L～p Error Estimates Of Some Two-grid Methods Of Expanded Mixed Finite Element Methods

The research of numerical method for the reaction-diffusion equation in models of groundwater through porous media had a great deal of effect upon society and economic profits. Many scientists pay much attention and have done many works on it. In this paper, we consider the non-linear reaction-diffusion equation, the resulting algebraic equations are large non-linear system with the expanded mixed finite element method. It is necessary to study highly efficient and highly accurate algorithms for non-linear systems.Some two-grid methods of expanded mixed finite element methods of two dimensional non-linear reaction-diffusion equation with small coefficient were introduced by Wu and Chen. The main idea derives from the early research of Xu about two-grid methods of standard finite element methods, it allows one to solve the discrete equation of original non-linear problem on coarse grid, then linearize the discrete equation with Newton iteration method on the fine grid.In this paper, we give the L^p error estimate to the approximate discrete solution of pressure in three two-grid methods presented by Wu and Chen. The first method is a two-steps algorithm, first we solve the original problem on the coarse grid, then we use Newton iteration on the fine grid. The second method is three-steps algorithm, making a correction on the coarse grid based on the two-steps algorithm. The third method is also three-steps algorithm, the difference between the second method and the third method is making a correction on the fine grid. The results which we get are the basis of the convergence analysis about the three two-grid methods. In the analysis, we present a new approximation-elliptic-mixed method projection and use the superconvergence property of mixed finite element method. From the results, we know the algorithms achieve asymptotically optimal approximation of discrete solution about the pressure in L^p norm as long as the mesh sizes satisfying H= O(h^1/2) , H = O(h^1/3) or H =O(h^1/4).