NAME¶

riesz - integrate a function by using quadrature formulae

DESCRIPTION¶

The function riesz implements the approximation of an integral by using quadrature formulae.

SYNOPSYS¶

template <class Function>
field riesz (const space& Xh, const Function& f, quadrature_option_type qopt = default_value);
template <class Function>
field riesz (const space& Xh, const Function& f, const geo& domain, quadrature_option_type qopt = default_value);

EXAMPLE¶

The following code compute the Riesz representant, denoted by lh of f(x), and the integral of f over the domain omega:

  Float f(const point& x);
  ...
  space Xh (omega_h, "P1");
  field lh = riesz (Xh, f);
  Float int_f = dot(lh, 1);
 

The Riesz representer is the lh vector of values:

  lh(i) = integrate f(x) phi_i(x) dx
 

where phi_i is the i-th basis function in Xh and the integral is evaluated by using a quadrature formulae. By default the quadrature formule is the Gauss one with the order equal to the polynomial order of Xh. Alternative quadrature formulae and order is available by passing an optional variable to riesz.

OPTIONS¶

An optional argument specifies the quadrature formulae used for the computation of the integral. The domain of integration is by default the mesh associated to the finite element space. An alternative domain dom, e.g. a part of the boundary can be supplied as an extra argument. This domain can be also a band associated to the banded level set method.

IMPLEMENTATION¶

template <class T, class M, class Function>
field_basic<T,M>
riesz (
    const space_basic<T,M>&       Xh,
    const Function&               f,
    const quadrature_option_type& qopt
       = quadrature_option_type(quadrature_option_type::max_family,0))
 

IMPLEMENTATION¶

template <class T, class M, class Function>
field_basic<T,M>
riesz (
    const space_basic<T,M>&       Xh,
    const Function&               f,
    const geo_basic<T,M>&         dom,
    const quadrature_option_type& qopt
       = quadrature_option_type(quadrature_option_type::max_family,0))
 

IMPLEMENTATION¶

template <class T, class M, class Function>
field_basic<T,M>
riesz (
    const space_basic<T,M>&       Xh,
    const Function&               f,
    std::string                   dom_name,
    const quadrature_option_type& qopt
       = quadrature_option_type(quadrature_option_type::max_family,0))
 

IMPLEMENTATION¶

template <class T, class M, class Function>
field_basic<T,M>
riesz (
    const space_basic<T,M>&       Xh,
    const Function&               f,
    const band_basic<T,M>&        gh,
    const quadrature_option_type& qopt
       = quadrature_option_type(quadrature_option_type::max_family,0))