Optimization of a parameterized inexact Uzawa method for saddle point problems

ABSTRACT

For large sparse saddle point problems, Cao et al. studied a modified generalized parameterized inexact Uzawa (MGPIU) method (see [Y. Cao, M.Q. Jiang, L.Q. Yao, New choices of preconditioning matrices for generalized inexact parameterized iterative methods, J. Comput. Appl. Math. 235 (1) (2010) 263–269]). For iterative methods of this type, the choice of the relaxation parameter is crucial for the methods to achieve their best performance. In this paper, for an example of 2D Stokes equations, we derive the optimal relaxation parameter for the continuous version of the MGPIU method, by minimizing the corresponding convergence factor that is obtained using Fourier analysis. In addition, we find that the MGPIU method is mesh parameter independent, however, it depends asymptotically linearly on the viscosity ν, which suggests that the numerical methods for Stokes equations should be investigated with the presence of the viscosity ν, though it can be scaled out from the equations in advance. We use numerical experiments to validate our theoretical findings.

KEYWORDS:

• Inexact Uzawa method
• optimal relaxation parameter
• saddle point problem
• Stokes equation
• Fourier analysis

2010 AMS SUBJECT CLASSIFICATIONS:

• 65F10
• 65N22

Acknowledgments

We thank the anonymous referees for their constructive suggestions and useful comments, which have substantially improved the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The research is partially supported by National Natural Science Foundation of China [11671074,11471047].