.\"
.de Id
..
.de Sp
.if n .sp
.if t .sp 0.4
..
.TH mixed_solver 4rheolef "rheolef-6.7" "rheolef-6.7" "rheolef-6.7"
.\" label: /*Class:mixed_solver
.SH NAME
\fBpcg_abtb\fP, \fBpcg_abtbc\fP, \fBpminres_abtb\fP, \fBpminres_abtbc\fP -- solvers for mixed linear problems
.\" skip: @findex pcg
.\" skip: @findex pminres
.\" skip: @findex pcg\_abtb
.\" skip: @findex pcg\_abtbc
.\" skip: @findex pminres\_abtb
.\" skip: @findex pminres\_abtbc
.\" skip: @cindex mixed linear problem
.\" skip: @cindex conjugate gradien algorithm
.\" skip: @cindex finite element method
.\" skip: @cindex stabilized mixed finite element method
.\" skip: @cindex Stokes problem
.\" skip: @cindex incompresible elasticity
.SH SYNOPSIS

.\" begin_example
.Sp
.nf
    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");
.PP
    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");
.Sp
.fi
.\" end_example
The synopsis is the same with the pminres algorithm.
.PP
.SH EXAMPLES

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

Preconditioned conjugate gradient algorithm on the pressure p applied to
the stabilized stokes problem:
.\" begin_example
.Sp
.nf
       [ A  B^T ] [ u ]    [ Mf ]
       [        ] [   ]  = [    ]
       [ B  -C  ] [ p ]    [ Mg ]
.Sp
.fi
.\" end_example
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:
.\" begin_example
.Sp
.nf
     inv(S1)*(B*inv(A)*B^T)*p = inv(S1)*(B*inv(A)*Mf - Mg)
.Sp
.fi
.\" end_example
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.
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.\" END