A shape function set of \( H(\text{div}) \) conforming functions. More...

#include <nedelecshapefunctions.hh>

Detailed Description

template<class ctype, int dimension, class T = double>
class Kaskade::HdivSimplexShapeFunctionSet< ctype, dimension, T >

A shape function set of \( H(\text{div}) \) conforming functions.

For each Lagrangian interpolation point \( \xi_i \) of each face, there is one shape function with non-vanishing normal component of size 1 exactly at this point. Those shape functions are associated to the corresponding face. The remaining basis functions that complete the polynomial space all have vanishing normal components and are associated to the cell.

Only the normal components at the face interpolation points are specified, the tangential components are well-defined but unspecified (implementation detail).

Definition at line 446 of file nedelecshapefunctions.hh.

Inheritance diagram for Kaskade::HdivSimplexShapeFunctionSet< ctype, dimension, T >:

Public Types
typedef HdivSimplexShapeFunction< ctype, dim, T >	value_type

typedef double	Scalar
	Scalar field type. More...

typedef DynamicMatrix< Dune::FieldMatrix< double, 1, 1 > >	Matrix
	A matrix type mapping one-component coefficient vectors. More...

typedef std::vector< Dune::FieldVector< ctype, dimension > >	InterpolationNodes
	A container type for holding interpolation points in the reference elements. More...

typedef DynamicMatrix< Dune::FieldMatrix< double, comp, 1 > >	SfValueArray

Public Member Functions
	HdivSimplexShapeFunctionSet (int order)
	Constructs a \( H(\text{div}) \) conforming shape function set of given polynomial order. More...

virtual value_type const &	operator[] (int i) const
	Random access to a shape function. More...

virtual Dune::GeometryType	type () const

virtual void	interpolate (typename ShapeFunctionSet< ctype, dim, T, dim >::SfValueArray const &A, DynamicMatrix< Dune::FieldMatrix< T, 1, 1 > > &IA) const
	Left-multiplies the provided matrix with the interpolation matrix of the shape function set. More...

virtual int	size () const
	Number of shape functions in the set. More...

int	order () const
	Maximal polynomial order of shape functions. More...

auto	referenceElement () const

void	initHierarchicalProjection (ShapeFunctionSet< ctype, dimension, double, comp > const *sfl)
	Initialize the hierarchical projection matrix based on the given lower order shape function set. More...

InterpolationNodes const &	interpolationNodes () const
	Interpolation points. More...

virtual void	interpolate (SfValueArray const &A, Matrix &IA) const=0
	Left-multiplies the provided matrix with the interpolation matrix of the shape function set. More...

void	evaluate (InterpolationNodes const &iNodes, SfValueArray &phi) const
	Evaluate shape function set at a set of points. In notation of the LocalToGlobalMapperConcept, this gives the matrix \( \Phi \): the entry Phi[i][j] is the value of shape function j evaluated at iNodes[i]. More...

Matrix const &	hierarchicProjection () const
	Returns a square matrix that projects shape function coefficients to a subspace spanned by shape functions of lower order. This is intended to be used to implement embedded error estimators. The actual definition of this subspace depends on the shape function set specified in the call to initHierarchicalProjection(). More...

virtual void	removeShapeFunction (size_t index)

Static Public Attributes
static int const	comps
	Number of components of the shape function values. More...

Protected Attributes
InterpolationNodes	iNodes

Matrix	projection

int	order_

int	size_

Member Typedef Documentation

◆ InterpolationNodes

typedef std::vector<Dune::FieldVector<ctype,dimension> > Kaskade::ShapeFunctionSet< ctype, dimension, double , comp >::InterpolationNodes

inherited

A container type for holding interpolation points in the reference elements.

Definition at line 164 of file pshapefunctions.hh.

◆ Matrix

typedef DynamicMatrix<Dune::FieldMatrix<double ,1,1> > Kaskade::ShapeFunctionSet< ctype, dimension, double , comp >::Matrix

inherited

A matrix type mapping one-component coefficient vectors.

Definition at line 119 of file pshapefunctions.hh.

◆ Scalar

typedef double Kaskade::ShapeFunctionSet< ctype, dimension, double , comp >::Scalar

inherited

Scalar field type.

Definition at line 112 of file pshapefunctions.hh.

◆ SfValueArray

typedef DynamicMatrix<Dune::FieldMatrix<double ,comp,1> > Kaskade::ShapeFunctionSet< ctype, dimension, double , comp >::SfValueArray

inherited

A twodimensional array type for holding shape function values evaluated at a set of nodes.

Definition at line 168 of file pshapefunctions.hh.

◆ value_type

template<class ctype , int dimension, class T = double>

typedef HdivSimplexShapeFunction<ctype,dim,T> Kaskade::HdivSimplexShapeFunctionSet< ctype, dimension, T >::value_type

Definition at line 452 of file nedelecshapefunctions.hh.

Constructor & Destructor Documentation

◆ HdivSimplexShapeFunctionSet()

template<class ctype , int dimension, class T = double>

Kaskade::HdivSimplexShapeFunctionSet< ctype, dimension, T >::HdivSimplexShapeFunctionSet ( int order )

Constructs a \( H(\text{div}) \) conforming shape function set of given polynomial order.

Parameters

order the polynomial order of the shape functions (at least 1)

Member Function Documentation

◆ evaluate()

void Kaskade::ShapeFunctionSet< ctype, dimension, double , comp >::evaluate	(	InterpolationNodes const &	iNodes,
		SfValueArray &	phi
	)		const

inlineinherited

Evaluate shape function set at a set of points. In notation of the LocalToGlobalMapperConcept, this gives the matrix \( \Phi \): the entry Phi[i][j] is the value of shape function j evaluated at iNodes[i].

Parameters

iNodes	the points at which the shape functions are to be evaluated.
phi	the array that is filled with shape function values. The array will be resized if needed.

Definition at line 253 of file pshapefunctions.hh.

◆ hierarchicProjection()

Matrix const & Kaskade::ShapeFunctionSet< ctype, dimension, double , comp >::hierarchicProjection ( ) const

inlineinherited

Returns a square matrix that projects shape function coefficients to a subspace spanned by shape functions of lower order. This is intended to be used to implement embedded error estimators. The actual definition of this subspace depends on the shape function set specified in the call to initHierarchicalProjection().

Definition at line 276 of file pshapefunctions.hh.

◆ initHierarchicalProjection()

void Kaskade::ShapeFunctionSet< ctype, dimension, double , comp >::initHierarchicalProjection ( ShapeFunctionSet< ctype, dimension, double , comp > const * sfl )

inlineinherited

Initialize the hierarchical projection matrix based on the given lower order shape function set.

Definition at line 174 of file pshapefunctions.hh.

◆ interpolate() [1/2]

virtual void Kaskade::ShapeFunctionSet< ctype, dimension, double , comp >::interpolate	(	SfValueArray const &	A,
		Matrix &	IA
	)		const

pure virtualinherited

Left-multiplies the provided matrix with the interpolation matrix of the shape function set.

Each column of A is interpreted as values of some function evaluated at this shape function set's interpolation points (see below). The columns of the output array IA then contain the shape function coefficients such that the corresponding linearcombination of shape functions "interpolates" that function in the interpolation points. What "interpolation" means is up to the actual implementation.

IA is automatically resized if needed.

Storage order: A[i][j] contains the value of function j at interpolation node i.

◆ interpolate() [2/2]

template<class ctype , int dimension, class T = double>

virtual void Kaskade::HdivSimplexShapeFunctionSet< ctype, dimension, T >::interpolate	(	typename ShapeFunctionSet< ctype, dim, T, dim >::SfValueArray const &	A,
		DynamicMatrix< Dune::FieldMatrix< T, 1, 1 > > &	IA
	)		const

inlinevirtual

Left-multiplies the provided matrix with the interpolation matrix of the shape function set.

Each column of A is interpreted as values of some function evaluated at this shape function set's interpolation points (edge midpoints). The columns of the output matrix IA then contain the shape function coefficients such that the corresponding linear combination of shape functions "interpolates" that function in the interpolation points. "Interpolation" means equal tangential components.

Definition at line 475 of file nedelecshapefunctions.hh.

◆ interpolationNodes()

InterpolationNodes const & Kaskade::ShapeFunctionSet< ctype, dimension, double , comp >::interpolationNodes ( ) const

inlineinherited

Interpolation points.

Provides interpolation points such that the shape function coefficients can be computed from function values at interpolation nodes by multiplication by this matrix. The interpolation points are guaranteed to be inside the reference element associated to this shape function set.

Definition at line 218 of file pshapefunctions.hh.

◆ operator[]()

template<class ctype , int dimension, class T = double>

virtual value_type const & Kaskade::HdivSimplexShapeFunctionSet< ctype, dimension, T >::operator[] ( int i ) const

inlinevirtual

Random access to a shape function.

Implements Kaskade::ShapeFunctionSet< ctype, dimension, double, dimension >.

Definition at line 460 of file nedelecshapefunctions.hh.

◆ order()

int Kaskade::ShapeFunctionSet< ctype, dimension, double , comp >::order ( ) const

inlineinherited

Maximal polynomial order of shape functions.

Definition at line 149 of file pshapefunctions.hh.

◆ referenceElement()

auto Kaskade::ShapeFunctionSet< ctype, dimension, double , comp >::referenceElement ( ) const

inlineinherited

Returns a reference to the reference element on which this shape function set is defined.

Definition at line 158 of file pshapefunctions.hh.

◆ removeShapeFunction()

virtual void Kaskade::ShapeFunctionSet< ctype, dimension, double , comp >::removeShapeFunction ( size_t index )

inlinevirtualinherited

Definition at line 284 of file pshapefunctions.hh.

◆ size()

virtual int Kaskade::ShapeFunctionSet< ctype, dimension, double , comp >::size ( ) const

inlinevirtualinherited

Number of shape functions in the set.

Definition at line 144 of file pshapefunctions.hh.

◆ type()

template<class ctype , int dimension, class T = double>

virtual Dune::GeometryType Kaskade::HdivSimplexShapeFunctionSet< ctype, dimension, T >::type ( ) const

inlinevirtual

Definition at line 462 of file nedelecshapefunctions.hh.

Member Data Documentation

◆ comps

int const Kaskade::ShapeFunctionSet< ctype, dimension, double , comp >::comps

staticinherited

Number of components of the shape function values.

Definition at line 132 of file pshapefunctions.hh.

◆ iNodes

InterpolationNodes Kaskade::ShapeFunctionSet< ctype, dimension, double , comp >::iNodes

protectedinherited

Definition at line 287 of file pshapefunctions.hh.

◆ order_

int Kaskade::ShapeFunctionSet< ctype, dimension, double , comp >::order_

protectedinherited

Definition at line 289 of file pshapefunctions.hh.

◆ projection

Matrix Kaskade::ShapeFunctionSet< ctype, dimension, double , comp >::projection

protectedinherited

Definition at line 288 of file pshapefunctions.hh.

◆ size_

int Kaskade::ShapeFunctionSet< ctype, dimension, double , comp >::size_

protectedinherited

Definition at line 290 of file pshapefunctions.hh.

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

nedelecshapefunctions.hh

Detailed Description

Public Types

Public Member Functions

Static Public Attributes

Protected Attributes

Member Typedef Documentation

◆ InterpolationNodes

◆ Matrix

◆ Scalar

◆ SfValueArray

◆ value_type

Constructor & Destructor Documentation

◆ HdivSimplexShapeFunctionSet()

Member Function Documentation

◆ evaluate()

◆ hierarchicProjection()

◆ initHierarchicalProjection()

◆ interpolate() [1/2]

◆ interpolate() [2/2]

◆ interpolationNodes()

◆ operator[]()

◆ order()

◆ referenceElement()

◆ removeShapeFunction()

◆ size()

◆ type()

Member Data Documentation

◆ comps

◆ iNodes

◆ order_

◆ projection

◆ size_