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 τ 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)τ,iτ], 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.