Local convergence analysis of L1/finite element scheme for a constant delay reaction-subdiffusion equation with uniform time mesh

TL;DR

This paper develops refined error estimates for an L1/finite element scheme applied to a reaction-subdiffusion equation with delay, introducing a novel discrete fractional Grönwall inequality that improves understanding of convergence behavior.

Contribution

The paper introduces a new discrete fractional Grönwall inequality for delayed reaction-subdiffusion equations, enabling sharper error analysis without Mittag-Leffler functions.

Findings

Error estimates show convergence improves over time intervals.

The new Grönwall inequality simplifies analysis by removing Mittag-Leffler functions.

Numerical tests confirm the theoretical error bounds.

Abstract

The aim of this paper is to develop a refined error estimate of L1/finite element scheme for a reaction-subdiffusion equation with constant delay $\tau$ and uniform time mesh. Under the non-uniform multi-singularity assumption of exact solution in time, the local truncation errors of the L1 scheme with uniform mesh is investigated. Then we introduce a fully discrete finite element scheme of the considered problem. Next, a novel discrete fractional Gr\"onwall inequality with constant delay term is proposed, which does not include the increasing Mittag-Leffler function comparing with some popular other cases. By applying this Gr\"onwall inequality, we obtain the pointwise-in-time and piecewise-in-time error estimates of the finite element scheme without the Mittag-Leffler function. In particular, the latter shows that, for the considered interval $((i-1)\tau,i\tau]$, although the convergence in time is low for $i=1$, it will be improved as the increasing $i$, which is consistent with the factual assumption that the smoothness of the solution will be improved as the increasing $i$. Finally, we present some numerical tests to verify the developed theory.