**Memorandum 1786**

*A class of nonsymmetric preconditioners for saddle point problems*

M.A. Botchev & G.H. Golub

**Abstract:**

For the iterative solution of saddle point problems,
a nonsymmetric preconditioner
is studied which, with respect to the upper-left block of the system matrix,
can be seen as a variant of SSOR. An idealized situation where
SSOR is taken with respect to the skew-symmetric part plus
the diagonal part of the upper-left block
is analyzed in detail. Since action of the preconditioner
involves solution of a Schur complement system, an inexact form
of the preconditioner can be of interest. This results in an
inner-outer iterative process.
Numerical experiments with solution of linearized Navier-Stokes
equations demonstrate the efficiency of the new preconditioner,
especially when the left-upper block is far from symmetric.

**Keywords:**
Saddle point problem, iterative method, preconditioning method,
nonsymmetric indefinite linear systems, SSOR, constraint preconditioner,
skew-symmetric preconditioner,
inner-outer iterations, Navier-Stokes equations

**Mathematics Subject Classification:**
65F10, 65F22, 65F35, 65N22, 65K10

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