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 pressure 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-6.5                                        rheolef-6.5                             solver_abtb(2rheolef)