Asymptotically Compatible Fractional Gr\"onwall Inequality and its Applications

TL;DR

This paper develops an asymptotically compatible fractional Gr"onwall inequality by estimating discrete convolution kernels, enabling precise stability and error analysis of difference schemes on non-uniform meshes, validated through numerical experiments.

Contribution

It introduces a new inequality for fractional calculus that improves error bounds for schemes on non-uniform grids, with explicit pointwise error estimates.

Findings

Validated error bounds for difference schemes on graded meshes

Numerical experiments confirm theoretical stability and accuracy

Provides a framework for analyzing fractional schemes on non-uniform meshes

Abstract

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.