Provided by: librheolef-dev_6.6-1build2_amd64 #### 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.
```