riesz  approximate a Riesz representer
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 oldfashioned code:
The
riesz function is now obsolete: it has been now suppersetted by the
integrate function see integrate(4). The oldfashioned 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.
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 ith
degree of freedom is:
/

dual(lh,vh) =  f(x) phi_i(x) dx

/ Omega
where phi_i is the ith 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*k1 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, classfunction or linear or nonlinear
expressions mixing fields and continuous functions.
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);
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.
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())
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())
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())
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())
integrate(4), integrate(4)