A LOCAL DISCONTINUOUS GALERKIN METHOD FOR TIME-FRACTIONAL DIFFUSION EQUATIONS*

doi:10.1007/s10473-023-0219-z

数学物理学报(英文版) ›› 2023, Vol. 43 ›› Issue (2): 839-854.doi: 10.1007/s10473-023-0219-z

A LOCAL DISCONTINUOUS GALERKIN METHOD FOR TIME-FRACTIONAL DIFFUSION EQUATIONS^*

Zhankuan Zeng, Yanping CHEN^†

School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China

收稿日期:2021-10-22 修回日期:2022-02-17 出版日期:2023-03-25 发布日期:2023-04-12
通讯作者: †Yanping CHEN, E-mail: yanpingchen@scnu.edu.cn .
作者简介:Zhankuan Zeng, E-mail: broadenzeng@gmail.com
基金资助:
This work was supported by the State Key Program of National Natural Science Foundation of China (11931003) and the National Natural Science Foundation of China (41974133).

A LOCAL DISCONTINUOUS GALERKIN METHOD FOR TIME-FRACTIONAL DIFFUSION EQUATIONS^*

Zhankuan Zeng, Yanping CHEN^†

School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China

Received:2021-10-22 Revised:2022-02-17 Online:2023-03-25 Published:2023-04-12
Contact: †Yanping CHEN, E-mail: yanpingchen@scnu.edu.cn .
About author:Zhankuan Zeng, E-mail: broadenzeng@gmail.com
Supported by:
This work was supported by the State Key Program of National Natural Science Foundation of China (11931003) and the National Natural Science Foundation of China (41974133).

摘要/Abstract

摘要： In this paper, a local discontinuous Galerkin (LDG) scheme for the time-fractional diffusion equation is proposed and analyzed. The Caputo time-fractional derivative (of order $\alpha$, with $0< \alpha <1$) is approximated by a finite difference method with an accuracy of order $3-\alpha$, and the space discretization is based on the LDG method. For the finite difference method, we summarize and supplement some previous work by others, and apply it to the analysis of the convergence and stability of the proposed scheme. The optimal error estimate is obtained in the $L^2$ norm, indicating that the scheme has temporal $(3 -\alpha)$ th-order accuracy and spatial $(k+1)$ th-order accuracy, where $k$ denotes the highest degree of a piecewise polynomial in discontinuous finite element space. The numerical results are also provided to verify the accuracy and efficiency of the considered scheme.

关键词: local discontinuous Galerkin method, time fractional diffusion equations, stability, convergence

Abstract: In this paper, a local discontinuous Galerkin (LDG) scheme for the time-fractional diffusion equation is proposed and analyzed. The Caputo time-fractional derivative (of order $\alpha$, with $0< \alpha <1$) is approximated by a finite difference method with an accuracy of order $3-\alpha$, and the space discretization is based on the LDG method. For the finite difference method, we summarize and supplement some previous work by others, and apply it to the analysis of the convergence and stability of the proposed scheme. The optimal error estimate is obtained in the $L^2$ norm, indicating that the scheme has temporal $(3 -\alpha)$ th-order accuracy and spatial $(k+1)$ th-order accuracy, where $k$ denotes the highest degree of a piecewise polynomial in discontinuous finite element space. The numerical results are also provided to verify the accuracy and efficiency of the considered scheme.

Key words: local discontinuous Galerkin method, time fractional diffusion equations, stability, convergence

Zhankuan Zeng, Yanping CHEN. A LOCAL DISCONTINUOUS GALERKIN METHOD FOR TIME-FRACTIONAL DIFFUSION EQUATIONS^*[J]. 数学物理学报(英文版), 2023, 43(2): 839-854.

Zhankuan Zeng, Yanping CHEN. A LOCAL DISCONTINUOUS GALERKIN METHOD FOR TIME-FRACTIONAL DIFFUSION EQUATIONS^*[J]. Acta mathematica scientia,Series B, 2023, 43(2): 839-854.

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A LOCAL DISCONTINUOUS GALERKIN METHOD FOR TIME-FRACTIONAL DIFFUSION EQUATIONS^*

A LOCAL DISCONTINUOUS GALERKIN METHOD FOR TIME-FRACTIONAL DIFFUSION EQUATIONS^*

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