NAME

       cg_abtb, cg_abtbc, minres_abtb, minres_abtbc -- solvers for mixed linear problems

SYNOPSIS

           template <class Matrix, class Vector, class Solver, class Preconditioner>
           int cg_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, const solver_option& sopt = solver_option());

           template <class Matrix, class Vector, class Solver, class Preconditioner>
           int cg_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, const solver_option& sopt = solver_option());

       The synopsis is the same with the minres 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.  Formally 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 stopping 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.

COPYRIGHT

       Copyright (C) 2000-2018 Pierre Saramito <Pierre.Saramito@imag.fr> GPLv3+: GNU GPL  version
       3  or  later  <http://gnu.org/licenses/gpl.html>.   This is free software: you are free to
       change and redistribute it.  There is NO WARRANTY, to the extent permitted by law.