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A New Two-Grid Method for Expanded Mixed Finite Element Solution of Nonlinear Reaction Diffusion Equations

Part of:Partial differential equations, initial value and time-dependent initial-boundary value problemsPartial differential equations, boundary value problems

Published online by Cambridge University Press:  17 January 2017

Shang Liu and
Yanping Chen
Shang Liu*
Affiliation:
School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, Hunan, ChinaSchool of Mathematics and Computational Science, Changsha University of Science and Technology, Changsha 410114, Hunan, China
Yanping Chen*
Affiliation:
School of Mathematical Science, South China Normal University, Guangzhou 520631, Guangdong, China
*
*Corresponding author. Email:liushangw@163.com (S. Liu), yanpingchen@scnu.edu.cn (Y. P. Chen)
*Corresponding author. Email:liushangw@163.com (S. Liu), yanpingchen@scnu.edu.cn (Y. P. Chen)

Abstract

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In the paper, we present an efficient two-grid method for the approximation of two-dimensional nonlinear reaction-diffusion equations using a expanded mixed finite-element method. We transfer the nonlinear reaction diffusion equation into first order nonlinear equations. The solution of the nonlinear system on the fine space is reduced to the solutions of two small (one linear and one non-linear) systems on the coarse space and a linear system on the fine space. Moreover, we obtain the error estimation for the two-grid algorithm. It is showed that coarse space can be extremely coarse and achieve asymptotically optimal approximation as long as the mesh sizes satisfy . An numerical example is also given to illustrate the effectiveness of the algorithm.

Keywords

Error estimationmixed finite elementsreaction-diffusion equationstwo-grid methods

MSC classification

Secondary:65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods65N15: Error bounds65M12: Stability and convergence of numerical methods

Information

Type
Research Article
Information
Advances in Applied Mathematics and Mechanics , Volume 9 , Issue 3 , June 2017, pp. 757 - 774
Copyright
Copyright © Global-Science Press 2017 

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