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

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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 \(L^2\)-convergence of the scheme are proved for all variable-order \(\alpha (t)\in (0,1)\). The proposed method is of accuracy-order \(O(\tau ^3+h^{k+1})\) , where \(\tau\), 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.

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Acknowledgements

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).

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Authors and Affiliations

  1. College of Science, Henan University of Technology, Zhengzhou, 450001, China

    Leilei Wei

  2. School of Mathematical Sciences, Huaqiao University, Quanzhou, 362021, Fujian Province, China

    Shuying Zhai

  3. College of Mathematics Sciences, **njiang Normal University, Urumqi, 830054, China

    **ndong Zhang

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  1. Leilei Wei
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  2. Shuying Zhai
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  3. **ndong Zhang
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Correspondence to Leilei Wei.

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Wei, L., Zhai, S. & Zhang, X. Error Estimate of a Fully Discrete Local Discontinuous Galerkin Method for Variable-Order Time-Fractional Diffusion Equations. Commun. Appl. Math. Comput. 3, 429–443 (2021). https://doi.org/10.1007/s42967-020-00081-7

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