Representation of a finite element bilinear form
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)).
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();
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.
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.
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>&);
// 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 form_option_type& fopt,
bool is_on_band);
template <class Expr>
void assembly (
const geo_basic<T,M>& domain,
const Expr& expr,
const form_option_type& fopt);
template <class Expr>
void assembly (
const band_basic<T,M>& gh,
const Expr& expr,
const form_option_type& fopt);
// backward compat: named forms
form_basic (const space_type& X, const space_type& Y,
const std::string& name = "",
const 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,
const quadrature_option_type& qopt = quadrature_option_type(quadrature_option_type::max_family,0));
template<class Function>
form_basic (const space_type& X, const space_type& Y,
const std::string& name,
Function weight,
const 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,
const 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,
const field_basic<T,M>& weight,
const quadrature_option_type& qopt = quadrature_option_type(quadrature_option_type::max_family,0));
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_type& qopt = quadrature_option_type(quadrature_option_type::max_family,0));
protected:
// backward compat: named forms (cont.)
template<class WeightFunction>
void form_init (
const std::string& name,
bool has_weight,
WeightFunction weight,
const quadrature_option_type& 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_type& 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;
field(2), csr(2), space(2), csr(2), field(2)