Reference documentation for deal.II version Git 040c6ad7d4 2020-09-26 18:01:03 +0200
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Public Types | Public Member Functions | Static Public Member Functions | Public Attributes | Static Public Attributes | Protected Member Functions | Static Protected Member Functions | Protected Attributes | Private Member Functions | Static Private Member Functions | Private Attributes | Friends | List of all members
FE_Nedelec< dim > Class Template Referenceabstract

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

Inheritance diagram for FE_Nedelec< dim >:
[legend]

Public Types

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

Public Member Functions

 FE_Nedelec (const unsigned int order)
 
virtual std::string get_name () const override
 
virtual bool has_support_on_face (const unsigned int shape_index, const unsigned int face_index) const override
 
virtual bool hp_constraints_are_implemented () const override
 
virtual FiniteElementDomination::Domination compare_for_domination (const FiniteElement< dim > &fe_other, const unsigned int codim=0) const override final
 
virtual std::vector< std::pair< unsigned int, unsigned int > > hp_vertex_dof_identities (const FiniteElement< dim > &fe_other) const override
 
virtual std::vector< std::pair< unsigned int, unsigned int > > hp_line_dof_identities (const FiniteElement< dim > &fe_other) const override
 
virtual std::vector< std::pair< unsigned int, unsigned int > > hp_quad_dof_identities (const FiniteElement< dim > &fe_other, const unsigned int face_no=0) const override
 
virtual void get_face_interpolation_matrix (const FiniteElement< dim > &source, FullMatrix< double > &matrix, const unsigned int face_no=0) const override
 
virtual void get_subface_interpolation_matrix (const FiniteElement< dim > &source, const unsigned int subface, FullMatrix< double > &matrix, const unsigned int face_no=0) const override
 
virtual const FullMatrix< double > & get_restriction_matrix (const unsigned int child, const RefinementCase< dim > &refinement_case=RefinementCase< dim >::isotropic_refinement) const override
 
virtual const FullMatrix< double > & get_prolongation_matrix (const unsigned int child, const RefinementCase< dim > &refinement_case=RefinementCase< dim >::isotropic_refinement) const override
 
virtual void convert_generalized_support_point_values_to_dof_values (const std::vector< Vector< double >> &support_point_values, std::vector< double > &nodal_values) const override
 
virtual std::pair< Table< 2, bool >, std::vector< unsigned int > > get_constant_modes () const override
 
virtual std::size_t memory_consumption () const override
 
virtual std::unique_ptr< FiniteElement< dim, dim > > clone () const override
 
template<>
void get_subface_interpolation_matrix (const FiniteElement< 1, 1 > &, const unsigned int, FullMatrix< double > &, const unsigned int) const
 
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 intoperator^ (const unsigned int multiplicity) const
 
const FiniteElement< dim, spacedim > & operator[] (const unsigned int fe_index) const
 
virtual bool operator== (const FiniteElement< dim, spacedim > &fe) const
 
bool operator== (const FiniteElementData &) const
 
bool operator!= (const FiniteElement< dim, spacedim > &) const
 
template<class Archive >
void serialize (Archive &ar, const unsigned int version)
 
ReferenceCell::Type reference_cell_type () const
 
unsigned int n_unique_quads () 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 max_dofs_per_quad () 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_quad_index (const unsigned int quad_no=0) 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
 
Transfer and constraint matrices
bool prolongation_is_implemented () const
 
bool isotropic_prolongation_is_implemented () const
 
bool restriction_is_implemented () const
 
bool isotropic_restriction_is_implemented () const
 
bool restriction_is_additive (const unsigned int index) const
 
const FullMatrix< double > & constraints (const ::internal::SubfaceCase< dim > &subface_case=::internal::SubfaceCase< dim >::case_isotropic) const
 
bool constraints_are_implemented (const ::internal::SubfaceCase< dim > &subface_case=::internal::SubfaceCase< dim >::case_isotropic) const
 
virtual void get_interpolation_matrix (const FiniteElement< dim, spacedim > &source, FullMatrix< double > &matrix) const
 
Functions to support hp
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
 
Index computations
std::pair< unsigned int, unsigned intsystem_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 intface_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
 
const FiniteElement< dim, spacedim > & get_sub_fe (const ComponentMask &mask) 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 intsystem_to_base_index (const unsigned int index) const
 
std::pair< std::pair< unsigned int, unsigned int >, unsigned intface_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 intcomponent_to_base_index (const unsigned int component) const
 
std::pair< unsigned int, unsigned intblock_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
ComponentMask component_mask (const FEValuesExtractors::Scalar &scalar) const
 
ComponentMask component_mask (const FEValuesExtractors::Vector &vector) const
 
ComponentMask component_mask (const FEValuesExtractors::SymmetricTensor< 2 > &sym_tensor) const
 
ComponentMask component_mask (const BlockMask &block_mask) const
 
BlockMask block_mask (const FEValuesExtractors::Scalar &scalar) const
 
BlockMask block_mask (const FEValuesExtractors::Vector &vector) const
 
BlockMask block_mask (const FEValuesExtractors::SymmetricTensor< 2 > &sym_tensor) const
 
BlockMask block_mask (const ComponentMask &component_mask) const
 
Support points and interpolation
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
 
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
 

Static Public Member Functions

static ::ExceptionBaseExcShapeFunctionNotPrimitive (int arg1)
 
static ::ExceptionBaseExcFENotPrimitive ()
 
static ::ExceptionBaseExcUnitShapeValuesDoNotExist ()
 
static ::ExceptionBaseExcFEHasNoSupportPoints ()
 
static ::ExceptionBaseExcEmbeddingVoid ()
 
static ::ExceptionBaseExcProjectionVoid ()
 
static ::ExceptionBaseExcWrongInterfaceMatrixSize (int arg1, int arg2)
 
static ::ExceptionBaseExcInterpolationNotImplemented ()
 
static ::ExceptionBaseExcInUse (int arg1, std::string arg2, std::string arg3)
 
static ::ExceptionBaseExcNoSubscriber (std::string arg1, std::string arg2)
 

Public Attributes

const unsigned int dofs_per_vertex
 
const unsigned int dofs_per_line
 
const unsigned int dofs_per_quad
 
const unsigned int dofs_per_hex
 
const unsigned int first_line_index
 
const unsigned int first_quad_index
 
const unsigned int first_hex_index
 
const unsigned int first_face_line_index
 
const unsigned int first_face_quad_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 const unsigned int space_dimension
 
static const unsigned int dimension = dim
 

Protected Member Functions

bool single_mapping_kind () const
 
MappingKind get_mapping_kind (const unsigned int i) const
 
virtual std::unique_ptr< typename FiniteElement< dim, dim >::InternalDataBaseget_data (const UpdateFlags update_flags, const Mapping< dim, dim > &, const Quadrature< dim > &quadrature, ::internal::FEValuesImplementation::FiniteElementRelatedData< dim, dim > &) const override
 
virtual std::unique_ptr< InternalDataBaseget_data (const UpdateFlags update_flags, const Mapping< dim, spacedim > &mapping, const Quadrature< dim > &quadrature, ::internal::FEValuesImplementation::FiniteElementRelatedData< dim, spacedim > &output_data) const=0
 
virtual void fill_fe_values (const typename Triangulation< dim, dim >::cell_iterator &cell, const CellSimilarity::Similarity cell_similarity, const Quadrature< dim > &quadrature, const Mapping< dim, dim > &mapping, const typename Mapping< dim, dim >::InternalDataBase &mapping_internal, const ::internal::FEValuesImplementation::MappingRelatedData< dim, dim > &mapping_data, const typename FiniteElement< dim, dim >::InternalDataBase &fe_internal, ::internal::FEValuesImplementation::FiniteElementRelatedData< dim, dim > &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 InternalDataBase &fe_internal, ::internal::FEValuesImplementation::FiniteElementRelatedData< dim, spacedim > &output_data) const=0
 
virtual void fill_fe_face_values (const typename Triangulation< dim, dim >::cell_iterator &cell, const unsigned int face_no, const Quadrature< dim - 1 > &quadrature, const Mapping< dim, dim > &mapping, const typename Mapping< dim, dim >::InternalDataBase &mapping_internal, const ::internal::FEValuesImplementation::MappingRelatedData< dim, dim > &mapping_data, const typename FiniteElement< dim, dim >::InternalDataBase &fe_internal, ::internal::FEValuesImplementation::FiniteElementRelatedData< dim, dim > &output_data) const override
 
virtual void fill_fe_face_values (const typename Triangulation< dim, spacedim >::cell_iterator &cell, const unsigned int face_no, const Quadrature< dim - 1 > &quadrature, const Mapping< dim, spacedim > &mapping, const typename Mapping< dim, spacedim >::InternalDataBase &mapping_internal, const ::internal::FEValuesImplementation::MappingRelatedData< dim, spacedim > &mapping_data, const InternalDataBase &fe_internal, ::internal::FEValuesImplementation::FiniteElementRelatedData< dim, spacedim > &output_data) const=0
 
virtual void fill_fe_subface_values (const typename Triangulation< dim, dim >::cell_iterator &cell, const unsigned int face_no, const unsigned int sub_no, const Quadrature< dim - 1 > &quadrature, const Mapping< dim, dim > &mapping, const typename Mapping< dim, dim >::InternalDataBase &mapping_internal, const ::internal::FEValuesImplementation::MappingRelatedData< dim, dim > &mapping_data, const typename FiniteElement< dim, dim >::InternalDataBase &fe_internal, ::internal::FEValuesImplementation::FiniteElementRelatedData< dim, dim > &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 InternalDataBase &fe_internal, ::internal::FEValuesImplementation::FiniteElementRelatedData< dim, spacedim > &output_data) const=0
 
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 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
 

Static Protected Member Functions

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

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
 
std::vector< Tensor< 3, dim > > cached_grad_grads
 
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< Table< 2, int > > adjust_quad_dof_index_for_face_orientation_table
 
std::vector< intadjust_line_dof_index_for_line_orientation_table
 
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< boolrestriction_is_additive_flags
 
const std::vector< ComponentMasknonzero_components
 
const std::vector< unsigned intn_nonzero_components_table
 
const bool cached_primitivity
 

Private Member Functions

void initialize_support_points (const unsigned int order)
 
void initialize_restriction ()
 
template<>
void initialize_support_points (const unsigned int)
 
template<>
void initialize_support_points (const unsigned int order)
 
template<>
void initialize_support_points (const unsigned int order)
 
template<>
void initialize_restriction ()
 

Static Private Member Functions

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

Private Attributes

Table< 2, doubleboundary_weights
 
Threads::Mutex mutex
 

Friends

template<int dim1>
class FE_Nedelec
 

Detailed Description

template<int dim>
class FE_Nedelec< dim >

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

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

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

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

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

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

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

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

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

Interpolation

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

Node values

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

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

Generalized support points

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

Definition at line 144 of file fe_nedelec.h.

Member Enumeration Documentation

◆ Conformity

template<int dim>
enum FiniteElementData::Conformity
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 241 of file fe_base.h.

Constructor & Destructor Documentation

◆ FE_Nedelec()

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

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

Definition at line 71 of file fe_nedelec.cc.

Member Function Documentation

◆ get_name()

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

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

Implements FiniteElement< dim, dim >.

Definition at line 216 of file fe_nedelec.cc.

◆ has_support_on_face()

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

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

Reimplemented from FiniteElement< dim, dim >.

Definition at line 2083 of file fe_nedelec.cc.

◆ hp_constraints_are_implemented()

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

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

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

Reimplemented from FiniteElement< dim, dim >.

Definition at line 2358 of file fe_nedelec.cc.

◆ compare_for_domination() [1/2]

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

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

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

Definition at line 2322 of file fe_nedelec.cc.

◆ hp_vertex_dof_identities() [1/2]

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

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

Definition at line 2365 of file fe_nedelec.cc.

◆ hp_line_dof_identities() [1/2]

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

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

Definition at line 2374 of file fe_nedelec.cc.

◆ hp_quad_dof_identities() [1/2]

template<int dim>
std::vector< std::pair< unsigned int, unsigned int > > FE_Nedelec< dim >::hp_quad_dof_identities ( const FiniteElement< dim > &  fe_other,
const unsigned int  face_no = 0 
) const
overridevirtual

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

Definition at line 2416 of file fe_nedelec.cc.

◆ get_face_interpolation_matrix() [1/2]

template<int dim>
void FE_Nedelec< dim >::get_face_interpolation_matrix ( const FiniteElement< dim > &  source,
FullMatrix< double > &  matrix,
const unsigned int  face_no = 0 
) const
overridevirtual

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

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

Definition at line 2470 of file fe_nedelec.cc.

◆ get_subface_interpolation_matrix() [1/3]

template<int dim>
void FE_Nedelec< dim >::get_subface_interpolation_matrix ( const FiniteElement< dim > &  source,
const unsigned int  subface,
FullMatrix< double > &  matrix,
const unsigned int  face_no = 0 
) const
overridevirtual

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

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

Definition at line 2577 of file fe_nedelec.cc.

◆ get_restriction_matrix()

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

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

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

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

Reimplemented from FiniteElement< dim, dim >.

Definition at line 3071 of file fe_nedelec.cc.

◆ get_prolongation_matrix()

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

Embedding matrix between grids.

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

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

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

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

Reimplemented from FiniteElement< dim, dim >.

Definition at line 3016 of file fe_nedelec.cc.

◆ convert_generalized_support_point_values_to_dof_values()

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

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

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

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

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

Parameters
[in]support_point_valuesAn 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_valuesAn array of size dofs_per_cell that contains the node functionals of the element applied to the given function.
Note
It is safe to call this function for (transformed) values on the real cell only for elements with trivial MappingKind. For all other elements (for example for H(curl), or H(div) conforming elements) vector values have to be transformed to the reference cell first.
Given what the function is supposed to do, the function clearly can only work for elements that actually implement (generalized) support points. Elements that do not have generalized support points – e.g., elements whose nodal functionals evaluate integrals or moments of functions (such as FE_Q_Hierarchical) – can in general not make sense of the operation that is required for this function. They consequently may not implement it.

Reimplemented from FiniteElement< dim, dim >.

Definition at line 3134 of file fe_nedelec.cc.

◆ get_constant_modes()

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

Return a list of constant modes of the element.

Reimplemented from FiniteElement< dim, dim >.

Definition at line 4087 of file fe_nedelec.cc.

◆ memory_consumption()

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

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

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

Reimplemented from FiniteElement< dim, dim >.

Definition at line 4103 of file fe_nedelec.cc.

◆ clone()

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

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

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

Implements FiniteElement< dim, dim >.

Definition at line 234 of file fe_nedelec.cc.

◆ get_dpo_vector()

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

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

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

Definition at line 2047 of file fe_nedelec.cc.

◆ initialize_support_points() [1/4]

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

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

◆ initialize_restriction() [1/2]

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

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

Definition at line 562 of file fe_nedelec.cc.

◆ initialize_support_points() [2/4]

template<>
void FE_Nedelec< 1 >::initialize_support_points ( const unsigned  int)
private

Definition at line 253 of file fe_nedelec.cc.

◆ initialize_support_points() [3/4]

template<>
void FE_Nedelec< 2 >::initialize_support_points ( const unsigned int  order)
private

Definition at line 262 of file fe_nedelec.cc.

◆ initialize_support_points() [4/4]

template<>
void FE_Nedelec< 3 >::initialize_support_points ( const unsigned int  order)
private

Definition at line 360 of file fe_nedelec.cc.

◆ initialize_restriction() [2/2]

template<>
void FE_Nedelec< 1 >::initialize_restriction ( )
private

Definition at line 549 of file fe_nedelec.cc.

◆ get_subface_interpolation_matrix() [2/3]

template<>
void FE_Nedelec< 1 >::get_subface_interpolation_matrix ( const FiniteElement< 1, 1 > &  ,
const unsigned  int,
FullMatrix< double > &  ,
const unsigned  int 
) const

Definition at line 2549 of file fe_nedelec.cc.

◆ requires_update_flags()

UpdateFlags FE_PolyTensor< dim, dim >::requires_update_flags ( const UpdateFlags  update_flags) const
overridevirtualinherited

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.

See also
UpdateFlags

Implements FiniteElement< dim, dim >.

Definition at line 2190 of file fe_poly_tensor.cc.

◆ shape_value()

double FE_PolyTensor< dim, dim >::shape_value ( const unsigned int  i,
const Point< dim > &  p 
) const
overridevirtualinherited

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 >.

Definition at line 229 of file fe_poly_tensor.cc.

◆ shape_value_component()

double FE_PolyTensor< dim, dim >::shape_value_component ( const unsigned int  i,
const Point< dim > &  p,
const unsigned int  component 
) const
overridevirtualinherited

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 >.

Definition at line 241 of file fe_poly_tensor.cc.

◆ shape_grad()

Tensor< 1, dim > FE_PolyTensor< dim, dim >::shape_grad ( const unsigned int  i,
const Point< dim > &  p 
) const
overridevirtualinherited

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 >.

Definition at line 275 of file fe_poly_tensor.cc.

◆ shape_grad_component()

Tensor< 1, dim > FE_PolyTensor< dim, dim >::shape_grad_component ( const unsigned int  i,
const Point< dim > &  p,
const unsigned int  component 
) const
overridevirtualinherited

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 >.

Definition at line 286 of file fe_poly_tensor.cc.

◆ shape_grad_grad()

Tensor< 2, dim > FE_PolyTensor< dim, dim >::shape_grad_grad ( const unsigned int  i,
const Point< dim > &  p 
) const
overridevirtualinherited

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 >.

Definition at line 321 of file fe_poly_tensor.cc.

◆ shape_grad_grad_component()

Tensor< 2, dim > FE_PolyTensor< dim, dim >::shape_grad_grad_component ( const unsigned int  i,
const Point< dim > &  p,
const unsigned int  component 
) const
overridevirtualinherited

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 >.

Definition at line 332 of file fe_poly_tensor.cc.

◆ single_mapping_kind()

bool FE_PolyTensor< dim, dim >::single_mapping_kind ( ) const
protectedinherited

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

Definition at line 207 of file fe_poly_tensor.cc.

◆ get_mapping_kind()

MappingKind FE_PolyTensor< dim, dim >::get_mapping_kind ( const unsigned int  i) const
protectedinherited

Returns MappingKind i for the finite element.

Definition at line 216 of file fe_poly_tensor.cc.

◆ get_data() [1/2]

virtual std::unique_ptr< typename FiniteElement<dim, dim >::InternalDataBase> FE_PolyTensor< dim, dim >::get_data ( const UpdateFlags  update_flags,
const Mapping< dim, dim > &  ,
const Quadrature< dim > &  quadrature,
::internal::FEValuesImplementation::FiniteElementRelatedData< dim, dim > &   
) const
inlineoverrideprotectedvirtualinherited

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_flagsA 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]mappingA reference to the mapping used for computing values and derivatives of shape functions.
[in]quadratureA reference to the object that describes where the shape functions should be evaluated.
[out]output_dataA 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.

Definition at line 238 of file fe_poly_tensor.h.

◆ get_data() [2/2]

virtual std::unique_ptr<InternalDataBase> FiniteElement< 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
protectedpure virtualinherited

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_flagsA 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]mappingA reference to the mapping used for computing values and derivatives of shape functions.
[in]quadratureA reference to the object that describes where the shape functions should be evaluated.
[out]output_dataA 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.

Implemented in FE_NedelecSZ< dim, spacedim >.

◆ fill_fe_values() [1/2]

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

Definition at line 370 of file fe_poly_tensor.cc.

◆ 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]cellThe cell of the triangulation for which this function is to compute a mapping from the reference cell to.
[in]cell_similarityWhether 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]quadratureA 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]mappingA 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_internalAn 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_dataThe 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_internalA 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_dataA 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() [1/2]

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

Definition at line 945 of file fe_poly_tensor.cc.

◆ fill_fe_face_values() [2/2]

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
protectedpure virtualinherited

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]cellThe cell of the triangulation for which this function is to compute a mapping from the reference cell to.
[in]face_noThe number of the face we are currently considering, indexed among the faces of the cell specified by the previous argument.
[in]quadratureA 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]mappingA 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_internalAn 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_dataThe 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_internalA 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_dataA 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_subface_values() [1/2]

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

Definition at line 1571 of file fe_poly_tensor.cc.

◆ 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]cellThe cell of the triangulation for which this function is to compute a mapping from the reference cell to.
[in]face_noThe number of the face we are currently considering, indexed among the faces of the cell specified by the previous argument.
[in]sub_noThe 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]quadratureA 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]mappingA 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_internalAn 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_dataThe 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_internalA 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_dataA 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.

◆ operator^()

std::pair< std::unique_ptr< FiniteElement< dim, spacedim > >, 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.

Definition at line 177 of file fe.cc.

◆ operator[]()

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

This operator returns a reference to the present object if the argument given equals to zero. While this does not seem particularly useful, it is helpful in writing code that works with both DoFHandler and the hp version hp::DoFHandler, since one can then write code like this:

dof_handler->get_fe()[cell->active_fe_index()].n_dofs_per_cell();

This code doesn't work in both situations without the present operator because DoFHandler::get_fe() returns a finite element, whereas hp::DoFHandler::get_fe() returns a collection of finite elements that doesn't offer a dofs_per_cell member variable: one first has to select which finite element to work on, which is done using the operator[]. Fortunately, cell->active_fe_index() also works for non-hp classes and simply returns zero in that case. The present operator[] accepts this zero argument, by returning the finite element with index zero within its collection (that, of course, consists only of the present finite element anyway).

Definition at line 3058 of file fe.h.

◆ shape_3rd_derivative()

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 FE_Poly< dim >.

Definition at line 256 of file fe.cc.

◆ shape_3rd_derivative_component()

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 FE_Poly< dim >.

Definition at line 267 of file fe.cc.

◆ shape_4th_derivative()

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 FE_Poly< dim >.

Definition at line 280 of file fe.cc.

◆ shape_4th_derivative_component()

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 FE_Poly< dim >.

Definition at line 291 of file fe.cc.

◆ 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.

Definition at line 729 of file fe.cc.

◆ 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.

Definition at line 785 of file fe.cc.

◆ 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.

Definition at line 757 of file fe.cc.

◆ 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.

Definition at line 813 of file fe.cc.

◆ restriction_is_additive()

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

Access the restriction_is_additive_flags field. See the discussion about restriction matrices in the general class documentation for more information.

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

Definition at line 3241 of file fe.h.

◆ constraints()

const FullMatrix< double > & FiniteElement< dim, spacedim >::constraints ( const ::internal::SubfaceCase< dim > &  subface_case = ::internal::SubfaceCase<dim>::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.

Definition at line 869 of file fe.cc.

◆ constraints_are_implemented()

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

Definition at line 841 of file fe.cc.

◆ get_interpolation_matrix()

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.

Definition at line 931 of file fe.cc.

◆ get_face_interpolation_matrix() [2/2]

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.

Definition at line 947 of file fe.cc.

◆ get_subface_interpolation_matrix() [3/3]

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.

Definition at line 964 of file fe.cc.

◆ hp_vertex_dof_identities() [2/2]

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

Definition at line 982 of file fe.cc.

◆ hp_line_dof_identities() [2/2]

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

Definition at line 993 of file fe.cc.

◆ hp_quad_dof_identities() [2/2]

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

Definition at line 1004 of file fe.cc.

◆ compare_for_domination() [2/2]

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.

Definition at line 1016 of file fe.cc.

◆ operator==() [1/2]

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.

Definition at line 1029 of file fe.cc.

◆ operator==() [2/2]

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

Comparison operator.

Definition at line 202 of file fe_data.cc.

◆ operator!=()

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

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

Definition at line 1044 of file fe.cc.

◆ system_to_component_index()

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

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.

Definition at line 3070 of file fe.h.

◆ component_to_system_index()

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

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.

Definition at line 3103 of file fe.h.

◆ face_system_to_component_index()

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

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.

Definition at line 3130 of file fe.h.

◆ adjust_quad_dof_index_for_face_orientation()

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.

Definition at line 660 of file fe.cc.

◆ face_to_cell_index()

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

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_indexThe index of the degree of freedom on a face. This index must be between zero and dofs_per_face.
faceThe number of the face this degree of freedom lives on. This number must be between zero and GeometryInfo::faces_per_cell.
face_orientationOne part of the description of the orientation of the face. See GlossFaceOrientation.
face_flipOne part of the description of the orientation of the face. See GlossFaceOrientation.
face_rotationOne 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.

Definition at line 568 of file fe.cc.

◆ adjust_line_dof_index_for_line_orientation()

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.

Definition at line 704 of file fe.cc.

◆ get_nonzero_components()

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

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).

Definition at line 3252 of file fe.h.

◆ n_nonzero_components()

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

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.

Definition at line 3262 of file fe.h.

◆ is_primitive() [1/2]

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

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.

Definition at line 3272 of file fe.h.

◆ is_primitive() [2/2]

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

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.

Definition at line 3281 of file fe.h.

◆ n_base_elements()

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

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.

Definition at line 3084 of file fe.h.

◆ base_element()

const FiniteElement< dim, spacedim > & 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.

Definition at line 1324 of file fe.cc.

◆ element_multiplicity()

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

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.

Definition at line 3093 of file fe.h.

◆ get_sub_fe() [1/2]

const FiniteElement< dim, spacedim > & 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);

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

ComponentMask Result Description
[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

Definition at line 1168 of file fe.cc.

◆ get_sub_fe() [2/2]

const FiniteElement< dim, spacedim > & 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.

Definition at line 1201 of file fe.cc.

◆ system_to_base_index()

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

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.

Definition at line 3165 of file fe.h.

◆ face_system_to_base_index()

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

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.

Definition at line 3176 of file fe.h.

◆ first_block_of_base()

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

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

Definition at line 3192 of file fe.h.

◆ component_to_base_index()

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

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.

Definition at line 3202 of file fe.h.

◆ block_to_base_index()

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

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

Definition at line 3214 of file fe.h.

◆ system_to_block_index()

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

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

Definition at line 3224 of file fe.h.

◆ 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.

Definition at line 384 of file fe.cc.

◆ component_mask() [1/4]

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
scalarAn 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.

Definition at line 396 of file fe.cc.

◆ component_mask() [2/4]

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
vectorAn 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.

Definition at line 414 of file fe.cc.

◆ component_mask() [3/4]

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_tensorAn 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.

Definition at line 436 of file fe.cc.

◆ component_mask() [4/4]

ComponentMask FiniteElement< dim, spacedim >::component_mask ( const BlockMask block_mask) const
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
block_maskThe mask that selects individual blocks of the finite element
Returns
A mask that selects those components corresponding to the selected blocks of the input argument.

Definition at line 461 of file fe.cc.

◆ block_mask() [1/4]

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
scalarAn 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.

Definition at line 482 of file fe.cc.

◆ block_mask() [2/4]

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
vectorAn 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.

Definition at line 493 of file fe.cc.

◆ block_mask() [3/4]

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_tensorAn 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.

Definition at line 504 of file fe.cc.

◆ block_mask() [4/4]

BlockMask FiniteElement< dim, spacedim >::block_mask ( const ComponentMask component_mask) const
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
component_maskThe mask that selects individual components of the finite element
Returns
A mask that selects those blocks corresponding to the selected blocks of the input argument.

Definition at line 516 of file fe.cc.

◆ get_unit_support_points()

const std::vector< Point< dim > > & 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.

Definition at line 1053 of file fe.cc.

◆ 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.

Definition at line 1069 of file fe.cc.

◆ unit_support_point()

Point< dim > 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.

Definition at line 1100 of file fe.cc.

◆ get_unit_face_support_points()

const std::vector< Point< dim - 1 > > & 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.

Definition at line 1112 of file fe.cc.

◆ 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.

For more information, see the documentation for the has_support_points() function.

Definition at line 1131 of file fe.cc.

◆ unit_face_support_point()

Point< dim - 1 > 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.

Definition at line 1142 of file fe.cc.

◆ get_generalized_support_points()

const std::vector< Point< dim > > & 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.

See the glossary entry on generalized support points for more information.

Definition at line 1078 of file fe.cc.

◆ 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.

See the glossary entry on generalized support points for more information.

Definition at line 1091 of file fe.cc.

◆ get_associated_geometry_primitive()

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

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_indexThe 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).

Definition at line 3304 of file fe.h.

◆ 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_onlyonly the restriction matrices required for isotropic refinement are reinited to the right size.
isotropic_prolongation_onlyonly the prolongation matrices required for isotropic refinement are reinited to the right size.

Definition at line 303 of file fe.cc.

◆ 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.

Definition at line 903 of file fe.cc.

◆ compute_n_nonzero_components()

std::vector< unsigned int > 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.

Definition at line 1271 of file fe.cc.

◆ get_face_data()

std::unique_ptr< typename FiniteElement< dim, spacedim >::InternalDataBase > 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

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_flagsA 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]mappingA reference to the mapping used for computing values and derivatives of shape functions.
[in]quadratureA reference to the object that describes where the shape functions should be evaluated.
[out]output_dataA 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.

Definition at line 1286 of file fe.cc.

◆ get_subface_data()

std::unique_ptr< typename FiniteElement< dim, spacedim >::InternalDataBase > 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_flagsA 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]mappingA reference to the mapping used for computing values and derivatives of shape functions.
[in]quadratureA reference to the object that describes where the shape functions should be evaluated.
[out]output_dataA 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.

Definition at line 1305 of file fe.cc.

◆ subscribe()

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

Subscribes a user of the object by storing the pointer validity. The subscriber may be identified by text supplied as identifier.

Definition at line 136 of file subscriptor.cc.

◆ unsubscribe()

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

Unsubscribes a user from the object.

Note
The identifier and the validity pointer must be the same as the one supplied to subscribe().

Definition at line 156 of file subscriptor.cc.

◆ n_subscriptions()

unsigned int Subscriptor::n_subscriptions ( ) const
inlineinherited

Return the present number of subscriptions to this object. This allows to use this class for reference counted lifetime determination where the last one to unsubscribe also deletes the object.

Definition at line 300 of file subscriptor.h.

◆ list_subscribers() [1/2]

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

List the subscribers to the input stream.

Definition at line 317 of file subscriptor.h.

◆ list_subscribers() [2/2]

void Subscriptor::list_subscribers ( ) const
inherited

List the subscribers to deallog.

Definition at line 204 of file subscriptor.cc.

◆ serialize()

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

Read or write the data of this object to or from a stream for the purpose of serialization.

This function does not actually serialize any of the member variables of this class. The reason is that what this class stores is only who subscribes to this object, but who does so at the time of storing the contents of this object does not necessarily have anything to do with who subscribes to the object when it is restored. Consequently, we do not want to overwrite the subscribers at the time of restoring, and then there is no reason to write the subscribers out in the first place.

Definition at line 309 of file subscriptor.h.

◆ reference_cell_type()

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

Return type of reference cell.

◆ n_unique_quads()

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.

◆ n_dofs_per_quad()

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.

◆ max_dofs_per_quad()

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

Number of components. See the glossary for more information.

◆ n_blocks()

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

Number of blocks. See the glossary for more information.

◆ 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.

◆ get_first_quad_index()

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.

◆ get_first_face_quad_index()

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.

Friends And Related Function Documentation

◆ FE_Nedelec

template<int dim>
template<int dim1>
friend class FE_Nedelec
friend

Definition at line 374 of file fe_nedelec.h.

Member Data Documentation

◆ boundary_weights

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

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

See the glossary entry on generalized support points for more information.

Definition at line 365 of file fe_nedelec.h.

◆ mutex

template<int dim>
Threads::Mutex FE_Nedelec< dim >::mutex
mutableprivate

Mutex for protecting initialization of restriction and embedding matrix.

Definition at line 370 of file fe_nedelec.h.

◆ mapping_kind

std::vector<MappingKind> FE_PolyTensor< dim, dim >::mapping_kind
protectedinherited

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.

◆ poly_space

const std::unique_ptr<const TensorPolynomialsBase<dim> > FE_PolyTensor< dim, dim >::poly_space
protectedinherited

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

Definition at line 475 of file fe_poly_tensor.h.

◆ inverse_node_matrix

FullMatrix<double> FE_PolyTensor< dim, dim >::inverse_node_matrix
protectedinherited

The inverse of the matrix aij of node values 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.

Definition at line 488 of file fe_poly_tensor.h.

◆ cache_mutex

std::mutex FE_PolyTensor< dim, dim >::cache_mutex
mutableprotectedinherited

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

Definition at line 493 of file fe_poly_tensor.h.

◆ cached_point

Point<dim> FE_PolyTensor< dim, dim >::cached_point
mutableprotectedinherited

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 500 of file fe_poly_tensor.h.

◆ cached_values

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

Cached shape function values after call to shape_value_component().

Definition at line 505 of file fe_poly_tensor.h.

◆ cached_grads

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

Cached shape function gradients after call to shape_grad_component().

Definition at line 510 of file fe_poly_tensor.h.

◆ cached_grad_grads

std::vector<Tensor<3, dim> > FE_PolyTensor< dim, dim >::cached_grad_grads
mutableprotectedinherited

Cached second derivatives of shape functions after call to shape_grad_grad_component().

Definition at line 516 of file fe_poly_tensor.h.

◆ space_dimension

const unsigned int FiniteElement< dim, spacedim >::space_dimension
staticinherited

The dimension of the image space, corresponding to Triangulation.

Definition at line 651 of file fe.h.

◆ restriction

std::vector<std::vector<FullMatrix<double> > > 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 2401 of file fe.h.

◆ prolongation

std::vector<std::vector<FullMatrix<double> > > 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 2415 of file fe.h.

◆ interface_constraints

FullMatrix<double> 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 2427 of file fe.h.

◆ unit_support_points

std::vector<Point<dim> > 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 2439 of file fe.h.

◆ unit_face_support_points

std::vector<std::vector<Point<dim - 1> > > 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 2446 of file fe.h.

◆ generalized_support_points

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

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

Definition at line 2452 of file fe.h.

◆ generalized_face_support_points

std::vector<std::vector<Point<dim - 1> > > FiniteElement< dim, spacedim >::generalized_face_support_points
protectedinherited

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

Definition at line 2458 of file fe.h.

◆ adjust_quad_dof_index_for_face_orientation_table

std::vector<Table<2, int> > FiniteElement< dim, spacedim >::adjust_quad_dof_index_for_face_orientation_table
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 2475 of file fe.h.

◆ adjust_line_dof_index_for_line_orientation_table

std::vector<int> FiniteElement< dim, spacedim >::adjust_line_dof_index_for_line_orientation_table
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 2490 of file fe.h.

◆ system_to_component_table

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

Store what system_to_component_index() will return.

Definition at line 2495 of file fe.h.

◆ face_system_to_component_table

std::vector<std::vector<std::pair<unsigned int, unsigned int> > > 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 2507 of file fe.h.

◆ system_to_base_table

std::vector<std::pair<std::pair<unsigned int, unsigned int>, 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 2526 of file fe.h.

◆ face_system_to_base_table

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

Likewise for the indices on faces.

Definition at line 2533 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 2539 of file fe.h.

◆ component_to_base_table

std::vector<std::pair<std::pair<unsigned int, unsigned int>, 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 2562 of file fe.h.

◆ restriction_is_additive_flags

const std::vector<bool> 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 2569 of file fe.h.

◆ nonzero_components

const std::vector<ComponentMask> 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 2578 of file fe.h.

◆ n_nonzero_components_table

const std::vector<unsigned int> 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 2587 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 2594 of file fe.h.

◆ dimension

template<int dim>
const unsigned int FiniteElementData< dim >::dimension = dim
staticinherited

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

Definition at line 281 of file fe_base.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 305 of file fe_base.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 311 of file fe_base.h.

◆ dofs_per_quad

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 325 of file fe_base.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 338 of file fe_base.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 343 of file fe_base.h.

◆ first_quad_index

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

First index of dof on a quad.

Definition at line 357 of file fe_base.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 362 of file fe_base.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 374 of file fe_base.h.

◆ first_face_quad_index

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 386 of file fe_base.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 401 of file fe_base.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 415 of file fe_base.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 425 of file fe_base.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 431 of file fe_base.h.

◆ conforming_space

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

Indicate the space this element conforms to.

Definition at line 436 of file fe_base.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 443 of file fe_base.h.


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