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\(\alpha \)-Robust Error Analysis of Two Nonuniform Schemes for Subdiffusion Equations with Variable-Order Derivatives

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Abstract

In this paper, we will consider the variable-order subdiffusion initial-boundary value problem with weakly singular solutions. By using the nonuniform L1 scheme and nonuniform Alikhanov scheme in time, two efficient numerical methods (which we call L1 FEM and Alikhanov FEM) are developed, where the finite element method is used in space. Firstly, an improved error analysis is given for the L1 FEM of Huang and Chen (Appl Math Lett 139:108559, 2023), and the derived error bounds remain valid as \(\alpha (t^*)\rightarrow 1^-\) for \(0\le t^*\le T\). To obtain the \(\alpha \)-robust optimal convergent analysis for Alikhanov FEM, the truncation error of the Alikhanov scheme for the variable-order Caputo derivative and an \(\alpha \)-robust bound on the complementary discrete kernels \(\mathbb {P}_j^{(n)}\) are presented. Combining these two results with an \(\alpha \)-robust discrete fractional Gronwall inequality, the optimal convergent results in \(L^\infty (L^2)\) norm and \(L^\infty (H^1)\) norm are derived. Furthermore, by adopting a simple postprocessing technique of the numerical solution, a higher convergence order in space is obtained. Finally, a numerical example is presented to verify the optimal theoretical convergent result.

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Data Availability

The datasets generated during the current study are available from the corresponding author on reasonable request.

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Acknowledgements

The research of Chaobao Huang is supported in part by the National Natural Science Foundation of China (under Grants 12101360 and 12171278), the Support Plan for Outstanding Youth Innovation Team in Shandong Higher Education Institutions under Grant 2022KJ184, and the Natural Science Foundation of Shandong Province (under Grants ZR2020QA031 and ZR2022MA068). The research of Hu Chen is supported in part by the National Natural Science Foundation of China under Grant 11801026, Natural Science Foundation of Shandong Province under Grant ZR2023MA077, and Fundamental Research Funds for the Central Universities (No. 202264006). The research of Xijun Yu is supported in part by the National Natural Science Foundation of China under Grant 12071046.

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Authors and Affiliations

  1. School of Statistics and Mathematics, Shandong University of Finance and Economics, Jinan, 250014, China

    Chaobao Huang

  2. School of Mathematics and Statistics, Shandong Normal University, Jinan, 250014, China

    Na An

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

    Hu Chen

  4. Laboratory of Computational Physics, Institute of Applied Physics and Computational Mathematics, Beijing, 100088, China

    Xijun Yu

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  1. Chaobao Huang
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Correspondence to Hu Chen.

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Huang, C., An, N., Chen, H. et al. \(\alpha \)-Robust Error Analysis of Two Nonuniform Schemes for Subdiffusion Equations with Variable-Order Derivatives. J Sci Comput 97, 43 (2023). https://doi.org/10.1007/s10915-023-02357-5

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