fe_nd.hpp

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// Copyright (c) 2010-2023, 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.

#ifndef MFEM_FE_ND
#define MFEM_FE_ND

#include "fe_base.hpp"
#include "fe_h1.hpp"

namespace mfem
{

/// Arbitrary order Nedelec elements in 3D on a cube
class ND_HexahedronElement : public VectorTensorFiniteElement
{
   static const double tk[18];
#ifndef MFEM_THREAD_SAFE
   mutable Vector shape_cx, shape_ox, shape_cy, shape_oy, shape_cz, shape_oz;
   mutable Vector dshape_cx, dshape_cy, dshape_cz;
#endif
   Array<int> dof2tk;
   const double *cp;

public:
   /** @brief Construct the ND_HexahedronElement of order @a p and closed and
       open BasisType @a cb_type and @a ob_type */
   ND_HexahedronElement(const int p,
                        const int cb_type = BasisType::GaussLobatto,
                        const int ob_type = BasisType::GaussLegendre);

   virtual void CalcVShape(const IntegrationPoint &ip,
                           DenseMatrix &shape) const;

   virtual void CalcVShape(ElementTransformation &Trans,
                           DenseMatrix &shape) const
   { CalcVShape_ND(Trans, shape); }

   virtual void CalcCurlShape(const IntegrationPoint &ip,
                              DenseMatrix &curl_shape) const;

   virtual void GetLocalInterpolation(ElementTransformation &Trans,
                                      DenseMatrix &I) const
   { LocalInterpolation_ND(*this, tk, dof2tk, Trans, I); }

   virtual void GetLocalRestriction(ElementTransformation &Trans,
                                    DenseMatrix &R) const
   { LocalRestriction_ND(tk, dof2tk, Trans, R); }

   virtual void GetTransferMatrix(const FiniteElement &fe,
                                  ElementTransformation &Trans,
                                  DenseMatrix &I) const
   { LocalInterpolation_ND(CheckVectorFE(fe), tk, dof2tk, Trans, I); }

   using FiniteElement::Project;

   virtual void Project(VectorCoefficient &vc,
                        ElementTransformation &Trans, Vector &dofs) const
   {
      if (obasis1d.IsIntegratedType()) { ProjectIntegrated(vc, Trans, dofs); }
      else { Project_ND(tk, dof2tk, vc, Trans, dofs); }
   }

   virtual void ProjectFromNodes(Vector &vc, ElementTransformation &Trans,
                                 Vector &dofs) const
   { Project_ND(tk, dof2tk, vc, Trans, dofs); }

   virtual void ProjectMatrixCoefficient(
      MatrixCoefficient &mc, ElementTransformation &T, Vector &dofs) const
   { ProjectMatrixCoefficient_ND(tk, dof2tk, mc, T, dofs); }

   virtual void Project(const FiniteElement &fe,
                        ElementTransformation &Trans,
                        DenseMatrix &I) const
   { Project_ND(tk, dof2tk, fe, Trans, I); }

   virtual void ProjectGrad(const FiniteElement &fe,
                            ElementTransformation &Trans,
                            DenseMatrix &grad) const
   { ProjectGrad_ND(tk, dof2tk, fe, Trans, grad); }

   virtual void ProjectCurl(const FiniteElement &fe,
                            ElementTransformation &Trans,
                            DenseMatrix &curl) const
   { ProjectCurl_ND(tk, dof2tk, fe, Trans, curl); }

   virtual void GetFaceMap(const int face_id, Array<int> &face_map) const;

protected:
   void ProjectIntegrated(VectorCoefficient &vc,
                          ElementTransformation &Trans,
                          Vector &dofs) const;
};


/// Arbitrary order Nedelec elements in 2D on a square
class ND_QuadrilateralElement : public VectorTensorFiniteElement
{
   static const double tk[8];

#ifndef MFEM_THREAD_SAFE
   mutable Vector shape_cx, shape_ox, shape_cy, shape_oy;
   mutable Vector dshape_cx, dshape_cy;
#endif
   Array<int> dof2tk;
   const double *cp;

public:
   /** @brief Construct the ND_QuadrilateralElement of order @a p and closed and
       open BasisType @a cb_type and @a ob_type */
   ND_QuadrilateralElement(const int p,
                           const int cb_type = BasisType::GaussLobatto,
                           const int ob_type = BasisType::GaussLegendre);
   virtual void CalcVShape(const IntegrationPoint &ip,
                           DenseMatrix &shape) const;
   virtual void CalcVShape(ElementTransformation &Trans,
                           DenseMatrix &shape) const
   { CalcVShape_ND(Trans, shape); }
   virtual void CalcCurlShape(const IntegrationPoint &ip,
                              DenseMatrix &curl_shape) const;
   virtual void GetLocalInterpolation(ElementTransformation &Trans,
                                      DenseMatrix &I) const
   { LocalInterpolation_ND(*this, tk, dof2tk, Trans, I); }
   virtual void GetLocalRestriction(ElementTransformation &Trans,
                                    DenseMatrix &R) const
   { LocalRestriction_ND(tk, dof2tk, Trans, R); }
   virtual void GetTransferMatrix(const FiniteElement &fe,
                                  ElementTransformation &Trans,
                                  DenseMatrix &I) const
   { LocalInterpolation_ND(CheckVectorFE(fe), tk, dof2tk, Trans, I); }
   using FiniteElement::Project;
   virtual void Project(VectorCoefficient &vc,
                        ElementTransformation &Trans, Vector &dofs) const
   {
      if (obasis1d.IsIntegratedType()) { ProjectIntegrated(vc, Trans, dofs); }
      else { Project_ND(tk, dof2tk, vc, Trans, dofs); }
   }
   virtual void ProjectFromNodes(Vector &vc, ElementTransformation &Trans,
                                 Vector &dofs) const
   { Project_ND(tk, dof2tk, vc, Trans, dofs); }
   virtual void ProjectMatrixCoefficient(
      MatrixCoefficient &mc, ElementTransformation &T, Vector &dofs) const
   { ProjectMatrixCoefficient_ND(tk, dof2tk, mc, T, dofs); }
   virtual void Project(const FiniteElement &fe,
                        ElementTransformation &Trans,
                        DenseMatrix &I) const
   { Project_ND(tk, dof2tk, fe, Trans, I); }
   virtual void ProjectGrad(const FiniteElement &fe,
                            ElementTransformation &Trans,
                            DenseMatrix &grad) const
   { ProjectGrad_ND(tk, dof2tk, fe, Trans, grad); }

   virtual void GetFaceMap(const int face_id, Array<int> &face_map) const;

protected:
   void ProjectIntegrated(VectorCoefficient &vc,
                          ElementTransformation &Trans,
                          Vector &dofs) const;
};


/// Arbitrary order Nedelec elements in 3D on a tetrahedron
class ND_TetrahedronElement : public VectorFiniteElement
{
   static const double tk[18], c;

#ifndef MFEM_THREAD_SAFE
   mutable Vector shape_x, shape_y, shape_z, shape_l;
   mutable Vector dshape_x, dshape_y, dshape_z, dshape_l;
   mutable DenseMatrix u;
#endif
   Array<int> dof2tk;
   DenseMatrixInverse Ti;

   mutable ND_TetStatelessDofTransformation doftrans;

public:
   /// Construct the ND_TetrahedronElement of order @a p
   ND_TetrahedronElement(const int p);
   virtual void CalcVShape(const IntegrationPoint &ip,
                           DenseMatrix &shape) const;
   virtual void CalcVShape(ElementTransformation &Trans,
                           DenseMatrix &shape) const
   { CalcVShape_ND(Trans, shape); }
   virtual void CalcCurlShape(const IntegrationPoint &ip,
                              DenseMatrix &curl_shape) const;
   virtual void GetLocalInterpolation(ElementTransformation &Trans,
                                      DenseMatrix &I) const
   { LocalInterpolation_ND(*this, tk, dof2tk, Trans, I); }
   virtual void GetLocalRestriction(ElementTransformation &Trans,
                                    DenseMatrix &R) const
   { LocalRestriction_ND(tk, dof2tk, Trans, R); }
   virtual void GetTransferMatrix(const FiniteElement &fe,
                                  ElementTransformation &Trans,
                                  DenseMatrix &I) const
   { LocalInterpolation_ND(CheckVectorFE(fe), tk, dof2tk, Trans, I); }
   virtual StatelessDofTransformation * GetDofTransformation() const
   { return &doftrans; }
   using FiniteElement::Project;
   virtual void Project(VectorCoefficient &vc,
                        ElementTransformation &Trans, Vector &dofs) const
   { Project_ND(tk, dof2tk, vc, Trans, dofs); }
   virtual void ProjectFromNodes(Vector &vc, ElementTransformation &Trans,
                                 Vector &dofs) const
   { Project_ND(tk, dof2tk, vc, Trans, dofs); }
   virtual void ProjectMatrixCoefficient(
      MatrixCoefficient &mc, ElementTransformation &T, Vector &dofs) const
   { ProjectMatrixCoefficient_ND(tk, dof2tk, mc, T, dofs); }
   virtual void Project(const FiniteElement &fe,
                        ElementTransformation &Trans,
                        DenseMatrix &I) const
   { Project_ND(tk, dof2tk, fe, Trans, I); }
   virtual void ProjectGrad(const FiniteElement &fe,
                            ElementTransformation &Trans,
                            DenseMatrix &grad) const
   { ProjectGrad_ND(tk, dof2tk, fe, Trans, grad); }

   virtual void ProjectCurl(const FiniteElement &fe,
                            ElementTransformation &Trans,
                            DenseMatrix &curl) const
   { ProjectCurl_ND(tk, dof2tk, fe, Trans, curl); }
};

/// Arbitrary order Nedelec elements in 2D on a triangle
class ND_TriangleElement : public VectorFiniteElement
{
   static const double tk[8], c;

#ifndef MFEM_THREAD_SAFE
   mutable Vector shape_x, shape_y, shape_l;
   mutable Vector dshape_x, dshape_y, dshape_l;
   mutable DenseMatrix u;
   mutable Vector curlu;
#endif
   Array<int> dof2tk;
   DenseMatrixInverse Ti;

   mutable ND_TriStatelessDofTransformation doftrans;

public:
   /// Construct the ND_TriangleElement of order @a p
   ND_TriangleElement(const int p);
   virtual void CalcVShape(const IntegrationPoint &ip,
                           DenseMatrix &shape) const;
   virtual void CalcVShape(ElementTransformation &Trans,
                           DenseMatrix &shape) const
   { CalcVShape_ND(Trans, shape); }
   virtual void CalcCurlShape(const IntegrationPoint &ip,
                              DenseMatrix &curl_shape) const;
   virtual void GetLocalInterpolation(ElementTransformation &Trans,
                                      DenseMatrix &I) const
   { LocalInterpolation_ND(*this, tk, dof2tk, Trans, I); }
   virtual void GetLocalRestriction(ElementTransformation &Trans,
                                    DenseMatrix &R) const
   { LocalRestriction_ND(tk, dof2tk, Trans, R); }
   virtual void GetTransferMatrix(const FiniteElement &fe,
                                  ElementTransformation &Trans,
                                  DenseMatrix &I) const
   { LocalInterpolation_ND(CheckVectorFE(fe), tk, dof2tk, Trans, I); }
   virtual StatelessDofTransformation * GetDofTransformation() const
   { return &doftrans; }
   using FiniteElement::Project;
   virtual void Project(VectorCoefficient &vc,
                        ElementTransformation &Trans, Vector &dofs) const
   { Project_ND(tk, dof2tk, vc, Trans, dofs); }
   virtual void ProjectFromNodes(Vector &vc, ElementTransformation &Trans,
                                 Vector &dofs) const
   { Project_ND(tk, dof2tk, vc, Trans, dofs); }
   virtual void ProjectMatrixCoefficient(
      MatrixCoefficient &mc, ElementTransformation &T, Vector &dofs) const
   { ProjectMatrixCoefficient_ND(tk, dof2tk, mc, T, dofs); }
   virtual void Project(const FiniteElement &fe,
                        ElementTransformation &Trans,
                        DenseMatrix &I) const
   { Project_ND(tk, dof2tk, fe, Trans, I); }
   virtual void ProjectGrad(const FiniteElement &fe,
                            ElementTransformation &Trans,
                            DenseMatrix &grad) const
   { ProjectGrad_ND(tk, dof2tk, fe, Trans, grad); }
};


/// Arbitrary order Nedelec elements in 1D on a segment
class ND_SegmentElement : public VectorTensorFiniteElement
{
   static const double tk[1];

   Array<int> dof2tk;

public:
   /** @brief Construct the ND_SegmentElement of order @a p and open
       BasisType @a ob_type */
   ND_SegmentElement(const int p, const int ob_type = BasisType::GaussLegendre);
   virtual void CalcShape(const IntegrationPoint &ip, Vector &shape) const
   { obasis1d.Eval(ip.x, shape); }
   virtual void CalcVShape(const IntegrationPoint &ip,
                           DenseMatrix &shape) const;
   virtual void CalcVShape(ElementTransformation &Trans,
                           DenseMatrix &shape) const
   { CalcVShape_ND(Trans, shape); }
   // virtual void CalcCurlShape(const IntegrationPoint &ip,
   //                            DenseMatrix &curl_shape) const;
   virtual void GetLocalInterpolation(ElementTransformation &Trans,
                                      DenseMatrix &I) const
   { LocalInterpolation_ND(*this, tk, dof2tk, Trans, I); }
   virtual void GetLocalRestriction(ElementTransformation &Trans,
                                    DenseMatrix &R) const
   { LocalRestriction_ND(tk, dof2tk, Trans, R); }
   virtual void GetTransferMatrix(const FiniteElement &fe,
                                  ElementTransformation &Trans,
                                  DenseMatrix &I) const
   { LocalInterpolation_ND(CheckVectorFE(fe), tk, dof2tk, Trans, I); }
   using FiniteElement::Project;
   virtual void Project(VectorCoefficient &vc,
                        ElementTransformation &Trans, Vector &dofs) const
   { Project_ND(tk, dof2tk, vc, Trans, dofs); }
   virtual void ProjectMatrixCoefficient(
      MatrixCoefficient &mc, ElementTransformation &T, Vector &dofs) const
   { ProjectMatrixCoefficient_ND(tk, dof2tk, mc, T, dofs); }
   virtual void Project(const FiniteElement &fe,
                        ElementTransformation &Trans,
                        DenseMatrix &I) const
   { Project_ND(tk, dof2tk, fe, Trans, I); }
   virtual void ProjectGrad(const FiniteElement &fe,
                            ElementTransformation &Trans,
                            DenseMatrix &grad) const
   { ProjectGrad_ND(tk, dof2tk, fe, Trans, grad); }
};

class ND_WedgeElement : public VectorFiniteElement
{
private:
   static const double tk[15];

#ifndef MFEM_THREAD_SAFE
   mutable Vector      t1_shape, s1_shape;
   mutable DenseMatrix tn_shape, sn_shape;
   mutable DenseMatrix t1_dshape, s1_dshape, tn_dshape;
#endif
   Array<int> dof2tk, t_dof, s_dof;

   mutable ND_WedgeStatelessDofTransformation doftrans;

   H1_TriangleElement H1TriangleFE;
   ND_TriangleElement NDTriangleFE;
   H1_SegmentElement  H1SegmentFE;
   ND_SegmentElement  NDSegmentFE;

public:
   ND_WedgeElement(const int p,
                   const int cb_type = BasisType::GaussLobatto,
                   const int ob_type = BasisType::GaussLegendre);

   virtual void CalcVShape(const IntegrationPoint &ip,
                           DenseMatrix &shape) const;

   virtual void CalcVShape(ElementTransformation &Trans,
                           DenseMatrix &shape) const
   { CalcVShape_ND(Trans, shape); }

   virtual void CalcCurlShape(const IntegrationPoint &ip,
                              DenseMatrix &curl_shape) const;

   virtual void GetLocalInterpolation(ElementTransformation &Trans,
                                      DenseMatrix &I) const
   { LocalInterpolation_ND(*this, tk, dof2tk, Trans, I); }
   virtual void GetLocalRestriction(ElementTransformation &Trans,
                                    DenseMatrix &R) const
   { LocalRestriction_ND(tk, dof2tk, Trans, R); }
   virtual void GetTransferMatrix(const FiniteElement &fe,
                                  ElementTransformation &Trans,
                                  DenseMatrix &I) const
   { LocalInterpolation_ND(CheckVectorFE(fe), tk, dof2tk, Trans, I); }

   virtual StatelessDofTransformation * GetDofTransformation() const
   { return &doftrans; }

   using FiniteElement::Project;

   virtual void Project(VectorCoefficient &vc,
                        ElementTransformation &Trans, Vector &dofs) const
   { Project_ND(tk, dof2tk, vc, Trans, dofs); }

   virtual void ProjectMatrixCoefficient(
      MatrixCoefficient &mc, ElementTransformation &T, Vector &dofs) const
   { ProjectMatrixCoefficient_ND(tk, dof2tk, mc, T, dofs); }

   virtual void Project(const FiniteElement &fe,
                        ElementTransformation &Trans,
                        DenseMatrix &I) const
   { Project_ND(tk, dof2tk, fe, Trans, I); }

   virtual void ProjectGrad(const FiniteElement &fe,
                            ElementTransformation &Trans,
                            DenseMatrix &grad) const
   { ProjectGrad_ND(tk, dof2tk, fe, Trans, grad); }

   virtual void ProjectCurl(const FiniteElement &fe,
                            ElementTransformation &Trans,
                            DenseMatrix &curl) const
   { ProjectCurl_ND(tk, dof2tk, fe, Trans, curl); }
};


/// A 0D Nedelec finite element for the boundary of a 1D domain
/** ND_R1D_PointElement provides a representation of the trace of a three
    component Nedelec basis restricted to 1D.
*/
class ND_R1D_PointElement : public VectorFiniteElement
{
   static const double tk[9];

public:
   /** @brief Construct the ND_R1D_PointElement */
   ND_R1D_PointElement(int p);

   using FiniteElement::CalcVShape;

   virtual void CalcVShape(const IntegrationPoint &ip,
                           DenseMatrix &shape) const;

   virtual void CalcVShape(ElementTransformation &Trans,
                           DenseMatrix &shape) const;
};

/// Arbitrary order, three component, Nedelec elements in 1D on a segment
/** ND_R1D_SegmentElement provides a representation of a three component Nedelec
    basis where the vector components vary along only one dimension.
*/
class ND_R1D_SegmentElement : public VectorFiniteElement
{
   static const double tk[9];
#ifndef MFEM_THREAD_SAFE
   mutable Vector shape_cx, shape_ox;
   mutable Vector dshape_cx;
#endif
   Array<int> dof_map, dof2tk;

   Poly_1D::Basis &cbasis1d, &obasis1d;

public:
   /** @brief Construct the ND_R1D_SegmentElement of order @a p and closed and
       open BasisType @a cb_type and @a ob_type */
   ND_R1D_SegmentElement(const int p,
                         const int cb_type = BasisType::GaussLobatto,
                         const int ob_type = BasisType::GaussLegendre);

   using FiniteElement::CalcVShape;
   using FiniteElement::CalcPhysCurlShape;

   virtual void CalcVShape(const IntegrationPoint &ip,
                           DenseMatrix &shape) const;

   virtual void CalcVShape(ElementTransformation &Trans,
                           DenseMatrix &shape) const;

   virtual void CalcCurlShape(const IntegrationPoint &ip,
                              DenseMatrix &curl_shape) const;

   virtual void CalcPhysCurlShape(ElementTransformation &Trans,
                                  DenseMatrix &curl_shape) const;

   virtual void GetLocalInterpolation(ElementTransformation &Trans,
                                      DenseMatrix &I) const
   { LocalInterpolation_ND(*this, tk, dof2tk, Trans, I); }

   virtual void GetLocalRestriction(ElementTransformation &Trans,
                                    DenseMatrix &R) const
   { LocalRestriction_ND(tk, dof2tk, Trans, R); }

   virtual void GetTransferMatrix(const FiniteElement &fe,
                                  ElementTransformation &Trans,
                                  DenseMatrix &I) const
   { LocalInterpolation_ND(CheckVectorFE(fe), tk, dof2tk, Trans, I); }

   using FiniteElement::Project;

   virtual void Project(VectorCoefficient &vc,
                        ElementTransformation &Trans, Vector &dofs) const;

   virtual void ProjectFromNodes(Vector &vc, ElementTransformation &Trans,
                                 Vector &dofs) const
   { Project_ND(tk, dof2tk, vc, Trans, dofs); }

   virtual void ProjectMatrixCoefficient(
      MatrixCoefficient &mc, ElementTransformation &T, Vector &dofs) const
   { ProjectMatrixCoefficient_ND(tk, dof2tk, mc, T, dofs); }

   virtual void Project(const FiniteElement &fe,
                        ElementTransformation &Trans,
                        DenseMatrix &I) const;

   virtual void ProjectGrad(const FiniteElement &fe,
                            ElementTransformation &Trans,
                            DenseMatrix &grad) const
   { ProjectGrad_ND(tk, dof2tk, fe, Trans, grad); }

   virtual void ProjectCurl(const FiniteElement &fe,
                            ElementTransformation &Trans,
                            DenseMatrix &curl) const
   { ProjectCurl_ND(tk, dof2tk, fe, Trans, curl); }
};


/** ND_R2D_SegmentElement provides a representation of a 3D Nedelec
    basis where the vector field is assumed constant in the third dimension.
*/
class ND_R2D_SegmentElement : public VectorFiniteElement
{
   static const double tk[4];
#ifndef MFEM_THREAD_SAFE
   mutable Vector shape_cx, shape_ox;
   mutable Vector dshape_cx;
#endif
   Array<int> dof_map, dof2tk;

   Poly_1D::Basis &cbasis1d, &obasis1d;

private:
   void LocalInterpolation(const VectorFiniteElement &cfe,
                           ElementTransformation &Trans,
                           DenseMatrix &I) const;

public:
   /** @brief Construct the ND_R2D_SegmentElement of order @a p and closed and
       open BasisType @a cb_type and @a ob_type */
   ND_R2D_SegmentElement(const int p,
                         const int cb_type = BasisType::GaussLobatto,
                         const int ob_type = BasisType::GaussLegendre);

   virtual void CalcVShape(const IntegrationPoint &ip,
                           DenseMatrix &shape) const;

   virtual void CalcVShape(ElementTransformation &Trans,
                           DenseMatrix &shape) const;

   virtual void CalcCurlShape(const IntegrationPoint &ip,
                              DenseMatrix &curl_shape) const;

   virtual void GetLocalInterpolation(ElementTransformation &Trans,
                                      DenseMatrix &I) const
   { LocalInterpolation(*this, Trans, I); }

   virtual void GetLocalRestriction(ElementTransformation &Trans,
                                    DenseMatrix &R) const
   { MFEM_ABORT("method is not overloaded"); }

   virtual void GetTransferMatrix(const FiniteElement &fe,
                                  ElementTransformation &Trans,
                                  DenseMatrix &I) const
   { LocalInterpolation(CheckVectorFE(fe), Trans, I); }

   using FiniteElement::Project;

   virtual void Project(VectorCoefficient &vc,
                        ElementTransformation &Trans, Vector &dofs) const;
};

class ND_R2D_FiniteElement : public VectorFiniteElement
{
protected:
   const double *tk;
   Array<int> dof_map, dof2tk;

   ND_R2D_FiniteElement(int p, Geometry::Type G, int Do, const double *tk_fe);

private:
   void LocalInterpolation(const VectorFiniteElement &cfe,
                           ElementTransformation &Trans,
                           DenseMatrix &I) const;

public:
   using FiniteElement::CalcVShape;
   using FiniteElement::CalcPhysCurlShape;

   virtual void CalcVShape(ElementTransformation &Trans,
                           DenseMatrix &shape) const;

   virtual void CalcPhysCurlShape(ElementTransformation &Trans,
                                  DenseMatrix &curl_shape) const;

   virtual void GetLocalInterpolation(ElementTransformation &Trans,
                                      DenseMatrix &I) const
   { LocalInterpolation(*this, Trans, I); }

   virtual void GetLocalRestriction(ElementTransformation &Trans,
                                    DenseMatrix &R) const;

   virtual void GetTransferMatrix(const FiniteElement &fe,
                                  ElementTransformation &Trans,
                                  DenseMatrix &I) const
   { LocalInterpolation(CheckVectorFE(fe), Trans, I); }

   using FiniteElement::Project;

   virtual void Project(VectorCoefficient &vc,
                        ElementTransformation &Trans, Vector &dofs) const;

   virtual void Project(const FiniteElement &fe, ElementTransformation &Trans,
                        DenseMatrix &I) const;

   virtual void ProjectGrad(const FiniteElement &fe,
                            ElementTransformation &Trans,
                            DenseMatrix &grad) const;
};

/// Arbitrary order Nedelec 3D elements in 2D on a triangle
class ND_R2D_TriangleElement : public ND_R2D_FiniteElement
{
private:
   static const double tk_t[15];

#ifndef MFEM_THREAD_SAFE
   mutable DenseMatrix nd_shape;
   mutable Vector      h1_shape;
   mutable DenseMatrix nd_dshape;
   mutable DenseMatrix h1_dshape;
#endif

   ND_TriangleElement ND_FE;
   H1_TriangleElement H1_FE;

public:
   /// Construct the ND_R2D_TriangleElement of order @a p
   ND_R2D_TriangleElement(const int p,
                          const int cb_type = BasisType::GaussLobatto);

   using ND_R2D_FiniteElement::CalcVShape;
   using ND_R2D_FiniteElement::CalcPhysCurlShape;

   virtual void CalcVShape(const IntegrationPoint &ip,
                           DenseMatrix &shape) const;
   virtual void CalcCurlShape(const IntegrationPoint &ip,
                              DenseMatrix &curl_shape) const;
};


/// Arbitrary order Nedelec 3D elements in 2D on a square
class ND_R2D_QuadrilateralElement : public ND_R2D_FiniteElement
{
   static const double tk_q[15];

#ifndef MFEM_THREAD_SAFE
   mutable Vector shape_cx, shape_ox, shape_cy, shape_oy;
   mutable Vector dshape_cx, dshape_cy;
#endif

   Poly_1D::Basis &cbasis1d, &obasis1d;

public:
   /** @brief Construct the ND_R2D_QuadrilateralElement of order @a p and
       closed and open BasisType @a cb_type and @a ob_type */
   ND_R2D_QuadrilateralElement(const int p,
                               const int cb_type = BasisType::GaussLobatto,
                               const int ob_type = BasisType::GaussLegendre);

   using ND_R2D_FiniteElement::CalcVShape;
   using ND_R2D_FiniteElement::CalcPhysCurlShape;

   virtual void CalcVShape(const IntegrationPoint &ip,
                           DenseMatrix &shape) const;
   virtual void CalcCurlShape(const IntegrationPoint &ip,
                              DenseMatrix &curl_shape) const;
};


} // namespace mfem

#endif