Hierarchic Finite Elements of arbitrary degree on quadrilaterals. More...

#include <lf/fe/fe.h>

Inheritance diagram for lf::fe::FeHierarchicQuad< SCALAR >:

Public Member Functions
	FeHierarchicQuad (const FeHierarchicQuad &)=default

	FeHierarchicQuad (FeHierarchicQuad &&) noexcept=default

FeHierarchicQuad &	operator= (const FeHierarchicQuad &)=default

FeHierarchicQuad &	operator= (FeHierarchicQuad &&) noexcept=default

	~FeHierarchicQuad () override=default

	FeHierarchicQuad (unsigned interior_degree, std::array< unsigned, 4 > edge_degrees, const quad::QuadRuleCache &qr_cache, nonstd::span< const lf::mesh::Orientation > rel_orient)

lf::base::RefEl	RefEl () const override
	Tells the type of reference cell underlying the parametric finite element. More...

unsigned	Degree () const override
	Request the maximal polynomial degree of the basis functions in this finite element. More...

lf::base::size_type	NumRefShapeFunctions () const override
	The local shape functions. More...

lf::base::size_type	NumRefShapeFunctions (dim_t codim) const override
	One shape function for each vertex, p-1 shape functions on the edges and (p-1)^2 shape functions on the quadrilateral. More...

lf::base::size_type	NumRefShapeFunctions (dim_t codim, sub_idx_t subidx) const override
	One shape function for each vertex, p-1 shape functions on the edges and (p-1)^2 shape functions on the quadrilateral. More...

Eigen::Matrix< SCALAR, Eigen::Dynamic, Eigen::Dynamic >	EvalReferenceShapeFunctions (const Eigen::MatrixXd &refcoords) const override
	Evaluation of all reference shape functions in a number of points. More...

Eigen::Matrix< SCALAR, Eigen::Dynamic, Eigen::Dynamic >	GradientsReferenceShapeFunctions (const Eigen::MatrixXd &refcoords) const override
	Computation of the gradients of all reference shape functions in a number of points. More...

Eigen::MatrixXd	EvaluationNodes () const override
	Evaluation nodes are the vertices, the points of a quadrature rule on each edge and the points of a quadrature rule on the interior of the quadrilateral. More...

lf::base::size_type	NumEvaluationNodes () const override
	Tell the number of evaluation (interpolation) nodes. More...

Eigen::Matrix< SCALAR, 1, Eigen::Dynamic >	NodalValuesToDofs (const Eigen::Matrix< SCALAR, 1, Eigen::Dynamic > &nodevals) const override
	Computes the linear combination of reference shape functions matching function values at evaluation nodes. More...

Public Member Functions inherited from lf::fe::ScalarReferenceFiniteElement< SCALAR >
virtual	~ScalarReferenceFiniteElement ()=default

virtual base::RefEl	RefEl () const =0
	Tells the type of reference cell underlying the parametric finite element. More...

virtual unsigned int	Degree () const =0
	Request the maximal polynomial degree of the basis functions in this finite element. More...

dim_t	Dimension () const
	Returns the spatial dimension of the reference cell. More...

virtual size_type	NumRefShapeFunctions () const
	Total number of reference shape functions associated with the reference cell. More...

virtual size_type	NumRefShapeFunctions (dim_t codim) const
	The number of interior reference shape functions for sub-entities of a particular co-dimension. More...

virtual size_type	NumRefShapeFunctions (dim_t codim, sub_idx_t subidx) const =0
	The number of interior reference shape functions for every sub-entity. More...

virtual Eigen::Matrix< SCALAR, Eigen::Dynamic, Eigen::Dynamic >	EvalReferenceShapeFunctions (const Eigen::MatrixXd &refcoords) const =0
	Evaluation of all reference shape functions in a number of points. More...

virtual Eigen::Matrix< SCALAR, Eigen::Dynamic, Eigen::Dynamic >	GradientsReferenceShapeFunctions (const Eigen::MatrixXd &refcoords) const =0
	Computation of the gradients of all reference shape functions in a number of points. More...

virtual Eigen::MatrixXd	EvaluationNodes () const =0
	Returns reference coordinates of "evaluation nodes" for evaluation of parametric degrees of freedom, nodal interpolation in the simplest case. More...

virtual size_type	NumEvaluationNodes () const =0
	Tell the number of evaluation (interpolation) nodes. More...

virtual Eigen::Matrix< SCALAR, 1, Eigen::Dynamic >	NodalValuesToDofs (const Eigen::Matrix< SCALAR, 1, Eigen::Dynamic > &nodvals) const
	Computes the linear combination of reference shape functions matching function values at evaluation nodes. More...

Private Attributes
unsigned	interior_degree_

std::array< unsigned, 4 >	edge_degrees_

std::array< const quad::QuadRule *, 4 >	qr_dual_edge_

FeHierarchicSegment< SCALAR >	fe1d_

nonstd::span< const lf::mesh::Orientation >	rel_orient_

Additional Inherited Members
Public Types inherited from lf::fe::ScalarReferenceFiniteElement< SCALAR >
using	Scalar = SCALAR
	The underlying scalar type. More...

Protected Member Functions inherited from lf::fe::ScalarReferenceFiniteElement< SCALAR >
	ScalarReferenceFiniteElement ()=default

	ScalarReferenceFiniteElement (const ScalarReferenceFiniteElement &)=default

	ScalarReferenceFiniteElement (ScalarReferenceFiniteElement &&) noexcept=default

ScalarReferenceFiniteElement &	operator= (const ScalarReferenceFiniteElement &)=default

ScalarReferenceFiniteElement &	operator= (ScalarReferenceFiniteElement &&) noexcept=default

Related Functions inherited from lf::fe::ScalarReferenceFiniteElement< SCALAR >
template<class SCALAR >
void	PrintInfo (std::ostream &o, const ScalarReferenceFiniteElement< SCALAR > &srfe, unsigned int ctrl=0)
	Print information about a ScalarReferenceFiniteElement to the given stream. More...

template<typename SCALAR >
std::ostream &	operator<< (std::ostream &o, const ScalarReferenceFiniteElement< SCALAR > &fe_desc)
	Stream output operator: just calls the ScalarReferenceFiniteElement::print() method. More...

Detailed Description

template<typename SCALAR>
class lf::fe::FeHierarchicQuad< SCALAR >

Hierarchic Finite Elements of arbitrary degree on quadrilaterals.

The basis functions on the quadrilateral has a tensor product structure and can thus be represented by products of basis functions on segments. The basis functions are therefore given by

\[ \widehat{b^{\square}}^{ij}(\widehat{x}, \widehat{y}) = \widehat{b}^i(\widehat{x})\widehat{b}^j(\widehat{y}) \]

where \(\widehat{b}^i(\widehat{x})\) and \(\widehat{b}^j(\widehat{y})\) are the basis functions on a segment.

The dual basis is therefore also quite simple, as we can recycle the one from the segments by first applying the dual basis along the \(x\)-axis and then apply the dual basis to the resulting 1d function.

Note: The Shape Functions are taken from the following paper: https://arxiv.org/pdf/1504.03025.pdf

See also: ScalarReferenceFiniteElement

Definition at line 1089 of file hierarchic_fe.h.

Constructor & Destructor Documentation

◆ FeHierarchicQuad() [1/3]

template<typename SCALAR >

lf::fe::FeHierarchicQuad< SCALAR >::FeHierarchicQuad ( const FeHierarchicQuad< SCALAR > & )

default

◆ FeHierarchicQuad() [2/3]

template<typename SCALAR >

lf::fe::FeHierarchicQuad< SCALAR >::FeHierarchicQuad ( FeHierarchicQuad< SCALAR > && )

defaultnoexcept

◆ ~FeHierarchicQuad()

template<typename SCALAR >

lf::fe::FeHierarchicQuad< SCALAR >::~FeHierarchicQuad ( )

overridedefault

◆ FeHierarchicQuad() [3/3]

template<typename SCALAR >

lf::fe::FeHierarchicQuad< SCALAR >::FeHierarchicQuad	(	unsigned	interior_degree,
		std::array< unsigned, 4 >	edge_degrees,
		const quad::QuadRuleCache &	qr_cache,
		nonstd::span< const lf::mesh::Orientation >	rel_orient
	)

inline

Definition at line 1097 of file hierarchic_fe.h.

References lf::fe::FeHierarchicQuad< SCALAR >::edge_degrees_, and lf::base::RefEl::kSegment().

Member Function Documentation

◆ Degree()

template<typename SCALAR >

unsigned lf::fe::FeHierarchicQuad< SCALAR >::Degree ( ) const

inlineoverridevirtual

Request the maximal polynomial degree of the basis functions in this finite element.

Implements lf::fe::ScalarReferenceFiniteElement< SCALAR >.

Definition at line 1127 of file hierarchic_fe.h.

References lf::fe::FeHierarchicQuad< SCALAR >::edge_degrees_, and lf::fe::FeHierarchicQuad< SCALAR >::interior_degree_.

Referenced by lf::fe::FeHierarchicQuad< SCALAR >::NodalValuesToDofs().

◆ EvalReferenceShapeFunctions()

template<typename SCALAR >

Eigen::Matrix< SCALAR, Eigen::Dynamic, Eigen::Dynamic > lf::fe::FeHierarchicQuad< SCALAR >::EvalReferenceShapeFunctions ( const Eigen::MatrixXd & refcoords ) const

inlineoverridevirtual

Evaluation of all reference shape functions in a number of points.

Parameters

refcoords coordinates of N points in the reference cell passed as columns of a matrix of size dim x N, where dim is the dimension of the reference element, that is =0 for points, =1 for edges, =2 for triangles and quadrilaterals

Returns: An Eigen Matrix of size NumRefShapeFunctions() x refcoords.cols() which contains the shape functions evaluated at every quadrature point.

Concerning the numbering of local shape functions, please consult the documentation of lf::assemble::DofHandler or the documentation of the class.

Note: shape functions are assumed to be real-valued.

Example: Triangular Linear Lagrangian finite elements

There are three reference shape functions \(\hat{b}^1,\hat{b}^2,\hat{b}^3\) associated with the vertices of the reference triangle. Let us assume that the refcoords argument is a 2x2 matrix \([\mathbf{x}_1\;\mathbf{x}_2]\), which corresponds to passing the coordinates of two points in the reference triangle. Then this method will return a 3x2 matrix:

\[ \begin{pmatrix}\hat{b}^1(\mathbf{x}_1) & \hat{b}^1(\mathbf{x}_2) \\ \hat{b}^2(\mathbf{x}_1) & \hat{b}^2(\mathbf{x}_2) \\ \hat{b}^3(\mathbf{x}_1)\ & \hat{b}^3(\mathbf{x}_2) \end{pmatrix} \]

Implements lf::fe::ScalarReferenceFiniteElement< SCALAR >.

Definition at line 1196 of file hierarchic_fe.h.

References lf::fe::FeHierarchicQuad< SCALAR >::edge_degrees_, lf::fe::FeHierarchicQuad< SCALAR >::fe1d_, lf::fe::FeHierarchicQuad< SCALAR >::interior_degree_, lf::fe::FeHierarchicQuad< SCALAR >::NumRefShapeFunctions(), lf::mesh::positive, and lf::fe::FeHierarchicQuad< SCALAR >::rel_orient_.

◆ EvaluationNodes()

template<typename SCALAR >

Eigen::MatrixXd lf::fe::FeHierarchicQuad< SCALAR >::EvaluationNodes ( ) const

inlineoverridevirtual

Evaluation nodes are the vertices, the points of a quadrature rule on each edge and the points of a quadrature rule on the interior of the quadrilateral.

Returns reference coordinates of "evaluation nodes" for evaluation of parametric degrees of freedom, nodal interpolation in the simplest case.

Returns: A d x N matrix, where d is the dimension of the reference cell, and N the number of interpolation nodes. The columns of this matrix contain their reference coordinates.

Every parametric scalar finite element implicitly defines a local interpolation operator by duality with the reference shape functions. This interpolation operator can be realized through evaluations at certain evaluation nodes, which are provided by this method.

Unisolvence

The evaluation points must satisfy the following requirement: If the values of a function belonging to the span of the reference shape functions are known in the evaluation nodes, then this function is uniquely determined. This entails that the number of evaluation nodes must be at least as big as the number of reference shape functions.

Note: It is not required that any vector a values at evaluation nodes can be produced by a suitable linear combination of reference shape functions. For instance, this will not be possible, if there are more evaluation points than reference shape functions. If both sets have the same size, however, the interpolation problem has a solution for any vector of values at the evluation points.

Example: Principal lattice

For triangular Lagrangian finite elements of degree p the evaluation nodes, usually called "interpolation nodes" in this context, can be chosen as \(\left(\frac{j}{p},\frac{k}{p}\right),\; 0\leq j,k \leq p, j+k\leq p\).

Moment-based degrees of freedom

For some finite element spaces the interpolation functional may be defined based on integrals over edges. In this case the evaluation nodes will be quadrature nodes for the approximate evaluation of these integrals.

The quadrature rule must be exact for the polynomials contained in the local finite element spaces.

Implements lf::fe::ScalarReferenceFiniteElement< SCALAR >.

Definition at line 1399 of file hierarchic_fe.h.

References lf::fe::FeHierarchicQuad< SCALAR >::edge_degrees_, lf::fe::FeHierarchicQuad< SCALAR >::fe1d_, lf::fe::FeHierarchicQuad< SCALAR >::interior_degree_, lf::mesh::positive, lf::fe::FeHierarchicQuad< SCALAR >::qr_dual_edge_, and lf::fe::FeHierarchicQuad< SCALAR >::rel_orient_.

◆ GradientsReferenceShapeFunctions()

template<typename SCALAR >

Eigen::Matrix< SCALAR, Eigen::Dynamic, Eigen::Dynamic > lf::fe::FeHierarchicQuad< SCALAR >::GradientsReferenceShapeFunctions ( const Eigen::MatrixXd & refcoords ) const

inlineoverridevirtual

Computation of the gradients of all reference shape functions in a number of points.

Parameters

refcoords coordinates of N points in the reference cell passed as columns of a matrix of size dim x N.

Returns: An Eigen Matrix of size NumRefShapeFunctions() x (dim * refcoords.cols()) where dim is the dimension of the reference finite element. The gradients are returned in chunks of rows of this matrix.

Concerning the numbering of local shape functions, please consult the documentation of lf::assemble::DofHandler.

Example: Triangular Linear Lagrangian finite elements

There are three reference shape functions \(\hat{b}^1,\hat{b}^2,\hat{b}^3\) associated with the vertices of the reference triangle. Let us assume that the refcoords argument is a 2x2 matrix \([\mathbf{x}_1\;\mathbf{x}_2]\), which corresponds to passing the coordinates of two points in the reference triangle. Then this method will return a 3x4 matrix:

\[ \begin{pmatrix} \grad^T\hat{b}^1(\mathbf{x}_1) & \grad^T\hat{b}^1(\mathbf{x}_2) \\ \grad^T\hat{b}^2(\mathbf{x}_1) & \grad^T\hat{b}^2(\mathbf{x}_2) \\ \grad^T\hat{b}^3(\mathbf{x}_1) & \grad^T\hat{b}^3(\mathbf{x}_2) \end{pmatrix} \]

Implements lf::fe::ScalarReferenceFiniteElement< SCALAR >.

Definition at line 1282 of file hierarchic_fe.h.

References lf::fe::FeHierarchicQuad< SCALAR >::edge_degrees_, lf::fe::FeHierarchicQuad< SCALAR >::fe1d_, lf::fe::FeHierarchicQuad< SCALAR >::interior_degree_, lf::fe::FeHierarchicQuad< SCALAR >::NumRefShapeFunctions(), lf::mesh::positive, and lf::fe::FeHierarchicQuad< SCALAR >::rel_orient_.

◆ NodalValuesToDofs()

template<typename SCALAR >

Eigen::Matrix< SCALAR, 1, Eigen::Dynamic > lf::fe::FeHierarchicQuad< SCALAR >::NodalValuesToDofs ( const Eigen::Matrix< SCALAR, 1, Eigen::Dynamic > & nodvals ) const

inlineoverridevirtual

Computes the linear combination of reference shape functions matching function values at evaluation nodes.

Parameters

nodvals row vector of function values at evaluation nodes The length of this vector must agree with NumEvaluationNodes().

Returns: The coefficients of the local "nodal interpolant" with respect to the reference shape functions. This is a row vector of length NumRefShapeFunctions().

If the evaluation nodes are interpolation nodes, that is, if the set of reference shape functions forms a cardinal basis with respect to these nodes, then we have NumEvaluationNodes() == NumRefShapeFunctions() and the linear mapping realized by NodalValuesToDofs() is the identity mapping.

Note: default implementation is the identity mapping

Requirement: reproduction of local finite element functions

If the vector of values at the evaluation nodes agrees with a vector of function values of a linear combination of reference shape functions at the evaluation nodes, then this method must return the very coefficients of the linear combination.

Reimplemented from lf::fe::ScalarReferenceFiniteElement< SCALAR >.

Definition at line 1491 of file hierarchic_fe.h.

References lf::fe::FeHierarchicQuad< SCALAR >::Degree(), lf::fe::FeHierarchicQuad< SCALAR >::edge_degrees_, lf::fe::FeHierarchicQuad< SCALAR >::fe1d_, lf::fe::ilegendre_dx(), lf::fe::FeHierarchicQuad< SCALAR >::interior_degree_, lf::fe::legendre_dx(), lf::fe::FeHierarchicQuad< SCALAR >::NumRefShapeFunctions(), lf::mesh::positive, lf::fe::FeHierarchicQuad< SCALAR >::qr_dual_edge_, and lf::fe::FeHierarchicQuad< SCALAR >::rel_orient_.

◆ NumEvaluationNodes()

template<typename SCALAR >

lf::base::size_type lf::fe::FeHierarchicQuad< SCALAR >::NumEvaluationNodes ( ) const

inlineoverridevirtual

Tell the number of evaluation (interpolation) nodes.

Implements lf::fe::ScalarReferenceFiniteElement< SCALAR >.

Definition at line 1477 of file hierarchic_fe.h.

References lf::fe::FeHierarchicQuad< SCALAR >::edge_degrees_, lf::fe::FeHierarchicQuad< SCALAR >::fe1d_, lf::fe::FeHierarchicQuad< SCALAR >::interior_degree_, and lf::fe::FeHierarchicQuad< SCALAR >::qr_dual_edge_.

◆ NumRefShapeFunctions() [1/3]

template<typename SCALAR >

lf::base::size_type lf::fe::FeHierarchicQuad< SCALAR >::NumRefShapeFunctions ( ) const

inlineoverridevirtual

The local shape functions.

Total number of reference shape functions associated with the reference cell.

Note: the total number of shape functions is the sum of the number of interior shape functions for all sub-entities and the entity itself.

Reimplemented from lf::fe::ScalarReferenceFiniteElement< SCALAR >.

Definition at line 1136 of file hierarchic_fe.h.

References lf::fe::FeHierarchicQuad< SCALAR >::NumRefShapeFunctions().

Referenced by lf::fe::FeHierarchicQuad< SCALAR >::EvalReferenceShapeFunctions(), lf::fe::FeHierarchicQuad< SCALAR >::GradientsReferenceShapeFunctions(), lf::fe::FeHierarchicQuad< SCALAR >::NodalValuesToDofs(), and lf::fe::FeHierarchicQuad< SCALAR >::NumRefShapeFunctions().

◆ NumRefShapeFunctions() [2/3]

template<typename SCALAR >

lf::base::size_type lf::fe::FeHierarchicQuad< SCALAR >::NumRefShapeFunctions ( dim_t codim ) const

inlineoverridevirtual

One shape function for each vertex, p-1 shape functions on the edges and (p-1)^2 shape functions on the quadrilateral.

The number of interior reference shape functions for sub-entities of a particular co-dimension.

Parameters

codim co-dimension of the subentity

Returns: number of interior reference shape function belonging to the specified sub-entity.

Note: this method will throw an exception when different numbers of reference shape functions are assigned to different sub-entities of the same co-dimension

Reimplemented from lf::fe::ScalarReferenceFiniteElement< SCALAR >.

Definition at line 1147 of file hierarchic_fe.h.

References lf::fe::FeHierarchicQuad< SCALAR >::edge_degrees_, and lf::fe::FeHierarchicQuad< SCALAR >::interior_degree_.

◆ NumRefShapeFunctions() [3/3]

template<typename SCALAR >

lf::base::size_type lf::fe::FeHierarchicQuad< SCALAR >::NumRefShapeFunctions	(	dim_t	codim,
		sub_idx_t	subidx
	)		const

inlineoverridevirtual

One shape function for each vertex, p-1 shape functions on the edges and (p-1)^2 shape functions on the quadrilateral.

The number of interior reference shape functions for every sub-entity.

Parameters

codim	do-dimension of the subentity
subidx	local index of the sub-entity

Returns: number of interior reference shape function belonging to the specified sub-entity.

Implements lf::fe::ScalarReferenceFiniteElement< SCALAR >.

Definition at line 1176 of file hierarchic_fe.h.

References lf::fe::FeHierarchicQuad< SCALAR >::edge_degrees_, and lf::fe::FeHierarchicQuad< SCALAR >::NumRefShapeFunctions().

◆ operator=() [1/2]

template<typename SCALAR >

FeHierarchicQuad & lf::fe::FeHierarchicQuad< SCALAR >::operator= ( const FeHierarchicQuad< SCALAR > & )

default

◆ operator=() [2/2]

template<typename SCALAR >

FeHierarchicQuad & lf::fe::FeHierarchicQuad< SCALAR >::operator= ( FeHierarchicQuad< SCALAR > && )

defaultnoexcept

◆ RefEl()

template<typename SCALAR >

lf::base::RefEl lf::fe::FeHierarchicQuad< SCALAR >::RefEl ( ) const

inlineoverridevirtual

Tells the type of reference cell underlying the parametric finite element.

Implements lf::fe::ScalarReferenceFiniteElement< SCALAR >.

Definition at line 1123 of file hierarchic_fe.h.

References lf::base::RefEl::kQuad().

Member Data Documentation

◆ edge_degrees_

template<typename SCALAR >

std::array<unsigned, 4> lf::fe::FeHierarchicQuad< SCALAR >::edge_degrees_

private

Definition at line 1618 of file hierarchic_fe.h.

Referenced by lf::fe::FeHierarchicQuad< SCALAR >::Degree(), lf::fe::FeHierarchicQuad< SCALAR >::EvalReferenceShapeFunctions(), lf::fe::FeHierarchicQuad< SCALAR >::EvaluationNodes(), lf::fe::FeHierarchicQuad< SCALAR >::FeHierarchicQuad(), lf::fe::FeHierarchicQuad< SCALAR >::GradientsReferenceShapeFunctions(), lf::fe::FeHierarchicQuad< SCALAR >::NodalValuesToDofs(), lf::fe::FeHierarchicQuad< SCALAR >::NumEvaluationNodes(), and lf::fe::FeHierarchicQuad< SCALAR >::NumRefShapeFunctions().

◆ fe1d_

template<typename SCALAR >

FeHierarchicSegment<SCALAR> lf::fe::FeHierarchicQuad< SCALAR >::fe1d_

private

Definition at line 1621 of file hierarchic_fe.h.

Referenced by lf::fe::FeHierarchicQuad< SCALAR >::EvalReferenceShapeFunctions(), lf::fe::FeHierarchicQuad< SCALAR >::EvaluationNodes(), lf::fe::FeHierarchicQuad< SCALAR >::GradientsReferenceShapeFunctions(), lf::fe::FeHierarchicQuad< SCALAR >::NodalValuesToDofs(), and lf::fe::FeHierarchicQuad< SCALAR >::NumEvaluationNodes().

◆ interior_degree_

template<typename SCALAR >

unsigned lf::fe::FeHierarchicQuad< SCALAR >::interior_degree_

private

Definition at line 1617 of file hierarchic_fe.h.

Referenced by lf::fe::FeHierarchicQuad< SCALAR >::Degree(), lf::fe::FeHierarchicQuad< SCALAR >::EvalReferenceShapeFunctions(), lf::fe::FeHierarchicQuad< SCALAR >::EvaluationNodes(), lf::fe::FeHierarchicQuad< SCALAR >::GradientsReferenceShapeFunctions(), lf::fe::FeHierarchicQuad< SCALAR >::NodalValuesToDofs(), lf::fe::FeHierarchicQuad< SCALAR >::NumEvaluationNodes(), and lf::fe::FeHierarchicQuad< SCALAR >::NumRefShapeFunctions().

◆ qr_dual_edge_

template<typename SCALAR >

std::array<const quad::QuadRule *, 4> lf::fe::FeHierarchicQuad< SCALAR >::qr_dual_edge_

private

Definition at line 1619 of file hierarchic_fe.h.

Referenced by lf::fe::FeHierarchicQuad< SCALAR >::EvaluationNodes(), lf::fe::FeHierarchicQuad< SCALAR >::NodalValuesToDofs(), and lf::fe::FeHierarchicQuad< SCALAR >::NumEvaluationNodes().

◆ rel_orient_

template<typename SCALAR >

nonstd::span<const lf::mesh::Orientation> lf::fe::FeHierarchicQuad< SCALAR >::rel_orient_

private

Definition at line 1622 of file hierarchic_fe.h.

Referenced by lf::fe::FeHierarchicQuad< SCALAR >::EvalReferenceShapeFunctions(), lf::fe::FeHierarchicQuad< SCALAR >::EvaluationNodes(), lf::fe::FeHierarchicQuad< SCALAR >::GradientsReferenceShapeFunctions(), and lf::fe::FeHierarchicQuad< SCALAR >::NodalValuesToDofs().

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

/home/nico/bildung/SemVI/thesis/lehrfempp/lib/lf/fe/hierarchic_fe.h

Public Member Functions

Private Attributes

Additional Inherited Members

Detailed Description

Constructor & Destructor Documentation

◆ FeHierarchicQuad() [1/3]

◆ FeHierarchicQuad() [2/3]

◆ ~FeHierarchicQuad()

◆ FeHierarchicQuad() [3/3]

Member Function Documentation

◆ Degree()

◆ EvalReferenceShapeFunctions()

Example: Triangular Linear Lagrangian finite elements

◆ EvaluationNodes()

Unisolvence

Example: Principal lattice

Moment-based degrees of freedom

◆ GradientsReferenceShapeFunctions()

Example: Triangular Linear Lagrangian finite elements

◆ NodalValuesToDofs()

Requirement: reproduction of local finite element functions

◆ NumEvaluationNodes()

◆ NumRefShapeFunctions() [1/3]

◆ NumRefShapeFunctions() [2/3]

◆ NumRefShapeFunctions() [3/3]

◆ operator=() [1/2]

◆ operator=() [2/2]

◆ RefEl()

Member Data Documentation

◆ edge_degrees_

◆ fe1d_

◆ interior_degree_

◆ qr_dual_edge_

◆ rel_orient_