On HSS-based sequential two-stage method for non-Hermitian saddle point problems☆
Introduction
A solution of large sparse non-Hermitian saddle point problems with the following form was considered:Here, is a non-Hermitian matrix and its Hermitian part is positive definite, is a matrix of full rank, , and . These assumptions guarantee the existence and uniqueness of the solution of linear systems (1.1); see [1], [2], [13], [12], [27].
Linear systems of the form (1.1) arises in a variety of scientific computing and engineering applications, including computational fluid dynamics [13], [19], constrained and weighted least squares optimization [26], [28], image reconstruction and registration [22], [23], [26], mixed finite element approximations of elliptic PDEs and Navier–Stokes problems [20], [17], [18] and so on; see [3], [5], [12], [13], [15] and reference therein.
In recent years, there has been a surge of interest in linear systems of the form (1.1), and a large number of iterative methods have been proposed because of their preservation of sparsity and lower requirement for storage. For example, Uzawa-type methods [16], [14], [31], preconditioned Krylov subspace iterative methods [13], [18], Hermitian and skew-Hermitian splitting (HSS) method and their accelerated variants [6], [8], [9], [1], [7], [24], [32], and restrictively preconditioned conjugate gradient methods [10], [29]. We refer to some comprehensive surveys [5], [13], [12], [4], [21] and the references therein for algebraic properties and solving methods for saddle point problems.
Recently, Li et al. proposed an efficient splitting iterative method for solving preconditioned saddle point problems with the symmetric positive definite (1, 1)-block [25]. Both theoretical results and numerical experiments have shown that this method is efficient and robust. In this paper, we focus on the numerical solution to the non-Hermitian saddle point problems and propose a new sequential two-stage method based on the HSS. Convergence properties are studied and numerical results are given to confirm the theoretical result.
The remainder of this paper is organized as follows: in Section 2, the new splitting iterative method is described and some of its convergence properties are studied. In Section 3, numerical experiments are provided to show the feasibility and effectiveness of the new method. Finally, in Section 4 we end this paper with some conclusions.
Section snippets
New iterative method
In this section, a new sequential two-stage method is proposed for solving non-Hermitian saddle point linear systems. To begin with, we introduce the preconditioning matrix presented in [30], [25].
Letwhere is a positive constant.
By preconditioning the non-Hermitian saddle point problem (1.1) from the left with , the following preconditioned linear system can be got:Here,
Numerical results
In this section, we will use two examples: one is a complex linear system modified from a real linear system and the other arises from the finite element discretization of the incompressible steady state Navier–Stokes problem, to assess the feasibility and effectiveness of our method in terms of both the number of iterations in stage I and the elapsing CPU time in seconds in two stages. All the tests are performed in MATLAB R2013a with machine precision .
The proposed method is compared
Conclusions
In this paper, a new sequential two-stage method is proposed for solving the non-Hermitian saddle point problems based on the Hermitian and skew-Hermitian splitting.
Theoretical analysis shows the proposed iterative method is convergent, and that the spectral radius of iterative matrix monotonically decreases and tends to zero as the iterative parameter approaches infinity. Numerical results further illustrate that the proposed iterative method is feasible and effective. Compared with the
Acknowledgment
The authors would like to thank the anonymous referee for his/her careful reading of the manuscript and useful comments and improvements.
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