On the semi-convergence of regularized HSS iteration methods for singular saddle point problems

Recently, Bai and Benzi proposed a class of regularized Hermitian and skew-Hermitian splitting (RHSS) iteration methods for solving the nonsingular saddle point problem. In this paper, we apply this method to solve the singular saddle point problem. In the process of the semi-convergence analysis, we get that the RHSS method and the HSS method are unconditionally semi-convergent, which has improved the previous results. Then some spectral properties of the corresponding preconditioned matrices and a class of improved preconditioned matrices are analyzed. Finally, some numerical experiments on linear systems arising from the discretization of the Stokes equation are presented to illustrate the feasibility and effectiveness of this method and the corresponding preconditioners.