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Linearized Galerkin FEMs for Nonlinear Time Fractional Parabolic Problems with Non-smooth Solutions in Time Direction

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Abstract

A Newton linearized Galerkin finite element method is proposed to solve nonlinear time fractional parabolic problems with non-smooth solutions in time direction. Iterative processes or corrected schemes become dispensable by the use of the Newton linearized method and graded meshes in the temporal direction. The optimal error estimate in the \(L^2\)-norm is obtained without any time step restrictions dependent on the spatial mesh size. Such unconditional convergence results are proved by including the initial time singularity into concern, while previous unconditional convergent results always require continuity and boundedness of the temporal derivative of the exact solution. Numerical experiments are conducted to confirm the theoretical results.

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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. Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong

    Chengda Wu

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

    Zhimin Zhang

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

    Zhimin Zhang

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  1. Dongfang Li
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  2. Chengda Wu
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  3. Zhimin Zhang
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Correspondence to Chengda Wu.

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This work is supported in part by the National Natural Science Foundation of China (NSFC) Grants Nos. 11771162, 11726603, 11871092, 11471031, 91430216, and U1530401.

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Li, D., Wu, C. & Zhang, Z. Linearized Galerkin FEMs for Nonlinear Time Fractional Parabolic Problems with Non-smooth Solutions in Time Direction. J Sci Comput 80, 403–419 (2019). https://doi.org/10.1007/s10915-019-00943-0

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  • DOI: https://doi.org/10.1007/s10915-019-00943-0

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