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.  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.

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).  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.

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(),
              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;

SEE ALSO

       mixed_solver(4)