NAME

       pcg_abtb, pcg_abtbc, pminres_abtb, pminres_abtbc -- solvers for mixed linear problems

SYNOPSIS

           template <class Matrix, class Vector, class Solver, class Preconditioner, class Size, class Real>
           int pcg_abtb (const Matrix& A, const Matrix& B, Vector& u, Vector& p,
             const Vector& Mf, const Vector& Mg, const Preconditioner& S1,
             const Solver& inner_solver_A, Size& max_iter, Real& tol,
             odiststream *p_derr = 0, std::string label = "pcg_abtb");

           template <class Matrix, class Vector, class Solver, class Preconditioner, class Size, class Real>
           int pcg_abtbc (const Matrix& A, const Matrix& B, const Matrix& C, Vector& u, Vector& p,
             const Vector& Mf, const Vector& Mg, const Preconditioner& S1,
             const Solver& inner_solver_A, Size& max_iter, Real& tol,
             odiststream *p_derr = 0, std::string label = "pcg_abtbc");

       The synopsis is the same with the pminres algorithm.

EXAMPLES

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

DESCRIPTION

       Preconditioned conjugate gradient algorithm on the pressure p applied to the stabilized stokes problem:

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

       where  A is symmetric positive definite and C is symmetric positive and semi-definite.  Such mixed linear
       problems appears for instance with the discretization of Stokes problems with stabilized  P1-P1  element,
       or  with nearly incompressible elasticity.  Formaly u = inv(A)*(Mf - B^T*p) and the reduced system writes
       for all non-singular matrix S1:

            inv(S1)*(B*inv(A)*B^T)*p = inv(S1)*(B*inv(A)*Mf - Mg)

       Uzawa or conjugate gradient algorithms  are  considered  on  the  reduced  problem.   Here,  S1  is  some
       preconditioner  for  the  Schur complement S=B*inv(A)*B^T.  Both direct or iterative solvers for S1*q = t
       are supported.  Application of inv(A) is performed via a call to a solver for systems such as  A*v  =  b.
       This  last  system  may be solved either by direct or iterative algorithms, thus, a general matrix solver
       class is submitted to the algorithm.  For most applications, such as the Stokes problem, the mass  matrix
       for  the  p  variable  is  a  good  S1  preconditioner for the Schur complement.  The stoping criteria is
       expressed using the S1 matrix, i.e. in L2 norm when this choice is considered.  It is scaled  by  the  L2
       norm of the right-hand side of the reduced system, also in S1 norm.

rheolef-6.7                                        rheolef-6.7                            mixed_solver(4rheolef)