TGMFE Algorithm Combined with Some Time Second-Order Schemes for Nonlinear Fourth-Order eaction Diffusion System

In this article, a two-grid mixed finite element (TGMFE) method with some second-order time discrete schemes is developed for numerically solving nonlinear fourth-order reaction diffusion equation. The two-grid MFE method is used to speed up the solution in space, and some second-order θ schemes are adopted to discretize the time derivative. The TGMFE method is formed by two main steps: a nonlinear MFE system based on the space coarse grid is firstly solved by the iterative algorithm, then a linearized MFE system on the fine grid is solved. Here, the stability and a priori error estimates in L2-norm for both nonlinear Galerkin MFE system and TGMFE scheme are proved. Finally, some convergence results are presented for both nonlinear Galerkin MFE system and TGMFE scheme to verify our theoretical analysis.