Local convergence analysis of L1/finite element scheme for a constant delay reaction-subdiffusion equation with uniform time mesh
Weiping Bu, Xin Zheng
TL;DR
The paper addresses local temporal convergence of an L1 time-stepping scheme combined with finite elements for a constant-delay reaction–subdiffusion model driven by the Caputo derivative $$_0^C D_t^{\alpha} u$$ with $0<\alpha<1$. It introduces a fully discrete scheme and, critically, a discrete fractional Grönwall inequality with constant delay that avoids Mittag–Leffler growth, enabling pointwise-in-time and piecewise-in-time error estimates on a uniform time mesh under multi-singularity in time. The main results show that on each interval $((i-1)\tau,i\tau]$, the temporal error scales as $\rho^{\min\{1,i\alpha\}}$ plus spatial error $h^2$, with local rate $i\alpha$ when $i\alpha\le 1$, reflecting increasing smoothness of the solution over time; numerical experiments validate these sharp estimates. Overall, the work provides sharper local convergence analysis for delayed subdiffusion models and presents a robust numerical approach free from problematic Mittag–Leffler growth terms.
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 $τ$ 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 Grönwall 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 Grönwall 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 $((i-1)τ,iτ]$, 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.