Analysis Of Some Two-grid Methods Of Expanded Mixed Finite Element Method

The reaction-diffusion equation have received a great deal of attention in models of groundwater through porous media. As we know, the resulting algebraic equations are large systems of non-linear equations with the expanded mixed finite element method for the equation. So, it is necessary to study highly efficient and highly accurate algorithms for non-linear systems. In the paper, we present some 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 . At first, we obtain some superconvergence estimates and error estimate of mixed finite element by using the projection properties. Then we offer a few algorithms of two-grid method for expanded mixed finite element solution of semi-linear and nonlinear reaction diffusion equations, and we make our efforts to prove the convergence of the algorithms. We know the algorithms achieve asymptotically optimal approximation applying the two-grid methods as long as the mesh sizes satisfy H = O(h^1/4 (or H = O(h^1/6)) for semi-linear problem and h = O(H^(3k+1)/(k+1) for non-linear reaction diffusion problem.In the end of the paper, we draw a conclusion of this work and make some comments on the prospect of the convergence analysis.