**solver_abtb** -- direct or iterative solver iterface for mixed linear
systems

solver_abtb stokes (a,b,mp);
solver_abtb elasticity (a,b,c,mp);

The

**solver_abtb** class provides direct or iterative algorithms for some
mixed problem:

[ A B^T ] [ u ] [ Mf ]
[ ] [ ] = [ ]
[ B -C ] [ p ] [ Mg ]

where A is symmetric positive definite and C is symmetric positive. By default,
iterative algorithms are considered for tridimensional problems and direct
methods otherwise. An optional argument can change this behavior. Such mixed
linear problems appears for instance with the discretization of Stokes
problems. The C matrix can be zero and then the corresponding argument can be
omitted when invoking the constructor. Non-zero C matrix appears for of Stokes
problems with stabilized P1-P1 element, or for nearly incompressible
elasticity problems.

When the kernel of

**B^T** is not reduced to zero, then the pressure p is
defined up to a constant and the system is singular. In the case of iterative
methods, this is not a problem. But when using direct method, the system is
then completed to impose a constraint on the pressure term and the whole
matrix is factored one time for all.

The preconditionned conjugate gradient algorithm is used, where the

**mp**
matrix is used as preconditionner. See see mixed_solver(4). The linear
sub-systems related to the

**A** matrix are also solved by an iterative
algorithm. Use a second optional argument to change this default behavior: a
factorization and a direct solver can be considered for these sub-systems.

See the user's manual for practical examples for the nearly incompressible
elasticity, the Stokes and the Navier-Stokes problems.

template <class T, class M = rheo_default_memory_model>
class solver_abtb_basic {
public:
// typedefs:
typedef typename csr<T,M>::size_type size_type;
// allocators:
solver_abtb_basic ();
solver_abtb_basic (const csr<T,M>& a, const csr<T,M>& b, const csr<T,M>& mp,
const solver_option_type& opt = solver_option_type(),
const solver_option_type& sub_opt = solver_option_type());
solver_abtb_basic (const csr<T,M>& a, const csr<T,M>& b, const csr<T,M>& c, const csr<T,M>& mp,
const solver_option_type& opt = solver_option_type(),
const solver_option_type& sub_opt = solver_option_type());
// accessors:
void solve (const vec<T,M>& f, const vec<T,M>& g, vec<T,M>& u, vec<T,M>& p) const;
protected:
// internal
void init();
// data:
mutable solver_option_type _opt;
mutable solver_option_type _sub_opt;
csr<T,M> _a;
csr<T,M> _b;
csr<T,M> _c;
csr<T,M> _mp;
solver_basic<T,M> _sA;
solver_basic<T,M> _sa;
solver_basic<T,M> _smp;
bool _need_constraint;
};
typedef solver_abtb_basic<Float,rheo_default_memory_model> solver_abtb;

mixed_solver(4)