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

TL;DR

This paper analyzes the semi-convergence of regularized HSS iteration methods for singular saddle point problems from Stokes equations, showing their unconditional semi-convergence and effectiveness through spectral analysis and numerical experiments.

Contribution

It extends the application of RHSS methods to singular saddle point problems from Stokes equations and proves their unconditional semi-convergence, improving previous results.

Findings

RHSS and HSS methods are unconditionally semi-convergent for these problems

Spectral properties of preconditioned matrices are characterized

Numerical experiments confirm method effectiveness

Abstract

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