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

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 $\tau$ based on the uniform time mesh. Under the non-uniform multi-singularity assumption of exact solution in time, the local truncation error of L1 scheme is investigated. Then we introduce a fully discrete finite element scheme of the considered problem. In order to investigate the error estimate, a novel discrete fractional Grönwall inequality with delay term is proposed, which does not include the increasing Mittag-Leffler function. By applying this Grönwall inequality, we obtain a local error estimate of the above fully discrete scheme without including the Mittag-Leffler function. In particular, the convergence result implies that, for the considered time interval $((i-1)\tau ,i\tau ]$, although the convergence rate in the sense of maximum time error is low for the first interval, i.e. $i=1$, 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.