Reference documentation for deal.II version GIT 6da2e5d553 2022-07-01 18:55:02+00:00
FE_PolyTensor< dim, spacedim > Class Template Referenceabstract

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

Inheritance diagram for FE_PolyTensor< dim, spacedim >:
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## Classes

class  InternalData

## Public Types

enum  Conformity {
unknown = 0x00 , L2 = 0x01 , Hcurl = 0x02 , Hdiv = 0x04 ,
H1 = Hcurl | Hdiv , H2 = 0x0e
}

## Public Member Functions

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 UpdateFlags requires_update_flags (const UpdateFlags update_flags) const override

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

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

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

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

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

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

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

virtual std::unique_ptr< FiniteElement< dim, spacedim > > clone () const=0

virtual std::string get_name () const=0

bool operator== (const FiniteElementData &) const

ReferenceCell reference_cell () const

unsigned int n_unique_faces () const

unsigned int n_dofs_per_vertex () const

unsigned int n_dofs_per_line () const

unsigned int n_dofs_per_quad (unsigned int face_no=0) const

unsigned int n_dofs_per_hex () const

unsigned int n_dofs_per_face (unsigned int face_no=0, unsigned int child=0) const

unsigned int max_dofs_per_face () const

unsigned int n_dofs_per_cell () const

template<int structdim>
unsigned int n_dofs_per_object (const unsigned int i=0) 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

unsigned int get_first_line_index () const

unsigned int get_first_hex_index () const

unsigned int get_first_face_line_index (const unsigned int face_no=0) const

unsigned int get_first_face_quad_index (const unsigned int face_no=0) const

Shape function access
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

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

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

virtual const FullMatrix< double > & get_prolongation_matrix (const unsigned int child, const RefinementCase< dim > &refinement_case=RefinementCase< dim >::isotropic_refinement) 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 bool hp_constraints_are_implemented () 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 unsigned int face_no=0) const

virtual void get_subface_interpolation_matrix (const FiniteElement< dim, spacedim > &source, const unsigned int subface, FullMatrix< double > &matrix, const unsigned int face_no=0) 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 unsigned int face_no=0) const

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

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

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

Index computations
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 face_no=0) const

unsigned int adjust_quad_dof_index_for_face_orientation (const unsigned int index, const unsigned int face_no, 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 unsigned int face_no=0) 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

Component and block matrices

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

## Public Attributes

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

static constexpr unsigned int space_dimension

static constexpr unsigned int dimension = dim

## Protected Member Functions

bool single_mapping_kind () const

bool adjust_quad_dof_sign_for_face_orientation (const unsigned int index, const unsigned int face_no, const bool face_orientation, const bool face_flip, const bool face_rotation) const

MappingKind get_mapping_kind (const unsigned int i) const

virtual std::unique_ptr< typename FiniteElement< dim, spacedim >::InternalDataBaseget_data (const UpdateFlags update_flags, const Mapping< dim, spacedim > &mapping, const Quadrature< dim > &quadrature, ::internal::FEValuesImplementation::FiniteElementRelatedData< dim, spacedim > &output_data) const override

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 typename FiniteElement< dim, spacedim >::InternalDataBase &fe_internal, ::internal::FEValuesImplementation::FiniteElementRelatedData< dim, spacedim > &output_data) const override

virtual void fill_fe_face_values (const typename Triangulation< dim, spacedim >::cell_iterator &cell, const unsigned int face_no, const hp::QCollection< 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 typename FiniteElement< dim, spacedim >::InternalDataBase &fe_internal, ::internal::FEValuesImplementation::FiniteElementRelatedData< dim, spacedim > &output_data) const override

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 typename FiniteElement< dim, spacedim >::InternalDataBase &fe_internal, ::internal::FEValuesImplementation::FiniteElementRelatedData< dim, spacedim > &output_data) const override

## Protected Attributes

std::vector< MappingKindmapping_kind

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

FullMatrix< doubleinverse_node_matrix

std::mutex cache_mutex

Point< dim > cached_point

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

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

## Private Attributes

const ReferenceCell reference_cell_kind

const unsigned int number_unique_faces

const std::vector< unsigned int > n_dofs_on_quad

const std::vector< unsigned int > first_index_of_quads

const std::vector< unsigned int > first_line_index_of_faces

const std::vector< unsigned int > first_quad_index_of_faces

const std::vector< unsigned int > n_dofs_on_face

const unsigned int dofs_per_face_max

## Support points and interpolation

std::vector< std::vector< FullMatrix< double > > > restriction

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

FullMatrix< doubleinterface_constraints

std::vector< Point< dim > > unit_support_points

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

std::vector< Point< dim > > generalized_support_points

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

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

std::vector< 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::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

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 unsigned int face_no=0) const

bool has_face_support_points (const unsigned int face_no=0) const

virtual Point< dim - 1 > unit_face_support_point (const unsigned int index, const unsigned int face_no=0) const

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

bool has_generalized_support_points () const

GeometryPrimitive get_associated_geometry_primitive (const unsigned int cell_dof_index) const

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

virtual std::size_t memory_consumption () const

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< InternalDataBaseget_face_data (const UpdateFlags update_flags, const Mapping< dim, spacedim > &mapping, const hp::QCollection< dim - 1 > &quadrature, ::internal::FEValuesImplementation::FiniteElementRelatedData< dim, spacedim > &output_data) const

virtual std::unique_ptr< InternalDataBaseget_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< InternalDataBaseget_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 hp::QCollection< 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

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

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 ::ExceptionBaseExcShapeFunctionNotPrimitive (int arg1)

static ::ExceptionBaseExcFENotPrimitive ()

static ::ExceptionBaseExcUnitShapeValuesDoNotExist ()

static ::ExceptionBaseExcFEHasNoSupportPoints ()

static ::ExceptionBaseExcEmbeddingVoid ()

static ::ExceptionBaseExcProjectionVoid ()

static ::ExceptionBaseExcWrongInterfaceMatrixSize (int arg1, int arg2)

static ::ExceptionBaseExcInterpolationNotImplemented ()

static std::vector< unsigned int > compute_n_nonzero_components (const std::vector< ComponentMask > &nonzero_components)

## Subscriptor functionality

Classes derived from Subscriptor provide a facility to subscribe to this object. This is mostly used by the SmartPointer class.

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)

static ::ExceptionBaseExcInUse (int arg1, std::string arg2, std::string arg3)

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

using map_value_type = decltype(counter_map)::value_type

using map_iterator = decltype(counter_map)::iterator

std::atomic< unsigned int > counter

std::map< std::string, unsigned int > counter_map

std::vector< std::atomic< bool > * > validity_pointers

const std::type_info * object_info

static std::mutex mutex

void check_no_subscribers () const noexcept

## Detailed Description

### template<int dim, int spacedim = dim> class FE_PolyTensor< dim, spacedim >

This class provides a unified framework for the implementation of FiniteElement classes based on tensor-valued polynomial spaces like PolynomialsBDM and PolynomialsRaviartThomas. In this, it is the tensor-valued equivalent of the FE_Poly class.

In essence, what this class requires is that a derived class describes to it a (vector-valued) polynomial space in which every polynomial has exactly dim vector components. The classes that provide such implementations are all derived from the TensorPolynomialsBase class, and an object of one of these derived types needs to be provided to the constructor of this class.

### Deriving classes

This class is not a fully implemented FiniteElement class, but implements some common features of vector valued elements based on vector valued polynomial classes. What's missing here in particular is information on the topological location of the node values (i.e., whether a degree of freedom is logically at a vertex, edge, face, or the interior of a cell – information that determines the continuity properties of the associated shape functions across cell interfaces), and derived classes need to provide this information.

Similarly, in many cases, node functionals depend on the shape of the mesh cell, since they evaluate normal or tangential components on the faces. In order to allow for a set of transformations, the variable mapping_kind has been introduced. It needs be set in the constructor of a derived class.

Any derived class must decide on the polynomial space to use. This polynomial space should be implemented simply as a set of vector valued polynomials like PolynomialsBDM and PolynomialsRaviartThomas. In order to facilitate this implementation, which basis the polynomial space chooses is not of importance to the current class – as described next, this class handles the transformation from the basis chosen by the polynomial space template argument to the basis we want to use for finite element computations internally.

#### Determining the correct basis

In most cases, the basis used by the class that describes the polynomial space, $$\{\tilde\varphi_j(\hat{\mathbf x})\}$$, does not match the one we want to use for the finite element description, $$\{\varphi_j(\hat{\mathbf x})\}$$. Rather, we need to express the finite element shape functions as a linear combination of the basis provided by the polynomial space:

\begin{align*} \varphi_j = \sum_k c_{jk} \tilde\varphi_j. \end{align*}

These expansion coefficients $$c_{jk}$$ are typically computed in the constructors of derived classes. To facilitate this, this class at first (unless told otherwise, see below), assumes that the shape functions should be exactly the ones provided by the polynomial space. In the constructor of the derived class, one then typically has code of the form

// Now compute the inverse node matrix, generating the correct
// basis functions from the raw ones. For a discussion of what
// exactly happens here, see FETools::compute_node_matrix.
this->inverse_node_matrix.reinit(n_dofs, n_dofs);
// From now on, the shape functions provided by FiniteElement::shape_value
// and similar functions will be the correct ones, not
// the raw shape functions from the polynomial space anymore.
FullMatrix< double > inverse_node_matrix
void invert(const FullMatrix< number2 > &M)
FullMatrix< double > compute_node_matrix(const FiniteElement< dim, spacedim > &fe)

The FETools::compute_node_matrix() function explains in more detail what exactly it computes, and how; in any case, the result is that inverse_node_matrix now contains the expansion coefficients $$c_{jk}$$, and the fact that this block of code now sets the matrix to a non-zero size indicates to the functions of the current class that it should from then on use the expanded basis, $$\{\varphi_j(\hat{\mathbf x})\}$$, and no longer the original, "raw" basis $$\{\tilde\varphi_j(\hat{\mathbf x})\}$$ when asked for values or derivatives of shape functions.

In order for this scheme to work, it is important to ensure that the size of the inverse_node_matrix be zero at the time when FETools::compute_node_matrix() is called; thus, the call to this function cannot be inlined into the last line – the result of the call really does need to be stored in the temporary object M.

#### Setting the transformation

In most cases, vector valued basis functions must be transformed when mapped from the reference cell to the actual grid cell. These transformations can be selected from the set MappingKind and stored in mapping_kind. Therefore, each constructor should contain a line like:

std::vector< MappingKind > mapping_kind
@ mapping_none
Definition: mapping.h:77

(in case no mapping is required) or using whatever value among the ones defined in MappingKind is appropriate for the element you are implementing. If each shape function may be mapped by different mappings, then mapping_kind may be a vector with the same number of elements as there are shape functions.

TensorPolynomialsBase

Definition at line 141 of file fe_poly_tensor.h.

## ◆ Conformity

template<int dim>
 inherited

Enumerator for the different types of continuity a finite element may have. Continuity is measured by the Sobolev space containing the constructed finite element space and is also called this way.

Note that certain continuities may imply others. For instance, a function in H1 is in Hcurl and Hdiv as well.

If you are interested in continuity in the classical sense, then the following relations hold:

1. H1 implies that the function is continuous over cell boundaries.

2. H2 implies that the function is continuously differentiable over cell boundaries.

3. L2 indicates that the element is discontinuous. Since discontinuous elements have no topological couplings between grid cells and code may actually depend on this property, L2 conformity is handled in a special way in the sense that it is not implied by any higher conformity.

In order to test if a finite element conforms to a certain space, use FiniteElementData<dim>::conforms().

Enumerator
unknown

Indicates incompatible continuities of a system.

L2

Discontinuous elements. See above!

Hcurl

Conformity with the space Hcurl (continuous tangential component of a vector field)

Hdiv

Conformity with the space Hdiv (continuous normal component of a vector field)

H1

Conformity with the space H1 (continuous)

H2

Conformity with the space H2 (continuously differentiable)

Definition at line 259 of file fe_data.h.

## ◆ FE_PolyTensor() [1/2]

template<int dim, int spacedim = dim>
 FE_PolyTensor< dim, spacedim >::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 )

Constructor. This constructor does a deep copy of the polynomials object via the TensorPolynomialsBase::clone() function and stores a pointer to the copy. As a consequence, the calling site can simply pass a temporary object as the first argument.

## ◆ FE_PolyTensor() [2/2]

template<int dim, int spacedim = dim>
 FE_PolyTensor< dim, spacedim >::FE_PolyTensor ( const FE_PolyTensor< dim, spacedim > & fe )

Copy constructor.

## ◆ requires_update_flags()

template<int dim, int spacedim = dim>
 virtual UpdateFlags FE_PolyTensor< dim, spacedim >::requires_update_flags ( const UpdateFlags update_flags ) const
overridevirtual

Given a set of update flags, compute which other quantities also need to be computed in order to satisfy the request by the given flags. Then return the combination of the original set of flags and those just computed.

As an example, if update_flags contains update_gradients a finite element class will typically require the computation of the inverse of the Jacobian matrix in order to rotate the gradient of shape functions on the reference cell to the real cell. It would then return not just update_gradients, but also update_covariant_transformation, the flag that makes the mapping class produce the inverse of the Jacobian matrix.

An extensive discussion of the interaction between this function and FEValues can be found in the How Mapping, FiniteElement, and FEValues work together documentation module.

UpdateFlags

Implements FiniteElement< dim, dim >.

## ◆ shape_value()

template<int dim, int spacedim = dim>
 virtual double FE_PolyTensor< dim, spacedim >::shape_value ( const unsigned int i, const Point< dim > & p ) const
overridevirtual

Compute the (scalar) value of shape function i at the given quadrature point p. Since the elements represented by this class are vector valued, there is no such scalar value and the function therefore throws an exception.

Reimplemented from FiniteElement< dim, dim >.

## ◆ shape_value_component()

template<int dim, int spacedim = dim>
 virtual double FE_PolyTensor< dim, spacedim >::shape_value_component ( const unsigned int i, const Point< dim > & p, const unsigned int component ) const
overridevirtual

Just like for shape_value(), but this function will be called when the shape function has more than one non-zero vector component. In that case, this function should return the value of the component-th vector component of the ith shape function at point p.

Reimplemented from FiniteElement< dim, dim >.

template<int dim, int spacedim = dim>
 virtual Tensor<1, dim> FE_PolyTensor< dim, spacedim >::shape_grad ( const unsigned int i, const Point< dim > & p ) const
overridevirtual

Compute the gradient of (scalar) shape function i at the given quadrature point p. Since the elements represented by this class are vector valued, there is no such scalar value and the function therefore throws an exception.

Reimplemented from FiniteElement< dim, dim >.

template<int dim, int spacedim = dim>
 virtual Tensor<1, dim> FE_PolyTensor< dim, spacedim >::shape_grad_component ( const unsigned int i, const Point< dim > & p, const unsigned int component ) const
overridevirtual

Just like for shape_grad(), but this function will be called when the shape function has more than one non-zero vector component. In that case, this function should return the gradient of the component-th vector component of the ith shape function at point p.

Reimplemented from FiniteElement< dim, dim >.

template<int dim, int spacedim = dim>
 virtual Tensor<2, dim> FE_PolyTensor< dim, spacedim >::shape_grad_grad ( const unsigned int i, const Point< dim > & p ) const
overridevirtual

Compute the Hessian of (scalar) shape function i at the given quadrature point p. Since the elements represented by this class are vector valued, there is no such scalar value and the function therefore throws an exception.

Reimplemented from FiniteElement< dim, dim >.

template<int dim, int spacedim = dim>
 virtual Tensor<2, dim> FE_PolyTensor< dim, spacedim >::shape_grad_grad_component ( const unsigned int i, const Point< dim > & p, const unsigned int component ) const
overridevirtual

Just like for shape_grad_grad(), but this function will be called when the shape function has more than one non-zero vector component. In that case, this function should return the gradient of the component-th vector component of the ith shape function at point p.

Reimplemented from FiniteElement< dim, dim >.

## ◆ single_mapping_kind()

template<int dim, int spacedim = dim>
 bool FE_PolyTensor< dim, spacedim >::single_mapping_kind ( ) const
protected

Returns a boolean that is true when the finite element uses a single mapping and false when the finite element uses multiple mappings.

template<int dim, int spacedim = dim>
 bool FE_PolyTensor< dim, spacedim >::adjust_quad_dof_sign_for_face_orientation ( const unsigned int index, const unsigned int face_no, const bool face_orientation, const bool face_flip, const bool face_rotation ) const
protected

For faces with non-standard face_orientation in 3D, the dofs on faces (quads) have to be permuted in order to be combined with the correct shape functions and additionally can change the sign. Given a local dof index on a quad, return the sign of the permuted shape function, if the face has non-standard face_orientation, face_flip or face_rotation. In 2D and 1D there is no need for permutation and consequently it does nothing in this case.

The permutation itself is returned by adjust_quad_dof_index_for_face_orientation implemented in the interface class FiniteElement<dim>.

## ◆ get_mapping_kind()

template<int dim, int spacedim = dim>
 MappingKind FE_PolyTensor< dim, spacedim >::get_mapping_kind ( const unsigned int i ) const
protected

Returns MappingKind i for the finite element.

## ◆ get_data()

template<int dim, int spacedim = dim>
 virtual std::unique_ptr< typename FiniteElement::InternalDataBase> FE_PolyTensor< dim, spacedim >::get_data ( const UpdateFlags update_flags, const Mapping< dim, spacedim > & mapping, const Quadrature< dim > & quadrature, ::internal::FEValuesImplementation::FiniteElementRelatedData< dim, spacedim > & output_data ) const
inlineoverrideprotectedvirtual

Create an internal data object and return a pointer to it of which the caller of this function then assumes ownership. This object will then be passed to the FiniteElement::fill_fe_values() every time the finite element shape functions and their derivatives are evaluated on a concrete cell. The object created here is therefore used by derived classes as a place for scratch objects that are used in evaluating shape functions, as well as to store information that can be pre-computed once and re-used on every cell (e.g., for evaluating the values and gradients of shape functions on the reference cell, for later re-use when transforming these values to a concrete cell).

This function is the first one called in the process of initializing a FEValues object for a given mapping and finite element object. The returned object will later be passed to FiniteElement::fill_fe_values() for a concrete cell, which will itself place its output into an object of type internal::FEValuesImplementation::FiniteElementRelatedData. Since there may be data that can already be computed in its final form on the reference cell, this function also receives a reference to the internal::FEValuesImplementation::FiniteElementRelatedData object as its last argument. This output argument is guaranteed to always be the same one when used with the InternalDataBase object returned by this function. In other words, the subdivision of scratch data and final data in the returned object and the output_data object is as follows: If data can be pre- computed on the reference cell in the exact form in which it will later be needed on a concrete cell, then this function should already emplace it in the output_data object. An example are the values of shape functions at quadrature points for the usual Lagrange elements which on a concrete cell are identical to the ones on the reference cell. On the other hand, if some data can be pre-computed to make computations on a concrete cell cheaper, then it should be put into the returned object for later re-use in a derive class's implementation of FiniteElement::fill_fe_values(). An example are the gradients of shape functions on the reference cell for Lagrange elements: to compute the gradients of the shape functions on a concrete cell, one has to multiply the gradients on the reference cell by the inverse of the Jacobian of the mapping; consequently, we cannot already compute the gradients on a concrete cell at the time the current function is called, but we can at least pre-compute the gradients on the reference cell, and store it in the object returned.

An extensive discussion of the interaction between this function and FEValues can be found in the How Mapping, FiniteElement, and FEValues work together documentation module. See also the documentation of the InternalDataBase class.

Parameters
 [in] update_flags A set of UpdateFlags values that describe what kind of information the FEValues object requests the finite element to compute. This set of flags may also include information that the finite element can not compute, e.g., flags that pertain to data produced by the mapping. An implementation of this function needs to set up all data fields in the returned object that are necessary to produce the finite- element related data specified by these flags, and may already pre- compute part of this information as discussed above. Elements may want to store these update flags (or a subset of these flags) in InternalDataBase::update_each so they know at the time when FiniteElement::fill_fe_values() is called what they are supposed to compute [in] mapping A reference to the mapping used for computing values and derivatives of shape functions. [in] quadrature A reference to the object that describes where the shape functions should be evaluated. [out] output_data A reference to the object that FEValues will use in conjunction with the object returned here and where an implementation of FiniteElement::fill_fe_values() will place the requested information. This allows the current function to already pre-compute pieces of information that can be computed on the reference cell, as discussed above. FEValues guarantees that this output object and the object returned by the current function will always be used together.
Returns
A pointer to an object of a type derived from InternalDataBase and that derived classes can use to store scratch data that can be pre- computed, or for scratch arrays that then only need to be allocated once. The calling site assumes ownership of this object and will delete it when it is no longer necessary.

Implements FiniteElement< dim, dim >.

Definition at line 278 of file fe_poly_tensor.h.

## ◆ fill_fe_values() [1/2]

template<int dim, int spacedim = dim>
 virtual void FE_PolyTensor< dim, spacedim >::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 typename FiniteElement< dim, spacedim >::InternalDataBase & fe_internal, ::internal::FEValuesImplementation::FiniteElementRelatedData< dim, spacedim > & output_data ) const
overrideprotectedvirtual

## ◆ fill_fe_face_values() [1/3]

template<int dim, int spacedim = dim>
 virtual void FE_PolyTensor< dim, spacedim >::fill_fe_face_values ( const typename Triangulation< dim, spacedim >::cell_iterator & cell, const unsigned int face_no, const hp::QCollection< 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 typename FiniteElement< dim, spacedim >::InternalDataBase & fe_internal, ::internal::FEValuesImplementation::FiniteElementRelatedData< dim, spacedim > & output_data ) const
overrideprotectedvirtual

## ◆ fill_fe_subface_values() [1/2]

template<int dim, int spacedim = dim>
 virtual void FE_PolyTensor< dim, spacedim >::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 typename FiniteElement< dim, spacedim >::InternalDataBase & fe_internal, ::internal::FEValuesImplementation::FiniteElementRelatedData< dim, spacedim > & output_data ) const
overrideprotectedvirtual

## ◆ operator^()

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

Creates information for creating a FESystem with this class as base element and with multiplicity multiplicity. In particular, the return type of this function can be used in the constructor for a FESystem object. This function calls clone() and hence creates a copy of the current object.

## ◆ clone()

 virtual std::unique_ptr > FiniteElement< dim, spacedim >::clone ( ) const
pure virtualinherited

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.

## ◆ get_name()

 virtual std::string FiniteElement< dim, spacedim >::get_name ( ) const
pure virtualinherited

Return a string that uniquely identifies a finite element. The general convention is that this is the class name, followed by the dimension in angle brackets, and the polynomial degree and whatever else is necessary in parentheses. For example, FE_Q<2>(3) is the value returned for a cubic element in 2d.

Systems of elements have their own naming convention, see the FESystem class.

## ◆ shape_3rd_derivative()

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

Return the tensor of third derivatives of the ith shape function at point p on the unit cell. The derivatives are derivatives on the unit cell with respect to unit cell coordinates. If the finite element is vector-valued, then return the value of the only non-zero component of the vector value of this shape function. If the shape function has more than one non-zero component (which we refer to with the term non-primitive), then derived classes implementing this function should throw an exception of type ExcShapeFunctionNotPrimitive. In that case, use the shape_3rd_derivative_component() function.

Implementations of this function should throw an exception of type ExcUnitShapeValuesDoNotExist if the shape functions of the FiniteElement under consideration depend on the shape of the cell in real space, i.e., if the shape functions are not defined by mapping from the reference cell. Some non-conforming elements are defined this way, as is the FE_DGPNonparametric class, to name just one example.

The default implementation of this virtual function does exactly this, i.e., it simply throws an exception of type ExcUnitShapeValuesDoNotExist.

Reimplemented in FESystem< dim, spacedim >, and FE_Poly< dim, spacedim >.

## ◆ shape_3rd_derivative_component()

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

Just like for shape_3rd_derivative(), but this function will be called when the shape function has more than one non-zero vector component. In that case, this function should return the gradient of the component- th vector component of the ith shape function at point p.

Reimplemented in FESystem< dim, spacedim >, and FE_Poly< dim, spacedim >.

## ◆ shape_4th_derivative()

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

Return the tensor of fourth derivatives of the ith shape function at point p on the unit cell. The derivatives are derivatives on the unit cell with respect to unit cell coordinates. If the finite element is vector-valued, then return the value of the only non-zero component of the vector value of this shape function. If the shape function has more than one non-zero component (which we refer to with the term non-primitive), then derived classes implementing this function should throw an exception of type ExcShapeFunctionNotPrimitive. In that case, use the shape_4th_derivative_component() function.

Implementations of this function should throw an exception of type ExcUnitShapeValuesDoNotExist if the shape functions of the FiniteElement under consideration depend on the shape of the cell in real space, i.e., if the shape functions are not defined by mapping from the reference cell. Some non-conforming elements are defined this way, as is the FE_DGPNonparametric class, to name just one example.

The default implementation of this virtual function does exactly this, i.e., it simply throws an exception of type ExcUnitShapeValuesDoNotExist.

Reimplemented in FESystem< dim, spacedim >, and FE_Poly< dim, spacedim >.

## ◆ shape_4th_derivative_component()

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

Just like for shape_4th_derivative(), but this function will be called when the shape function has more than one non-zero vector component. In that case, this function should return the gradient of the component- th vector component of the ith shape function at point p.

Reimplemented in FESystem< dim, spacedim >, and FE_Poly< dim, spacedim >.

## ◆ has_support_on_face()

 virtual bool FiniteElement< dim, spacedim >::has_support_on_face ( const unsigned int shape_index, const unsigned int face_index ) const
virtualinherited

This function returns true, if the shape function shape_index has non-zero function values somewhere on the face face_index. The function is typically used to determine whether some matrix elements resulting from face integrals can be assumed to be zero and may therefore be omitted from integration.

A default implementation is provided in this base class which always returns true. This is the safe way to go.

## ◆ get_restriction_matrix()

 virtual const FullMatrix& FiniteElement< dim, spacedim >::get_restriction_matrix ( const unsigned int child, const RefinementCase< dim > & refinement_case = RefinementCase::isotropic_refinement ) const
virtualinherited

Return the matrix that describes restricting a finite element field from the given child (as obtained by the given refinement_case) to the parent cell. The interpretation of the returned matrix depends on what restriction_is_additive() returns for each shape function.

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

If projection matrices are not implemented in the derived finite element class, this function aborts with an exception of type FiniteElement::ExcProjectionVoid. You can check whether this would happen by first calling the restriction_is_implemented() or the isotropic_restriction_is_implemented() function.

## ◆ get_prolongation_matrix()

 virtual const FullMatrix& FiniteElement< dim, spacedim >::get_prolongation_matrix ( const unsigned int child, const RefinementCase< dim > & refinement_case = RefinementCase::isotropic_refinement ) const
virtualinherited

Prolongation/embedding matrix between grids.

The identity operator from a coarse grid space into a fine grid space (where both spaces are identified as functions defined on the parent and child cells) is associated with a matrix P that maps the corresponding representations of these functions in terms of their nodal values. 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.

If prolongation matrices are not implemented in the derived finite element class, this function aborts with an exception of type FiniteElement::ExcEmbeddingVoid. You can check whether this would happen by first calling the prolongation_is_implemented() or the isotropic_prolongation_is_implemented() function.

## ◆ prolongation_is_implemented()

 bool FiniteElement< dim, spacedim >::prolongation_is_implemented ( ) const
inherited

Return whether this element implements its prolongation matrices. The return value also indicates whether a call to the get_prolongation_matrix() function will generate an error or not.

Note, that this function returns true only if the prolongation matrices of the isotropic and all anisotropic refinement cases are implemented. If you are interested in the prolongation matrices for isotropic refinement only, use the isotropic_prolongation_is_implemented function instead.

This function is mostly here in order to allow us to write more efficient test programs which we run on all kinds of weird elements, and for which we simply need to exclude certain tests in case something is not implemented. It will in general probably not be a great help in applications, since there is not much one can do if one needs these features and they are not implemented. This function could be used to check whether a call to get_prolongation_matrix() will succeed; however, one then still needs to cope with the lack of information this just expresses.

## ◆ isotropic_prolongation_is_implemented()

 bool FiniteElement< dim, spacedim >::isotropic_prolongation_is_implemented ( ) const
inherited

Return whether this element implements its prolongation matrices for isotropic children. The return value also indicates whether a call to the get_prolongation_matrix function will generate an error or not.

This function is mostly here in order to allow us to write more efficient test programs which we run on all kinds of weird elements, and for which we simply need to exclude certain tests in case something is not implemented. It will in general probably not be a great help in applications, since there is not much one can do if one needs these features and they are not implemented. This function could be used to check whether a call to get_prolongation_matrix() will succeed; however, one then still needs to cope with the lack of information this just expresses.

## ◆ restriction_is_implemented()

 bool FiniteElement< dim, spacedim >::restriction_is_implemented ( ) const
inherited

Return whether this element implements its restriction matrices. The return value also indicates whether a call to the get_restriction_matrix() function will generate an error or not.

Note, that this function returns true only if the restriction matrices of the isotropic and all anisotropic refinement cases are implemented. If you are interested in the restriction matrices for isotropic refinement only, use the isotropic_restriction_is_implemented() function instead.

This function is mostly here in order to allow us to write more efficient test programs which we run on all kinds of weird elements, and for which we simply need to exclude certain tests in case something is not implemented. It will in general probably not be a great help in applications, since there is not much one can do if one needs these features and they are not implemented. This function could be used to check whether a call to get_restriction_matrix() will succeed; however, one then still needs to cope with the lack of information this just expresses.

## ◆ isotropic_restriction_is_implemented()

 bool FiniteElement< dim, spacedim >::isotropic_restriction_is_implemented ( ) const
inherited

Return whether this element implements its restriction matrices for isotropic children. The return value also indicates whether a call to the get_restriction_matrix() function will generate an error or not.

This function is mostly here in order to allow us to write more efficient test programs which we run on all kinds of weird elements, and for which we simply need to exclude certain tests in case something is not implemented. It will in general probably not be a great help in applications, since there is not much one can do if one needs these features and they are not implemented. This function could be used to check whether a call to get_restriction_matrix() will succeed; however, one then still needs to cope with the lack of information this just expresses.

 bool FiniteElement< dim, spacedim >::restriction_is_additive ( const unsigned int index ) const
inherited

The index must be between zero and the number of shape functions of this element.

## ◆ constraints()

 const FullMatrix& FiniteElement< dim, spacedim >::constraints ( const ::internal::SubfaceCase< dim > & subface_case = ::internal::SubfaceCase::case_isotropic ) const
inherited

Return a read only reference to the matrix that describes the constraints at the interface between a refined and an unrefined cell.

Some finite elements do not (yet) implement hanging node constraints. If this is the case, then this function will generate an exception, since no useful return value can be generated. If you should have a way to live with this, then you might want to use the constraints_are_implemented() function to check up front whether this function will succeed or generate the exception.

## ◆ constraints_are_implemented()

 bool FiniteElement< dim, spacedim >::constraints_are_implemented ( const ::internal::SubfaceCase< dim > & subface_case = ::internal::SubfaceCase::case_isotropic ) const
inherited

Return whether this element implements its hanging node constraints. The return value also indicates whether a call to the constraints() function will generate an error or not.

This function is mostly here in order to allow us to write more efficient test programs which we run on all kinds of weird elements, and for which we simply need to exclude certain tests in case hanging node constraints are not implemented. It will in general probably not be a great help in applications, since there is not much one can do if one needs hanging node constraints and they are not implemented. This function could be used to check whether a call to constraints() will succeed; however, one then still needs to cope with the lack of information this just expresses.

## ◆ hp_constraints_are_implemented()

 virtual bool FiniteElement< dim, spacedim >::hp_constraints_are_implemented ( ) const
virtualinherited

Return whether this element implements its hanging node constraints in the new way, which has to be used to make elements "hp-compatible". That means, the element properly implements the get_face_interpolation_matrix and get_subface_interpolation_matrix methods. Therefore the return value also indicates whether a call to the get_face_interpolation_matrix() method and the get_subface_interpolation_matrix() method will generate an error or not.

Currently the main purpose of this function is to allow the make_hanging_node_constraints method to decide whether the new procedures, which are supposed to work in the hp-framework can be used, or if the old well verified but not hp-capable functions should be used. Once the transition to the new scheme for computing the interface constraints is complete, this function will be superfluous and will probably go away.

Derived classes should implement this function accordingly. The default assumption is that a finite element does not provide hp-capable face interpolation, and the default implementation therefore returns false.

## ◆ get_interpolation_matrix()

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

Return the matrix interpolating from the given finite element to the present one. The size of the matrix is then dofs_per_cell times source.dofs_per_cell.

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.

Reimplemented in FE_Nothing< dim, spacedim >, and FESystem< dim, spacedim >.

## ◆ get_face_interpolation_matrix()

 virtual void FiniteElement< dim, spacedim >::get_face_interpolation_matrix ( const FiniteElement< dim, spacedim > & source, FullMatrix< double > & matrix, const unsigned int face_no = 0 ) const
virtualinherited

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 ExcInterpolationNotImplemented.

## ◆ get_subface_interpolation_matrix()

 virtual void FiniteElement< dim, spacedim >::get_subface_interpolation_matrix ( const FiniteElement< dim, spacedim > & source, const unsigned int subface, FullMatrix< double > & matrix, const unsigned int face_no = 0 ) const
virtualinherited

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.

## ◆ hp_vertex_dof_identities()

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

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 n_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.

## ◆ hp_line_dof_identities()

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

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

 virtual std::vector > FiniteElement< dim, spacedim >::hp_quad_dof_identities ( const FiniteElement< dim, spacedim > & fe_other, const unsigned int face_no = 0 ) const
virtualinherited

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

## ◆ compare_for_domination()

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

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.

## ◆ operator==() [1/2]

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

Comparison operator.

The implementation in the current class checks for equality of the following pieces of information between the current object and the one given as argument, in this order:

• the dynamic type (i.e., the type of the most derived class) of the current object and of the given object,
• the name returned by get_name(),
• as all of the fields in FiniteElementData,
• constraint matrices.

This covers most cases where elements can differ, but there are cases of derived elements that are different and for which the current function still returns true. For these cases, derived classes should overload this function.

Note
This operator specifically does not check the following member variables of the current class:
• restriction matrices,
• prolongation matrices of this object and the argument. This is because these member variables may be initialized only on demand by derived classes, rather than being available immediately. Consequently, comparing these members would not only be costly because these are generally big arrays, but also because their computation may be expensive. On the other hand, derived classes for which these arrays may differ for two objects even though the above list compares as equal, will probably want to implement their own operator==() anyway.

## ◆ operator==() [2/2]

template<int dim>
 bool FiniteElementData< dim >::operator== ( const FiniteElementData< dim > & f ) const
inherited

Comparison operator.

Definition at line 196 of file fe_data.cc.

## ◆ operator!=()

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

Non-equality comparison operator. Defined in terms of the equality comparison operator.

## ◆ system_to_component_index()

 std::pair FiniteElement< dim, spacedim >::system_to_component_index ( const unsigned int index ) const
inherited

Compute vector component and index of this shape function within the shape functions corresponding to this component from the index of a shape function within this finite element.

If the element is scalar, then the component is always zero, and the index within this component is equal to the overall index.

If the shape function referenced has more than one non-zero component, then it cannot be associated with one vector component, and an exception of type ExcShapeFunctionNotPrimitive will be raised.

Note that if the element is composed of other (base) elements, and a base element has more than one component but all its shape functions are primitive (i.e. are non-zero in only one component), then this mapping contains valid information. However, the index of a shape function of this element within one component (i.e. the second number of the respective entry of this array) does not indicate the index of the respective shape function within the base element (since that has more than one vector-component). For this information, refer to the system_to_base_table field and the system_to_base_index() function.

See the class description above for an example of how this function is typically used.

The use of this function is explained extensively in the step-8 and step-20 tutorial programs as well as in the Handling vector valued problems module.

## ◆ component_to_system_index()

 unsigned int FiniteElement< dim, spacedim >::component_to_system_index ( const unsigned int component, const unsigned int index ) const
inherited

Compute the shape function for the given vector component and index.

If the element is scalar, then the component must be zero, and the index within this component is equal to the overall index.

This is the opposite operation from the system_to_component_index() function.

## ◆ face_system_to_component_index()

 std::pair FiniteElement< dim, spacedim >::face_system_to_component_index ( const unsigned int index, const unsigned int face_no = 0 ) const
inherited

Same as system_to_component_index(), but do it for shape functions and their indices on a face. The range of allowed indices is therefore 0..dofs_per_face.

You will rarely need this function in application programs, since almost all application codes only need to deal with cell indices, not face indices. The function is mainly there for use inside the library.

 unsigned int FiniteElement< dim, spacedim >::adjust_quad_dof_index_for_face_orientation ( const unsigned int index, const unsigned int face_no, const bool face_orientation, const bool face_flip, const bool face_rotation ) const
inherited

For faces with non-standard face_orientation in 3D, the dofs on faces (quads) have to be permuted in order to be combined with the correct shape functions. Given a local dof index on a quad, return the local index, if the face has non-standard face_orientation, face_flip or face_rotation. In 2D and 1D there is no need for permutation and consequently an exception is thrown.

## ◆ face_to_cell_index()

 virtual unsigned int FiniteElement< dim, spacedim >::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
virtualinherited

Given an index in the natural ordering of indices on a face, return the index of the same degree of freedom on the cell.

To explain the concept, consider the case where we would like to know whether a degree of freedom on a face, for example as part of an FESystem element, is primitive. Unfortunately, the is_primitive() function in the FiniteElement class takes a cell index, so we would need to find the cell index of the shape function that corresponds to the present face index. This function does that.

Code implementing this would then look like this:

for (i=0; i<dofs_per_face; ++i)
if (fe.is_primitive(fe.face_to_cell_index(i, some_face_no)))
... do whatever
const unsigned int dofs_per_face
Definition: fe_data.h:419

The function takes additional arguments that account for the fact that actual faces can be in their standard ordering with respect to the cell under consideration, or can be flipped, oriented, etc.

Parameters
 face_dof_index The index of the degree of freedom on a face. This index must be between zero and dofs_per_face. face The number of the face this degree of freedom lives on. This number must be between zero and GeometryInfo::faces_per_cell. face_orientation One part of the description of the orientation of the face. See GlossFaceOrientation. face_flip One part of the description of the orientation of the face. See GlossFaceOrientation. face_rotation One part of the description of the orientation of the face. See GlossFaceOrientation.
Returns
The index of this degree of freedom within the set of degrees of freedom on the entire cell. The returned value will be between zero and dofs_per_cell.
Note
This function exists in this class because that is where it was first implemented. However, it can't really work in the most general case without knowing what element we have. The reason is that when a face is flipped or rotated, we also need to know whether we need to swap the degrees of freedom on this face, or whether they are immune from this. For this, consider the situation of a $$Q_3$$ element in 2d. If face_flip is true, then we need to consider the two degrees of freedom on the edge in reverse order. On the other hand, if the element were a $$Q_1^2$$, then because the two degrees of freedom on this edge belong to different vector components, they should not be considered in reverse order. What all of this shows is that the function can't work if there are more than one degree of freedom per line or quad, and that in these cases the function will throw an exception pointing out that this functionality will need to be provided by a derived class that knows what degrees of freedom actually represent.

Reimplemented in FESystem< dim, spacedim >.

 unsigned int FiniteElement< dim, spacedim >::adjust_line_dof_index_for_line_orientation ( const unsigned int index, const bool line_orientation ) const
inherited

For lines with non-standard line_orientation in 3D, the dofs on lines have to be permuted in order to be combined with the correct shape functions. Given a local dof index on a line, return the local index, if the line has non-standard line_orientation. In 2D and 1D there is no need for permutation, so the given index is simply returned.

## ◆ get_nonzero_components()

 const ComponentMask& FiniteElement< dim, spacedim >::get_nonzero_components ( const unsigned int i ) const
inherited

Return in which of the vector components of this finite element the ith shape function is non-zero. The length of the returned array is equal to the number of vector components of this element.

For most finite element spaces, the result of this function will be a vector with exactly one element being true, since for most spaces the individual vector components are independent. In that case, the component with the single zero is also the first element of what system_to_component_index() returns.

Only for those spaces that couple the components, for example to make a shape function divergence free, will there be more than one true entry. Elements for which this is true are called non-primitive (see GlossPrimitive).

## ◆ n_nonzero_components()

 unsigned int FiniteElement< dim, spacedim >::n_nonzero_components ( const unsigned int i ) const
inherited

Return in how many vector components the ith shape function is non-zero. This value equals the number of entries equal to true in the result of the get_nonzero_components() function.

For most finite element spaces, the result will be equal to one. It is not equal to one only for those ansatz spaces for which vector-valued shape functions couple the individual components, for example in order to make them divergence-free.

## ◆ is_primitive() [1/2]

 bool FiniteElement< dim, spacedim >::is_primitive ( ) const
inherited

Return whether the entire finite element is primitive, in the sense that all its shape functions are primitive. If the finite element is scalar, then this is always the case.

Since this is an extremely common operation, the result is cached and returned by this function.

## ◆ is_primitive() [2/2]

 bool FiniteElement< dim, spacedim >::is_primitive ( const unsigned int i ) const
inherited

Return whether the ith shape function is primitive in the sense that the shape function is non-zero in only one vector component. Non- primitive shape functions would then, for example, be those of divergence free ansatz spaces, in which the individual vector components are coupled.

The result of the function is true if and only if the result of n_nonzero_components(i) is equal to one.

## ◆ n_base_elements()

 unsigned int FiniteElement< dim, spacedim >::n_base_elements ( ) const
inherited

Number of base elements in a mixed discretization.

Note that even for vector valued finite elements, the number of components needs not coincide with the number of base elements, since they may be reused. For example, if you create a FESystem with three identical finite element classes by using the constructor that takes one finite element and a multiplicity, then the number of base elements is still one, although the number of components of the finite element is equal to the multiplicity.

## ◆ base_element()

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

Access to base element objects. If the element is atomic, then base_element(0) is this.

Reimplemented in FESystem< dim, spacedim >, and FE_Enriched< dim, spacedim >.

## ◆ element_multiplicity()

 unsigned int FiniteElement< dim, spacedim >::element_multiplicity ( const unsigned int index ) const
inherited

This index denotes how often the base element index is used in a composed element. If the element is atomic, then the result is always equal to one. See the documentation for the n_base_elements() function for more details.

## ◆ get_sub_fe() [1/2]

 const FiniteElement& FiniteElement< dim, spacedim >::get_sub_fe ( const ComponentMask & mask ) const
inherited

Return a reference to a contained finite element that matches the components selected by the given ComponentMask mask.

For an arbitrarily nested FESystem, this function returns the inner-most FiniteElement that matches the given mask. The method fails if the mask does not exactly match one of the contained finite elements. It is most useful if the current object is an FESystem, as the return value can only be this in all other cases.

Note that the returned object can be an FESystem if the mask matches it but not any of the contained objects.

Let us illustrate the function with the an FESystem fe with 7 components:

FESystem<2> fe_velocity(FE_Q<2>(2), 2);
FE_Q<2> fe_pressure(1);
FE_DGP<2> fe_dg(0);
FE_BDM<2> fe_nonprim(1);
FESystem<2> fe(fe_velocity, 1, fe_pressure, 1, fe_dg, 2, fe_nonprim, 1);
Definition: fe_bdm.h:60
Definition: fe_dgp.h:310
Definition: fe_q.h:549

The following table lists all possible component masks you can use:

[true,true,true,true,true,true,true] FESystem<2>[FESystem<2>[FE_Q<2>(2)^2]-FE_Q<2>(1)-FE_DGP<2>(0)^2-FE_BDM<2>(1)] fe itself, the whole FESystem
[true,true,false,false,false,false,false] FESystem<2>[FE_Q<2>(2)^2] just the fe_velocity
[true,false,false,false,false,false,false] FE_Q<2>(2) The first component in fe_velocity
[false,true,false,false,false,false,false] FE_Q<2>(2) The second component in fe_velocity
[false,false,true,false,false,false,false] FE_Q<2>(1) fe_pressure
[false,false,false,true,false,false,false] FE_DGP<2>(0) first copy of fe_dg
[false,false,false,false,true,false,false] FE_DGP<2>(0) second copy of fe_dg
[false,false,false,false,false,true,true] FE_BDM<2>(1) both components of fe_nonprim

## ◆ get_sub_fe() [2/2]

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

Return a reference to a contained finite element that matches the components n_selected_components components starting at component with index first_component.

See the other get_sub_fe() function above for more details.

Reimplemented in FESystem< dim, spacedim >.

## ◆ system_to_base_index()

 std::pair, unsigned int> FiniteElement< dim, spacedim >::system_to_base_index ( const unsigned int index ) const
inherited

Return for shape function index the base element it belongs to, the number of the copy of this base element (which is between zero and the multiplicity of this element), and the index of this shape function within this base element.

If the element is not composed of others, then base and instance are always zero, and the index is equal to the number of the shape function. If the element is composed of single instances of other elements (i.e. all with multiplicity one) all of which are scalar, then base values and dof indices within this element are equal to the system_to_component_table. It differs only in case the element is composed of other elements and at least one of them is vector-valued itself.

See the class documentation above for an example of how this function is typically used.

This function returns valid values also in the case of vector-valued (i.e. non-primitive) shape functions, in contrast to the system_to_component_index() function.

## ◆ face_system_to_base_index()

 std::pair, unsigned int> FiniteElement< dim, spacedim >::face_system_to_base_index ( const unsigned int index, const unsigned int face_no = 0 ) const
inherited

Same as system_to_base_index(), but for degrees of freedom located on a face. The range of allowed indices is therefore 0..dofs_per_face.

You will rarely need this function in application programs, since almost all application codes only need to deal with cell indices, not face indices. The function is mainly there for use inside the library.

## ◆ first_block_of_base()

 types::global_dof_index FiniteElement< dim, spacedim >::first_block_of_base ( const unsigned int b ) const
inherited

Given a base element number, return the first block of a BlockVector it would generate.

## ◆ component_to_base_index()

 std::pair FiniteElement< dim, spacedim >::component_to_base_index ( const unsigned int component ) const
inherited

For each vector component, return which base element implements this component and which vector component in this base element this is. This information is only of interest for vector-valued finite elements which are composed of several sub-elements. In that case, one may want to obtain information about the element implementing a certain vector component, which can be done using this function and the FESystem::base_element() function.

If this is a scalar finite element, then the return value is always equal to a pair of zeros.

## ◆ block_to_base_index()

 std::pair FiniteElement< dim, spacedim >::block_to_base_index ( const unsigned int block ) const
inherited

Return the base element for this block and the number of the copy of the base element.

## ◆ system_to_block_index()

 std::pair FiniteElement< dim, spacedim >::system_to_block_index ( const unsigned int component ) const
inherited

The vector block and the index inside the block for this shape function.

## ◆ component_to_block_index()

 unsigned int FiniteElement< dim, spacedim >::component_to_block_index ( const unsigned int component ) const
inherited

The vector block for this component.

 ComponentMask FiniteElement< dim, spacedim >::component_mask ( const FEValuesExtractors::Scalar & scalar ) const
inherited

Return a component mask with as many elements as this object has vector components and of which exactly the one component is true that corresponds to the given argument. See the glossary for more information.

Parameters
 scalar An object that represents a single scalar vector component of this finite element.
Returns
A component mask that is false in all components except for the one that corresponds to the argument.

 ComponentMask FiniteElement< dim, spacedim >::component_mask ( const FEValuesExtractors::Vector & vector ) const
inherited

Return a component mask with as many elements as this object has vector components and of which exactly the dim components are true that correspond to the given argument. See the glossary for more information.

Parameters
 vector An object that represents dim vector components of this finite element.
Returns
A component mask that is false in all components except for the ones that corresponds to the argument.

 ComponentMask FiniteElement< dim, spacedim >::component_mask ( const FEValuesExtractors::SymmetricTensor< 2 > & sym_tensor ) const
inherited

Return a component mask with as many elements as this object has vector components and of which exactly the dim*(dim+1)/2 components are true that correspond to the given argument. See the glossary for more information.

Parameters
 sym_tensor An object that represents dim*(dim+1)/2 components of this finite element that are jointly to be interpreted as forming a symmetric tensor.
Returns
A component mask that is false in all components except for the ones that corresponds to the argument.

inherited

Given a block mask (see this glossary entry), produce a component mask (see this glossary entry) that represents the components that correspond to the blocks selected in the input argument. This is essentially a conversion operator from BlockMask to ComponentMask.

Parameters
Returns
A mask that selects those components corresponding to the selected blocks of the input argument.

 BlockMask FiniteElement< dim, spacedim >::block_mask ( const FEValuesExtractors::Scalar & scalar ) const
inherited

Return a block mask with as many elements as this object has blocks and of which exactly the one component is true that corresponds to the given argument. See the glossary for more information.

Note
This function will only succeed if the scalar referenced by the argument encompasses a complete block. In other words, if, for example, you pass an extractor for the single $$x$$ velocity and this object represents an FE_RaviartThomas object, then the single scalar object you selected is part of a larger block and consequently there is no block mask that would represent it. The function will then produce an exception.
Parameters
 scalar An object that represents a single scalar vector component of this finite element.
Returns
A component mask that is false in all components except for the one that corresponds to the argument.

 BlockMask FiniteElement< dim, spacedim >::block_mask ( const FEValuesExtractors::Vector & vector ) const
inherited

Return a component mask with as many elements as this object has vector components and of which exactly the dim components are true that correspond to the given argument. See the glossary for more information.

Note
The same caveat applies as to the version of the function above: The extractor object passed as argument must be so that it corresponds to full blocks and does not split blocks of this element.
Parameters
 vector An object that represents dim vector components of this finite element.
Returns
A component mask that is false in all components except for the ones that corresponds to the argument.

 BlockMask FiniteElement< dim, spacedim >::block_mask ( const FEValuesExtractors::SymmetricTensor< 2 > & sym_tensor ) const
inherited

Return a component mask with as many elements as this object has vector components and of which exactly the dim*(dim+1)/2 components are true that correspond to the given argument. See the glossary for more information.

Note
The same caveat applies as to the version of the function above: The extractor object passed as argument must be so that it corresponds to full blocks and does not split blocks of this element.
Parameters
 sym_tensor An object that represents dim*(dim+1)/2 components of this finite element that are jointly to be interpreted as forming a symmetric tensor.
Returns
A component mask that is false in all components except for the ones that corresponds to the argument.

inherited

Given a component mask (see this glossary entry), produce a block mask (see this glossary entry) that represents the blocks that correspond to the components selected in the input argument. This is essentially a conversion operator from ComponentMask to BlockMask.

Note
This function will only succeed if the components referenced by the argument encompasses complete blocks. In other words, if, for example, you pass an component mask for the single $$x$$ velocity and this object represents an FE_RaviartThomas object, then the single component you selected is part of a larger block and consequently there is no block mask that would represent it. The function will then produce an exception.
Parameters
Returns
A mask that selects those blocks corresponding to the selected blocks of the input argument.

## ◆ get_constant_modes()

 virtual std::pair, std::vector > FiniteElement< dim, spacedim >::get_constant_modes ( ) const
virtualinherited

Return a list of constant modes of the element. The number of rows in the resulting table depends on the elements in use. For standard elements, the table has as many rows as there are components in the element and dofs_per_cell columns. To each component of the finite element, the row in the returned table contains a basis representation of the constant function 1 on the element. However, there are some scalar elements where there is more than one constant mode, e.g. the element FE_Q_DG0.

In order to match the constant modes to the actual components in the element, the returned data structure also returns a vector with as many components as there are constant modes on the element that contains the component number.

## ◆ get_unit_support_points()

 const std::vector >& FiniteElement< dim, spacedim >::get_unit_support_points ( ) const
inherited

Return the support points of the trial functions on the unit cell, if the derived finite element defines them. Finite elements that allow some kind of interpolation operation usually have support points. On the other hand, elements that define their degrees of freedom by, for example, moments on faces, or as derivatives, don't have support points. In that case, the returned field is empty.

If the finite element defines support points, then their number equals the number of degrees of freedom of the element. The order of points in the array matches that returned by the cell->get_dof_indices function.

See the class documentation for details on support points.

Note
Finite elements' implementation of this function returns these points in the same order as shape functions. The order of shape functions is typically documented in the class documentation of the various finite element classes. In particular, shape functions (and consequently the mapped quadrature points discussed in the class documentation of this class) will then traverse first those shape functions located on vertices, then on lines, then on quads, etc.
If this element implements support points, then it will return one such point per shape function. Since multiple shape functions may be defined at the same location, the support points returned here may be duplicated. An example would be an element of the kind FESystem(FE_Q(1),3) for which each support point would appear three times in the returned array.

## ◆ has_support_points()

 bool FiniteElement< dim, spacedim >::has_support_points ( ) const
inherited

Return whether a finite element has defined support points. If the result is true, then a call to the get_unit_support_points() yields a non-empty array.

The result may be false if an element is not defined by interpolating shape functions, for example by P-elements on quadrilaterals. It will usually only be true if the element constructs its shape functions by the requirement that they be one at a certain point and zero at all the points associated with the other shape functions.

In composed elements (i.e. for the FESystem class), the result will be true if all the base elements have defined support points. FE_Nothing is a special case in FESystems, because it has 0 support points and has_support_points() is false, but an FESystem containing an FE_Nothing among other elements will return true.

## ◆ unit_support_point()

 virtual Point FiniteElement< dim, spacedim >::unit_support_point ( const unsigned int index ) const
virtualinherited

Return the position of the support point of the indexth shape function. If it does not exist, raise an exception.

The default implementation simply returns the respective element from the array you get from get_unit_support_points(), but derived elements may overload this function. In particular, note that the FESystem class overloads it so that it can return the support points of individual base elements, if not all the base elements define support points. In this way, you can still ask for certain support points, even if get_unit_support_points() only returns an empty array.

Reimplemented in FESystem< dim, spacedim >.

## ◆ get_unit_face_support_points()

 const std::vector >& FiniteElement< dim, spacedim >::get_unit_face_support_points ( const unsigned int face_no = 0 ) const
inherited

Return the support points of the trial functions on the unit face, if the derived finite element defines some. Finite elements that allow some kind of interpolation operation usually have support points. On the other hand, elements that define their degrees of freedom by, for example, moments on faces, or as derivatives, don't have support points. In that case, the returned field is empty

Note that elements that have support points need not necessarily have some on the faces, even if the interpolation points are located physically on a face. For example, the discontinuous elements have interpolation points on the vertices, and for higher degree elements also on the faces, but they are not defined to be on faces since in that case degrees of freedom from both sides of a face (or from all adjacent elements to a vertex) would be identified with each other, which is not what we would like to have). Logically, these degrees of freedom are therefore defined to belong to the cell, rather than the face or vertex. In that case, the returned element would therefore have length zero.

If the finite element defines support points, then their number equals the number of degrees of freedom on the face (dofs_per_face). The order of points in the array matches that returned by the cell->face(face)->get_dof_indices function.

See the class documentation for details on support points.

## ◆ has_face_support_points()

 bool FiniteElement< dim, spacedim >::has_face_support_points ( const unsigned int face_no = 0 ) const
inherited

Return whether a finite element has defined support points on faces. If the result is true, then a call to the get_unit_face_support_points() yields a non-empty vector.

## ◆ unit_face_support_point()

 virtual Point FiniteElement< dim, spacedim >::unit_face_support_point ( const unsigned int index, const unsigned int face_no = 0 ) const
virtualinherited

The function corresponding to the unit_support_point() function, but for faces. See there for more information.

Reimplemented in FESystem< dim, spacedim >.

## ◆ get_generalized_support_points()

 const std::vector >& FiniteElement< dim, spacedim >::get_generalized_support_points ( ) const
inherited

Return a vector of generalized support points.

Note
The vector returned by this function is always a minimal set of unique support points. This is in contrast to the behavior of get_unit_support_points() that returns a repeated list of unit support points for an FESystem of numerous (Lagrangian) base elements.

## ◆ has_generalized_support_points()

 bool FiniteElement< dim, spacedim >::has_generalized_support_points ( ) const
inherited

Return whether a finite element has defined generalized support points. If the result is true, then a call to the get_generalized_support_points() yields a non-empty vector.

## ◆ get_associated_geometry_primitive()

 GeometryPrimitive FiniteElement< dim, spacedim >::get_associated_geometry_primitive ( const unsigned int cell_dof_index ) const
inherited

For a given degree of freedom, return whether it is logically associated with a vertex, line, quad or hex.

For instance, for continuous finite elements this coincides with the lowest dimensional object the support point of the degree of freedom lies on. To give an example, for $$Q_1$$ elements in 3d, every degree of freedom is defined by a shape function that we get by interpolating using support points that lie on the vertices of the cell. The support of these points of course extends to all edges connected to this vertex, as well as the adjacent faces and the cell interior, but we say that logically the degree of freedom is associated with the vertex as this is the lowest- dimensional object it is associated with. Likewise, for $$Q_2$$ elements in 3d, the degrees of freedom with support points at edge midpoints would yield a value of GeometryPrimitive::line from this function, whereas those on the centers of faces in 3d would return GeometryPrimitive::quad.

To make this more formal, the kind of object returned by this function represents the object so that the support of the shape function corresponding to the degree of freedom, (i.e., that part of the domain where the function "lives") is the union of all of the cells sharing this object. To return to the example above, for $$Q_2$$ in 3d, the shape function with support point at an edge midpoint has support on all cells that share the edge and not only the cells that share the adjacent faces, and consequently the function will return GeometryPrimitive::line.

On the other hand, for discontinuous elements of type $$DGQ_2$$, a degree of freedom associated with an interpolation polynomial that has its support point physically located at a line bounding a cell, but is nonzero only on one cell. Consequently, it is logically associated with the interior of that cell (i.e., with a GeometryPrimitive::quad in 2d and a GeometryPrimitive::hex in 3d).

Parameters
 [in] cell_dof_index The index of a shape function or degree of freedom. This index must be in the range [0,dofs_per_cell).
Note
The integer value of the object returned by this function equals the dimensionality of the object it describes, and can consequently be used in generic programming paradigms. For example, if a degree of freedom is associated with a vertex, then this function returns GeometryPrimitive::vertex, which has a numeric value of zero (the dimensionality of a vertex).

## ◆ convert_generalized_support_point_values_to_dof_values()

 virtual void FiniteElement< dim, spacedim >::convert_generalized_support_point_values_to_dof_values ( const std::vector< Vector< double >> & support_point_values, std::vector< double > & nodal_values ) const
virtualinherited

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.

## ◆ memory_consumption()

 virtual std::size_t FiniteElement< dim, spacedim >::memory_consumption ( ) const
virtualinherited

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.

## ◆ ExcShapeFunctionNotPrimitive()

 static ::ExceptionBase& FiniteElement< dim, spacedim >::ExcShapeFunctionNotPrimitive ( int arg1 )
staticinherited

Exception

Note
The message that will be printed by this exception reads:
<< "The shape function with index " << arg1 << " is not primitive, i.e. it is vector-valued and " << "has more than one non-zero vector component. This " << "function cannot be called for these shape functions. " << "Maybe you want to use the same function with the " << "_component suffix?"

## ◆ ExcFENotPrimitive()

 static ::ExceptionBase& FiniteElement< dim, spacedim >::ExcFENotPrimitive ( )
staticinherited

Exception

## ◆ ExcUnitShapeValuesDoNotExist()

 static ::ExceptionBase& FiniteElement< dim, spacedim >::ExcUnitShapeValuesDoNotExist ( )
staticinherited

Exception

Note
The message that will be printed by this exception reads:
"You are trying to access the values or derivatives of shape functions " "on the reference cell of an element that does not define its shape " "functions through mapping from the reference cell. Consequently, " "you cannot ask for shape function values or derivatives on the " "reference cell."

## ◆ ExcFEHasNoSupportPoints()

 static ::ExceptionBase& FiniteElement< dim, spacedim >::ExcFEHasNoSupportPoints ( )
staticinherited

Attempt to access support points of a finite element that is not Lagrangian.

Note
The message that will be printed by this exception reads:
"You are trying to access the support points of a finite " "element that either has no support points at all, or for " "which the corresponding tables have not been implemented."

## ◆ ExcEmbeddingVoid()

 static ::ExceptionBase& FiniteElement< dim, spacedim >::ExcEmbeddingVoid ( )
staticinherited

Attempt to access embedding matrices of a finite element that did not implement these matrices.

Note
The message that will be printed by this exception reads:
"You are trying to access the matrices that describe how " "to embed a finite element function on one cell into the " "finite element space on one of its children (i.e., the " "'embedding' or 'prolongation' matrices). However, the " "current finite element can either not define this sort of " "operation, or it has not yet been implemented."

## ◆ ExcProjectionVoid()

 static ::ExceptionBase& FiniteElement< dim, spacedim >::ExcProjectionVoid ( )
staticinherited

Attempt to access restriction matrices of a finite element that did not implement these matrices.

Exception

Note
The message that will be printed by this exception reads:
"You are trying to access the matrices that describe how " "to restrict a finite element function from the children " "of one cell to the finite element space defined on their " "parent (i.e., the 'restriction' or 'projection' matrices). " "However, the current finite element can either not define " "this sort of operation, or it has not yet been " "implemented."

## ◆ ExcWrongInterfaceMatrixSize()

 static ::ExceptionBase& FiniteElement< dim, spacedim >::ExcWrongInterfaceMatrixSize ( int arg1, int arg2 )
staticinherited

Exception

Note
The message that will be printed by this exception reads:
<< "The interface matrix has a size of " << arg1 << 'x' << arg2 << ", which is not reasonable for the current element " "in the present dimension."

## ◆ ExcInterpolationNotImplemented()

 static ::ExceptionBase& FiniteElement< dim, spacedim >::ExcInterpolationNotImplemented ( )
staticinherited

Exception

## ◆ reinit_restriction_and_prolongation_matrices()

 void FiniteElement< dim, spacedim >::reinit_restriction_and_prolongation_matrices ( const bool isotropic_restriction_only = false, const bool isotropic_prolongation_only = false )
protectedinherited

Reinit the vectors of restriction and prolongation matrices to the right sizes: For every refinement case, except for RefinementCase::no_refinement, and for every child of that refinement case the space of one restriction and prolongation matrix is allocated, see the documentation of the restriction and prolongation vectors for more detail on the actual vector sizes.

Parameters
 isotropic_restriction_only only the restriction matrices required for isotropic refinement are reinited to the right size. isotropic_prolongation_only only the prolongation matrices required for isotropic refinement are reinited to the right size.

## ◆ interface_constraints_size()

 TableIndices<2> FiniteElement< dim, spacedim >::interface_constraints_size ( ) const
protectedinherited

Return the size of interface constraint matrices. Since this is needed in every derived finite element class when initializing their size, it is placed into this function, to avoid having to recompute the dimension- dependent size of these matrices each time.

Note that some elements do not implement the interface constraints for certain polynomial degrees. In this case, this function still returns the size these matrices should have when implemented, but the actual matrices are empty.

## ◆ compute_n_nonzero_components()

 static std::vector FiniteElement< dim, spacedim >::compute_n_nonzero_components ( const std::vector< ComponentMask > & nonzero_components )
staticprotectedinherited

Given the pattern of nonzero components for each shape function, compute for each entry how many components are non-zero for each shape function. This function is used in the constructor of this class.

## ◆ get_face_data() [1/2]

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

Like get_data(), but return an object that will later be used for evaluating shape function information at quadrature points on faces of cells. The object will then be used in calls to implementations of FiniteElement::fill_fe_face_values(). See the documentation of get_data() for more information.

The default implementation of this function converts the face quadrature into a cell quadrature with appropriate quadrature point locations, and with that calls the get_data() function above that has to be implemented in derived classes.

Parameters
 [in] update_flags A set of UpdateFlags values that describe what kind of information the FEValues object requests the finite element to compute. This set of flags may also include information that the finite element can not compute, e.g., flags that pertain to data produced by the mapping. An implementation of this function needs to set up all data fields in the returned object that are necessary to produce the finite- element related data specified by these flags, and may already pre- compute part of this information as discussed above. Elements may want to store these update flags (or a subset of these flags) in InternalDataBase::update_each so they know at the time when FiniteElement::fill_fe_face_values() is called what they are supposed to compute [in] mapping A reference to the mapping used for computing values and derivatives of shape functions. [in] quadrature A reference to the object that describes where the shape functions should be evaluated. [out] output_data A reference to the object that FEValues will use in conjunction with the object returned here and where an implementation of FiniteElement::fill_fe_face_values() will place the requested information. This allows the current function to already pre-compute pieces of information that can be computed on the reference cell, as discussed above. FEValues guarantees that this output object and the object returned by the current function will always be used together.
Returns
A pointer to an object of a type derived from InternalDataBase and that derived classes can use to store scratch data that can be pre- computed, or for scratch arrays that then only need to be allocated once. The calling site assumes ownership of this object and will delete it when it is no longer necessary.

Reimplemented in FESystem< dim, spacedim >, and FE_Enriched< dim, spacedim >.

## ◆ get_face_data() [2/2]

 virtual std::unique_ptr FiniteElement< dim, spacedim >::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
protectedvirtualinherited
Deprecated:
Use the version taking a hp::QCollection argument.

## ◆ get_subface_data()

 virtual std::unique_ptr FiniteElement< dim, spacedim >::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
protectedvirtualinherited

Like get_data(), but return an object that will later be used for evaluating shape function information at quadrature points on children of faces of cells. The object will then be used in calls to implementations of FiniteElement::fill_fe_subface_values(). See the documentation of get_data() for more information.

The default implementation of this function converts the face quadrature into a cell quadrature with appropriate quadrature point locations, and with that calls the get_data() function above that has to be implemented in derived classes.

Parameters
 [in] update_flags A set of UpdateFlags values that describe what kind of information the FEValues object requests the finite element to compute. This set of flags may also include information that the finite element can not compute, e.g., flags that pertain to data produced by the mapping. An implementation of this function needs to set up all data fields in the returned object that are necessary to produce the finite- element related data specified by these flags, and may already pre- compute part of this information as discussed above. Elements may want to store these update flags (or a subset of these flags) in InternalDataBase::update_each so they know at the time when FiniteElement::fill_fe_subface_values() is called what they are supposed to compute [in] mapping A reference to the mapping used for computing values and derivatives of shape functions. [in] quadrature A reference to the object that describes where the shape functions should be evaluated. [out] output_data A reference to the object that FEValues will use in conjunction with the object returned here and where an implementation of FiniteElement::fill_fe_subface_values() will place the requested information. This allows the current function to already pre-compute pieces of information that can be computed on the reference cell, as discussed above. FEValues guarantees that this output object and the object returned by the current function will always be used together.
Returns
A pointer to an object of a type derived from InternalDataBase and that derived classes can use to store scratch data that can be pre- computed, or for scratch arrays that then only need to be allocated once. The calling site assumes ownership of this object and will delete it when it is no longer necessary.

Reimplemented in FESystem< dim, spacedim >, and FE_Enriched< dim, spacedim >.

## ◆ fill_fe_values() [2/2]

 virtual void FiniteElement< dim, spacedim >::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
protectedpure virtualinherited

Compute information about the shape functions on the cell denoted by the first argument. Derived classes will have to implement this function based on the kind of element they represent. It is called by FEValues::reinit().

Conceptually, this function evaluates shape functions and their derivatives at the quadrature points represented by the mapped locations of those described by the quadrature argument to this function. In many cases, computing derivatives of shape functions (and in some cases also computing values of shape functions) requires making use of the mapping from the reference to the real cell; this information can either be taken from the mapping_data object that has been filled for the current cell before this function is called, or by calling the member functions of a Mapping object with the mapping_internal object that also corresponds to the current cell.

The information computed by this function is used to fill the various member variables of the output argument of this function. Which of the member variables of that structure should be filled is determined by the update flags stored in the FiniteElement::InternalDataBase::update_each field of the object passed to this function. These flags are typically set by FiniteElement::get_data(), FiniteElement::get_face_date() and FiniteElement::get_subface_data() (or, more specifically, implementations of these functions in derived classes).

An extensive discussion of the interaction between this function and FEValues can be found in the How Mapping, FiniteElement, and FEValues work together documentation module.

Parameters
 [in] cell The cell of the triangulation for which this function is to compute a mapping from the reference cell to. [in] cell_similarity Whether or not the cell given as first argument is simply a translation, rotation, etc of the cell for which this function was called the most recent time. This information is computed simply by matching the vertices (as stored by the Triangulation) between the previous and the current cell. The value passed here may be modified by implementations of this function and should then be returned (see the discussion of the return value of this function). [in] quadrature A reference to the quadrature formula in use for the current evaluation. This quadrature object is the same as the one used when creating the internal_data object. The current object is then responsible for evaluating shape functions at the mapped locations of the quadrature points represented by this object. [in] mapping A reference to the mapping object used to map from the reference cell to the current cell. This object was used to compute the information in the mapping_data object before the current function was called. It is also the mapping object that created the mapping_internal object via Mapping::get_data(). You will need the reference to this mapping object most often to call Mapping::transform() to transform gradients and higher derivatives from the reference to the current cell. [in] mapping_internal An object specific to the mapping object. What the mapping chooses to store in there is of no relevance to the current function, but you may have to pass a reference to this object to certain functions of the Mapping class (e.g., Mapping::transform()) if you need to call them from the current function. [in] mapping_data The output object into which the Mapping::fill_fe_values() function wrote the mapping information corresponding to the current cell. This includes, for example, Jacobians of the mapping that may be of relevance to the current function, as well as other information that FEValues::reinit() requested from the mapping. [in] fe_internal A reference to an object previously created by get_data() and that may be used to store information the mapping can compute once on the reference cell. See the documentation of the FiniteElement::InternalDataBase class for an extensive description of the purpose of these objects. [out] output_data A reference to an object whose member variables should be computed. Not all of the members of this argument need to be filled; which ones need to be filled is determined by the update flags stored inside the fe_internal object.
Note
FEValues ensures that this function is always called with the same pair of fe_internal and output_data objects. In other words, if an implementation of this function knows that it has written a piece of data into the output argument in a previous call, then there is no need to copy it there again in a later call if the implementation knows that this is the same value.

## ◆ fill_fe_face_values() [2/3]

 virtual void FiniteElement< dim, spacedim >::fill_fe_face_values ( const typename Triangulation< dim, spacedim >::cell_iterator & cell, const unsigned int face_no, const hp::QCollection< 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
protectedvirtualinherited

This function is the equivalent to FiniteElement::fill_fe_values(), but for faces of cells. See there for an extensive discussion of its purpose. It is called by FEFaceValues::reinit().

Parameters
 [in] cell The cell of the triangulation for which this function is to compute a mapping from the reference cell to. [in] face_no The number of the face we are currently considering, indexed among the faces of the cell specified by the previous argument. [in] quadrature A reference to the quadrature formula in use for the current evaluation. This quadrature object is the same as the one used when creating the internal_data object. The current object is then responsible for evaluating shape functions at the mapped locations of the quadrature points represented by this object. [in] mapping A reference to the mapping object used to map from the reference cell to the current cell. This object was used to compute the information in the mapping_data object before the current function was called. It is also the mapping object that created the mapping_internal object via Mapping::get_data(). You will need the reference to this mapping object most often to call Mapping::transform() to transform gradients and higher derivatives from the reference to the current cell. [in] mapping_internal An object specific to the mapping object. What the mapping chooses to store in there is of no relevance to the current function, but you may have to pass a reference to this object to certain functions of the Mapping class (e.g., Mapping::transform()) if you need to call them from the current function. [in] mapping_data The output object into which the Mapping::fill_fe_values() function wrote the mapping information corresponding to the current cell. This includes, for example, Jacobians of the mapping that may be of relevance to the current function, as well as other information that FEValues::reinit() requested from the mapping. [in] fe_internal A reference to an object previously created by get_data() and that may be used to store information the mapping can compute once on the reference cell. See the documentation of the FiniteElement::InternalDataBase class for an extensive description of the purpose of these objects. [out] output_data A reference to an object whose member variables should be computed. Not all of the members of this argument need to be filled; which ones need to be filled is determined by the update flags stored inside the fe_internal object.

## ◆ fill_fe_face_values() [3/3]

 virtual void FiniteElement< dim, spacedim >::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
protectedvirtualinherited
Deprecated:
Use the version taking a hp::QCollection argument.

## ◆ fill_fe_subface_values() [2/2]

 virtual void FiniteElement< dim, spacedim >::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
protectedpure virtualinherited

This function is the equivalent to FiniteElement::fill_fe_values(), but for the children of faces of cells. See there for an extensive discussion of its purpose. It is called by FESubfaceValues::reinit().

Parameters
 [in] cell The cell of the triangulation for which this function is to compute a mapping from the reference cell to. [in] face_no The number of the face we are currently considering, indexed among the faces of the cell specified by the previous argument. [in] sub_no The number of the subface, i.e., the number of the child of a face, that we are currently considering, indexed among the children of the face specified by the previous argument. [in] quadrature A reference to the quadrature formula in use for the current evaluation. This quadrature object is the same as the one used when creating the internal_data object. The current object is then responsible for evaluating shape functions at the mapped locations of the quadrature points represented by this object. [in] mapping A reference to the mapping object used to map from the reference cell to the current cell. This object was used to compute the information in the mapping_data object before the current function was called. It is also the mapping object that created the mapping_internal object via Mapping::get_data(). You will need the reference to this mapping object most often to call Mapping::transform() to transform gradients and higher derivatives from the reference to the current cell. [in] mapping_internal An object specific to the mapping object. What the mapping chooses to store in there is of no relevance to the current function, but you may have to pass a reference to this object to certain functions of the Mapping class (e.g., Mapping::transform()) if you need to call them from the current function. [in] mapping_data The output object into which the Mapping::fill_fe_values() function wrote the mapping information corresponding to the current cell. This includes, for example, Jacobians of the mapping that may be of relevance to the current function, as well as other information that FEValues::reinit() requested from the mapping. [in] fe_internal A reference to an object previously created by get_data() and that may be used to store information the mapping can compute once on the reference cell. See the documentation of the FiniteElement::InternalDataBase class for an extensive description of the purpose of these objects. [out] output_data A reference to an object whose member variables should be computed. Not all of the members of this argument need to be filled; which ones need to be filled is determined by the update flags stored inside the fe_internal object.

## ◆ reference_cell()

template<int dim>
 ReferenceCell FiniteElementData< dim >::reference_cell ( ) const
inherited

Return the kind of reference cell this element is defined on: For example, whether the element's reference cell is a square or triangle, or similar choices in higher dimensions.

template<int dim>
 unsigned int FiniteElementData< dim >::n_unique_quads ( ) const
inherited

Number of unique quads. If all quads have the same type, the value is one; else it equals the number of quads.

## ◆ n_unique_faces()

template<int dim>
 unsigned int FiniteElementData< dim >::n_unique_faces ( ) const
inherited

Number of unique faces. If all faces have the same type, the value is one; else it equals the number of faces.

## ◆ n_dofs_per_vertex()

template<int dim>
 unsigned int FiniteElementData< dim >::n_dofs_per_vertex ( ) const
inherited

Number of dofs per vertex.

## ◆ n_dofs_per_line()

template<int dim>
 unsigned int FiniteElementData< dim >::n_dofs_per_line ( ) const
inherited

Number of dofs per line. Not including dofs on lower dimensional objects.

template<int dim>
 unsigned int FiniteElementData< dim >::n_dofs_per_quad ( unsigned int face_no = 0 ) const
inherited

Number of dofs per quad. Not including dofs on lower dimensional objects.

template<int dim>
 unsigned int FiniteElementData< dim >::max_dofs_per_quad ( ) const
inherited

Maximum number of dofs per quad. Not including dofs on lower dimensional objects.

## ◆ n_dofs_per_hex()

template<int dim>
 unsigned int FiniteElementData< dim >::n_dofs_per_hex ( ) const
inherited

Number of dofs per hex. Not including dofs on lower dimensional objects.

## ◆ n_dofs_per_face()

template<int dim>
 unsigned int FiniteElementData< dim >::n_dofs_per_face ( unsigned int face_no = 0, unsigned int child = 0 ) const
inherited

Number of dofs per face, accumulating degrees of freedom of all lower dimensional objects.

## ◆ max_dofs_per_face()

template<int dim>
 unsigned int FiniteElementData< dim >::max_dofs_per_face ( ) const
inherited

Maximum number of dofs per face, accumulating degrees of freedom of all lower dimensional objects.

## ◆ n_dofs_per_cell()

template<int dim>
 unsigned int FiniteElementData< dim >::n_dofs_per_cell ( ) const
inherited

Number of dofs per cell, accumulating degrees of freedom of all lower dimensional objects.

## ◆ n_dofs_per_object()

template<int dim>
template<int structdim>
 unsigned int FiniteElementData< dim >::n_dofs_per_object ( const unsigned int i = 0 ) const
inherited

Return the number of degrees per structdim-dimensional object. For structdim==0, the function therefore returns dofs_per_vertex, for structdim==1 dofs_per_line, etc. This function is mostly used to allow some template trickery for functions that should work on all sorts of objects without wanting to use the different names (vertex, line, ...) associated with these objects.

## ◆ n_components()

template<int dim>
 unsigned int FiniteElementData< dim >::n_components ( ) const
inherited

## ◆ n_blocks()

template<int dim>
 unsigned int FiniteElementData< dim >::n_blocks ( ) const
inherited

## ◆ block_indices()

template<int dim>
 const BlockIndices& FiniteElementData< dim >::block_indices ( ) const
inherited

Detailed information on block sizes.

## ◆ tensor_degree()

template<int dim>
 unsigned int FiniteElementData< dim >::tensor_degree ( ) const
inherited

Maximal polynomial degree of a shape function in a single coordinate direction.

This function can be used to determine the optimal quadrature rule.

## ◆ conforms()

template<int dim>
 bool FiniteElementData< dim >::conforms ( const Conformity ) const
inherited

Test whether a finite element space conforms to a certain Sobolev space.

Note
This function will return a true value even if the finite element space has higher regularity than asked for.

## ◆ get_first_line_index()

template<int dim>
 unsigned int FiniteElementData< dim >::get_first_line_index ( ) const
inherited

Return first index of dof on a line.

template<int dim>
 unsigned int FiniteElementData< dim >::get_first_quad_index ( const unsigned int quad_no = 0 ) const
inherited

Return first index of dof on a quad.

## ◆ get_first_hex_index()

template<int dim>
 unsigned int FiniteElementData< dim >::get_first_hex_index ( ) const
inherited

Return first index of dof on a hexahedron.

## ◆ get_first_face_line_index()

template<int dim>
 unsigned int FiniteElementData< dim >::get_first_face_line_index ( const unsigned int face_no = 0 ) const
inherited

Return first index of dof on a line for face data.

template<int dim>
 unsigned int FiniteElementData< dim >::get_first_face_quad_index ( const unsigned int face_no = 0 ) const
inherited

Return first index of dof on a quad for face data.

## ◆ mapping_kind

template<int dim, int spacedim = dim>
 std::vector FE_PolyTensor< dim, spacedim >::mapping_kind
protected

The mapping type to be used to map shape functions from the reference cell to the mesh cell. If this vector is length one, the same mapping will be applied to all shape functions. If the vector size is equal to the finite element dofs per cell, then each shape function will be mapped according to the corresponding entry in the vector.

Definition at line 218 of file fe_poly_tensor.h.

template<int dim, int spacedim = dim>
protected

For faces with non-standard face_orientation in 3D, the dofs on faces (quads) need not only to be permuted in order to be combined with the correct shape functions. Additionally they may change their sign.

The constructor of this class fills this table with 'false' values, i.e., no sign change at all. Derived finite element classes have to fill this Table with the correct values, see the documentation in GeometryInfo<dim> and this glossary entry on face orientation.

The table must be filled in finite element classes derived from FE_PolyTensor in a meaningful way since the permutation pattern and the pattern of sign changes depends on how the finite element distributes the local dofs on the faces. An example is the function initialize_quad_dof_index_permutation_and_sign_change() in the FE_RaviartThomas class that fills this table.

Definition at line 265 of file fe_poly_tensor.h.

## ◆ poly_space

template<int dim, int spacedim = dim>
 const std::unique_ptr > FE_PolyTensor< dim, spacedim >::poly_space
protected

A copy of the object passed to the constructor that describes the polynomial space.

Definition at line 519 of file fe_poly_tensor.h.

## ◆ inverse_node_matrix

template<int dim, int spacedim = dim>
 FullMatrix FE_PolyTensor< dim, spacedim >::inverse_node_matrix
protected

The inverse of the matrix aij of node functionals Ni applied to polynomial pj. This matrix is used to convert polynomials in the "raw" basis provided in poly_space to the basis dual to the node functionals on the reference cell.

This object is not filled by FE_PolyTensor, but is a chance for a derived class to allow for reorganization of the basis functions. If it is left empty, the basis in poly_space is used, i.e., it is assumed that poly_space already satisfies the Kronecker delta property with regard to the node functionals.

Definition at line 534 of file fe_poly_tensor.h.

## ◆ cache_mutex

template<int dim, int spacedim = dim>
 std::mutex FE_PolyTensor< dim, spacedim >::cache_mutex
mutableprotected

A mutex to be used to guard access to the variables below.

Definition at line 539 of file fe_poly_tensor.h.

## ◆ cached_point

template<int dim, int spacedim = dim>
 Point FE_PolyTensor< dim, spacedim >::cached_point
mutableprotected

If a shape function is computed at a single point, we must compute all of them to apply inverse_node_matrix. In order to avoid too much overhead, we cache the point and the function values for the next evaluation.

Definition at line 546 of file fe_poly_tensor.h.

## ◆ cached_values

template<int dim, int spacedim = dim>
 std::vector > FE_PolyTensor< dim, spacedim >::cached_values
mutableprotected

Cached shape function values after call to shape_value_component().

Definition at line 551 of file fe_poly_tensor.h.

template<int dim, int spacedim = dim>
 std::vector > FE_PolyTensor< dim, spacedim >::cached_grads
mutableprotected

Definition at line 556 of file fe_poly_tensor.h.

template<int dim, int spacedim = dim>
mutableprotected

Definition at line 562 of file fe_poly_tensor.h.

## ◆ space_dimension

 constexpr unsigned int FiniteElement< dim, spacedim >::space_dimension
staticconstexprinherited

The dimension of the image space, corresponding to Triangulation.

Definition at line 660 of file fe.h.

## ◆ restriction

 std::vector > > FiniteElement< dim, spacedim >::restriction
protectedinherited

Vector of projection matrices. See get_restriction_matrix() above. The constructor initializes these matrices to zero dimensions, which can be changed by derived classes implementing them.

Note, that restriction[refinement_case-1][child] includes the restriction matrix of child child for the RefinementCase refinement_case. Here, we use refinement_case-1 instead of refinement_case as for RefinementCase::no_refinement(=0) there are no restriction matrices available.

Definition at line 2386 of file fe.h.

## ◆ prolongation

 std::vector > > FiniteElement< dim, spacedim >::prolongation
protectedinherited

Vector of embedding matrices. See get_prolongation_matrix() above. The constructor initializes these matrices to zero dimensions, which can be changed by derived classes implementing them.

Note, that prolongation[refinement_case-1][child] includes the prolongation matrix of child child for the RefinementCase refinement_case. Here, we use refinement_case-1 instead of refinement_case as for RefinementCase::no_refinement(=0) there are no prolongation matrices available.

Definition at line 2400 of file fe.h.

## ◆ interface_constraints

 FullMatrix FiniteElement< dim, spacedim >::interface_constraints
protectedinherited

Specify the constraints which the dofs on the two sides of a cell interface underlie if the line connects two cells of which one is refined once.

For further details see the general description of the derived class.

This field is obviously useless in one dimension and has there a zero size.

Definition at line 2412 of file fe.h.

## ◆ unit_support_points

 std::vector > FiniteElement< dim, spacedim >::unit_support_points
protectedinherited

List of support points on the unit cell, in case the finite element has any. The constructor leaves this field empty, derived classes may write in some contents.

Finite elements that allow some kind of interpolation operation usually have support points. On the other hand, elements that define their degrees of freedom by, for example, moments on faces, or as derivatives, don't have support points. In that case, this field remains empty.

Definition at line 2424 of file fe.h.

## ◆ unit_face_support_points

 std::vector > > FiniteElement< dim, spacedim >::unit_face_support_points
protectedinherited

Same for the faces. See the description of the get_unit_face_support_points() function for a discussion of what contributes a face support point.

Definition at line 2431 of file fe.h.

## ◆ generalized_support_points

 std::vector > FiniteElement< dim, spacedim >::generalized_support_points
protectedinherited

Support points used for interpolation functions of non-Lagrangian elements.

Definition at line 2437 of file fe.h.

## ◆ generalized_face_support_points

 std::vector > > FiniteElement< dim, spacedim >::generalized_face_support_points
protectedinherited

Face support points used for interpolation functions of non-Lagrangian elements.

Definition at line 2443 of file fe.h.

protectedinherited

For faces with non-standard face_orientation in 3D, the dofs on faces (quads) have to be permuted in order to be combined with the correct shape functions. Given a local dof index on a quad, return the shift in the local index, if the face has non-standard face_orientation, i.e. old_index + shift = new_index. In 2D and 1D there is no need for permutation so the vector is empty. In 3D it has the size of  dofs_per_quad * 8 , where 8 is the number of orientations, a face can be in (all combinations of the three bool flags face_orientation, face_flip and face_rotation).

The constructor of this class fills this table with zeros, i.e., no permutation at all. Derived finite element classes have to fill this Table with the correct values.

Definition at line 2460 of file fe.h.

protectedinherited

For lines with non-standard line_orientation in 3D, the dofs on lines have to be permuted in order to be combined with the correct shape functions. Given a local dof index on a line, return the shift in the local index, if the line has non-standard line_orientation, i.e. old_index + shift = new_index. In 2D and 1D there is no need for permutation so the vector is empty. In 3D it has the size of dofs_per_line.

The constructor of this class fills this table with zeros, i.e., no permutation at all. Derived finite element classes have to fill this vector with the correct values.

Definition at line 2475 of file fe.h.

## ◆ system_to_component_table

 std::vector > FiniteElement< dim, spacedim >::system_to_component_table
protectedinherited

Store what system_to_component_index() will return.

Definition at line 2480 of file fe.h.

## ◆ face_system_to_component_table

 std::vector > > FiniteElement< dim, spacedim >::face_system_to_component_table
protectedinherited

Map between linear dofs and component dofs on face. This is filled with default values in the constructor, but derived classes will have to overwrite the information if necessary.

By component, we mean the vector component, not the base element. The information thus makes only sense if a shape function is non-zero in only one component.

Definition at line 2492 of file fe.h.

## ◆ system_to_base_table

 std::vector, unsigned int> > FiniteElement< dim, spacedim >::system_to_base_table
protectedinherited

For each shape function, store to which base element and which instance of this base element (in case its multiplicity is greater than one) it belongs, and its index within this base element. If the element is not composed of others, then base and instance are always zero, and the index is equal to the number of the shape function. If the element is composed of single instances of other elements (i.e. all with multiplicity one) all of which are scalar, then base values and dof indices within this element are equal to the system_to_component_table. It differs only in case the element is composed of other elements and at least one of them is vector-valued itself.

This array has valid values also in the case of vector-valued (i.e. non-primitive) shape functions, in contrast to the system_to_component_table.

Definition at line 2511 of file fe.h.

## ◆ face_system_to_base_table

 std::vector< std::vector, unsigned int> > > FiniteElement< dim, spacedim >::face_system_to_base_table
protectedinherited

Likewise for the indices on faces.

Definition at line 2518 of file fe.h.

## ◆ base_to_block_indices

 BlockIndices FiniteElement< dim, spacedim >::base_to_block_indices
protectedinherited

For each base element, store the number of blocks generated by the base and the first block in a block vector it will generate.

Definition at line 2524 of file fe.h.

## ◆ component_to_base_table

 std::vector, unsigned int> > FiniteElement< dim, spacedim >::component_to_base_table
protectedinherited

The base element establishing a component.

For each component number c, the entries have the following meaning:

table[c].first.first
Number of the base element for c. This is the index you can pass to base_element().
table[c].first.second
Component within the base element for c. This value is between 0 and the n_components() of this base element.
table[c].second
Index of the multiple of the base element that contains c. This value is between 0 and the element_multiplicity() of this base element.

This variable is set to the correct size by the constructor of this class, but needs to be initialized by derived classes, unless its size is one and the only entry is a zero, which is the case for scalar elements. In that case, the initialization by the base class is sufficient.

Note
This table is filled by FETools::Compositing::build_cell_tables().

Definition at line 2547 of file fe.h.

 const std::vector FiniteElement< dim, spacedim >::restriction_is_additive_flags
protectedinherited

A flag determining whether restriction matrices are to be concatenated or summed up. See the discussion about restriction matrices in the general class documentation for more information.

Definition at line 2554 of file fe.h.

## ◆ nonzero_components

 const std::vector FiniteElement< dim, spacedim >::nonzero_components
protectedinherited

For each shape function, give a vector of bools (with size equal to the number of vector components which this finite element has) indicating in which component each of these shape functions is non-zero.

For primitive elements, there is only one non-zero component.

Definition at line 2563 of file fe.h.

## ◆ n_nonzero_components_table

 const std::vector FiniteElement< dim, spacedim >::n_nonzero_components_table
protectedinherited

This array holds how many values in the respective entry of the nonzero_components element are non-zero. The array is thus a short-cut to allow faster access to this information than if we had to count the non-zero entries upon each request for this information. The field is initialized in the constructor of this class.

Definition at line 2572 of file fe.h.

## ◆ cached_primitivity

 const bool FiniteElement< dim, spacedim >::cached_primitivity
protectedinherited

Store whether all shape functions are primitive. Since finding this out is a very common operation, we cache the result, i.e. compute the value in the constructor for simpler access.

Definition at line 2579 of file fe.h.

## ◆ dimension

template<int dim>
 constexpr unsigned int FiniteElementData< dim >::dimension = dim
staticconstexprinherited

The dimension of the finite element, which is the template parameter dim

Definition at line 299 of file fe_data.h.

## ◆ reference_cell_kind

template<int dim>
 const ReferenceCell FiniteElementData< dim >::reference_cell_kind
privateinherited

Reference cell type.

Definition at line 305 of file fe_data.h.

template<int dim>
 const unsigned int FiniteElementData< dim >::number_unique_quads
privateinherited

Number of unique quads. If all quads have the same type, the value is one; else it equals the number of quads.

Definition at line 311 of file fe_data.h.

## ◆ number_unique_faces

template<int dim>
 const unsigned int FiniteElementData< dim >::number_unique_faces
privateinherited

Number of unique faces. If all faces have the same type, the value is one; else it equals the number of faces.

Definition at line 317 of file fe_data.h.

## ◆ dofs_per_vertex

template<int dim>
 const unsigned int FiniteElementData< dim >::dofs_per_vertex
inherited

Number of degrees of freedom on a vertex.

Definition at line 323 of file fe_data.h.

## ◆ dofs_per_line

template<int dim>
 const unsigned int FiniteElementData< dim >::dofs_per_line
inherited

Number of degrees of freedom in a line; not including the degrees of freedom on the vertices of the line.

Definition at line 329 of file fe_data.h.

template<int dim>
privateinherited

Number of degrees of freedom on quads. If all quads have the same number of degrees freedoms the values equal dofs_per_quad.

Definition at line 336 of file fe_data.h.

template<int dim>
 const unsigned int FiniteElementData< dim >::dofs_per_quad
inherited

Number of degrees of freedom in a quadrilateral; not including the degrees of freedom on the lines and vertices of the quadrilateral.

Definition at line 343 of file fe_data.h.

template<int dim>
 const unsigned int FiniteElementData< dim >::dofs_per_quad_max
privateinherited

Maximum number of degrees of freedom on any quad.

Definition at line 349 of file fe_data.h.

## ◆ dofs_per_hex

template<int dim>
 const unsigned int FiniteElementData< dim >::dofs_per_hex
inherited

Number of degrees of freedom in a hexahedron; not including the degrees of freedom on the quadrilaterals, lines and vertices of the hexahedron.

Definition at line 356 of file fe_data.h.

## ◆ first_line_index

template<int dim>
 const unsigned int FiniteElementData< dim >::first_line_index
inherited

First index of dof on a line.

Definition at line 361 of file fe_data.h.

template<int dim>
privateinherited

First index of a quad. If all quads have the same number of degrees of freedom, only the first index of the first quad is stored since the indices of the others can be simply recomputed.

Definition at line 369 of file fe_data.h.

template<int dim>
 const unsigned int FiniteElementData< dim >::first_quad_index
inherited

First index of dof on a quad.

Definition at line 375 of file fe_data.h.

## ◆ first_hex_index

template<int dim>
 const unsigned int FiniteElementData< dim >::first_hex_index
inherited

First index of dof on a hexahedron.

Definition at line 380 of file fe_data.h.

## ◆ first_line_index_of_faces

template<int dim>
 const std::vector FiniteElementData< dim >::first_line_index_of_faces
privateinherited

Index of the first line of all faces.

Definition at line 386 of file fe_data.h.

## ◆ first_face_line_index

template<int dim>
 const unsigned int FiniteElementData< dim >::first_face_line_index
inherited

First index of dof on a line for face data.

Definition at line 392 of file fe_data.h.

template<int dim>
privateinherited

Index of the first quad of all faces.

Definition at line 398 of file fe_data.h.

template<int dim>
 const unsigned int FiniteElementData< dim >::first_face_quad_index
inherited

First index of dof on a quad for face data.

Definition at line 404 of file fe_data.h.

## ◆ n_dofs_on_face

template<int dim>
 const std::vector FiniteElementData< dim >::n_dofs_on_face
privateinherited

Number of degrees of freedom on faces. If all faces have the same number of degrees freedoms the values equal dofs_per_quad.

Definition at line 411 of file fe_data.h.

## ◆ dofs_per_face

template<int dim>
 const unsigned int FiniteElementData< dim >::dofs_per_face
inherited

Number of degrees of freedom on a face. This is the accumulated number of degrees of freedom on all the objects of dimension up to dim-1 constituting a face.

Definition at line 419 of file fe_data.h.

## ◆ dofs_per_face_max

template<int dim>
 const unsigned int FiniteElementData< dim >::dofs_per_face_max
privateinherited

Maximum number of degrees of freedom on any face.

Definition at line 425 of file fe_data.h.

## ◆ dofs_per_cell

template<int dim>
 const unsigned int FiniteElementData< dim >::dofs_per_cell
inherited

Total number of degrees of freedom on a cell. This is the accumulated number of degrees of freedom on all the objects of dimension up to dim constituting a cell.

Definition at line 433 of file fe_data.h.

## ◆ components

template<int dim>
 const unsigned int FiniteElementData< dim >::components
inherited

Number of vector components of this finite element, and dimension of the image space. For vector-valued finite elements (i.e. when this number is greater than one), the number of vector components is in many cases equal to the number of base elements glued together with the help of the FESystem class. However, for elements like the Nedelec element, the number is greater than one even though we only have one base element.

Definition at line 443 of file fe_data.h.

## ◆ degree

template<int dim>
 const unsigned int FiniteElementData< dim >::degree
inherited

Maximal polynomial degree of a shape function in a single coordinate direction.

Definition at line 449 of file fe_data.h.

## ◆ conforming_space

template<int dim>
 const Conformity FiniteElementData< dim >::conforming_space
inherited

Indicate the space this element conforms to.

Definition at line 454 of file fe_data.h.

## ◆ block_indices_data

template<int dim>
 const BlockIndices FiniteElementData< dim >::block_indices_data
inherited

Storage for an object describing the sizes of each block of a compound element. For an element which is not an FESystem, this contains only a single block with length dofs_per_cell.

Definition at line 461 of file fe_data.h.

The documentation for this class was generated from the following file: