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）逼近米塔格-莱夫勒函数的快速算法，并将该算法应用于粘弹性波传播模型，其中空间离散采用混合有限元方法，时间二阶导数采用Newmark-beta格式进行离散。与传统分数阶导数的L1格式相比，我们的快速算法将内存复杂度从$\mathcal O(N_sN)$降低至$\mathcal O(N_sN_{exp})$，计算复杂度从$\mathcal O(N_sN^2)$降低至$\mathcal O(N_sN_{exp}N)$，其中$N$表示时间网格点总数，$N_{exp}$为SOE中指数项的数量，$N_s$代表与空间离散相关的内存与计算复杂度。数值实验验证了理论结果。