Reference documentation for deal.II version Git 38fac49 2014-09-30 14:58:30 -0500
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QGaussLogR< dim > Class Template Reference

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

Inheritance diagram for QGaussLogR< dim >:
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Public Member Functions

 QGaussLogR (const unsigned int n, const Point< dim > x0=Point< dim >(), const double alpha=1, const bool factor_out_singular_weight=false)
 
template<>
 QGaussLogR (const unsigned int n, const Point< 1 > x0, const double alpha, const bool flag)
 
- 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 (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 ()
 
Quadratureoperator= (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 &)
 
virtual ~Subscriptor ()
 
Subscriptoroperator= (const Subscriptor &)
 
void subscribe (const char *identifier=0) const
 
void unsubscribe (const char *identifier=0) const
 
unsigned int n_subscriptions () const
 
void list_subscribers () const
 
 DeclException3 (ExcInUse, int, char *, std::string &,<< "Object of class "<< arg2<< " is still used by "<< arg1<< " other objects.\n"<< "(Additional information: "<< arg3<< ")\n"<< "Note the entry in the Frequently Asked Questions of "<< "deal.II (linked to from http://www.dealii.org/) for "<< "more information on what this error means.")
 
 DeclException2 (ExcNoSubscriber, char *, char *,<< "No subscriber with identifier \""<< arg2<< "\" did subscribe to this object of class "<< arg1)
 
template<class Archive >
void serialize (Archive &ar, const unsigned int version)
 

Protected Attributes

const double fraction
 
- Protected Attributes inherited from Quadrature< dim >
std::vector< Point< dim > > quadrature_points
 
std::vector< doubleweights
 

Additional Inherited Members

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

Detailed Description

template<int dim>
class QGaussLogR< dim >

A class for Gauss quadrature with arbitrary logarithmic weighting function. This formula is used to to integrate \(\ln(|x-x_0|/\alpha)\;f(x)\) on the interval \([0,1]\), where \(f\) is a smooth function without singularities, and \(x_0\) and \(\alpha\) are given at construction time, and are the location of the singularity \(x_0\) and an arbitrary scaling factor in the singularity.

You have to make sure that the point \(x_0\) is not one of the Gauss quadrature points of order \(N\), otherwise an exception is thrown, since the quadrature weights cannot be computed correctly.

This quadrature formula is rather expensive, since it uses internally two Gauss quadrature formulas of order n to integrate the nonsingular part of the factor, and two GaussLog quadrature formulas to integrate on the separate segments \([0,x_0]\) and \([x_0,1]\). If the singularity is one of the extremes and the factor alpha is 1, then this quadrature is the same as QGaussLog.

The last argument from the constructor allows you to use this quadrature rule in one of two possible ways:

\[ \int_0^1 g(x) dx = \int_0^1 f(x) \ln\left(\frac{|x-x_0|}{\alpha}\right) dx = \sum_{i=0}^N w_i g(q_i) = \sum_{i=0}^N \bar{w}_i f(q_i) \]

Which one of the two sets of weights is provided, can be selected by the factor_out_singular_weight parameter. If it is false (the default), then the \(\bar{w}_i\) weigths are computed, and you should provide only the smooth function \(f(x)\), since the singularity is included inside the quadrature. If the parameter is set to true, then the singularity is factored out of the quadrature formula, and you should provide a function \(g(x)\), which should at least be similar to \(\ln(|x-x_0|/\alpha)\).

Notice that this quadrature rule is worthless if you try to use it for regular functions once you factored out the singularity.

The weights and functions have been tabulated up to order 12.

Definition at line 314 of file quadrature_lib.h.

Constructor & Destructor Documentation

template<int dim>
QGaussLogR< dim >::QGaussLogR ( const unsigned int  n,
const Point< dim >  x0 = Point< dim >(),
const double  alpha = 1,
const bool  factor_out_singular_weight = false 
)

The constructor takes four arguments: the order of the gauss formula on each of the segments \([0,x_0]\) and \([x_0,1]\), the actual location of the singularity, the scale factor inside the logarithmic function and a flag that decides whether the singularity is left inside the quadrature formula or it is factored out, to be included in the integrand.

Member Data Documentation

template<int dim>
const double QGaussLogR< dim >::fraction
protected

This is the length of interval \((0,origin)\), or 1 if either of the two extremes have been selected.

Definition at line 340 of file quadrature_lib.h.


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