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

TL;DR

This paper introduces a two-grid mixed finite element method combined with second-order time schemes to efficiently solve nonlinear fourth-order reaction diffusion equations, providing stability, error estimates, and improved computational efficiency.

Contribution

The paper develops a novel TGMFE method with second-order time schemes for nonlinear fourth-order equations, including stability analysis and convergence verification.

Findings

Convergence rate of second-order schemes is close to 2.

TGMFE method reduces CPU time compared to nonlinear Galerkin MFE.

Theoretical error estimates are validated through numerical results.

Abstract

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.