#include <deal.II/base/quadrature_lib.h>

Inheritance diagram for QGaussLobatto< dim >:

Public Member Functions
	QGaussLobatto (const unsigned int n)

Public Member Functions inherited from Quadrature< dim >
	Quadrature (const unsigned int n_quadrature_points=0)

	Quadrature (const SubQuadrature &, const Quadrature< 1 > &)

	Quadrature (const Quadrature< dim !=1 ? 1 :0 > &quadrature_1d)

	Quadrature (const Quadrature< dim > &q)

	Quadrature (Quadrature< dim > &&)=default

	Quadrature (const std::vector< Point< dim > > &points, const std::vector< double > &weights)

	Quadrature (const std::vector< Point< dim > > &points)

	Quadrature (const Point< dim > &point)

virtual	~Quadrature ()

Quadrature &	operator= (const Quadrature< dim > &)

bool	operator== (const Quadrature< dim > &p) const

void	initialize (const std::vector< Point< dim > > &points, const std::vector< double > &weights)

unsigned int	size () const

const Point< dim > &	point (const unsigned int i) const

const std::vector< Point< dim > > &	get_points () const

double	weight (const unsigned int i) const

const std::vector< double > &	get_weights () const

std::size_t	memory_consumption () const

template<class Archive >
void	serialize (Archive &ar, const unsigned int version)

Public Member Functions inherited from Subscriptor
	Subscriptor ()

	Subscriptor (const Subscriptor &)

	Subscriptor (Subscriptor &&)

virtual	~Subscriptor ()

Subscriptor &	operator= (const Subscriptor &)

Subscriptor &	operator= (Subscriptor &&)

void	subscribe (const char *identifier=0) const

void	unsubscribe (const char *identifier=0) const

unsigned int	n_subscriptions () const

void	list_subscribers () const

template<class Archive >
void	serialize (Archive &ar, const unsigned int version)

Protected Member Functions
std::vector< long double >	compute_quadrature_points (const unsigned int q, const int alpha, const int beta) const

std::vector< long double >	compute_quadrature_weights (const std::vector< long double > &x, const int alpha, const int beta) const

long double	JacobiP (const long double x, const int alpha, const int beta, const unsigned int n) const

Static Protected Member Functions
static long double	gamma (const unsigned int n)

Additional Inherited Members
Public Types inherited from Quadrature< dim >
typedef Quadrature< dim-1 >	SubQuadrature

Static Public Member Functions inherited from Subscriptor
static ::ExceptionBase &	ExcInUse (int arg1, char *arg2, std::string &arg3)

static ::ExceptionBase &	ExcNoSubscriber (char arg1, char arg2)

Protected Attributes inherited from Quadrature< dim >
std::vector< Point< dim > >	quadrature_points

std::vector< double >	weights

Detailed Description

template<int dim>
class QGaussLobatto< dim >

The Gauss-Lobatto family of quadrature rules for numerical integration.

This modification of the Gauss quadrature uses the two interval end points as well. Being exact for polynomials of degree 2n-3, this formula is suboptimal by two degrees.

The quadrature points are interval end points plus the roots of the derivative of the Legendre polynomial P_n-1 of degree n-1. The quadrature weights are 2/(n(n-1)(P_n-1(x_i)²).

Note: This implementation has not been optimized concerning numerical stability and efficiency. It can be easily adapted to the general case of Gauss-Lobatto-Jacobi-Bouzitat quadrature with arbitrary parameters $\alpha$ , $\beta$ , of which the Gauss-Lobatto-Legendre quadrature ( $\alpha = \beta = 0$ ) is a special case.

See also: http://en.wikipedia.org/wiki/Handbook_of_Mathematical_Functions; Karniadakis, G.E. and Sherwin, S.J.: Spectral/hp element methods for computational fluid dynamics. Oxford: Oxford University Press, 2005

Author: Guido Kanschat, 2005, 2006; F. Prill, 2006

Definition at line 75 of file quadrature_lib.h.

Constructor & Destructor Documentation

◆ QGaussLobatto()

template<int dim>

QGaussLobatto< dim >::QGaussLobatto ( const unsigned int n )

Generate a formula with n quadrature points (in each space direction).

Definition at line 970 of file quadrature_lib.cc.

Member Function Documentation

◆ compute_quadrature_points()

template<int dim>

std::vector<long double> QGaussLobatto< dim >::compute_quadrature_points	(	const unsigned int	q,
		const int	alpha,
		const int	beta
	)		const

protected

Compute Legendre-Gauss-Lobatto quadrature points in the interval $[-1, +1]$ . They are equal to the roots of the corresponding Jacobi polynomial (specified by alpha, beta). q is the number of points.

Returns: Vector containing nodes.

◆ compute_quadrature_weights()

template<int dim>

std::vector<long double> QGaussLobatto< dim >::compute_quadrature_weights	(	const std::vector< long double > &	x,
		const int	alpha,
		const int	beta
	)		const

protected

Compute Legendre-Gauss-Lobatto quadrature weights. The quadrature points and weights are related to Jacobi polynomial specified by alpha, beta. x denotes the quadrature points.

Returns: Vector containing weights.

◆ JacobiP()

template<int dim>

long double QGaussLobatto< dim >::JacobiP	(	const long double	x,
		const int	alpha,
		const int	beta,
		const unsigned int	n
	)		const

protected

Evaluate a Jacobi polynomial $P^{\alpha, \beta}_n(x)$ specified by the parameters alpha, beta, n. Note: The Jacobi polynomials are not orthonormal and defined on the interval $[-1, +1]$ . x is the point of evaluation.

◆ gamma()

template<int dim>

static long double QGaussLobatto< dim >::gamma ( const unsigned int n )

staticprotected

Evaluate the Gamma function $\Gamma(n) = (n-1)!$ .

Parameters

n	point of evaluation (integer).

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

deal.II/base/quadrature_lib.h
/build/deal.ii-IXg7Js/deal.ii-8.5.1/source/base/quadrature_lib.cc

Public Member Functions

Protected Member Functions

Static Protected Member Functions

Additional Inherited Members

Detailed Description

template<int dim> class QGaussLobatto< dim >

Constructor & Destructor Documentation

◆ QGaussLobatto()

Member Function Documentation

◆ compute_quadrature_points()

◆ compute_quadrature_weights()

◆ JacobiP()

◆ gamma()

template<int dim>
class QGaussLobatto< dim >