A note on the positive stable block triangular preconditioner for generalized saddle point problems
Introduction
We consider the following generalized saddle point linear systemwhere is symmetric and positive definite, has full rank, is symmetric and positive semi-definite, and . Problem (1.1) arises in a variety of problems, such as constrained quadratic programming, constrained least squares problems, mixed finite element approximations of elliptic PDEs, computational fluid dynamics, and so on. We refer the reader to [9] for a general discussion.
When the matrix blocks and are large and sparse, iterative methods become more attractive than direct methods for solving the saddle point problems (1.1). Many iterative methods are proposed to solve the saddle point problems (1.1), such as generalized successive overrelaxation (GSOR) method [7], parameterized inexact Uzawa methods [8], [10], local Hermitian and skew-Hermitian splitting method [13] and so on. A very good survey on useful iterative methods was presented in [9]. In particular, Krylov subspace methods might be used. It is often advantageous to use a preconditioner with such iterative methods. The preconditioner should reduce the number of iterations required for convergence but not significantly increase the amount of computation required at each iteration. Preconditioning for system (1.1) has been studied in many papers, such as block triangular preconditioners [1], [2], [12], [14], [19], [21], constraint preconditioners [3], [15], HSS preconditioners [4], [5], matrix splitting preconditioners [6], [11], [16], [17], [20] and so on.
Recently, Cao [12] studied the application of the block triangular preconditionerwhere and are symmetric and positive definite. Cao has showed that the preconditioned matrix is indefinite with all eigenvalues being real and the estimate for the interval containing these real eigenvalues has been studied. However, the interval containing these real eigenvalues is somewhat rough. It is necessary and very important to estimate that eigenvalues of preconditioned matrix are far from the origin for the numerical stability and convergence. This is main motivation of our work in this note. In fact, once given a preconditioner, we should ensure that the eigenvalues are away from the origin [19]. In this paper, we give a better estimate on the bounds for the eigenvalues of the preconditioned matrix under the following conditionwhich is a natural assumption. In this case, the intervals containing these real eigenvalues given in this paper remove the origin. It is benefit for further study, such as the choices of the preconditioning matrices and , the parameterized case [14] and so on.
The reminder of the paper is organized as follows. In Section 2, the positive stable block triangular preconditioner is recalled and a sharper bound for the eigenvalues of the preconditioned matrix than that presented in [12] is analyzed. In Section 3, numerical experiment of a model Stokes problem is presented to show the estimate and the effectiveness of the positive stable block triangular preconditioner .
Throughout this paper, means that is symmetric and positive definite. Let denote the spectrum of A. I represents the (appropriately dimensioned) identity matrix.
Section snippets
Eigenvalue analysis
We consider the eigenvalue problemConsider additionally the block diagonal matrixThen we can rewrite the eigenvalue problem (2.1) aswhere and .
It has been studied in [12] that all the eigenvalues of the preconditioned matrix are real. We now study the bounds for the eigenvalues of or, equivalently, the bounds for the eigenvalues of .
Assume first
Numerical experiment
In this section, we will test the new bound for eigenvalues of the preconditioned matrix and the effectiveness of the positive stable block triangular preconditioner . The generalized saddle point matrix is arisen from the classical incompressible steady Stokes problemwith suitable boundary conditions on . In (3.1), and p stand for viscosity, Laplace operator, velocity and pressure of the fluid, respectively.
The test problem is a “leaky” two-dimensional
Acknowledgments
The authors would like to express their great thankfulness to the referees for the constructive suggestions, which are valuable in improving the quality of our manuscript. This work is supported by the National Natural Science Pre-Research Foundation of Soochow University (SDY2011B01), the College Postgraduate Research and Innovation Project of Jiangsu province (No. CX10B_029Z) and the Nominated Excellent Thesis for PHD Candidates Program of Soochow University (No. 23320957).
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