Reconstruction-Based a Posteriori Error Estimates for the L1 Method for Time Fractional Parabolic Problems
Year: 2025
Author: Jiliang Cao, Aiguo Xiao, Wansheng Wang
Journal of Computational Mathematics, Vol. 43 (2025), Iss. 2 : pp. 345–368
Abstract
In this paper, we study a posteriori error estimates of the L1 scheme for time discretizations of time fractional parabolic differential equations, whose solutions have generally the initial singularity. To derive optimal order a posteriori error estimates, the quadratic reconstruction for the L1 method and the necessary fractional integral reconstruction for the first-step integration are introduced. By using these continuous, piecewise time reconstructions, the upper and lower error bounds depending only on the discretization parameters and the data of the problems are derived. Various numerical experiments for the one-dimensional linear fractional parabolic equations with smooth or nonsmooth exact solution are used to verify and complement our theoretical results, with the convergence of $α$ order for the nonsmooth case on a uniform mesh. To recover the optimal convergence order $2−α$ on a nonuniform mesh, we further develop a time adaptive algorithm by means of barrier function recently introduced. The numerical implementations are performed on nonsmooth case again and verify that the true error and a posteriori error can achieve the optimal convergence order in adaptive mesh.
You do not have full access to this article.
Already a Subscriber? Sign in as an individual or via your institution
Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jcm.2210-m2022-0085
Journal of Computational Mathematics, Vol. 43 (2025), Iss. 2 : pp. 345–368
Published online: 2025-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 24
Keywords: Time fractional parabolic differential equations A posteriori error estimates L1 method Fractional integral reconstruction Quadratic reconstruction.
Author Details
-
Adaptive fast L1 − 2 scheme for solving time fractional parabolic problems
Cao, Jiliang
Wang, Wansheng
Xiao, Aiguo
Computers & Mathematics with Applications, Vol. 179 (2025), Iss. P.59
https://doi.org/10.1016/j.camwa.2024.12.003 [Citations: 0]