Generalized skew-Hermitian triangular splitting iteration methods for saddle-point linear systems

Abstract

A generalized skew-Hermitian triangular splitting iteration method is presented for solving non-Hermitian linear systems with strong skew-Hermitian parts. We study the convergence of the generalized skew-Hermitian triangular splitting iteration methods for non-Hermitian positive definite linear systems, as well as spectrum distribution of the preconditioned matrix with respect to the preconditioner induced from the generalized skew-Hermitian triangular splitting. Then the generalized skew-Hermitian triangular splitting iteration method is applied to non-Hermitian positive semidefinite saddle-point linear systems, and we prove its convergence under suitable restrictions on the iteration parameters. By specially choosing the values of the iteration parameters, we obtain a few of the existing iteration methods in the literature. Numerical results show that the generalized skew-Hermitian triangular splitting iteration methods are effective for solving non-Hermitian saddle-point linear systems with strong skew-Hermitian parts.

A New GSOR Method for Generalised Saddle Point Problems

“…In recent years, many iterative methods have been introduced to solve the problem (1.1), including Uzawa-type schemes [14,20,23,25,33,34,51,53], iterative projection methods [3], block and approximate Schur complement preconditioners [17,19,22,35,40,42,43], iterative null space methods [1,26,48], splitting methods [4,7,[9][10][11][12][13]18,29,30,36,38,39,41,46,50], indefinite preconditioning [31,37], and preconditioning methods based on approximate factorisation of the coefficient matrix [6,8,28,44]. A classical approach to solve (1.1) is the successive overrelaxation (SOR) iteration method [49], which can involve relatively low computation per iterative step.…”

A block alternating splitting iteration method for a class of block two-by-two complex linear systems

“…Jiang and Cao in [11] presented the local HSS (LHSS) and modified LHSS iteration methods. In [12], Krukier et al gave the generalized skew-Hermitian triangular splitting (GSTS) iteration method and Pan et al in [13] proposed a preconditioned-GMRES method for this class of problems. For more details, we refer to [14][15][16].…”

Variants of the accelerated parameterized inexact Uzawa method for saddle-point problems

“…For example, the matrix splitting iterative methods [7,11,18,36,40,45], Uzawa-type methods [12,17,21,22,25], HSS method and its variants [2,[4][5][6]8,9,28,29,38], Krylov subspace methods [1,10,[33][34][35]42], null space methods [24] and so on. When the saddle-point problem (1) is singular, there are also many relaxation iteration methods which have been established, e.g., the Uzawa-type methods [30,46,48,49] and the HSS-like methods [3,20,39].…”