On SOR-like iteration methods for solving weakly nonlinear systems

ABSTRACT

In this paper, we introduce a class of SOR-like iteration methods for solving the systems of the weakly nonlinear equation, which is by reformulating equivalently the weakly nonlinear equation as a two-by-two block nonlinear equation. Two types of the global convergence theorems are given under suitable choices of the involved splitting matrix and parameter. Numerical results for the three-dimensional nonlinear convection-diffusion equation and the linear complementarity problem show that the proposed iteration methods are feasible and efficient for solving the weakly nonlinear equations.

KEYWORDS:

• Weakly nonlinear equation
• matrix splitting
• SOR
• convergence theory
• nonlinear convection-diffusion equation
• linear complementarity problem

AMS SUBJECT CLASSIFICATIONS:

• 65H10
• 65F10

Acknowledgments

The authors would like to express their great thankfulness to the referees for the comments and constructive suggestions very much, which are valuable in improving the quality of the original paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research was supported by the National Natural Science Foundation of China (Nos. 11901098 and U1839207), the Strategic Priority Research Program of Chinese Academy of Sciences (No. XDC01040100), and the National Key Research and Development Program of China (Nos. 2019YFC0312003 and 2018YFC1504200).

Notes on contributors

Yifen Ke

Yifen Ke received a PhD degree from Fujian Normal University in 2017 and spent two years as a Postdoctoral Researcher at University of Chinese Academy of Sciences from 2017 until 2019. She is currently an Associate Professor in the College of Mathematics and Informatics at Fujian Normal University. Her research focuses on the design, analysis, and implementation of numerical methods for solving large-scale linear and nonlinear problems.

Changfeng Ma

Changfeng Ma graduated from the Academy of Mathematics and Systems Science, Chinese Academy of Sciences in 2003. He received his PhD degree in Computational Mathematics from the Chinese Academy of Sciences. Presently, he is a professor at Fujian Normal University.