In this work, we will give proper estimates for the discrete convolution complementary (DCC) kernels, which leads to the asymptotically compatible fractional Gr\"onwall inequality. The consequence can be applied in the analysis of the stability and pointwise-in-time error of difference-type schemes on a non-uniform mesh. The pointwise error is explicitly bound when a non-uniform time grid is given by a specific scale function e.g. graded mesh, can be given directly. Numerical experiments towards the conclusion of this work validate the error analysis.

翻译：本文给出了离散卷积互补核的适当估计，由此推导出渐近相容分数阶格朗沃尔不等式。该结论可用于非均匀网格上差分格式的稳定性分析与逐点时间误差分析。当非均匀时间网格由特定尺度函数（如分级网格）给出时，可直接得到显式逐点误差界。本文数值实验验证了误差分析结果的正确性。