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We present the Wavelet-based Edge Multiscale Parareal (WEMP) Algorithm,recently proposed in [Li and Hu, {\it J. Comput. Phys.}, 2021], for efficientlysolving subdiffusion equations with heterogeneous coefficients in long time.This algorithm combines the benefits of multiscale methods, which can handleheterogeneity in the spatial domain, and the strength of parareal algorithmsfor speeding up time evolution problems when sufficient processors areavailable. Our algorithm overcomes the challenge posed by the nonlocality ofthe fractional derivative in previous parabolic problem work by constructing anauxiliary problem on each coarse temporal subdomain to completely uncouple thetemporal variable. We prove the approximation properties of the correctionoperator and derive a new summation of exponential to generate a single-steptime 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 ouralgorithm in terms of the mesh size in the spatial domain, the level parameterused in the multiscale method, the coarse-scale time step size, and thefine-scale time step size. Finally, we present several numerical tests thatdemonstrate the effectiveness of our algorithm and validate our theoreticalresults.

Wavelet-based Edge Multiscale Parareal Algorithm for subdiffusion equations with heterogeneous coefficients in a large time domain