$L2-1_\sigma$ Finite Element Method for Time-Fractional Diffusion Problems with Discontinuous Coefficients
Year: 2023
Author: Yanping Chen, Xuejiao Tan, Yunqing Huang
East Asian Journal on Applied Mathematics, Vol. 13 (2023), Iss. 4 : pp. 813–834
Abstract
A time-fractional diffusion equation with an interface problem caused by discontinuous coefficients is considered. To solve it, in the temporal direction Alikhanov’s $L2-1_σ$ method with graded mesh is presented to deal with the weak singularity at $t = 0,$ while in the spatial direction a finite element method with uniform mesh is employed to handle the discontinuous coefficients. Then, with the help of discrete fractional Grönwall inequality and the robustness theory of $α → 1^−,$ we show that the method has stable error bounds at $α → 1^−,$ the fully discrete schemes $L^2(Ω)$ norm and $H^1(Ω)$ semi-norm are unconditionally stable, and the optimal convergence order is $\mathscr{O}(h^2 + N^{−{\rm min}\{rα,2\}})$ and $\mathscr{O}(h + N^{−{\rm min}\{rα,2\}}),$ respectively, where, $h,$ $N,$ $α,$ $r$ is the total number of spatial parameter, the time-fractional order coefficient, and the time grid constant. Finally, three numerical examples are provided to illustrate our theoretical results.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/eajam.2022-178.101022
East Asian Journal on Applied Mathematics, Vol. 13 (2023), Iss. 4 : pp. 813–834
Published online: 2023-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 22
Keywords: Time-fractional interface problems finite element $L2-1_σ$ method weak singularity.