L～p Error Estimate Of An Efficient Four-step Scheme Two-grid Method For Semi-linear Reaction-diffusion Equations Using An 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.Li Wu and 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 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 take some superconvergence estimates of mixed finite element method and give the L^p error estimate of a four-step scheme two-grid method for semi-linear reaction diffusion equations using an expanded mixed finite element method.This procedure involves four steps:solving a non-linear system from the full-discretized of expanded mixed finite element on the coarse grid;solving a linear system from Newton iteration on the fine grid; making one more correction on the coarse grid;solving one more linear system from Newton iteration on the fine grid.The key to prove is offering a new approximation-elliptic-mixed method pro-jection.Prom the results,we know that the L^p error estimate have the same order of accuracy to the L² error estimate and this two-grid method can be achieved asymptotically optimal approximation of the pressure as long as the mesh size satisfying H = O(h^1/6) for semi-linear problem.