Finite element on a point. More...

#include <lf/fe/fe.h>

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

Public Member Functions
	FePoint (unsigned degree)
	Create a new FePoint by specifying the degree of the shape functions.

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.

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

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 &) const override
	Computation of the gradients of all reference shape functions in a number of points.

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

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

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

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

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

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

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.

Private Attributes
unsigned	degree_

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<class SCALAR>
class lf::fe::FePoint< SCALAR >

Finite element on a point.

This is a specialization of ScalarReferenceFiniteElement for an entity of dimension 0, which is exactly one scalar value. It is an ingredient of all scalar finite element spaces.

Definition at line 25 of file fe_point.h.

Constructor & Destructor Documentation

◆ FePoint()

template<class SCALAR >

lf::fe::FePoint< SCALAR >::FePoint ( unsigned degree )

inlineexplicit

Create a new FePoint by specifying the degree of the shape functions.

Parameters

degree The degree of the shape function.

Definition at line 32 of file fe_point.h.

Member Function Documentation

◆ Degree()

template<class SCALAR >

unsigned lf::fe::FePoint< 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 37 of file fe_point.h.

References lf::fe::FePoint< SCALAR >::degree_.

◆ EvalReferenceShapeFunctions()

template<class SCALAR >

Eigen::Matrix< SCALAR, Eigen::Dynamic, Eigen::Dynamic > lf::fe::FePoint< 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 45 of file fe_point.h.

◆ EvaluationNodes()

template<class SCALAR >

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

inlineoverridevirtual

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 54 of file fe_point.h.

◆ GradientsReferenceShapeFunctions()

template<class SCALAR >

Eigen::Matrix< SCALAR, Eigen::Dynamic, Eigen::Dynamic > lf::fe::FePoint< 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 50 of file fe_point.h.

◆ NumEvaluationNodes()

template<class SCALAR >

size_type lf::fe::FePoint< SCALAR >::NumEvaluationNodes ( ) const

inlineoverridevirtual

Tell the number of evaluation (interpolation) nodes.

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

Definition at line 57 of file fe_point.h.

◆ NumRefShapeFunctions()

template<class SCALAR >

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

inlineoverridevirtual

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 38 of file fe_point.h.

◆ RefEl()

template<class SCALAR >

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

inlineoverridevirtual

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

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

Definition at line 34 of file fe_point.h.

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

Member Data Documentation

◆ degree_

template<class SCALAR >

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

private

Definition at line 60 of file fe_point.h.

Referenced by lf::fe::FePoint< SCALAR >::Degree().

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

/__w/lehrfempp/lehrfempp/lib/lf/fe/fe_point.h

Public Member Functions

Private Attributes

Additional Inherited Members

Detailed Description

Constructor & Destructor Documentation

◆ FePoint()

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

◆ NumEvaluationNodes()

◆ NumRefShapeFunctions()

◆ RefEl()

Member Data Documentation

◆ degree_