.\"
.de Id
..
.de Sp
.if n .sp
.if t .sp 0.4
..
.TH form 2rheolef "rheolef-6.1" "rheolef-6.1" "rheolef-6.1"
.\" 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
.SH ALGABRA

 Forms, as matrices (see csr(2)), support linear algebra:
 Adding or substracting 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 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 (const space_type& X, const space_type& Y,
        const std::string& name = "",
        quadrature_option_type  qopt = quadrature_option_type(quadrature_option_type::max_family,0));

    form_basic (const space_type& X, const space_type& Y,
        const std::string& name,
        const geo_basic<T,M>& gamma,
        quadrature_option_type  qopt = quadrature_option_type(quadrature_option_type::max_family,0));

    form_basic (const space_type& X, const space_type& Y,
        const std::string& name,
        const field_basic<T,M>& weight,
        quadrature_option_type  qopt = quadrature_option_type(quadrature_option_type::max_family,0));

    form_basic (const space_type& X, const space_type& Y,
        const std::string& name,
        const band_basic<T,M>& bh,
        quadrature_option_type  qopt = quadrature_option_type(quadrature_option_type::max_family,0));

// 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 T& lambda);
    field_basic<T,M> operator*  (const field_basic<T,M>& xh) const;
#ifdef TO_CLEAN
    template <class Expr>
    field_basic<T,M> operator*  (const field_expr<Expr>& xh) const;
#endif // TO_CLEAN
    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:
    void assembly (const form_element<T,M>&   form_e,
                   const geo_basic<T,M>&      X_geo,
                   const geo_basic<T,M>&      Y_geo,
                   bool  X_geo_is_background = true);
    void form_init (
                   const std::string&      name,
                   bool                    has_weight,
                   const field_basic<T,M>& weight,
                   quadrature_option_type  qopt);

};
template<class T, class M> form_basic<T,M> trans (const form_basic<T,M>& a);
typedef form_basic<Float,rheo_default_memory_model> form;
.Sp
.fi
.\" end_example

.\" LENGTH = 4
.SH SEE ALSO
field(2), csr(2), space(2), csr(2)