Reference documentation for deal.II version 9.2.0
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#include <deal.II/fe/fe_q_dg0.h>
Public Member Functions | |
FE_Q_DG0 (const unsigned int p) | |
FE_Q_DG0 (const Quadrature< 1 > &points) | |
virtual std::string | get_name () const override |
virtual void | convert_generalized_support_point_values_to_dof_values (const std::vector< Vector< double >> &support_point_values, std::vector< double > &nodal_values) const override |
virtual void | get_interpolation_matrix (const FiniteElement< dim, spacedim > &source, FullMatrix< double > &matrix) const override |
virtual bool | has_support_on_face (const unsigned int shape_index, const unsigned int face_index) const override |
virtual std::pair< Table< 2, bool >, std::vector< unsigned int > > | get_constant_modes () const override |
virtual std::unique_ptr< FiniteElement< dim, spacedim > > | clone () const override |
virtual FiniteElementDomination::Domination | compare_for_domination (const FiniteElement< dim, spacedim > &fe_other, const unsigned int codim=0) const override final |
Public Member Functions inherited from FE_Q_Base< TensorProductPolynomialsConst< dim >, dim, dim > | |
FE_Q_Base (const TensorProductPolynomialsConst< dim > &poly_space, const FiniteElementData< dim > &fe_data, const std::vector< bool > &restriction_is_additive_flags) | |
virtual void | get_face_interpolation_matrix (const FiniteElement< dim, spacedim > &source, FullMatrix< double > &matrix) const override |
virtual void | get_subface_interpolation_matrix (const FiniteElement< dim, spacedim > &source, const unsigned int subface, FullMatrix< double > &matrix) const override |
virtual const FullMatrix< double > & | get_restriction_matrix (const unsigned int child, const RefinementCase< dim > &refinement_case=RefinementCase< dim >::isotropic_refinement) const override |
virtual const FullMatrix< double > & | get_prolongation_matrix (const unsigned int child, const RefinementCase< dim > &refinement_case=RefinementCase< dim >::isotropic_refinement) const override |
virtual unsigned int | face_to_cell_index (const unsigned int face_dof_index, const unsigned int face, const bool face_orientation=true, const bool face_flip=false, const bool face_rotation=false) const override |
virtual std::pair< Table< 2, bool >, std::vector< unsigned int > > | get_constant_modes () const override |
virtual bool | hp_constraints_are_implemented () const override |
virtual std::vector< std::pair< unsigned int, unsigned int > > | hp_vertex_dof_identities (const FiniteElement< dim, spacedim > &fe_other) const override |
virtual std::vector< std::pair< unsigned int, unsigned int > > | hp_line_dof_identities (const FiniteElement< dim, spacedim > &fe_other) const override |
virtual std::vector< std::pair< unsigned int, unsigned int > > | hp_quad_dof_identities (const FiniteElement< dim, spacedim > &fe_other) const override |
Public Member Functions inherited from FE_Poly< TensorProductPolynomialsConst< dim >, dim, spacedim > | |
FE_Poly (const TensorProductPolynomialsConst< 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 |
virtual UpdateFlags | requires_update_flags (const UpdateFlags update_flags) const override |
std::vector< unsigned int > | get_poly_space_numbering () const |
std::vector< unsigned int > | get_poly_space_numbering_inverse () const |
virtual double | shape_value (const unsigned int i, const Point< dim > &p) const override |
virtual double | shape_value_component (const unsigned int i, const Point< dim > &p, const unsigned int component) const override |
virtual Tensor< 1, dim > | shape_grad (const unsigned int i, const Point< dim > &p) const override |
virtual Tensor< 1, dim > | shape_grad_component (const unsigned int i, const Point< dim > &p, const unsigned int component) const override |
virtual Tensor< 2, dim > | shape_grad_grad (const unsigned int i, const Point< dim > &p) const override |
virtual Tensor< 2, dim > | shape_grad_grad_component (const unsigned int i, const Point< dim > &p, const unsigned int component) const override |
virtual Tensor< 3, dim > | shape_3rd_derivative (const unsigned int i, const Point< dim > &p) const override |
virtual Tensor< 3, dim > | shape_3rd_derivative_component (const unsigned int i, const Point< dim > &p, const unsigned int component) const override |
virtual Tensor< 4, dim > | shape_4th_derivative (const unsigned int i, const Point< dim > &p) const override |
virtual Tensor< 4, dim > | shape_4th_derivative_component (const unsigned int i, const Point< dim > &p, const unsigned int component) const override |
virtual std::size_t | memory_consumption () const override |
Public Member Functions inherited from FiniteElement< dim, spacedim > | |
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 |
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 FiniteElementDomination::Domination | compare_for_face_domination (const FiniteElement< dim, spacedim > &fe_other) const final |
virtual bool | operator== (const FiniteElement< dim, spacedim > &fe) const |
bool | operator!= (const FiniteElement< dim, spacedim > &) 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 |
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 |
GeometryPrimitive | get_associated_geometry_primitive (const unsigned int cell_dof_index) const |
virtual std::size_t | memory_consumption () const |
Public Member Functions inherited from Subscriptor | |
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) |
Public Member Functions inherited from FiniteElementData< dim > | |
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 |
Static Private Member Functions | |
static std::vector< bool > | get_riaf_vector (const unsigned int degree) |
static std::vector< unsigned int > | get_dpo_vector (const unsigned int degree) |
Additional Inherited Members | |
Public Types inherited from FiniteElementData< dim > | |
enum | Conformity { unknown = 0x00, L2 = 0x01, Hcurl = 0x02, Hdiv = 0x04, H1 = Hcurl | Hdiv, H2 = 0x0e } |
Static Public Member Functions inherited from FE_Q_Base< TensorProductPolynomialsConst< dim >, dim, dim > | |
static ::ExceptionBase & | ExcFEQCannotHaveDegree0 () |
Static Public Member Functions inherited from FiniteElement< dim, spacedim > | |
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 Public Member Functions inherited from Subscriptor | |
static ::ExceptionBase & | ExcInUse (int arg1, std::string arg2, std::string arg3) |
static ::ExceptionBase & | ExcNoSubscriber (std::string arg1, std::string arg2) |
Public Attributes inherited from FiniteElementData< dim > | |
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 inherited from FiniteElement< dim, spacedim > | |
static const unsigned int | space_dimension = spacedim |
Static Public Attributes inherited from FiniteElementData< dim > | |
static const unsigned int | dimension = dim |
Protected Member Functions inherited from FE_Q_Base< TensorProductPolynomialsConst< dim >, dim, dim > | |
void | initialize (const std::vector< Point< 1 >> &support_points_1d) |
void | initialize_constraints (const std::vector< Point< 1 >> &points) |
void | initialize_unit_support_points (const std::vector< Point< 1 >> &points) |
void | initialize_unit_face_support_points (const std::vector< Point< 1 >> &points) |
void | initialize_quad_dof_index_permutation () |
Protected Member Functions inherited from FE_Poly< TensorProductPolynomialsConst< dim >, dim, spacedim > | |
virtual std::unique_ptr< typename FiniteElement< dim, spacedim >::InternalDataBase > | get_data (const UpdateFlags update_flags, const Mapping< dim, spacedim > &, const Quadrature< dim > &quadrature, ::internal::FEValuesImplementation::FiniteElementRelatedData< dim, spacedim > &output_data) const override |
virtual void | fill_fe_values (const typename Triangulation< dim, spacedim >::cell_iterator &cell, const CellSimilarity::Similarity cell_similarity, const Quadrature< dim > &quadrature, const Mapping< dim, spacedim > &mapping, const typename Mapping< dim, spacedim >::InternalDataBase &mapping_internal, const ::internal::FEValuesImplementation::MappingRelatedData< dim, spacedim > &mapping_data, const typename FiniteElement< dim, spacedim >::InternalDataBase &fe_internal, ::internal::FEValuesImplementation::FiniteElementRelatedData< dim, spacedim > &output_data) const override |
void | fill_fe_values (const Triangulation< 1, 2 >::cell_iterator &, const CellSimilarity::Similarity cell_similarity, const Quadrature< 1 > &quadrature, const Mapping< 1, 2 > &mapping, const Mapping< 1, 2 >::InternalDataBase &mapping_internal, const ::internal::FEValuesImplementation::MappingRelatedData< 1, 2 > &mapping_data, const FiniteElement< 1, 2 >::InternalDataBase &fe_internal,::internal::FEValuesImplementation::FiniteElementRelatedData< 1, 2 > &output_data) const |
void | fill_fe_values (const Triangulation< 2, 3 >::cell_iterator &, const CellSimilarity::Similarity cell_similarity, const Quadrature< 2 > &quadrature, const Mapping< 2, 3 > &mapping, const Mapping< 2, 3 >::InternalDataBase &mapping_internal, const ::internal::FEValuesImplementation::MappingRelatedData< 2, 3 > &mapping_data, const FiniteElement< 2, 3 >::InternalDataBase &fe_internal,::internal::FEValuesImplementation::FiniteElementRelatedData< 2, 3 > &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 typename FiniteElement< dim, spacedim >::InternalDataBase &fe_internal, ::internal::FEValuesImplementation::FiniteElementRelatedData< dim, spacedim > &output_data) const override |
virtual void | fill_fe_subface_values (const typename Triangulation< dim, spacedim >::cell_iterator &cell, const unsigned int face_no, const unsigned int sub_no, const Quadrature< dim - 1 > &quadrature, const Mapping< dim, spacedim > &mapping, const typename Mapping< dim, spacedim >::InternalDataBase &mapping_internal, const ::internal::FEValuesImplementation::MappingRelatedData< dim, spacedim > &mapping_data, const typename FiniteElement< dim, spacedim >::InternalDataBase &fe_internal, ::internal::FEValuesImplementation::FiniteElementRelatedData< dim, spacedim > &output_data) const override |
void | correct_hessians (internal::FEValuesImplementation::FiniteElementRelatedData< dim, spacedim > &output_data, const internal::FEValuesImplementation::MappingRelatedData< dim, spacedim > &mapping_data, const unsigned int n_q_points) const |
void | correct_third_derivatives (internal::FEValuesImplementation::FiniteElementRelatedData< dim, spacedim > &output_data, const internal::FEValuesImplementation::MappingRelatedData< dim, spacedim > &mapping_data, const unsigned int n_q_points) const |
Protected Member Functions inherited from FiniteElement< dim, spacedim > | |
void | reinit_restriction_and_prolongation_matrices (const bool isotropic_restriction_only=false, const bool isotropic_prolongation_only=false) |
TableIndices< 2 > | interface_constraints_size () const |
virtual std::unique_ptr< InternalDataBase > | get_face_data (const UpdateFlags update_flags, const Mapping< dim, spacedim > &mapping, const 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 |
Static Protected Member Functions inherited from FE_Q_Base< TensorProductPolynomialsConst< dim >, dim, dim > | |
static std::vector< unsigned int > | get_dpo_vector (const unsigned int degree) |
Static Protected Member Functions inherited from FiniteElement< dim, spacedim > | |
static std::vector< unsigned int > | compute_n_nonzero_components (const std::vector< ComponentMask > &nonzero_components) |
Protected Attributes inherited from FE_Poly< TensorProductPolynomialsConst< dim >, dim, spacedim > | |
TensorProductPolynomialsConst< dim > | poly_space |
Protected Attributes inherited from FiniteElement< dim, spacedim > | |
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 a scalar Lagrange finite element Qp+DG0
that yields the finite element space of continuous, piecewise polynomials of degree p
in each coordinate direction plus the space of locally constant functions. This class is realized using tensor product polynomials based on equidistant or given support points.
The standard constructor of this class takes the degree p
of this finite element. Alternatively, it can take a quadrature formula points
defining the support points of the Lagrange interpolation in one coordinate direction.
For more information about the spacedim
template parameter check the documentation of FiniteElement or the one of Triangulation.
For more information regarding this element see: Boffi, D., et al. "Local Mass Conservation of Stokes Finite Elements." Journal of Scientific Computing (2012): 1-18.
The constructor creates a TensorProductPolynomials object that includes the tensor product of LagrangeEquidistant
polynomials of degree p
plus the locally constant function. This TensorProductPolynomialsConst
object provides all values and derivatives of the shape functions. In case a quadrature rule is given, the constructor creates a TensorProductPolynomialsConst object that includes the tensor product of Lagrange
polynomials with the support points from points
and a locally constant function.
Furthermore the constructor fills the interface_constrains
, the prolongation
(embedding) and the restriction
matrices.
The original ordering of the shape functions represented by the TensorProductPolynomialsConst 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. Finally there is a support point for the discontinuous shape function in the middle of the cell. To be explicit, these numberings are listed in the following:
1D case:
* 0---2---1 *
2D case:
* 2-------3 * | | * | 5 | * | | * 0-------1 *
3D case:
* 6-------7 6-------7 * /| | / /| * / | | / / | * / | | / / | * 4 | 8 | 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]
; [1/2, 1/2, 1/2]
; 1D case:
* 0---2---1 *
Index 3 has the same coordinates as index 2
2D case:
* 2---7---3 * | | * 4 8 5 * | | * 0---6---1 *
Index 9 has the same coordinates as index 2
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 vertices have number 26 and 27.
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]
; [1/2, 1/2, 1/2]
; 1D case:
* 0--2-4-3--1 *
* 2--10-11-3 * | | * 5 14 15 7 * | 16 | * 4 12 13 6 * | | * 0--8--9--1 *
1D case:
* 0--2--3--4--1 *
Index 5 has the same coordinates as index 3
* 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 *Index 21 has the same coordinates as index 20
Definition at line 239 of file fe_q_dg0.h.
Constructor for tensor product polynomials of degree p
plus locally constant functions.
Definition at line 36 of file fe_q_dg0.cc.
FE_Q_DG0< dim, spacedim >::FE_Q_DG0 | ( | const Quadrature< 1 > & | points | ) |
Constructor for tensor product polynomials with support points points
plus locally constant functions based on a one-dimensional quadrature formula. The degree of the finite element is points.size()-1
. Note that the first point has to be 0 and the last one 1.
Definition at line 64 of file fe_q_dg0.cc.
|
overridevirtual |
Return a string that uniquely identifies a finite element. This class returns FE_Q_DG0<dim>(degree)
, with dim
and degree
replaced by appropriate values.
Implements FiniteElement< dim, spacedim >.
Definition at line 95 of file fe_q_dg0.cc.
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overridevirtual |
Given the values of a function \(f(\mathbf x)\) at the (generalized) support points of the reference cell, this function then computes what the nodal values of the element are, i.e., \(\Psi_i[f]\), where \(\Psi_i\) are the node functionals of the element (see also Node values or node functionals). The values \(\Psi_i[f]\) are then the expansion coefficients for the shape functions of the finite element function that interpolates the given function \(f(x)\), i.e., \( f_h(\mathbf x) = \sum_i \Psi_i[f] \varphi_i(\mathbf x) \) is the finite element interpolant of \(f\) with the current element. The operation described here is used, for example, in the FETools::compute_node_matrix() function.
In more detail, let us assume that the generalized support points (see this glossary entry ) of the current element are \(\hat{\mathbf x}_i\) and that the node functionals associated with the current element are \(\Psi_i[\cdot]\). Then, the fact that the element is based on generalized support points, implies that if we apply \(\Psi_i\) to a (possibly vector-valued) finite element function \(\varphi\), the result must have the form \(\Psi_i[\varphi] = f_i(\varphi(\hat{\mathbf x}_i))\) – in other words, the value of the node functional \(\Psi_i\) applied to \(\varphi\) only depends on the values of \(\varphi\) at \(\hat{\mathbf x}_i\) and not on values anywhere else, or integrals of \(\varphi\), or any other kind of information.
The exact form of \(f_i\) depends on the element. For example, for scalar Lagrange elements, we have that in fact \(\Psi_i[\varphi] = \varphi(\hat{\mathbf x}_i)\). If you combine multiple scalar Lagrange elements via an FESystem object, then \(\Psi_i[\varphi] = \varphi(\hat{\mathbf x}_i)_{c(i)}\) where \(c(i)\) is the result of the FiniteElement::system_to_component_index() function's return value's first component. In these two cases, \(f_i\) is therefore simply the identity (in the scalar case) or a function that selects a particular vector component of its argument. On the other hand, for Raviart-Thomas elements, one would have that \(f_i(\mathbf y) = \mathbf y \cdot \mathbf n_i\) where \(\mathbf n_i\) is the normal vector of the face at which the shape function is defined.
Given all of this, what this function does is the following: If you input a list of values of a function \(\varphi\) at all generalized support points (where each value is in fact a vector of values with as many components as the element has), then this function returns a vector of values obtained by applying the node functionals to these values. In other words, if you pass in \(\{\varphi(\hat{\mathbf x}_i)\}_{i=0}^{N-1}\) then you will get out a vector \(\{\Psi[\varphi]\}_{i=0}^{N-1}\) where \(N\) equals dofs_per_cell
.
[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. |
Reimplemented from FiniteElement< dim, spacedim >.
Definition at line 183 of file fe_q_dg0.cc.
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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
.
These matrices are only available if the source element is also a FE_Q_DG0
element. Otherwise, an exception of type FiniteElement<dim,spacedim>::ExcInterpolationNotImplemented is thrown.
Reimplemented from FE_Q_Base< TensorProductPolynomialsConst< dim >, dim, dim >.
Definition at line 211 of file fe_q_dg0.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 FE_Q_Base< TensorProductPolynomialsConst< dim >, dim, dim >.
Definition at line 262 of file fe_q_dg0.cc.
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Return a list of constant modes of the element. For this element, there are two constant modes despite the element is scalar: The first constant mode is all ones for the usual FE_Q basis and the second one only using the discontinuous part.
Reimplemented from FiniteElement< dim, spacedim >.
Definition at line 278 of file fe_q_dg0.cc.
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A sort of virtual copy constructor, this function returns a copy of the finite element object. Derived classes need to override the function here in this base class and return an object of the same type as the derived class.
Some places in the library, for example the constructors of FESystem as well as the hp::FECollection class, need to make copies of finite elements without knowing their exact type. They do so through this function.
Implements FiniteElement< dim, spacedim >.
Definition at line 174 of file fe_q_dg0.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.
Reimplemented from FiniteElement< dim, spacedim >.
Definition at line 297 of file fe_q_dg0.cc.
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Return the restriction_is_additive flags. Only the last component is true.
Definition at line 237 of file fe_q_dg0.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 248 of file fe_q_dg0.cc.