An extension of the positive-definite and skew-Hermitian splitting method for preconditioning of generalized saddle point problems

Abstract

We extend the positive-definite and skew-Hermitian splitting (PSS) iterative method to EPSS (for extended PSS) for solving non-Hermitian positive-definite systems of linear equations. In this kind of extension, the shift matrix is replaced by a Hermitian positive-definite matrix Σ. It is proved that the method is unconditionally convergent. Then, the EPSS method with a 2 × 2-block diagonal matrix Σ as the shift matrix is applied to generalized saddle point problems and the convergence of the method is verified. Naturally, the induced preconditioner can be applied to generalized saddle point problems. In some special cases, this preconditioner coincides with some existing preconditioners. A special case of the EPSS preconditioner, say SEPSS, is used to accelerate the convergence of GMRES. Effectiveness of this preconditioner is investigated by some numerical experiments.