solver_abtb(2rheolef)						    rheolef-6.1 					     solver_abtb(2rheolef)

NAME

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

SYNOPSIS

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

DESCRIPTION

       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.  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.

DIRECT ALGORITHM

       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 pres-
       sure term and the whole matrix is factored one time for all.

ITERATIVE ALGORITHM

       The preconditionned conjugate gradient algorithm is used, where the mp matrix is used as preconditionner. See see mixed_solver(4).

EXAMPLES

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

IMPLEMENTATION

       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());
	 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());

       // 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;
	 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;

SEE ALSO

       mixed_solver(4)

rheolef-6.1							    rheolef-6.1 					     solver_abtb(2rheolef)