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Sharp error estimate of a Grünwald–Letnikov scheme for reaction-subdiffusion equations

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Abstract

The sharp error estimate of a Grünwald–Letnikov (GL) scheme on uniform mesh for reaction-subdiffusion equations with weakly singular solutions is considered, where the spatial domain is a square in R2 which is discretized by Legendre Galerkin spectral method. A discrete fractional Grönwall inequality is shown by constructing a family of discrete complementary convolution (DCC) kernels for the discrete convolution coefficients of GL scheme, which is used to get the stability and convergence of the fully discrete scheme. By a delicate analysis of the convolution sum of DCC kernels and truncation errors, we also show that the error estimate is sharp and α-robust as the fractional order \(\alpha \rightarrow 1^{-}\), which avoids the so-called coefficient blow-up phenomenon. Numerical examples are provided to verify the sharpness of our theoretical analysis.

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Funding

The research is supported in part by the National Natural Science Foundation of China (Nos. 11801026, 11771035, 11971416), sponsored by OUC Scientific Research Starting Fund of Introduced Talent, NSAF U1930402, the Natural Science Foundation of Hubei Province (No. 2019CFA007), Xiangtan University 2018ICIP01, the Program for Scientific and Technological Innovation Talents in Universities of Henan Province (No. 19HASTIT025) and the Foundation for University Key Young Teacher of Henan Province (No. 2019GGJS214)

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Authors and Affiliations

  1. School of Mathematical Sciences, Ocean University of China, 266100, Qingdao, China

    Hu Chen

  2. School of Science, Xuchang University, 461000, Xuchang, China

    Yanhua Shi & Yanmin Zhao

  3. School of Mathematics and Statistics, and Hubei Key Laboratory of Computational Science, Wuhan Univeristy, 430072, Wuhan, China

    Jiwei Zhang

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  1. Hu Chen
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  2. Yanhua Shi
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  3. Jiwei Zhang
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  4. Yanmin Zhao
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Correspondence to Yanmin Zhao.

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Chen, H., Shi, Y., Zhang, J. et al. Sharp error estimate of a Grünwald–Letnikov scheme for reaction-subdiffusion equations. Numer Algor 89, 1465–1477 (2022). https://doi.org/10.1007/s11075-021-01161-2

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  • DOI: https://doi.org/10.1007/s11075-021-01161-2

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