Asymptotically Compatible Fractional Gr\"onwall Inequality and its Applications

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.