Reference documentation for deal.II version Git 34859fde61 2019-12-10 11:26:22 -0700
FE_Nedelec< dim > Class Template Reference

#include <deal.II/fe/fe_nedelec.h>

Inheritance diagram for FE_Nedelec< dim >:
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## Public Member Functions

FE_Nedelec (const unsigned int order)

virtual std::string get_name () const override

virtual bool has_support_on_face (const unsigned int shape_index, const unsigned int face_index) const override

virtual bool hp_constraints_are_implemented () const override

virtual FiniteElementDomination::Domination compare_for_domination (const FiniteElement< dim > &fe_other, const unsigned int codim=0) const override final

virtual std::vector< std::pair< unsigned int, unsigned int > > hp_vertex_dof_identities (const FiniteElement< dim > &fe_other) const override

virtual std::vector< std::pair< unsigned int, unsigned int > > hp_line_dof_identities (const FiniteElement< dim > &fe_other) const override

virtual std::vector< std::pair< unsigned int, unsigned int > > hp_quad_dof_identities (const FiniteElement< dim > &fe_other) const override

virtual void get_face_interpolation_matrix (const FiniteElement< dim > &source, FullMatrix< double > &matrix) const override

virtual void get_subface_interpolation_matrix (const FiniteElement< dim > &source, const unsigned int subface, FullMatrix< double > &matrix) const override

virtual const FullMatrix< double > & get_restriction_matrix (const unsigned int child, const RefinementCase< dim > &refinement_case=RefinementCase< dim >::isotropic_refinement) const override

virtual const FullMatrix< double > & get_prolongation_matrix (const unsigned int child, const RefinementCase< dim > &refinement_case=RefinementCase< dim >::isotropic_refinement) const override

virtual void convert_generalized_support_point_values_to_dof_values (const std::vector< Vector< double >> &support_point_values, std::vector< double > &nodal_values) const override

virtual std::pair< Table< 2, bool >, std::vector< unsigned int > > get_constant_modes () const override

virtual std::size_t memory_consumption () const override

virtual std::unique_ptr< FiniteElement< dim, dim > > clone () const override

Public Member Functions inherited from FE_PolyTensor< dim >
FE_PolyTensor (const TensorPolynomialsBase< dim > &polynomials, const FiniteElementData< dim > &fe_data, const std::vector< bool > &restriction_is_additive_flags, const std::vector< ComponentMask > &nonzero_components)

FE_PolyTensor (const FE_PolyTensor &fe)

virtual double shape_value (const unsigned int i, const Point< dim > &p) const override

virtual Tensor< 1, dim > shape_grad (const unsigned int i, const Point< dim > &p) const override

virtual Tensor< 2, dim > shape_grad_grad (const unsigned int i, const Point< dim > &p) const override

Public Member Functions inherited from FiniteElement< dim, dim >
FiniteElement (const FiniteElementData< dim > &fe_data, const std::vector< bool > &restriction_is_additive_flags, const std::vector< ComponentMask > &nonzero_components)

FiniteElement (FiniteElement< dim, spacedim > &&)=default

FiniteElement (const FiniteElement< dim, spacedim > &)=default

virtual ~FiniteElement () override=default

std::pair< std::unique_ptr< FiniteElement< dim, spacedim > >, unsigned int > operator^ (const unsigned int multiplicity) const

const FiniteElement< dim, spacedim > & operator[] (const unsigned int fe_index) const

virtual bool operator== (const FiniteElement< dim, spacedim > &fe) const

bool operator!= (const FiniteElement< dim, spacedim > &) const

virtual Tensor< 3, dim > shape_3rd_derivative (const unsigned int i, const Point< dim > &p) const

virtual Tensor< 3, dim > shape_3rd_derivative_component (const unsigned int i, const Point< dim > &p, const unsigned int component) const

virtual Tensor< 4, dim > shape_4th_derivative (const unsigned int i, const Point< dim > &p) const

virtual Tensor< 4, dim > shape_4th_derivative_component (const unsigned int i, const Point< dim > &p, const unsigned int component) const

bool prolongation_is_implemented () const

bool isotropic_prolongation_is_implemented () const

bool restriction_is_implemented () const

bool isotropic_restriction_is_implemented () const

bool restriction_is_additive (const unsigned int index) const

const FullMatrix< double > & constraints (const ::internal::SubfaceCase< dim > &subface_case=::internal::SubfaceCase< dim >::case_isotropic) const

bool constraints_are_implemented (const ::internal::SubfaceCase< dim > &subface_case=::internal::SubfaceCase< dim >::case_isotropic) const

virtual void get_interpolation_matrix (const FiniteElement< dim, spacedim > &source, FullMatrix< double > &matrix) const

virtual void get_face_interpolation_matrix (const FiniteElement< dim, spacedim > &source, FullMatrix< double > &matrix) const

virtual void get_subface_interpolation_matrix (const FiniteElement< dim, spacedim > &source, const unsigned int subface, FullMatrix< double > &matrix) const

virtual std::vector< std::pair< unsigned int, unsigned int > > hp_vertex_dof_identities (const FiniteElement< dim, spacedim > &fe_other) const

virtual std::vector< std::pair< unsigned int, unsigned int > > hp_line_dof_identities (const FiniteElement< dim, spacedim > &fe_other) const

virtual std::vector< std::pair< unsigned int, unsigned int > > hp_quad_dof_identities (const FiniteElement< dim, spacedim > &fe_other) const

virtual FiniteElementDomination::Domination compare_for_face_domination (const FiniteElement< dim, spacedim > &fe_other) const final

virtual FiniteElementDomination::Domination compare_for_domination (const FiniteElement< dim, spacedim > &fe_other, const unsigned int codim=0) const

std::pair< unsigned int, unsigned int > system_to_component_index (const unsigned int index) const

unsigned int component_to_system_index (const unsigned int component, const unsigned int index) const

std::pair< unsigned int, unsigned int > face_system_to_component_index (const unsigned int index) const

unsigned int adjust_quad_dof_index_for_face_orientation (const unsigned int index, const bool face_orientation, const bool face_flip, const bool face_rotation) const

virtual unsigned int face_to_cell_index (const unsigned int face_dof_index, const unsigned int face, const bool face_orientation=true, const bool face_flip=false, const bool face_rotation=false) const

unsigned int adjust_line_dof_index_for_line_orientation (const unsigned int index, const bool line_orientation) const

const ComponentMaskget_nonzero_components (const unsigned int i) const

unsigned int n_nonzero_components (const unsigned int i) const

bool is_primitive () const

bool is_primitive (const unsigned int i) const

unsigned int n_base_elements () const

virtual const FiniteElement< dim, spacedim > & base_element (const unsigned int index) const

unsigned int element_multiplicity (const unsigned int index) const

virtual const FiniteElement< dim, spacedim > & get_sub_fe (const unsigned int first_component, const unsigned int n_selected_components) const

std::pair< std::pair< unsigned int, unsigned int >, unsigned int > system_to_base_index (const unsigned int index) const

std::pair< std::pair< unsigned int, unsigned int >, unsigned int > face_system_to_base_index (const unsigned int index) const

types::global_dof_index first_block_of_base (const unsigned int b) const

std::pair< unsigned int, unsigned int > component_to_base_index (const unsigned int component) const

std::pair< unsigned int, unsigned int > block_to_base_index (const unsigned int block) const

std::pair< unsigned int, types::global_dof_indexsystem_to_block_index (const unsigned int component) const

unsigned int component_to_block_index (const unsigned int component) const

const std::vector< Point< dim > > & get_unit_support_points () const

bool has_support_points () const

virtual Point< dim > unit_support_point (const unsigned int index) const

const std::vector< Point< dim - 1 > > & get_unit_face_support_points () const

bool has_face_support_points () const

virtual Point< dim - 1 > unit_face_support_point (const unsigned int index) const

const std::vector< Point< dim > > & get_generalized_support_points () const

bool has_generalized_support_points () const

const std::vector< Point< dim - 1 > > & get_generalized_face_support_points () const

bool has_generalized_face_support_points () const

GeometryPrimitive get_associated_geometry_primitive (const unsigned int cell_dof_index) const

Public Member Functions inherited from Subscriptor
Subscriptor ()

Subscriptor (const Subscriptor &)

Subscriptor (Subscriptor &&) noexcept

virtual ~Subscriptor ()

Subscriptoroperator= (const Subscriptor &)

Subscriptoroperator= (Subscriptor &&) noexcept

void subscribe (std::atomic< bool > *const validity, const std::string &identifier="") const

void unsubscribe (std::atomic< bool > *const validity, const std::string &identifier="") const

unsigned int n_subscriptions () const

template<typename StreamType >
void list_subscribers (StreamType &stream) const

void list_subscribers () const

template<class Archive >
void serialize (Archive &ar, const unsigned int version)

Public Member Functions inherited from FiniteElementData< dim >
FiniteElementData (const std::vector< unsigned int > &dofs_per_object, const unsigned int n_components, const unsigned int degree, const Conformity conformity=unknown, const BlockIndices &block_indices=BlockIndices())

unsigned int n_dofs_per_vertex () const

unsigned int n_dofs_per_line () const

unsigned int n_dofs_per_hex () const

unsigned int n_dofs_per_face () const

unsigned int n_dofs_per_cell () const

template<int structdim>
unsigned int n_dofs_per_object () const

unsigned int n_components () const

unsigned int n_blocks () const

const BlockIndicesblock_indices () const

unsigned int tensor_degree () const

bool conforms (const Conformity) const

bool operator== (const FiniteElementData &) const

## Private Member Functions

void initialize_support_points (const unsigned int order)

void initialize_restriction ()

## Static Private Member Functions

static std::vector< unsigned int > get_dpo_vector (const unsigned int degree, bool dg=false)

## Private Attributes

Table< 2, double > boundary_weights

## Friends

template<int dim1>
class FE_Nedelec

Public Types inherited from FiniteElementData< dim >
enum  Conformity {
unknown = 0x00, L2 = 0x01, Hcurl = 0x02, Hdiv = 0x04,
H1 = Hcurl | Hdiv, H2 = 0x0e
}

Static Public Member Functions inherited from FiniteElement< dim, dim >
static ::ExceptionBaseExcShapeFunctionNotPrimitive (int arg1)

static ::ExceptionBaseExcFENotPrimitive ()

static ::ExceptionBaseExcUnitShapeValuesDoNotExist ()

static ::ExceptionBaseExcFEHasNoSupportPoints ()

static ::ExceptionBaseExcEmbeddingVoid ()

static ::ExceptionBaseExcProjectionVoid ()

static ::ExceptionBaseExcWrongInterfaceMatrixSize (int arg1, int arg2)

static ::ExceptionBaseExcInterpolationNotImplemented ()

Static Public Member Functions inherited from Subscriptor
static ::ExceptionBaseExcInUse (int arg1, std::string arg2, std::string arg3)

static ::ExceptionBaseExcNoSubscriber (std::string arg1, std::string arg2)

Public Attributes inherited from FiniteElementData< dim >
const unsigned int dofs_per_vertex

const unsigned int dofs_per_line

const unsigned int dofs_per_hex

const unsigned int first_line_index

const unsigned int first_hex_index

const unsigned int first_face_line_index

const unsigned int dofs_per_face

const unsigned int dofs_per_cell

const unsigned int components

const unsigned int degree

const Conformity conforming_space

const BlockIndices block_indices_data

Static Public Attributes inherited from FiniteElement< dim, dim >
static const unsigned int space_dimension

Static Public Attributes inherited from FiniteElementData< dim >
static const unsigned int dimension = dim

Protected Member Functions inherited from FE_PolyTensor< dim >
bool single_mapping_kind () const

MappingKind get_mapping_kind (const unsigned int i) const

Protected Member Functions inherited from FiniteElement< dim, dim >
void reinit_restriction_and_prolongation_matrices (const bool isotropic_restriction_only=false, const bool isotropic_prolongation_only=false)

TableIndices< 2 > interface_constraints_size () const

virtual std::unique_ptr< InternalDataBase > get_data (const UpdateFlags update_flags, const Mapping< dim, spacedim > &mapping, const Quadrature< dim > &quadrature, ::internal::FEValuesImplementation::FiniteElementRelatedData< dim, spacedim > &output_data) const=0

virtual std::unique_ptr< InternalDataBase > get_face_data (const UpdateFlags update_flags, const Mapping< dim, spacedim > &mapping, const Quadrature< dim - 1 > &quadrature, ::internal::FEValuesImplementation::FiniteElementRelatedData< dim, spacedim > &output_data) const

virtual std::unique_ptr< InternalDataBase > get_subface_data (const UpdateFlags update_flags, const Mapping< dim, spacedim > &mapping, const Quadrature< dim - 1 > &quadrature, ::internal::FEValuesImplementation::FiniteElementRelatedData< dim, spacedim > &output_data) const

virtual void fill_fe_values (const typename Triangulation< dim, spacedim >::cell_iterator &cell, const CellSimilarity::Similarity cell_similarity, const Quadrature< dim > &quadrature, const Mapping< dim, spacedim > &mapping, const typename Mapping< dim, spacedim >::InternalDataBase &mapping_internal, const ::internal::FEValuesImplementation::MappingRelatedData< dim, spacedim > &mapping_data, const InternalDataBase &fe_internal, ::internal::FEValuesImplementation::FiniteElementRelatedData< dim, spacedim > &output_data) const=0

virtual void fill_fe_face_values (const typename Triangulation< dim, spacedim >::cell_iterator &cell, const unsigned int face_no, const Quadrature< dim - 1 > &quadrature, const Mapping< dim, spacedim > &mapping, const typename Mapping< dim, spacedim >::InternalDataBase &mapping_internal, const ::internal::FEValuesImplementation::MappingRelatedData< dim, spacedim > &mapping_data, const InternalDataBase &fe_internal, ::internal::FEValuesImplementation::FiniteElementRelatedData< dim, spacedim > &output_data) const=0

virtual void fill_fe_subface_values (const typename Triangulation< dim, spacedim >::cell_iterator &cell, const unsigned int face_no, const unsigned int sub_no, const Quadrature< dim - 1 > &quadrature, const Mapping< dim, spacedim > &mapping, const typename Mapping< dim, spacedim >::InternalDataBase &mapping_internal, const ::internal::FEValuesImplementation::MappingRelatedData< dim, spacedim > &mapping_data, const InternalDataBase &fe_internal, ::internal::FEValuesImplementation::FiniteElementRelatedData< dim, spacedim > &output_data) const=0

Static Protected Member Functions inherited from FiniteElement< dim, dim >
static std::vector< unsigned int > compute_n_nonzero_components (const std::vector< ComponentMask > &nonzero_components)

Protected Attributes inherited from FE_PolyTensor< dim >
std::vector< MappingKindmapping_kind

const std::unique_ptr< const TensorPolynomialsBase< dim > > poly_space

FullMatrix< double > inverse_node_matrix

Point< dim > cached_point

std::vector< Tensor< 1, dim > > cached_values

std::vector< Tensor< 2, dim > > cached_grads

Protected Attributes inherited from FiniteElement< dim, dim >
std::vector< std::vector< FullMatrix< double > > > restriction

std::vector< std::vector< FullMatrix< double > > > prolongation

FullMatrix< double > interface_constraints

std::vector< Point< dim > > unit_support_points

std::vector< Point< dim - 1 > > unit_face_support_points

std::vector< Point< dim > > generalized_support_points

std::vector< Point< dim - 1 > > generalized_face_support_points

std::vector< std::pair< unsigned int, unsigned int > > system_to_component_table

std::vector< std::pair< unsigned int, unsigned int > > face_system_to_component_table

std::vector< std::pair< std::pair< unsigned int, unsigned int >, unsigned int > > system_to_base_table

std::vector< std::pair< std::pair< unsigned int, unsigned int >, unsigned int > > face_system_to_base_table

BlockIndices base_to_block_indices

std::vector< std::pair< std::pair< unsigned int, unsigned int >, unsigned int > > component_to_base_table

const std::vector< unsigned int > n_nonzero_components_table

const bool cached_primitivity

## Detailed Description

### template<int dim> class FE_Nedelec< dim >

Warning
Several aspects of the implementation are experimental. For the moment, it is safe to use the element on globally refined meshes with consistent orientation of faces. See the todo entries below for more detailed caveats.

Implementation of Nédélec elements. The Nédélec space is designed to solve problems in which the solution only lives in the space $$H^\text{curl}=\{ {\mathbf u} \in L_2: \text{curl}\, {\mathbf u} \in L_2\}$$, rather than in the more commonly used space $$H^1=\{ u \in L_2: \nabla u \in L_2\}$$. In other words, the solution must be a vector field whose curl is square integrable, but for which the gradient may not be square integrable. The typical application for this space (and these elements) is to the Maxwell equations and corresponding simplifications, such as the reduced version of the Maxwell equation that only involves the electric field $$\mathbf E$$ which has to satisfy the equation $$\text{curl}\, \text{curl}\, {\mathbf E} = 0$$ in the time independent case when no currents are present, or the equation $$\text{curl}\,\text{curl}\,{\mathbf A} = 4\pi{\mathbf j}$$ that the magnetic vector potential $$\mathbf A$$ has to satisfy in the time independent case.

The defining characteristic of functions in $$H^\text{curl}$$ is that they are in general discontinuous – but that if you draw a line in 2d (or a surface in 3d), then the tangential component(s) of the vector field must be continuous across the line (or surface) even though the normal component may not be. As a consequence, the Nédélec element is constructed in such a way that (i) it is vector-valued, (ii) the shape functions are discontinuous, but (iii) the tangential component(s) of the vector field represented by each shape function are continuous across the faces of cells.

Other properties of the Nédélec element are that (i) it is not a primitive element; (ii) the shape functions are defined so that certain integrals over the faces are either zero or one, rather than the common case of certain point values being either zero or one.

We follow the commonly used – though confusing – definition of the "degree" of Nédélec elements. Specifically, the "degree" of the element denotes the polynomial degree of the largest complete polynomial subspace contained in the finite element space, even if the space may contain shape functions of higher polynomial degree. The lowest order element is consequently FE_Nedelec(0), i.e., the Raviart-Thomas element "of degree zero", even though the functions of this space are in general polynomials of degree one in each variable. This choice of "degree" implies that the approximation order of the function itself is degree+1, as with usual polynomial spaces. The numbering so chosen implies the sequence

$Q_{k+1} \stackrel{\text{grad}}{\rightarrow} \text{Nedelec}_k \stackrel{\text{curl}}{\rightarrow} \text{RaviartThomas}_k \stackrel{\text{div}}{\rightarrow} DGQ_{k}$

Note that this follows the convention of Brezzi and Raviart, though not the one used in the original paper by Nédélec.

This class is not implemented for the codimension one case (spacedim != dim).

Todo:
Even if this element is implemented for two and three space dimensions, the definition of the node values relies on consistently oriented faces in 3D. Therefore, care should be taken on complicated meshes.

### Interpolation

The interpolation operators associated with the Nédélec element are constructed such that interpolation and computing the curl are commuting operations on rectangular mesh cells. We require this from interpolating arbitrary functions as well as the restriction matrices.

#### Node values

The node values for an element of degree k on the reference cell are:

1. On edges: the moments of the tangential component with respect to polynomials of degree k.
2. On faces: the moments of the tangential components with respect to dim-1 dimensional FE_Nedelec polynomials of degree k-1.
3. In cells: the moments with respect to gradients of polynomials in FE_Q of degree k.

#### Generalized support points

The node values above rely on integrals, which will be computed by quadrature rules themselves. The generalized support points are a set of points such that this quadrature can be performed with sufficient accuracy. The points needed are those of QGaussk+1 on each edge and QGaussk+2 on each face and in the interior of the cell (or none for N1).

Date
2009, 2010, 2011

Definition at line 147 of file fe_nedelec.h.

## ◆ FE_Nedelec()

template<int dim>
 FE_Nedelec< dim >::FE_Nedelec ( const unsigned int order )

Constructor for the Nedelec element of given order. The maximal polynomial degree of the shape functions is order+1 (in each variable; the total polynomial degree may be higher). If order = 0, the element is linear and has degrees of freedom only on the edges. If order >=1 the element has degrees of freedom on the edges, faces and volume. For example the 3D version of FE_Nedelec has 12 degrees of freedom for order = 0 and 54 for degree = 1. It is important to have enough quadrature points in order to perform the quadrature with sufficient accuracy. For example QGauss<dim>(order + 2) can be used for the quadrature formula, where order is the order of FE_Nedelec.

Definition at line 71 of file fe_nedelec.cc.

## ◆ get_name()

template<int dim>
 std::string FE_Nedelec< dim >::get_name ( ) const
overridevirtual

Return a string that uniquely identifies a finite element. This class returns FE_Nedelec<dim>(degree), with dim and degree replaced by appropriate values.

Implements FiniteElement< dim, dim >.

Definition at line 207 of file fe_nedelec.cc.

## ◆ has_support_on_face()

template<int dim>
 bool FE_Nedelec< dim >::has_support_on_face ( const unsigned int shape_index, const unsigned int face_index ) const
overridevirtual

This function returns true, if the shape function shape_index has non-zero function values somewhere on the face face_index.

Reimplemented from FiniteElement< dim, dim >.

Definition at line 2036 of file fe_nedelec.cc.

## ◆ hp_constraints_are_implemented()

template<int dim>
 bool FE_Nedelec< dim >::hp_constraints_are_implemented ( ) const
overridevirtual

Return whether this element implements its hanging node constraints in the new way, which has to be used to make elements "hp compatible".

For the FE_Nedelec class the result is always true (independent of the degree of the element), as it implements the complete set of functions necessary for hp capability.

Reimplemented from FiniteElement< dim, dim >.

Definition at line 2313 of file fe_nedelec.cc.

## ◆ compare_for_domination()

template<int dim>
 FiniteElementDomination::Domination FE_Nedelec< dim >::compare_for_domination ( const FiniteElement< dim > & fe_other, const unsigned int codim = 0 ) const
finaloverridevirtual

Return whether this element dominates another one given as argument fe_other, whether it is the other way around, whether neither dominates, or if either could dominate. The codim parameter describes the codimension of the investigated subspace and specifies that it is subject to this comparison. For example, if codim==0 then this function compares which element dominates at the cell level. If codim==1, then the elements are compared at faces, i.e., the comparison happens between the function spaces of the two finite elements as restricted to a face. Larger values of codim work correspondingly.

For a definition of domination, see FiniteElementDomination::Domination and in particular the hp paper.

Definition at line 2277 of file fe_nedelec.cc.

## ◆ hp_vertex_dof_identities()

template<int dim>
 std::vector< std::pair< unsigned int, unsigned int > > FE_Nedelec< dim >::hp_vertex_dof_identities ( const FiniteElement< dim > & fe_other ) const
overridevirtual

If, on a vertex, several finite elements are active, the hp code first assigns the degrees of freedom of each of these FEs different global indices. It then calls this function to find out which of them should get identical values, and consequently can receive the same global DoF index. This function therefore returns a list of identities between DoFs of the present finite element object with the DoFs of fe_other, which is a reference to a finite element object representing one of the other finite elements active on this particular vertex. The function computes which of the degrees of freedom of the two finite element objects are equivalent, both numbered between zero and the corresponding value of dofs_per_vertex of the two finite elements. The first index of each pair denotes one of the vertex dofs of the present element, whereas the second is the corresponding index of the other finite element.

Definition at line 2320 of file fe_nedelec.cc.

## ◆ hp_line_dof_identities()

template<int dim>
 std::vector< std::pair< unsigned int, unsigned int > > FE_Nedelec< dim >::hp_line_dof_identities ( const FiniteElement< dim > & fe_other ) const
overridevirtual

Same as hp_vertex_dof_indices(), except that the function treats degrees of freedom on lines.

Definition at line 2329 of file fe_nedelec.cc.

template<int dim>
 std::vector< std::pair< unsigned int, unsigned int > > FE_Nedelec< dim >::hp_quad_dof_identities ( const FiniteElement< dim > & fe_other ) const
overridevirtual

Same as hp_vertex_dof_indices(), except that the function treats degrees of freedom on lines.

Definition at line 2371 of file fe_nedelec.cc.

## ◆ get_face_interpolation_matrix()

template<int dim>
 void FE_Nedelec< dim >::get_face_interpolation_matrix ( const FiniteElement< dim > & source, FullMatrix< double > & matrix ) const
overridevirtual

Return the matrix interpolating from a face of one element to the face of the neighboring element. The size of the matrix is then source.dofs_per_face times this->dofs_per_face.

Derived elements will have to implement this function. They may only provide interpolation matrices for certain source finite elements, for example those from the same family. If they don't implement interpolation from a given element, then they must throw an exception of type FiniteElement<dim>::ExcInterpolationNotImplemented.

Definition at line 2425 of file fe_nedelec.cc.

## ◆ get_subface_interpolation_matrix()

template<int dim>
 void FE_Nedelec< dim >::get_subface_interpolation_matrix ( const FiniteElement< dim > & source, const unsigned int subface, FullMatrix< double > & matrix ) const
overridevirtual

Return the matrix interpolating from a face of one element to the subface of the neighboring element. The size of the matrix is then source.dofs_per_face times this->dofs_per_face.

Derived elements will have to implement this function. They may only provide interpolation matrices for certain source finite elements, for example those from the same family. If they don't implement interpolation from a given element, then they must throw an exception of type ExcInterpolationNotImplemented.

Definition at line 2527 of file fe_nedelec.cc.

## ◆ get_restriction_matrix()

template<int dim>
 const FullMatrix< double > & FE_Nedelec< dim >::get_restriction_matrix ( const unsigned int child, const RefinementCase< dim > & refinement_case = RefinementCase::isotropic_refinement ) const
overridevirtual

Projection from a fine grid space onto a coarse grid space. If this projection operator is associated with a matrix P, then the restriction of this matrix P_i to a single child cell is returned here.

The matrix P is the concatenation or the sum of the cell matrices P_i, depending on the restriction_is_additive_flags. This distinguishes interpolation (concatenation) and projection with respect to scalar products (summation).

Row and column indices are related to coarse grid and fine grid spaces, respectively, consistent with the definition of the associated operator.

Reimplemented from FiniteElement< dim, dim >.

Definition at line 3018 of file fe_nedelec.cc.

## ◆ get_prolongation_matrix()

template<int dim>
 const FullMatrix< double > & FE_Nedelec< dim >::get_prolongation_matrix ( const unsigned int child, const RefinementCase< dim > & refinement_case = RefinementCase::isotropic_refinement ) const
overridevirtual

Embedding matrix between grids.

The identity operator from a coarse grid space into a fine grid space is associated with a matrix P. The restriction of this matrix P_i to a single child cell is returned here.

The matrix P is the concatenation, not the sum of the cell matrices P_i. That is, if the same non-zero entry j,k exists in two different child matrices P_i, the value should be the same in both matrices and it is copied into the matrix P only once.

Row and column indices are related to fine grid and coarse grid spaces, respectively, consistent with the definition of the associated operator.

These matrices are used by routines assembling the prolongation matrix for multi-level methods. Upon assembling the transfer matrix between cells using this matrix array, zero elements in the prolongation matrix are discarded and will not fill up the transfer matrix.

Reimplemented from FiniteElement< dim, dim >.

Definition at line 2958 of file fe_nedelec.cc.

## ◆ convert_generalized_support_point_values_to_dof_values()

template<int dim>
 void FE_Nedelec< dim >::convert_generalized_support_point_values_to_dof_values ( const std::vector< Vector< double >> & support_point_values, std::vector< double > & nodal_values ) const
overridevirtual

Given the values of a function $$f(\mathbf x)$$ at the (generalized) support points of the reference cell, this function then computes what the nodal values of the element are, i.e., $$\Psi_i[f]$$, where $$\Psi_i$$ are the node functionals of the element (see also Node values or node functionals). The values $$\Psi_i[f]$$ are then the expansion coefficients for the shape functions of the finite element function that interpolates the given function $$f(x)$$, i.e., $$f_h(\mathbf x) = \sum_i \Psi_i[f] \varphi_i(\mathbf x)$$ is the finite element interpolant of $$f$$ with the current element. The operation described here is used, for example, in the FETools::compute_node_matrix() function.

In more detail, let us assume that the generalized support points (see this glossary entry ) of the current element are $$\hat{\mathbf x}_i$$ and that the node functionals associated with the current element are $$\Psi_i[\cdot]$$. Then, the fact that the element is based on generalized support points, implies that if we apply $$\Psi_i$$ to a (possibly vector-valued) finite element function $$\varphi$$, the result must have the form $$\Psi_i[\varphi] = f_i(\varphi(\hat{\mathbf x}_i))$$ – in other words, the value of the node functional $$\Psi_i$$ applied to $$\varphi$$ only depends on the values of $$\varphi$$ at $$\hat{\mathbf x}_i$$ and not on values anywhere else, or integrals of $$\varphi$$, or any other kind of information.

The exact form of $$f_i$$ depends on the element. For example, for scalar Lagrange elements, we have that in fact $$\Psi_i[\varphi] = \varphi(\hat{\mathbf x}_i)$$. If you combine multiple scalar Lagrange elements via an FESystem object, then $$\Psi_i[\varphi] = \varphi(\hat{\mathbf x}_i)_{c(i)}$$ where $$c(i)$$ is the result of the FiniteElement::system_to_component_index() function's return value's first component. In these two cases, $$f_i$$ is therefore simply the identity (in the scalar case) or a function that selects a particular vector component of its argument. On the other hand, for Raviart-Thomas elements, one would have that $$f_i(\mathbf y) = \mathbf y \cdot \mathbf n_i$$ where $$\mathbf n_i$$ is the normal vector of the face at which the shape function is defined.

Given all of this, what this function does is the following: If you input a list of values of a function $$\varphi$$ at all generalized support points (where each value is in fact a vector of values with as many components as the element has), then this function returns a vector of values obtained by applying the node functionals to these values. In other words, if you pass in $$\{\varphi(\hat{\mathbf x}_i)\}_{i=0}^{N-1}$$ then you will get out a vector $$\{\Psi[\varphi]\}_{i=0}^{N-1}$$ where $$N$$ equals dofs_per_cell.

Parameters
 [in] support_point_values An array of size dofs_per_cell (which equals the number of points the get_generalized_support_points() function will return) where each element is a vector with as many entries as the element has vector components. This array should contain the values of a function at the generalized support points of the current element. [out] nodal_values An array of size dofs_per_cell that contains the node functionals of the element applied to the given function.
Note
It is safe to call this function for (transformed) values on the real cell only for elements with trivial MappingKind. For all other elements (for example for H(curl), or H(div) conforming elements) vector values have to be transformed to the reference cell first.
Given what the function is supposed to do, the function clearly can only work for elements that actually implement (generalized) support points. Elements that do not have generalized support points – e.g., elements whose nodal functionals evaluate integrals or moments of functions (such as FE_Q_Hierarchical) – can in general not make sense of the operation that is required for this function. They consequently may not implement it.

Reimplemented from FiniteElement< dim, dim >.

Definition at line 3087 of file fe_nedelec.cc.

## ◆ get_constant_modes()

template<int dim>
 std::pair< Table< 2, bool >, std::vector< unsigned int > > FE_Nedelec< dim >::get_constant_modes ( ) const
overridevirtual

Return a list of constant modes of the element.

Reimplemented from FiniteElement< dim, dim >.

Definition at line 4035 of file fe_nedelec.cc.

## ◆ memory_consumption()

template<int dim>
 std::size_t FE_Nedelec< dim >::memory_consumption ( ) const
overridevirtual

Determine an estimate for the memory consumption (in bytes) of this object.

This function is made virtual, since finite element objects are usually accessed through pointers to their base class, rather than the class itself.

Reimplemented from FiniteElement< dim, dim >.

Definition at line 4051 of file fe_nedelec.cc.

## ◆ clone()

template<int dim>
 std::unique_ptr< FiniteElement< dim, dim > > FE_Nedelec< dim >::clone ( ) const
overridevirtual

A sort of virtual copy constructor, this function returns a copy of the finite element object. Derived classes need to override the function here in this base class and return an object of the same type as the derived class.

Some places in the library, for example the constructors of FESystem as well as the hp::FECollection class, need to make copies of finite elements without knowing their exact type. They do so through this function.

Implements FiniteElement< dim, dim >.

Definition at line 225 of file fe_nedelec.cc.

## ◆ get_dpo_vector()

template<int dim>
 std::vector< unsigned int > FE_Nedelec< dim >::get_dpo_vector ( const unsigned int degree, bool dg = false )
staticprivate

Only for internal use. Its full name is get_dofs_per_object_vector function and it creates the dofs_per_object vector that is needed within the constructor to be passed to the constructor of FiniteElementData.

If the optional argument dg is true, the vector returned will have all degrees of freedom assigned to the cell, none on the faces and edges.

Definition at line 2000 of file fe_nedelec.cc.

## ◆ initialize_support_points()

template<int dim>
 void FE_Nedelec< dim >::initialize_support_points ( const unsigned int order )
private

Initialize the generalized_support_points field of the FiniteElement class and fill the tables with interpolation weights (boundary_weights and interior_weights). Called from the constructor.

## ◆ initialize_restriction()

template<int dim>
 void FE_Nedelec< dim >::initialize_restriction ( )
private

Initialize the interpolation from functions on refined mesh cells onto the father cell. According to the philosophy of the Nédélec element, this restriction operator preserves the curl of a function weakly.

Definition at line 516 of file fe_nedelec.cc.

## ◆ boundary_weights

template<int dim>
 Table<2, double> FE_Nedelec< dim >::boundary_weights
private

These are the factors multiplied to a function in the generalized_face_support_points when computing the integration.

Definition at line 364 of file fe_nedelec.h.

## ◆ mutex

template<int dim>