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Nonconforming Virtual Element Method for the Time Fractional Reaction–Subdiffusion Equation with Non-smooth Data

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Abstract

In this paper, we consider the nonconforming virtual element method (VEM) for the approximation of the time fractional reaction–subdiffusion equation involving the Caputo fractional derivative. For the numerical discrete method of the Caputo fractional derivative, we permit the use of nonuniform time steps, since they are helpful to deal with the non-smooth system. Meanwhile, the nonconforming VEM, which is constructed for any order of accuracy and for very general shaped polygonal and polyhedral meshes, is adopted for the discretization of the spatial direction. By introducing a new Ritz projection operator and using two extended ty\(L^2\)-normpes of continuous and discrete fractional Grönwall inequalities, the optimal error estimates for the spatial semi-discrete and temporal-spatial fully discrete systems are proved detailedly. Besides, the fully discrete scheme is proved to be unconditionally stable with regard to the \(L^2\)- and \(H^1\)-norms, respectively. Finally, some numerical calculations are implemented to verify the theoretical results.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, 450001, People’s Republic of China

    Meng Li, Jikun Zhao & Shaochun Chen

  2. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, People’s Republic of China

    Chengming Huang

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  1. Meng Li
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Correspondence to Jikun Zhao.

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This work was supported by NSF of China (Nos. 11801527, 11701522, 11701523, 11771163, 11671160), China Postdoctoral Science Foundation (No. 2018M632791), Key Scientific Research Projects of Higher Eduction of Henan (19A110034), and NSF of Anhui Higher Education Institutions of China (No. KJ2017A704).

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Li, M., Zhao, J., Huang, C. et al. Nonconforming Virtual Element Method for the Time Fractional Reaction–Subdiffusion Equation with Non-smooth Data. J Sci Comput 81, 1823–1859 (2019). https://doi.org/10.1007/s10915-019-01064-4

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