Analysis Of Improved Two-grid Method For Semi-linear Reaction-diffusion Equations By Expanded Mixed Finite Element Method

Reaction-diffusion equations have received a great deal of application in real life, such as groundwater problem, bio-geochemical phenomena, environment contamination and the reasonable exploit of petroleum reservoir and so on. Scientists have done a great deal of research on it's numerical methods.Professor Li Wu and Professor Yanping Chen have presented a few two-grid methods for semi-linear reaction diffusion equations with small diffusion coefficient by expanded mixed finite element method. These two-grid ideas are from Professor Xu's work on standard finite element method. The basic idea is using Newton iteration to linearize the non-linear algebra systems, then making correction to improve the accuracy.In this paper, we present an improved two-grid method for two dimensional semi-linear reaction-diffusion equations by expanded mixed finite element methods, on the basis of the two-grid methods presented by Wu and Chen by virtue of an interpolation postprocessing technique. Firstly, we solve the original problem on the coarse grid. Then we postprocess the solution of the coarse grid. Finally, we use Newton iteration on the fine grid. In the analysis, we use the property of the interpolation postprocessing operator and use the superconvergence property of mixed finite element method. According to the result, we know that coarse gird can be extremely coarse to improve the efficiency without affecting the accurate.