  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 approximate spatial direction, and some second-order $\theta$ schemes
formulated at time $t_{k-\theta}$ are considered to discretize the time
direction. TGMFE method covers two main steps: a nonlinear MFE system based on
the space coarse grid is solved by the iterative algorithm and a coarse
solution is arrived at, then a linearized MFE system with fine grid is
considered and a TGMFE solution is obtained. Here, the stability and a priori
error estimates in $L^2$-norm for both nonlinear Galerkin MFE system and TGMFE
scheme are derived. Finally, some convergence results are computed for both
nonlinear Galerkin MFE system and TGMFE scheme to verify our theoretical
analysis, which show that the convergence rate of the time second-order
$\theta$ scheme including Crank-Nicolson scheme and second-order backward
difference scheme is close to $2$, and that with the comparison to the
computing time of nonlinear Galerkin MFE method, the CPU-time by using TGMFE
method can be saved.
