Provided by: librheolef-dev_6.6-1build2_amd64 #### NAME

```       riesz - approximate a Riesz representer

```

#### SYNOPSYS

```        template <class Function>
field riesz (const space& Xh, const Expr& expr,

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

```

#### NOTE

```       This  function  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 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>
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,
```       integrate(4)