In this paper, two efficient two-grid algorithms with $L1$ scheme are presented for solving two-dimensional nonlinear time fractional diffusion equations. The classical $L1$ scheme is considered in the time direction, and the two-grid FE method is used to approximate spatial direction. To linearize the discrete equations, the Newton iteration approach and correction technique are applied. The two-grid algorithms reduce the solution of the nonlinear fractional problem on a fine grid to one linear equation on the same fine grid and an original nonlinear problem on a much coarser grid. As a result, our algorithms save total computational cost. Theoretical analysis shows that the two-grid algorithms maintain asymptotically optimal accuracy. Moreover, the numerical experiment presented further confirms the theoretical results.

本文提出了两种有效的两网格算法 $\mathrm{大号}\mathrm{1个}$提出了求解二维非线性时间分数阶扩散方程的方法。古典$\mathrm{大号}\mathrm{1个}$在时间方向上考虑该方案，并且使用两网格有限元方法来近似空间方向。为了线性化离散方程，应用了牛顿迭代法和校正技术。两网格算法将细网格上的非线性分数问题的解决方案简化为同一细网格上的一个线性方程组，而将粗网格上的原始非线性问题简化为一个线性方程组。结果，我们的算法节省了总的计算成本。理论分析表明，两网格算法保持了渐近最优精度。此外，提出的数值实验进一步证实了理论结果。