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A Novel Discrete Fractional Grönwall-Type Inequality and Its Application in Pointwise-in-Time Error Estimates

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Abstract

We present a family of fully-discrete schemes for numerically solving nonlinear sub-diffusion equations, taking the weak regularity of the exact solutions into account. Using a novel discrete fractional Grönwall inequality, we obtain pointwise-in-time error estimates of the time-stepping methods. It is proved that as \(t\rightarrow 0\), the convergence orders can be \(\sigma _{k}\), where \(\sigma _{k}\) is the regularity parameter. The initial convergence results are sharp. As t is far away from 0, the schemes give a better convergence results. Numerical experiments are given 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 & Mianfu She

  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, University of Macau, Macao, China

    Hai-wei Sun

  4. School of Mathematics and Computational Science & Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan University, Hunan, 411105, China

    Xiaoqiang Yan

  5. College of Liberal Arts and Sciences, National University of Defense Technology, Changsha, 410073, China

    Xiaoqiang Yan

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  1. Dongfang Li
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Correspondence to Xiaoqiang Yan.

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This work is supported in part by NSFC (No. 11771162), research grants of the Science and Technology Development Fund, Macau SAR (File No. 0122/2020/A3), and MYRG2020-00224-FST from University of Macau, the Fundamental Research Funds for the Central Universities (HUST: 2021JYCXJJ012) and the science and technology innovation Program of Hunan Province (No.2020RC2039).

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Li, D., She, M., Sun, Hw. et al. A Novel Discrete Fractional Grönwall-Type Inequality and Its Application in Pointwise-in-Time Error Estimates. J Sci Comput 91, 27 (2022). https://doi.org/10.1007/s10915-022-01803-0

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  • DOI: https://doi.org/10.1007/s10915-022-01803-0

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