.\"
.de Id
..
.de Sp
.if n .sp
.if t .sp 0.4
..
.TH form 2rheolef "rheolef-7.0" "rheolef-7.0" "rheolef-7.0"
.\" label: /*Class:form
.SH NAME
\fBform\fP - representation of a finite element bilinear form
.SH DESCRIPTION

The form class groups four sparse matrix, associated to
a bilinear form on two finite element spaces:
.\" begin_example
.Sp
.nf
       a: U*V   ----> IR
         (u,v)  |---> a(u,v)
.Sp
.fi
.\" end_example
The operator \fBA\fP associated to the bilinear form is
defined by:
.\" begin_example
.Sp
.nf
       A: U  ----> V'
          u  |---> A(u)
.Sp
.fi
.\" end_example
where \fBu\fP and \fBv\fP are fields (see field(2)),
and \fBA(u)\fP is such that \fBa(u,v)=<A(u),v>\fP
for all u in U and v in V and where \fB<.,.>\fP denotes
the duality product between V and V'.
Since V is a finite dimensional spaces, the duality product is
the euclidian product in IR^dim(V).
.PP
Since both U and V are finite dimensional spaces,
the linear operator can be represented by a matrix.
The \fBform\fP class is represented by four sparse matrix
in \fBcsr\fP format (see csr(2)),
associated to unknown and blocked degrees of freedom
of origin and destination spaces (see space(2)).
.SH EXAMPLE

.PP
The operator A associated to a bilinear form a(.,.) by
the relation (Au,v) = a(u,v) could be applied by using
a sample matrix notation A*u, as shown by the following code:
.\" begin_example
.Sp
.nf
      geo omega("square");
      space V (omega,"P1");
      form a (V,V,"grad_grad");
      field uh = interpolate (fct, V);
      field vh = a*uh;
      cout << v;
.Sp
.fi
.\" end_example
.PP
The form-field \fBvh=a*uh\fP operation is equivalent to
the following matrix-vector operations:
.\" begin_example
.Sp
.nf
     vh.set_u() = a.uu()*uh.u() + a.ub()*uh.b();
     vh.set_b() = a.bu()*uh.u() + a.bb()*uh.b();
.Sp
.fi
.\" end_example
.PP
.SH ALGEBRA

Forms, as matrices (see csr(2)), support linear algebra:
Adding or subtracting two forms writes \fBa+b\fP and \fBa-b\fP, respectively,
and multiplying a form by a field \fBuh\fP writes \fBa*uh\fP.
Thus, any linear combination of forms is available.
.PP
.\" skip: @cindex quarature formula
.SH WEIGHTED FORM

A weighted form is a form with an extra weight function \fBw(x)\fP, e.g.:
.\" begin_example
.Sp
.nf
                   /
                  |
       a(uh,vh) = |       grad(uh).grad(vh) w(x) dx
                  |
                 / Omega
.Sp
.fi
.\" end_example
In the present implementation, \fBw\fP can be any field, function or class-function
or any nonlinear field expression (see field(2)).
As the integration cannot be performed exactly in general, a quadrature formula
can be supplied.
This feature is extensively used when solving nonlinear problems.
.PP
.\" skip start:SEE ALSO:
.\" skip start:AUTHOR:
.\" skip start:DATE:
.\" skip start:METHODS:
.\" END
.SH IMPLEMENTATION
.\" begin_example
.Sp
.nf
template<class T, class M>
class form_basic {
public :
// typedefs:

    typedef typename csr<T,M>::size_type    size_type;
    typedef T                               value_type;
    typedef typename scalar_traits<T>::type float_type;
    typedef geo_basic<float_type,M>         geo_type;
    typedef space_basic<float_type,M>       space_type;

// allocator/deallocator:

    form_basic ();
    form_basic (const form_basic<T,M>&);
    form_basic<T,M>& operator= (const form_basic<T,M>&);

// allocators from initializer list (c++ 2011):

#ifdef _RHEOLEF_HAVE_STD_INITIALIZER_LIST
    form_basic (const std::initializer_list<form_concat_value<T,M> >& init_list);
    form_basic (const std::initializer_list<form_concat_line <T,M> >& init_list);
#endif // _RHEOLEF_HAVE_STD_INITIALIZER_LIST

// accessors:

    const space_type& get_first_space() const;
    const space_type& get_second_space() const;
    const geo_type&   get_geo() const;

    const communicator& comm() const;

// linear algebra:

    form_basic<T,M>  operator+  (const form_basic<T,M>& b) const;
    form_basic<T,M>  operator-  (const form_basic<T,M>& b) const;
    form_basic<T,M>  operator*  (const form_basic<T,M>& b) const;
    form_basic<T,M>& operator*= (const T& lambda);
    field_basic<T,M> operator*  (const field_basic<T,M>& xh) const;
    field_basic<T,M> trans_mult (const field_basic<T,M>& yh) const;
    float_type operator () (const field_basic<T,M>& uh, const field_basic<T,M>& vh) const;

// io:

    odiststream& put (odiststream& ops, bool show_partition = true) const;
    void dump (std::string name) const;

// accessors & modifiers to unknown & blocked parts:

    const csr<T,M>&     uu() const { return _uu; }
    const csr<T,M>&     ub() const { return _ub; }
    const csr<T,M>&     bu() const { return _bu; }
    const csr<T,M>&     bb() const { return _bb; }
          csr<T,M>& set_uu()       { return _uu; }
          csr<T,M>& set_ub()       { return _ub; }
          csr<T,M>& set_bu()       { return _bu; }
          csr<T,M>& set_bb()       { return _bb; }

// data
protected:
    space_type  _X;
    space_type  _Y;
    csr<T,M>    _uu;
    csr<T,M>    _ub;
    csr<T,M>    _bu;
    csr<T,M>    _bb;

// internals:
public:
    // with vf expression arg
    template <class Expr>
    void assembly_internal (
        const geo_basic<T,M>&         dom,
        const geo_basic<T,M>&         band,
        const band_basic<T,M>&        gh,
        const Expr&                   expr,
        const integrate_option&       fopt,
        bool                          is_on_band);
    template <class Expr>
    void assembly (
        const geo_basic<T,M>&         domain,
        const Expr&                   expr,
        const integrate_option&       fopt);
    template <class Expr>
    void assembly (
        const band_basic<T,M>&        gh,
        const Expr&                   expr,
        const integrate_option&       fopt);

    // backward compat: named forms
    form_basic (const space_type& X, const space_type& Y,
        const std::string& name = "",
        const quadrature_option& qopt = quadrature_option());

    form_basic (const space_type& X, const space_type& Y,
        const std::string& name,
        const field_basic<T,M>& weight,
        const quadrature_option& qopt = quadrature_option());

    template<class Function>
    form_basic (const space_type& X, const space_type& Y,
        const std::string& name,
        Function weight,
        const quadrature_option& qopt = quadrature_option());

    form_basic (const space_type& X, const space_type& Y,
        const std::string& name,
        const geo_basic<T,M>& gamma,
        const quadrature_option& qopt = quadrature_option());

    form_basic (const space_type& X, const space_type& Y,
        const std::string& name,
        const geo_basic<T,M>& gamma,
        const field_basic<T,M>& weight,
        const quadrature_option& qopt = quadrature_option());

    template<class Function>
    form_basic (
        const space_type& X,
        const space_type& Y,
        const std::string& name,
        const geo_basic<T,M>& gamma,
        Function weight,
        const quadrature_option& qopt = quadrature_option());
protected:
    // backward compat: named forms (cont.)
    template<class WeightFunction>
    void form_init (
                   const std::string&      name,
                   bool                    has_weight,
                   WeightFunction          weight,
                   const quadrature_option& qopt);
    template<class WeightFunction>
    void form_init_on_domain (
                   const std::string&      name,
                   const geo_basic<T,M>&   gamma,
                   bool                    has_weight,
                   WeightFunction          weight,
                   const geo_basic<T,M>&   w_omega, // the domain where the fct weight is defined
                   const quadrature_option& qopt);
};
template<class T, class M> form_basic<T,M> trans (const form_basic<T,M>& a);
template<class T, class M> field_basic<T,M> diag (const form_basic<T,M>& a);
template<class T, class M> form_basic<T,M>  diag (const field_basic<T,M>& dh);
typedef form_basic<Float,rheo_default_memory_model> form;
.Sp
.fi
.\" end_example

.\" LENGTH = 5
.SH SEE ALSO
field(2), csr(2), space(2), csr(2), field(2)
.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.