We propose a fast scheme for approximating the Mittag-Leffler function by an efficient sum-of-exponentials (SOE), and apply the scheme to the viscoelastic model of wave propagation with mixed finite element methods for the spatial discretization and the Newmark-beta scheme for the second-order temporal derivative. Compared with traditional L1 scheme for fractional derivative, our fast scheme reduces the memory complexity from $\mathcal O(N_sN) $ to $\mathcal O(N_sN_{exp})$ and the computation complexity from $\mathcal O(N_sN^2)$ to $\mathcal O(N_sN_{exp}N)$, where $N$ denotes the total number of temporal grid points, $N_{exp}$ is the number of exponentials in SOE, and $N_s$ represents the complexity of memory and computation related to the spatial discretization. Numerical experiments are provided to verify the theoretical results.

翻译：我们提出了一种通过高效指数和（SOE）逼近Mittag-Leffler函数的快速格式，并将该格式应用于波传播的粘弹性模型，其中空间离散采用混合有限元方法，二阶时间导数采用Newmark-beta格式。与传统的分数阶导数L1格式相比，我们的快速格式将内存复杂度从$\mathcal O(N_sN)$降低到$\mathcal O(N_sN_{exp})$，将计算复杂度从$\mathcal O(N_sN^2)$降低到$\mathcal O(N_sN_{exp}N)$，其中$N$表示时间网格点的总数，$N_{exp}$是SOE中指数项的数量，$N_s$表示与空间离散相关的内存和计算复杂度。数值实验验证了理论结果。