A New Two-Grid Method for Expanded Mixed Finite Element Solution of Nonlinear Reaction Diffusion Equations (Advances in Applied Mathematics and Mechanics, 2018)
In the paper, we present an efficient two-grid method for the approximationof two-dimensional nonlinear reaction-diffusion equations using an expanded mixedfinite-element method. We transfer the nonlinear reaction diffusion equation into firstorder nonlinear equations. The solution of the nonlinear system on the fine space isreduced to the solutions of two small (one linear and one non-linear) systems on thecoarse space and a linear system on the fine space. Moreover, we obtain the error estimation for the two-grid algorithm. It is showed that coarse space can be extremelycoarse and achieve asymptotically optimal approximation as long as the mesh sizessatisfy $h^{k+1}=\mathcal{O}(H^{3k+1})$. A numerical example is also given to illustrate the effectiveness of the algorithm.