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Unconditionally Optimal Error Estimates of a Linearized Galerkin Method for Nonlinear Time Fractional Reaction–Subdiffusion Equations

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Abstract

This paper is concerned with unconditionally optimal error estimates of linearized Galerkin finite element methods to numerically solve some multi-dimensional fractional reaction–subdiffusion equations, while the classical analysis for numerical approximation of multi-dimensional nonlinear parabolic problems usually require a restriction on the time-step, which is dependent on the spatial grid size. To obtain the unconditionally optimal error estimates, the key point is to obtain the boundedness of numerical solutions in the \(L^\infty \)-norm. For this, we introduce a time-discrete elliptic equation, construct an energy function for the nonlocal problem, and handle the error summation properly. Compared with integer-order nonlinear problems, the nonlocal convolution in the time fractional derivative causes much difficulties in developing and analyzing numerical schemes. Numerical examples are given to validate our theoretical results.

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Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, China

    Dongfang Li

  2. Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Huazhong University of Science and Technology, Wuhan, 430074, China

    Dongfang Li

  3. Beijing Computational Science Research Center, Beijing, 100193, China

    Jiwei Zhang & Zhimin Zhang

  4. Department of Mathematics, Wayne State University, Detroit, MI, 48202, USA

    Zhimin Zhang

Authors
  1. Dongfang Li
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  2. Jiwei Zhang
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  3. Zhimin Zhang
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Correspondence to Jiwei Zhang.

Additional information

This work is partially supported by NSFC under Grant Nos. 11771035, 11771162, 11571128, 91430216 and U1530401.

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Li, D., Zhang, J. & Zhang, Z. Unconditionally Optimal Error Estimates of a Linearized Galerkin Method for Nonlinear Time Fractional Reaction–Subdiffusion Equations. J Sci Comput 76, 848–866 (2018). https://doi.org/10.1007/s10915-018-0642-9

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  • DOI: https://doi.org/10.1007/s10915-018-0642-9

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