 riesz(4) 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,

```

## 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,

```

## 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,

```

## 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,