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.

翻译：本文旨在针对具有常时滞$\tau$及均匀时间网格的反应-亚扩散方程，建立其L1/有限元格式的精细误差估计。在精确解具有时间非均匀多重奇异性假设下，研究了均匀网格L1格式的局部截断误差。随后，我们为所考虑问题引入了全离散有限元格式。接着，提出了一种新颖的含常时滞项的离散分数阶Gr\"onwall不等式，与一些常见的其他情形相比，该不等式不包含递增的Mittag-Leffler函数。通过应用此Gr\"onwall不等式，我们得到了不含Mittag-Leffler函数的有限元格式的逐时间点及分段逐时间误差估计。特别地，后者表明，对于所考虑的区间$((i-1)\tau,i\tau]$，尽管当$i=1$时时间收敛阶较低，但随着$i$增大收敛性将得到改善，这与解的光滑性随$i$增大而提高的实际假设相一致。最后，我们给出了一些数值实验以验证所发展的理论。