GSTS-Uzawa method for a class of complex singular saddle point problems
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GSTS-Uzawa method for a class of complex singular saddle point problems Jin-Song Xiong1 · Xing-Bao Gao1
Received: 23 April 2017 / Revised: 25 October 2017 / Accepted: 28 October 2017 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2017
Abstract In this paper, we propose GSTS–Uzawa method for solving a class of complex singular saddle point problems based on generalized skew-Hermitian triangular splitting (GSTS) iteration method and classical Uzawa method. We research on its semi-convergence properties and the eigenvalues distributions of its preconditioned matrix. The resulting GSTS– Uzawa preconditioner is used to precondition Krylov subspace methods such as the restarted generalized minimal residual (GMRES) method for solving the equivalent formulation of the complex singular saddle point problems. The theoretical results and effectiveness of the GSTS–Uzawa method are supported by a numerical example. Keywords Complex singular saddle point problems · GSTS-Uzawa method · Iterative method · Semi-convergence Mathematics Subject Classification 65F10 · 65F08
1 Introduction Consider the complex singular saddle point problems x f A E = , AX = y −g −E ∗ 0
(1)
where A ∈ Cm×m is a large sparse nonsingular and non-Hermitian matrix, E ∈ Cm×n is rank-deficient with m ≤ n, E ∗ is the conjugate transpose of E, f ∈ Cm , g ∈ Cn .
Communicated by José Mario Martínez.
B
Jin-Song Xiong [email protected] Xing-Bao Gao [email protected]
1
College of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710119, China
123
J.-S. Xiong, X.-B. Gao
Such saddle point problems arise in a variety of scientific computing and engineering applications, including fast Fourier transform-based solution of certain time-dependent partial differential equations (Bertaccini 2004), computational fluid dynamics (Benzi et al. 2005; Elman et al. 2002, 2007; Elman and Golub 1994), mixed finite element approximation of elliptic partial differential equations (Benzi et al. 2005; Perugia and Simoncini 2000), constrained optimization (Benzi et al. 2005; Betts 2001; Liesen et al. 2001), weighted leastsquares problem (Benzi et al. 2005; Yuan 1993; Yuan et al. 1996; Yuan and Iusem 1996) and so on. In recent decades, there has been a surge of interest in solving saddle point problems (1), and a large number of iterative methods have been proposed. In 1958, in order to solve the optimization problem in economics, the classical Uzawa method was proposed by Arrow et al. (1958). Its specific form is as follows: (k+1) Ax = f − E ∗ y (k) , (k+1) y = y (k) + μ(E x (k+1) − g). Owing to the method is simple in form and easy to implement by computer, many researchers have paid close attention to it and proposed a lot of Uzawa-type methods (Bai and Wang 2008; Bai et al. 2005a; Cao 2003, 2004; Bramble et al. 2000; Huang and Su 2017; Krukier et al. 2014; Shao and Li 2015; Benzi and Liu 2007; Zheng et al. 2009; Liang and Zhang 2014, 2017; Ma and Zhang 2011; Li et al. 2014). But the method requires the solution
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