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Reconstruction-Based a Posteriori Error Estimates for the L1 Method for Time Fractional Parabolic Problems

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.

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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

Jiliang Cao

Aiguo Xiao

Wansheng Wang

  1. 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]