Error Estimate of a Fully Discrete Local Discontinuous Galerkin Method for Variable-Order Time-Fractional Diffusion Equations

doi:10.1007/s42967-020-00081-7

Communications on Applied Mathematics and Computation ›› 2021, Vol. 3 ›› Issue (3): 429-444.doi: 10.1007/s42967-020-00081-7

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Error Estimate of a Fully Discrete Local Discontinuous Galerkin Method for Variable-Order Time-Fractional Diffusion Equations

Leilei Wei¹, Shuying Zhai², Xindong Zhang³

1 College of Science, Henan University of Technology, Zhengzhou 450001, China;
2 School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, Fujian Province, China;
3 College of Mathematics Sciences, Xinjiang Normal University, Urumqi 830054, China

收稿日期:2020-03-06 修回日期:2020-05-13 出版日期:2021-09-20 发布日期:2021-09-16
通讯作者: Leilei Wei E-mail:leileiwei09@gmail.com,leileiwei@haut.edu.cn
基金资助:
This work was supported by the Fundamental Research Funds for the Henan Provincial Colleges and Universities in Henan University of Technology (2018RCJH10), the Training Plan of Young Backbone Teachers in Henan University of Technology (21420049), the Training Plan of Young Backbone Teachers in Colleges and Universities of Henan Province (2019GGJS094), the innovative Funds Plan of Henan University of Technology, Foundation of Henan Educational Committee (19A110005), and the National Natural Science Foundation of China (11861068).

Error Estimate of a Fully Discrete Local Discontinuous Galerkin Method for Variable-Order Time-Fractional Diffusion Equations

Leilei Wei¹, Shuying Zhai², Xindong Zhang³

1 College of Science, Henan University of Technology, Zhengzhou 450001, China;
2 School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, Fujian Province, China;
3 College of Mathematics Sciences, Xinjiang Normal University, Urumqi 830054, China

Received:2020-03-06 Revised:2020-05-13 Online:2021-09-20 Published:2021-09-16
Contact: Leilei Wei E-mail:leileiwei09@gmail.com,leileiwei@haut.edu.cn
Supported by:
This work was supported by the Fundamental Research Funds for the Henan Provincial Colleges and Universities in Henan University of Technology (2018RCJH10), the Training Plan of Young Backbone Teachers in Henan University of Technology (21420049), the Training Plan of Young Backbone Teachers in Colleges and Universities of Henan Province (2019GGJS094), the innovative Funds Plan of Henan University of Technology, Foundation of Henan Educational Committee (19A110005), and the National Natural Science Foundation of China (11861068).

摘要/Abstract

摘要： The aim of this paper is to develop a fully discrete local discontinuous Galerkin method to solve a class of variable-order fractional diffusion problems. The scheme is discretized by a weighted-shifted Grünwald formula in the temporal discretization and a local discontinuous Galerkin method in the spatial direction. The stability and the L²-convergence of the scheme are proved for all variable-order α(t) ∈ (0, 1). The proposed method is of accuracy-order O(τ³ + h^k+1), where τ, h, and k are the temporal step size, the spatial step size, and the degree of piecewise P^k polynomials, respectively. Some numerical tests are provided to illustrate the accuracy and the capability of the scheme.

关键词: Variable-order derivative, Discontinuous Galerkin method, Stability, Error estimates

Abstract: The aim of this paper is to develop a fully discrete local discontinuous Galerkin method to solve a class of variable-order fractional diffusion problems. The scheme is discretized by a weighted-shifted Grünwald formula in the temporal discretization and a local discontinuous Galerkin method in the spatial direction. The stability and the L²-convergence of the scheme are proved for all variable-order α(t) ∈ (0, 1). The proposed method is of accuracy-order O(τ³ + h^k+1), where τ, h, and k are the temporal step size, the spatial step size, and the degree of piecewise P^k polynomials, respectively. Some numerical tests are provided to illustrate the accuracy and the capability of the scheme.

Key words: Variable-order derivative, Discontinuous Galerkin method, Stability, Error estimates

中图分类号:

65M06

Leilei Wei, Shuying Zhai, Xindong Zhang. Error Estimate of a Fully Discrete Local Discontinuous Galerkin Method for Variable-Order Time-Fractional Diffusion Equations[J]. Communications on Applied Mathematics and Computation, 2021, 3(3): 429-444.

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Error Estimate of a Fully Discrete Local Discontinuous Galerkin Method for Variable-Order Time-Fractional Diffusion Equations

Error Estimate of a Fully Discrete Local Discontinuous Galerkin Method for Variable-Order Time-Fractional Diffusion Equations

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