Due to the nonlocal feature of fractional differential operators, the numerical solution to fractional partial differential equations usually requires expensive memory and computation costs. This paper develops a fast scheme for fractional viscoelastic models of wave propagation. We first apply the Laplace transform to convert the time-fractional constitutive equation into an integro-differential form that involves the Mittag-Leffler function as a convolution kernel. Then we construct an efficient sum-of-exponentials (SOE) approximation for the Mittag-Leffler function. We use mixed finite elements for the spatial discretization and the Newmark scheme for the temporal discretization of the second time-derivative of the displacement variable in the kinematical equation and finally obtain the fast algorithm. Compared with the traditional L1 scheme for time 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 confirm the theoretical results.

翻译：由于分数阶微分算子的非局部特性，分数阶偏微分方程的数值求解通常需要高昂的内存和计算成本。本文针对分数阶粘弹性波传播模型提出了一种快速数值格式。我们首先应用拉普拉斯变换将时间分数阶本构方程转化为包含米塔格-莱弗勒函数作为卷积核的积分-微分形式。随后为米塔格-莱弗勒函数构建了高效的多指数和逼近。在运动学方程中，我们采用混合有限元方法进行空间离散化，并应用纽马克格式对位移变量的二阶时间导数进行时间离散化，最终获得快速算法。与传统的时间分数阶导数L1格式相比，本快速算法将内存复杂度从$\mathcal O(N_sN)$降低至$\mathcal O(N_sN_{exp})$，将计算复杂度从$\mathcal O(N_sN^2)$降低至$\mathcal O(N_sN_{exp}N)$，其中$N$表示时间网格点总数，$N_{exp}$为多指数和中的指数项数量，$N_s$代表与空间离散化相关的内存和计算复杂度。数值实验验证了理论结果。