The nonlocality of the fractional operator causes numerical difficulties for long time computation of the time-fractional evolution equations. This paper develops a high-order fast time-stepping discontinuous Galerkin finite element method for the time-fractional diffusion equations, which saves storage and computational time. The optimal error estimate $O(N^{-p-1} + h^{m+1} + \varepsilon N^{r\alpha})$ of the current time-stepping discontinuous Galerkin method is rigorous proved, where $N$ denotes the number of time intervals, $p$ is the degree of polynomial approximation on each time subinterval, $h$ is the maximum space step, $r\ge1$, $m$ is the order of finite element space, and $\varepsilon>0$ can be arbitrarily small. Numerical simulations verify the theoretical analysis.

翻译：分数阶算子的非局部性给时间分数阶演化方程的长时间计算带来了数值困难。本文针对时间分数阶扩散方程，发展了一种高阶快速时间步长间断伽辽金有限元方法，该方法能够节省存储空间和计算时间。严格证明了当前时间步长间断伽辽金方法的最优误差估计$O(N^{-p-1} + h^{m+1} + \varepsilon N^{r\alpha})$，其中$N$表示时间区间数，$p$为每个时间子区间上多项式逼近的阶数，$h$为最大空间步长，$r\ge1$，$m$为有限元空间的阶数，$\varepsilon>0$可任意小。数值模拟验证了理论分析结果。