Discontinuous constant finite element on a triangular reference element. More...

Inheritance diagram for projects::dpg::FeDiscontinuousO0Tria< SCALAR >:

Public Member Functions
	FeDiscontinuousO0Tria (const FeDiscontinuousO0Tria &)=default

	FeDiscontinuousO0Tria (FeDiscontinuousO0Tria &&) noexcept=default

FeDiscontinuousO0Tria &	operator= (const FeDiscontinuousO0Tria &)=default

FeDiscontinuousO0Tria &	operator= (FeDiscontinuousO0Tria &&) noexcept=default

	FeDiscontinuousO0Tria ()=default

	~FeDiscontinuousO0Tria () override=default

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...

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

size_type	NumRefShapeFunctions (dim_t codim) const override

size_type	NumRefShapeFunctions (dim_t codim, sub_idx_t) const override

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
	Only evaluation node is the barycenter of the reference triangle. More...

size_type	NumEvaluationNodes () const override
	One evaluation node. 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...

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 projects::dpg::FeDiscontinuousO0Tria< SCALAR >

Discontinuous constant finite element on a triangular reference element.

This is a specialization of lf::fe::ScalarReferenceFiniteElement. Refer to its documentation.

The only reference shape function is constant and associated with the barycenter of the reference triangle, it is an interior shape function.

Definition at line 33 of file discontinuous_fe_constant.h.

Constructor & Destructor Documentation

◆ FeDiscontinuousO0Tria() [1/3]

template<typename SCALAR >

projects::dpg::FeDiscontinuousO0Tria< SCALAR >::FeDiscontinuousO0Tria ( const FeDiscontinuousO0Tria< SCALAR > & )

default

◆ FeDiscontinuousO0Tria() [2/3]

template<typename SCALAR >

projects::dpg::FeDiscontinuousO0Tria< SCALAR >::FeDiscontinuousO0Tria ( FeDiscontinuousO0Tria< SCALAR > && )

defaultnoexcept

◆ FeDiscontinuousO0Tria() [3/3]

template<typename SCALAR >

projects::dpg::FeDiscontinuousO0Tria< SCALAR >::FeDiscontinuousO0Tria ( )

default

◆ ~FeDiscontinuousO0Tria()

template<typename SCALAR >

projects::dpg::FeDiscontinuousO0Tria< SCALAR >::~FeDiscontinuousO0Tria ( )

overridedefault

Member Function Documentation

◆ Degree()

template<typename SCALAR >

unsigned projects::dpg::FeDiscontinuousO0Tria< 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 47 of file discontinuous_fe_constant.h.

◆ EvalReferenceShapeFunctions()

template<typename SCALAR >

Eigen::Matrix< SCALAR, Eigen::Dynamic, Eigen::Dynamic > projects::dpg::FeDiscontinuousO0Tria< 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 72 of file discontinuous_fe_constant.h.

◆ EvaluationNodes()

template<typename SCALAR >

Eigen::MatrixXd projects::dpg::FeDiscontinuousO0Tria< SCALAR >::EvaluationNodes ( ) const

inlineoverridevirtual

Only evaluation node is the barycenter of the reference triangle.

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 99 of file discontinuous_fe_constant.h.

◆ GradientsReferenceShapeFunctions()

template<typename SCALAR >

Eigen::Matrix< SCALAR, Eigen::Dynamic, Eigen::Dynamic > projects::dpg::FeDiscontinuousO0Tria< 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 85 of file discontinuous_fe_constant.h.