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

Inheritance diagram for FE_Q< dim, spacedim >:

Public Member Functions
	FE_Q (const unsigned int p)

	FE_Q (const Quadrature< 1 > &points)

virtual std::string	get_name () const override

virtual std::unique_ptr< FiniteElement< dim, spacedim > >	clone () 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 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< TensorProductPolynomials< dim >, dim, dim >
	FE_Q_Base (const TensorProductPolynomials< dim > &poly_space, const FiniteElementData< dim > &fe_data, const std::vector< bool > &restriction_is_additive_flags)

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 bool	has_support_on_face (const unsigned int shape_index, const unsigned int face_index) 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< 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

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

virtual std::pair< Table< 2, bool >, std::vector< unsigned int > >	get_constant_modes () 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

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

Implementation of a scalar Lagrange finite element Qp that yields the finite element space of continuous, piecewise polynomials of degree p in each coordinate direction. This class is realized using tensor product polynomials based on 1D Lagrange polynomials with equidistant (degree up to 2), Gauss-Lobatto (starting from degree 3), 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.

Implementation

The constructor creates a TensorProductPolynomials object that includes the tensor product of LagrangeEquidistant polynomials of degree p. This TensorProductPolynomials object provides all values and derivatives of the shape functions. In case a quadrature rule is given, the constructor creates a TensorProductPolynomials object that includes the tensor product of Lagrange polynomials with the support points from points.

Furthermore the constructor fills the interface_constraints, the prolongation (embedding) and the restriction matrices. These are implemented only up to a certain degree and may not be available for very high polynomial degree.

Unit support point distribution and conditioning of interpolation

When constructing an FE_Q 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.

If combined with Gauss-Lobatto quadrature, FE_Q based on the default support points gives diagonal mass matrices. This case is demonstrated in step-48. However, this element can be combined with arbitrary quadrature rules through the usual FEValues approach, including full Gauss quadrature. In the general case, the mass matrix is non-diagonal.

Numbering of the degrees of freedom (DoFs)

The original ordering of the shape functions represented by the TensorProductPolynomials is a tensor product numbering. However, the shape functions on a cell are renumbered beginning with the shape functions whose support points are at the vertices, then on the line, on the quads, and finally (for 3d) on the hexes. To be explicit, these numberings are listed in the following:

Q1 elements

1D case:
```
*      0-------1
*   
```

2D case:

*      2-------3
*      |       |
*      |       |
*      |       |
*      0-------1
*

3D case:

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

The respective coordinate values of the support points of the shape functions are as follows:

Shape function 0: [0, 0, 0];
Shape function 1: [1, 0, 0];
Shape function 2: [0, 1, 0];
Shape function 3: [1, 1, 0];
Shape function 4: [0, 0, 1];
Shape function 5: [1, 0, 1];
Shape function 6: [0, 1, 1];
Shape function 7: [1, 1, 1];

In 2d, these shape functions look as follows:


\(Q_1\) element, shape function 0	\(Q_1\) element, shape function 1

\(Q_1\) element, shape function 2	\(Q_1\) element, shape function 3

Q2 elements

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

2D case:

*      2---7---3
*      |       |
*      4   8   5
*      |       |
*      0---6---1
*

3D case:
```
*         6--15---7        6--15---7
*        /|       |       /       /|
*      12 |       19     12      1319
*      /  18      |     /       /  |
*     4   |       |    4---14--5   |
*     |   2---11--3    |       |   3
*     |  /       /     |      17  /
*    16 8       9     16       | 9
*     |/       /       |       |/
*     0---10--1        0---10--1
*
*         *-------*        *-------*
*        /|       |       /       /|
*       / |  23   |      /  25   / |
*      /  |       |     /       /  |
*     *   |       |    *-------*   |
*     |20 *-------*    |       |21 *
*     |  /       /     |   22  |  /
*     | /  24   /      |       | /
*     |/       /       |       |/
*     *-------*        *-------*
*   
```
The center vertex has number 26.

The respective coordinate values of the support points of the shape functions are as follows:
- Shape function 0: [0, 0, 0];
- Shape function 1: [1, 0, 0];
- Shape function 2: [0, 1, 0];
- Shape function 3: [1, 1, 0];
- Shape function 4: [0, 0, 1];
- Shape function 5: [1, 0, 1];
- Shape function 6: [0, 1, 1];
- Shape function 7: [1, 1, 1];
- Shape function 8: [0, 1/2, 0];
- Shape function 9: [1, 1/2, 0];
- Shape function 10: [1/2, 0, 0];
- Shape function 11: [1/2, 1, 0];
- Shape function 12: [0, 1/2, 1];
- Shape function 13: [1, 1/2, 1];
- Shape function 14: [1/2, 0, 1];
- Shape function 15: [1/2, 1, 1];
- Shape function 16: [0, 0, 1/2];
- Shape function 17: [1, 0, 1/2];
- Shape function 18: [0, 1, 1/2];
- Shape function 19: [1, 1, 1/2];
- Shape function 20: [0, 1/2, 1/2];
- Shape function 21: [1, 1/2, 1/2];
- Shape function 22: [1/2, 0, 1/2];
- Shape function 23: [1/2, 1, 1/2];
- Shape function 24: [1/2, 1/2, 0];
- Shape function 25: [1/2, 1/2, 1];
- Shape function 26: [1/2, 1/2, 1/2];

In 2d, these shape functions look as follows (the black plane corresponds to zero; negative shape function values may not be visible):


\(Q_2\) element, shape function 0	\(Q_2\) element, shape function 1

\(Q_2\) element, shape function 2	\(Q_2\) element, shape function 3

\(Q_2\) element, shape function 4	\(Q_2\) element, shape function 5

\(Q_2\) element, shape function 6	\(Q_2\) element, shape function 7

\(Q_2\) element, shape function 8

Q3 elements

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

2D case:

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

In 2d, these shape functions look as follows (the black plane corresponds to zero; negative shape function values may not be visible):


\(Q_3\) element, shape function 0	\(Q_3\) element, shape function 1

\(Q_3\) element, shape function 2	\(Q_3\) element, shape function 3

\(Q_3\) element, shape function 4	\(Q_3\) element, shape function 5

\(Q_3\) element, shape function 6	\(Q_3\) element, shape function 7

\(Q_3\) element, shape function 8	\(Q_3\) element, shape function 9

\(Q_3\) element, shape function 10	\(Q_3\) element, shape function 11

\(Q_3\) element, shape function 12	\(Q_3\) element, shape function 13

\(Q_3\) element, shape function 14	\(Q_3\) element, shape function 15

Q4 elements

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

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
*

In 2d, these shape functions look as follows (the black plane corresponds to zero; negative shape function values may not be visible):


\(Q_4\) element, shape function 0	\(Q_4\) element, shape function 1

\(Q_4\) element, shape function 2	\(Q_4\) element, shape function 3

\(Q_4\) element, shape function 4	\(Q_4\) element, shape function 5

\(Q_4\) element, shape function 6	\(Q_4\) element, shape function 7

\(Q_4\) element, shape function 8	\(Q_4\) element, shape function 9

\(Q_4\) element, shape function 10	\(Q_4\) element, shape function 11

\(Q_4\) element, shape function 12	\(Q_4\) element, shape function 13

\(Q_4\) element, shape function 14	\(Q_4\) element, shape function 15

\(Q_4\) element, shape function 16	\(Q_4\) element, shape function 17

\(Q_4\) element, shape function 18	\(Q_4\) element, shape function 19

\(Q_4\) element, shape function 20	\(Q_4\) element, shape function 21

\(Q_4\) element, shape function 22	\(Q_4\) element, shape function 23

\(Q_4\) element, shape function 24

Author: Wolfgang Bangerth, 1998, 2003; Guido Kanschat, 2001; Ralf Hartmann, 2001, 2004, 2005; Oliver Kayser-Herold, 2004; Katharina Kormann, 2008; Martin Kronbichler, 2008

Definition at line 554 of file fe_q.h.

Constructor & Destructor Documentation

◆ FE_Q() [1/2]

template<int dim, int spacedim>

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

Constructor for tensor product polynomials of degree p based on Gauss-Lobatto support (node) points. For polynomial degrees of one and two, these are the usual equidistant points.

Definition at line 57 of file fe_q.cc.

◆ FE_Q() [2/2]

template<int dim, int spacedim>

FE_Q< dim, spacedim >::FE_Q ( const Quadrature< 1 > & points )

Constructor for tensor product polynomials with support points points 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. Constructing FE_Q<dim>(QGaussLobatto<1>(fe_degree+1)) is equivalent to the constructor that specifies the polynomial degree only. For selecting equidistant nodes at fe_degree > 2, construct FE_Q<dim>(QIterated<1>(QTrapez<1>(),fe_degree)).

Definition at line 74 of file fe_q.cc.

Member Function Documentation

◆ get_name()

template<int dim, int spacedim>

std::string FE_Q< dim, spacedim >::get_name

overridevirtual

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

Implements FiniteElement< dim, spacedim >.

Definition at line 91 of file fe_q.cc.

◆ clone()

template<int dim, int spacedim>

std::unique_ptr< FiniteElement< dim, spacedim > > FE_Q< 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 170 of file fe_q.cc.

◆ convert_generalized_support_point_values_to_dof_values()

template<int dim, int spacedim>

void FE_Q< 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 >.

Definition at line 149 of file fe_q.cc.

◆ compare_for_domination()

template<int dim, int spacedim>

FiniteElementDomination::Domination FE_Q< 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 179 of file fe_q.cc.

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

include/deal.II/fe/fe_q.h
source/fe/fe_q.cc

Public Member Functions

Additional Inherited Members

Detailed Description

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

Implementation

Unit support point distribution and conditioning of interpolation

Numbering of the degrees of freedom (DoFs)

Q1 elements

Q2 elements

Q3 elements

Q4 elements

Constructor & Destructor Documentation

◆ FE_Q() [1/2]

◆ FE_Q() [2/2]

Member Function Documentation

◆ get_name()

◆ clone()

◆ convert_generalized_support_point_values_to_dof_values()

◆ compare_for_domination()

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