Can graded meshes yield more accurate numerical solution than uniform meshes? A time-dependent nonlocal diffusion problem with a weakly singular kernel is considered using collocation method. For its steady-state counterpart, under the sufficiently smooth solution, we first clarify that the standard graded meshes are worse than uniform meshes and may even lead to divergence; instead, an optimal convergence rate arises in so-called anomalous graded meshes. Furthermore, under low regularity solutions, it may suffer from a severe order reduction in (Chen, Qi, Shi and Wu, IMA J. Numer. Anal., 41 (2021) 3145--3174). In this case, conversely, a sharp error estimates appears in standard graded meshes, but offering far less than first-order accuracy. For the time-dependent case, however, second-order convergence can be achieved on graded meshes. The related analysis are easily extended for certain multidimensional problems. Numerical results are provided that confirm the sharpness of the error estimates.

翻译：分片网格能否比均匀网格产生更精确的数值解？本文采用配置方法研究了具有弱奇异核的时变非局部扩散问题。对于其稳态情形，在解充分光滑的条件下，我们首先阐明了标准分片网格的精度低于均匀网格，甚至可能导致发散；相反，在所谓的反常分片网格中可获得最优收敛速率。此外，在低正则性解的情况下，（Chen, Qi, Shi and Wu, IMA J. Numer. Anal., 41 (2021) 3145--3174）中可能出现严重的阶数降低。此时，标准分片网格反而能提供尖锐的误差估计，但精度远低于一阶。然而在时变情形下，分片网格可实现二阶收敛。相关分析可轻松推广至某些多维问题。数值结果验证了误差估计的尖锐性。