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

Inheritance diagram for FE_DGQ< dim, spacedim >:

Public Member Functions
	FE_DGQ (const unsigned int p)

virtual std::string	get_name () const override

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

virtual void	get_face_interpolation_matrix (const FiniteElement< dim, spacedim > &source, FullMatrix< double > &matrix) const 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 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 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 std::size_t	memory_consumption () const override

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

Functions to support hp
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

virtual bool	hp_constraints_are_implemented () 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_Poly< TensorProductPolynomials< dim >, dim, spacedim >
	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

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

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

bool	operator!= (const FiniteElement< dim, spacedim > &) 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 FiniteElementDomination::Domination	compare_for_face_domination (const FiniteElement< dim, spacedim > &fe_other) const final

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

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

Protected Member Functions
	FE_DGQ (const std::vector< Polynomials::Polynomial< double >> &polynomials)

Protected Member Functions inherited from FE_Poly< TensorProductPolynomials< dim >, dim, spacedim >
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 unsigned int dof) 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

Private Member Functions
void	rotate_indices (std::vector< unsigned int > &indices, const char direction) const

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

Friends
template<int dim1, int spacedim1>
class	FE_DGQ

template<int dim1, int spacedim1>
class	MappingQ

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 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

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< TensorProductPolynomials< dim >, dim, spacedim >
TensorProductPolynomials< 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

Detailed Description

template<int dim, int spacedim = dim>
class FE_DGQ< dim, spacedim >

Implementation of scalar, discontinuous tensor product elements based on equidistant support points.

This is a discontinuous finite element based on tensor products of Lagrangian polynomials. The shape functions are Lagrangian interpolants of an equidistant grid of points on the unit cell. The points are numbered in lexicographical order, with x running fastest, then y, then z (if these coordinates are present for a given space dimension at all). For example, these are the node orderings for FE_DGQ(1) in 3d:

*         6-------7        6-------7
*        /|       |       /       /|
*       / |       |      /       / |
*      /  |       |     /       /  |
*     4   |       |    4-------5   |
*     |   2-------3    |       |   3
*     |  /       /     |       |  /
*     | /       /      |       | /
*     |/       /       |       |/
*     0-------1        0-------1
*

and FE_DGQ(2):

*         24--25--26       24--25--26
*        /|       |       /       /|
*      21 |       |     21  22  23 |
*      /  15  16  17    /       /  17
*    18   |       |   18--19--20   |
*     |12 6---7---8    |       |14 8
*     9  /       /     9  10  11  /
*     | 3   4   5      |       | 5
*     |/       /       |       |/
*     0---1---2        0---1---2
*

with node 13 being placed in the interior of the hex.

Note, however, that these are just the Lagrange interpolation points of the shape functions. Even though they may physically be on the boundary of the cell, they are logically in the interior since there are no continuity requirements for these shape functions across cell boundaries. While discontinuous, when restricted to a single cell the shape functions of this element are exactly the same as those of the FE_Q element where they are shown visually.

Unit support point distribution and conditioning of interpolation

When constructing an FE_DGQ element at polynomial degrees one or two, equidistant support points at 0 and 1 (linear case) or 0, 0.5, and 1 (quadratic case) are used. The unit support or nodal points x_i are those points where the jth Lagrange polynomial satisfies the \(\delta_{ij}\) property, i.e., where one polynomial is one and all the others are zero. For higher polynomial degrees, the support points are non-equidistant by default, and chosen to be the support points of the (degree+1)-order Gauss-Lobatto quadrature rule. This point distribution yields well-conditioned Lagrange interpolation at arbitrary polynomial degrees. By contrast, polynomials based on equidistant points get increasingly ill-conditioned as the polynomial degree increases. In interpolation, this effect is known as the Runge phenomenon. For Galerkin methods, the Runge phenomenon is typically not visible in the solution quality but rather in the condition number of the associated system matrices. For example, the elemental mass matrix of equidistant points at degree 10 has condition number 2.6e6, whereas the condition number for Gauss-Lobatto points is around 400.

The Gauss-Lobatto points in 1D include the end points 0 and +1 of the unit interval. The interior points are shifted towards the end points, which gives a denser point distribution close to the element boundary.

Author: Ralf Hartmann, Guido Kanschat 2001, 2004

Definition at line 109 of file fe_dgq.h.

Constructor & Destructor Documentation

◆ FE_DGQ() [1/2]

template<int dim, int spacedim>

FE_DGQ< dim, spacedim >::FE_DGQ ( const unsigned int p )

Constructor for tensor product polynomials of degree p. The shape functions created using this constructor correspond to Lagrange interpolation polynomials for Gauss-Lobatto support (node) points in each coordinate direction.

Definition at line 62 of file fe_dgq.cc.

◆ FE_DGQ() [2/2]

template<int dim, int spacedim>

FE_DGQ< dim, spacedim >::FE_DGQ ( const std::vector< Polynomials::Polynomial< double >> & polynomials )

protected

Constructor for tensor product polynomials based on an arbitrary vector of polynomials. This constructor is used in derived classes to construct e.g. elements with arbitrary nodes or elements based on Legendre polynomials.

The degree of these polynomials is polynomials.size()-1.

Definition at line 94 of file fe_dgq.cc.

Member Function Documentation

◆ get_name()

template<int dim, int spacedim>

std::string FE_DGQ< dim, spacedim >::get_name ( ) const

overridevirtual

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

Implements FiniteElement< dim, spacedim >.

Reimplemented in FE_DGQHermite< dim, spacedim >, FE_DGQLegendre< dim, spacedim >, and FE_DGQArbitraryNodes< dim, spacedim >.

Definition at line 127 of file fe_dgq.cc.

◆ get_interpolation_matrix()

template<int dim, int spacedim>

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

overridevirtual

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_DGQ element. Otherwise, an exception of type FiniteElement<dim>::ExcInterpolationNotImplemented is thrown.

Reimplemented from FiniteElement< dim, spacedim >.

Definition at line 267 of file fe_dgq.cc.

◆ get_face_interpolation_matrix()

template<int dim, int spacedim>

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

overridevirtual

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.

Reimplemented from FiniteElement< dim, spacedim >.

Definition at line 346 of file fe_dgq.cc.

◆ get_subface_interpolation_matrix()

template<int dim, int spacedim>

void FE_DGQ< dim, spacedim >::get_subface_interpolation_matrix	(	const FiniteElement< dim, spacedim > &	source,
		const unsigned int	subface,
		FullMatrix< double > &	matrix
	)		const

overridevirtual

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.

Reimplemented from FiniteElement< dim, spacedim >.

Definition at line 372 of file fe_dgq.cc.

◆ get_restriction_matrix()

template<int dim, int spacedim>

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

overridevirtual

Projection from a fine grid space onto a coarse grid space. Overrides the respective method in FiniteElement, implementing lazy evaluation (initialize when requested).

If this projection operator is associated with a matrix P, then the restriction of this matrix P_i to a single child cell is returned here.

The matrix P is the concatenation or the sum of the cell matrices P_i, depending on the restriction_is_additive_flags. This distinguishes interpolation (concatenation) and projection with respect to scalar products (summation).

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

Reimplemented from FiniteElement< dim, spacedim >.

Definition at line 479 of file fe_dgq.cc.

◆ get_prolongation_matrix()

template<int dim, int spacedim>

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

overridevirtual

Embedding matrix between grids. Overrides the respective method in FiniteElement, implementing lazy evaluation (initialize when queried).

The identity operator from a coarse grid space into a fine grid space is associated with a matrix P. The restriction of this matrix P_i to a single child cell is returned here.

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

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

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

Reimplemented from FiniteElement< dim, spacedim >.

Definition at line 399 of file fe_dgq.cc.

◆ hp_vertex_dof_identities()

template<int dim, int spacedim>

std::vector< std::pair< unsigned int, unsigned int > > FE_DGQ< dim, spacedim >::hp_vertex_dof_identities ( const FiniteElement< dim, spacedim > & fe_other ) const

overridevirtual

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.

This being a discontinuous element, the set of such constraints is of course empty.

Reimplemented from FiniteElement< dim, spacedim >.

Definition at line 568 of file fe_dgq.cc.

◆ hp_line_dof_identities()

template<int dim, int spacedim>

std::vector< std::pair< unsigned int, unsigned int > > FE_DGQ< dim, spacedim >::hp_line_dof_identities ( const FiniteElement< dim, spacedim > & fe_other ) const

overridevirtual

Same as hp_vertex_dof_indices(), except that the function treats degrees of freedom on lines.

This being a discontinuous element, the set of such constraints is of course empty.

Reimplemented from FiniteElement< dim, spacedim >.

Definition at line 582 of file fe_dgq.cc.

◆ hp_quad_dof_identities()

template<int dim, int spacedim>

std::vector< std::pair< unsigned int, unsigned int > > FE_DGQ< dim, spacedim >::hp_quad_dof_identities ( const FiniteElement< dim, spacedim > & fe_other ) const

overridevirtual

Same as hp_vertex_dof_indices(), except that the function treats degrees of freedom on quads.

This being a discontinuous element, the set of such constraints is of course empty.

Reimplemented from FiniteElement< dim, spacedim >.

Definition at line 596 of file fe_dgq.cc.

◆ hp_constraints_are_implemented()

template<int dim, int spacedim>

bool FE_DGQ< dim, spacedim >::hp_constraints_are_implemented ( ) const

overridevirtual

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_DGQ class the result is always true (independent of the degree of the element), as it has no hanging nodes (being a discontinuous element).

Reimplemented from FiniteElement< dim, spacedim >.

Definition at line 559 of file fe_dgq.cc.

◆ compare_for_domination()

template<int dim, int spacedim>

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

finaloverridevirtual

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

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

Reimplemented from FiniteElement< dim, spacedim >.

Definition at line 610 of file fe_dgq.cc.

◆ has_support_on_face()

template<int dim, int spacedim>

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

overridevirtual

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, spacedim >.

Definition at line 719 of file fe_dgq.cc.

◆ get_constant_modes()

template<int dim, int spacedim>

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

overridevirtual

Return a list of constant modes of the element. For this element, it simply returns one row with all entries set to true.

Reimplemented from FiniteElement< dim, spacedim >.

Reimplemented in FE_DGQLegendre< dim, spacedim >.

Definition at line 812 of file fe_dgq.cc.

◆ convert_generalized_support_point_values_to_dof_values()

template<int dim, int spacedim>

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

overridevirtual

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

Reimplemented from FiniteElement< dim, spacedim >.

Reimplemented in FE_DGQArbitraryNodes< dim, spacedim >.

Definition at line 143 of file fe_dgq.cc.

◆ memory_consumption()

template<int dim, int spacedim>

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

overridevirtual

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, spacedim >.

Definition at line 824 of file fe_dgq.cc.

◆ clone()

template<int dim, int spacedim>

std::unique_ptr< FiniteElement< dim, spacedim > > FE_DGQ< dim, spacedim >::clone ( ) const

overridevirtual

A sort of virtual copy constructor, this function returns a copy of the finite element object. Derived classes need to override the function here in this base class and return an object of the same type as the derived class.

Some places in the library, for example the constructors of FESystem as well as the hp::FECollection class, need to make copies of finite elements without knowing their exact type. They do so through this function.

Implements FiniteElement< dim, spacedim >.

Reimplemented in FE_DGQHermite< dim, spacedim >, FE_DGQLegendre< dim, spacedim >, and FE_DGQArbitraryNodes< dim, spacedim >.

Definition at line 164 of file fe_dgq.cc.

◆ get_dpo_vector()

template<int dim, int spacedim>

std::vector< unsigned int > FE_DGQ< dim, spacedim >::get_dpo_vector ( const unsigned int degree )

staticprivate

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 177 of file fe_dgq.cc.

◆ rotate_indices()

template<int dim, int spacedim>

void FE_DGQ< dim, spacedim >::rotate_indices	(	std::vector< unsigned int > &	indices,
		const char	direction
	)		const

private

Compute renumbering for rotation of degrees of freedom.

This function rotates a tensor product numbering of degrees of freedom by 90 degrees. It is used to compute the transfer matrices of the children by using only the matrix for the first child.

The direction parameter determines the type of rotation. It is one character of xXyYzZ. The character determines the axis of rotation, case determines the direction. Lower case is counter-clockwise seen in direction of the axis.

Since rotation around the y-axis is not used, it is not implemented either.

Definition at line 190 of file fe_dgq.cc.

Friends And Related Function Documentation

◆ FE_DGQ

template<int dim, int spacedim = dim>

template<int dim1, int spacedim1>

friend class FE_DGQ

friend

Allow access from other dimensions.

Definition at line 384 of file fe_dgq.h.

◆ MappingQ

template<int dim, int spacedim = dim>

template<int dim1, int spacedim1>

friend class MappingQ

friend

Allow MappingQ class to access to build_renumbering function.

Definition at line 390 of file fe_dgq.h.

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

deal.II/fe/fe_dgq.h
source/fe/fe_dgq.cc

Public Member Functions

Protected Member Functions

Private Member Functions

Static Private Member Functions

Friends

Additional Inherited Members

Detailed Description

template<int dim, int spacedim = dim> class FE_DGQ< dim, spacedim >

Unit support point distribution and conditioning of interpolation

Constructor & Destructor Documentation

◆ FE_DGQ() [1/2]

◆ FE_DGQ() [2/2]

Member Function Documentation

◆ get_name()

◆ get_interpolation_matrix()

◆ get_face_interpolation_matrix()

◆ get_subface_interpolation_matrix()

◆ get_restriction_matrix()

◆ get_prolongation_matrix()

◆ hp_vertex_dof_identities()

◆ hp_line_dof_identities()

◆ hp_quad_dof_identities()

◆ hp_constraints_are_implemented()

◆ compare_for_domination()

◆ has_support_on_face()

◆ get_constant_modes()

◆ convert_generalized_support_point_values_to_dof_values()

◆ memory_consumption()

◆ clone()

◆ get_dpo_vector()

◆ rotate_indices()

Friends And Related Function Documentation

◆ FE_DGQ

◆ MappingQ

template<int dim, int spacedim = dim>
class FE_DGQ< dim, spacedim >