TY - JOUR
T1 - Reconstruction-Based a Posteriori Error Estimates for the L1 Method for Time Fractional Parabolic Problems
AU - Cao , Jiliang
AU - Xiao , Aiguo
AU - Wang , Wansheng
JO - Journal of Computational Mathematics
VL - 2
SP - 345
EP - 368
PY - 2024
DA - 2024/11
SN - 43
DO - http://doi.org/10.4208/jcm.2210-m2022-0085
UR - https://global-sci.org/intro/article_detail/jcm/23541.html
KW - Time fractional parabolic differential equations, A posteriori error estimates, L1 method, Fractional integral reconstruction, Quadratic reconstruction.
AB - <p style="text-align: justify;">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.</p>