fespace.hpp

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// Copyright (c) 2010-2022, 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_FESPACE
#define MFEM_FESPACE

#include "../config/config.hpp"
#include "../linalg/sparsemat.hpp"
#include "../mesh/mesh.hpp"
#include "fe_coll.hpp"
#include "doftrans.hpp"
#include "restriction.hpp"
#include <iostream>
#include <unordered_map>

namespace mfem
{

/** @brief The ordering method used when the number of unknowns per mesh node
    (vector dimension) is bigger than 1. */
class Ordering
{
public:
   /// %Ordering methods:
   enum Type
   {
      byNODES, /**< loop first over the nodes (inner loop) then over the vector
                    dimension (outer loop); symbolically it can be represented
                    as: XXX...,YYY...,ZZZ... */
      byVDIM   /**< loop first over the vector dimension (inner loop) then over
                    the nodes (outer loop); symbolically it can be represented
                    as: XYZ,XYZ,XYZ,... */
   };

   template <Type Ord>
   static inline int Map(int ndofs, int vdim, int dof, int vd);

   template <Type Ord>
   static void DofsToVDofs(int ndofs, int vdim, Array<int> &dofs);
};

/// @brief Type describing possible layouts for Q-vectors.
/// @sa QuadratureInterpolator and FaceQuadratureInterpolator.
enum class QVectorLayout
{
   byNODES,  ///< NQPT x VDIM x NE (values) / NQPT x VDIM x DIM x NE (grads)
   byVDIM    ///< VDIM x NQPT x NE (values) / VDIM x DIM x NQPT x NE (grads)
};

template <> inline int
Ordering::Map<Ordering::byNODES>(int ndofs, int vdim, int dof, int vd)
{
   MFEM_ASSERT(dof < ndofs && -1-dof < ndofs && 0 <= vd && vd < vdim, "");
   return (dof >= 0) ? dof+ndofs*vd : dof-ndofs*vd;
}

template <> inline int
Ordering::Map<Ordering::byVDIM>(int ndofs, int vdim, int dof, int vd)
{
   MFEM_ASSERT(dof < ndofs && -1-dof < ndofs && 0 <= vd && vd < vdim, "");
   return (dof >= 0) ? vd+vdim*dof : -1-(vd+vdim*(-1-dof));
}


/// Constants describing the possible orderings of the DOFs in one element.
enum class ElementDofOrdering
{
   /// Native ordering as defined by the FiniteElement.
   /** This ordering can be used by tensor-product elements when the
       interpolation from the DOFs to quadrature points does not use the
       tensor-product structure. */
   NATIVE,
   /// Lexicographic ordering for tensor-product FiniteElements.
   /** This ordering can be used only with tensor-product elements. */
   LEXICOGRAPHIC
};

// Forward declarations
class NURBSExtension;
class BilinearFormIntegrator;
class QuadratureSpace;
class QuadratureInterpolator;
class FaceQuadratureInterpolator;


/** @brief Class FiniteElementSpace - responsible for providing FEM view of the
    mesh, mainly managing the set of degrees of freedom. */
class FiniteElementSpace
{
   friend class InterpolationGridTransfer;
   friend class PRefinementTransferOperator;
   friend void Mesh::Swap(Mesh &, bool);
   friend class LORBase;

protected:
   /// The mesh that FE space lives on (not owned).
   Mesh *mesh;

   /// Associated FE collection (not owned).
   const FiniteElementCollection *fec;

   /// %Vector dimension (number of unknowns per degree of freedom).
   int vdim;

   /** Type of ordering of the vector dofs when #vdim > 1.
       - Ordering::byNODES - first nodes, then vector dimension,
       - Ordering::byVDIM  - first vector dimension, then nodes */
   Ordering::Type ordering;

   /// Number of degrees of freedom. Number of unknowns is #ndofs * #vdim.
   int ndofs;

   /** Polynomial order for each element. If empty, all elements are assumed
       to be of the default order (fec->GetOrder()). */
   Array<char> elem_order;

   int nvdofs, nedofs, nfdofs, nbdofs;
   int uni_fdof; ///< # of single face DOFs if all faces uniform; -1 otherwise
   int *bdofs; ///< internal DOFs of elements if mixed/var-order; NULL otherwise

   /** Variable order spaces only: DOF assignments for edges and faces, see
       docs in MakeDofTable. For constant order spaces the tables are empty. */
   Table var_edge_dofs;
   Table var_face_dofs; ///< NOTE: also used for spaces with mixed faces

   /** Additional data for the var_*_dofs tables: individual variant orders
       (these are basically alternate J arrays for var_edge/face_dofs). */
   Array<char> var_edge_orders, var_face_orders;

   // precalculated DOFs for each element, boundary element, and face
   mutable Table *elem_dof; // owned (except in NURBS FE space)
   mutable Table *elem_fos; // face orientations by element index
   mutable Table *bdr_elem_dof; // owned (except in NURBS FE space)
   mutable Table *bdr_elem_fos; // bdr face orientations by bdr element index
   mutable Table *face_dof; // owned; in var-order space contains variant 0 DOFs

   Array<int> dof_elem_array, dof_ldof_array;

   NURBSExtension *NURBSext;
   int own_ext;
   mutable Array<int> face_to_be; // NURBS FE space only

   Array<DofTransformation*> DoFTrans;
   mutable VDofTransformation VDoFTrans;

   /** Matrix representing the prolongation from the global conforming dofs to
       a set of intermediate partially conforming dofs, e.g. the dofs associated
       with a "cut" space on a non-conforming mesh. */
   mutable SparseMatrix *cP; // owned
   /// Conforming restriction matrix such that cR.cP=I.
   mutable SparseMatrix *cR; // owned
   /// A version of the conforming restriction matrix for variable-order spaces.
   mutable SparseMatrix *cR_hp; // owned
   mutable bool cP_is_set;

   /// Transformation to apply to GridFunctions after space Update().
   OperatorHandle Th;

   /// The element restriction operators, see GetElementRestriction().
   mutable OperatorHandle L2E_nat, L2E_lex;
   /// The face restriction operators, see GetFaceRestriction().
   using key_face = std::tuple<bool, ElementDofOrdering, FaceType, L2FaceValues>;
   struct key_hash
   {
      std::size_t operator()(const key_face& k) const
      {
         return std::get<0>(k)
                + 2 * (int)std::get<1>(k)
                + 4 * (int)std::get<2>(k)
                + 8 * (int)std::get<3>(k);
      }
   };
   using map_L2F = std::unordered_map<const key_face,FaceRestriction*,key_hash>;
   mutable map_L2F L2F;

   mutable Array<QuadratureInterpolator*> E2Q_array;
   mutable Array<FaceQuadratureInterpolator*> E2IFQ_array;
   mutable Array<FaceQuadratureInterpolator*> E2BFQ_array;

   /** Update counter, incremented every time the space is constructed/updated.
       Used by GridFunctions to check if they are up to date with the space. */
   long sequence;

   /** Mesh sequence number last seen when constructing the space. The space
       needs updating if Mesh::GetSequence() is larger than this. */
   long mesh_sequence;

   /// True if at least one element order changed (variable-order space only).
   bool orders_changed;

   bool relaxed_hp; // see SetRelaxedHpConformity()

   void UpdateNURBS();

   void Construct();
   void Destroy();

   void ConstructDoFTrans();
   void DestroyDoFTrans();

   void BuildElementToDofTable() const;
   void BuildBdrElementToDofTable() const;
   void BuildFaceToDofTable() const;

   /** @brief  Generates partial face_dof table for a NURBS space.

       The table is only defined for exterior faces that coincide with a
       boundary. */
   void BuildNURBSFaceToDofTable() const;

   /// Bit-mask representing a set of orders needed by an edge/face.
   typedef std::uint64_t VarOrderBits;
   static constexpr int MaxVarOrder = 8*sizeof(VarOrderBits) - 1;

   /// Return the minimum order (least significant bit set) in the bit mask.
   static int MinOrder(VarOrderBits bits);

   /// Return element order: internal version of GetElementOrder without checks.
   int GetElementOrderImpl(int i) const;

   /** In a variable order space, calculate a bitmask of polynomial orders that
       need to be represented on each edge and face. */
   void CalcEdgeFaceVarOrders(Array<VarOrderBits> &edge_orders,
                              Array<VarOrderBits> &face_orders) const;

   /** Build the table var_edge_dofs (or var_face_dofs) in a variable order
       space; return total edge/face DOFs. */
   int MakeDofTable(int ent_dim, const Array<int> &entity_orders,
                    Table &entity_dofs, Array<char> *var_ent_order);

   /// Search row of a DOF table for a DOF set of size 'ndof', return first DOF.
   int FindDofs(const Table &var_dof_table, int row, int ndof) const;

   /** In a variable order space, return edge DOFs associated with a polynomial
       order that has 'ndof' degrees of freedom. */
   int FindEdgeDof(int edge, int ndof) const
   { return FindDofs(var_edge_dofs, edge, ndof); }

   /// Similar to FindEdgeDof, but used for mixed meshes too.
   int FindFaceDof(int face, int ndof) const
   { return FindDofs(var_face_dofs, face, ndof); }

   int FirstFaceDof(int face, int variant = 0) const
   { return uni_fdof >= 0 ? face*uni_fdof : var_face_dofs.GetRow(face)[variant];}

   /// Return number of possible DOF variants for edge/face (var. order spaces).
   int GetNVariants(int entity, int index) const;

   /// Helper to encode a sign flip into a DOF index (for Hcurl/Hdiv shapes).
   static inline int EncodeDof(int entity_base, int idx)
   { return (idx >= 0) ? (entity_base + idx) : (-1-(entity_base + (-1-idx))); }

   /// Helpers to remove encoded sign from a DOF
   static inline int DecodeDof(int dof)
   { return (dof >= 0) ? dof : (-1 - dof); }

   static inline int DecodeDof(int dof, double& sign)
   { return (dof >= 0) ? (sign = 1, dof) : (sign = -1, (-1 - dof)); }

   /// Helper to get vertex, edge or face DOFs (entity=0,1,2 resp.).
   int GetEntityDofs(int entity, int index, Array<int> &dofs,
                     Geometry::Type master_geom = Geometry::INVALID,
                     int variant = 0) const;

   // Get degenerate face DOFs: see explanation in method implementation.
   int GetDegenerateFaceDofs(int index, Array<int> &dofs,
                             Geometry::Type master_geom, int variant) const;

   int GetNumBorderDofs(Geometry::Type geom, int order) const;

   /// Calculate the cP and cR matrices for a nonconforming mesh.
   void BuildConformingInterpolation() const;

   static void AddDependencies(SparseMatrix& deps, Array<int>& master_dofs,
                               Array<int>& slave_dofs, DenseMatrix& I,
                               int skipfirst = 0);

   static bool DofFinalizable(int dof, const Array<bool>& finalized,
                              const SparseMatrix& deps);

   void AddEdgeFaceDependencies(SparseMatrix &deps, Array<int>& master_dofs,
                                const FiniteElement *master_fe,
                                Array<int> &slave_dofs, int slave_face,
                                const DenseMatrix *pm) const;

   /// Replicate 'mat' in the vector dimension, according to vdim ordering mode.
   void MakeVDimMatrix(SparseMatrix &mat) const;

   /// GridFunction interpolation operator applicable after mesh refinement.
   class RefinementOperator : public Operator
   {
      const FiniteElementSpace* fespace;
      DenseTensor localP[Geometry::NumGeom];
      Table* old_elem_dof; // Owned.
      Table* old_elem_fos; // Owned.

      Array<DofTransformation*> old_DoFTrans;
      mutable VDofTransformation old_VDoFTrans;

      void ConstructDoFTrans();

   public:
      /** Construct the operator based on the elem_dof table of the original
          (coarse) space. The class takes ownership of the table. */
      RefinementOperator(const FiniteElementSpace* fespace,
                         Table *old_elem_dof/*takes ownership*/,
                         Table *old_elem_fos/*takes ownership*/, int old_ndofs);
      RefinementOperator(const FiniteElementSpace *fespace,
                         const FiniteElementSpace *coarse_fes);
      virtual void Mult(const Vector &x, Vector &y) const;
      virtual void MultTranspose(const Vector &x, Vector &y) const;
      virtual ~RefinementOperator();
   };

   /// Derefinement operator, used by the friend class InterpolationGridTransfer.
   class DerefinementOperator : public Operator
   {
      const FiniteElementSpace *fine_fes; // Not owned.
      DenseTensor localR[Geometry::NumGeom];
      Table *coarse_elem_dof; // Owned.
      // Table *coarse_elem_fos; // Owned.
      Table coarse_to_fine;
      Array<int> coarse_to_ref_type;
      Array<Geometry::Type> ref_type_to_geom;
      Array<int> ref_type_to_fine_elem_offset;

   public:
      DerefinementOperator(const FiniteElementSpace *f_fes,
                           const FiniteElementSpace *c_fes,
                           BilinearFormIntegrator *mass_integ);
      virtual void Mult(const Vector &x, Vector &y) const;
      virtual ~DerefinementOperator();
   };

   /** This method makes the same assumptions as the method:
       void GetLocalRefinementMatrices(
           const FiniteElementSpace &coarse_fes, Geometry::Type geom,
           DenseTensor &localP) const
       which is defined below. It also assumes that the coarse fes and this have
       the same vector dimension, vdim. */
   SparseMatrix *RefinementMatrix_main(const int coarse_ndofs,
                                       const Table &coarse_elem_dof,
                                       const Table *coarse_elem_fos,
                                       const DenseTensor localP[]) const;

   void GetLocalRefinementMatrices(Geometry::Type geom,
                                   DenseTensor &localP) const;
   void GetLocalDerefinementMatrices(Geometry::Type geom,
                                     DenseTensor &localR) const;

   /** Calculate explicit GridFunction interpolation matrix (after mesh
       refinement). NOTE: consider using the RefinementOperator class instead
       of the fully assembled matrix, which can take a lot of memory. */
   SparseMatrix* RefinementMatrix(int old_ndofs, const Table* old_elem_dof,
                                  const Table* old_elem_fos);

   /// Calculate GridFunction restriction matrix after mesh derefinement.
   SparseMatrix* DerefinementMatrix(int old_ndofs, const Table* old_elem_dof,
                                    const Table* old_elem_fos);

   /** @brief Return in @a localP the local refinement matrices that map
       between fespaces after mesh refinement. */
   /** This method assumes that this->mesh is a refinement of coarse_fes->mesh
       and that the CoarseFineTransformations of this->mesh are set accordingly.
       Another assumption is that the FEs of this use the same MapType as the FEs
       of coarse_fes. Finally, it assumes that the spaces this and coarse_fes are
       NOT variable-order spaces. */
   void GetLocalRefinementMatrices(const FiniteElementSpace &coarse_fes,
                                   Geometry::Type geom,
                                   DenseTensor &localP) const;

   /// Help function for constructors + Load().
   void Constructor(Mesh *mesh, NURBSExtension *ext,
                    const FiniteElementCollection *fec,
                    int vdim = 1, int ordering = Ordering::byNODES);

   /// Updates the internal mesh pointer. @warning @a new_mesh must be
   /// <b>topologically identical</b> to the existing mesh. Used if the address
   /// of the Mesh object has changed, e.g. in @a Mesh::Swap.
   virtual void UpdateMeshPointer(Mesh *new_mesh);

   /// Resize the elem_order array on mesh change.
   void UpdateElementOrders();

   /// @brief Copies the prolongation and restriction matrices from @a fes.
   ///
   /// Used for low order preconditioning on non-conforming meshes. If the DOFs
   /// require a permutation, it will be supplied by non-NULL @a perm. NULL @a
   /// perm indicates that no permutation is required.
   virtual void CopyProlongationAndRestriction(const FiniteElementSpace &fes,
                                               const Array<int> *perm);

public:
   /** @brief Default constructor: the object is invalid until initialized using
       the method Load(). */
   FiniteElementSpace();

   /** @brief Copy constructor: deep copy all data from @a orig except the Mesh,
       the FiniteElementCollection, and some derived data. */
   /** If the @a mesh or @a fec pointers are NULL (default), then the new
       FiniteElementSpace will reuse the respective pointers from @a orig. If
       any of these pointers is not NULL, the given pointer will be used instead
       of the one used by @a orig.

       @note The objects pointed to by the @a mesh and @a fec parameters must be
       either the same objects as the ones used by @a orig, or copies of them.
       Otherwise, the behavior is undefined.

       @note Derived data objects, such as the conforming prolongation and
       restriction matrices, and the update operator, will not be copied, even
       if they are created in the @a orig object. */
   FiniteElementSpace(const FiniteElementSpace &orig, Mesh *mesh = NULL,
                      const FiniteElementCollection *fec = NULL);

   FiniteElementSpace(Mesh *mesh,
                      const FiniteElementCollection *fec,
                      int vdim = 1, int ordering = Ordering::byNODES)
   { Constructor(mesh, NULL, fec, vdim, ordering); }

   /// Construct a NURBS FE space based on the given NURBSExtension, @a ext.
   /** @note If the pointer @a ext is NULL, this constructor is equivalent to
       the standard constructor with the same arguments minus the
       NURBSExtension, @a ext. */
   FiniteElementSpace(Mesh *mesh, NURBSExtension *ext,
                      const FiniteElementCollection *fec,
                      int vdim = 1, int ordering = Ordering::byNODES)
   { Constructor(mesh, ext, fec, vdim, ordering); }

   /// Copy assignment not supported
   FiniteElementSpace& operator=(const FiniteElementSpace&) = delete;

   /// Returns the mesh
   inline Mesh *GetMesh() const { return mesh; }

   const NURBSExtension *GetNURBSext() const { return NURBSext; }
   NURBSExtension *GetNURBSext() { return NURBSext; }
   NURBSExtension *StealNURBSext();

   bool Conforming() const { return mesh->Conforming() && cP == NULL; }
   bool Nonconforming() const { return mesh->Nonconforming() || cP != NULL; }

   /// Sets the order of the i'th finite element.
   /** By default, all elements are assumed to be of fec->GetOrder(). Once
       SetElementOrder is called, the space becomes a variable order space. */
   void SetElementOrder(int i, int p);

   /// Returns the order of the i'th finite element.
   int GetElementOrder(int i) const;

   /// Return the maximum polynomial order.
   int GetMaxElementOrder() const
   { return IsVariableOrder() ? elem_order.Max() : fec->GetOrder(); }

   /// Returns true if the space contains elements of varying polynomial orders.
   bool IsVariableOrder() const { return elem_order.Size(); }

   /// The returned SparseMatrix is owned by the FiniteElementSpace.
   const SparseMatrix *GetConformingProlongation() const;

   /// The returned SparseMatrix is owned by the FiniteElementSpace.
   const SparseMatrix *GetConformingRestriction() const;

   /** Return a version of the conforming restriction matrix for variable-order
       spaces with complex hp interfaces, where some true DOFs are not owned by
       any elements and need to be interpolated from higher order edge/face
       variants (see also @a SetRelaxedHpConformity()). */
   /// The returned SparseMatrix is owned by the FiniteElementSpace.
   const SparseMatrix *GetHpConformingRestriction() const;

   /// The returned Operator is owned by the FiniteElementSpace.
   virtual const Operator *GetProlongationMatrix() const
   { return GetConformingProlongation(); }

   /// Return an operator that performs the transpose of GetRestrictionOperator
   /** The returned operator is owned by the FiniteElementSpace. In serial this
       is the same as GetProlongationMatrix() */
   virtual const Operator *GetRestrictionTransposeOperator() const
   { return GetConformingProlongation(); }

   /// An abstract operator that performs the same action as GetRestrictionMatrix
   /** In some cases this is an optimized matrix-free implementation. The
       returned operator is owned by the FiniteElementSpace. */
   virtual const Operator *GetRestrictionOperator() const
   { return GetConformingRestriction(); }

   /// The returned SparseMatrix is owned by the FiniteElementSpace.
   virtual const SparseMatrix *GetRestrictionMatrix() const
   { return GetConformingRestriction(); }

   /// The returned SparseMatrix is owned by the FiniteElementSpace.
   virtual const SparseMatrix *GetHpRestrictionMatrix() const
   { return GetHpConformingRestriction(); }

   /// Return an Operator that converts L-vectors to E-vectors.
   /** An L-vector is a vector of size GetVSize() which is the same size as a
       GridFunction. An E-vector represents the element-wise discontinuous
       version of the FE space.

       The layout of the E-vector is: ND x VDIM x NE, where ND is the number of
       degrees of freedom, VDIM is the vector dimension of the FE space, and NE
       is the number of the mesh elements.

       The parameter @a e_ordering describes how the local DOFs in each element
       should be ordered, see ElementDofOrdering.

       For discontinuous spaces, the element restriction corresponds to a
       permutation of the degrees of freedom, implemented by the
       L2ElementRestriction class.

       The returned Operator is owned by the FiniteElementSpace. */
   const ElementRestrictionOperator *GetElementRestriction(
      ElementDofOrdering e_ordering) const;

   /// Return an Operator that converts L-vectors to E-vectors on each face.
   virtual const FaceRestriction *GetFaceRestriction(
      ElementDofOrdering e_ordering, FaceType,
      L2FaceValues mul = L2FaceValues::DoubleValued) const;

   /** @brief Return a QuadratureInterpolator that interpolates E-vectors to
       quadrature point values and/or derivatives (Q-vectors). */
   /** An E-vector represents the element-wise discontinuous version of the FE
       space and can be obtained, for example, from a GridFunction using the
       Operator returned by GetElementRestriction().

       All elements will use the same IntegrationRule, @a ir as the target
       quadrature points.

       @note The returned pointer is shared. A good practice, before using it,
       is to set all its properties to their expected values, as other parts of
       the code may also change them. That is, it's good to call
       SetOutputLayout() and DisableTensorProducts() before interpolating. */
   const QuadratureInterpolator *GetQuadratureInterpolator(
      const IntegrationRule &ir) const;

   /** @brief Return a QuadratureInterpolator that interpolates E-vectors to
       quadrature point values and/or derivatives (Q-vectors). */
   /** An E-vector represents the element-wise discontinuous version of the FE
       space and can be obtained, for example, from a GridFunction using the
       Operator returned by GetElementRestriction().

       The target quadrature points in the elements are described by the given
       QuadratureSpace, @a qs.

       @note The returned pointer is shared. A good practice, before using it,
       is to set all its properties to their expected values, as other parts of
       the code may also change them. That is, it's good to call
       SetOutputLayout() and DisableTensorProducts() before interpolating. */
   const QuadratureInterpolator *GetQuadratureInterpolator(
      const QuadratureSpace &qs) const;

   /** @brief Return a FaceQuadratureInterpolator that interpolates E-vectors to
       quadrature point values and/or derivatives (Q-vectors).

       @note The returned pointer is shared. A good practice, before using it,
       is to set all its properties to their expected values, as other parts of
       the code may also change them. That is, it's good to call
       SetOutputLayout() and DisableTensorProducts() before interpolating. */
   const FaceQuadratureInterpolator *GetFaceQuadratureInterpolator(
      const IntegrationRule &ir, FaceType type) const;

   /// Returns the polynomial degree of the i'th finite element.
   /** NOTE: it is recommended to use GetElementOrder in new code. */
   int GetOrder(int i) const { return GetElementOrder(i); }

   /** Return the order of an edge. In a variable order space, return the order
       of a specific variant, or -1 if there are no more variants. */
   int GetEdgeOrder(int edge, int variant = 0) const;

   /// Returns the polynomial degree of the i'th face finite element
   int GetFaceOrder(int face, int variant = 0) const;

   /// Returns vector dimension.
   inline int GetVDim() const { return vdim; }

   /// Returns number of degrees of freedom.
   inline int GetNDofs() const { return ndofs; }

   /// Return the number of vector dofs, i.e. GetNDofs() x GetVDim().
   inline int GetVSize() const { return vdim * ndofs; }

   /// Return the number of vector true (conforming) dofs.
   virtual int GetTrueVSize() const { return GetConformingVSize(); }

   /// Returns the number of conforming ("true") degrees of freedom
   /// (if the space is on a nonconforming mesh with hanging nodes).
   int GetNConformingDofs() const;

   int GetConformingVSize() const { return vdim * GetNConformingDofs(); }

   /// Return the ordering method.
   inline Ordering::Type GetOrdering() const { return ordering; }

   const FiniteElementCollection *FEColl() const { return fec; }

   /// Number of all scalar vertex dofs
   int GetNVDofs() const { return nvdofs; }
   /// Number of all scalar edge-interior dofs
   int GetNEDofs() const { return nedofs; }
   /// Number of all scalar face-interior dofs
   int GetNFDofs() const { return nfdofs; }

   /// Returns number of vertices in the mesh.
   inline int GetNV() const { return mesh->GetNV(); }

   /// Returns number of elements in the mesh.
   inline int GetNE() const { return mesh->GetNE(); }

   /// Returns number of faces (i.e. co-dimension 1 entities) in the mesh.
   /** The co-dimension 1 entities are those that have dimension 1 less than the
       mesh dimension, e.g. for a 2D mesh, the faces are the 1D entities, i.e.
       the edges. */
   inline int GetNF() const { return mesh->GetNumFaces(); }

   /// Returns number of boundary elements in the mesh.
   inline int GetNBE() const { return mesh->GetNBE(); }

   /// Returns the number of faces according to the requested type.
   /** If type==Boundary returns only the "true" number of boundary faces
       contrary to GetNBE() that returns "fake" boundary faces associated to
       visualization for GLVis.
       Similarly, if type==Interior, the "fake" boundary faces associated to
       visualization are counted as interior faces. */
   inline int GetNFbyType(FaceType type) const
   { return mesh->GetNFbyType(type); }

   /// Returns the type of element i.
   inline int GetElementType(int i) const
   { return mesh->GetElementType(i); }

   /// Returns the vertices of element i.
   inline void GetElementVertices(int i, Array<int> &vertices) const
   { mesh->GetElementVertices(i, vertices); }

   /// Returns the type of boundary element i.
   inline int GetBdrElementType(int i) const
   { return mesh->GetBdrElementType(i); }

   /// Returns ElementTransformation for the @a i-th element.
   ElementTransformation *GetElementTransformation(int i) const
   { return mesh->GetElementTransformation(i); }

   /** @brief Returns the transformation defining the @a i-th element in the
       user-defined variable @a ElTr. */
   void GetElementTransformation(int i, IsoparametricTransformation *ElTr)
   { mesh->GetElementTransformation(i, ElTr); }

   /// Returns ElementTransformation for the @a i-th boundary element.
   ElementTransformation *GetBdrElementTransformation(int i) const
   { return mesh->GetBdrElementTransformation(i); }

   int GetAttribute(int i) const { return mesh->GetAttribute(i); }

   int GetBdrAttribute(int i) const { return mesh->GetBdrAttribute(i); }

   /// Returns indices of degrees of freedom of element 'elem'.
   virtual DofTransformation *GetElementDofs(int elem, Array<int> &dofs) const;

   /// Returns indices of degrees of freedom for boundary element 'bel'.
   virtual DofTransformation *GetBdrElementDofs(int bel,
                                                Array<int> &dofs) const;

   /** @brief Returns the indices of the degrees of freedom for the specified
       face, including the DOFs for the edges and the vertices of the face. */
   /** In variable order spaces, multiple variants of DOFs can be returned.
       See @a GetEdgeDofs for more details.
       @return Order of the selected variant, or -1 if there are no more
       variants.*/
   virtual int GetFaceDofs(int face, Array<int> &dofs, int variant = 0) const;

   /** @brief Returns the indices of the degrees of freedom for the specified
       edge, including the DOFs for the vertices of the edge. */
   /** In variable order spaces, multiple sets of DOFs may exist on an edge,
       corresponding to the different polynomial orders of incident elements.
       The 'variant' parameter is the zero-based index of the desired DOF set.
       The variants are ordered from lowest polynomial degree to the highest.
       @return Order of the selected variant, or -1 if there are no more
       variants. */
   int GetEdgeDofs(int edge, Array<int> &dofs, int variant = 0) const;

   void GetVertexDofs(int i, Array<int> &dofs) const;

   void GetElementInteriorDofs(int i, Array<int> &dofs) const;

   void GetFaceInteriorDofs(int i, Array<int> &dofs) const;

   int GetNumElementInteriorDofs(int i) const;

   void GetEdgeInteriorDofs(int i, Array<int> &dofs) const;

   /** @brief Returns the indices of all of the VDofs for the specified
       dimension 'vd'. */
   /** The 'ndofs' parameter defines the number of Dofs in the
       FiniteElementSpace. If 'ndofs' is -1 (the default value), then the
       number of Dofs is determined by the FiniteElementSpace. */
   void GetVDofs(int vd, Array<int> &dofs, int ndofs = -1) const;

   void DofsToVDofs(Array<int> &dofs, int ndofs = -1) const;

   void DofsToVDofs(int vd, Array<int> &dofs, int ndofs = -1) const;

   int DofToVDof(int dof, int vd, int ndofs = -1) const;

   int VDofToDof(int vdof) const
   { return (ordering == Ordering::byNODES) ? (vdof%ndofs) : (vdof/vdim); }

   static void AdjustVDofs(Array<int> &vdofs);

   /// Returns indexes of degrees of freedom in array dofs for i'th element.
   DofTransformation *GetElementVDofs(int i, Array<int> &vdofs) const;

   /// Returns indexes of degrees of freedom for i'th boundary element.
   DofTransformation *GetBdrElementVDofs(int i, Array<int> &vdofs) const;

   /// Returns indexes of degrees of freedom for i'th face element (2D and 3D).
   void GetFaceVDofs(int i, Array<int> &vdofs) const;

   /// Returns indexes of degrees of freedom for i'th edge.
   void GetEdgeVDofs(int i, Array<int> &vdofs) const;

   void GetVertexVDofs(int i, Array<int> &vdofs) const;

   void GetElementInteriorVDofs(int i, Array<int> &vdofs) const;

   void GetEdgeInteriorVDofs(int i, Array<int> &vdofs) const;

   /// (@deprecated) Use the Update() method if the space or mesh changed.
   MFEM_DEPRECATED void RebuildElementToDofTable();

   /** @brief Reorder the scalar DOFs based on the element ordering.

       The new ordering is constructed as follows: 1) loop over all elements as
       ordered in the Mesh; 2) for each element, assign new indices to all of
       its current DOFs that are still unassigned; the new indices we assign are
       simply the sequence `0,1,2,...`; if there are any signed DOFs their sign
       is preserved. */
   void ReorderElementToDofTable();

   const Table *GetElementToFaceOrientationTable() const { return elem_fos; }

   /** @brief Return a reference to the internal Table that stores the lists of
       scalar dofs, for each mesh element, as returned by GetElementDofs(). */
   const Table &GetElementToDofTable() const { return *elem_dof; }

   /** @brief Return a reference to the internal Table that stores the lists of
       scalar dofs, for each boundary mesh element, as returned by
       GetBdrElementDofs(). */
   const Table &GetBdrElementToDofTable() const
   { if (!bdr_elem_dof) { BuildBdrElementToDofTable(); } return *bdr_elem_dof; }

   /** @brief Return a reference to the internal Table that stores the lists of
       scalar dofs, for each face in the mesh, as returned by GetFaceDofs(). In
       this context, "face" refers to a (dim-1)-dimensional mesh entity. */
   /** @note In the case of a NURBS space, the rows corresponding to interior
       faces will be empty. */
   const Table &GetFaceToDofTable() const
   { if (!face_dof) { BuildFaceToDofTable(); } return *face_dof; }

   /** @brief Initialize internal data that enables the use of the methods
       GetElementForDof() and GetLocalDofForDof(). */
   void BuildDofToArrays();

   /// Return the index of the first element that contains dof @a i.
   /** This method can be called only after setup is performed using the method
       BuildDofToArrays(). */
   int GetElementForDof(int i) const { return dof_elem_array[i]; }
   /// Return the local dof index in the first element that contains dof @a i.
   /** This method can be called only after setup is performed using the method
       BuildDofToArrays(). */
   int GetLocalDofForDof(int i) const { return dof_ldof_array[i]; }

   /** @brief Returns pointer to the FiniteElement in the FiniteElementCollection
        associated with i'th element in the mesh object. */
   virtual const FiniteElement *GetFE(int i) const;

   /** @brief Returns pointer to the FiniteElement in the FiniteElementCollection
        associated with i'th boundary face in the mesh object. */
   const FiniteElement *GetBE(int i) const;

   /** @brief Returns pointer to the FiniteElement in the FiniteElementCollection
        associated with i'th face in the mesh object.  Faces in this case refer
        to the MESHDIM-1 primitive so in 2D they are segments and in 1D they are
        points.*/
   const FiniteElement *GetFaceElement(int i) const;

   /** @brief Returns pointer to the FiniteElement in the FiniteElementCollection
        associated with i'th edge in the mesh object. */
   const FiniteElement *GetEdgeElement(int i, int variant = 0) const;

   /// Return the trace element from element 'i' to the given 'geom_type'
   const FiniteElement *GetTraceElement(int i, Geometry::Type geom_type) const;

   /** @brief Mark degrees of freedom associated with boundary elements with
       the specified boundary attributes (marked in 'bdr_attr_is_ess').
       For spaces with 'vdim' > 1, the 'component' parameter can be used
       to restricts the marked vDOFs to the specified component. */
   virtual void GetEssentialVDofs(const Array<int> &bdr_attr_is_ess,
                                  Array<int> &ess_vdofs,
                                  int component = -1) const;

   /** @brief Get a list of essential true dofs, ess_tdof_list, corresponding to the
       boundary attributes marked in the array bdr_attr_is_ess.
       For spaces with 'vdim' > 1, the 'component' parameter can be used
       to restricts the marked tDOFs to the specified component. */
   virtual void GetEssentialTrueDofs(const Array<int> &bdr_attr_is_ess,
                                     Array<int> &ess_tdof_list,
                                     int component = -1);

   /** @brief Get a list of all boundary true dofs, @a boundary_dofs. For spaces
       with 'vdim' > 1, the 'component' parameter can be used to restricts the
       marked tDOFs to the specified component. Equivalent to
       FiniteElementSpace::GetEssentialTrueDofs with all boundary attributes
       marked as essential. */
   void GetBoundaryTrueDofs(Array<int> &boundary_dofs, int component = -1);

   /// Convert a Boolean marker array to a list containing all marked indices.
   static void MarkerToList(const Array<int> &marker, Array<int> &list);

   /** @brief Convert an array of indices (list) to a Boolean marker array where all
       indices in the list are marked with the given value and the rest are set
       to zero. */
   static void ListToMarker(const Array<int> &list, int marker_size,
                            Array<int> &marker, int mark_val = -1);

   /** @brief For a partially conforming FE space, convert a marker array (nonzero
       entries are true) on the partially conforming dofs to a marker array on
       the conforming dofs. A conforming dofs is marked iff at least one of its
       dependent dofs is marked. */
   void ConvertToConformingVDofs(const Array<int> &dofs, Array<int> &cdofs);

   /** @brief For a partially conforming FE space, convert a marker array (nonzero
       entries are true) on the conforming dofs to a marker array on the
       (partially conforming) dofs. A dof is marked iff it depends on a marked
       conforming dofs, where dependency is defined by the ConformingRestriction
       matrix; in other words, a dof is marked iff it corresponds to a marked
       conforming dof. */
   void ConvertFromConformingVDofs(const Array<int> &cdofs, Array<int> &dofs);

   /** @brief Generate the global restriction matrix from a discontinuous
       FE space to the continuous FE space of the same polynomial degree. */
   SparseMatrix *D2C_GlobalRestrictionMatrix(FiniteElementSpace *cfes);

   /** @brief Generate the global restriction matrix from a discontinuous
       FE space to the piecewise constant FE space. */
   SparseMatrix *D2Const_GlobalRestrictionMatrix(FiniteElementSpace *cfes);

   /** @brief Construct the restriction matrix from the FE space given by
       (*this) to the lower degree FE space given by (*lfes) which
       is defined on the same mesh. */
   SparseMatrix *H2L_GlobalRestrictionMatrix(FiniteElementSpace *lfes);

   /** @brief Construct and return an Operator that can be used to transfer
       GridFunction data from @a coarse_fes, defined on a coarse mesh, to @a
       this FE space, defined on a refined mesh. */
   /** It is assumed that the mesh of this FE space is a refinement of the mesh
       of @a coarse_fes and the CoarseFineTransformations returned by the method
       Mesh::GetRefinementTransforms() of the refined mesh are set accordingly.
       The Operator::Type of @a T can be set to request an Operator of the set
       type. Currently, only Operator::MFEM_SPARSEMAT and Operator::ANY_TYPE
       (matrix-free) are supported. When Operator::ANY_TYPE is requested, the
       choice of the particular Operator sub-class is left to the method.  This
       method also works in parallel because the transfer operator is local to
       the MPI task when the input is a synchronized ParGridFunction. */
   void GetTransferOperator(const FiniteElementSpace &coarse_fes,
                            OperatorHandle &T) const;

   /** @brief Construct and return an Operator that can be used to transfer
       true-dof data from @a coarse_fes, defined on a coarse mesh, to @a this FE
       space, defined on a refined mesh.

       This method calls GetTransferOperator() and multiplies the result by the
       prolongation operator of @a coarse_fes on the right, and by the
       restriction operator of this FE space on the left.

       The Operator::Type of @a T can be set to request an Operator of the set
       type. In serial, the supported types are: Operator::MFEM_SPARSEMAT and
       Operator::ANY_TYPE (matrix-free). In parallel, the supported types are:
       Operator::Hypre_ParCSR and Operator::ANY_TYPE. Any other type is treated
       as Operator::ANY_TYPE: the operator representation choice is made by this
       method. */
   virtual void GetTrueTransferOperator(const FiniteElementSpace &coarse_fes,
                                        OperatorHandle &T) const;

   /** @brief Reflect changes in the mesh: update number of DOFs, etc. Also, calculate
       GridFunction transformation operator (unless want_transform is false).
       Safe to call multiple times, does nothing if space already up to date. */
   virtual void Update(bool want_transform = true);

   /// Get the GridFunction update operator.
   const Operator* GetUpdateOperator() { Update(); return Th.Ptr(); }

   /// Return the update operator in the given OperatorHandle, @a T.
   void GetUpdateOperator(OperatorHandle &T) { T = Th; }

   /** @brief Set the ownership of the update operator: if set to false, the
       Operator returned by GetUpdateOperator() must be deleted outside the
       FiniteElementSpace. */
   /** The update operator ownership is automatically reset to true when a new
       update operator is created by the Update() method. */
   void SetUpdateOperatorOwner(bool own) { Th.SetOperatorOwner(own); }

   /// Specify the Operator::Type to be used by the update operators.
   /** The default type is Operator::ANY_TYPE which leaves the choice to this
       class. The other currently supported option is Operator::MFEM_SPARSEMAT
       which is only guaranteed to be honored for a refinement update operator.
       Any other type will be treated as Operator::ANY_TYPE.
       @note This operation destroys the current update operator (if owned). */
   void SetUpdateOperatorType(Operator::Type tid) { Th.SetType(tid); }

   /// Free the GridFunction update operator (if any), to save memory.
   virtual void UpdatesFinished() { Th.Clear(); }

   /** Return update counter, similar to Mesh::GetSequence(). Used by
       GridFunction to check if it is up to date with the space. */
   long GetSequence() const { return sequence; }

   /// Return whether or not the space is discontinuous (L2)
   bool IsDGSpace() const
   {
      return dynamic_cast<const L2_FECollection*>(fec) != NULL;
   }

   /** In variable order spaces on nonconforming (NC) meshes, this function
       controls whether strict conformity is enforced in cases where coarse
       edges/faces have higher polynomial order than their fine NC neighbors.
       In the default (strict) case, the coarse side polynomial order is
       reduced to that of the lowest order fine edge/face, so all fine
       neighbors can interpolate the coarse side exactly. If relaxed == true,
       some discontinuities in the solution in such cases are allowed and the
       coarse side is not restricted. For an example, see
       https://github.com/mfem/mfem/pull/1423#issuecomment-621340392 */
   void SetRelaxedHpConformity(bool relaxed = true)
   {
      relaxed_hp = relaxed;
      orders_changed = true; // force update
      Update(false);
   }

   /// Save finite element space to output stream @a out.
   void Save(std::ostream &out) const;

   /** @brief Read a FiniteElementSpace from a stream. The returned
       FiniteElementCollection is owned by the caller. */
   FiniteElementCollection *Load(Mesh *m, std::istream &input);

   virtual ~FiniteElementSpace();
};

/// @brief Return true if the mesh contains only one topology and the elements are tensor elements.
inline bool UsesTensorBasis(const FiniteElementSpace& fes)
{
   Mesh & mesh = *fes.GetMesh();
   const bool mixed = mesh.GetNumGeometries(mesh.Dimension()) > 1;
   // Potential issue: empty local mesh --> no element 0.
   return !mixed &&
          dynamic_cast<const mfem::TensorBasisElement *>(fes.GetFE(0))!=nullptr;
}

}

#endif