The generalized Uzawa-SHSS method for non-Hermitian saddle-point problems
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The generalized Uzawa-SHSS method for non-Hermitian saddle-point problems Zhengge Huang1 · Ligong Wang1 · Zhong Xu1 · Jingjing Cui1
Received: 10 May 2016 / Revised: 28 September 2016 / Accepted: 9 October 2016 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2016
Abstract Recently, Li and Wu (2015) proposed the single-step Hermitian and skewHermitian splitting (SHSS) method for solving the non-Hermitian positive definite linear systems. Based on the single-step Hermitian and skew-Hermitian splitting of the (1,1) part of the saddle-point coefficient matrix, a new Uzawa-type method is proposed for solving a class of saddle-point problems with non-Hermitian positive definite (1,1) parts. Convergence (Semi-convergence) properties of this new method for nonsingular (singular) are derived under suitable conditions. Numerical examples are implemented to confirm the theoretical results and verify that this new method is more feasibility and robustness than the new HSSlike (NHSS-like), the Uzawa-HSS and the parameterized Uzawa-skew-Hermitian triangular splitting (PU-STS) methods for solving both the nonsingular and the singular saddle-point problems with non-Hermitian positive definite and Hermitian dominant (1,1) parts. Keywords Saddle-point problem · Uzawa method · Single-step Hermitian and Skew-Hermitian splitting · Convergence · Semi-convergence
Communicated by Andreas Fischer. This work is supported by the National Natural Science Foundations of China (No. 11171273) and sponsored by Innovation Foundation for Doctor Dissertation of Northwestern Polytechnical University (No. CX201628).
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Ligong Wang [email protected] Zhengge Huang [email protected] Zhong Xu [email protected] Jingjing Cui [email protected]
1
Department of Applied Mathematics, School of Science, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, People’s Republic of China
123
Z. Huang et al.
Mathematics Subject Classification 65F10
1 Introduction Consider the following large and sparse saddle-point problem Au =
A B −B ∗ 0
x f = ≡ b, y −g
(1)
where A ∈ Cm×m is a non-Hermitian positive definite matrix, B ∈ Cm×n , f ∈ Cm and g ∈ Cn with m ≥ n. Here, B ∗ denotes the conjugate transpose of B. Linear system (1) arises in a number of scientific and engineering applications, such as computational fluid dynamics, mixed finite element approximations of elliptic partial differential equations, constrained optimizations, Navier–Stokes equations and so forth; see Benzi et al. (2005), Bai (2006), Bai et al. (2005), Brezzi and Fortin (1991), Golub and Wathen (1998), Haber and Modersitzki (2004) and the references therein. When B is of full column rank, the saddle-point problem (1) is nonsingular. A number of effective iterative methods based on matrix splitting have been proposed for solving nonsingular saddle-point problem (1), including the Uzawa-type methods (Bramble et al. 1997; Elman and Golub 1994; Bai and Wang 2008; Cao et al. 2011), the Hermitian and skew-Hermitian splitting (HSS
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