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, $N_{exp}$ the number of exponentials in SOE, and $N_s$ 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与计算成本。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。随后构造了米塔格-莱弗勒函数的高效指数和逼近。在空间离散化中采用混合有限元方法，在运动学方程中对位移变量的二阶时间导数采用纽马克格式进行时间离散，最终得到快速算法。与传统时间分数阶导数的L1格式相比，本快速算法将内存复杂度从$\mathcal O(N_sN)$降至$\mathcal O(N_sN_{exp})$，计算复杂度从$\mathcal O(N_sN^2)$降至$\mathcal O(N_sN_{exp}N)$，其中$N$表示时间网格点总数，$N_{exp}$为指数和逼近中的指数项数量，$N_s$为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。