2024/07/06 by Weiping Bu, Xin Zheng, Bu, Weiping +1 · 1 citation
Engineering · Mathematics · #Advanced Numerical Methods in Computational Mathematics #Differential Equations and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods for differential equations
paper · pdf · doi:10.48550/arxiv.2407.05041
openalex publication_date 2024/07/06 · openalex created_date 2024/07/10 · openalex updated_date 2026/07/28
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.