The generalized double shift-splitting preconditioner for nonsymmetric generalized saddle point problems from the steady
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The generalized double shift-splitting preconditioner for nonsymmetric generalized saddle point problems from the steady Navier–Stokes equations Hong-Tao Fan1,2 · Xin-Yun Zhu3 · Bing Zheng1
Received: 14 June 2016 / Revised: 16 August 2017 / Accepted: 1 September 2017 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2017
Abstract In this paper, a generalized double shift-splitting (GDSS) preconditioner induced by a new matrix splitting method is proposed and implemented for nonsymmetric generalized saddle point problems having a nonsymmetric positive definite (1,1)-block and a positive definite (2,2)-block. Detailed theoretical analysis of the iteration matrix is provided to show the GDSS method, which corresponds to the GDSS preconditioner, is unconditionally convergent. Additionally, a deteriorated GDSS (DGDSS) method is proposed. It is shown that, with suitable choice of parameter matrix, the DGDSS preconditioned matrix has an eigenvalue at 1 with multiplicity n, and the other m eigenvalues are of the form 1−λ with |λ| < 1, independently of the Schur complement matrix related. Finally, numerical experiments arising from a model Navier–Stokes problem are provided to validate and illustrate the effectiveness of the proposed preconditioner, with which a faster convergence for Krylov subspace iteration methods can be achieved. Keywords Nonsymmetric generalized saddle point problem · Generalized double shift-splitting · Krylov subspace method · Convergence Mathematics Subject Classification 65F10 · 65F50
Communicated by Paul Cizmas.
B
Hong-Tao Fan [email protected]
1
School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China
2
Institute of Applied Mathematics, College of Science, Northwest A&F University, Yangling, Shaanxi 712100, China
3
Department of Mathematics, University of Texas of the Permian Basin, Odessa, TX 79762, USA
123
H.-T. Fan et al.
1 Introduction Consider the following two-by-two block linear systems of the form
A BT Au = −B C
x f = ≡ b, y −g
(1.1)
where A ∈ Rn×n is positive definite, B ∈ Rm×n is of full rank and C ∈ Rm×m is positive definite with m ≤ n; B T denotes the transpose of the matrix B; and, x, f ∈ Rn and y, g ∈ Rm . Here, it should be mentioned that the matrix E is positive definite or positive semi-definite means its symmetric part H = 21 (E + E T ) is positive definite or positive semi-definite, respectively. These assumptions guarantee the existence and uniqueness of the solution the system of linear equation (1.1). The two-by-two block linear system (1.1) is known as a nonsymmetric generalized saddle point problem when C = 0 and a nonsymmetric saddle point problem when C = 0. Such problems are important in many scientific and engineering applications, arising in fields such as computational fluid dynamics (Elman et al. 2005), mixed finite element approximations of elliptic partial differential equations (Brezzi and Fortin 1991), restrictively preconditioned conjugate gradient methods (Bai and Li 2003), and
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