On semi-convergence of generalized skew-Hermitian triangular splitting iteration methods for singular saddle-point problems

Abstract

Recently, Krukier et al. (2014) [13] proposed an efficient generalized skew-Hermitian triangular splitting (GSTS) iteration method for nonsingular saddle-point linear systems with strong skew-Hermitian parts. In this work, we further use the GSTS method to solve singular saddle-point problems. The semi-convergence properties of GSTS method are analyzed by using singular value decomposition and Moore-Penrose inverse, under suitable restrictions on the involved iteration parameters. Numerical results are presented to demonstrate the feasibility and efficiency of the GSTS iteration methods, both used as solvers and preconditioners for GMRES method.

On semi-convergence of the Uzawa–HSS method for singular saddle-point problems

“…We call (1.1) the singular saddle-point problem. A number of effective methods have been proposed in the literature to solve the singular saddle-point problems, such as the Uzawa-type methods [8][9][10][11], Krylov subspace methods [12,13] and matrix splitting iteration methods [14][15][16][17][18] Based on the above splitting, Bai et al [19] proposed an HSS iteration method for solving non-Hermitian positive definite system of linear equations. The iteration scheme of the HSS method used for solving Au ¼ q can be written as ðaI þ HÞu ðkþ1=2Þ ¼ ðaI À SÞu ðkÞ þ q; ðaI þ SÞu ðkþ1Þ ¼ ðaI À HÞu ðkþ1=2Þ þ q; where a is a positive iteration parameter and TðaÞ ¼ ðaI þ SÞ À1 ðaI À HÞðaI þ HÞ À1 ðaI À SÞ ¼ ðaI þ SÞ À1 ðaI þ HÞ À1 ðaI À HÞðaI À SÞ;…”