In this article, an efficient two-grid finite element method is proposed for solving the nonlinear time fractional variable coefficient diffusion equations. This algorithm firstly solves a nonlinear system to get the numerical solution $u_H^n$ on the coarse grid with size $H$, then based on the initial iterative solution $u_H^n$ on the coarse grid, the linearized finite element system is solved on the fine grid with size $h$ to get the numerical solution $U_h^n$, in which the temporal direction is approximated by the $L2-1_{\sigma }$ scheme. Besides, the stability and priori error estimates of standard finite element method and two-grid method are given. Finally, the validity and efficiency of the two-grid algorithm are verified by two numerical experiments.

本文提出了一种求解非线性时间分数变系数扩散方程的高效双网格有限元方法。该算法首先求解一个非线性系统，得到数值解${你}_H^n$在具有大小的粗网格上$H$，然后根据初始迭代解${你}_H^n$在粗网格上，线性化有限元系统在具有尺寸的细网格上求解$H$得到数值解${ü}_H^n$，其中时间方向近似于$\mathrm{大号}2-1_{\sigma }$方案。此外，还给出了标准有限元法和两格法的稳定性和先验误差估计。最后通过两个数值实验验证了两网格算法的有效性和效率。