NAME¶
riesz - approximate a Riesz representer
SYNOPSYS¶
The riesz function is now obsolete: it has been now suppersetted by the
integrate function see integrate(4).
template <class Expr>
field riesz (space, Expr expr);
field riesz (space, Expr expr, quadrature_option_type);
field riesz (space, Expr expr, domain);
field riesz (space, Expr expr, domain, quadrature_option_type);
The domain can be also provided by its name as a string. The old-fashioned code:
NOTE¶
The riesz function is now obsolete: it has been now suppersetted by the
integrate function see integrate(4). The old-fashioned code:
field l1h = riesz (Xh, f);
field l2h = riesz (Xh, f, "boundary");
writes now:
test v (Xh);
field l1h = integrate (f*v);
field l2h = integrate ("boundary", f*v);
The riesz function is still present in the library for backward
compatibility purpose.
DESCRIPTION¶
Let f be any continuous function, its Riesz representer in the finite
element space Xh on the domain Omega is defind by:
/
|
dual(lh,vh) = | f(x) vh(x) dx
|
/ Omega
for all vh in Xh, where dual denotes the duality between
Xh and its dual. As Xh is a finite dimensional space, its dual
is identified as Xh and the duality product as the Euclidian one. The
Riesz representer is thus the lh field of Xh where its i-th
degree of freedom is:
/
|
dual(lh,vh) = | f(x) phi_i(x) dx
|
/ Omega
where phi_i is the i-th basis function in Xh. The integral is evaluated
by using a quadrature formula. By default the quadrature formule is the Gauss
one with the order equal to 2*k-1 where $k is the polynomial
degree in Xh. Alternative quadrature formula and order is available by
passing an optional variable to riesz.
The function riesz implements the approximation of the
Riesz representer by using some quadrature formula for the evaluation of the
integrals. Its argument can be any function, class-function or linear or
nonlinear expressions mixing fields and continuous functions.
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 = dual(lh, 1);
OPTIONS¶
An optional argument specifies the quadrature formula 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>
inline
field_basic<T,M>
riesz (
const space_basic<T,M>& Xh,
const Function& f,
const quadrature_option_type& qopt
= quadrature_option_type())
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())
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())
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())