Abstract In this paper, two-grid algorithms based on expanded mixed finite element method are presented for solving two-dimensional semilinear time fractional reaction-diffusion equations. To obtain the fully discrete scheme, the classical L1 scheme is considered in the time direction, and the expanded mixed finite element method is used to approximate spatial direction. Then the error estimates and stability of fully discrete scheme are derived. To linearize the nonlinear system, the two-grid method based on Newton iteration are constructed. 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 is presented further confirm the theoretical results.

中文翻译：

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用扩展混合有限元法求解半线性时间分数阶反应扩散方程的二格法

摘要 本文提出了基于扩展混合有限元法求解二维半线性时间分数阶反应扩散方程的二维网格算法。为了获得完全离散的方案，在时间方向考虑经典的L1方案，并使用扩展混合有限元方法来逼近空间方向。然后推导出全离散方案的误差估计和稳定性。为了对非线性系统进行线性化，构造了基于牛顿迭代的二网格法。双网格算法将细网格上的非线性分数问题的解简化为同一细网格上的一个线性方程和粗得多的网格上的原始非线性问题。因此，我们的算法节省了总计算成本。理论分析表明，双网格算法保持渐近最优精度。此外，数值实验的提出进一步证实了理论结果。