We present the Wavelet-based Edge Multiscale Parareal (WEMP) Algorithm, recently proposed in [Li and Hu, {\it J. Comput. Phys.}, 2021], for efficiently solving subdiffusion equations with heterogeneous coefficients in long time. This algorithm combines the benefits of multiscale methods, which can handle heterogeneity in the spatial domain, and the strength of parareal algorithms for speeding up time evolution problems when sufficient processors are available. Our algorithm overcomes the challenge posed by the nonlocality of the fractional derivative in previous parabolic problem work by constructing an auxiliary problem on each coarse temporal subdomain to completely uncouple the temporal variable. We prove the approximation properties of the correction operator and derive a new summation of exponential to generate a single-step time stepping scheme, with the number of terms of $\mathcal{O}(|\log{\tau_f}|^2)$ independent of the final time, where $\tau_f$ is the fine-scale time step size. We establish the convergence rate of our algorithm in terms of the mesh size in the spatial domain, the level parameter used in the multiscale method, the coarse-scale time step size, and the fine-scale time step size. Finally, we present several numerical tests that demonstrate the effectiveness of our algorithm and validate our theoretical results.

翻译：我们提出了近期在[Li and Hu, J. Comput. Phys., 2021]中提出的基于小波的边缘多尺度并行(WEMP)算法，用于高效求解长时间内具有异质系数的次扩散方程。该算法结合了多尺度方法（能够处理空间域中的异质性）和并行算法（在拥有足够处理器时可加速时间演化问题）的优势。我们通过在每个粗时间子域上构造辅助问题，完全解耦时间变量，克服了先前抛物型问题研究中分数阶导数的非局部性带来的挑战。我们证明了校正算子的逼近性质，并推导出一种新的指数求和形式，生成单步时间推进方案，其项数为$\mathcal{O}(|\log{\tau_f}|^2)$，与最终时间无关，其中$\tau_f$为细尺度时间步长。我们从空间域网格尺寸、多尺度方法中使用的层级参数、粗尺度时间步长以及细尺度时间步长角度，建立了算法的收敛速率。最后，我们通过若干数值实验验证了算法的有效性及其理论结果。