Reference documentation for deal.II version 9.3.0

#include <deal.II/fe/fe_q.h>
Public Types  
enum  Conformity { unknown = 0x00, L2 = 0x01, Hcurl = 0x02, Hdiv = 0x04, H1 = Hcurl  Hdiv, H2 = 0x0e } 
Public Member Functions  
FE_Q (const unsigned int p)  
FE_Q (const Quadrature< 1 > &points)  
virtual std::string  get_name () const override 
virtual std::unique_ptr< FiniteElement< dim, spacedim > >  clone () const override 
virtual void  convert_generalized_support_point_values_to_dof_values (const std::vector< Vector< double >> &support_point_values, std::vector< double > &nodal_values) const override 
virtual FiniteElementDomination::Domination  compare_for_domination (const FiniteElement< dim, spacedim > &fe_other, const unsigned int codim=0) const override final 
virtual void  get_interpolation_matrix (const FiniteElement< dim, spacedim > &source, FullMatrix< double > &matrix) const override 
virtual void  get_face_interpolation_matrix (const FiniteElement< dim, spacedim > &source, FullMatrix< double > &matrix, const unsigned int face_no=0) const override 
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 override 
virtual bool  has_support_on_face (const unsigned int shape_index, const unsigned int face_index) const override 
virtual const FullMatrix< double > &  get_restriction_matrix (const unsigned int child, const RefinementCase< dim > &refinement_case=RefinementCase< dim >::isotropic_refinement) const override 
virtual const FullMatrix< double > &  get_prolongation_matrix (const unsigned int child, const RefinementCase< dim > &refinement_case=RefinementCase< dim >::isotropic_refinement) const override 
virtual unsigned int  face_to_cell_index (const unsigned int face_dof_index, const unsigned int face, const bool face_orientation=true, const bool face_flip=false, const bool face_rotation=false) const override 
virtual std::pair< Table< 2, bool >, std::vector< unsigned int > >  get_constant_modes () const override 
unsigned int  get_degree () const 
virtual UpdateFlags  requires_update_flags (const UpdateFlags update_flags) const override 
const ScalarPolynomialsBase< dim > &  get_poly_space () const 
std::vector< unsigned int >  get_poly_space_numbering () const 
std::vector< unsigned int >  get_poly_space_numbering_inverse () const 
virtual double  shape_value (const unsigned int i, const Point< dim > &p) const override 
virtual double  shape_value_component (const unsigned int i, const Point< dim > &p, const unsigned int component) const override 
virtual Tensor< 1, dim >  shape_grad (const unsigned int i, const Point< dim > &p) const override 
virtual Tensor< 1, dim >  shape_grad_component (const unsigned int i, const Point< dim > &p, const unsigned int component) const override 
virtual Tensor< 2, dim >  shape_grad_grad (const unsigned int i, const Point< dim > &p) const override 
virtual Tensor< 2, dim >  shape_grad_grad_component (const unsigned int i, const Point< dim > &p, const unsigned int component) const override 
virtual Tensor< 3, dim >  shape_3rd_derivative (const unsigned int i, const Point< dim > &p) const override 
virtual Tensor< 3, dim >  shape_3rd_derivative_component (const unsigned int i, const Point< dim > &p, const unsigned int component) const override 
virtual Tensor< 4, dim >  shape_4th_derivative (const unsigned int i, const Point< dim > &p) const override 
virtual Tensor< 4, dim >  shape_4th_derivative_component (const unsigned int i, const Point< dim > &p, const unsigned int component) const override 
virtual std::size_t  memory_consumption () const override 
std::pair< std::unique_ptr< FiniteElement< dim, spacedim > >, unsigned int >  operator^ (const unsigned int multiplicity) const 
const FiniteElement< dim, spacedim > &  operator[] (const unsigned int fe_index) const 
virtual bool  operator== (const FiniteElement< dim, spacedim > &fe) const 
bool  operator== (const FiniteElementData &) const 
bool  operator!= (const FiniteElement< dim, spacedim > &) const 
template<class Archive >  
void  serialize (Archive &ar, const unsigned int version) 
ReferenceCell  reference_cell () const 
unsigned int  n_unique_quads () const 
unsigned int  n_unique_faces () const 
unsigned int  n_dofs_per_vertex () const 
unsigned int  n_dofs_per_line () const 
unsigned int  n_dofs_per_quad (unsigned int face_no=0) const 
unsigned int  max_dofs_per_quad () const 
unsigned int  n_dofs_per_hex () const 
unsigned int  n_dofs_per_face (unsigned int face_no=0, unsigned int child=0) const 
unsigned int  max_dofs_per_face () const 
unsigned int  n_dofs_per_cell () const 
template<int structdim>  
unsigned int  n_dofs_per_object (const unsigned int i=0) const 
unsigned int  n_components () const 
unsigned int  n_blocks () const 
const 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 
Functions to support hp  
virtual bool  hp_constraints_are_implemented () const override 
virtual std::vector< std::pair< unsigned int, unsigned int > >  hp_vertex_dof_identities (const FiniteElement< dim, spacedim > &fe_other) const override 
virtual std::vector< std::pair< unsigned int, unsigned int > >  hp_line_dof_identities (const FiniteElement< dim, spacedim > &fe_other) const override 
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 override 
Transfer and constraint matrices  
bool  prolongation_is_implemented () const 
bool  isotropic_prolongation_is_implemented () const 
bool  restriction_is_implemented () const 
bool  isotropic_restriction_is_implemented () const 
bool  restriction_is_additive (const unsigned int index) const 
const FullMatrix< double > &  constraints (const ::internal::SubfaceCase< dim > &subface_case=::internal::SubfaceCase< dim >::case_isotropic) const 
bool  constraints_are_implemented (const ::internal::SubfaceCase< dim > &subface_case=::internal::SubfaceCase< dim >::case_isotropic) const 
Index computations  
std::pair< unsigned int, unsigned int >  system_to_component_index (const unsigned int index) const 
unsigned int  component_to_system_index (const unsigned int component, const unsigned int index) const 
std::pair< unsigned int, unsigned int >  face_system_to_component_index (const unsigned int index, const unsigned int face_no=0) const 
unsigned int  adjust_quad_dof_index_for_face_orientation (const unsigned int index, const unsigned int face_no, const bool face_orientation, const bool face_flip, const bool face_rotation) const 
unsigned int  adjust_line_dof_index_for_line_orientation (const unsigned int index, const bool line_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 
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 
Support points and interpolation  
const std::vector< Point< dim > > &  get_unit_support_points () const 
bool  has_support_points () const 
virtual Point< dim >  unit_support_point (const unsigned int index) const 
const std::vector< Point< dim  1 > > &  get_unit_face_support_points (const unsigned int face_no=0) const 
bool  has_face_support_points (const unsigned int face_no=0) const 
virtual Point< dim  1 >  unit_face_support_point (const unsigned int index, const unsigned int face_no=0) const 
const std::vector< Point< dim > > &  get_generalized_support_points () const 
bool  has_generalized_support_points () const 
GeometryPrimitive  get_associated_geometry_primitive (const unsigned int cell_dof_index) const 
Subscriptor functionality  
Classes derived from Subscriptor provide a facility to subscribe to this object. This is mostly used by the SmartPointer class.  
void  subscribe (std::atomic< bool > *const validity, const std::string &identifier="") const 
void  unsubscribe (std::atomic< bool > *const validity, const std::string &identifier="") const 
unsigned int  n_subscriptions () const 
template<typename StreamType >  
void  list_subscribers (StreamType &stream) const 
void  list_subscribers () const 
Static Public Member Functions  
static ::ExceptionBase &  ExcFEQCannotHaveDegree0 () 
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 const unsigned int  space_dimension = spacedim 
static const unsigned int  dimension = dim 
Protected Member Functions  
void  initialize (const std::vector< Point< 1 >> &support_points_1d) 
void  initialize_constraints (const std::vector< Point< 1 >> &points) 
void  initialize_unit_support_points (const std::vector< Point< 1 >> &points) 
void  initialize_unit_face_support_points (const std::vector< Point< 1 >> &points) 
void  initialize_quad_dof_index_permutation () 
virtual std::unique_ptr< typename FiniteElement< dim, spacedim >::InternalDataBase >  get_data (const UpdateFlags update_flags, const Mapping< dim, spacedim > &mapping, const Quadrature< dim > &quadrature, ::internal::FEValuesImplementation::FiniteElementRelatedData< dim, spacedim > &output_data) const override 
virtual void  fill_fe_values (const typename Triangulation< dim, spacedim >::cell_iterator &cell, const CellSimilarity::Similarity cell_similarity, const Quadrature< dim > &quadrature, const Mapping< dim, spacedim > &mapping, const typename Mapping< dim, spacedim >::InternalDataBase &mapping_internal, const ::internal::FEValuesImplementation::MappingRelatedData< dim, spacedim > &mapping_data, const typename FiniteElement< dim, spacedim >::InternalDataBase &fe_internal, ::internal::FEValuesImplementation::FiniteElementRelatedData< dim, spacedim > &output_data) const override 
virtual void  fill_fe_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 typename FiniteElement< dim, spacedim >::InternalDataBase &fe_internal, ::internal::FEValuesImplementation::FiniteElementRelatedData< dim, spacedim > &output_data) const override 
virtual void  fill_fe_face_values (const typename Triangulation< dim, spacedim >::cell_iterator &cell, const unsigned int face_no, const hp::QCollection< dim  1 > &quadrature, const Mapping< dim, spacedim > &mapping, const typename Mapping< dim, spacedim >::InternalDataBase &mapping_internal, const ::internal::FEValuesImplementation::MappingRelatedData< dim, spacedim > &mapping_data, const 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 typename FiniteElement< dim, spacedim >::InternalDataBase &fe_internal, ::internal::FEValuesImplementation::FiniteElementRelatedData< dim, spacedim > &output_data) const override 
virtual void  fill_fe_subface_values (const typename Triangulation< dim, spacedim >::cell_iterator &cell, const unsigned int face_no, const unsigned int sub_no, const Quadrature< dim  1 > &quadrature, const Mapping< dim, spacedim > &mapping, const typename Mapping< dim, spacedim >::InternalDataBase &mapping_internal, const ::internal::FEValuesImplementation::MappingRelatedData< dim, spacedim > &mapping_data, const InternalDataBase &fe_internal, ::internal::FEValuesImplementation::FiniteElementRelatedData< dim, spacedim > &output_data) const =0 
void  correct_hessians (internal::FEValuesImplementation::FiniteElementRelatedData< dim, spacedim > &output_data, const internal::FEValuesImplementation::MappingRelatedData< dim, spacedim > &mapping_data, const unsigned int n_q_points) const 
void  correct_third_derivatives (internal::FEValuesImplementation::FiniteElementRelatedData< dim, spacedim > &output_data, const internal::FEValuesImplementation::MappingRelatedData< dim, spacedim > &mapping_data, const unsigned int n_q_points) const 
void  reinit_restriction_and_prolongation_matrices (const bool isotropic_restriction_only=false, const bool isotropic_prolongation_only=false) 
TableIndices< 2 >  interface_constraints_size () const 
virtual std::unique_ptr< 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 
Static Protected Member Functions  
static std::vector< unsigned int >  get_dpo_vector (const unsigned int degree) 
static std::vector< unsigned int >  compute_n_nonzero_components (const std::vector< ComponentMask > &nonzero_components) 
Protected Attributes  
const std::unique_ptr< ScalarPolynomialsBase< dim > >  poly_space 
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 
Implementation of a scalar Lagrange finite element Qp
that yields the finite element space of continuous, piecewise polynomials of degree p
in each coordinate direction. This class is realized using tensor product polynomials based on 1D Lagrange polynomials with equidistant (degree up to 2), GaussLobatto (starting from degree 3), or given support points.
The standard constructor of this class takes the degree p
of this finite element. Alternatively, it can take a quadrature formula points
defining the support points of the Lagrange interpolation in one coordinate direction.
For more information about the spacedim
template parameter check the documentation of FiniteElement or the one of Triangulation.
The constructor creates a TensorProductPolynomials object that includes the tensor product of LagrangeEquidistant
polynomials of degree p
. This TensorProductPolynomials
object provides all values and derivatives of the shape functions. In case a quadrature rule is given, the constructor creates a TensorProductPolynomials object that includes the tensor product of Lagrange
polynomials with the support points from points
.
Furthermore the constructor fills the interface_constraints
, the prolongation
(embedding) and the restriction
matrices. These are implemented only up to a certain degree and may not be available for very high polynomial degree.
When constructing an FE_Q element at polynomial degrees one or two, equidistant support points at 0 and 1 (linear case) or 0, 0.5, and 1 (quadratic case) are used. The unit support or nodal points x_{i} are those points where the jth Lagrange polynomial satisfies the \(\delta_{ij}\) property, i.e., where one polynomial is one and all the others are zero. For higher polynomial degrees, the support points are nonequidistant by default, and chosen to be the support points of the (degree+1)
order GaussLobatto quadrature rule. This point distribution yields wellconditioned Lagrange interpolation at arbitrary polynomial degrees. By contrast, polynomials based on equidistant points get increasingly illconditioned as the polynomial degree increases. In interpolation, this effect is known as the Runge phenomenon. For Galerkin methods, the Runge phenomenon is typically not visible in the solution quality but rather in the condition number of the associated system matrices. For example, the elemental mass matrix of equidistant points at degree 10 has condition number 2.6e6, whereas the condition number for GaussLobatto points is around 400.
The GaussLobatto points in 1D include the end points 0 and +1 of the unit interval. The interior points are shifted towards the end points, which gives a denser point distribution close to the element boundary.
If combined with GaussLobatto quadrature, FE_Q based on the default support points gives diagonal mass matrices. This case is demonstrated in step48. However, this element can be combined with arbitrary quadrature rules through the usual FEValues approach, including full Gauss quadrature. In the general case, the mass matrix is nondiagonal.
The original ordering of the shape functions represented by the TensorProductPolynomials is a tensor product numbering. However, the shape functions on a cell are renumbered beginning with the shape functions whose support points are at the vertices, then on the line, on the quads, and finally (for 3d) on the hexes. To be explicit, these numberings are listed in the following:
1D case:
* 01 *
2D case:
* 23 *   *   *   * 01 *
3D case:
* 67 67 * /  / / * /   / /  * /   / /  * 4   45  *  23   3 *  / /   / *  / /   / * / /  / * 01 01 *
The respective coordinate values of the support points of the shape functions are as follows:
[0, 0, 0]
; [1, 0, 0]
; [0, 1, 0]
; [1, 1, 0]
; [0, 0, 1]
; [1, 0, 1]
; [0, 1, 1]
; [1, 1, 1]
; In 2d, these shape functions look as follows:
\(Q_1\) element, shape function 0  \(Q_1\) element, shape function 1 
\(Q_1\) element, shape function 2  \(Q_1\) element, shape function 3 
1D case:
* 021 *
2D case:
* 273 *   * 4 8 5 *   * 061 *
3D case:
* 6157 6157 * /  / / * 12  19 12 1319 * / 18  / /  * 4   4145  *  2113   3 *  / /  17 / * 16 8 9 16  9 * / /  / * 0101 0101 * * ** ** * /  / / * /  23  / 25 /  * /   / /  * *   **  * 20 **  21 * *  / /  22  / *  / 24 /   / * / /  / * ** ** *
The center vertex has number 26.
The respective coordinate values of the support points of the shape functions are as follows:
[0, 0, 0]
; [1, 0, 0]
; [0, 1, 0]
; [1, 1, 0]
; [0, 0, 1]
; [1, 0, 1]
; [0, 1, 1]
; [1, 1, 1]
; [0, 1/2, 0]
; [1, 1/2, 0]
; [1/2, 0, 0]
; [1/2, 1, 0]
; [0, 1/2, 1]
; [1, 1/2, 1]
; [1/2, 0, 1]
; [1/2, 1, 1]
; [0, 0, 1/2]
; [1, 0, 1/2]
; [0, 1, 1/2]
; [1, 1, 1/2]
; [0, 1/2, 1/2]
; [1, 1/2, 1/2]
; [1/2, 0, 1/2]
; [1/2, 1, 1/2]
; [1/2, 1/2, 0]
; [1/2, 1/2, 1]
; [1/2, 1/2, 1/2]
; In 2d, these shape functions look as follows (the black plane corresponds to zero; negative shape function values may not be visible):
\(Q_2\) element, shape function 0  \(Q_2\) element, shape function 1 
\(Q_2\) element, shape function 2  \(Q_2\) element, shape function 3 
\(Q_2\) element, shape function 4  \(Q_2\) element, shape function 5 
\(Q_2\) element, shape function 6  \(Q_2\) element, shape function 7 
\(Q_2\) element, shape function 8 
1D case:
* 0231 *
* 210113 *   * 5 14 15 7 *   * 4 12 13 6 *   * 0891 *
In 2d, these shape functions look as follows (the black plane corresponds to zero; negative shape function values may not be visible):
\(Q_3\) element, shape function 0  \(Q_3\) element, shape function 1 
\(Q_3\) element, shape function 2  \(Q_3\) element, shape function 3 
\(Q_3\) element, shape function 4  \(Q_3\) element, shape function 5 
\(Q_3\) element, shape function 6  \(Q_3\) element, shape function 7 
\(Q_3\) element, shape function 8  \(Q_3\) element, shape function 9 
\(Q_3\) element, shape function 10  \(Q_3\) element, shape function 11 
\(Q_3\) element, shape function 12  \(Q_3\) element, shape function 13 
\(Q_3\) element, shape function 14  \(Q_3\) element, shape function 15 
1D case:
* 02341 *
* 21314153 *   * 6 22 23 24 9 *   * 5 19 20 21 8 *   * 4 16 17 18 7 *   * 01011121 *
In 2d, these shape functions look as follows (the black plane corresponds to zero; negative shape function values may not be visible):
\(Q_4\) element, shape function 0  \(Q_4\) element, shape function 1 
\(Q_4\) element, shape function 2  \(Q_4\) element, shape function 3 
\(Q_4\) element, shape function 4  \(Q_4\) element, shape function 5 
\(Q_4\) element, shape function 6  \(Q_4\) element, shape function 7 
\(Q_4\) element, shape function 8  \(Q_4\) element, shape function 9 
\(Q_4\) element, shape function 10  \(Q_4\) element, shape function 11 
\(Q_4\) element, shape function 12  \(Q_4\) element, shape function 13 
\(Q_4\) element, shape function 14  \(Q_4\) element, shape function 15 
\(Q_4\) element, shape function 16  \(Q_4\) element, shape function 17 
\(Q_4\) element, shape function 18  \(Q_4\) element, shape function 19 
\(Q_4\) element, shape function 20  \(Q_4\) element, shape function 21 
\(Q_4\) element, shape function 22  \(Q_4\) element, shape function 23 
\(Q_4\) element, shape function 24 

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 H^{1} is in H^{curl} and H^{div} as well.
If you are interested in continuity in the classical sense, then the following relations hold:
H^{1} implies that the function is continuous over cell boundaries.
H^{2} implies that the function is continuously differentiable over cell boundaries.
In order to test if a finite element conforms to a certain space, use FiniteElementData<dim>::conforms().
FE_Q< dim, spacedim >::FE_Q  (  const Quadrature< 1 > &  points  ) 
Constructor for tensor product polynomials with support points points
based on a onedimensional quadrature formula. The degree of the finite element is points.size()1
. Note that the first point has to be 0 and the last one 1. Constructing FE_Q<dim>(QGaussLobatto<1>(fe_degree+1))
is equivalent to the constructor that specifies the polynomial degree only. For selecting equidistant nodes at fe_degree > 2
, construct FE_Q<dim>(QIterated<1>(QTrapezoid<1>(),fe_degree))
.
The space created by this constructor is the same as if you had called FE_Q<dim>(points.size()1)
, but what is different are the basis functions of this space. This is useful in contexts where one wants to achieve certain properties of the matrices that result from integrating bilinear forms of these shape functions. For example, when computing the mass matrix on cell \(K\),
\[ M_{ij}^K = \int_K \varphi_i(\mathbf x) \varphi_j(\mathbf x) \; dx \]
one generally applies a quadrature formula and approximates the real mass matrix by the following:
\[ M_{ij}^K = \sum_q \varphi_i(\mathbf x_q) \varphi_j(\mathbf x_q) w_q, \]
where the locations of the quadrature points \(\mathbf x_q\) and the weights \(w_q\) depend on the cell \(K\). If the quadrature formula used to define these points \(\mathbf x_q\) is the same as the one passed to this constructor (or, in higher dimensions, is generated by a tensor product of the one used to the constructor), then \(\varphi_i(\mathbf x_q) = \delta_{iq}\), and the matrix reduces to
\[ M_{ij}^K = \sum_q \delta_{iq} \delta_{jq} w_q = \delta_{ij} w_i, \]
i.e., a diagonal matrix with the weights \(w_i\) on the diagonal. Such constructions are quite useful when using explicit time stepping methods because, for example, one then only has to invert a diagonal mass matrix in each time step when solving the linear system.

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

overridevirtual 
A sort of virtual copy constructor, this function returns a copy of the finite element object. Derived classes need to override the function here in this base class and return an object of the same type as the derived class.
Some places in the library, for example the constructors of FESystem as well as the hp::FECollection class, need to make copies of finite elements without knowing their exact type. They do so through this function.
Implements FiniteElement< dim, spacedim >.

overridevirtual 
Implementation of the corresponding function in the FiniteElement class. Since the current element is interpolatory, the nodal values are exactly the support point values. Furthermore, since the current element is scalar, the support point values need to be vectors of length 1.
Reimplemented from FiniteElement< dim, spacedim >.

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

overridevirtualinherited 
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.n_dofs_per_cell()
.
These matrices are only available if the source element is also a FE_Q
element. Otherwise, an exception of type FiniteElement<dim,spacedim>::ExcInterpolationNotImplemented is thrown.
Reimplemented from FiniteElement< dim, spacedim >.
Reimplemented in FE_Q_DG0< dim, spacedim >, FE_Q_Bubbles< dim, spacedim >, and FE_Bernstein< dim, spacedim >.
Definition at line 511 of file fe_q_base.cc.

overridevirtualinherited 
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
. The FE_Q element family only provides interpolation matrices for elements of the same type and FE_Nothing. For all other elements, an exception of type FiniteElement<dim,spacedim>::ExcInterpolationNotImplemented is thrown.
Reimplemented from FiniteElement< dim, spacedim >.
Reimplemented in FE_Bernstein< dim, spacedim >.
Definition at line 609 of file fe_q_base.cc.

overridevirtualinherited 
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
. The FE_Q element family only provides interpolation matrices for elements of the same type and FE_Nothing. For all other elements, an exception of type FiniteElement<dim,spacedim>::ExcInterpolationNotImplemented is thrown.
Reimplemented from FiniteElement< dim, spacedim >.
Reimplemented in FE_Bernstein< dim, spacedim >.
Definition at line 625 of file fe_q_base.cc.

overridevirtualinherited 
This function returns true
, if the shape function shape_index
has nonzero function values somewhere on the face face_index
.
Reimplemented from FiniteElement< dim, spacedim >.
Reimplemented in FE_Q_DG0< dim, spacedim >, and FE_Q_Bubbles< dim, spacedim >.
Definition at line 1583 of file fe_q_base.cc.

overridevirtualinherited 
Projection from a fine grid space onto a coarse grid space. Overrides the respective method in FiniteElement, implementing lazy evaluation (initialize when requested).
If this projection operator is associated with a matrix P
, then the restriction of this matrix P_i
to a single child cell is returned here.
The matrix P
is the concatenation or the sum of the cell matrices P_i
, depending on the restriction_is_additive_flags. This distinguishes interpolation (concatenation) and projection with respect to scalar products (summation).
Row and column indices are related to coarse grid and fine grid spaces, respectively, consistent with the definition of the associated operator.
If projection matrices are not implemented in the derived finite element class, this function aborts with ExcProjectionVoid. You can check whether this is the case by calling the restriction_is_implemented() or the isotropic_restriction_is_implemented() function.
Reimplemented from FiniteElement< dim, spacedim >.
Reimplemented in FE_Q_Bubbles< dim, spacedim >, and FE_Bernstein< dim, spacedim >.
Definition at line 1439 of file fe_q_base.cc.

overridevirtualinherited 
Embedding matrix between grids. Overrides the respective method in FiniteElement, implementing lazy evaluation (initialize when queried).
The identity operator from a coarse grid space into a fine grid space is associated with a matrix P
. The restriction of this matrix P_i
to a single child cell is returned here.
The matrix P
is the concatenation, not the sum of the cell matrices P_i
. That is, if the same nonzero 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 multilevel 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 projection matrices are not implemented in the derived finite element class, this function aborts with ExcEmbeddingVoid. You can check whether this is the case by calling the prolongation_is_implemented() or the isotropic_prolongation_is_implemented() function.
Reimplemented from FiniteElement< dim, spacedim >.
Reimplemented in FE_Q_Bubbles< dim, spacedim >, and FE_Bernstein< dim, spacedim >.
Definition at line 1240 of file fe_q_base.cc.

overridevirtualinherited 
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:
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.
face_dof_index  The index of the degree of freedom on a face. This index must be between zero and dofs_per_face. 
face  The number of the face this degree of freedom lives on. This number must be between zero and GeometryInfo::faces_per_cell. 
face_orientation  One part of the description of the orientation of the face. See GlossFaceOrientation. 
face_flip  One part of the description of the orientation of the face. See GlossFaceOrientation. 
face_rotation  One part of the description of the orientation of the face. See GlossFaceOrientation. 
Reimplemented from FiniteElement< dim, spacedim >.
Definition at line 1101 of file fe_q_base.cc.

overridevirtualinherited 
Return a list of constant modes of the element. For this element, the list consists of true arguments for all components.
Reimplemented from FiniteElement< dim, spacedim >.
Reimplemented in FE_Q_DG0< dim, spacedim >.
Definition at line 1685 of file fe_q_base.cc.

overridevirtualinherited 
Return whether this element implements its hanging node constraints in the new way, which has to be used to make elements "hpcompatible".
For the FE_Q class the result is always true (independent of the degree of the element), as it implements the complete set of functions necessary for hpcapability.
Reimplemented from FiniteElement< dim, spacedim >.
Reimplemented in FE_Bernstein< dim, spacedim >.
Definition at line 729 of file fe_q_base.cc.

overridevirtualinherited 
If, on a vertex, several finite elements are active, the hpcode 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.
Reimplemented from FiniteElement< dim, spacedim >.
Reimplemented in FE_Bernstein< dim, spacedim >.
Definition at line 738 of file fe_q_base.cc.

overridevirtualinherited 
Same as hp_vertex_dof_indices(), except that the function treats degrees of freedom on lines.
Reimplemented from FiniteElement< dim, spacedim >.
Reimplemented in FE_Bernstein< dim, spacedim >.
Definition at line 782 of file fe_q_base.cc.

overridevirtualinherited 
Same as hp_vertex_dof_indices(), except that the function treats degrees of freedom on quads.
Reimplemented from FiniteElement< dim, spacedim >.
Reimplemented in FE_Bernstein< dim, spacedim >.
Definition at line 877 of file fe_q_base.cc.

staticprotectedinherited 
Only for internal use. Its full name is get_dofs_per_object_vector
function and it creates the dofs_per_object
vector that is needed within the constructor to be passed to the constructor of FiniteElementData
.
Definition at line 1216 of file fe_q_base.cc.

protectedinherited 
Perform the initialization of the element based on 1D support points, i.e., sets renumbering, initializes unit support points, initializes constraints as well as restriction and prolongation matrices.
Definition at line 435 of file fe_q_base.cc.

protectedinherited 
Initialize the hanging node constraints matrices. Called from initialize().
Definition at line 1230 of file fe_q_base.cc.

protectedinherited 
Initialize the unit_support_points
field of the FiniteElement class. Called from initialize().
Definition at line 953 of file fe_q_base.cc.

protectedinherited 
Initialize the unit_face_support_points
field of the FiniteElement class. Called from initialize().
Definition at line 978 of file fe_q_base.cc.

protectedinherited 
Initialize the adjust_quad_dof_index_for_face_orientation_table
field of the FiniteElement class. Called from initialize().
Definition at line 1016 of file fe_q_base.cc.

inherited 
Return the polynomial degree of this finite element, i.e. the value passed to the constructor.

overridevirtualinherited 
Given a set of update flags, compute which other quantities also need to be computed in order to satisfy the request by the given flags. Then return the combination of the original set of flags and those just computed.
As an example, if update_flags
contains update_gradients a finite element class will typically require the computation of the inverse of the Jacobian matrix in order to rotate the gradient of shape functions on the reference cell to the real cell. It would then return not just update_gradients, but also update_covariant_transformation, the flag that makes the mapping class produce the inverse of the Jacobian matrix.
An extensive discussion of the interaction between this function and FEValues can be found in the How Mapping, FiniteElement, and FEValues work together documentation module.
Implements FiniteElement< dim, spacedim >.

inherited 
Return the underlying polynomial space.

inherited 
Return the numbering of the underlying polynomial space compared to lexicographic ordering of the basis functions. Returns PolynomialType::get_numbering().

inherited 
Return the inverse numbering of the underlying polynomial space. Returns PolynomialType::get_numbering_inverse().

overridevirtualinherited 
Return the value of the i
th shape function at the point p
. See the FiniteElement base class for more information about the semantics of this function.
Reimplemented from FiniteElement< dim, spacedim >.

overridevirtualinherited 
Return the value of the component
th vector component of the i
th shape function at the point p
. See the FiniteElement base class for more information about the semantics of this function.
Since this element is scalar, the returned value is the same as if the function without the _component
suffix were called, provided that the specified component is zero.
Reimplemented from FiniteElement< dim, spacedim >.

overridevirtualinherited 
Return the gradient of the i
th shape function at the point p
. See the FiniteElement base class for more information about the semantics of this function.
Reimplemented from FiniteElement< dim, spacedim >.

overridevirtualinherited 
Return the gradient of the component
th vector component of the i
th shape function at the point p
. See the FiniteElement base class for more information about the semantics of this function.
Since this element is scalar, the returned value is the same as if the function without the _component
suffix were called, provided that the specified component is zero.
Reimplemented from FiniteElement< dim, spacedim >.

overridevirtualinherited 
Return the tensor of second derivatives of the i
th shape function at point p
on the unit cell. See the FiniteElement base class for more information about the semantics of this function.
Reimplemented from FiniteElement< dim, spacedim >.

overridevirtualinherited 
Return the second derivative of the component
th vector component of the i
th shape function at the point p
. See the FiniteElement base class for more information about the semantics of this function.
Since this element is scalar, the returned value is the same as if the function without the _component
suffix were called, provided that the specified component is zero.
Reimplemented from FiniteElement< dim, spacedim >.

overridevirtualinherited 
Return the tensor of third derivatives of the i
th shape function at point p
on the unit cell. See the FiniteElement base class for more information about the semantics of this function.
Reimplemented from FiniteElement< dim, spacedim >.

overridevirtualinherited 
Return the third derivative of the component
th vector component of the i
th shape function at the point p
. See the FiniteElement base class for more information about the semantics of this function.
Since this element is scalar, the returned value is the same as if the function without the _component
suffix were called, provided that the specified component is zero.
Reimplemented from FiniteElement< dim, spacedim >.

overridevirtualinherited 
Return the tensor of fourth derivatives of the i
th shape function at point p
on the unit cell. See the FiniteElement base class for more information about the semantics of this function.
Reimplemented from FiniteElement< dim, spacedim >.

overridevirtualinherited 
Return the fourth derivative of the component
th vector component of the i
th shape function at the point p
. See the FiniteElement base class for more information about the semantics of this function.
Since this element is scalar, the returned value is the same as if the function without the _component
suffix were called, provided that the specified component is zero.
Reimplemented from FiniteElement< dim, spacedim >.

overridevirtualinherited 
Return an estimate (in bytes) for the memory consumption of this object.
Reimplemented from FiniteElement< dim, spacedim >.
Reimplemented in FE_Q_Hierarchical< dim >, FE_DGP< dim, spacedim >, and FE_DGPMonomial< dim >.

inlineoverrideprotectedvirtualinherited 
Create an internal data object and return a pointer to it of which the caller of this function then assumes ownership. This object will then be passed to the FiniteElement::fill_fe_values() every time the finite element shape functions and their derivatives are evaluated on a concrete cell. The object created here is therefore used by derived classes as a place for scratch objects that are used in evaluating shape functions, as well as to store information that can be precomputed once and reused on every cell (e.g., for evaluating the values and gradients of shape functions on the reference cell, for later reuse 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 precomputed to make computations on a concrete cell cheaper, then it should be put into the returned object for later reuse 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 precompute the gradients on the reference cell, and store it in the object returned.
An extensive discussion of the interaction between this function and FEValues can be found in the How Mapping, FiniteElement, and FEValues work together documentation module. See also the documentation of the InternalDataBase class.
[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 precompute 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. 
Implements FiniteElement< dim, spacedim >.

overrideprotectedvirtualinherited 

protectedpure virtualinherited 
Compute information about the shape functions on the cell denoted by the first argument. Derived classes will have to implement this function based on the kind of element they represent. It is called by FEValues::reinit().
Conceptually, this function evaluates shape functions and their derivatives at the quadrature points represented by the mapped locations of those described by the quadrature argument to this function. In many cases, computing derivatives of shape functions (and in some cases also computing values of shape functions) requires making use of the mapping from the reference to the real cell; this information can either be taken from the mapping_data
object that has been filled for the current cell before this function is called, or by calling the member functions of a Mapping object with the mapping_internal
object that also corresponds to the current cell.
The information computed by this function is used to fill the various member variables of the output argument of this function. Which of the member variables of that structure should be filled is determined by the update flags stored in the FiniteElement::InternalDataBase::update_each field of the object passed to this function. These flags are typically set by FiniteElement::get_data(), FiniteElement::get_face_date() and FiniteElement::get_subface_data() (or, more specifically, implementations of these functions in derived classes).
An extensive discussion of the interaction between this function and FEValues can be found in the How Mapping, FiniteElement, and FEValues work together documentation module.
[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. 
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.

overrideprotectedvirtualinherited 

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().
[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. 

protectedvirtualinherited 

overrideprotectedvirtualinherited 

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().
[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. 

protectedinherited 
Correct the shape Hessians by subtracting the terms corresponding to the Jacobian pushed forward gradient.
Before the correction, the Hessians would be given by
\[ D_{ijk} = \frac{d^2\phi_i}{d \hat x_J d \hat x_K} (J_{jJ})^{1} (J_{kK})^{1}, \]
where \(J_{iI}=\frac{d x_i}{d \hat x_I}\). After the correction, the correct Hessians would be given by
\[ \frac{d^2 \phi_i}{d x_j d x_k} = D_{ijk}  H_{mjk} \frac{d \phi_i}{d x_m}, \]
where \(H_{ijk}\) is the Jacobian pushedforward derivative.

protectedinherited 
Correct the shape third derivatives by subtracting the terms corresponding to the Jacobian pushed forward gradient and second derivative.
Before the correction, the third derivatives would be given by
\[ D_{ijkl} = \frac{d^3\phi_i}{d \hat x_J d \hat x_K d \hat x_L} (J_{jJ})^{1} (J_{kK})^{1} (J_{lL})^{1}, \]
where \(J_{iI}=\frac{d x_i}{d \hat x_I}\). After the correction, the correct third derivative would be given by
\[ \frac{d^3\phi_i}{d x_j d x_k d x_l} = D_{ijkl}  H_{mjl} \frac{d^2 \phi_i}{d x_k d x_m}  H_{mkl} \frac{d^2 \phi_i}{d x_j d x_m}  H_{mjk} \frac{d^2 \phi_i}{d x_l d x_m}  K_{mjkl} \frac{d \phi_i}{d x_m}, \]
where \(H_{ijk}\) is the Jacobian pushedforward derivative and \(K_{ijkl}\) is the Jacobian pushedforward second derivative.

inherited 

inlineinherited 
This operator returns a reference to the present object if the argument given equals to zero. While this does not seem particularly useful, it is helpful in writing code that works with both DoFHandler and the hp version hp::DoFHandler, since one can then write code like this:
This code doesn't work in both situations without the present operator because DoFHandler::get_fe() returns a finite element, whereas hp::DoFHandler::get_fe() returns a collection of finite elements that doesn't offer a dofs_per_cell
member variable: one first has to select which finite element to work on, which is done using the operator[]. Fortunately, cell>active_fe_index()
also works for nonhpclasses and simply returns zero in that case. The present operator[] accepts this zero argument, by returning the finite element with index zero within its collection (that, of course, consists only of the present finite element anyway).

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.

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.

inherited 
Return whether this element implements its restriction matrices. The return value also indicates whether a call to the get_restriction_matrix() function will generate an error or not.
Note, that this function returns true
only if the restriction matrices of the isotropic and all anisotropic refinement cases are implemented. If you are interested in the restriction matrices for isotropic refinement only, use the isotropic_restriction_is_implemented() function instead.
This function is mostly here in order to allow us to write more efficient test programs which we run on all kinds of weird elements, and for which we simply need to exclude certain tests in case something is not implemented. It will in general probably not be a great help in applications, since there is not much one can do if one needs these features and they are not implemented. This function could be used to check whether a call to get_restriction_matrix()
will succeed; however, one then still needs to cope with the lack of information this just expresses.

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

inlineinherited 
Access the restriction_is_additive_flags field. See the discussion about restriction matrices in the general class documentation for more information.
The index must be between zero and the number of shape functions of this element.

inherited 
Return a read only reference to the matrix that describes the constraints at the interface between a refined and an unrefined cell.
Some finite elements do not (yet) implement hanging node constraints. If this is the case, then this function will generate an exception, since no useful return value can be generated. If you should have a way to live with this, then you might want to use the constraints_are_implemented() function to check up front whether this function will succeed or generate the exception.

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

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:
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.

inherited 
Comparison operator.
Definition at line 180 of file fe_data.cc.

inherited 

inlineinherited 
Compute vector component and index of this shape function within the shape functions corresponding to this component from the index of a shape function within this finite element.
If the element is scalar, then the component is always zero, and the index within this component is equal to the overall index.
If the shape function referenced has more than one nonzero 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 nonzero 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 vectorcomponent). 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 step8 and step20 tutorial programs as well as in the Handling vector valued problems module.

inlineinherited 
Compute the shape function for the given vector component and index.
If the element is scalar, then the component must be zero, and the index within this component is equal to the overall index.
This is the opposite operation from the system_to_component_index() function.

inlineinherited 
Same as system_to_component_index(), but do it for shape functions and their indices on a face. The range of allowed indices is therefore 0..dofs_per_face.
You will rarely need this function in application programs, since almost all application codes only need to deal with cell indices, not face indices. The function is mainly there for use inside the library.

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

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

inlineinherited 
Return in which of the vector components of this finite element the ith
shape function is nonzero. 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 nonprimitive (see GlossPrimitive).

inlineinherited 
Return in how many vector components the ith
shape function is non zero. This value equals the number of entries equal to true
in the result of the get_nonzero_components() function.
For most finite element spaces, the result will be equal to one. It is not equal to one only for those ansatz spaces for which vectorvalued shape functions couple the individual components, for example in order to make them divergencefree.

inlineinherited 
Return whether the entire finite element is primitive, in the sense that all its shape functions are primitive. If the finite element is scalar, then this is always the case.
Since this is an extremely common operation, the result is cached and returned by this function.

inlineinherited 
Return whether the ith
shape function is primitive in the sense that the shape function is nonzero 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.

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

virtualinherited 
Access to base element objects. If the element is atomic, then base_element(0)
is this
.
Reimplemented in FESystem< dim, spacedim >, and FE_Enriched< dim, spacedim >.

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

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 innermost 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:
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)^2FE_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 

virtualinherited 
Return a reference to a contained finite element that matches the components n_selected_components
components starting at component with index first_component
.
See the other get_sub_fe() function above for more details.
Reimplemented in FESystem< dim, spacedim >.

inlineinherited 
Return for shape function index
the base element it belongs to, the number of the copy of this base element (which is between zero and the multiplicity of this element), and the index of this shape function within this base element.
If the element is not composed of others, then base and instance are always zero, and the index is equal to the number of the shape function. If the element is composed of single instances of other elements (i.e. all with multiplicity one) all of which are scalar, then base values and dof indices within this element are equal to the system_to_component_table. It differs only in case the element is composed of other elements and at least one of them is vectorvalued 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 vectorvalued (i.e. nonprimitive) shape functions, in contrast to the system_to_component_index() function.

inlineinherited 
Same as system_to_base_index(), but for degrees of freedom located on a face. The range of allowed indices is therefore 0..dofs_per_face.
You will rarely need this function in application programs, since almost all application codes only need to deal with cell indices, not face indices. The function is mainly there for use inside the library.

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

inlineinherited 
For each vector component, return which base element implements this component and which vector component in this base element this is. This information is only of interest for vectorvalued finite elements which are composed of several subelements. 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.

inlineinherited 

inlineinherited 

inherited 

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

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

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.
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. 

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.
block_mask  The mask that selects individual blocks of the finite element 

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

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

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.
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. 

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.
component_mask  The mask that selects individual components of the finite element 

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.
FESystem(FE_Q(1),3)
for which each support point would appear three times in the returned array.

inherited 
Return whether a finite element has defined support points. If the result is true, then a call to the get_unit_support_points() yields a nonempty array.
The result may be false if an element is not defined by interpolating shape functions, for example by Pelements on quadrilaterals. It will usually only be true if the element constructs its shape functions by the requirement that they be one at a certain point and zero at all the points associated with the other shape functions.
In composed elements (i.e. for the FESystem class), the result will be true if all the base elements have defined support points. FE_Nothing is a special case in FESystems, because it has 0 support points and has_support_points() is false, but an FESystem containing an FE_Nothing among other elements will return true.

virtualinherited 
Return the position of the support point of the indexth
shape function. If it does not exist, raise an exception.
The default implementation simply returns the respective element from the array you get from get_unit_support_points(), but derived elements may overload this function. In particular, note that the FESystem class overloads it so that it can return the support points of individual base elements, if not all the base elements define support points. In this way, you can still ask for certain support points, even if get_unit_support_points() only returns an empty array.
Reimplemented in FESystem< dim, spacedim >.

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.

inherited 
Return whether a finite element has defined support points on faces. If the result is true, then a call to the get_unit_face_support_points() yields a nonempty vector.
For more information, see the documentation for the has_support_points() function.

virtualinherited 
The function corresponding to the unit_support_point() function, but for faces. See there for more information.
Reimplemented in FESystem< dim, spacedim >.

inherited 
Return a vector of generalized support points.
See the glossary entry on generalized support points for more information.

inherited 
Return whether a finite element has defined generalized support points. If the result is true, then a call to the get_generalized_support_points() yields a nonempty vector.
See the glossary entry on generalized support points for more information.

inlineinherited 
For a given degree of freedom, return whether it is logically associated with a vertex, line, quad or hex.
For instance, for continuous finite elements this coincides with the lowest dimensional object the support point of the degree of freedom lies on. To give an example, for \(Q_1\) elements in 3d, every degree of freedom is defined by a shape function that we get by interpolating using support points that lie on the vertices of the cell. The support of these points of course extends to all edges connected to this vertex, as well as the adjacent faces and the cell interior, but we say that logically the degree of freedom is associated with the vertex as this is the lowest dimensional object it is associated with. Likewise, for \(Q_2\) elements in 3d, the degrees of freedom with support points at edge midpoints would yield a value of GeometryPrimitive::line from this function, whereas those on the centers of faces in 3d would return GeometryPrimitive::quad.
To make this more formal, the kind of object returned by this function represents the object so that the support of the shape function corresponding to the degree of freedom, (i.e., that part of the domain where the function "lives") is the union of all of the cells sharing this object. To return to the example above, for \(Q_2\) in 3d, the shape function with support point at an edge midpoint has support on all cells that share the edge and not only the cells that share the adjacent faces, and consequently the function will return GeometryPrimitive::line.
On the other hand, for discontinuous elements of type \(DGQ_2\), a degree of freedom associated with an interpolation polynomial that has its support point physically located at a line bounding a cell, but is nonzero only on one cell. Consequently, it is logically associated with the interior of that cell (i.e., with a GeometryPrimitive::quad in 2d and a GeometryPrimitive::hex in 3d).
[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) . 

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.
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. 

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.

staticprotectedinherited 

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.
[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 precompute 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. 
Reimplemented in FESystem< dim, spacedim >, FE_Enriched< dim, spacedim >, FE_PolyFace< PolynomialType, dim, spacedim >, FE_PolyFace< TensorProductPolynomials< dim  1 >, dim, spacedim >, and FE_PolyFace< PolynomialSpace< dim  1 >, dim, spacedim >.

protectedvirtualinherited 

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.
[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 precompute 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. 
Reimplemented in FESystem< dim, spacedim >, FE_Enriched< dim, spacedim >, FE_PolyFace< PolynomialType, dim, spacedim >, FE_PolyFace< TensorProductPolynomials< dim  1 >, dim, spacedim >, and FE_PolyFace< PolynomialSpace< dim  1 >, dim, spacedim >.

inherited 
Subscribes a user of the object by storing the pointer validity
. The subscriber may be identified by text supplied as identifier
.
Definition at line 136 of file subscriptor.cc.

inherited 
Unsubscribes a user from the object.
identifier
and the validity
pointer must be the same as the one supplied to subscribe(). Definition at line 156 of file subscriptor.cc.

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 301 of file subscriptor.h.

inlineinherited 
List the subscribers to the input stream
.
Definition at line 318 of file subscriptor.h.

inherited 
List the subscribers to deallog
.
Definition at line 204 of file subscriptor.cc.

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 310 of file subscriptor.h.

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

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

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

inherited 
Number of dofs per vertex.

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

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

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

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

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

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

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

inherited 
Return the number of degrees per structdimdimensional 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.

inherited 
Number of components. See the glossary for more information.

inherited 
Number of blocks. See the glossary for more information.

inherited 
Detailed information on block sizes.

inherited 
Maximal polynomial degree of a shape function in a single coordinate direction.
This function can be used to determine the optimal quadrature rule.

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

inherited 
Return first index of dof on a line.

inherited 
Return first index of dof on a quad.

inherited 
Return first index of dof on a hexahedron.

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

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

protectedinherited 

staticinherited 
The dimension of the image space, corresponding to Triangulation.

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_case1][child]
includes the restriction matrix of child child
for the RefinementCase refinement_case
. Here, we use refinement_case1
instead of refinement_case
as for RefinementCase::no_refinement(=0) there are no restriction matrices available.

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_case1][child]
includes the prolongation matrix of child child
for the RefinementCase refinement_case
. Here, we use refinement_case1
instead of refinement_case
as for RefinementCase::no_refinement(=0) there are no prolongation matrices available.

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.

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.

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.

protectedinherited 

protectedinherited 

protectedinherited 
For faces with nonstandard face_orientation in 3D, the dofs on faces (quads) have to be permuted in order to be combined with the correct shape functions. Given a local dof index
on a quad, return the shift in the local index, if the face has nonstandard face_orientation, i.e. old_index + shift = new_index
. In 2D and 1D there is no need for permutation so the vector is empty. In 3D it has the size of dofs_per_quad * 8
, where 8 is the number of orientations, a face can be in (all combinations of the three bool flags face_orientation, face_flip and face_rotation).
The constructor of this class fills this table with zeros, i.e., no permutation at all. Derived finite element classes have to fill this Table with the correct values.

protectedinherited 
For lines with nonstandard line_orientation in 3D, the dofs on lines have to be permuted in order to be combined with the correct shape functions. Given a local dof index
on a line, return the shift in the local index, if the line has nonstandard line_orientation, i.e. old_index + shift = new_index
. In 2D and 1D there is no need for permutation so the vector is empty. In 3D it has the size of dofs_per_line.
The constructor of this class fills this table with zeros, i.e., no permutation at all. Derived finite element classes have to fill this vector with the correct values.

protectedinherited 
Store what system_to_component_index() will return.

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 nonzero in only one component.

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

protectedinherited 

protectedinherited 

protectedinherited 
The base element establishing a component.
For each component number c
, the entries have the following meaning:
table[c].first.first
c
. This is the index you can pass to base_element(). table[c].first.second
c
. This value is between 0 and the n_components() of this base element. table[c].second
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.

protectedinherited 

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 nonzero.
For primitive elements, there is only one nonzero component.

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

protectedinherited 

staticinherited 

inherited 

inherited 

inherited 

inherited 

inherited 

inherited 

inherited 

inherited 

inherited 

inherited 

inherited 

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

inherited 

inherited 

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.