.\"
.de Id
..
.de Sp
.if n .sp
.if t .sp 0.4
..
.TH riesz 4rheolef "rheolef-7.0" "rheolef-7.0" "rheolef-7.0"
.\" label: /*Class:riesz
.SH NAME
\fBriesz\fP - approximate a Riesz representer
.\" skip: @findex  riesz
.\" skip: @cindex  riesz representer
.\" skip: @cindex  quadrature formula
.\" skip: @clindex space
.\" skip: @clindex field
.PP
.SH SYNOPSIS

The \fBriesz\fP function is now obsolete: it has been now suppersetted by the \fBintegrate\fP function
see integrate(4).
.\" begin_example
.Sp
.nf
 template <class Expr>
 field riesz (space, Expr expr);
 field riesz (space, Expr expr, quadrature_option);
 field riesz (space, Expr expr, domain);
 field riesz (space, Expr expr, domain, quadrature_option);
.Sp
.fi
.\" end_example
The domain can be also provided by its name as a string.
The old-fashioned code:
.SH NOTE

The \fBriesz\fP function is now obsolete: it has been now suppersetted by the \fBintegrate\fP function
see integrate(4).
The old-fashioned code:
.\" begin_example
.Sp
.nf
     field l1h = riesz (Xh, f);
     field l2h = riesz (Xh, f, "boundary");
.Sp
.fi
.\" end_example
writes now:
.\" begin_example
.Sp
.nf
     test v (Xh);
     field l1h = integrate (f*v);
     field l2h = integrate ("boundary", f*v);
.Sp
.fi
.\" end_example
The \fBriesz\fP function is still present in the library for backward compatibility purpose.
.SH DESCRIPTION

Let \fBf\fP be any continuous function, its Riesz representer in the finite
element space \fBXh\fP on the domain \fBOmega\fP is defind by:
.\" begin_example
.Sp
.nf
                /
                |
  dual(lh,vh) = |      f(x) vh(x) dx
                |
               / Omega
.Sp
.fi
.\" end_example
for all \fBvh\fP in \fBXh\fP, where \fBdual\fP denotes the duality
between \fBXh\fP and its dual. As \fBXh\fP is a finite dimensional space,
its dual is identified as \fBXh\fP and the duality product as the Euclidian one.
The Riesz representer is thus the \fBlh\fP field of \fBXh\fP where
its i-th degree of freedom is:
.\" begin_example
.Sp
.nf
                /
                |
  dual(lh,vh) = |      f(x) phi_i(x) dx
                |
               / Omega
.Sp
.fi
.\" end_example
where phi_i is the i-th basis function in \fBXh\fP.
The integral is evaluated by using a quadrature formula.
By default the quadrature formule is the Gauss one with
the order equal to \fB2*k-1\fP where $\fBk\fP is the polynomial degree in \fBXh\fP.
Alternative quadrature formula and order is available
by passing an optional variable to riesz.
.PP
The function \fBriesz\fP 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.
.PP
.\" skip: @cindex  integrate
.\" skip: @findex  integrate
.SH EXAMPLE

The following code compute the Riesz representant, denoted
by \fBlh\fP of f(x), and the integral of f over the domain omega:
.\" begin_example
.Sp
.nf
  Float f(const point& x);
  ...
  space Xh (omega_h, "P1");
  field lh = riesz (Xh, f);
  Float int_f = dual(lh, 1);
.Sp
.fi
.\" end_example
.SH 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 \fBdom\fP, e.g. a part of
the boundary can be supplied as an extra argument.
This domain can be also a \fBband\fP associated to
the banded level set method.
.\" END
.SH IMPLEMENTATION
.\" begin_example
.Sp
.nf
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& qopt
       = quadrature_option())
.Sp
.fi
.\" end_example
.SH IMPLEMENTATION
.\" begin_example
.Sp
.nf
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& qopt
       = quadrature_option())
.Sp
.fi
.\" end_example
.SH IMPLEMENTATION
.\" begin_example
.Sp
.nf
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& qopt
       = quadrature_option())
.Sp
.fi
.\" end_example
.SH IMPLEMENTATION
.\" begin_example
.Sp
.nf
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& qopt
       = quadrature_option())
.Sp
.fi
.\" end_example

.\" LENGTH = 2
.SH SEE ALSO
integrate(4), integrate(4)
.SH COPYRIGHT
Copyright (C) 2000-2018 Pierre Saramito <Pierre.Saramito@imag.fr> GPLv3+: GNU GPL version 3 or later <http://gnu.org/licenses/gpl.html>.
This is free software: you are free to change and redistribute it.  There is NO WARRANTY, to the extent permitted by law.