Reference documentation for deal.II version Git 24041777c5 20211202 20:43:25 0700

#include <deal.II/fe/fe_p1nc.h>
Public Types  
enum  Conformity { unknown = 0x00, L2 = 0x01, Hcurl = 0x02, Hdiv = 0x04, H1 = Hcurl  Hdiv, H2 = 0x0e } 
Public Member Functions  
FE_P1NC ()  
virtual std::string  get_name () const override 
virtual UpdateFlags  requires_update_flags (const UpdateFlags flags) const override 
virtual std::unique_ptr< FiniteElement< 2, 2 > >  clone () const override 
virtual  ~FE_P1NC () override=default 
std::pair< std::unique_ptr< FiniteElement< dim, spacedim > >, unsigned int >  operator^ (const unsigned int multiplicity) const 
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 
virtual std::size_t  memory_consumption () const 
template<class Archive >  
void  serialize (Archive &ar, const unsigned int version) 
ReferenceCell  reference_cell () const 
unsigned int  n_unique_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 
Shape function access  
virtual double  shape_value (const unsigned int i, const Point< dim > &p) const 
virtual double  shape_value_component (const unsigned int i, const Point< dim > &p, const unsigned int component) const 
virtual Tensor< 1, dim >  shape_grad (const unsigned int i, const Point< dim > &p) const 
virtual Tensor< 1, dim >  shape_grad_component (const unsigned int i, const Point< dim > &p, const unsigned int component) const 
virtual Tensor< 2, dim >  shape_grad_grad (const unsigned int i, const Point< dim > &p) const 
virtual Tensor< 2, dim >  shape_grad_grad_component (const unsigned int i, const Point< dim > &p, const unsigned int component) const 
virtual Tensor< 3, dim >  shape_3rd_derivative (const unsigned int i, const Point< dim > &p) const 
virtual Tensor< 3, dim >  shape_3rd_derivative_component (const unsigned int i, const Point< dim > &p, const unsigned int component) const 
virtual Tensor< 4, dim >  shape_4th_derivative (const unsigned int i, const Point< dim > &p) const 
virtual Tensor< 4, dim >  shape_4th_derivative_component (const unsigned int i, const Point< dim > &p, const unsigned int component) const 
virtual bool  has_support_on_face (const unsigned int shape_index, const unsigned int face_index) const 
Transfer and constraint matrices  
virtual const FullMatrix< double > &  get_restriction_matrix (const unsigned int child, const RefinementCase< dim > &refinement_case=RefinementCase< dim >::isotropic_refinement) const 
virtual const FullMatrix< double > &  get_prolongation_matrix (const unsigned int child, const RefinementCase< dim > &refinement_case=RefinementCase< dim >::isotropic_refinement) const 
bool  prolongation_is_implemented () const 
bool  isotropic_prolongation_is_implemented () const 
bool  restriction_is_implemented () const 
bool  isotropic_restriction_is_implemented () const 
bool  restriction_is_additive (const unsigned int index) const 
const FullMatrix< double > &  constraints (const ::internal::SubfaceCase< dim > &subface_case=::internal::SubfaceCase< dim >::case_isotropic) const 
bool  constraints_are_implemented (const ::internal::SubfaceCase< dim > &subface_case=::internal::SubfaceCase< dim >::case_isotropic) const 
virtual bool  hp_constraints_are_implemented () const 
virtual void  get_interpolation_matrix (const FiniteElement< dim, spacedim > &source, FullMatrix< double > &matrix) const 
Functions to support hpadaptivity  
virtual void  get_face_interpolation_matrix (const FiniteElement< dim, spacedim > &source, FullMatrix< double > &matrix, const unsigned int face_no=0) const 
virtual void  get_subface_interpolation_matrix (const FiniteElement< dim, spacedim > &source, const unsigned int subface, FullMatrix< double > &matrix, const unsigned int face_no=0) const 
virtual std::vector< std::pair< unsigned int, unsigned int > >  hp_vertex_dof_identities (const FiniteElement< dim, spacedim > &fe_other) const 
virtual std::vector< std::pair< unsigned int, unsigned int > >  hp_line_dof_identities (const FiniteElement< dim, spacedim > &fe_other) const 
virtual std::vector< std::pair< unsigned int, unsigned int > >  hp_quad_dof_identities (const FiniteElement< dim, spacedim > &fe_other, const unsigned int face_no=0) const 
virtual FiniteElementDomination::Domination  compare_for_domination (const FiniteElement< dim, spacedim > &fe_other, const unsigned int codim=0) const 
Index computations  
std::pair< unsigned int, unsigned int >  system_to_component_index (const unsigned int index) const 
unsigned int  component_to_system_index (const unsigned int component, const unsigned int index) const 
std::pair< unsigned int, unsigned int >  face_system_to_component_index (const unsigned int index, const unsigned int face_no=0) const 
unsigned int  adjust_quad_dof_index_for_face_orientation (const unsigned int index, const unsigned int face_no, const bool face_orientation, const bool face_flip, const bool face_rotation) const 
virtual unsigned int  face_to_cell_index (const unsigned int face_dof_index, const unsigned int face, const bool face_orientation=true, const bool face_flip=false, const bool face_rotation=false) const 
unsigned int  adjust_line_dof_index_for_line_orientation (const unsigned int index, const bool line_orientation) const 
const 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 
virtual std::pair< Table< 2, bool >, std::vector< unsigned int > >  get_constant_modes () const 
Support points and interpolation  
const std::vector< Point< dim > > &  get_unit_support_points () const 
bool  has_support_points () const 
virtual Point< dim >  unit_support_point (const unsigned int index) const 
const std::vector< Point< dim  1 > > &  get_unit_face_support_points (const unsigned int face_no=0) const 
bool  has_face_support_points (const unsigned int face_no=0) const 
virtual Point< dim  1 >  unit_face_support_point (const unsigned int index, const unsigned int face_no=0) const 
const std::vector< Point< dim > > &  get_generalized_support_points () const 
bool  has_generalized_support_points () const 
GeometryPrimitive  get_associated_geometry_primitive (const unsigned int cell_dof_index) const 
virtual void  convert_generalized_support_point_values_to_dof_values (const std::vector< Vector< double >> &support_point_values, std::vector< double > &nodal_values) const 
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 &  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 
static const unsigned int  dimension = dim 
Protected Member Functions  
void  reinit_restriction_and_prolongation_matrices (const bool isotropic_restriction_only=false, const bool isotropic_prolongation_only=false) 
TableIndices< 2 >  interface_constraints_size () const 
virtual std::unique_ptr< InternalDataBase >  get_data (const UpdateFlags update_flags, const Mapping< dim, spacedim > &mapping, const Quadrature< dim > &quadrature, ::internal::FEValuesImplementation::FiniteElementRelatedData< dim, spacedim > &output_data) const=0 
virtual std::unique_ptr< InternalDataBase >  get_face_data (const UpdateFlags update_flags, const Mapping< dim, spacedim > &mapping, const hp::QCollection< dim  1 > &quadrature, ::internal::FEValuesImplementation::FiniteElementRelatedData< dim, spacedim > &output_data) const 
virtual std::unique_ptr< InternalDataBase >  get_face_data (const UpdateFlags update_flags, const Mapping< dim, spacedim > &mapping, const Quadrature< dim  1 > &quadrature, internal::FEValuesImplementation::FiniteElementRelatedData< dim, spacedim > &output_data) const 
virtual std::unique_ptr< InternalDataBase >  get_subface_data (const UpdateFlags update_flags, const Mapping< dim, spacedim > &mapping, const Quadrature< dim  1 > &quadrature, ::internal::FEValuesImplementation::FiniteElementRelatedData< dim, spacedim > &output_data) const 
virtual void  fill_fe_values (const typename Triangulation< dim, spacedim >::cell_iterator &cell, const CellSimilarity::Similarity cell_similarity, const Quadrature< dim > &quadrature, const Mapping< dim, spacedim > &mapping, const typename Mapping< dim, spacedim >::InternalDataBase &mapping_internal, const ::internal::FEValuesImplementation::MappingRelatedData< dim, spacedim > &mapping_data, const InternalDataBase &fe_internal, ::internal::FEValuesImplementation::FiniteElementRelatedData< dim, spacedim > &output_data) const=0 
virtual void  fill_fe_face_values (const typename Triangulation< dim, spacedim >::cell_iterator &cell, const unsigned int face_no, const hp::QCollection< dim  1 > &quadrature, const Mapping< dim, spacedim > &mapping, const typename Mapping< dim, spacedim >::InternalDataBase &mapping_internal, const ::internal::FEValuesImplementation::MappingRelatedData< dim, spacedim > &mapping_data, const InternalDataBase &fe_internal, ::internal::FEValuesImplementation::FiniteElementRelatedData< dim, spacedim > &output_data) const 
virtual void  fill_fe_face_values (const typename Triangulation< dim, spacedim >::cell_iterator &cell, const unsigned int face_no, const Quadrature< dim  1 > &quadrature, const Mapping< dim, spacedim > &mapping, const typename Mapping< dim, spacedim >::InternalDataBase &mapping_internal, const internal::FEValuesImplementation::MappingRelatedData< dim, spacedim > &mapping_data, const InternalDataBase &fe_internal, internal::FEValuesImplementation::FiniteElementRelatedData< dim, spacedim > &output_data) const 
virtual void  fill_fe_subface_values (const typename Triangulation< dim, spacedim >::cell_iterator &cell, const unsigned int face_no, const unsigned int sub_no, const Quadrature< dim  1 > &quadrature, const Mapping< dim, spacedim > &mapping, const typename Mapping< dim, spacedim >::InternalDataBase &mapping_internal, const ::internal::FEValuesImplementation::MappingRelatedData< dim, spacedim > &mapping_data, const InternalDataBase &fe_internal, ::internal::FEValuesImplementation::FiniteElementRelatedData< dim, spacedim > &output_data) const=0 
Static Protected Member Functions  
static std::vector< unsigned int >  compute_n_nonzero_components (const std::vector< ComponentMask > &nonzero_components) 
Protected Attributes  
std::vector< std::vector< FullMatrix< double > > >  restriction 
std::vector< std::vector< FullMatrix< double > > >  prolongation 
FullMatrix< double >  interface_constraints 
std::vector< Point< dim > >  unit_support_points 
std::vector< std::vector< Point< dim  1 > > >  unit_face_support_points 
std::vector< Point< dim > >  generalized_support_points 
std::vector< std::vector< Point< dim  1 > > >  generalized_face_support_points 
std::vector< Table< 2, int > >  adjust_quad_dof_index_for_face_orientation_table 
std::vector< int >  adjust_line_dof_index_for_line_orientation_table 
std::vector< std::pair< unsigned int, unsigned int > >  system_to_component_table 
std::vector< std::vector< std::pair< unsigned int, unsigned int > > >  face_system_to_component_table 
std::vector< std::pair< std::pair< unsigned int, unsigned int >, unsigned int > >  system_to_base_table 
std::vector< std::vector< std::pair< std::pair< unsigned int, unsigned int >, unsigned int > > >  face_system_to_base_table 
BlockIndices  base_to_block_indices 
std::vector< std::pair< std::pair< unsigned int, unsigned int >, unsigned int > >  component_to_base_table 
const std::vector< bool >  restriction_is_additive_flags 
const std::vector< ComponentMask >  nonzero_components 
const std::vector< unsigned int >  n_nonzero_components_table 
const bool  cached_primitivity 
Private Member Functions  
virtual std::unique_ptr< FiniteElement< 2, 2 >::InternalDataBase >  get_data (const UpdateFlags update_flags, const Mapping< 2, 2 > &, const Quadrature< 2 > &quadrature, ::internal::FEValuesImplementation::FiniteElementRelatedData< 2, 2 > &output_data) const override 
virtual std::unique_ptr< FiniteElement< 2, 2 >::InternalDataBase >  get_face_data (const UpdateFlags update_flags, const Mapping< 2, 2 > &, const hp::QCollection< 1 > &quadrature, ::internal::FEValuesImplementation::FiniteElementRelatedData< 2, 2 > &output_data) const override 
virtual std::unique_ptr< FiniteElement< 2, 2 >::InternalDataBase >  get_subface_data (const UpdateFlags update_flags, const Mapping< 2, 2 > &, const Quadrature< 1 > &quadrature, ::internal::FEValuesImplementation::FiniteElementRelatedData< 2, 2 > &output_data) const override 
virtual void  fill_fe_values (const Triangulation< 2, 2 >::cell_iterator &cell, const CellSimilarity::Similarity cell_similarity, const Quadrature< 2 > &quadrature, const Mapping< 2, 2 > &mapping, const Mapping< 2, 2 >::InternalDataBase &mapping_internal, const internal::FEValuesImplementation::MappingRelatedData< 2, 2 > &mapping_data, const FiniteElement< 2, 2 >::InternalDataBase &fe_internal, internal::FEValuesImplementation::FiniteElementRelatedData< 2, 2 > &output_data) const override 
virtual void  fill_fe_face_values (const Triangulation< 2, 2 >::cell_iterator &cell, const unsigned int face_no, const hp::QCollection< 1 > &quadrature, const Mapping< 2, 2 > &mapping, const Mapping< 2, 2 >::InternalDataBase &mapping_internal, const ::internal::FEValuesImplementation::MappingRelatedData< 2, 2 > &mapping_data, const InternalDataBase &fe_internal, ::internal::FEValuesImplementation::FiniteElementRelatedData< 2, 2 > &output_data) const override 
virtual void  fill_fe_subface_values (const Triangulation< 2, 2 >::cell_iterator &cell, const unsigned int face_no, const unsigned int sub_no, const Quadrature< 1 > &quadrature, const Mapping< 2, 2 > &mapping, const Mapping< 2, 2 >::InternalDataBase &mapping_internal, const ::internal::FEValuesImplementation::MappingRelatedData< 2, 2 > &mapping_data, const InternalDataBase &fe_internal, ::internal::FEValuesImplementation::FiniteElementRelatedData< 2, 2 > &output_data) const override 
void  initialize_constraints () 
Static Private Member Functions  
static std::vector< unsigned int >  get_dpo_vector () 
static ndarray< double, 4, 3 >  get_linear_shape_coefficients (const Triangulation< 2, 2 >::cell_iterator &cell) 
Implementation of the scalar version of the P1 nonconforming finite element, a piecewise linear element on quadrilaterals in 2D. This implementation is only for 2D cells in a 2D space (i.e., codimension 0).
Unlike the usual continuous, \(H^1\) conforming finite elements, the P1 nonconforming element does not enforce continuity across edges. However, it requires the continuity in an integral sense: any function in the space should have the same integral values on two sides of the common edge shared by two adjacent elements.
Thus, each function in the nonconforming element space can be discontinuous, and consequently not included in \(H^1_0\), just like the basis functions in Discontinuous Galerkin (DG) finite element spaces. On the other hand, basis functions in DG spaces are completely discontinuous across edges without any relation between the values from both sides. This is a reason why usual weak formulations for DG schemes contain additional penalty terms for jump across edges to control discontinuity. However, nonconforming elements usually do not need additional terms in their weak formulations because their integrals along edges are the same from both sides, i.e., there is some level of continuity.
Since any function in the P1 nonconforming space is piecewise linear on each element, the function value at the midpoint of each edge is same as the mean value on the edge. Thus the continuity of the integral value across each edge is equivalent to the continuity of the midpoint value of each edge in this case.
Thus for the P1 nonconforming element, the function values at midpoints on edges of a cell are important. The first attempt to define (local) degrees of freedom (DoFs) on a quadrilateral is by using midpoint values of a function.
However, these 4 functionals are not linearly independent because a linear function on 2D is uniquely determined by only 3 independent values. A simple observation reads that any linear function on a quadrilateral should satisfy the 'dice rule': the sum of two function values at the midpoints of the edge pair on opposite sides of a cell is equal to the sum of those at the midpoints of the other edge pair. This is called the 'dice rule' because the number of points on opposite sides of a dice always adds up to the same number as well (in the case of dice, to seven).
In formulas, the dice rule is written as \(\phi(m_0) + \phi(m_1) = \phi(m_2) + \phi(m_3)\) for all \(\phi\) in the function space where \(m_j\) is the midpoint of the edge \(e_j\). Here, we assume the standard numbering convention for edges used in deal.II and described in class GeometryInfo.
Conversely if 4 values at midpoints satisfying the dice rule are given, then there always exists the unique linear function which coincides with 4 midpoints values.
Due to the dice rule, three values at any three midpoints can determine the last value at the last midpoint. It means that the number of independent local functionals on a cell is 3, and this is also the dimension of the linear polynomial space on a cell in 2D.
Before introducing the degrees of freedom, we present 4 local shape functions on a cell. Due to the dice rule, we need a special construction for shape functions. Although the following 4 shape functions are not linearly independent within a cell, they are helpful to define the global basis functions which are linearly independent on the whole domain. Again, we assume the standard numbering for vertices used in deal.II.
* 23 *   *   *   *   *   *   *   *   *   * 01 *
For each vertex \(v_j\) of given cell, there are two edges of which \(v_j\) is one of end points. Consider a linear function such that it has value 0.5 at the midpoints of two adjacent edges, and 0.0 at the two midpoints of the other edges. Note that the set of these values satisfies the dice rule which is described above. We denote such a function associated with vertex \(v_j\) by \(\phi_j\). Then the set of 4 shape functions is a partition of unity on a cell: \(\sum_{j=0}^{3} \phi_j = 1\). (This is easy to see: at each edge midpoint, the sum of the four function adds up to one because two functions have value 0.5 and the other value 0.0. Because the function is globally linear, the only function that can have value 1 at four points must also be globally equal to one.)
The following figures represent \(\phi_j\) for \(j=0,\cdots,3\) with their midpoint values:
shape function \(\phi_0\):
* +0.0+ *   *   *   *   * 0.5 0.0 *   *   *   *   * +0.5+ *
shape function \(\phi_1\):
* +0.0+ *   *   *   *   * 0.0 0.5 *   *   *   *   * +0.5+ *
shape function \(\phi_2\):
* +0.5+ *   *   *   *   * 0.5 0.0 *   *   *   *   * +0.0+ *
shape function \(\phi_3\):
* +0.5+ *   *   *   *   * 0.0 0.5 *   *   *   *   * +0.0+ *
The local DoFs are defined by the coefficients of the shape functions associated with vertices, respectively. Although these 4 local DoFs are not linearly independent within a single cell, this definition is a good start point for the definition of the global DoFs.
We want to emphasize that the shape functions are constructed on each cell, not on the reference cell only. Usual finite elements are defined based on a 'parametric' concept. It means that a function space for a finite element is defined on one reference cell, and it is transformed into each cell via a mapping from the reference cell. However the P1 nonconforming element does not follow such concept. It defines a function space with linear shape functions on each cell without any help of a function space on the reference cell. In other words, the element is defined in real space, not via a mapping from a reference cell. In this, it is similar to the FE_DGPNonparametric element.
Thus this implementation does not have to compute shape values on the reference cell. Rather, the shape values are computed by construction of the shape functions on each cell independently.
We next have to consider the global basis functions for the element because the system of equations which we ultimately have to solve is for a global system, not local. The global basis functions associated with a node are defined by a cellwise composition of local shape functions associated with the node on each element.
There is a theoretical result about the linear independence of the global basis functions depending on the type of the boundary condition we consider.
When homogeneous Dirichlet boundary conditions are given, the global basis functions associated with interior nodes are linearly independent. Then, the number of DoFs is equal to the number of interior nodes, and consequently the same as the number of DoFs for the standard bilinear \(Q_1\) finite element.
When Neumann boundary conditions are given, the global basis functions associated with all nodes (including boundary nodes) are actually not linearly independent. There exists one redundancy. Thus in this case, the number of DoFs is equal to the number of all nodes minus 1. This is, again as for the regular \(Q_1\) element.
For a smooth function, we construct a piecewise linear function which belongs to the element space by using its nodal values as DoF values.
Note that for the P1 nonconforming element, two nodal values of a smooth function and its interpolant do not coincide in general, in contrast to ordinary Lagrange finite elements. Of course, it is meaningless to refer 'nodal value' because the element space has nonconformity. But it is also true even though the single global basis function associated with a node is considered the unique 'nodal value' at the node. For instance, consider the basis function associated with a node. Consider two lines representing the level sets for value 0.5 and 0, respectively, by connecting two midpoints. Then we cut the quad into two subtriangles by the diagonal which is placed along those two lines. It gives another level set for value 0.25 which coincides with the cutting diagonal. Therefore these three level sets are all parallel in the quad and it gives the value 0.75 at the base node, not value 1. This is true whether the quadrilateral is a rectangle, parallelogram, or any other shape.
The original paper for the P1 nonconforming element by Park and Sheen is accessible at https://doi.org/10.1137/S0036142902404923 , see [105] .

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_P1NC::FE_P1NC  (  ) 
Constructor for the P1 nonconforming element. It is only for 2D and codimension = 0.
Definition at line 24 of file fe_p1nc.cc.

overridevirtualdefault 
Destructor.

overridevirtual 
Return a string that uniquely identifies a finite element. The general convention is that this is the class name, followed by the dimension in angle brackets, and the polynomial degree and whatever else is necessary in parentheses. For example, FE_Q<2>(3)
is the value returned for a cubic element in 2d.
Systems of elements have their own naming convention, see the FESystem class.
Implements FiniteElement< 2, 2 >.
Definition at line 42 of file fe_p1nc.cc.

overridevirtual 
Given a set of update flags, compute which other quantities also need to be computed in order to satisfy the request by the given flags. Then return the combination of the original set of flags and those just computed.
As an example, if update_flags
contains update_gradients a finite element class will typically require the computation of the inverse of the Jacobian matrix in order to rotate the gradient of shape functions on the reference cell to the real cell. It would then return not just update_gradients, but also update_covariant_transformation, the flag that makes the mapping class produce the inverse of the Jacobian matrix.
An extensive discussion of the interaction between this function and FEValues can be found in the How Mapping, FiniteElement, and FEValues work together documentation module.
Implements FiniteElement< 2, 2 >.
Definition at line 50 of file fe_p1nc.cc.

overridevirtual 
A sort of virtual copy constructor, this function returns a copy of the finite element object. Derived classes need to override the function here in this base class and return an object of the same type as the derived class.
Some places in the library, for example the constructors of FESystem as well as the hp::FECollection class, need to make copies of finite elements without knowing their exact type. They do so through this function.
Implements FiniteElement< 2, 2 >.
Definition at line 69 of file fe_p1nc.cc.

staticprivate 
Return the vector consists of the numbers of degrees of freedom per objects.
Definition at line 77 of file fe_p1nc.cc.

staticprivate 
Return the coefficients of 4 local linear shape functions \(\phi_j(x,y) = a x + b y + c\) on given cell. For each local shape function, the array consists of three coefficients is in order of a,b and c.
Definition at line 89 of file fe_p1nc.cc.

overrideprivatevirtual 
Do the work which is needed before cellwise data computation. Since the shape functions are constructed independently on each cell, the data on the reference cell is not necessary. It returns an empty variable type of @ InternalDataBase and updates @ update_flags, and computes trivially zero Hessian for each cell if it is needed.
Definition at line 136 of file fe_p1nc.cc.

overrideprivatevirtual 
Definition at line 160 of file fe_p1nc.cc.

overrideprivatevirtual 
Definition at line 186 of file fe_p1nc.cc.

overrideprivatevirtual 
Compute the data on the current cell.
Definition at line 210 of file fe_p1nc.cc.

overrideprivatevirtual 
Compute the data on the face of the current cell.
Definition at line 247 of file fe_p1nc.cc.

overrideprivatevirtual 
Compute the data on the subface of the current cell.
Definition at line 294 of file fe_p1nc.cc.

private 
Create the constraints matrix for hanging edges.
Definition at line 340 of file fe_p1nc.cc.

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).

virtualinherited 
Return the value of the ith
shape function at the point p
. p
is a point on the reference element. If the finite element is vectorvalued, then return the value of the only nonzero component of the vector value of this shape function. If the shape function has more than one nonzero component (which we refer to with the term nonprimitive), then derived classes implementing this function should throw an exception of type ExcShapeFunctionNotPrimitive. In that case, use the shape_value_component() function.
Implementations of this function should throw an exception of type ExcUnitShapeValuesDoNotExist if the shape functions of the FiniteElement under consideration depend on the shape of the cell in real space, i.e., if the shape functions are not defined by mapping from the reference cell. Some nonconforming elements are defined this way, as is the FE_DGPNonparametric class, to name just one example.
The default implementation of this virtual function does exactly this, i.e., it simply throws an exception of type ExcUnitShapeValuesDoNotExist.

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

virtualinherited 
Return the gradient of the ith
shape function at the point p
. p
is a point on the reference element, and likewise the gradient is the gradient on the unit cell with respect to unit cell coordinates. If the finite element is vectorvalued, then return the value of the only nonzero component of the vector value of this shape function. If the shape function has more than one nonzero component (which we refer to with the term nonprimitive), then derived classes implementing this function should throw an exception of type ExcShapeFunctionNotPrimitive. In that case, use the shape_grad_component() function.
Implementations of this function should throw an exception of type ExcUnitShapeValuesDoNotExist if the shape functions of the FiniteElement under consideration depend on the shape of the cell in real space, i.e., if the shape functions are not defined by mapping from the reference cell. Some nonconforming elements are defined this way, as is the FE_DGPNonparametric class, to name just one example.
The default implementation of this virtual function does exactly this, i.e., it simply throws an exception of type ExcUnitShapeValuesDoNotExist.

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

virtualinherited 
Return the tensor of second derivatives of the ith
shape function at point p
on the unit cell. The derivatives are derivatives on the unit cell with respect to unit cell coordinates. If the finite element is vectorvalued, then return the value of the only nonzero component of the vector value of this shape function. If the shape function has more than one nonzero component (which we refer to with the term nonprimitive), then derived classes implementing this function should throw an exception of type ExcShapeFunctionNotPrimitive. In that case, use the shape_grad_grad_component() function.
Implementations of this function should throw an exception of type ExcUnitShapeValuesDoNotExist if the shape functions of the FiniteElement under consideration depend on the shape of the cell in real space, i.e., if the shape functions are not defined by mapping from the reference cell. Some nonconforming elements are defined this way, as is the FE_DGPNonparametric class, to name just one example.
The default implementation of this virtual function does exactly this, i.e., it simply throws an exception of type ExcUnitShapeValuesDoNotExist.

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

virtualinherited 
Return the tensor of third derivatives of the ith
shape function at point p
on the unit cell. The derivatives are derivatives on the unit cell with respect to unit cell coordinates. If the finite element is vectorvalued, then return the value of the only nonzero component of the vector value of this shape function. If the shape function has more than one nonzero component (which we refer to with the term nonprimitive), then derived classes implementing this function should throw an exception of type ExcShapeFunctionNotPrimitive. In that case, use the shape_3rd_derivative_component() function.
Implementations of this function should throw an exception of type ExcUnitShapeValuesDoNotExist if the shape functions of the FiniteElement under consideration depend on the shape of the cell in real space, i.e., if the shape functions are not defined by mapping from the reference cell. Some nonconforming elements are defined this way, as is the FE_DGPNonparametric class, to name just one example.
The default implementation of this virtual function does exactly this, i.e., it simply throws an exception of type ExcUnitShapeValuesDoNotExist.

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

virtualinherited 
Return the tensor of fourth derivatives of the ith
shape function at point p
on the unit cell. The derivatives are derivatives on the unit cell with respect to unit cell coordinates. If the finite element is vectorvalued, then return the value of the only nonzero component of the vector value of this shape function. If the shape function has more than one nonzero component (which we refer to with the term nonprimitive), then derived classes implementing this function should throw an exception of type ExcShapeFunctionNotPrimitive. In that case, use the shape_4th_derivative_component() function.
Implementations of this function should throw an exception of type ExcUnitShapeValuesDoNotExist if the shape functions of the FiniteElement under consideration depend on the shape of the cell in real space, i.e., if the shape functions are not defined by mapping from the reference cell. Some nonconforming elements are defined this way, as is the FE_DGPNonparametric class, to name just one example.
The default implementation of this virtual function does exactly this, i.e., it simply throws an exception of type ExcUnitShapeValuesDoNotExist.

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

virtualinherited 
This function returns true
, if the shape function shape_index
has nonzero function values somewhere on the face face_index
. The function is typically used to determine whether some matrix elements resulting from face integrals can be assumed to be zero and may therefore be omitted from integration.
A default implementation is provided in this base class which always returns true
. This is the safe way to go.

virtualinherited 
Return the matrix that describes restricting a finite element field from the given child
(as obtained by the given refinement_case
) to the parent cell. The interpretation of the returned matrix depends on what restriction_is_additive() returns for each shape function.
Row and column indices are related to coarse grid and fine grid spaces, respectively, consistent with the definition of the associated operator.
If projection matrices are not implemented in the derived finite element class, this function aborts with an exception of type FiniteElement::ExcProjectionVoid. You can check whether this would happen by first calling the restriction_is_implemented() or the isotropic_restriction_is_implemented() function.

virtualinherited 
Prolongation/embedding matrix between grids.
The identity operator from a coarse grid space into a fine grid space (where both spaces are identified as functions defined on the parent and child cells) is associated with a matrix P
that maps the corresponding representations of these functions in terms of their nodal values. The restriction of this matrix P_i
to a single child cell is returned here.
The matrix P
is the concatenation, not the sum of the cell matrices P_i
. That is, if the same 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 prolongation matrices are not implemented in the derived finite element class, this function aborts with an exception of type FiniteElement::ExcEmbeddingVoid. You can check whether this would happen by first calling the prolongation_is_implemented() or the isotropic_prolongation_is_implemented() function.

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 
Return whether this element implements its hanging node constraints in the new way, which has to be used to make elements "hpcompatible". That means, the element properly implements the get_face_interpolation_matrix and get_subface_interpolation_matrix methods. Therefore the return value also indicates whether a call to the get_face_interpolation_matrix() method and the get_subface_interpolation_matrix() method will generate an error or not.
Currently the main purpose of this function is to allow the make_hanging_node_constraints method to decide whether the new procedures, which are supposed to work in the hpframework can be used, or if the old well verified but not hpcapable functions should be used. Once the transition to the new scheme for computing the interface constraints is complete, this function will be superfluous and will probably go away.
Derived classes should implement this function accordingly. The default assumption is that a finite element does not provide hpcapable face interpolation, and the default implementation therefore returns false
.

virtualinherited 
Return the matrix interpolating from the given finite element to the present one. The size of the matrix is then dofs_per_cell times source.dofs_per_cell
.
Derived elements will have to implement this function. They may only provide interpolation matrices for certain source finite elements, for example those from the same family. If they don't implement interpolation from a given element, then they must throw an exception of type ExcInterpolationNotImplemented.

virtualinherited 
Return the matrix interpolating from a face of one element to the face of the neighboring element. The size of the matrix is then source.dofs_per_face
times this>dofs_per_face
.
Derived elements will have to implement this function. They may only provide interpolation matrices for certain source finite elements, for example those from the same family. If they don't implement interpolation from a given element, then they must throw an exception of type ExcInterpolationNotImplemented.

virtualinherited 
Return the matrix interpolating from a face of one element to the subface of the neighboring element. The size of the matrix is then source.dofs_per_face
times this>dofs_per_face
.
Derived elements will have to implement this function. They may only provide interpolation matrices for certain source finite elements, for example those from the same family. If they don't implement interpolation from a given element, then they must throw an exception of type ExcInterpolationNotImplemented.

virtualinherited 
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.

virtualinherited 

virtualinherited 

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

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.

virtualinherited 
Given an index in the natural ordering of indices on a face, return the index of the same degree of freedom on the cell.
To explain the concept, consider the case where we would like to know whether a degree of freedom on a face, for example as part of an FESystem element, is primitive. Unfortunately, the is_primitive() function in the FiniteElement class takes a cell index, so we would need to find the cell index of the shape function that corresponds to the present face index. This function does that.
Code implementing this would then look like this:
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. 

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 nonzero. 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 

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.

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 

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

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.

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.

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) . 

virtualinherited 
Given the values of a function \(f(\mathbf x)\) at the (generalized) support points of the reference cell, this function then computes what the nodal values of the element are, i.e., \(\Psi_i[f]\), where \(\Psi_i\) are the node functionals of the element (see also Node values or node functionals). The values \(\Psi_i[f]\) are then the expansion coefficients for the shape functions of the finite element function that interpolates the given function \(f(x)\), i.e., \( f_h(\mathbf x) = \sum_i \Psi_i[f] \varphi_i(\mathbf x) \) is the finite element interpolant of \(f\) with the current element. The operation described here is used, for example, in the FETools::compute_node_matrix() function.
In more detail, let us assume that the generalized support points (see this glossary entry ) of the current element are \(\hat{\mathbf x}_i\) and that the node functionals associated with the current element are \(\Psi_i[\cdot]\). Then, the fact that the element is based on generalized support points, implies that if we apply \(\Psi_i\) to a (possibly vectorvalued) finite element function \(\varphi\), the result must have the form \(\Psi_i[\varphi] = f_i(\varphi(\hat{\mathbf x}_i))\) – in other words, the value of the node functional \(\Psi_i\) applied to \(\varphi\) only depends on the values of \(\varphi\) at \(\hat{\mathbf x}_i\) and not on values anywhere else, or integrals of \(\varphi\), or any other kind of information.
The exact form of \(f_i\) depends on the element. For example, for scalar Lagrange elements, we have that in fact \(\Psi_i[\varphi] = \varphi(\hat{\mathbf x}_i)\). If you combine multiple scalar Lagrange elements via an FESystem object, then \(\Psi_i[\varphi] = \varphi(\hat{\mathbf x}_i)_{c(i)}\) where \(c(i)\) is the result of the FiniteElement::system_to_component_index() function's return value's first component. In these two cases, \(f_i\) is therefore simply the identity (in the scalar case) or a function that selects a particular vector component of its argument. On the other hand, for RaviartThomas elements, one would have that \(f_i(\mathbf y) = \mathbf y \cdot \mathbf n_i\) where \(\mathbf n_i\) is the normal vector of the face at which the shape function is defined.
Given all of this, what this function does is the following: If you input a list of values of a function \(\varphi\) at all generalized support points (where each value is in fact a vector of values with as many components as the element has), then this function returns a vector of values obtained by applying the node functionals to these values. In other words, if you pass in \(\{\varphi(\hat{\mathbf x}_i)\}_{i=0}^{N1}\) then you will get out a vector \(\{\Psi[\varphi]\}_{i=0}^{N1}\) where \(N\) equals dofs_per_cell
.
[in]  support_point_values  An array of size dofs_per_cell (which equals the number of points the get_generalized_support_points() function will return) where each element is a vector with as many entries as the element has vector components. This array should contain the values of a function at the generalized support points of the current element. 
[out]  nodal_values  An array of size dofs_per_cell that contains the node functionals of the element applied to the given function. 

virtualinherited 

staticinherited 
Exception

staticinherited 
Exception

staticinherited 
Exception

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

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

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

staticinherited 
Exception

staticinherited 
Exception

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 

protectedpure virtualinherited 
Create an internal data object and return a pointer to it of which the caller of this function then assumes ownership. This object will then be passed to the FiniteElement::fill_fe_values() every time the finite element shape functions and their derivatives are evaluated on a concrete cell. The object created here is therefore used by derived classes as a place for scratch objects that are used in evaluating shape functions, as well as to store information that can be 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. 

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. 

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. 

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.

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 

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. 

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.

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. nonprimitive) 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.