#include <deal.II/fe/fe_q_dg0.h>

Inheritance diagram for FE_Q_DG0< dim, spacedim >:

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, spacedim >
	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 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

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

virtual std::size_t	memory_consumption () 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

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

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, spacedim >
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, spacedim >
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 >
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

Static Protected Member Functions inherited from FE_Q_Base< TensorProductPolynomialsConst< dim >, dim, spacedim >
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

Detailed Description

template<int dim, int spacedim = dim>
class FE_Q_DG0< dim, spacedim >

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.

Implementation

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.

Numbering of the degrees of freedom (DoFs)

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:

Q1 elements

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:

Index 0: [ 0, 0, 0];
Index 1: [ 1, 0, 0];
Index 2: [ 0, 1, 0];
Index 3: [ 1, 1, 0];
Index 4: [ 0, 0, 1];
Index 5: [ 1, 0, 1];
Index 6: [ 0, 1, 1];
Index 7: [ 1, 1, 1];
Index 8: [1/2, 1/2, 1/2];

Q2 elements

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:

Index 0: [0, 0, 0];
Index 1: [1, 0, 0];
Index 2: [0, 1, 0];
Index 3: [1, 1, 0];
Index 4: [0, 0, 1];
Index 5: [1, 0, 1];
Index 6: [0, 1, 1];
Index 7: [1, 1, 1];
Index 8: [0, 1/2, 0];
Index 9: [1, 1/2, 0];
Index 10: [1/2, 0, 0];
Index 11: [1/2, 1, 0];
Index 12: [0, 1/2, 1];
Index 13: [1, 1/2, 1];
Index 14: [1/2, 0, 1];
Index 15: [1/2, 1, 1];
Index 16: [0, 0, 1/2];
Index 17: [1, 0, 1/2];
Index 18: [0, 1, 1/2];
Index 19: [1, 1, 1/2];
Index 20: [0, 1/2, 1/2];
Index 21: [1, 1/2, 1/2];
Index 22: [1/2, 0, 1/2];
Index 23: [1/2, 1, 1/2];
Index 24: [1/2, 1/2, 0];
Index 25: [1/2, 1/2, 1];
Index 26: [1/2, 1/2, 1/2];
Index 27: [1/2, 1/2, 1/2];

Q3 elements

1D case:
```
*      0--2-4-3--1
*   
```

2D case:

*      2--10-11-3
*      |        |
*      5  14 15 7
*      |    16  |
*      4  12 13 6
*      |        |
*      0--8--9--1
*

Q4 elements

1D case:
```
*      0--2--3--4--1
*   
```
Index 5 has the same coordinates as index 3

2D case:

*      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 & Destructor Documentation

◆ FE_Q_DG0() [1/2]

template<int dim, int spacedim>

FE_Q_DG0< dim, spacedim >::FE_Q_DG0 ( const unsigned int p )

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() [2/2]

template<int dim, int spacedim>

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.

Member Function Documentation

◆ get_name()

template<int dim, int spacedim>

std::string FE_Q_DG0< dim, spacedim >::get_name ( ) const

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.

◆ convert_generalized_support_point_values_to_dof_values()

template<int dim, int spacedim>

void FE_Q_DG0< 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

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.

Parameters

[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.

Note: It is safe to call this function for (transformed) values on the real cell only for elements with trivial MappingType. For all other elements (for example for H(curl), or H(div) conforming elements) vector values have to be transformed to the reference cell first.; Given what the function is supposed to do, the function clearly can only work for elements that actually implement (generalized) support points. Elements that do not have generalized support points – e.g., elements whose nodal functionals evaluate integrals or moments of functions (such as FE_Q_Hierarchical) – can in general not make sense of the operation that is required for this function. They consequently may not implement it.

Reimplemented from FiniteElement< dim, spacedim >.

Definition at line 183 of file fe_q_dg0.cc.

◆ get_interpolation_matrix()

template<int dim, int spacedim>

void FE_Q_DG0< 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_Q_DG0 element. Otherwise, an exception of type FiniteElement<dim,spacedim>::ExcInterpolationNotImplemented is thrown.

Reimplemented from FE_Q_Base< TensorProductPolynomialsConst< dim >, dim, spacedim >.

Definition at line 211 of file fe_q_dg0.cc.

◆ has_support_on_face()

template<int dim, int spacedim>

bool FE_Q_DG0< 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 FE_Q_Base< TensorProductPolynomialsConst< dim >, dim, spacedim >.

Definition at line 262 of file fe_q_dg0.cc.

◆ get_constant_modes()

template<int dim, int spacedim>

std::pair< Table< 2, bool >, std::vector< unsigned int > > FE_Q_DG0< dim, spacedim >::get_constant_modes ( ) const

overridevirtual

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 FE_Q_Base< TensorProductPolynomialsConst< dim >, dim, spacedim >.

Definition at line 278 of file fe_q_dg0.cc.

◆ clone()

template<int dim, int spacedim>

std::unique_ptr< FiniteElement< dim, spacedim > > FE_Q_DG0< 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 >.

Definition at line 174 of file fe_q_dg0.cc.

◆ compare_for_domination()

template<int dim, int spacedim>

FiniteElementDomination::Domination FE_Q_DG0< 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 297 of file fe_q_dg0.cc.

◆ get_riaf_vector()

template<int dim, int spacedim>

std::vector< bool > FE_Q_DG0< dim, spacedim >::get_riaf_vector ( const unsigned int degree )

staticprivate

Return the restriction_is_additive flags. Only the last component is true.

Definition at line 237 of file fe_q_dg0.cc.

◆ get_dpo_vector()

template<int dim, int spacedim>

std::vector< unsigned int > FE_Q_DG0< 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 248 of file fe_q_dg0.cc.

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

deal.II/fe/fe_q_dg0.h
source/fe/fe_q_dg0.cc

Public Member Functions

Static Private Member Functions

Additional Inherited Members

Detailed Description

template<int dim, int spacedim = dim> class FE_Q_DG0< dim, spacedim >

Implementation

Numbering of the degrees of freedom (DoFs)

Q1 elements

Q2 elements

Q3 elements

Q4 elements

Constructor & Destructor Documentation

◆ FE_Q_DG0() [1/2]

◆ FE_Q_DG0() [2/2]

Member Function Documentation

◆ get_name()

◆ convert_generalized_support_point_values_to_dof_values()

◆ get_interpolation_matrix()

◆ has_support_on_face()

◆ get_constant_modes()

◆ clone()

◆ compare_for_domination()

◆ get_riaf_vector()

◆ get_dpo_vector()

template<int dim, int spacedim = dim>
class FE_Q_DG0< dim, spacedim >