In this paper, the well-known shifted Grünwald–Letnikov formula is revisited to solve the two-dimensional fractional boundary value problems (FBVPs) with non-smooth solution. Both stability analysis and error estimates of the scheme are carried out in the maximum norm. An improved algorithm is presented based on the extrapolation technique to obtain high-order accuracy for the problems with low regularity. Various numerical experiments are given to support the theoretical finding and verify the effectiveness of the improved algorithm. It is illustrated that, by using the proposed algorithm, both accuracy and convergence rate can be significantly improved for solving the two-dimensional FBVPs as well as fractional diffusion equations with non-smooth solution. Especially, the convergence rate of corrected numerical solutions even can reach second-order in the case of solving problems involved one-sided fractional derivatives.

在本文中，重新审视了著名的移位 Grünwald-Letnikov 公式，以解决具有非光滑解的二维分数边值问题 (FBVP)。该方案的稳定性分析和误差估计均在最大范数下进行。针对低规律性问题，提出了一种基于外推技术的改进算法，以获得高阶精度。给出了各种数值实验来支持理论发现并验证改进算法的有效性。说明该算法在求解二维 FBVP 以及非光滑解的分数扩散方程时，精度和收敛速度均得到显着提高。尤其，