dune-pdelab: Dune::PB::OrthonormalPolynomialBasis< FieldType, k, d, bt, ComputationFieldType, basisType > Class Template Reference

dune-pdelab 2.5-dev

Integrate monomials over the reference element. More...

#include <dune/pdelab/finiteelement/l2orthonormal.hh>

Inheritance diagram for Dune::PB::OrthonormalPolynomialBasis< FieldType, k, d, bt, ComputationFieldType, basisType >:

Inheritance graph

Public Types
enum	{ n = Traits::template Size<k,d>::value }

typedef Dune::FieldMatrix< FieldType, n, n >	LowprecMat

typedef Dune::FieldMatrix< ComputationFieldType, n, n >	HighprecMat

Public Member Functions
	OrthonormalPolynomialBasis ()

template<class LFE >
	OrthonormalPolynomialBasis (const LFE &lfe)

int	size (int l)

template<typename Point , typename Result >
void	evaluateFunction (const Point &x, Result &r) const

template<typename Point , typename Result >
void	evaluateJacobian (const Point &x, Result &r) const

template<typename Point , typename Result >
void	evaluateFunction (int l, const Point &x, Result &r) const

template<typename Point , typename Result >
void	evaluateJacobian (int l, const Point &x, Result &r) const

Detailed Description

template<typename FieldType, int k, int d, Dune::GeometryType::BasicType bt, typename ComputationFieldType = FieldType, BasisType basisType = BasisType::Pk>
class Dune::PB::OrthonormalPolynomialBasis< FieldType, k, d, bt, ComputationFieldType, basisType >

Integrate monomials over the reference element.

Computes an L_2 orthonormal basis of P_k on the given reference element. The basis polynomials are stored in a monomial representation. With the matrix coeffs private to this class we have

$phi_i(x) = \sum_{j=0}{n_k-1} c[i][j] x^{\alpha_j} \qquad (1)$

with n_k : the dimension of P_k alpha_j : the exponents of the j-th monomial

The class can be used to evaluate polynomials with any degree l smaller or equal to the compile-time parameter k.

Calculating derivatives. From (1) we have

$\begin{align*} \partial_s \phi_i(x) &= \sum_{j=0}{n_k-1} c[i][j] \partial_s x^{(\alpha_{j1},...,\alpha_{jd})} \\ &= \sum_{j=0}{n_k-1} c[i][j] * \alpha_js * x^{\beta_j} \end{align*}$

where beta_jr = alpha_jr-1 if r=s and alpha_jr else.

Template Parameters

FieldType	Type to represent coefficients after computation.
k	The polynomial degreee.
d	The space dimension.
GeometryType::BasicType	The reference element
ComputationFieldType	Type to do computations with. Might be high precission.
basisType	Type of the polynomial basis. eiter Pk or Qk

Member Typedef Documentation

◆ HighprecMat

template<typename FieldType , int k, int d, Dune::GeometryType::BasicType bt, typename ComputationFieldType = FieldType, BasisType basisType = BasisType::Pk>

typedef Dune::FieldMatrix<ComputationFieldType,n,n> Dune::PB::OrthonormalPolynomialBasis< FieldType, k, d, bt, ComputationFieldType, basisType >::HighprecMat

◆ LowprecMat

template<typename FieldType , int k, int d, Dune::GeometryType::BasicType bt, typename ComputationFieldType = FieldType, BasisType basisType = BasisType::Pk>

typedef Dune::FieldMatrix<FieldType,n,n> Dune::PB::OrthonormalPolynomialBasis< FieldType, k, d, bt, ComputationFieldType, basisType >::LowprecMat

Member Enumeration Documentation

◆ anonymous enum

template<typename FieldType , int k, int d, Dune::GeometryType::BasicType bt, typename ComputationFieldType = FieldType, BasisType basisType = BasisType::Pk>

anonymous enum

Enumerator
n

Constructor & Destructor Documentation

◆ OrthonormalPolynomialBasis() [1/2]

template<typename FieldType , int k, int d, Dune::GeometryType::BasicType bt, typename ComputationFieldType = FieldType, BasisType basisType = BasisType::Pk>

Dune::PB::OrthonormalPolynomialBasis< FieldType, k, d, bt, ComputationFieldType, basisType >::OrthonormalPolynomialBasis ( )

inline

◆ OrthonormalPolynomialBasis() [2/2]

template<typename FieldType , int k, int d, Dune::GeometryType::BasicType bt, typename ComputationFieldType = FieldType, BasisType basisType = BasisType::Pk>

template<class LFE >

Dune::PB::OrthonormalPolynomialBasis< FieldType, k, d, bt, ComputationFieldType, basisType >::OrthonormalPolynomialBasis ( const LFE & lfe )

inline

Member Function Documentation

◆ evaluateFunction() [1/2]

template<typename FieldType , int k, int d, Dune::GeometryType::BasicType bt, typename ComputationFieldType = FieldType, BasisType basisType = BasisType::Pk>

template<typename Point , typename Result >

void Dune::PB::OrthonormalPolynomialBasis< FieldType, k, d, bt, ComputationFieldType, basisType >::evaluateFunction	(	const Point &	x,
		Result &	r
	)		const

inline

◆ evaluateFunction() [2/2]

template<typename FieldType , int k, int d, Dune::GeometryType::BasicType bt, typename ComputationFieldType = FieldType, BasisType basisType = BasisType::Pk>

template<typename Point , typename Result >

void Dune::PB::OrthonormalPolynomialBasis< FieldType, k, d, bt, ComputationFieldType, basisType >::evaluateFunction	(	int	l,
		const Point &	x,
		Result &	r
	)		const

inline

◆ evaluateJacobian() [1/2]

template<typename FieldType , int k, int d, Dune::GeometryType::BasicType bt, typename ComputationFieldType = FieldType, BasisType basisType = BasisType::Pk>

template<typename Point , typename Result >

void Dune::PB::OrthonormalPolynomialBasis< FieldType, k, d, bt, ComputationFieldType, basisType >::evaluateJacobian	(	const Point &	x,
		Result &	r
	)		const

inline

◆ evaluateJacobian() [2/2]

template<typename FieldType , int k, int d, Dune::GeometryType::BasicType bt, typename ComputationFieldType = FieldType, BasisType basisType = BasisType::Pk>

template<typename Point , typename Result >

void Dune::PB::OrthonormalPolynomialBasis< FieldType, k, d, bt, ComputationFieldType, basisType >::evaluateJacobian	(	int	l,
		const Point &	x,
		Result &	r
	)		const

inline

◆ size()

template<typename FieldType , int k, int d, Dune::GeometryType::BasicType bt, typename ComputationFieldType = FieldType, BasisType basisType = BasisType::Pk>

int Dune::PB::OrthonormalPolynomialBasis< FieldType, k, d, bt, ComputationFieldType, basisType >::size ( int l )

inline

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

l2orthonormal.hh