#include <deal.II/fe/fe_p1nc.h>

Inheritance diagram for FE_P1NC:

[legend]

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

Public Member Functions
	FE_P1NC ()

virtual std::string	get_name () const override

virtual UpdateFlags	requires_update_flags (const UpdateFlags flags) const override

virtual std::unique_ptr< FiniteElement< 2, 2 > >	clone () const override

virtual	~FE_P1NC () override=default

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

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

bool	operator== (const FiniteElementData &) const

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

virtual std::size_t	memory_consumption () const

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

ReferenceCell	reference_cell () const

unsigned int	n_unique_2d_subobjects () 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 BlockIndices &	block_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 double	shape_value (const unsigned int i, const Point< dim > &p) const

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

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

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

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

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

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

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

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

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

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

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

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

bool	prolongation_is_implemented () const

bool	isotropic_prolongation_is_implemented () const

bool	restriction_is_implemented () const

bool	isotropic_restriction_is_implemented () const

bool	restriction_is_additive (const unsigned int index) const

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

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

virtual bool	hp_constraints_are_implemented () const

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

Functions to support hp-adaptivity
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 int >	system_to_component_index (const unsigned int index) const

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

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

unsigned int	adjust_quad_dof_index_for_face_orientation (const unsigned int index, const unsigned int face_no, const unsigned char combined_orientation) const

virtual unsigned int	face_to_cell_index (const unsigned int face_dof_index, const unsigned int face, const unsigned char combined_orientation=ReferenceCell::default_combined_face_orientation()) const

unsigned int	adjust_line_dof_index_for_line_orientation (const unsigned int index, const unsigned char combined_orientation) const

const ComponentMask &	get_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

virtual const Table< 2, bool > &	get_local_dof_sparsity_pattern () 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 int >	system_to_base_index (const unsigned int index) const

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

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

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

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

std::pair< unsigned int, types::global_dof_index >	system_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

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

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

EnableObserverPointer functionality
Classes derived from EnableObserverPointer provide a facility to subscribe to this object. This is mostly used by the ObserverPointer 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 ::ExceptionBase &	ExcShapeFunctionNotPrimitive (int arg1)

static ::ExceptionBase &	ExcFENotPrimitive ()

static ::ExceptionBase &	ExcUnitShapeValuesDoNotExist ()

static ::ExceptionBase &	ExcFEHasNoSupportPoints ()

static ::ExceptionBase &	ExcEmbeddingVoid ()

static ::ExceptionBase &	ExcProjectionVoid ()

static ::ExceptionBase &	ExcWrongInterfaceMatrixSize (int arg1, int arg2)

static ::ExceptionBase &	ExcInterpolationNotImplemented ()

static ::ExceptionBase &	ExcInUse (int arg1, std::string arg2, std::string arg3)

static ::ExceptionBase &	ExcNoSubscriber (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 constexpr unsigned int	space_dimension

static constexpr unsigned int	dimension = dim

Protected Member Functions
void	reinit_restriction_and_prolongation_matrices (const bool isotropic_restriction_only=false, const bool isotropic_prolongation_only=false)

TableIndices< 2 >	interface_constraints_size () const

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

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

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

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

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

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

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

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

Static Protected Member Functions
static std::vector< unsigned int >	compute_n_nonzero_components (const std::vector< ComponentMask > &nonzero_components)

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

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

FullMatrix< double >	interface_constraints

std::vector< Point< dim > >	unit_support_points

std::vector< 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< int >	adjust_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< bool >	restriction_is_additive_flags

const std::vector< ComponentMask >	nonzero_components

const std::vector< unsigned int >	n_nonzero_components_table

const bool	cached_primitivity

Table< 2, bool >	local_dof_sparsity_pattern

Private Types
using	map_value_type = decltype(counter_map)::value_type

using	map_iterator = decltype(counter_map)::iterator

Private Member Functions
virtual std::unique_ptr< FiniteElement< 2, 2 >::InternalDataBase >	get_data (const UpdateFlags update_flags, const Mapping< 2, 2 > &, const Quadrature< 2 > &quadrature, ::internal::FEValuesImplementation::FiniteElementRelatedData< 2, 2 > &output_data) const override

virtual std::unique_ptr< FiniteElement< 2, 2 >::InternalDataBase >	get_face_data (const UpdateFlags update_flags, const Mapping< 2, 2 > &, const hp::QCollection< 1 > &quadrature, ::internal::FEValuesImplementation::FiniteElementRelatedData< 2, 2 > &output_data) const override

virtual std::unique_ptr< FiniteElement< 2, 2 >::InternalDataBase >	get_subface_data (const UpdateFlags update_flags, const Mapping< 2, 2 > &, const Quadrature< 1 > &quadrature, ::internal::FEValuesImplementation::FiniteElementRelatedData< 2, 2 > &output_data) const override

virtual void	fill_fe_values (const Triangulation< 2, 2 >::cell_iterator &cell, const CellSimilarity::Similarity cell_similarity, const Quadrature< 2 > &quadrature, const Mapping< 2, 2 > &mapping, const Mapping< 2, 2 >::InternalDataBase &mapping_internal, const internal::FEValuesImplementation::MappingRelatedData< 2, 2 > &mapping_data, const FiniteElement< 2, 2 >::InternalDataBase &fe_internal, internal::FEValuesImplementation::FiniteElementRelatedData< 2, 2 > &output_data) const override

virtual void	fill_fe_face_values (const Triangulation< 2, 2 >::cell_iterator &cell, const unsigned int face_no, const hp::QCollection< 1 > &quadrature, const Mapping< 2, 2 > &mapping, const Mapping< 2, 2 >::InternalDataBase &mapping_internal, const internal::FEValuesImplementation::MappingRelatedData< 2, 2 > &mapping_data, const InternalDataBase &fe_internal, ::internal::FEValuesImplementation::FiniteElementRelatedData< 2, 2 > &output_data) const override

virtual void	fill_fe_subface_values (const Triangulation< 2, 2 >::cell_iterator &cell, const unsigned int face_no, const unsigned int sub_no, const Quadrature< 1 > &quadrature, const Mapping< 2, 2 > &mapping, const Mapping< 2, 2 >::InternalDataBase &mapping_internal, const internal::FEValuesImplementation::MappingRelatedData< 2, 2 > &mapping_data, const InternalDataBase &fe_internal, ::internal::FEValuesImplementation::FiniteElementRelatedData< 2, 2 > &output_data) const override

void	initialize_constraints ()

void	check_no_subscribers () const noexcept

Static Private Member Functions
static std::vector< unsigned int >	get_dpo_vector ()

static ndarray< double, 4, 3 >	get_linear_shape_coefficients (const Triangulation< 2, 2 >::cell_iterator &cell)

Private Attributes
std::atomic< unsigned int >	counter

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

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

const std::type_info *	object_info

const ReferenceCell	reference_cell_kind

const unsigned int	number_of_unique_2d_subobjects

const unsigned int	number_unique_faces

const std::vector< unsigned int >	n_dofs_on_quad

const unsigned int	dofs_per_quad_max

const std::vector< unsigned int >	first_index_of_quads

const std::vector< unsigned int >	first_line_index_of_faces

const std::vector< unsigned int >	first_quad_index_of_faces

const std::vector< unsigned int >	n_dofs_on_face

const unsigned int	dofs_per_face_max

Static Private Attributes
static std::mutex	mutex

Detailed Description

Implementation of the scalar version of the P1 nonconforming finite element, a piecewise linear element on quadrilaterals in 2d. This implementation is only for 2d cells in a 2d space (i.e., codimension 0).

Unlike the usual continuous, \(H^1\) conforming finite elements, the P1 nonconforming element does not enforce continuity across edges. However, it requires the continuity in an integral sense: any function in the space should have the same integral values on two sides of the common edge shared by two adjacent elements.

Thus, each function in the nonconforming element space can be discontinuous, and consequently not included in \(H^1_0\), just like the basis functions in Discontinuous Galerkin (DG) finite element spaces. On the other hand, basis functions in DG spaces are completely discontinuous across edges without any relation between the values from both sides. This is a reason why usual weak formulations for DG schemes contain additional penalty terms for jump across edges to control discontinuity. However, nonconforming elements usually do not need additional terms in their weak formulations because their integrals along edges are the same from both sides, i.e., there is some level of continuity.

Dice Rule

Since any function in the P1 nonconforming space is piecewise linear on each element, the function value at the midpoint of each edge is same as the mean value on the edge. Thus the continuity of the integral value across each edge is equivalent to the continuity of the midpoint value of each edge in this case.

Thus for the P1 nonconforming element, the function values at midpoints on edges of a cell are important. The first attempt to define (local) degrees of freedom (DoFs) on a quadrilateral is by using midpoint values of a function.

However, these 4 functionals are not linearly independent because a linear function on 2d is uniquely determined by only 3 independent values. A simple observation reads that any linear function on a quadrilateral should satisfy the 'dice rule': the sum of two function values at the midpoints of the edge pair on opposite sides of a cell is equal to the sum of those at the midpoints of the other edge pair. This is called the 'dice rule' because the number of points on opposite sides of a dice always adds up to the same number as well (in the case of dice, to seven).

In formulas, the dice rule is written as \(\phi(m_0) + \phi(m_1) = \phi(m_2) + \phi(m_3)\) for all \(\phi\) in the function space where \(m_j\) is the midpoint of the edge \(e_j\). Here, we assume the standard numbering convention for edges used in deal.II and described in class GeometryInfo.

Conversely if 4 values at midpoints satisfying the dice rule are given, then there always exists the unique linear function which coincides with 4 midpoints values.

Due to the dice rule, three values at any three midpoints can determine the last value at the last midpoint. It means that the number of independent local functionals on a cell is 3, and this is also the dimension of the linear polynomial space on a cell in 2d.

Shape functions

Before introducing the degrees of freedom, we present 4 local shape functions on a cell. Due to the dice rule, we need a special construction for shape functions. Although the following 4 shape functions are not linearly independent within a cell, they are helpful to define the global basis functions which are linearly independent on the whole domain. Again, we assume the standard numbering for vertices used in deal.II.

*  2---------|---------3
*  |                   |
*  |                   |
*  |                   |
*  |                   |
*  -                   -
*  |                   |
*  |                   |
*  |                   |
*  |                   |
*  0---------|---------1
*

For each vertex \(v_j\) of given cell, there are two edges of which \(v_j\) is one of end points. Consider a linear function such that it has value 0.5 at the midpoints of two adjacent edges, and 0.0 at the two midpoints of the other edges. Note that the set of these values satisfies the dice rule which is described above. We denote such a function associated with vertex \(v_j\) by \(\phi_j\). Then the set of 4 shape functions is a partition of unity on a cell: \(\sum_{j=0}^{3} \phi_j = 1\). (This is easy to see: at each edge midpoint, the sum of the four function adds up to one because two functions have value 0.5 and the other value 0.0. Because the function is globally linear, the only function that can have value 1 at four points must also be globally equal to one.)

The following figures represent \(\phi_j\) for \(j=0,\cdots,3\) with their midpoint values:

shape function \(\phi_0\):

*  +--------0.0--------+
*  |                   |
*  |                   |
*  |                   |
*  |                   |
* 0.5                 0.0
*  |                   |
*  |                   |
*  |                   |
*  |                   |
*  +--------0.5--------+
*

shape function \(\phi_1\):

*  +--------0.0--------+
*  |                   |
*  |                   |
*  |                   |
*  |                   |
* 0.0                 0.5
*  |                   |
*  |                   |
*  |                   |
*  |                   |
*  +--------0.5--------+
*

shape function \(\phi_2\):

*  +--------0.5--------+
*  |                   |
*  |                   |
*  |                   |
*  |                   |
* 0.5                 0.0
*  |                   |
*  |                   |
*  |                   |
*  |                   |
*  +--------0.0--------+
*

shape function \(\phi_3\):

*  +--------0.5--------+
*  |                   |
*  |                   |
*  |                   |
*  |                   |
* 0.0                 0.5
*  |                   |
*  |                   |
*  |                   |
*  |                   |
*  +--------0.0--------+
*

The local DoFs are defined by the coefficients of the shape functions associated with vertices, respectively. Although these 4 local DoFs are not linearly independent within a single cell, this definition is a good start point for the definition of the global DoFs.

We want to emphasize that the shape functions are constructed on each cell, not on the reference cell only. Usual finite elements are defined based on a 'parametric' concept. It means that a function space for a finite element is defined on one reference cell, and it is transformed into each cell via a mapping from the reference cell. However the P1 nonconforming element does not follow such concept. It defines a function space with linear shape functions on each cell without any help of a function space on the reference cell. In other words, the element is defined in real space, not via a mapping from a reference cell. In this, it is similar to the FE_DGPNonparametric element.

Thus this implementation does not have to compute shape values on the reference cell. Rather, the shape values are computed by construction of the shape functions on each cell independently.

Degrees of freedom

We next have to consider the global basis functions for the element because the system of equations which we ultimately have to solve is for a global system, not local. The global basis functions associated with a node are defined by a cell-wise composition of local shape functions associated with the node on each element.

There is a theoretical result about the linear independence of the global basis functions depending on the type of the boundary condition we consider.

When homogeneous Dirichlet boundary conditions are given, the global basis functions associated with interior nodes are linearly independent. Then, the number of DoFs is equal to the number of interior nodes, and consequently the same as the number of DoFs for the standard bilinear \(Q_1\) finite element.

When Neumann boundary conditions are given, the global basis functions associated with all nodes (including boundary nodes) are actually not linearly independent. There exists one redundancy. Thus in this case, the number of DoFs is equal to the number of all nodes minus 1. This is, again as for the regular \(Q_1\) element.

Unit support points

For a smooth function, we construct a piecewise linear function which belongs to the element space by using its nodal values as DoF values.

Note that for the P1 nonconforming element, two nodal values of a smooth function and its interpolant do not coincide in general, in contrast to ordinary Lagrange finite elements. Of course, it is meaningless to refer 'nodal value' because the element space has nonconformity. But it is also true even though the single global basis function associated with a node is considered the unique 'nodal value' at the node. For instance, consider the basis function associated with a node. Consider two lines representing the level sets for value 0.5 and 0, respectively, by connecting two midpoints. Then we cut the quad into two sub-triangles by the diagonal which is placed along those two lines. It gives another level set for value 0.25 which coincides with the cutting diagonal. Therefore these three level sets are all parallel in the quad and it gives the value 0.75 at the base node, not value 1. This is true whether the quadrilateral is a rectangle, parallelogram, or any other shape.

Reference

The original paper for the P1 nonconforming element by Park and Sheen is accessible at https://doi.org/10.1137/S0036142902404923 , see [174] .

Definition at line 258 of file fe_p1nc.h.

Member Typedef Documentation

◆ map_value_type

using EnableObserverPointer::map_value_type = decltype(counter_map)::value_type

privateinherited

The data type used in counter_map.

Definition at line 238 of file enable_observer_pointer.h.

◆ map_iterator

using EnableObserverPointer::map_iterator = decltype(counter_map)::iterator

privateinherited

The iterator type used in counter_map.

Definition at line 243 of file enable_observer_pointer.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 H^curl and H^div as well.

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

H¹ implies that the function is continuous over cell boundaries.
H² implies that the function is continuously differentiable over cell boundaries.
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 H^curl (continuous tangential component of a vector field)
Hdiv	Conformity with the space H^div (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 261 of file fe_data.h.

Constructor & Destructor Documentation

◆ FE_P1NC()

FE_P1NC::FE_P1NC ( )

Constructor for the P1 nonconforming element. It is only for 2d and codimension = 0.

Definition at line 23 of file fe_p1nc.cc.

◆ ~FE_P1NC()

virtual FE_P1NC::~FE_P1NC ( )

overridevirtualdefault

Destructor.

Member Function Documentation

◆ get_name()

std::string FE_P1NC::get_name ( ) const

overridevirtual

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

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

Implements FiniteElement< 2, 2 >.

Definition at line 41 of file fe_p1nc.cc.

◆ requires_update_flags()

UpdateFlags FE_P1NC::requires_update_flags ( const UpdateFlags update_flags ) const

overridevirtual

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

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

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

See also: The interplay of UpdateFlags, Mapping, and FiniteElement in FEValues

Implements FiniteElement< 2, 2 >.

Definition at line 49 of file fe_p1nc.cc.

◆ clone()

std::unique_ptr< FiniteElement< 2, 2 > > FE_P1NC::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< 2, 2 >.

Definition at line 68 of file fe_p1nc.cc.

◆ get_dpo_vector()

std::vector< unsigned int > FE_P1NC::get_dpo_vector ( )

staticprivate

Return the vector consists of the numbers of degrees of freedom per objects.

Definition at line 76 of file fe_p1nc.cc.

◆ get_linear_shape_coefficients()

ndarray< double, 4, 3 > FE_P1NC::get_linear_shape_coefficients ( const Triangulation< 2, 2 >::cell_iterator & cell )

staticprivate

Return the coefficients of 4 local linear shape functions \(\phi_j(x,y) = a x + b y + c\) on given cell. For each local shape function, the array consists of three coefficients is in order of a,b and c.

Definition at line 88 of file fe_p1nc.cc.

◆ get_data() [1/2]

std::unique_ptr< FiniteElement< 2, 2 >::InternalDataBase > FE_P1NC::get_data	(	const UpdateFlags	update_flags,
		const Mapping< 2, 2 > &	,
		const Quadrature< 2 > &	quadrature,
		::internal::FEValuesImplementation::FiniteElementRelatedData< 2, 2 > &	output_data
	)		const

overrideprivatevirtual

Do the work which is needed before cellwise data computation. Since the shape functions are constructed independently on each cell, the data on the reference cell is not necessary. It returns an empty variable of type InternalDataBase and stores update_flags, and computes trivially zero Hessian for each cell if it is needed.

Definition at line 135 of file fe_p1nc.cc.

◆ get_face_data() [1/3]

std::unique_ptr< FiniteElement< 2, 2 >::InternalDataBase > FE_P1NC::get_face_data	(	const UpdateFlags	update_flags,
		const Mapping< 2, 2 > &	,
		const hp::QCollection< 1 > &	quadrature,
		::internal::FEValuesImplementation::FiniteElementRelatedData< 2, 2 > &	output_data
	)		const

overrideprivatevirtual

Definition at line 159 of file fe_p1nc.cc.

◆ get_subface_data() [1/2]

std::unique_ptr< FiniteElement< 2, 2 >::InternalDataBase > FE_P1NC::get_subface_data	(	const UpdateFlags	update_flags,
		const Mapping< 2, 2 > &	,
		const Quadrature< 1 > &	quadrature,
		::internal::FEValuesImplementation::FiniteElementRelatedData< 2, 2 > &	output_data
	)		const

overrideprivatevirtual

Definition at line 185 of file fe_p1nc.cc.

◆ fill_fe_values() [1/2]

void FE_P1NC::fill_fe_values	(	const Triangulation< 2, 2 >::cell_iterator &	cell,
		const CellSimilarity::Similarity	cell_similarity,
		const Quadrature< 2 > &	quadrature,
		const Mapping< 2, 2 > &	mapping,
		const Mapping< 2, 2 >::InternalDataBase &	mapping_internal,
		const internal::FEValuesImplementation::MappingRelatedData< 2, 2 > &	mapping_data,
		const FiniteElement< 2, 2 >::InternalDataBase &	fe_internal,
		internal::FEValuesImplementation::FiniteElementRelatedData< 2, 2 > &	output_data
	)		const

overrideprivatevirtual

Compute the data on the current cell.

Definition at line 209 of file fe_p1nc.cc.

◆ fill_fe_face_values() [1/3]

void FE_P1NC::fill_fe_face_values	(	const Triangulation< 2, 2 >::cell_iterator &	cell,
		const unsigned int	face_no,
		const hp::QCollection< 1 > &	quadrature,
		const Mapping< 2, 2 > &	mapping,
		const Mapping< 2, 2 >::InternalDataBase &	mapping_internal,
		const internal::FEValuesImplementation::MappingRelatedData< 2, 2 > &	mapping_data,
		const InternalDataBase &	fe_internal,
		::internal::FEValuesImplementation::FiniteElementRelatedData< 2, 2 > &	output_data
	)		const

overrideprivatevirtual

Compute the data on the face of the current cell.

Definition at line 246 of file fe_p1nc.cc.

◆ fill_fe_subface_values() [1/2]

void FE_P1NC::fill_fe_subface_values	(	const Triangulation< 2, 2 >::cell_iterator &	cell,
		const unsigned int	face_no,
		const unsigned int	sub_no,
		const Quadrature< 1 > &	quadrature,
		const Mapping< 2, 2 > &	mapping,
		const Mapping< 2, 2 >::InternalDataBase &	mapping_internal,
		const internal::FEValuesImplementation::MappingRelatedData< 2, 2 > &	mapping_data,
		const InternalDataBase &	fe_internal,
		::internal::FEValuesImplementation::FiniteElementRelatedData< 2, 2 > &	output_data
	)		const

overrideprivatevirtual

Compute the data on the subface of the current cell.

Definition at line 293 of file fe_p1nc.cc.

◆ initialize_constraints()

void FE_P1NC::initialize_constraints ( )

private

Create the constraints matrix for hanging edges.

Definition at line 339 of file fe_p1nc.cc.

◆ 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.

◆ shape_value()

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

virtualinherited

Return the value of the ith shape function at the point p. p is a point on the reference element. 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_value_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.

◆ shape_value_component()

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

virtualinherited

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.

◆ shape_grad()

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

virtualinherited

Return the gradient of the ith shape function at the point p. p is a point on the reference element, and likewise the gradient is the gradient 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_grad_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.

◆ shape_grad_component()

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

virtualinherited

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.

◆ shape_grad_grad()

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

virtualinherited

Return the tensor of second 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_grad_grad_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.

◆ shape_grad_grad_component()

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

virtualinherited

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.

◆ shape_3rd_derivative()

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

virtualinherited

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

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

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

◆ shape_3rd_derivative_component()

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

virtualinherited

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

◆ shape_4th_derivative()

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

virtualinherited

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

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

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

◆ shape_4th_derivative_component()

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

virtualinherited

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

◆ has_support_on_face()

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

virtualinherited

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

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

◆ get_restriction_matrix()

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

virtualinherited

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

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

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

Note: The term "restriction" is also used in the definition of multigrid methods, but in a different context. For the current function, we are interested in the interpolation of a finite element function from child cells to the parent cell, and this results in a situation where you have a function that is constant on a fine mesh, the function "restricted" to the coarse mesh will also be constant and have the same value. In contrast, in the multigrid context, the restriction operation is applied to residual (rather than solution) vectors, and is usually chosen as the transpose of the prolongation operation. For the multigrid restriction operation, one would consequently not typically expect that it maps a constant function to a constant function. In other words, the meaning of the term "restriction" in the current context and in the multigrid context is quite different, and the two should not be confused.

◆ get_prolongation_matrix()

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

virtualinherited

Prolongation/embedding matrix between grids.

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

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

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

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

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

◆ prolongation_is_implemented()

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

inherited

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

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

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

◆ isotropic_prolongation_is_implemented()

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

inherited

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

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

◆ restriction_is_implemented()

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

◆ get_interpolation_matrix()

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

virtualinherited

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

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

◆ get_face_interpolation_matrix()

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

virtualinherited

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

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

◆ get_subface_interpolation_matrix()

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

virtualinherited

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

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

◆ hp_vertex_dof_identities()

virtual std::vector< 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.

◆ hp_line_dof_identities()

virtual 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.

◆ hp_quad_dof_identities()

virtual 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.

◆ compare_for_domination()

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

virtualinherited

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

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

◆ operator==() [1/2]

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

virtualinherited

Comparison operator.

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

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

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

Note

This operator specifically does not check the following member variables of the current class:

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

◆ operator==() [2/2]

template<int dim>

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

inherited

Comparison operator.

Definition at line 195 of file fe_data.cc.

◆ operator!=()

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

inherited

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

◆ system_to_component_index()

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

inherited

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

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

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

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

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

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

◆ component_to_system_index()

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

inherited

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

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

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

◆ face_system_to_component_index()

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

inherited

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

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

◆ 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 unsigned char	combined_orientation
	)		const

inherited

Given a local dof index on a quad, return the local index accounting for the face orientation combined_orientation. This is only necessary in 3d: consequently, if this function is called in 1d or 2d then an exception is thrown.

◆ face_to_cell_index()

virtual unsigned int FiniteElement< dim, spacedim >::face_to_cell_index	(	const unsigned int	face_dof_index,
		const unsigned int	face,
		const unsigned char	combined_orientation = `ReferenceCell::default_combined_face_orientation()`
	)		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_index	The index of the degree of freedom on a face. This index must be between zero and dofs_per_face.
face	The number of the face this degree of freedom lives on. This number must be between zero and GeometryInfo::faces_per_cell.
combined_orientation	The combined orientation flag containing the orientation, rotation, and flip 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.

◆ adjust_line_dof_index_for_line_orientation()

unsigned int FiniteElement< dim, spacedim >::adjust_line_dof_index_for_line_orientation	(	const unsigned int	index,
		const unsigned char	combined_orientation
	)		const

inherited

Given a local dof index on a line and the orientation combined_orientation of that line, return the local dof which accounts for combined_orientation.

Note: In both 1d and 2d all-quadrilateral meshes all lines have the standard orientation. However, since 2d meshes may contain both quadrilaterals and triangles, this assumption cannot be made in this class.

◆ get_nonzero_components()

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

inherited

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

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

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

◆ n_nonzero_components()

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

inherited

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

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

◆ is_primitive() [1/2]

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

inherited

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

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

◆ is_primitive() [2/2]

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

inherited

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

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

◆ get_local_dof_sparsity_pattern()

virtual const Table< 2, bool > & FiniteElement< dim, spacedim >::get_local_dof_sparsity_pattern ( ) const

virtualinherited

Return a table (or sparsity pattern) with information which of the local DoFs per cell couple with each other.

If the FiniteElement returns an empty (0 by 0) table, which is the default implementation for any class that does not override this function, all DoFs are assumed to couple.

Otherwise, this function returns a square table with n_dofs_per_cell rows and columns. The entry (i,j) then denotes if the DoF i and j are coupled. This should be true if the support of the shape functions have a non-empty intersection.

An example for an element that does make use of this feature is FE_Q_iso_Q1.

◆ n_base_elements()

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

inherited

Number of base elements in a mixed discretization.

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

◆ base_element()

virtual const FiniteElement< 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.

◆ element_multiplicity()

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

inherited

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

◆ get_sub_fe() [1/2]

const FiniteElement< 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`

◆ get_sub_fe() [2/2]

virtual 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.

◆ 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

inherited

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

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

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

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

◆ face_system_to_base_index()

std::pair< 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

inherited

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

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

◆ first_block_of_base()

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

inherited

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

◆ component_to_base_index()

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

inherited

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

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

◆ block_to_base_index()

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

inherited

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

◆ system_to_block_index()

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

inherited

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

◆ component_to_block_index()

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

inherited

The vector block for this component.

◆ 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

scalar An object that represents a single scalar vector component of this finite element.

Returns: A component mask that is false in all components except for the one that corresponds to the argument.

◆ 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

vector An object that represents dim vector components of this finite element.

Returns: A component mask that is false in all components except for the ones that corresponds to the argument.

◆ 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_tensor An object that represents dim*(dim+1)/2 components of this finite element that are jointly to be interpreted as forming a symmetric tensor.

Returns: A component mask that is false in all components except for the ones that corresponds to the argument.

◆ 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_mask The 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.

◆ 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

scalar An object that represents a single scalar vector component of this finite element.

Returns: A component mask that is false in all components except for the one that corresponds to the argument.

◆ 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

vector An object that represents dim vector components of this finite element.

Returns: A component mask that is false in all components except for the ones that corresponds to the argument.

◆ 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_tensor An object that represents dim*(dim+1)/2 components of this finite element that are jointly to be interpreted as forming a symmetric tensor.

Returns: A component mask that is false in all components except for the ones that corresponds to the argument.

◆ 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_mask The 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.

◆ get_constant_modes()

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

virtualinherited

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

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

◆ get_unit_support_points()

const std::vector< 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.

◆ 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 an array with dofs_per_cell entries. Note that the function's name is poorly chosen: The function does not return whether an element has support points, but whether its implementation is able to provide a list of support points via get_unit_support_points().

The result may be false if a finite element defines itself not by interpolating shape functions, but by other means. A typical example are discontinuous \(P\)-type elements on quadrilaterals (rather than the common \(Q\)-type elements on quadrilaterals). Elements will generally only true if they construct their shape functions by the requirement that they be nonzero at a certain point and zero at all the points associated with the other shape functions. In other words, if the "node functionals" that form the dual space of the finite element space and are used to define the shape functions are point evaluations.

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 since it does not have support point (it has no shape functions, after all), and so returns an empty array from its get_unit_support_points() function. Nonetheless, because that array has the right size, it reports true in the current function.

◆ unit_support_point()

virtual 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.

◆ 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.

◆ 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 an array with dofs_per_face entries. Note that the function's name is poorly chosen: The function does not return whether an element has support points, but whether its implementation is able to provide a list of support points via get_unit_face_support_points().

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

◆ unit_face_support_point()

virtual 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.

◆ 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. As a consequence, it is possible to have fewer generalized support points than degrees of freedom in the element. An example is the element FESystem<dim>(FE_Q<dim>(1), 2), which has two copies of the \(Q_1\) element. In 2d, each copy has 4 degrees of freedom, and each copy has its support points in the four vertices of the cell. While the get_support_points() function would return a vector of size 8 in which each of the vertices is listed twice, this function strips out the duplicates and returns a vector of length 4 in which each vertex is listed only once. This is possible because the purpose of this function is to return a list of points so that it is possible to interpolate an arbitrary function onto the finite element space, and this is possible by knowing the two components of the function in question at the four vertices of the cell – it is not necessary to ask for this information twice at each vertex.

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

◆ 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 an array with minimal set of unique support points. Note that the function's name is poorly chosen: The function does not return whether an element has support points, but whether its implementation is able to provide a list of support points via get_unit_support_points().

See the glossary entry on generalized support points or the documentation for the has_support_points() function for more information.

◆ get_associated_geometry_primitive()

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

inherited

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

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

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

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

Parameters

[in] cell_dof_index The index of a shape function or degree of freedom. This index must be in the range [0,dofs_per_cell).

Note: The integer value of the object returned by this function equals the dimensionality of the object it describes, and can consequently be used in generic programming paradigms. For example, if a degree of freedom is associated with a vertex, then this function returns GeometryPrimitive::vertex, which has a numeric value of zero (the dimensionality of a vertex).

◆ convert_generalized_support_point_values_to_dof_values()

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

virtualinherited

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

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

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

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

Parameters

[in]	support_point_values	An array of size `dofs_per_cell` (which equals the number of points the get_generalized_support_points() function will return) where each element is a vector with as many entries as the element has vector components. This array should contain the values of a function at the generalized support points of the current element.
[out]	nodal_values	An array of size `dofs_per_cell` that contains the node functionals of the element applied to the given function.

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

◆ memory_consumption()

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

virtualinherited

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

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

◆ ExcShapeFunctionNotPrimitive()

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

staticinherited

Exception

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

◆ ExcFENotPrimitive()

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

staticinherited

Exception

◆ ExcUnitShapeValuesDoNotExist()

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

staticinherited

Exception

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

◆ ExcFEHasNoSupportPoints()

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

staticinherited

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

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

◆ ExcEmbeddingVoid()

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

staticinherited

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

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

◆ ExcProjectionVoid()

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

staticinherited

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

Exception

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

◆ ExcWrongInterfaceMatrixSize()

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

staticinherited

Exception

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

◆ ExcInterpolationNotImplemented()

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

staticinherited

Exception

◆ reinit_restriction_and_prolongation_matrices()

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

protectedinherited

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

Parameters

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

◆ interface_constraints_size()

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

protectedinherited

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

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

◆ compute_n_nonzero_components()

static std::vector< 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.

◆ 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 topic. See also the documentation of the InternalDataBase class.

Parameters

[in]	update_flags	A set of UpdateFlags values that describe what kind of information the FEValues object requests the finite element to compute. This set of flags may also include information that the finite element can not compute, e.g., flags that pertain to data produced by the mapping. An implementation of this function needs to set up all data fields in the returned object that are necessary to produce the finite-element related data specified by these flags, and may already pre-compute part of this information as discussed above. Elements may want to store these update flags (or a subset of these flags) in InternalDataBase::update_each so they know at the time when FiniteElement::fill_fe_values() is called what they are supposed to compute
[in]	mapping	A reference to the mapping used for computing values and derivatives of shape functions.
[in]	quadrature	A reference to the object that describes where the shape functions should be evaluated.
[out]	output_data	A reference to the object that FEValues will use in conjunction with the object returned here and where an implementation of FiniteElement::fill_fe_values() will place the requested information. This allows the current function to already pre-compute pieces of information that can be computed on the reference cell, as discussed above. FEValues guarantees that this output object and the object returned by the current function will always be used together.

Returns: A pointer to an object of a type derived from InternalDataBase and that derived classes can use to store scratch data that can be pre-computed, or for scratch arrays that then only need to be allocated once. The calling site assumes ownership of this object and will delete it when it is no longer necessary.

◆ get_face_data() [2/3]

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

protectedvirtualinherited

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

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

Parameters

[in]	update_flags	A set of UpdateFlags values that describe what kind of information the FEValues object requests the finite element to compute. This set of flags may also include information that the finite element can not compute, e.g., flags that pertain to data produced by the mapping. An implementation of this function needs to set up all data fields in the returned object that are necessary to produce the finite-element related data specified by these flags, and may already pre-compute part of this information as discussed above. Elements may want to store these update flags (or a subset of these flags) in InternalDataBase::update_each so they know at the time when FiniteElement::fill_fe_face_values() is called what they are supposed to compute
[in]	mapping	A reference to the mapping used for computing values and derivatives of shape functions.
[in]	quadrature	A reference to the object that describes where the shape functions should be evaluated.
[out]	output_data	A reference to the object that FEValues will use in conjunction with the object returned here and where an implementation of FiniteElement::fill_fe_face_values() will place the requested information. This allows the current function to already pre-compute pieces of information that can be computed on the reference cell, as discussed above. FEValues guarantees that this output object and the object returned by the current function will always be used together.

Returns: A pointer to an object of a type derived from InternalDataBase and that derived classes can use to store scratch data that can be pre-computed, or for scratch arrays that then only need to be allocated once. The calling site assumes ownership of this object and will delete it when it is no longer necessary.

◆ get_face_data() [3/3]

virtual std::unique_ptr< 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

Deprecated:: Use the version taking a hp::QCollection argument.

◆ get_subface_data() [2/2]

virtual std::unique_ptr< 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_flags	A set of UpdateFlags values that describe what kind of information the FEValues object requests the finite element to compute. This set of flags may also include information that the finite element can not compute, e.g., flags that pertain to data produced by the mapping. An implementation of this function needs to set up all data fields in the returned object that are necessary to produce the finite-element related data specified by these flags, and may already pre-compute part of this information as discussed above. Elements may want to store these update flags (or a subset of these flags) in InternalDataBase::update_each so they know at the time when FiniteElement::fill_fe_subface_values() is called what they are supposed to compute
[in]	mapping	A reference to the mapping used for computing values and derivatives of shape functions.
[in]	quadrature	A reference to the object that describes where the shape functions should be evaluated.
[out]	output_data	A reference to the object that FEValues will use in conjunction with the object returned here and where an implementation of FiniteElement::fill_fe_subface_values() will place the requested information. This allows the current function to already pre-compute pieces of information that can be computed on the reference cell, as discussed above. FEValues guarantees that this output object and the object returned by the current function will always be used together.

Returns: A pointer to an object of a type derived from InternalDataBase and that derived classes can use to store scratch data that can be pre-computed, or for scratch arrays that then only need to be allocated once. The calling site assumes ownership of this object and will delete it when it is no longer necessary.

◆ 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 topic.

Parameters

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

Note: FEValues ensures that this function is always called with the same pair of fe_internal and output_data objects. In other words, if an implementation of this function knows that it has written a piece of data into the output argument in a previous call, then there is no need to copy it there again in a later call if the implementation knows that this is the same value.

◆ fill_fe_face_values() [2/3]

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

protectedvirtualinherited

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

Parameters

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

◆ fill_fe_face_values() [3/3]

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

protectedvirtualinherited

Deprecated:: Use the version taking a hp::QCollection argument.

◆ fill_fe_subface_values() [2/2]

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

protectedpure virtualinherited

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

Parameters

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

◆ subscribe()

void EnableObserverPointer::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 131 of file enable_observer_pointer.cc.

◆ unsubscribe()

void EnableObserverPointer::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 151 of file enable_observer_pointer.cc.

◆ n_subscriptions()

unsigned int EnableObserverPointer::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 322 of file enable_observer_pointer.h.

◆ list_subscribers() [1/2]

template<typename StreamType >

void EnableObserverPointer::list_subscribers ( StreamType & stream ) const

inlineinherited

List the subscribers to the input stream.

Definition at line 339 of file enable_observer_pointer.h.

◆ list_subscribers() [2/2]

void EnableObserverPointer::list_subscribers ( ) const

inherited

List the subscribers to deallog.

Definition at line 199 of file enable_observer_pointer.cc.

◆ serialize()

template<class Archive >

void EnableObserverPointer::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 using the BOOST serialization library.

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 331 of file enable_observer_pointer.h.

◆ check_no_subscribers()

void EnableObserverPointer::check_no_subscribers ( ) const

privatenoexceptinherited

Check that there are no objects subscribing to this object. If this check passes then it is safe to destroy the current object. It this check fails then this function will either abort or print an error message to deallog (by using the AssertNothrow mechanism), but will not throw an exception.

Note: Since this function is just a consistency check it does nothing in release mode.; If this function is called when there is an uncaught exception then, rather than aborting, this function prints an error message to the standard error stream and returns.

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.

Definition at line 661 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 2450 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 2464 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 2476 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 2488 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 2495 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 2501 of file fe.h.

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 2542 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 2554 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 2573 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 2580 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 2586 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 2609 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 2616 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 2625 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 2634 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 2641 of file fe.h.

◆ local_dof_sparsity_pattern

Table<2, bool> FiniteElement< dim, spacedim >::local_dof_sparsity_pattern

protectedinherited

This table is returned by get_local_dof_sparsity_pattern(). Its meaning is described there in detail. This variable has to be filled by any class derived from FiniteElement that does not want to keep the default empty table (meaning all DoFs couple within the cell).

Definition at line 2649 of file fe.h.

◆ counter

std::atomic<unsigned int> EnableObserverPointer::counter

mutableprivateinherited

Store the number of objects which subscribed to this object. Initially, this number is zero, and upon destruction it shall be zero again (i.e. all objects which subscribed should have unsubscribed again).

The creator (and owner) of an object is counted in the map below if HE manages to supply identification.

We use the mutable keyword in order to allow subscription to constant objects also.

This counter may be read from and written to concurrently in multithreaded code: hence we use the std::atomic class template.

Definition at line 227 of file enable_observer_pointer.h.

◆ counter_map

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

mutableprivateinherited

In this map, we count subscriptions for each different identification string supplied to subscribe().

Definition at line 233 of file enable_observer_pointer.h.

◆ validity_pointers

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

mutableprivateinherited

In this vector, we store pointers to the validity bool in the ObserverPointer objects that subscribe to this class.

Definition at line 249 of file enable_observer_pointer.h.

◆ object_info

const std::type_info* EnableObserverPointer::object_info

mutableprivateinherited

Pointer to the typeinfo object of this object, from which we can later deduce the class name. Since this information on the derived class is neither available in the destructor, nor in the constructor, we obtain it in between and store it here.

Definition at line 257 of file enable_observer_pointer.h.

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 464 of file fe_data.h.

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

include/deal.II/fe/fe_p1nc.h
source/fe/fe_p1nc.cc

Public Types

Public Member Functions

Static Public Member Functions

Public Attributes

Static Public Attributes

Protected Member Functions

Static Protected Member Functions

Protected Attributes

Private Types

Private Member Functions

Static Private Member Functions

Private Attributes

Static Private Attributes

Detailed Description

Dice Rule

Shape functions

Degrees of freedom

Unit support points

Reference

Member Typedef Documentation

◆ map_value_type

◆ map_iterator

Member Enumeration Documentation

◆ Conformity

Constructor & Destructor Documentation

◆ FE_P1NC()

◆ ~FE_P1NC()

Member Function Documentation

◆ get_name()

◆ requires_update_flags()

◆ clone()

◆ get_dpo_vector()

◆ get_linear_shape_coefficients()

◆ get_data() [1/2]

◆ get_face_data() [1/3]

◆ get_subface_data() [1/2]

◆ fill_fe_values() [1/2]

◆ fill_fe_face_values() [1/3]

◆ fill_fe_subface_values() [1/2]

◆ initialize_constraints()

◆ operator^()

◆ shape_value()

◆ shape_value_component()

◆ shape_grad()

◆ shape_grad_component()

◆ shape_grad_grad()

◆ shape_grad_grad_component()

◆ shape_3rd_derivative()

◆ shape_3rd_derivative_component()

◆ shape_4th_derivative()

◆ shape_4th_derivative_component()

◆ has_support_on_face()

◆ get_restriction_matrix()

◆ get_prolongation_matrix()

◆ prolongation_is_implemented()

◆ isotropic_prolongation_is_implemented()

◆ restriction_is_implemented()

◆ isotropic_restriction_is_implemented()

◆ restriction_is_additive()

◆ constraints()

◆ constraints_are_implemented()

◆ hp_constraints_are_implemented()

◆ get_interpolation_matrix()

◆ get_face_interpolation_matrix()

◆ get_subface_interpolation_matrix()

◆ hp_vertex_dof_identities()

◆ hp_line_dof_identities()

◆ hp_quad_dof_identities()

◆ compare_for_domination()

◆ operator==() [1/2]

◆ operator==() [2/2]

◆ operator!=()

◆ system_to_component_index()

◆ component_to_system_index()

◆ face_system_to_component_index()

◆ adjust_quad_dof_index_for_face_orientation()

◆ face_to_cell_index()

◆ adjust_line_dof_index_for_line_orientation()

◆ get_nonzero_components()

◆ n_nonzero_components()