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form

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);