Reference documentation for deal.II version Git ab1cc5b 2017-03-24 06:23:49 -0600
QGaussOneOverR< dim > Class Template Reference

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

Inheritance diagram for QGaussOneOverR< dim >:
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## Public Member Functions

QGaussOneOverR (const unsigned int n, const Point< dim > singularity, const bool factor_out_singular_weight=false)

QGaussOneOverR (const unsigned int n, const unsigned int vertex_index, const bool factor_out_singular_weight=false)

Public Member Functions inherited from Quadrature< dim >

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

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

Quadrature (const Point< dim > &point)

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 ()

Subscriptoroperator= (const Subscriptor &)

Subscriptoroperator= (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)

## Static Private Member Functions

static unsigned int quad_size (const Point< dim > singularity, const unsigned int n)

Public Types inherited from Quadrature< dim >

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 QGaussOneOverR< dim >

A class for Gauss quadrature with $$1/R$$ weighting function. This formula can be used to integrate $$1/R \ f(x)$$ on the reference element $$[0,1]^2$$, where $$f$$ is a smooth function without singularities, and $$R$$ is the distance from the point $$x$$ to the vertex $$\xi$$, given at construction time by specifying its index. Notice that this distance is evaluated in the reference element.

This quadrature formula is obtained from two QGauss quadrature formulas, upon transforming them into polar coordinate system centered at the singularity, and then again into another reference element. This allows for the singularity to be cancelled by part of the Jacobian of the transformation, which contains $$R$$. In practice the reference element is transformed into a triangle by collapsing one of the sides adjacent to the singularity. The Jacobian of this transformation contains $$R$$, which is removed before scaling the original quadrature, and this process is repeated for the next half element.

Upon construction it is possible to specify whether we want the singularity removed, or not. In other words, this quadrature can be used to integrate $$g(x) = 1/R\ f(x)$$, or simply $$f(x)$$, with the $$1/R$$ factor already included in the quadrature weights.

Definition at line 339 of file quadrature_lib.h.

## Constructor & Destructor Documentation

template<int dim>
 QGaussOneOverR< dim >::QGaussOneOverR ( const unsigned int n, const Point< dim > singularity, const bool factor_out_singular_weight = false )

This constructor takes three arguments: the order of the Gauss formula, the point of the reference element in which the singularity is located, and whether we include the weighting singular function inside the quadrature, or we leave it in the user function to be integrated.

Traditionally, quadrature formulas include their weighting function, and the last argument is set to false by default. There are cases, however, where this is undesirable (for example when you only know that your singularity has the same order of 1/R, but cannot be written exactly in this way).

In other words, you can use this function in either of the following way, obtaining the same result:

// This will produce the integral of f(x)/R
// And the same here
// This also will produce the integral of f(x)/R, but 1/R has to
// be specified.
for(unsigned int i=0; i<singular_quad.size(); ++i) {
}
template<int dim>
 QGaussOneOverR< dim >::QGaussOneOverR ( const unsigned int n, const unsigned int vertex_index, const bool factor_out_singular_weight = false )

The constructor takes three arguments: the order of the Gauss formula, the index of the vertex where the singularity is located, and whether we include the weighting singular function inside the quadrature, or we leave it in the user function to be integrated. Notice that this is a specialized version of the previous constructor which works only for the vertices of the quadrilateral.

Traditionally, quadrature formulas include their weighting function, and the last argument is set to false by default. There are cases, however, where this is undesirable (for example when you only know that your singularity has the same order of 1/R, but cannot be written exactly in this way).

In other words, you can use this function in either of the following way, obtaining the same result:

// This will produce the integral of f(x)/R
// And the same here
// This also will produce the integral of f(x)/R, but 1/R has to
// be specified.
for(unsigned int i=0; i<singular_quad.size(); ++i) {