New modified shift-splitting preconditioners for non-symmetric saddle point problems
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Arabian Journal of Mathematics
Mahin Ardeshiry · Hossein Sadeghi Goughery Hossein Noormohammadi Pour
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New modified shift-splitting preconditioners for non-symmetric saddle point problems
Received: 30 November 2018 / Accepted: 7 May 2019 © The Author(s) 2019
Abstract Zhou et al. and Huang et al. have proposed the modified shift-splitting (MSS) preconditioner and the generalized modified shift-splitting (GMSS) for non-symmetric saddle point problems, respectively. They have used symmetric positive definite and skew-symmetric splitting of the (1, 1)-block in a saddle point problem. In this paper, we use positive definite and skew-symmetric splitting instead and present new modified shiftsplitting (NMSS) method for solving large sparse linear systems in saddle point form with a dominant positive definite part in (1, 1)-block. We investigate the convergence and semi-convergence properties of this method for nonsingular and singular saddle point problems. We also use the NMSS method as a preconditioner for GMRES method. The numerical results show that if the (1, 1)-block has a positive definite dominant part, the NMSS-preconditioned GMRES method can cause better performance results compared to other preconditioned GMRES methods such as GMSS, MSS, Uzawa-HSS and PU-STS. Meanwhile, the NMSS preconditioner is made for non-symmetric saddle point problems with symmetric and non-symmetric (1, 1)-blocks. Mathematics Subject Classification
65F10 · 65F08
1 Introduction Consider the following non-symmetric saddle point linear system A B f x = b, = AU = −g y −B T 0
(1)
where A ∈ Rn×n is positive definite (symmetric or non-symmetric); B ∈ Rn×m (m ≤ n) is a rectangular matrix of rank r ≤ m; f ∈ Rn and g ∈ Rm are the given vectors. In general, matrices A and B in A are large and sparse. System (1) is important and arises in a variety of scientific and engineering applications, such as computational fluid dynamics, constrained optimization, mixed or hybrid finite elements approximations of second-order elliptic problems, see [1,7,15]. M. Ardeshiry · H. S. Goughery (B) Department of Mathematics, Kerman Branch, Islamic Azad University, Kerman, Iran E-mail: [email protected]; [email protected] M. Ardeshiry E-mail: [email protected] H. N. Pour Department of Mathematics, Anar Branch, Islamic Azad University, Anar, Kerman, Iran E-mail: [email protected]
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In recent years, many studies have focused on solving large linear systems in saddle point form. Iterative methods are used for solving saddle point problems (1), when matrix blocks A and B are large and sparse. Some of these methods, such as Uzawa [7], inexact Uzawa [16] and the Hermitian and skew-Hermitian splitting method [2,18,21] have been presented. In reality, these methods use much less memory compared to Krylov subspace methods, but Krylov subspace methods are very efficient. Unfortunately, for solving saddle point problems (1), Krylov subspace methods work very slowly and they require good preconditioners to increase the
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