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LA 0.1 (Default branch)
03-07-2008
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form(2rheolef) rheolef-6.1 form(2rheolef) 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 dual- ity 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(); ALGABRA
Forms, as matrices (see csr(2)), support linear algebra: Adding or substracting 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. 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 (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; SEE ALSO
field(2), csr(2), space(2), csr(2) rheolef-6.1 rheolef-6.1 form(2rheolef)