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

#include <lf/fe/fe.h>

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

Public Member Functions
	FeHierarchicSegment (const FeHierarchicSegment &)=default

	FeHierarchicSegment (FeHierarchicSegment &&) noexcept=default

FeHierarchicSegment &	operator= (const FeHierarchicSegment &)=default

FeHierarchicSegment &	operator= (FeHierarchicSegment &&) noexcept=default

	~FeHierarchicSegment () override=default

	FeHierarchicSegment (unsigned degree, const quad::QuadRuleCache &qr_cache)

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

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

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

lf::base::size_type	NumRefShapeFunctions (dim_t codim) const override
	One shape function for each vertex, p-1 shape functions for the segment.

lf::base::size_type	NumRefShapeFunctions (dim_t codim, sub_idx_t subidx) const override
	One shape function for each vertex, p-1 shape functions for the segment.

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.

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.

Eigen::MatrixXd	EvaluationNodes () const override
	Evaluation nodes are the endpoints of the segment and the Chebyshev nodes of degree p-1 on the segment.

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

Eigen::Matrix< SCALAR, 1, Eigen::Dynamic >	NodalValuesToDofs (const Eigen::Matrix< SCALAR, 1, Eigen::Dynamic > &nodevals) const override
	Maps function evaluations to basis function coefficients.

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

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

Private Attributes
unsigned	degree_

const lf::quad::QuadRule *	qr_dual_

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

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 Symbols 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.

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

Detailed Description

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

Hierarchic Finite Elements of arbitrary degree on segments.

The shape functions associated with the vertices are given by

\[ \begin{align*} \widehat{b^\cdot}^1(x) &:= 1 - x \\ \widehat{b^\cdot}^2(x) &:= x \end{align*} \]

and the interior basis functions associated with the segment itself are given by the integrated shifted Legendre polynomials

\[ \widehat{b^-}^i(x) = L_i(x) = \int_0^{x}\! P_{i-1}(\xi) \,\mathrm{d}\xi \quad\mbox{ for }\quad i= 2, \cdots, p \]

where \(P_i : [0, 1] \to \mathbb{R}\) is the shifted Legendre polynomial of degree \(i\).

To compute the basis function coefficients from point evaluations of a function, we make use of the dual basis given by

\[ \lambda_i^-[f] = \begin{cases} f(0) &\mbox{ for } i = 0 \\ f(1) &\mbox{ for } i = 1 \\ \frac{1}{2i - 1} \left [P_{i-1}(1)f(1) - P_{i-1}(0)f(0) - \int_0^1\! P_{i-1}'(x)f(x) \,\mathrm{d}x \right] &\mbox{ for } i \geq 2 \end{cases} \]

Attention: Note that the local coordinate \(\widehat{x}\) may be flipped by applying an affine transformation \(\widehat{x} \mapsto 1 - \widehat{x}\). This must be done if the relative orientation of the edge is lf::mesh::Orientation::negative in order to keep a global ordering of DOFs on the cell interfaces.

A complete description of the basis functions and dual basis can be found here.

See also: ScalarReferenceFiniteElement

Definition at line 242 of file hierarchic_fe.h.

Constructor & Destructor Documentation

◆ FeHierarchicSegment() [1/3]

template<typename SCALAR>

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

default

References FeHierarchicSegment().

Referenced by FeHierarchicSegment(), FeHierarchicSegment(), FeHierarchicSegment(), operator=(), operator=(), and ~FeHierarchicSegment().

◆ FeHierarchicSegment() [2/3]

template<typename SCALAR>

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

defaultnoexcept

References FeHierarchicSegment().

◆ ~FeHierarchicSegment()

template<typename SCALAR>

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

overridedefault

References FeHierarchicSegment().

◆ FeHierarchicSegment() [3/3]

template<typename SCALAR>

lf::fe::FeHierarchicSegment< SCALAR >::FeHierarchicSegment	(	unsigned	degree,
		const quad::QuadRuleCache &	qr_cache )

inline

Definition at line 250 of file hierarchic_fe.h.

References degree_, FeHierarchicSegment(), qr_dual_, RefEl(), and lf::fe::ScalarReferenceFiniteElement< SCALAR >::ScalarReferenceFiniteElement().

Member Function Documentation

◆ Degree()

template<typename SCALAR>

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

inlinenodiscardoverridevirtual

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

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

Definition at line 259 of file hierarchic_fe.h.

References degree_.

◆ EvalReferenceShapeFunctions()

template<typename SCALAR>

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

inlinenodiscardoverridevirtual

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 295 of file hierarchic_fe.h.

References degree_, lf::fe::ILegendre(), and NumRefShapeFunctions().

◆ EvaluationNodes()

template<typename SCALAR>

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

inlinenodiscardoverridevirtual

Evaluation nodes are the endpoints of the segment and the Chebyshev nodes of degree p-1 on the segment.

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 335 of file hierarchic_fe.h.

References NumEvaluationNodes(), and qr_dual_.

◆ GradientsReferenceShapeFunctions()

template<typename SCALAR>

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

inlinenodiscardoverridevirtual

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 312 of file hierarchic_fe.h.

References degree_, lf::fe::ILegendreDx(), and NumRefShapeFunctions().

◆ NodalValuesToDofs()

template<typename SCALAR>

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

inlinenodiscardoverridevirtual

Maps function evaluations to basis function coefficients.

Parameters

nodevals The value of the function at the evaluation nodes

This function computes the basis function coefficients using the dual basis of the basis functions on the segment. It is given by

\[ \lambda_i^-[f] = \begin{cases} f(0) &\mbox{ for } i = 0 \\ f(1) &\mbox{ for } i = 1 \\ \frac{1}{2i - 1} \left [P_{i-1}(1)f(1) - P_{i-1}(0)f(0) - \int_0^1\! P_{i-1}'(x)f(x) \,\mathrm{d}x \right] &\mbox{ for } i \geq 2 \end{cases} \]

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

Definition at line 367 of file hierarchic_fe.h.

References lf::fe::ILegendreDx(), lf::fe::LegendreDx(), NumRefShapeFunctions(), qr_dual_, and lf::fe::transpose().

◆ NumEvaluationNodes()

template<typename SCALAR>

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

inlinenodiscardoverridevirtual

Tell the number of evaluation (interpolation) nodes.

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

Definition at line 348 of file hierarchic_fe.h.

References qr_dual_.

Referenced by EvaluationNodes().

◆ NumRefShapeFunctions() [1/3]

template<typename SCALAR>

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

inlinenodiscardoverridevirtual

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 265 of file hierarchic_fe.h.

References degree_.

Referenced by EvalReferenceShapeFunctions(), GradientsReferenceShapeFunctions(), NodalValuesToDofs(), and NumRefShapeFunctions().

◆ NumRefShapeFunctions() [2/3]

template<typename SCALAR>

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

inlinenodiscardoverridevirtual

One shape function for each vertex, p-1 shape functions for the segment.

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 274 of file hierarchic_fe.h.

References degree_.

◆ NumRefShapeFunctions() [3/3]

template<typename SCALAR>

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

inlinenodiscardoverridevirtual

One shape function for each vertex, p-1 shape functions for the segment.

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 287 of file hierarchic_fe.h.

References NumRefShapeFunctions().

◆ operator=() [1/2]

template<typename SCALAR>

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

default

References FeHierarchicSegment().

◆ operator=() [2/2]

template<typename SCALAR>

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

defaultnoexcept

References FeHierarchicSegment().

◆ RefEl()

template<typename SCALAR>

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

inlinenodiscardoverridevirtual

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

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

Definition at line 255 of file hierarchic_fe.h.

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

Referenced by FeHierarchicSegment().

Member Data Documentation

◆ degree_

template<typename SCALAR>

unsigned lf::fe::FeHierarchicSegment< SCALAR >::degree_

private

Definition at line 392 of file hierarchic_fe.h.

Referenced by Degree(), EvalReferenceShapeFunctions(), FeHierarchicSegment(), GradientsReferenceShapeFunctions(), NumRefShapeFunctions(), and NumRefShapeFunctions().

◆ qr_dual_

template<typename SCALAR>

const lf::quad::QuadRule* lf::fe::FeHierarchicSegment< SCALAR >::qr_dual_

private

Definition at line 393 of file hierarchic_fe.h.

Referenced by EvaluationNodes(), FeHierarchicSegment(), NodalValuesToDofs(), and NumEvaluationNodes().

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

/home/runner/work/lehrfempp/lehrfempp/lib/lf/fe/hierarchic_fe.h

Public Member Functions

Private Attributes

Additional Inherited Members

Detailed Description

Constructor & Destructor Documentation

◆ FeHierarchicSegment() [1/3]

◆ FeHierarchicSegment() [2/3]

◆ ~FeHierarchicSegment()

◆ FeHierarchicSegment() [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()

◆ NumEvaluationNodes()

◆ NumRefShapeFunctions() [1/3]

◆ NumRefShapeFunctions() [2/3]

◆ NumRefShapeFunctions() [3/3]

◆ operator=() [1/2]

◆ operator=() [2/2]

◆ RefEl()

Member Data Documentation

◆ degree_

◆ qr_dual_