QGaussLobatto< dim > Class Template Reference

#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).

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

• deal.II/base/quadrature_lib.h
• source/base/quadrature_lib.cc