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Reference documentation for deal.II version 9.1.1
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#include <deal.II/fe/fe_q_hierarchical.h>
Public Member Functions | |
FE_Q_Hierarchical (const unsigned int p) | |
virtual std::string | get_name () const override |
virtual std::unique_ptr< FiniteElement< dim, dim > > | clone () const override |
virtual bool | has_support_on_face (const unsigned int shape_index, const unsigned int face_index) const override |
virtual void | get_face_interpolation_matrix (const FiniteElement< dim > &source, FullMatrix< double > &matrix) const override |
virtual void | get_subface_interpolation_matrix (const FiniteElement< dim > &source, const unsigned int subface, FullMatrix< double > &matrix) const override |
virtual std::size_t | memory_consumption () const override |
std::vector< unsigned int > | get_embedding_dofs (const unsigned int sub_degree) const |
virtual std::pair< Table< 2, bool >, std::vector< unsigned int > > | get_constant_modes () const override |
template<> | |
bool | has_support_on_face (const unsigned int, const unsigned int) const |
template<> | |
bool | has_support_on_face (const unsigned int shape_index, const unsigned int face_index) const |
Functions to support hp | |
virtual bool | hp_constraints_are_implemented () const override |
virtual void | get_interpolation_matrix (const FiniteElement< dim > &source, FullMatrix< double > &matrix) const override |
virtual const FullMatrix< double > & | get_prolongation_matrix (const unsigned int child, const RefinementCase< dim > &refinement_case=RefinementCase< dim >::isotropic_refinement) const override |
virtual std::vector< std::pair< unsigned int, unsigned int > > | hp_vertex_dof_identities (const FiniteElement< dim > &fe_other) const override |
virtual std::vector< std::pair< unsigned int, unsigned int > > | hp_line_dof_identities (const FiniteElement< dim > &fe_other) const override |
virtual std::vector< std::pair< unsigned int, unsigned int > > | hp_quad_dof_identities (const FiniteElement< dim > &fe_other) const override |
virtual FiniteElementDomination::Domination | compare_for_domination (const FiniteElement< dim > &fe_other, const unsigned int codim=0) const override final |
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FE_Poly (const TensorProductPolynomials< dim > &poly_space, const FiniteElementData< dim > &fe_data, const std::vector< bool > &restriction_is_additive_flags, const std::vector< ComponentMask > &nonzero_components) | |
unsigned int | get_degree () const |
std::vector< unsigned int > | get_poly_space_numbering () const |
std::vector< unsigned int > | get_poly_space_numbering_inverse () const |
virtual double | shape_value (const unsigned int i, const Point< dim > &p) const override |
virtual double | shape_value_component (const unsigned int i, const Point< dim > &p, const unsigned int component) const override |
virtual Tensor< 1, dim > | shape_grad (const unsigned int i, const Point< dim > &p) const override |
virtual Tensor< 1, dim > | shape_grad_component (const unsigned int i, const Point< dim > &p, const unsigned int component) const override |
virtual Tensor< 2, dim > | shape_grad_grad (const unsigned int i, const Point< dim > &p) const override |
virtual Tensor< 2, dim > | shape_grad_grad_component (const unsigned int i, const Point< dim > &p, const unsigned int component) const override |
virtual Tensor< 3, dim > | shape_3rd_derivative (const unsigned int i, const Point< dim > &p) const override |
virtual Tensor< 3, dim > | shape_3rd_derivative_component (const unsigned int i, const Point< dim > &p, const unsigned int component) const override |
virtual Tensor< 4, dim > | shape_4th_derivative (const unsigned int i, const Point< dim > &p) const override |
virtual Tensor< 4, dim > | shape_4th_derivative_component (const unsigned int i, const Point< dim > &p, const unsigned int component) const override |
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FiniteElement (const FiniteElementData< dim > &fe_data, const std::vector< bool > &restriction_is_additive_flags, const std::vector< ComponentMask > &nonzero_components) | |
FiniteElement (FiniteElement< dim, spacedim > &&)=default | |
FiniteElement (const FiniteElement< dim, spacedim > &)=default | |
virtual | ~FiniteElement () 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 FiniteElement< dim, spacedim > &) const |
virtual const FullMatrix< double > & | get_restriction_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 void | get_interpolation_matrix (const FiniteElement< dim, spacedim > &source, FullMatrix< double > &matrix) const |
virtual void | get_face_interpolation_matrix (const FiniteElement< dim, spacedim > &source, FullMatrix< double > &matrix) const |
virtual void | get_subface_interpolation_matrix (const FiniteElement< dim, spacedim > &source, const unsigned int subface, FullMatrix< double > &matrix) 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 |
virtual FiniteElementDomination::Domination | compare_for_face_domination (const FiniteElement< dim, spacedim > &fe_other) const final |
virtual FiniteElementDomination::Domination | compare_for_domination (const FiniteElement< dim, spacedim > &fe_other, const unsigned int codim=0) const |
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 | adjust_quad_dof_index_for_face_orientation (const unsigned int index, 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 |
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 |
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 |
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 |
bool | has_face_support_points () const |
virtual Point< dim - 1 > | unit_face_support_point (const unsigned int index) const |
const std::vector< Point< dim > > & | get_generalized_support_points () const |
bool | has_generalized_support_points () const |
const std::vector< Point< dim - 1 > > & | get_generalized_face_support_points () const |
bool | has_generalized_face_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 |
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Subscriptor () | |
Subscriptor (const Subscriptor &) | |
Subscriptor (Subscriptor &&) noexcept | |
virtual | ~Subscriptor () |
Subscriptor & | operator= (const Subscriptor &) |
Subscriptor & | operator= (Subscriptor &&) noexcept |
void | subscribe (std::atomic< bool > *const validity, const std::string &identifier="") const |
void | unsubscribe (std::atomic< bool > *const validity, const std::string &identifier="") const |
unsigned int | n_subscriptions () const |
template<typename StreamType > | |
void | list_subscribers (StreamType &stream) const |
void | list_subscribers () const |
template<class Archive > | |
void | serialize (Archive &ar, const unsigned int version) |
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FiniteElementData (const std::vector< unsigned int > &dofs_per_object, const unsigned int n_components, const unsigned int degree, const Conformity conformity=unknown, const BlockIndices &block_indices=BlockIndices()) | |
unsigned int | n_dofs_per_vertex () const |
unsigned int | n_dofs_per_line () const |
unsigned int | n_dofs_per_quad () const |
unsigned int | n_dofs_per_hex () const |
unsigned int | n_dofs_per_face () const |
unsigned int | n_dofs_per_cell () const |
template<int structdim> | |
unsigned int | n_dofs_per_object () 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 |
bool | operator== (const FiniteElementData &) const |
Private Member Functions | |
void | build_dofs_cell (std::vector< FullMatrix< double >> &dofs_cell, std::vector< FullMatrix< double >> &dofs_subcell) const |
void | initialize_constraints (const std::vector< FullMatrix< double >> &dofs_subcell) |
void | initialize_embedding_and_restriction (const std::vector< FullMatrix< double >> &dofs_cell, const std::vector< FullMatrix< double >> &dofs_subcell) |
void | initialize_generalized_support_points () |
void | initialize_generalized_face_support_points () |
Static Private Member Functions | |
static std::vector< unsigned int > | get_dpo_vector (const unsigned int degree) |
static std::vector< unsigned int > | hierarchic_to_fe_q_hierarchical_numbering (const FiniteElementData< dim > &fe) |
static std::vector< unsigned int > | face_fe_q_hierarchical_to_hierarchic_numbering (const unsigned int degree) |
Private Attributes | |
const std::vector< unsigned int > | face_renumber |
Friends | |
template<int dim1> | |
class | FE_Q_Hierarchical |
Additional Inherited Members | |
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enum | Conformity { unknown = 0x00, L2 = 0x01, Hcurl = 0x02, Hdiv = 0x04, H1 = Hcurl | Hdiv, H2 = 0x0e } |
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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 () |
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static ::ExceptionBase & | ExcInUse (int arg1, std::string arg2, std::string arg3) |
static ::ExceptionBase & | ExcNoSubscriber (std::string arg1, std::string arg2) |
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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 |
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static const unsigned int | space_dimension |
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static const unsigned int | dimension = dim |
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void | correct_third_derivatives (internal::FEValuesImplementation::FiniteElementRelatedData< dim, dim > &output_data, const internal::FEValuesImplementation::MappingRelatedData< dim, dim > &mapping_data, const unsigned int n_q_points, const unsigned int dof) const |
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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 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 Quadrature< dim - 1 > &quadrature, const Mapping< dim, spacedim > &mapping, const typename Mapping< dim, spacedim >::InternalDataBase &mapping_internal, const ::internal::FEValuesImplementation::MappingRelatedData< dim, spacedim > &mapping_data, const InternalDataBase &fe_internal, ::internal::FEValuesImplementation::FiniteElementRelatedData< dim, spacedim > &output_data) const=0 |
virtual void | fill_fe_subface_values (const typename Triangulation< dim, 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 |
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static std::vector< unsigned int > | compute_n_nonzero_components (const std::vector< ComponentMask > &nonzero_components) |
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TensorProductPolynomials< dim > | poly_space |
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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< Point< dim - 1 > > | unit_face_support_points |
std::vector< Point< dim > > | generalized_support_points |
std::vector< Point< dim - 1 > > | generalized_face_support_points |
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::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::pair< std::pair< unsigned int, unsigned int >, unsigned int > > | face_system_to_base_table |
BlockIndices | base_to_block_indices |
std::vector< std::pair< std::pair< unsigned int, unsigned int >, unsigned int > > | component_to_base_table |
const std::vector< bool > | restriction_is_additive_flags |
const std::vector< ComponentMask > | nonzero_components |
const std::vector< unsigned int > | n_nonzero_components_table |
const bool | cached_primitivity |
Implementation of hierarchical Qp
shape functions that yield the finite element space of continuous, piecewise polynomials of degree p
. This class is realized using tensor product polynomials based on a hierarchical basis Polynomials::Hierarchical on the interval [0,1]
which is suitable for building an hp
tensor product finite element if we assume that each element has a single degree.
The constructor of this class takes the degree p
of this finite element.
This class is not implemented for the codimension one case (spacedim != dim
).
The constructor creates a TensorProductPolynomials object that includes the tensor product of Hierarchical
polynomials of degree p
. This TensorProductPolynomials
object provides all values and derivatives of the shape functions.
The original ordering of the shape functions represented by the TensorProductPolynomials is a tensor product numbering. However, the shape functions on a cell are renumbered beginning with the shape functions whose support points are at the vertices, then on the line, on the quads, and finally (for 3d) on the hexes. To be explicit, these numberings are listed in the following:
The \(Q_1^H\) element is of polynomial degree one and, consequently, is exactly the same as the \(Q_1\) element in class FE_Q. In particular, the shape function are defined in the exact same way:
1D case:
* 0-------1 *
2D case:
* 2-------3 * | | * | | * | | * 0-------1 *
3D case:
* 6-------7 6-------7 * /| | / /| * / | | / / | * / | | / / | * 4 | | 4-------5 | * | 2-------3 | | 3 * | / / | | / * | / / | | / * |/ / | |/ * 0-------1 0-------1 *
The respective coordinate values of the support points of the degrees of freedom are as follows:
[0, 0, 0]
; [1, 0, 0]
; [0, 1, 0]
; [1, 1, 0]
; [0, 0, 1]
; [1, 0, 1]
; [0, 1, 1]
; [1, 1, 1]
; In 2d, these shape functions look as follows:
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\(Q_1^H\) element, shape function 0 | \(Q_1^H\) element, shape function 1 |
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\(Q_1^H\) element, shape function 2 | \(Q_1^H\) element, shape function 3 |
1D case:
* 0---2---1 *
2D case:
* 2---7---3 * | | * 4 8 5 * | | * 0---6---1 *
3D case:
* 6--15---7 6--15---7 * /| | / /| * 12 | 19 12 1319 * / 18 | / / | * 4 | | 4---14--5 | * | 2---11--3 | | 3 * | / / | 17 / * 16 8 9 16 | 9 * |/ / | |/ * 0---10--1 0---8---1 * * *-------* *-------* * /| | / /| * / | 23 | / 25 / | * / | | / / | * * | | *-------* | * |20 *-------* | |21 * * | / / | 22 | / * | / 24 / | | / * |/ / | |/ * *-------* *-------* *
The center vertex has number 26.
The respective coordinate values of the support points of the degrees of freedom are as follows:
[0, 0, 0]
; [1, 0, 0]
; [0, 1, 0]
; [1, 1, 0]
; [0, 0, 1]
; [1, 0, 1]
; [0, 1, 1]
; [1, 1, 1]
; [0, 1/2, 0]
; [1, 1/2, 0]
; [1/2, 0, 0]
; [1/2, 1, 0]
; [0, 1/2, 1]
; [1, 1/2, 1]
; [1/2, 0, 1]
; [1/2, 1, 1]
; [0, 0, 1/2]
; [1, 0, 1/2]
; [0, 1, 1/2]
; [1, 1, 1/2]
; [0, 1/2, 1/2]
; [1, 1/2, 1/2]
; [1/2, 0, 1/2]
; [1/2, 1, 1/2]
; [1/2, 1/2, 0]
; [1/2, 1/2, 1]
; [1/2, 1/2, 1/2]
; In 2d, these shape functions look as follows (the black plane corresponds to zero; negative shape function values may not be visible):
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\(Q_2^H\) element, shape function 0 | \(Q_2^H\) element, shape function 1 |
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\(Q_2^H\) element, shape function 2 | \(Q_2^H\) element, shape function 3 |
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\(Q_2^H\) element, shape function 4 | \(Q_2^H\) element, shape function 5 |
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\(Q_2^H\) element, shape function 6 | \(Q_2^H\) element, shape function 7 |
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\(Q_2^H\) element, shape function 8 |
1D case:
* 0--2--3--1 *
* 2--10-11-3 * | | * 5 14 15 7 * | | * 4 12 13 6 * | | * 0--8--9--1 *
In 2d, these shape functions look as follows (the black plane corresponds to zero; negative shape function values may not be visible):
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\(Q_3^H\) element, shape function 0 | \(Q_3^H\) element, shape function 1 |
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\(Q_3^H\) element, shape function 2 | \(Q_3^H\) element, shape function 3 |
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\(Q_3^H\) element, shape function 4 | \(Q_3^H\) element, shape function 5 |
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\(Q_3^H\) element, shape function 6 | \(Q_3^H\) element, shape function 7 |
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\(Q_3^H\) element, shape function 8 | \(Q_3^H\) element, shape function 9 |
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\(Q_3^H\) element, shape function 10 | \(Q_3^H\) element, shape function 11 |
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\(Q_3^H\) element, shape function 12 | \(Q_3^H\) element, shape function 13 |
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\(Q_3^H\) element, shape function 14 | \(Q_3^H\) element, shape function 15 |
1D case:
* 0--2--3--4--1 *
* 2--13-14-15-3 * | | * 6 22 23 24 9 * | | * 5 19 20 21 8 * | | * 4 16 17 18 7 * | | * 0--10-11-12-1 *
In 2d, these shape functions look as follows (the black plane corresponds to zero; negative shape function values may not be visible):
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\(Q_4^H\) element, shape function 0 | \(Q_4^H\) element, shape function 1 |
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\(Q_4^H\) element, shape function 2 | \(Q_4^H\) element, shape function 3 |
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\(Q_4^H\) element, shape function 4 | \(Q_4^H\) element, shape function 5 |
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\(Q_4^H\) element, shape function 6 | \(Q_4^H\) element, shape function 7 |
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\(Q_4^H\) element, shape function 8 | \(Q_4^H\) element, shape function 9 |
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\(Q_4^H\) element, shape function 10 | \(Q_4^H\) element, shape function 11 |
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\(Q_4^H\) element, shape function 12 | \(Q_4^H\) element, shape function 13 |
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\(Q_4^H\) element, shape function 14 | \(Q_4^H\) element, shape function 15 |
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\(Q_4^H\) element, shape function 16 | \(Q_4^H\) element, shape function 17 |
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\(Q_4^H\) element, shape function 18 | \(Q_4^H\) element, shape function 19 |
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\(Q_4^H\) element, shape function 20 | \(Q_4^H\) element, shape function 21 |
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\(Q_4^H\) element, shape function 22 | \(Q_4^H\) element, shape function 23 |
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\(Q_4^H\) element, shape function 24 |
Definition at line 543 of file fe_q_hierarchical.h.
FE_Q_Hierarchical< dim >::FE_Q_Hierarchical | ( | const unsigned int | p | ) |
Constructor for tensor product polynomials of degree p
.
Definition at line 61 of file fe_q_hierarchical.cc.
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overridevirtual |
Return a string that uniquely identifies a finite element. This class returns FE_Q_Hierarchical<dim>(degree)
, with dim
and degree
replaced by appropriate values.
Implements FiniteElement< dim, dim >.
Definition at line 121 of file fe_q_hierarchical.cc.
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overridevirtual |
A sort of virtual copy constructor, this function returns a copy of the finite element object. Derived classes need to override the function here in this base class and return an object of the same type as the derived class.
Some places in the library, for example the constructors of FESystem as well as the hp::FECollection class, need to make copies of finite elements without knowing their exact type. They do so through this function.
Implements FiniteElement< dim, dim >.
Definition at line 140 of file fe_q_hierarchical.cc.
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This function returns true
, if the shape function shape_index
has non-zero function values somewhere on the face face_index
.
Reimplemented from FiniteElement< dim, dim >.
Definition at line 2166 of file fe_q_hierarchical.cc.
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Return whether this element implements its hanging node constraints in the new way, which has to be used to make elements "hp compatible".
For the FE_Q_Hierarchical class the result is always true (independent of the degree of the element), as it implements the complete set of functions necessary for hp capability.
Reimplemented from FiniteElement< dim, dim >.
Definition at line 220 of file fe_q_hierarchical.cc.
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Return the matrix interpolating from the given finite element to the present one. Interpolation only between FE_Q_Hierarchical is supported.
Definition at line 149 of file fe_q_hierarchical.cc.
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Embedding matrix between grids. Only isotropic refinement is supported.
Reimplemented from FiniteElement< dim, dim >.
Definition at line 200 of file fe_q_hierarchical.cc.
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If, on a vertex, several finite elements are active, the hp code first assigns the degrees of freedom of each of these FEs different global indices. It then calls this function to find out which of them should get identical values, and consequently can receive the same global DoF index. This function therefore returns a list of identities between DoFs of the present finite element object with the DoFs of fe_other
, which is a reference to a finite element object representing one of the other finite elements active on this particular vertex. The function computes which of the degrees of freedom of the two finite element objects are equivalent, both numbered between zero and the corresponding value of dofs_per_vertex of the two finite elements. The first index of each pair denotes one of the vertex dofs of the present element, whereas the second is the corresponding index of the other finite element.
Definition at line 228 of file fe_q_hierarchical.cc.
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Same as above but for lines.
Definition at line 261 of file fe_q_hierarchical.cc.
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Same as above but for faces.
Definition at line 299 of file fe_q_hierarchical.cc.
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Return whether this element dominates another one given as argument fe_other
, whether it is the other way around, whether neither dominates, or if either could dominate. The codim
parameter describes the codimension of the investigated subspace and specifies that it is subject to this comparison. For example, if codim==0
then this function compares which element dominates at the cell level. If codim==1
, then the elements are compared at faces, i.e., the comparison happens between the function spaces of the two finite elements as restricted to a face. Larger values of codim
work correspondingly.
For a definition of domination, see FiniteElementDomination::Domination and in particular the hp paper.
Definition at line 337 of file fe_q_hierarchical.cc.
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Return the matrix interpolating from a face of one element to the face of the neighboring element. The size of the matrix is then source.dofs_per_face
times this->dofs_per_face
.
Derived elements will have to implement this function. They may only provide interpolation matrices for certain source finite elements, for example those from the same family. If they don't implement interpolation from a given element, then they must throw an exception of type FiniteElement<dim>::ExcInterpolationNotImplemented
.
Definition at line 915 of file fe_q_hierarchical.cc.
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Return the matrix interpolating from a face of one element to the subface of the neighboring element. The size of the matrix is then source.dofs_per_face
times this->dofs_per_face
.
Derived elements will have to implement this function. They may only provide interpolation matrices for certain source finite elements, for example those from the same family. If they don't implement interpolation from a given element, then they must throw an exception of type ExcInterpolationNotImplemented
.
Definition at line 1001 of file fe_q_hierarchical.cc.
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Determine an estimate for the memory consumption (in bytes) of this object.
This function is made virtual, since finite element objects are usually accessed through pointers to their base class, rather than the class itself.
Reimplemented from FiniteElement< dim, dim >.
Definition at line 2411 of file fe_q_hierarchical.cc.
std::vector< unsigned int > FE_Q_Hierarchical< dim >::get_embedding_dofs | ( | const unsigned int | sub_degree | ) | const |
For a finite element of degree sub_degree
< degree
, we return a vector which maps the numbering on an FE of degree sub_degree
into the numbering on this element.
Definition at line 2254 of file fe_q_hierarchical.cc.
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Return a list of constant modes of the element. For this element, the list consists of true arguments for the first vertex shape functions and false for the remaining ones.
Reimplemented from FiniteElement< dim, dim >.
Definition at line 2394 of file fe_q_hierarchical.cc.
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Only for internal use. Its full name is get_dofs_per_object_vector
function and it creates the dofs_per_object
vector that is needed within the constructor to be passed to the constructor of FiniteElementData
.
Definition at line 1934 of file fe_q_hierarchical.cc.
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The numbering of the degrees of freedom in continuous finite elements is hierarchic, i.e. in such a way that we first number the vertex dofs, in the order of the vertices as defined by the triangulation, then the line dofs in the order and respecting the direction of the lines, then the dofs on quads, etc.
The dofs associated with 1d hierarchical polynomials are ordered with the vertices first ( \(phi_0(x)=1-x\) and \(phi_1(x)=x\)) and then the line dofs (the higher degree polynomials). The 2d and 3d hierarchical polynomials originate from the 1d hierarchical polynomials by tensor product. In the following, the resulting numbering of dofs will be denoted by fe_q_hierarchical numbering
.
This function constructs a table which fe_q_hierarchical index each degree of freedom in the hierarchic numbering would have.
This function is analogous to the FETools::hierarchic_to_lexicographic_numbering() function. However, in contrast to the fe_q_hierarchical numbering defined above, the lexicographic numbering originates from the tensor products of consecutive numbered dofs (like for LagrangeEquidistant).
It is assumed that the size of the output argument already matches the correct size, which is equal to the number of degrees of freedom in the finite element.
Definition at line 1946 of file fe_q_hierarchical.cc.
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This is an analogon to the previous function, but working on faces.
Definition at line 2120 of file fe_q_hierarchical.cc.
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Initialize two auxiliary fields that will be used in setting up the various matrices in the constructor.
Definition at line 389 of file fe_q_hierarchical.cc.
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Initialize the hanging node constraints matrices. Called from the constructor.
Definition at line 503 of file fe_q_hierarchical.cc.
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Initialize the embedding matrices. Called from the constructor.
Definition at line 673 of file fe_q_hierarchical.cc.
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Initialize the generalized_support_points
field of the FiniteElement class. Called from the constructor.
Definition at line 808 of file fe_q_hierarchical.cc.
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Initialize the generalized_face_support_points
field of the FiniteElement class. Called from the constructor.
Definition at line 1884 of file fe_q_hierarchical.cc.
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This function returns true
, if the shape function shape_index
has non-zero function values somewhere on the face face_index
. The function is typically used to determine whether some matrix elements resulting from face integrals can be assumed to be zero and may therefore be omitted from integration.
A default implementation is provided in this base class which always returns true
. This is the safe way to go.
Reimplemented from FiniteElement< dim, dim >.
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This function returns true
, if the shape function shape_index
has non-zero function values somewhere on the face face_index
. The function is typically used to determine whether some matrix elements resulting from face integrals can be assumed to be zero and may therefore be omitted from integration.
A default implementation is provided in this base class which always returns true
. This is the safe way to go.
Reimplemented from FiniteElement< dim, dim >.
Definition at line 2143 of file fe_q_hierarchical.cc.
Allow access from other dimensions. We need this since we want to call the functions get_dpo_vector
and lexicographic_to_hierarchic_numbering
for the faces of the finite element of dimension dim+1.
Definition at line 795 of file fe_q_hierarchical.h.
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Mapping from lexicographic to shape function numbering on first face.
Definition at line 786 of file fe_q_hierarchical.h.