<tt>form</tt>

finite element bilinear form

Description

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

   a: Uh*Vh   ----> IR
     (uh,vh)  +---> a(uh,vh)

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

   A: Uh  ----> Vh
      uh  +---> A*uh

where uh is a field, and vh=A*uh in Vh is such that a(uh,vh)=dual(A*uh,vh) for all vh in Vh and where dual(.,.) denotes the duality product between Vh and its dual. Since Vh is a finite dimensional space, its dual is identified to Vh itself and the duality product is the euclidean product in IR^dim(Vh). Also, the linear operator can be represented by a matrix.

In practice, bilinear forms are created by using the integrate function.

Algebra

Forms, as matrix, support standard algebra. Adding or subtracting two forms writes a+b and a-b, respectively, while multiplying by a scalar lambda writes lambda*a and multiplying two forms writes a*b. Also, multiplying a form by a field uh writes a*uh. The form inversion is not as direct as e.g. as inv(a), since forms are very large matrix in practice: form inversion can be obtained via the solver class. A notable exception is the case of block-diagonal forms at the element level: in that case, a direct inversion is possible during the assembly process, see integrate_option.

Representation

The degrees of freedom (see space) are splited between unknowns and blocked, i.e. uh=[uh.u,uh.b] for any field uh in Uh. Conversely, vh=[vh.u,vh.b] for any field vh in Vh. Then, the form-field vh=a*uh operation is formally equivalent to the following matrix-vector block operations:

    [ vh.u ]   [ a.uu  a.ub ] [ uh.u ]
    [      ] = [            ] [      ]
    [ vh.b ]   [ a.bu  a.bb ] [ uh.n ]

or, after expansion:

    vh.u = a.uu*uh.u + a.ub*vh.b
    vh.b = a.bu*uh.b + a.bb*vh.b

i.e. the A matrix also admits a 2x2 block structure. Then, the form class is represented by four sparse matrix and the csr compressed format is used. Note that the previous formal relations for vh=a*uh writes equivalently within the Rheolef library as:

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

Implementation

This documentation has been generated from file main/lib/form.h

The form class is simply an alias to the form_basic class

typedef form_basic<Float,rheo_default_memory_model> form;

The form_basic class provides an interface to four sparse matrix:

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>&);
 
    template<class Expr, class Sfinae = typename std::enable_if<details::is_form_lazy<Expr>::value, Expr>::type>
    form_basic (const Expr&);
 
    template<class Expr, class Sfinae = typename std::enable_if<details::is_form_lazy<Expr>::value, Expr>::type>
    form_basic<T,M>& operator= (const Expr&);
 
// allocators from initializer list (c++ 2011):
 
    form_basic (const std::initializer_list<details::form_concat_value<T,M> >& init_list);
    form_basic (const std::initializer_list<details::form_concat_line <T,M> >& init_list);
 
// accessors:
 
    const space_type& get_first_space() const;
    const space_type& get_second_space() const;
    const geo_type&   get_geo() const;
    bool is_symmetric() const;
    void set_symmetry (bool is_symm = true) const;
    bool is_definite_positive() const;
    void set_definite_positive (bool is_dp = true) const;
    bool is_symmetric_definite_positive() const;
    void set_symmetric_definite_positive() 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; }

};
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);