT0QuadFiniteElement::GetLocalInterpolation (
   ElementTransformation &Trans, DenseMatrix &I) const
{
   int k, j;
#ifdef MFEM_THREAD_SAFE
   DenseMatrix vshape(Dof, Dim);
   DenseMatrix Jinv(Dim);
#endif

#ifdef MFEM_DEBUG
   for (k = 0; k < 4; k++)
   {
      CalcVShape (Nodes.IntPoint(k), vshape);
      for (j = 0; j < 4; j++)
      {
         double d = vshape(j,0)*nk[k][0]+vshape(j,1)*nk[k][1];
         if (j == k) { d -= 1.0; }
         if (fabs(d) > 1.0e-12)
         {
            mfem::err << "RT0QuadFiniteElement::GetLocalInterpolation (...)\n"
                      " k = " << k << ", j = " << j << ", d = " << d << endl;
            mfem_error();
         }
      }
   }
#endif

   IntegrationPoint ip;
   ip.x = ip.y = 0.0;
   Trans.SetIntPoint (&ip);
   // Trans must be linear (more to have embedding?)
   // set Jinv = |J| J^{-t} = adj(J)^t
   CalcAdjugateTranspose (Trans.Jacobian(), Jinv);
   double vk[2];
   Vector xk (vk, 2);

   for (k = 0; k < 4; k++)
   {
      Trans.Transform (Nodes.IntPoint (k), xk);
      ip.x = vk[0]; ip.y = vk[1];
      CalcVShape (ip, vshape);
      //  vk = |J| J^{-t} nk
      vk[0] = Jinv(0,0)*nk[k][0]+Jinv(0,1)*nk[k][1];
      vk[1] = Jinv(1,0)*nk[k][0]+Jinv(1,1)*nk[k][1];
      for (j = 0; j < 4; j++)
         if (fabs (I(k,j) = vshape(j,0)*vk[0]+vshape(j,1)*vk[1]) < 1.0e-12)
         {
            I(k,j) = 0.0;
         }
   }
}

void RT0QuadFiniteElement::Project (
   VectorCoefficient &vc, ElementTransformation &Trans,
   Vector &dofs) const
{
   double vk[2];
   Vector xk (vk, 2);
#ifdef MFEM_THREAD_SAFE
   DenseMatrix Jinv(Dim);
#endif

   for (int k = 0; k < 4; k++)
   {
      Trans.SetIntPoint (&Nodes.IntPoint (k));
      // set Jinv = |J| J^{-t} = adj(J)^t
      CalcAdjugateTranspose (Trans.Jacobian(), Jinv);

      vc.Eval (xk, Trans, Nodes.IntPoint (k));
      //  xk^t |J| J^{-t} nk
      dofs(k) = (vk[0] * ( Jinv(0,0)*nk[k][0]+Jinv(0,1)*nk[k][1] ) +
                 vk[1] * ( Jinv(1,0)*nk[k][0]+Jinv(1,1)*nk[k][1] ));
   }
}

RT1TriangleFiniteElement::RT1TriangleFiniteElement()
   : VectorFiniteElement(2, Geometry::TRIANGLE, 8, 2, H_DIV)
{
   Nodes.IntPoint(0).x = 0.33333333333333333333;
   Nodes.IntPoint(0).y = 0.0;
   Nodes.IntPoint(1).x = 0.66666666666666666667;
   Nodes.IntPoint(1).y = 0.0;
   Nodes.IntPoint(2).x = 0.66666666666666666667;
   Nodes.IntPoint(2).y = 0.33333333333333333333;
   Nodes.IntPoint(3).x = 0.33333333333333333333;
   Nodes.IntPoint(3).y = 0.66666666666666666667;
   Nodes.IntPoint(4).x = 0.0;
   Nodes.IntPoint(4).y = 0.66666666666666666667;
   Nodes.IntPoint(5).x = 0.0;
   Nodes.IntPoint(5).y = 0.33333333333333333333;
   Nodes.IntPoint(6).x = 0.33333333333333333333;
   Nodes.IntPoint(6).y = 0.33333333333333333333;
   Nodes.IntPoint(7).x = 0.33333333333333333333;
   Nodes.IntPoint(7).y = 0.33333333333333333333;
}

void RT1TriangleFiniteElement::CalcVShape(const IntegrationPoint &ip,
                                          DenseMatrix &shape) const
{
   double x = ip.x, y = ip.y;

   shape(0,0) = -2 * x * (-1 + x + 2 * y);
   shape(0,1) = -2 * (-1 + y) * (-1 + x + 2 * y);
   shape(1,0) =  2 * x * (x - y);
   shape(1,1) =  2 * (x - y) * (-1 + y);
   shape(2,0) =  2 * x * (-1 + 2 * x + y);
   shape(2,1) =  2 * y * (-1 + 2 * x + y);
   shape(3,0) =  2 * x * (-1 + x + 2 * y);
   shape(3,1) =  2 * y * (-1 + x + 2 * y);
   shape(4,0) = -2 * (-1 + x) * (x - y);
   shape(4,1) =  2 * y * (-x + y);
   shape(5,0) = -2 * (-1 + x) * (-1 + 2 * x + y);
   shape(5,1) = -2 * y * (-1 + 2 * x + y);
   shape(6,0) = -3 * x * (-2 + 2 * x + y);
   shape(6,1) = -3 * y * (-1 + 2 * x + y);
   shape(7,0) = -3 * x * (-1 + x + 2 * y);
   shape(7,1) = -3 * y * (-2 + x + 2 * y);
}

void RT1TriangleFiniteElement::CalcDivShape(const IntegrationPoint &ip,
                                            Vector &divshape) const
{
   double x = ip.x, y = ip.y;

   divshape(0) = -2 * (-4 + 3 * x + 6 * y);
   divshape(1) =  2 + 6 * x - 6 * y;
   divshape(2) = -4 + 12 * x + 6 * y;
   divshape(3) = -4 + 6 * x + 12 * y;
   divshape(4) =  2 - 6 * x + 6 * y;
   divshape(5) = -2 * (-4 + 6 * x + 3 * y);
   divshape(6) = -9 * (-1 + 2 * x + y);
   divshape(7) = -9 * (-1 + x + 2 * y);
}

const double RT1TriangleFiniteElement::nk[8][2] =
{
   { 0,-1}, { 0,-1},
   { 1, 1}, { 1, 1},
   {-1, 0}, {-1, 0},
   { 1, 0}, { 0, 1}
};

void RT1TriangleFiniteElement::GetLocalInterpolation (
   ElementTransformation &Trans, DenseMatrix &I) const
{
   int k, j;
#ifdef MFEM_THREAD_SAFE
   DenseMatrix vshape(Dof, Dim);
   DenseMatrix Jinv(Dim);
#endif

#ifdef MFEM_DEBUG
   for (k = 0; k < 8; k++)
   {
      CalcVShape (Nodes.IntPoint(k), vshape);
      for (j = 0; j < 8; j++)
      {
         double d = vshape(j,0)*nk[k][0]+vshape(j,1)*nk[k][1];
         if (j == k) { d -= 1.0; }
         if (fabs(d) > 1.0e-12)
         {
            mfem::err << "RT1QuadFiniteElement::GetLocalInterpolation (...)\n"
                      " k = " << k << ", j = " << j << ", d = " << d << endl;
            mfem_error();
         }
      }
   }
#endif

   IntegrationPoint ip;
   ip.x = ip.y = 0.0;
   Trans.SetIntPoint (&ip);
   // Trans must be linear (more to have embedding?)
   // set Jinv = |J| J^{-t} = adj(J)^t
   CalcAdjugateTranspose (Trans.Jacobian(), Jinv);
   double vk[2];
   Vector xk (vk, 2);

   for (k = 0; k < 8; k++)
   {
      Trans.Transform (Nodes.IntPoint (k), xk);
      ip.x = vk[0]; ip.y = vk[1];
      CalcVShape (ip, vshape);
      //  vk = |J| J^{-t} nk
      vk[0] = Jinv(0,0)*nk[k][0]+Jinv(0,1)*nk[k][1];
      vk[1] = Jinv(1,0)*nk[k][0]+Jinv(1,1)*nk[k][1];
      for (j = 0; j < 8; j++)
         if (fabs (I(k,j) = vshape(j,0)*vk[0]+vshape(j,1)*vk[1]) < 1.0e-12)
         {
            I(k,j) = 0.0;
         }
   }
}

void RT1TriangleFiniteElement::Project (
   VectorCoefficient &vc, ElementTransformation &Trans, Vector &dofs) const
{
   double vk[2];
   Vector xk (vk, 2);
#ifdef MFEM_THREAD_SAFE
   DenseMatrix Jinv(Dim);
#endif

   for (int k = 0; k < 8; k++)
   {
      Trans.SetIntPoint (&Nodes.IntPoint (k));
      // set Jinv = |J| J^{-t} = adj(J)^t
      CalcAdjugateTranspose (Trans.Jacobian(), Jinv);

      vc.Eval (xk, Trans, Nodes.IntPoint (k));
      //  xk^t |J| J^{-t} nk
      dofs(k) = (vk[0] * ( Jinv(0,0)*nk[k][0]+Jinv(0,1)*nk[k][1] ) +
                 vk[1] * ( Jinv(1,0)*nk[k][0]+Jinv(1,1)*nk[k][1] ));
      dofs(k) *= 0.5;
   }
}

RT1QuadFiniteElement::RT1QuadFiniteElement()
   : VectorFiniteElement(2, Geometry::SQUARE, 12, 2, H_DIV, FunctionSpace::Qk)
{
   // y = 0
   Nodes.IntPoint(0).x  = 1./3.;
   Nodes.IntPoint(0).y  = 0.0;
   Nodes.IntPoint(1).x  = 2./3.;
   Nodes.IntPoint(1).y  = 0.0;
   // x = 1
   Nodes.IntPoint(2).x  = 1.0;
   Nodes.IntPoint(2).y  = 1./3.;
   Nodes.IntPoint(3).x  = 1.0;
   Nodes.IntPoint(3).y  = 2./3.;
   // y = 1
   Nodes.IntPoint(4).x  = 2./3.;
   Nodes.IntPoint(4).y  = 1.0;
   Nodes.IntPoint(5).x  = 1./3.;
   Nodes.IntPoint(5).y  = 1.0;
   // x = 0
   Nodes.IntPoint(6).x  = 0.0;
   Nodes.IntPoint(6).y  = 2./3.;
   Nodes.IntPoint(7).x  = 0.0;
   Nodes.IntPoint(7).y  = 1./3.;
   // x = 0.5 (interior)
   Nodes.IntPoint(8).x  = 0.5;
   Nodes.IntPoint(8).y  = 1./3.;
   Nodes.IntPoint(9).x  = 0.5;
   Nodes.IntPoint(9).y  = 2./3.;
   // y = 0.5 (interior)
   Nodes.IntPoint(10).x = 1./3.;
   Nodes.IntPoint(10).y = 0.5;
   Nodes.IntPoint(11).x = 2./3.;
   Nodes.IntPoint(11).y = 0.5;
}

void RT1QuadFiniteElement::CalcVShape(const IntegrationPoint &ip,
                                      DenseMatrix &shape) const
{
   double x = ip.x, y = ip.y;

   // y = 0
   shape(0,0)  = 0;
   shape(0,1)  = -( 1. - 3.*y + 2.*y*y)*( 2. - 3.*x);
   shape(1,0)  = 0;
   shape(1,1)  = -( 1. - 3.*y + 2.*y*y)*(-1. + 3.*x);
   // x = 1
   shape(2,0)  = (-x + 2.*x*x)*( 2. - 3.*y);
   shape(2,1)  = 0;
   shape(3,0)  = (-x + 2.*x*x)*(-1. + 3.*y);
   shape(3,1)  = 0;
   // y = 1
   shape(4,0)  = 0;
   shape(4,1)  = (-y + 2.*y*y)*(-1. + 3.*x);
   shape(5,0)  = 0;
   shape(5,1)  = (-y + 2.*y*y)*( 2. - 3.*x);
   // x = 0
   shape(6,0)  = -(1. - 3.*x + 2.*x*x)*(-1. + 3.*y);
   shape(6,1)  = 0;
   shape(7,0)  = -(1. - 3.*x + 2.*x*x)*( 2. - 3.*y);
   shape(7,1)  = 0;
   // x = 0.5 (interior)
   shape(8,0)  = (4.*x - 4.*x*x)*( 2. - 3.*y);
   shape(8,1)  = 0;
   shape(9,0)  = (4.*x - 4.*x*x)*(-1. + 3.*y);
   shape(9,1)  = 0;
   // y = 0.5 (interior)
   shape(10,0) = 0;
   shape(10,1) = (4.*y - 4.*y*y)*( 2. - 3.*x);
   shape(11,0) = 0;
   shape(11,1) = (4.*y - 4.*y*y)*(-1. + 3.*x);
}

void RT1QuadFiniteElement::CalcDivShape(const IntegrationPoint &ip,
                                        Vector &divshape) const
{
   double x = ip.x, y = ip.y;

   divshape(0)  = -(-3. + 4.*y)*( 2. - 3.*x);
   divshape(1)  = -(-3. + 4.*y)*(-1. + 3.*x);
   divshape(2)  = (-1. + 4.*x)*( 2. - 3.*y);
   divshape(3)  = (-1. + 4.*x)*(-1. + 3.*y);
   divshape(4)  = (-1. + 4.*y)*(-1. + 3.*x);
   divshape(5)  = (-1. + 4.*y)*( 2. - 3.*x);
   divshape(6)  = -(-3. + 4.*x)*(-1. + 3.*y);
   divshape(7)  = -(-3. + 4.*x)*( 2. - 3.*y);
   divshape(8)  = ( 4. - 8.*x)*( 2. - 3.*y);
   divshape(9)  = ( 4. - 8.*x)*(-1. + 3.*y);
   divshape(10) = ( 4. - 8.*y)*( 2. - 3.*x);
   divshape(11) = ( 4. - 8.*y)*(-1. + 3.*x);
}

const double RT1QuadFiniteElement::nk[12][2] =
{
   // y = 0
   {0,-1}, {0,-1},
   // X = 1
   {1, 0}, {1, 0},
   // y = 1
   {0, 1}, {0, 1},
   // x = 0
   {-1,0}, {-1,0},
   // x = 0.5 (interior)
   {1, 0}, {1, 0},
   // y = 0.5 (interior)
   {0, 1}, {0, 1}
};

void RT1QuadFiniteElement::GetLocalInterpolation (
   ElementTransformation &Trans, DenseMatrix &I) const
{
   int k, j;
#ifdef MFEM_THREAD_SAFE
   DenseMatrix vshape(Dof, Dim);
   DenseMatrix Jinv(Dim);
#endif

#ifdef MFEM_DEBUG
   for (k = 0; k < 12; k++)
   {
      CalcVShape (Nodes.IntPoint(k), vshape);
      for (j = 0; j < 12; j++)
      {
         double d = vshape(j,0)*nk[k][0]+vshape(j,1)*nk[k][1];
         if (j == k) { d -= 1.0; }
         if (fabs(d) > 1.0e-12)
         {
            mfem::err << "RT1QuadFiniteElement::GetLocalInterpolation (...)\n"
                      " k = " << k << ", j = " << j << ", d = " << d << endl;
            mfem_error();
         }
      }
   }
#endif

   IntegrationPoint ip;
   ip.x = ip.y = 0.0;
   Trans.SetIntPoint (&ip);
   // Trans must be linear (more to have embedding?)
   // set Jinv = |J| J^{-t} = adj(J)^t
   CalcAdjugateTranspose (Trans.Jacobian(), Jinv);
   double vk[2];
   Vector xk (vk, 2);

   for (k = 0; k < 12; k++)
   {
      Trans.Transform (Nodes.IntPoint (k), xk);
      ip.x = vk[0]; ip.y = vk[1];
      CalcVShape (ip, vshape);
      //  vk = |J| J^{-t} nk
      vk[0] = Jinv(0,0)*nk[k][0]+Jinv(0,1)*nk[k][1];
      vk[1] = Jinv(1,0)*nk[k][0]+Jinv(1,1)*nk[k][1];
      for (j = 0; j < 12; j++)
         if (fabs (I(k,j) = vshape(j,0)*vk[0]+vshape(j,1)*vk[1]) < 1.0e-12)
         {
            I(k,j) = 0.0;
         }
   }
}

void RT1QuadFiniteElement::Project (
   VectorCoefficient &vc, ElementTransformation &Trans, Vector &dofs) const
{
   double vk[2];
   Vector xk (vk, 2);
#ifdef MFEM_THREAD_SAFE
   DenseMatrix Jinv(Dim);
#endif

   for (int k = 0; k < 12; k++)
   {
      Trans.SetIntPoint (&Nodes.IntPoint (k));
      // set Jinv = |J| J^{-t} = adj(J)^t
      CalcAdjugateTranspose (Trans.Jacobian(), Jinv);

      vc.Eval (xk, Trans, Nodes.IntPoint (k));
      //  xk^t |J| J^{-t} nk
      dofs(k) = (vk[0] * ( Jinv(0,0)*nk[k][0]+Jinv(0,1)*nk[k][1] ) +
                 vk[1] * ( Jinv(1,0)*nk[k][0]+Jinv(1,1)*nk[k][1] ));
   }
}

const double RT2TriangleFiniteElement::M[15][15] =
{
   {
      0, -5.3237900077244501311, 5.3237900077244501311, 16.647580015448900262,
      0, 24.442740046346700787, -16.647580015448900262, -12.,
      -19.118950038622250656, -47.237900077244501311, 0, -34.414110069520051180,
      12., 30.590320061795601049, 15.295160030897800524
   },
   {
      0, 1.5, -1.5, -15., 0, 2.625, 15., 15., -4.125, 30., 0, -14.625, -15.,
      -15., 10.5
   },
   {
      0, -0.67620999227554986889, 0.67620999227554986889, 7.3524199845510997378,
      0, -3.4427400463467007866, -7.3524199845510997378, -12.,
      4.1189500386222506555, -0.76209992275549868892, 0, 7.4141100695200511800,
      12., -6.5903200617956010489, -3.2951600308978005244
   },
   {
      0, 0, 1.5, 0, 0, 1.5, -11.471370023173350393, 0, 2.4713700231733503933,
      -11.471370023173350393, 0, 2.4713700231733503933, 15.295160030897800524,
      0, -3.2951600308978005244
   },
   {
      0, 0, 4.875, 0, 0, 4.875, -16.875, 0, -16.875, -16.875, 0, -16.875, 10.5,
      36., 10.5
   },
   {
      0, 0, 1.5, 0, 0, 1.5, 2.4713700231733503933, 0, -11.471370023173350393,
      2.4713700231733503933, 0, -11.471370023173350393, -3.2951600308978005244,
      0, 15.295160030897800524
   },
   {
      -0.67620999227554986889, 0, -3.4427400463467007866, 0,
      7.3524199845510997378, 0.67620999227554986889, 7.4141100695200511800, 0,
      -0.76209992275549868892, 4.1189500386222506555, -12.,
      -7.3524199845510997378, -3.2951600308978005244, -6.5903200617956010489,
      12.
   },
   {
      1.5, 0, 2.625, 0, -15., -1.5, -14.625, 0, 30., -4.125, 15., 15., 10.5,
      -15., -15.
   },
   {
      -5.3237900077244501311, 0, 24.442740046346700787, 0, 16.647580015448900262,
      5.3237900077244501311, -34.414110069520051180, 0, -47.237900077244501311,
      -19.118950038622250656, -12., -16.647580015448900262, 15.295160030897800524,
      30.590320061795601049, 12.
   },
   { 0, 0, 18., 0, 0, 6., -42., 0, -30., -26., 0, -14., 24., 32., 8.},
   { 0, 0, 6., 0, 0, 18., -14., 0, -26., -30., 0, -42., 8., 32., 24.},
   { 0, 0, -6., 0, 0, -4., 30., 0, 4., 22., 0, 4., -24., -16., 0},
   { 0, 0, -4., 0, 0, -8., 20., 0, 8., 36., 0, 8., -16., -32., 0},
   { 0, 0, -8., 0, 0, -4., 8., 0, 36., 8., 0, 20., 0, -32., -16.},
   { 0, 0, -4., 0, 0, -6., 4., 0, 22., 4., 0, 30., 0, -16., -24.}
};

RT2TriangleFiniteElement::RT2TriangleFiniteElement()
   : VectorFiniteElement(2, Geometry::TRIANGLE, 15, 3, H_DIV)
{
   const double p = 0.11270166537925831148;

   Nodes.IntPoint(0).x = p;
   Nodes.IntPoint(0).y = 0.0;
   Nodes.IntPoint(1).x = 0.5;
   Nodes.IntPoint(1).y = 0.0;
   Nodes.IntPoint(2).x = 1.-p;
   Nodes.IntPoint(2).y = 0.0;
   Nodes.IntPoint(3).x = 1.-p;
   Nodes.IntPoint(3).y = p;
   Nodes.IntPoint(4).x = 0.5;
   Nodes.IntPoint(4).y = 0.5;
   Nodes.IntPoint(5).x = p;
   Nodes.IntPoint(5).y = 1.-p;
   Nodes.IntPoint(6).x = 0.0;
   Nodes.IntPoint(6).y = 1.-p;
   Nodes.IntPoint(7).x = 0.0;
   Nodes.IntPoint(7).y = 0.5;
   Nodes.IntPoint(8).x = 0.0;
   Nodes.IntPoint(8).y = p;
   Nodes.IntPoint(9).x  = 0.25;
   Nodes.IntPoint(9).y  = 0.25;
   Nodes.IntPoint(10).x = 0.25;
   Nodes.IntPoint(10).y = 0.25;
   Nodes.IntPoint(11).x = 0.5;
   Nodes.IntPoint(11).y = 0.25;
   Nodes.IntPoint(12).x = 0.5;
   Nodes.IntPoint(12).y = 0.25;
   Nodes.IntPoint(13).x = 0.25;
   Nodes.IntPoint(13).y = 0.5;
   Nodes.IntPoint(14).x = 0.25;
   Nodes.IntPoint(14).y = 0.5;
}

void RT2TriangleFiniteElement::CalcVShape(const IntegrationPoint &ip,
                                          DenseMatrix &shape) const
{
   double x = ip.x, y = ip.y;

   double Bx[15] = {1., 0., x, 0., y, 0., x*x, 0., x*y, 0., y*y, 0., x*x*x,
                    x*x*y, x*y*y
                   };
   double By[15] = {0., 1., 0., x, 0., y, 0., x*x, 0., x*y, 0., y*y,
                    x*x*y, x*y*y, y*y*y
                   };

   for (int i = 0; i < 15; i++)
   {
      double cx = 0.0, cy = 0.0;
      for (int j = 0; j < 15; j++)
      {
         cx += M[i][j] * Bx[j];
         cy += M[i][j] * By[j];
      }
      shape(i,0) = cx;
      shape(i,1) = cy;
   }
}

void RT2TriangleFiniteElement::CalcDivShape(const IntegrationPoint &ip,
                                            Vector &divshape) const
{
   double x = ip.x, y = ip.y;

   double DivB[15] = {0., 0., 1., 0., 0., 1., 2.*x, 0., y, x, 0., 2.*y,
                      4.*x*x, 4.*x*y, 4.*y*y
                     };

   for (int i = 0; i < 15; i++)
   {
      double div = 0.0;
      for (int j = 0; j < 15; j++)
      {
         div += M[i][j] * DivB[j];
      }
      divshape(i) = div;
   }
}

const double RT2QuadFiniteElement::pt[4] = {0.,1./3.,2./3.,1.};

const double RT2QuadFiniteElement::dpt[3] = {0.25,0.5,0.75};

RT2QuadFiniteElement::RT2QuadFiniteElement()
   : VectorFiniteElement(2, Geometry::SQUARE, 24, 3, H_DIV, FunctionSpace::Qk)
{
   // y = 0 (pt[0])
   Nodes.IntPoint(0).x  = dpt[0];  Nodes.IntPoint(0).y  =  pt[0];
   Nodes.IntPoint(1).x  = dpt[1];  Nodes.IntPoint(1).y  =  pt[0];
   Nodes.IntPoint(2).x  = dpt[2];  Nodes.IntPoint(2).y  =  pt[0];
   // x = 1 (pt[3])
   Nodes.IntPoint(3).x  =  pt[3];  Nodes.IntPoint(3).y  = dpt[0];
   Nodes.IntPoint(4).x  =  pt[3];  Nodes.IntPoint(4).y  = dpt[1];
   Nodes.IntPoint(5).x  =  pt[3];  Nodes.IntPoint(5).y  = dpt[2];
   // y = 1 (pt[3])
   Nodes.IntPoint(6).x  = dpt[2];  Nodes.IntPoint(6).y  =  pt[3];
   Nodes.IntPoint(7).x  = dpt[1];  Nodes.IntPoint(7).y  =  pt[3];
   Nodes.IntPoint(8).x  = dpt[0];  Nodes.IntPoint(8).y  =  pt[3];
   // x = 0 (pt[0])
   Nodes.IntPoint(9).x  =  pt[0];  Nodes.IntPoint(9).y  = dpt[2];
   Nodes.IntPoint(10).x =  pt[0];  Nodes.IntPoint(10).y = dpt[1];
   Nodes.IntPoint(11).x =  pt[0];  Nodes.IntPoint(11).y = dpt[0];
   // x = pt[1] (interior)
   Nodes.IntPoint(12).x =  pt[1];  Nodes.IntPoint(12).y = dpt[0];
   Nodes.IntPoint(13).x =  pt[1];  Nodes.IntPoint(13).y = dpt[1];
   Nodes.IntPoint(14).x =  pt[1];  Nodes.IntPoint(14).y = dpt[2];
   // x = pt[2] (interior)
   Nodes.IntPoint(15).x =  pt[2];  Nodes.IntPoint(15).y = dpt[0];
   Nodes.IntPoint(16).x =  pt[2];  Nodes.IntPoint(16).y = dpt[1];
   Nodes.IntPoint(17).x =  pt[2];  Nodes.IntPoint(17).y = dpt[2];
   // y = pt[1] (interior)
   Nodes.IntPoint(18).x = dpt[0];  Nodes.IntPoint(18).y =  pt[1];
   Nodes.IntPoint(19).x = dpt[1];  Nodes.IntPoint(19).y =  pt[1];
   Nodes.IntPoint(20).x = dpt[2];  Nodes.IntPoint(20).y =  pt[1];
   // y = pt[2] (interior)
   Nodes.IntPoint(21).x = dpt[0];  Nodes.IntPoint(21).y =  pt[2];
   Nodes.IntPoint(22).x = dpt[1];  Nodes.IntPoint(22).y =  pt[2];
   Nodes.IntPoint(23).x = dpt[2];  Nodes.IntPoint(23).y =  pt[2];
}

void RT2QuadFiniteElement::CalcVShape(const IntegrationPoint &ip,
                                      DenseMatrix &shape) const
{
   double x = ip.x, y = ip.y;

   double ax0 =  pt[0] - x;
   double ax1 =  pt[1] - x;
   double ax2 =  pt[2] - x;
   double ax3 =  pt[3] - x;

   double by0 = dpt[0] - y;
   double by1 = dpt[1] - y;
   double by2 = dpt[2] - y;

   double ay0 =  pt[0] - y;
   double ay1 =  pt[1] - y;
   double ay2 =  pt[2] - y;
   double ay3 =  pt[3] - y;

   double bx0 = dpt[0] - x;
   double bx1 = dpt[1] - x;
   double bx2 = dpt[2] - x;

   double A01 =  pt[0] -  pt[1];
   double A02 =  pt[0] -  pt[2];
   double A12 =  pt[1] -  pt[2];
   double A03 =  pt[0] -  pt[3];
   double A13 =  pt[1] -  pt[3];
   double A23 =  pt[2] -  pt[3];

   double B01 = dpt[0] - dpt[1];
   double B02 = dpt[0] - dpt[2];
   double B12 = dpt[1] - dpt[2];

   double tx0 =  (bx1*bx2)/(B01*B02);
   double tx1 = -(bx0*bx2)/(B01*B12);
   double tx2 =  (bx0*bx1)/(B02*B12);

   double ty0 =  (by1*by2)/(B01*B02);
   double ty1 = -(by0*by2)/(B01*B12);
   double ty2 =  (by0*by1)/(B02*B12);

   // y = 0 (p[0])
   shape(0,  0) =  0;
   shape(0,  1) =  (ay1*ay2*ay3)/(A01*A02*A03)*tx0;
   shape(1,  0) =  0;
   shape(1,  1) =  (ay1*ay2*ay3)/(A01*A02*A03)*tx1;
   shape(2,  0) =  0;
   shape(2,  1) =  (ay1*ay2*ay3)/(A01*A02*A03)*tx2;
   // x = 1 (p[3])
   shape(3,  0) =  (ax0*ax1*ax2)/(A03*A13*A23)*ty0;
   shape(3,  1) =  0;
   shape(4,  0) =  (ax0*ax1*ax2)/(A03*A13*A23)*ty1;
   shape(4,  1) =  0;
   shape(5,  0) =  (ax0*ax1*ax2)/(A03*A13*A23)*ty2;
   shape(5,  1) =  0;
   // y = 1 (p[3])
   shape(6,  0) =  0;
   shape(6,  1) =  (ay0*ay1*ay2)/(A03*A13*A23)*tx2;
   shape(7,  0) =  0;
   shape(7,  1) =  (ay0*ay1*ay2)/(A03*A13*A23)*tx1;
   shape(8,  0) =  0;
   shape(8,  1) =  (ay0*ay1*ay2)/(A03*A13*A23)*tx0;
   // x = 0 (p[0])
   shape(9,  0) =  (ax1*ax2*ax3)/(A01*A02*A03)*ty2;
   shape(9,  1) =  0;
   shape(10, 0) =  (ax1*ax2*ax3)/(A01*A02*A03)*ty1;
   shape(10, 1) =  0;
   shape(11, 0) =  (ax1*ax2*ax3)/(A01*A02*A03)*ty0;
   shape(11, 1) =  0;
   // x = p[1] (interior)
   shape(12, 0) =  (ax0*ax2*ax3)/(A01*A12*A13)*ty0;
   shape(12, 1) =  0;
   shape(13, 0) =  (ax0*ax2*ax3)/(A01*A12*A13)*ty1;
   shape(13, 1) =  0;
   shape(14, 0) =  (ax0*ax2*ax3)/(A01*A12*A13)*ty2;
   shape(14, 1) =  0;
   // x = p[2] (interior)
   shape(15, 0) = -(ax0*ax1*ax3)/(A02*A12*A23)*ty0;
   shape(15, 1) =  0;
   shape(16, 0) = -(ax0*ax1*ax3)/(A02*A12*A23)*ty1;
   shape(16, 1) =  0;
   shape(17, 0) = -(ax0*ax1*ax3)/(A02*A12*A23)*ty2;
   shape(17, 1) =  0;
   // y = p[1] (interior)
   shape(18, 0) =  0;
   shape(18, 1) =  (ay0*ay2*ay3)/(A01*A12*A13)*tx0;
   shape(19, 0) =  0;
   shape(19, 1) =  (ay0*ay2*ay3)/(A01*A12*A13)*tx1;
   shape(20, 0) =  0;
   shape(20, 1) =  (ay0*ay2*ay3)/(A01*A12*A13)*tx2;
   // y = p[2] (interior)
   shape(21, 0) =  0;
   shape(21, 1) = -(ay0*ay1*ay3)/(A02*A12*A23)*tx0;
   shape(22, 0) =  0;
   shape(22, 1) = -(ay0*ay1*ay3)/(A02*A12*A23)*tx1;
   shape(23, 0) =  0;
   shape(23, 1) = -(ay0*ay1*ay3)/(A02*A12*A23)*tx2;
}

void RT2QuadFiniteElement::CalcDivShape(const IntegrationPoint &ip,
                                        Vector &divshape) const
{
   double x = ip.x, y = ip.y;

   double a01 =  pt[0]*pt[1];
   double a02 =  pt[0]*pt[2];
   double a12 =  pt[1]*pt[2];
   double a03 =  pt[0]*pt[3];
   double a13 =  pt[1]*pt[3];
   double a23 =  pt[2]*pt[3];

   double bx0 = dpt[0] - x;
   double bx1 = dpt[1] - x;
   double bx2 = dpt[2] - x;

   double by0 = dpt[0] - y;
   double by1 = dpt[1] - y;
   double by2 = dpt[2] - y;

   double A01 =  pt[0] -  pt[1];
   double A02 =  pt[0] -  pt[2];
   double A12 =  pt[1] -  pt[2];
   double A03 =  pt[0] -  pt[3];
   double A13 =  pt[1] -  pt[3];
   double A23 =  pt[2] -  pt[3];

   double A012 = pt[0] + pt[1] + pt[2];
   double A013 = pt[0] + pt[1] + pt[3];
   double A023 = pt[0] + pt[2] + pt[3];
   double A123 = pt[1] + pt[2] + pt[3];

   double B01 = dpt[0] - dpt[1];
   double B02 = dpt[0] - dpt[2];
   double B12 = dpt[1] - dpt[2];

   double tx0 =  (bx1*bx2)/(B01*B02);
   double tx1 = -(bx0*bx2)/(B01*B12);
   double tx2 =  (bx0*bx1)/(B02*B12);

   double ty0 =  (by1*by2)/(B01*B02);
   double ty1 = -(by0*by2)/(B01*B12);
   double ty2 =  (by0*by1)/(B02*B12);

   // y = 0 (p[0])
   divshape(0)  = -(a12 + a13 + a23 - 2.*A123*y + 3.*y*y)/(A01*A02*A03)*tx0;
   divshape(1)  = -(a12 + a13 + a23 - 2.*A123*y + 3.*y*y)/(A01*A02*A03)*tx1;
   divshape(2)  = -(a12 + a13 + a23 - 2.*A123*y + 3.*y*y)/(A01*A02*A03)*tx2;
   // x = 1 (p[3])
   divshape(3)  = -(a01 + a02 + a12 - 2.*A012*x + 3.*x*x)/(A03*A13*A23)*ty0;
   divshape(4)  = -(a01 + a02 + a12 - 2.*A012*x + 3.*x*x)/(A03*A13*A23)*ty1;
   divshape(5)  = -(a01 + a02 + a12 - 2.*A012*x + 3.*x*x)/(A03*A13*A23)*ty2;
   // y = 1 (p[3])
   divshape(6)  = -(a01 + a02 + a12 - 2.*A012*y + 3.*y*y)/(A03*A13*A23)*tx2;
   divshape(7)  = -(a01 + a02 + a12 - 2.*A012*y + 3.*y*y)/(A03*A13*A23)*tx1;
   divshape(8)  = -(a01 + a02 + a12 - 2.*A012*y + 3.*y*y)/(A03*A13*A23)*tx0;
   // x = 0 (p[0])
   divshape(9)  = -(a12 + a13 + a23 - 2.*A123*x + 3.*x*x)/(A01*A02*A03)*ty2;
   divshape(10) = -(a12 + a13 + a23 - 2.*A123*x + 3.*x*x)/(A01*A02*A03)*ty1;
   divshape(11) = -(a12 + a13 + a23 - 2.*A123*x + 3.*x*x)/(A01*A02*A03)*ty0;
   // x = p[1] (interior)
   divshape(12) = -(a02 + a03 + a23 - 2.*A023*x + 3.*x*x)/(A01*A12*A13)*ty0;
   divshape(13) = -(a02 + a03 + a23 - 2.*A023*x + 3.*x*x)/(A01*A12*A13)*ty1;
   divshape(14) = -(a02 + a03 + a23 - 2.*A023*x + 3.*x*x)/(A01*A12*A13)*ty2;
   // x = p[2] (interior)
   divshape(15) =  (a01 + a03 + a13 - 2.*A013*x + 3.*x*x)/(A02*A12*A23)*ty0;
   divshape(16) =  (a01 + a03 + a13 - 2.*A013*x + 3.*x*x)/(A02*A12*A23)*ty1;
   divshape(17) =  (a01 + a03 + a13 - 2.*A013*x + 3.*x*x)/(A02*A12*A23)*ty2;
   // y = p[1] (interior)
   divshape(18) = -(a02 + a03 + a23 - 2.*A023*y + 3.*y*y)/(A01*A12*A13)*tx0;
   divshape(19) = -(a02 + a03 + a23 - 2.*A023*y + 3.*y*y)/(A01*A12*A13)*tx1;
   divshape(20) = -(a02 + a03 + a23 - 2.*A023*y + 3.*y*y)/(A01*A12*A13)*tx2;
   // y = p[2] (interior)
   divshape(21) =  (a01 + a03 + a13 - 2.*A013*y + 3.*y*y)/(A02*A12*A23)*tx0;
   divshape(22) =  (a01 + a03 + a13 - 2.*A013*y + 3.*y*y)/(A02*A12*A23)*tx1;
   divshape(23) =  (a01 + a03 + a13 - 2.*A013*y + 3.*y*y)/(A02*A12*A23)*tx2;
}

const double RT2QuadFiniteElement::nk[24][2] =
{
   // y = 0
   {0,-1}, {0,-1}, {0,-1},
   // x = 1
   {1, 0}, {1, 0}, {1, 0},
   // y = 1
   {0, 1}, {0, 1}, {0, 1},
   // x = 0
   {-1,0}, {-1,0}, {-1,0},
   // x = p[1] (interior)
   {1, 0}, {1, 0}, {1, 0},
   // x = p[2] (interior)
   {1, 0}, {1, 0}, {1, 0},
   // y = p[1] (interior)
   {0, 1}, {0, 1}, {0, 1},
   // y = p[1] (interior)
   {0, 1}, {0, 1}, {0, 1}
};

void RT2QuadFiniteElement::GetLocalInterpolation (
   ElementTransformation &Trans, DenseMatrix &I) const
{
   int k, j;
#ifdef MFEM_THREAD_SAFE
   DenseMatrix vshape(Dof, Dim);
   DenseMatrix Jinv(Dim);
#endif

#ifdef MFEM_DEBUG
   for (k = 0; k < 24; k++)
   {
      CalcVShape (Nodes.IntPoint(k), vshape);
      for (j = 0; j < 24; j++)
      {
         double d = vshape(j,0)*nk[k][0]+vshape(j,1)*nk[k][1];
         if (j == k) { d -= 1.0; }
         if (fabs(d) > 1.0e-12)
         {
            mfem::err << "RT2QuadFiniteElement::GetLocalInterpolation (...)\n"
                      " k = " << k << ", j = " << j << ", d = " << d << endl;
            mfem_error();
         }
      }
   }
#endif

   IntegrationPoint ip;
   ip.x = ip.y = 0.0;
   Trans.SetIntPoint (&ip);
   // Trans must be linear (more to have embedding?)
   // set Jinv = |J| J^{-t} = adj(J)^t
   CalcAdjugateTranspose (Trans.Jacobian(), Jinv);
   double vk[2];
   Vector xk (vk, 2);

   for (k = 0; k < 24; k++)
   {
      Trans.Transform (Nodes.IntPoint (k), xk);
      ip.x = vk[0]; ip.y = vk[1];
      CalcVShape (ip, vshape);
      //  vk = |J| J^{-t} nk
      vk[0] = Jinv(0,0)*nk[k][0]+Jinv(0,1)*nk[k][1];
      vk[1] = Jinv(1,0)*nk[k][0]+Jinv(1,1)*nk[k][1];
      for (j = 0; j < 24; j++)
         if (fabs (I(k,j) = vshape(j,0)*vk[0]+vshape(j,1)*vk[1]) < 1.0e-12)
         {
            I(k,j) = 0.0;
         }
   }
}

void RT2QuadFiniteElement::Project (
   VectorCoefficient &vc, ElementTransformation &Trans, Vector &dofs) const
{
   double vk[2];
   Vector xk (vk, 2);
#ifdef MFEM_THREAD_SAFE
   DenseMatrix Jinv(Dim);
#endif

   for (int k = 0; k < 24; k++)
   {
      Trans.SetIntPoint (&Nodes.IntPoint (k));
      // set Jinv = |J| J^{-t} = adj(J)^t
      CalcAdjugateTranspose (Trans.Jacobian(), Jinv);

      vc.Eval (xk, Trans, Nodes.IntPoint (k));
      //  xk^t |J| J^{-t} nk
      dofs(k) = (vk[0] * ( Jinv(0,0)*nk[k][0]+Jinv(0,1)*nk[k][1] ) +
                 vk[1] * ( Jinv(1,0)*nk[k][0]+Jinv(1,1)*nk[k][1] ));
   }
}

P1SegmentFiniteElement::P1SegmentFiniteElement()
   : NodalFiniteElement(1, Geometry::SEGMENT, 2, 1)
{
   Nodes.IntPoint(0).x = 0.33333333333333333333;
   Nodes.IntPoint(1).x = 0.66666666666666666667;
}

void P1SegmentFiniteElement::CalcShape(const IntegrationPoint &ip,
                                       Vector &shape) const
{
   double x = ip.x;

   shape(0) = 2. - 3. * x;
   shape(1) = 3. * x - 1.;
}

void P1SegmentFiniteElement::CalcDShape(const IntegrationPoint &ip,
                                        DenseMatrix &dshape) const
{
   dshape(0,0) = -3.;
   dshape(1,0) =  3.;
}


P2SegmentFiniteElement::P2SegmentFiniteElement()
   : NodalFiniteElement(1, Geometry::SEGMENT, 3, 2)
{
   const double p = 0.11270166537925831148;

   Nodes.IntPoint(0).x = p;
   Nodes.IntPoint(1).x = 0.5;
   Nodes.IntPoint(2).x = 1.-p;
}

void P2SegmentFiniteElement::CalcShape(const IntegrationPoint &ip,
                                       Vector &shape) const
{
   const double p = 0.11270166537925831148;
   const double w = 1./((1-2*p)*(1-2*p));
   double x = ip.x;

   shape(0) = (2*x-1)*(x-1+p)*w;
   shape(1) = 4*(x-1+p)*(p-x)*w;
   shape(2) = (2*x-1)*(x-p)*w;
}

void P2SegmentFiniteElement::CalcDShape(const IntegrationPoint &ip,
                                        DenseMatrix &dshape) const
{
   const double p = 0.11270166537925831148;
   const double w = 1./((1-2*p)*(1-2*p));
   double x = ip.x;

   dshape(0,0) = (-3+4*x+2*p)*w;
   dshape(1,0) = (4-8*x)*w;
   dshape(2,0) = (-1+4*x-2*p)*w;
}


Lagrange1DFiniteElement::Lagrange1DFiniteElement(int degree)
   : NodalFiniteElement(1, Geometry::SEGMENT, degree+1, degree)
{
   int i, m = degree;

   Nodes.IntPoint(0).x = 0.0;
   Nodes.IntPoint(1).x = 1.0;
   for (i = 1; i < m; i++)
   {
      Nodes.IntPoint(i+1).x = double(i) / m;
   }

   rwk.SetSize(degree+1);
#ifndef MFEM_THREAD_SAFE
   rxxk.SetSize(degree+1);
#endif

   rwk(0) = 1.0;
   for (i = 1; i <= m; i++)
   {
      rwk(i) = rwk(i-1) * ( (double)(m) / (double)(i) );
   }
   for (i = 0; i < m/2+1; i++)
   {
      rwk(m-i) = ( rwk(i) *= rwk(m-i) );
   }
   for (i = m-1; i >= 0; i -= 2)
   {
      rwk(i) = -rwk(i);
   }
}

void Lagrange1DFiniteElement::CalcShape(const IntegrationPoint &ip,
                                        Vector &shape) const
{
   double w, wk, x = ip.x;
   int i, k, m = GetOrder();

#ifdef MFEM_THREAD_SAFE
   Vector rxxk(m+1);
#endif

   k = (int) floor ( m * x + 0.5 );
   k = k > m ? m : k < 0 ? 0 : k; // clamp k to [0,m]

   wk = 1.0;
   for (i = 0; i <= m; i++)
      if (i != k)
      {
         wk *= ( rxxk(i) = x - (double)(i) / m );
      }
   w = wk * ( rxxk(k) = x - (double)(k) / m );

   if (k != 0)
   {
      shape(0) = w * rwk(0) / rxxk(0);
   }
   else
   {
      shape(0) = wk * rwk(0);
   }
   if (k != m)
   {
      shape(1) = w * rwk(m) / rxxk(m);
   }
   else
   {
      shape(1) = wk * rwk(k);
   }
   for (i = 1; i < m; i++)
      if (i != k)
      {
         shape(i+1) = w * rwk(i) / rxxk(i);
      }
      else
      {
         shape(k+1) = wk * rwk(k);
      }
}

void Lagrange1DFiniteElement::CalcDShape(const IntegrationPoint &ip,
                                         DenseMatrix &dshape) const
{
   double s, srx, w, wk, x = ip.x;
   int i, k, m = GetOrder();

#ifdef MFEM_THREAD_SAFE
   Vector rxxk(m+1);
#endif

   k = (int) floor ( m * x + 0.5 );
   k = k > m ? m : k < 0 ? 0 : k; // clamp k to [0,m]

   wk = 1.0;
   for (i = 0; i <= m; i++)
      if (i != k)
      {
         wk *= ( rxxk(i) = x - (double)(i) / m );
      }
   w = wk * ( rxxk(k) = x - (double)(k) / m );

   for (i = 0; i <= m; i++)
   {
      rxxk(i) = 1.0 / rxxk(i);
   }
   srx = 0.0;
   for (i = 0; i <= m; i++)
      if (i != k)
      {
         srx += rxxk(i);
      }
   s = w * srx + wk;

   if (k != 0)
   {
      dshape(0,0) = (s - w * rxxk(0)) * rwk(0) * rxxk(0);
   }
   else
   {
      dshape(0,0) = wk * srx * rwk(0);
   }
   if (k != m)
   {
      dshape(1,0) = (s - w * rxxk(m)) * rwk(m) * rxxk(m);
   }
   else
   {
      dshape(1,0) = wk * srx * rwk(k);
   }
   for (i = 1; i < m; i++)
      if (i != k)
      {
         dshape(i+1,0) = (s - w * rxxk(i)) * rwk(i) * rxxk(i);
      }
      else
      {
         dshape(k+1,0) = wk * srx * rwk(k);
      }
}


P1TetNonConfFiniteElement::P1TetNonConfFiniteElement()
   : NodalFiniteElement(3, Geometry::TETRAHEDRON, 4, 1)
{
   Nodes.IntPoint(0).x = 0.33333333333333333333;
   Nodes.IntPoint(0).y = 0.33333333333333333333;
   Nodes.IntPoint(0).z = 0.33333333333333333333;

   Nodes.IntPoint(1).x = 0.0;
   Nodes.IntPoint(1).y = 0.33333333333333333333;
   Nodes.IntPoint(1).z = 0.33333333333333333333;

   Nodes.IntPoint(2).x = 0.33333333333333333333;
   Nodes.IntPoint(2).y = 0.0;
   Nodes.IntPoint(2).z = 0.33333333333333333333;

   Nodes.IntPoint(3).x = 0.33333333333333333333;
   Nodes.IntPoint(3).y = 0.33333333333333333333;
   Nodes.IntPoint(3).z = 0.0;

}

void P1TetNonConfFiniteElement::CalcShape(const IntegrationPoint &ip,
                                          Vector &shape) const
{
   double L0, L1, L2, L3;

   L1 = ip.x;  L2 = ip.y;  L3 = ip.z;  L0 = 1.0 - L1 - L2 - L3;
   shape(0) = 1.0 - 3.0 * L0;
   shape(1) = 1.0 - 3.0 * L1;
   shape(2) = 1.0 - 3.0 * L2;
   shape(3) = 1.0 - 3.0 * L3;
}

void P1TetNonConfFiniteElement::CalcDShape(const IntegrationPoint &ip,
                                           DenseMatrix &dshape) const
{
   dshape(0,0) =  3.0; dshape(0,1) =  3.0; dshape(0,2) =  3.0;
   dshape(1,0) = -3.0; dshape(1,1) =  0.0; dshape(1,2) =  0.0;
   dshape(2,0) =  0.0; dshape(2,1) = -3.0; dshape(2,2) =  0.0;
   dshape(3,0) =  0.0; dshape(3,1) =  0.0; dshape(3,2) = -3.0;
}


P0TetFiniteElement::P0TetFiniteElement()
   : NodalFiniteElement(3, Geometry::TETRAHEDRON, 1, 0)
{
   Nodes.IntPoint(0).x = 0.25;
   Nodes.IntPoint(0).y = 0.25;
   Nodes.IntPoint(0).z = 0.25;
}

void P0TetFiniteElement::CalcShape(const IntegrationPoint &ip,
                                   Vector &shape) const
{
   shape(0) = 1.0;
}

void P0TetFiniteElement::CalcDShape(const IntegrationPoint &ip,
                                    DenseMatrix &dshape) const
{
   dshape(0,0) =  0.0; dshape(0,1) =  0.0; dshape(0,2) = 0.0;
}


P0HexFiniteElement::P0HexFiniteElement()
   : NodalFiniteElement(3, Geometry::CUBE, 1, 0, FunctionSpace::Qk)
{
   Nodes.IntPoint(0).x = 0.5;
   Nodes.IntPoint(0).y = 0.5;
   Nodes.IntPoint(0).z = 0.5;
}

void P0HexFiniteElement::CalcShape(const IntegrationPoint &ip,
                                   Vector &shape) const
{
   shape(0) = 1.0;
}

void P0HexFiniteElement::CalcDShape(const IntegrationPoint &ip,
                                    DenseMatrix &dshape) const
{
   dshape(0,0) =  0.0; dshape(0,1) =  0.0; dshape(0,2) = 0.0;
}


LagrangeHexFiniteElement::LagrangeHexFiniteElement (int degree)
   : NodalFiniteElement(3, Geometry::CUBE, (degree+1)*(degree+1)*(degree+1),
                        degree, FunctionSpace::Qk)
{
   if (degree == 2)
   {
      I = new int[Dof];
      J = new int[Dof];
      K = new int[Dof];
      // nodes
      I[ 0] = 0; J[ 0] = 0; K[ 0] = 0;
      I[ 1] = 1; J[ 1] = 0; K[ 1] = 0;
      I[ 2] = 1; J[ 2] = 1; K[ 2] = 0;
      I[ 3] = 0; J[ 3] = 1; K[ 3] = 0;
      I[ 4] = 0; J[ 4] = 0; K[ 4] = 1;
      I[ 5] = 1; J[ 5] = 0; K[ 5] = 1;
      I[ 6] = 1; J[ 6] = 1; K[ 6] = 1;
      I[ 7] = 0; J[ 7] = 1; K[ 7] = 1;
      // edges
      I[ 8] = 2; J[ 8] = 0; K[ 8] = 0;
      I[ 9] = 1; J[ 9] = 2; K[ 9] = 0;
      I[10] = 2; J[10] = 1; K[10] = 0;
      I[11] = 0; J[11] = 2; K[11] = 0;
      I[12] = 2; J[12] = 0; K[12] = 1;
      I[13] = 1; J[13] = 2; K[13] = 1;
      I[14] = 2; J[14] = 1; K[14] = 1;
      I[15] = 0; J[15] = 2; K[15] = 1;
      I[16] = 0; J[16] = 0; K[16] = 2;
      I[17] = 1; J[17] = 0; K[17] = 2;
      I[18] = 1; J[18] = 1; K[18] = 2;
      I[19] = 0; J[19] = 1; K[19] = 2;
      // faces
      I[20] = 2; J[20] = 2; K[20] = 0;
      I[21] = 2; J[21] = 0; K[21] = 2;
      I[22] = 1; J[22] = 2; K[22] = 2;
      I[23] = 2; J[23] = 1; K[23] = 2;
      I[24] = 0; J[24] = 2; K[24] = 2;
      I[25] = 2; J[25] = 2; K[25] = 1;
      // element
      I[26] = 2; J[26] = 2; K[26] = 2;
   }
   else if (degree == 3)
   {
      I = new int[Dof];
      J = new int[Dof];
      K = new int[Dof];
      // nodes
      I[ 0] = 0; J[ 0] = 0; K[ 0] = 0;
      I[ 1] = 1; J[ 1] = 0; K[ 1] = 0;
      I[ 2] = 1; J[ 2] = 1; K[ 2] = 0;
      I[ 3] = 0; J[ 3] = 1; K[ 3] = 0;
      I[ 4] = 0; J[ 4] = 0; K[ 4] = 1;
      I[ 5] = 1; J[ 5] = 0; K[ 5] = 1;
      I[ 6] = 1; J[ 6] = 1; K[ 6] = 1;
      I[ 7] = 0; J[ 7] = 1; K[ 7] = 1;
      // edges
      I[ 8] = 2; J[ 8] = 0; K[ 8] = 0;
      I[ 9] = 3; J[ 9] = 0; K[ 9] = 0;
      I[10] = 1; J[10] = 2; K[10] = 0;
      I[11] = 1; J[11] = 3; K[11] = 0;
      I[12] = 2; J[12] = 1; K[12] = 0;
      I[13] = 3; J[13] = 1; K[13] = 0;
      I[14] = 0; J[14] = 2; K[14] = 0;
      I[15] = 0; J[15] = 3; K[15] = 0;
      I[16] = 2; J[16] = 0; K[16] = 1;
      I[17] = 3; J[17] = 0; K[17] = 1;
      I[18] = 1; J[18] = 2; K[18] = 1;
      I[19] = 1; J[19] = 3; K[19] = 1;
      I[20] = 2; J[20] = 1; K[20] = 1;
      I[21] = 3; J[21] = 1; K[21] = 1;
      I[22] = 0; J[22] = 2; K[22] = 1;
      I[23] = 0; J[23] = 3; K[23] = 1;
      I[24] = 0; J[24] = 0; K[24] = 2;
      I[25] = 0; J[25] = 0; K[25] = 3;
      I[26] = 1; J[26] = 0; K[26] = 2;
      I[27] = 1; J[27] = 0; K[27] = 3;
      I[28] = 1; J[28] = 1; K[28] = 2;
      I[29] = 1; J[29] = 1; K[29] = 3;
      I[30] = 0; J[30] = 1; K[30] = 2;
      I[31] = 0; J[31] = 1; K[31] = 3;
      // faces
      I[32] = 2; J[32] = 3; K[32] = 0;
      I[33] = 3; J[33] = 3; K[33] = 0;
      I[34] = 2; J[34] = 2; K[34] = 0;
      I[35] = 3; J[35] = 2; K[35] = 0;
      I[36] = 2; J[36] = 0; K[36] = 2;
      I[37] = 3; J[37] = 0; K[37] = 2;
      I[38] = 2; J[38] = 0; K[38] = 3;
      I[39] = 3; J[39] = 0; K[39] = 3;
      I[40] = 1; J[40] = 2; K[40] = 2;
      I[41] = 1; J[41] = 3; K[41] = 2;
      I[42] = 1; J[42] = 2; K[42] = 3;
      I[43] = 1; J[43] = 3; K[43] = 3;
      I[44] = 3; J[44] = 1; K[44] = 2;
      I[45] = 2; J[45] = 1; K[45] = 2;
      I[46] = 3; J[46] = 1; K[46] = 3;
      I[47] = 2; J[47] = 1; K[47] = 3;
      I[48] = 0; J[48] = 3; K[48] = 2;
      I[49] = 0; J[49] = 2; K[49] = 2;
      I[50] = 0; J[50] = 3; K[50] = 3;
      I[51] = 0; J[51] = 2; K[51] = 3;
      I[52] = 2; J[52] = 2; K[52] = 1;
      I[53] = 3; J[53] = 2; K[53] = 1;
      I[54] = 2; J[54] = 3; K[54] = 1;
      I[55] = 3; J[55] = 3; K[55] = 1;
      // element
      I[56] = 2; J[56] = 2; K[56] = 2;
      I[57] = 3; J[57] = 2; K[57] = 2;
      I[58] = 3; J[58] = 3; K[58] = 2;
      I[59] = 2; J[59] = 3; K[59] = 2;
      I[60] = 2; J[60] = 2; K[60] = 3;
      I[61] = 3; J[61] = 2; K[61] = 3;
      I[62] = 3; J[62] = 3; K[62] = 3;
      I[63] = 2; J[63] = 3; K[63] = 3;
   }
   else
   {
      mfem_error ("LagrangeHexFiniteElement::LagrangeHexFiniteElement");
   }

   fe1d = new Lagrange1DFiniteElement(degree);
   dof1d = fe1d -> GetDof();

#ifndef MFEM_THREAD_SAFE
   shape1dx.SetSize(dof1d);
   shape1dy.SetSize(dof1d);
   shape1dz.SetSize(dof1d);

   dshape1dx.SetSize(dof1d,1);
   dshape1dy.SetSize(dof1d,1);
   dshape1dz.SetSize(dof1d,1);
#endif

   for (int n = 0; n < Dof; n++)
   {
      Nodes.IntPoint(n).x = fe1d -> GetNodes().IntPoint(I[n]).x;
      Nodes.IntPoint(n).y = fe1d -> GetNodes().IntPoint(J[n]).x;
      Nodes.IntPoint(n).z = fe1d -> GetNodes().IntPoint(K[n]).x;
   }
}

void LagrangeHexFiniteElement::CalcShape(const IntegrationPoint &ip,
                                         Vector &shape) const
{
   IntegrationPoint ipy, ipz;
   ipy.x = ip.y;
   ipz.x = ip.z;

#ifdef MFEM_THREAD_SAFE
   Vector shape1dx(dof1d), shape1dy(dof1d), shape1dz(dof1d);
#endif

   fe1d -> CalcShape(ip,  shape1dx);
   fe1d -> CalcShape(ipy, shape1dy);
   fe1d -> CalcShape(ipz, shape1dz);

   for (int n = 0; n < Dof; n++)
   {
      shape(n) = shape1dx(I[n]) *  shape1dy(J[n]) * shape1dz(K[n]);
   }
}

void LagrangeHexFiniteElement::CalcDShape(const IntegrationPoint &ip,
                                          DenseMatrix &dshape) const
{
   IntegrationPoint ipy, ipz;
   ipy.x = ip.y;
   ipz.x = ip.z;

#ifdef MFEM_THREAD_SAFE
   Vector shape1dx(dof1d), shape1dy(dof1d), shape1dz(dof1d);
   DenseMatrix dshape1dx(dof1d,1), dshape1dy(dof1d,1), dshape1dz(dof1d,1);
#endif

   fe1d -> CalcShape(ip,  shape1dx);
   fe1d -> CalcShape(ipy, shape1dy);
   fe1d -> CalcShape(ipz, shape1dz);

   fe1d -> CalcDShape(ip,  dshape1dx);
   fe1d -> CalcDShape(ipy, dshape1dy);
   fe1d -> CalcDShape(ipz, dshape1dz);

   for (int n = 0; n < Dof; n++)
   {
      dshape(n,0) = dshape1dx(I[n],0) * shape1dy(J[n])    * shape1dz(K[n]);
      dshape(n,1) = shape1dx(I[n])    * dshape1dy(J[n],0) * shape1dz(K[n]);
      dshape(n,2) = shape1dx(I[n])    * shape1dy(J[n])    * dshape1dz(K[n],0);
   }
}

LagrangeHexFiniteElement::~LagrangeHexFiniteElement ()
{
   delete fe1d;

   delete [] I;
   delete [] J;
   delete [] K;
}


RefinedLinear1DFiniteElement::RefinedLinear1DFiniteElement()
   : NodalFiniteElement(1, Geometry::SEGMENT, 3, 4)
{
   Nodes.IntPoint(0).x = 0.0;
   Nodes.IntPoint(1).x = 1.0;
   Nodes.IntPoint(2).x = 0.5;
}

void RefinedLinear1DFiniteElement::CalcShape(const IntegrationPoint &ip,
                                             Vector &shape) const
{
   double x = ip.x;

   if (x <= 0.5)
   {
      shape(0) = 1.0 - 2.0 * x;
      shape(1) = 0.0;
      shape(2) = 2.0 * x;
   }
   else
   {
      shape(0) = 0.0;
      shape(1) = 2.0 * x - 1.0;
      shape(2) = 2.0 - 2.0 * x;
   }
}

void RefinedLinear1DFiniteElement::CalcDShape(const IntegrationPoint &ip,
                                              DenseMatrix &dshape) const
{
   double x = ip.x;

   if (x <= 0.5)
   {
      dshape(0,0) = - 2.0;
      dshape(1,0) =   0.0;
      dshape(2,0) =   2.0;
   }
   else
   {
      dshape(0,0) =   0.0;
      dshape(1,0) =   2.0;
      dshape(2,0) = - 2.0;
   }
}

RefinedLinear2DFiniteElement::RefinedLinear2DFiniteElement()
   : NodalFiniteElement(2, Geometry::TRIANGLE, 6, 5)
{
   Nodes.IntPoint(0).x = 0.0;
   Nodes.IntPoint(0).y = 0.0;
   Nodes.IntPoint(1).x = 1.0;
   Nodes.IntPoint(1).y = 0.0;
   Nodes.IntPoint(2).x = 0.0;
   Nodes.IntPoint(2).y = 1.0;
   Nodes.IntPoint(3).x = 0.5;
   Nodes.IntPoint(3).y = 0.0;
   Nodes.IntPoint(4).x = 0.5;
   Nodes.IntPoint(4).y = 0.5;
   Nodes.IntPoint(5).x = 0.0;
   Nodes.IntPoint(5).y = 0.5;
}

void RefinedLinear2DFiniteElement::CalcShape(const IntegrationPoint &ip,
                                             Vector &shape) const
{
   int i;

   double L0, L1, L2;
   L0 = 2.0 * ( 1. - ip.x - ip.y );
   L1 = 2.0 * ( ip.x );
   L2 = 2.0 * ( ip.y );

   // The reference triangle is split in 4 triangles as follows:
   //
   // T0 - 0,3,5
   // T1 - 1,3,4
   // T2 - 2,4,5
   // T3 - 3,4,5

   for (i = 0; i < 6; i++)
   {
      shape(i) = 0.0;
   }

   if (L0 >= 1.0)   // T0
   {
      shape(0) = L0 - 1.0;
      shape(3) =       L1;
      shape(5) =       L2;
   }
   else if (L1 >= 1.0)   // T1
   {
      shape(3) =       L0;
      shape(1) = L1 - 1.0;
      shape(4) =       L2;
   }
   else if (L2 >= 1.0)   // T2
   {
      shape(5) =       L0;
      shape(4) =       L1;
      shape(2) = L2 - 1.0;
   }
   else   // T3
   {
      shape(3) = 1.0 - L2;
      shape(4) = 1.0 - L0;
      shape(5) = 1.0 - L1;
   }
}

void RefinedLinear2DFiniteElement::CalcDShape(const IntegrationPoint &ip,
                                              DenseMatrix &dshape) const
{
   int i,j;

   double L0, L1, L2;
   L0 = 2.0 * ( 1. - ip.x - ip.y );
   L1 = 2.0 * ( ip.x );
   L2 = 2.0 * ( ip.y );

   double DL0[2], DL1[2], DL2[2];
   DL0[0] = -2.0; DL0[1] = -2.0;
   DL1[0] =  2.0; DL1[1] =  0.0;
   DL2[0] =  0.0; DL2[1] =  2.0;

   for (i = 0; i < 6; i++)
      for (j = 0; j < 2; j++)
      {
         dshape(i,j) = 0.0;
      }

   if (L0 >= 1.0)   // T0
   {
      for (j = 0; j < 2; j++)
      {
         dshape(0,j) = DL0[j];
         dshape(3,j) = DL1[j];
         dshape(5,j) = DL2[j];
      }
   }
   else if (L1 >= 1.0)   // T1
   {
      for (j = 0; j < 2; j++)
      {
         dshape(3,j) = DL0[j];
         dshape(1,j) = DL1[j];
         dshape(4,j) = DL2[j];
      }
   }
   else if (L2 >= 1.0)   // T2
   {
      for (j = 0; j < 2; j++)
      {
         dshape(5,j) = DL0[j];
         dshape(4,j) = DL1[j];
         dshape(2,j) = DL2[j];
      }
   }
   else   // T3
   {
      for (j = 0; j < 2; j++)
      {
         dshape(3,j) = - DL2[j];
         dshape(4,j) = - DL0[j];
         dshape(5,j) = - DL1[j];
      }
   }
}

RefinedLinear3DFiniteElement::RefinedLinear3DFiniteElement()
   : NodalFiniteElement(3, Geometry::TETRAHEDRON, 10, 4)
{
   Nodes.IntPoint(0).x = 0.0;
   Nodes.IntPoint(0).y = 0.0;
   Nodes.IntPoint(0).z = 0.0;
   Nodes.IntPoint(1).x = 1.0;
   Nodes.IntPoint(1).y = 0.0;
   Nodes.IntPoint(1).z = 0.0;
   Nodes.IntPoint(2).x = 0.0;
   Nodes.IntPoint(2).y = 1.0;
   Nodes.IntPoint(2).z = 0.0;
   Nodes.IntPoint(3).x = 0.0;
   Nodes.IntPoint(3).y = 0.0;
   Nodes.IntPoint(3).z = 1.0;
   Nodes.IntPoint(4).x = 0.5;
   Nodes.IntPoint(4).y = 0.0;
   Nodes.IntPoint(4).z = 0.0;
   Nodes.IntPoint(5).x = 0.0;
   Nodes.IntPoint(5).y = 0.5;
   Nodes.IntPoint(5).z = 0.0;
   Nodes.IntPoint(6).x = 0.0;
   Nodes.IntPoint(6).y = 0.0;
   Nodes.IntPoint(6).z = 0.5;
   Nodes.IntPoint(7).x = 0.5;
   Nodes.IntPoint(7).y = 0.5;
   Nodes.IntPoint(7).z = 0.0;
   Nodes.IntPoint(8).x = 0.5;
   Nodes.IntPoint(8).y = 0.0;
   Nodes.IntPoint(8).z = 0.5;
   Nodes.IntPoint(9).x = 0.0;
   Nodes.IntPoint(9).y = 0.5;
   Nodes.IntPoint(9).z = 0.5;
}

void RefinedLinear3DFiniteElement::CalcShape(const IntegrationPoint &ip,
                                             Vector &shape) const
{
   int i;

   double L0, L1, L2, L3, L4, L5;
   L0 = 2.0 * ( 1. - ip.x - ip.y - ip.z );
   L1 = 2.0 * ( ip.x );
   L2 = 2.0 * ( ip.y );
   L3 = 2.0 * ( ip.z );
   L4 = 2.0 * ( ip.x + ip.y );
   L5 = 2.0 * ( ip.y + ip.z );

   // The reference tetrahedron is split in 8 tetrahedra as follows:
   //
   // T0 - 0,4,5,6
   // T1 - 1,4,7,8
   // T2 - 2,5,7,9
   // T3 - 3,6,8,9
   // T4 - 4,5,6,8
   // T5 - 4,5,7,8
   // T6 - 5,6,8,9
   // T7 - 5,7,8,9

   for (i = 0; i < 10; i++)
   {
      shape(i) = 0.0;
   }

   if (L0 >= 1.0)   // T0
   {
      shape(0) = L0 - 1.0;
      shape(4) =       L1;
      shape(5) =       L2;
      shape(6) =       L3;
   }
   else if (L1 >= 1.0)   // T1
   {
      shape(4) =       L0;
      shape(1) = L1 - 1.0;
      shape(7) =       L2;
      shape(8) =       L3;
   }
   else if (L2 >= 1.0)   // T2
   {
      shape(5) =       L0;
      shape(7) =       L1;
      shape(2) = L2 - 1.0;
      shape(9) =       L3;
   }
   else if (L3 >= 1.0)   // T3
   {
      shape(6) =       L0;
      shape(8) =       L1;
      shape(9) =       L2;
      shape(3) = L3 - 1.0;
   }
   else if ((L4 <= 1.0) && (L5 <= 1.0))   // T4
   {
      shape(4) = 1.0 - L5;
      shape(5) =       L2;
      shape(6) = 1.0 - L4;
      shape(8) = 1.0 - L0;
   }
   else if ((L4 >= 1.0) && (L5 <= 1.0))   // T5
   {
      shape(4) = 1.0 - L5;
      shape(5) = 1.0 - L1;
      shape(7) = L4 - 1.0;
      shape(8) =       L3;
   }
   else if ((L4 <= 1.0) && (L5 >= 1.0))   // T6
   {
      shape(5) = 1.0 - L3;
      shape(6) = 1.0 - L4;
      shape(8) =       L1;
      shape(9) = L5 - 1.0;
   }
   else if ((L4 >= 1.0) && (L5 >= 1.0))   // T7
   {
      shape(5) =       L0;
      shape(7) = L4 - 1.0;
      shape(8) = 1.0 - L2;
      shape(9) = L5 - 1.0;
   }
}

void RefinedLinear3DFiniteElement::CalcDShape(const IntegrationPoint &ip,
                                              DenseMatrix &dshape) const
{
   int i,j;

   double L0, L1, L2, L3, L4, L5;
   L0 = 2.0 * ( 1. - ip.x - ip.y - ip.z );
   L1 = 2.0 * ( ip.x );
   L2 = 2.0 * ( ip.y );
   L3 = 2.0 * ( ip.z );
   L4 = 2.0 * ( ip.x + ip.y );
   L5 = 2.0 * ( ip.y + ip.z );

   double DL0[3], DL1[3], DL2[3], DL3[3], DL4[3], DL5[3];
   DL0[0] = -2.0; DL0[1] = -2.0; DL0[2] = -2.0;
   DL1[0] =  2.0; DL1[1] =  0.0; DL1[2] =  0.0;
   DL2[0] =  0.0; DL2[1] =  2.0; DL2[2] =  0.0;
   DL3[0] =  0.0; DL3[1] =  0.0; DL3[2] =  2.0;
   DL4[0] =  2.0; DL4[1] =  2.0; DL4[2] =  0.0;
   DL5[0] =  0.0; DL5[1] =  2.0; DL5[2] =  2.0;

   for (i = 0; i < 10; i++)
      for (j = 0; j < 3; j++)
      {
         dshape(i,j) = 0.0;
      }

   if (L0 >= 1.0)   // T0
   {
      for (j = 0; j < 3; j++)
      {
         dshape(0,j) = DL0[j];
         dshape(4,j) = DL1[j];
         dshape(5,j) = DL2[j];
         dshape(6,j) = DL3[j];
      }
   }
   else if (L1 >= 1.0)   // T1
   {
      for (j = 0; j < 3; j++)
      {
         dshape(4,j) = DL0[j];
         dshape(1,j) = DL1[j];
         dshape(7,j) = DL2[j];
         dshape(8,j) = DL3[j];
      }
   }
   else if (L2 >= 1.0)   // T2
   {
      for (j = 0; j < 3; j++)
      {
         dshape(5,j) = DL0[j];
         dshape(7,j) = DL1[j];
         dshape(2,j) = DL2[j];
         dshape(9,j) = DL3[j];
      }
   }
   else if (L3 >= 1.0)   // T3
   {
      for (j = 0; j < 3; j++)
      {
         dshape(6,j) = DL0[j];
         dshape(8,j) = DL1[j];
         dshape(9,j) = DL2[j];
         dshape(3,j) = DL3[j];
      }
   }
   else if ((L4 <= 1.0) && (L5 <= 1.0))   // T4
   {
      for (j = 0; j < 3; j++)
      {
         dshape(4,j) = - DL5[j];
         dshape(5,j) =   DL2[j];
         dshape(6,j) = - DL4[j];
         dshape(8,j) = - DL0[j];
      }
   }
   else if ((L4 >= 1.0) && (L5 <= 1.0))   // T5
   {
      for (j = 0; j < 3; j++)
      {
         dshape(4,j) = - DL5[j];
         dshape(5,j) = - DL1[j];
         dshape(7,j) =   DL4[j];
         dshape(8,j) =   DL3[j];
      }
   }
   else if ((L4 <= 1.0) && (L5 >= 1.0))   // T6
   {
      for (j = 0; j < 3; j++)
      {
         dshape(5,j) = - DL3[j];
         dshape(6,j) = - DL4[j];
         dshape(8,j) =   DL1[j];
         dshape(9,j) =   DL5[j];
      }
   }
   else if ((L4 >= 1.0) && (L5 >= 1.0))   // T7
   {
      for (j = 0; j < 3; j++)
      {
         dshape(5,j) =   DL0[j];
         dshape(7,j) =   DL4[j];
         dshape(8,j) = - DL2[j];
         dshape(9,j) =   DL5[j];
      }
   }
}


RefinedBiLinear2DFiniteElement::RefinedBiLinear2DFiniteElement()
   : NodalFiniteElement(2, Geometry::SQUARE, 9, 1, FunctionSpace::rQk)
{
   Nodes.IntPoint(0).x = 0.0;
   Nodes.IntPoint(0).y = 0.0;
   Nodes.IntPoint(1).x = 1.0;
   Nodes.IntPoint(1).y = 0.0;
   Nodes.IntPoint(2).x = 1.0;
   Nodes.IntPoint(2).y = 1.0;
   Nodes.IntPoint(3).x = 0.0;
   Nodes.IntPoint(3).y = 1.0;
   Nodes.IntPoint(4).x = 0.5;
   Nodes.IntPoint(4).y = 0.0;
   Nodes.IntPoint(5).x = 1.0;
   Nodes.IntPoint(5).y = 0.5;
   Nodes.IntPoint(6).x = 0.5;
   Nodes.IntPoint(6).y = 1.0;
   Nodes.IntPoint(7).x = 0.0;
   Nodes.IntPoint(7).y = 0.5;
   Nodes.IntPoint(8).x = 0.5;
   Nodes.IntPoint(8).y = 0.5;
}

void RefinedBiLinear2DFiniteElement::CalcShape(const IntegrationPoint &ip,
                                               Vector &shape) const
{
   int i;
   double x = ip.x, y = ip.y;
   double Lx, Ly;
   Lx = 2.0 * ( 1. - x );
   Ly = 2.0 * ( 1. - y );

   // The reference square is split in 4 squares as follows:
   //
   // T0 - 0,4,7,8
   // T1 - 1,4,5,8
   // T2 - 2,5,6,8
   // T3 - 3,6,7,8

   for (i = 0; i < 9; i++)
   {
      shape(i) = 0.0;
   }

   if ((x <= 0.5) && (y <= 0.5))   // T0
   {
      shape(0) = (Lx - 1.0) * (Ly - 1.0);
      shape(4) = (2.0 - Lx) * (Ly - 1.0);
      shape(8) = (2.0 - Lx) * (2.0 - Ly);
      shape(7) = (Lx - 1.0) * (2.0 - Ly);
   }
   else if ((x >= 0.5) && (y <= 0.5))   // T1
   {
      shape(4) =        Lx  * (Ly - 1.0);
      shape(1) = (1.0 - Lx) * (Ly - 1.0);
      shape(5) = (1.0 - Lx) * (2.0 - Ly);
      shape(8) =        Lx  * (2.0 - Ly);
   }
   else if ((x >= 0.5) && (y >= 0.5))   // T2
   {
      shape(8) =        Lx  *        Ly ;
      shape(5) = (1.0 - Lx) *        Ly ;
      shape(2) = (1.0 - Lx) * (1.0 - Ly);
      shape(6) =        Lx  * (1.0 - Ly);
   }
   else if ((x <= 0.5) && (y >= 0.5))   // T3
   {
      shape(7) = (Lx - 1.0) *        Ly ;
      shape(8) = (2.0 - Lx) *        Ly ;
      shape(6) = (2.0 - Lx) * (1.0 - Ly);
      shape(3) = (Lx - 1.0) * (1.0 - Ly);
   }
}

void RefinedBiLinear2DFiniteElement::CalcDShape(const IntegrationPoint &ip,
                                                DenseMatrix &dshape) const
{
   int i,j;
   double x = ip.x, y = ip.y;
   double Lx, Ly;
   Lx = 2.0 * ( 1. - x );
   Ly = 2.0 * ( 1. - y );

   for (i = 0; i < 9; i++)
      for (j = 0; j < 2; j++)
      {
         dshape(i,j) = 0.0;
      }

   if ((x <= 0.5) && (y <= 0.5))   // T0
   {
      dshape(0,0) =  2.0 * (1.0 - Ly);
      dshape(0,1) =  2.0 * (1.0 - Lx);

      dshape(4,0) =  2.0 * (Ly - 1.0);
      dshape(4,1) = -2.0 * (2.0 - Lx);

      dshape(8,0) =  2.0 * (2.0 - Ly);
      dshape(8,1) =  2.0 * (2.0 - Lx);

      dshape(7,0) = -2.0 * (2.0 - Ly);
      dshape(7,0) =  2.0 * (Lx - 1.0);
   }
   else if ((x >= 0.5) && (y <= 0.5))   // T1
   {
      dshape(4,0) = -2.0 * (Ly - 1.0);
      dshape(4,1) = -2.0 * Lx;

      dshape(1,0) =  2.0 * (Ly - 1.0);
      dshape(1,1) = -2.0 * (1.0 - Lx);

      dshape(5,0) =  2.0 * (2.0 - Ly);
      dshape(5,1) =  2.0 * (1.0 - Lx);

      dshape(8,0) = -2.0 * (2.0 - Ly);
      dshape(8,1) =  2.0 * Lx;
   }
   else if ((x >= 0.5) && (y >= 0.5))   // T2
   {
      dshape(8,0) = -2.0 * Ly;
      dshape(8,1) = -2.0 * Lx;

      dshape(5,0) =  2.0 * Ly;
      dshape(5,1) = -2.0 * (1.0 - Lx);

      dshape(2,0) =  2.0 * (1.0 - Ly);
      dshape(2,1) =  2.0 * (1.0 - Lx);

      dshape(6,0) = -2.0 * (1.0 - Ly);
      dshape(6,1) =  2.0 * Lx;
   }
   else if ((x <= 0.5) && (y >= 0.5))   // T3
   {
      dshape(7,0) = -2.0 * Ly;
      dshape(7,1) = -2.0 * (Lx - 1.0);

      dshape(8,0) =  2.0 * Ly ;
      dshape(8,1) = -2.0 * (2.0 - Lx);

      dshape(6,0) = 2.0 * (1.0 - Ly);
      dshape(6,1) = 2.0 * (2.0 - Lx);

      dshape(3,0) = -2.0 * (1.0 - Ly);
      dshape(3,1) =  2.0 * (Lx - 1.0);
   }
}

RefinedTriLinear3DFiniteElement::RefinedTriLinear3DFiniteElement()
   : NodalFiniteElement(3, Geometry::CUBE, 27, 2, FunctionSpace::rQk)
{
   double I[27];
   double J[27];
   double K[27];
   // nodes
   I[ 0] = 0.0; J[ 0] = 0.0; K[ 0] = 0.0;
   I[ 1] = 1.0; J[ 1] = 0.0; K[ 1] = 0.0;
   I[ 2] = 1.0; J[ 2] = 1.0; K[ 2] = 0.0;
   I[ 3] = 0.0; J[ 3] = 1.0; K[ 3] = 0.0;
   I[ 4] = 0.0; J[ 4] = 0.0; K[ 4] = 1.0;
   I[ 5] = 1.0; J[ 5] = 0.0; K[ 5] = 1.0;
   I[ 6] = 1.0; J[ 6] = 1.0; K[ 6] = 1.0;
   I[ 7] = 0.0; J[ 7] = 1.0; K[ 7] = 1.0;
   // edges
   I[ 8] = 0.5; J[ 8] = 0.0; K[ 8] = 0.0;
   I[ 9] = 1.0; J[ 9] = 0.5; K[ 9] = 0.0;
   I[10] = 0.5; J[10] = 1.0; K[10] = 0.0;
   I[11] = 0.0; J[11] = 0.5; K[11] = 0.0;
   I[12] = 0.5; J[12] = 0.0; K[12] = 1.0;
   I[13] = 1.0; J[13] = 0.5; K[13] = 1.0;
   I[14] = 0.5; J[14] = 1.0; K[14] = 1.0;
   I[15] = 0.0; J[15] = 0.5; K[15] = 1.0;
   I[16] = 0.0; J[16] = 0.0; K[16] = 0.5;
   I[17] = 1.0; J[17] = 0.0; K[17] = 0.5;
   I[18] = 1.0; J[18] = 1.0; K[18] = 0.5;
   I[19] = 0.0; J[19] = 1.0; K[19] = 0.5;
   // faces
   I[20] = 0.5; J[20] = 0.5; K[20] = 0.0;
   I[21] = 0.5; J[21] = 0.0; K[21] = 0.5;
   I[22] = 1.0; J[22] = 0.5; K[22] = 0.5;
   I[23] = 0.5; J[23] = 1.0; K[23] = 0.5;
   I[24] = 0.0; J[24] = 0.5; K[24] = 0.5;
   I[25] = 0.5; J[25] = 0.5; K[25] = 1.0;
   // element
   I[26] = 0.5; J[26] = 0.5; K[26] = 0.5;

   for (int n = 0; n < 27; n++)
   {
      Nodes.IntPoint(n).x = I[n];
      Nodes.IntPoint(n).y = J[n];
      Nodes.IntPoint(n).z = K[n];
   }
}

void RefinedTriLinear3DFiniteElement::CalcShape(const IntegrationPoint &ip,
                                                Vector &shape) const
{
   int i, N[8];
   double Lx, Ly, Lz;
   double x = ip.x, y = ip.y, z = ip.z;

   for (i = 0; i < 27; i++)
   {
      shape(i) = 0.0;
   }

   if ((x <= 0.5) && (y <= 0.5) && (z <= 0.5))   // T0
   {
      Lx = 1.0 - 2.0 * x;
      Ly = 1.0 - 2.0 * y;
      Lz = 1.0 - 2.0 * z;

      N[0] =  0;
      N[1] =  8;
      N[2] = 20;
      N[3] = 11;
      N[4] = 16;
      N[5] = 21;
      N[6] = 26;
      N[7] = 24;
   }
   else if ((x >= 0.5) && (y <= 0.5) && (z <= 0.5))   // T1
   {
      Lx = 2.0 - 2.0 * x;
      Ly = 1.0 - 2.0 * y;
      Lz = 1.0 - 2.0 * z;

      N[0] =  8;
      N[1] =  1;
      N[2] =  9;
      N[3] = 20;
      N[4] = 21;
      N[5] = 17;
      N[6] = 22;
      N[7] = 26;
   }
   else if ((x <= 0.5) && (y >= 0.5) && (z <= 0.5))   // T2
   {
      Lx = 2.0 - 2.0 * x;
      Ly = 2.0 - 2.0 * y;
      Lz = 1.0 - 2.0 * z;

      N[0] = 20;
      N[1] =  9;
      N[2] =  2;