Elsevier

Applied Mathematics and Computation

Volume 242, 1 September 2014, Pages 907-916
Applied Mathematics and Computation

On HSS-based sequential two-stage method for non-Hermitian saddle point problems

https://doi.org/10.1016/j.amc.2014.06.083Get rights and content

Abstract

For large sparse saddle point problems with symmetric positive definite (1, 1)-block, Li et al. studied an efficient iterative method (see Li et al. (2011)) [25]. By making use of the same preconditioning technique and a new matrix splitting based on the Hermitian and skew-Hermitian splitting (HSS) of the (1, 1)-block of the preconditioned non-Hermitian saddle point systems, an efficient sequential two-stage method is proposed for solving the non-Hermitian saddle point problems. Theoretical analysis shows the proposed iterative method is convergent, and that the spectral radius of iterative matrix monotonically decreases and tends to 0 as the iterative parameter α approaches infinity. Numerical experiments arising from Naiver–Stokes problem are provided to show that the new iterative method is feasible, effective and robust.

Introduction

A solution of large sparse non-Hermitian saddle point problems with the following form was considered:ABB0xy=fg,orAu=b.Here, ACn×n is a non-Hermitian matrix and its Hermitian part H=12(A+A) is positive definite, BCm×n is a matrix of full rank, x,fCn,y,gCm, and mn. 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].

LetP(α)=In-B(BB)-10ImIn0B-αBB=In-B(BB)-1BαBB-αBB,where α is a positive constant.

By preconditioning the non-Hermitian saddle point problem (1.1) from the left with P(α), the following preconditioned linear system can be got:A10A3A2xy=b1b2.Here, A1=(I-B(BB)-1B)A+αBB,A2=BB,A3=BA-α(BB)B,b1=f-B

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 10-16.

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 α 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.

References (32)

  • Z.-Z. Bai

    Structured preconditioners for nonsingular matrices of block two-by-two structures

    Math. Comput.

    (2006)
  • Z.-Z. Bai

    Block alternating splitting implicit iteration methods for saddle-point problems from time-harmonic eddy current models

    Numer. Linear Algebra Appl.

    (2012)
  • Z.-Z. Bai

    Optimal parameters in the HSS-like methods for saddle-point problems

    Numer. Linear Algebra Appl.

    (2009)
  • Z.-Z. Bai

    Several splittings for non-Hermitian linear systems

    Sci. Chin. Ser. A: Math.

    (2008)
  • Z.-Z. Bai et al.

    Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems

    SIAM J. Matrix Anal. Appl.

    (2003)
  • Z.-Z. Bai et al.

    Accelerated Hermitian and skew-Hermitian splitting iteration methods for saddle-point problems

    IMA J. Numer. Anal.

    (2007)
  • Cited by (0)

    This work was supported by the National Natural Science Foundation of China (11271174) and the Research Foundation of the Higher Education Institutions of Gansu Province, China (2014A-111).

    View full text