Local convergence analysis of L1/finite element scheme for a constant delay reaction-subdiffusion equation with uniform time mesh
<|reference_start|>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 $\tau$ 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.<|reference_end|>
@article{bu2024local,
    title={Local convergence analysis of L1/finite element scheme for a constant
  delay reaction-subdiffusion equation with uniform time mesh},
    author={Weiping Bu and Xin Zheng},
    journal={arXiv preprint arXiv:2407.05041},
    year={2024},
    archivePrefix={arXiv},
    eprint={2407.05041},
    primaryClass={math.NA cs.NA}
}