| The reaction-diffusion equation has received a great deal of attention in models of groundwater through porous media. Introducing the other variable,the resulting algebraic equations are large system of non-linear equations with the mixed finite element method for the equation.So,it is meaningful to study highly efficient and highly accurate algorithms for non-linear systems. In the paper, we present two two-grid methods for solving two-dimensional reaction-diffusion using expanded mixed finite element method. The key feature of the two-grid method is that it allows one to execute all the nonlinear iteration on a system associated with a coarse grid, then we solve linear systems based on Newton iteration on the fine grid or on the coarse grid, but without sacrificing the order of accuracy of the fine-grid solution. We offer a algorithm of two-grid method for expanded mixed finite element solution of nonlinear reaction diffusion equations using two Newton iterations, and we make our efforts to prove the convergence of the algorithm. We know the algorithm achieves asymptotically optimal approximationapplying the two-grid methods as long as the mesh size satisfy H = O(hk+1/(3k+1)) for nonlinear reaction diffusion problem. We give another algorithm that achieves asymptotically optimal approximation applying the two-grid methods as long as the mesh size satisfy H = O(hk+1/(3k+1)) for nonlinear reaction diffusion problem.This paper is divided into 5 sections . The first chapter contains the history , development and actuality of two-grid methods . In section 2 , we introduce some concepts,some convergence properties and results which we'll use in the sequel . In section 3 , we introduce a two-grid method for expanded mixed finite element of nonlinear reaction-diffusion equations with the convergence. In section 4 , we propose two algorithms and give their error estimates . Finally , we present an outline of the article and some prospects in the field .
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