Provided by: librheolef-dev_7.0-3_amd64 bug

NAME

       form - representation of a finite element bilinear form

DESCRIPTION

       The  form  class  groups  four  sparse matrix, associated to a bilinear form on two finite
       element spaces:

              a: U*V   ----> IR
                (u,v)  |---> a(u,v)

       The operator A associated to the bilinear form is defined by:

              A: U  ----> V'
                 u  |---> A(u)

       where u and v are fields (see field(2)), and A(u) is such that a(u,v)=<A(u),v> for  all  u
       in  U and v in V and where <.,.> 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).

       Since both U and V are finite dimensional spaces, the linear operator can  be  represented
       by  a  matrix.   The  form  class  is represented by four sparse matrix in csr format (see
       csr(2)), associated to unknown and blocked degrees of freedom of  origin  and  destination
       spaces (see space(2)).

EXAMPLE

       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:

             geo omega("square");
             space V (omega,"P1");
             form a (V,V,"grad_grad");
             field uh = interpolate (fct, V);
             field vh = a*uh;
             cout << v;

       The form-field vh=a*uh operation is equivalent to the following matrix-vector operations:

            vh.set_u() = a.uu()*uh.u() + a.ub()*uh.b();
            vh.set_b() = a.bu()*uh.u() + a.bb()*uh.b();

ALGEBRA

       Forms, as matrices (see csr(2)), support linear algebra: Adding or subtracting  two  forms
       writes a+b and a-b, respectively, and multiplying a form by a field uh writes a*uh.  Thus,
       any linear combination of forms is available.

WEIGHTED FORM

       A weighted form is a form with an extra weight function w(x), e.g.:

                          /
                         |
              a(uh,vh) = |       grad(uh).grad(vh) w(x) dx
                         |
                        / Omega

       In the present implementation, w 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.

IMPLEMENTATION

       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;

SEE ALSO

       field(2), csr(2), space(2), csr(2), field(2)

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.