## 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 | / Omegafor 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 | / Omegawhere 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())