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\(H^{1}\)-norm error analysis of a robust ADI method on graded mesh for three-dimensional subdiffusion problems

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Abstract

This work proposes a robust ADI scheme on graded mesh for solving three-dimensional subdiffusion problems. The Caputo fractional derivative is discretized by L1 scheme, where the graded mesh is used to eliminate the weak singular behavior of the exact solution at the initial time \(t=0\). The spatial derivatives are approximated by the finite difference method. Based on the improved discrete fractional \(Gr\ddot{o}nwall\) inequality, we prove the stability and \(\alpha \)-robust \(H^{1}\)-norm convergence, in which the error bound does not blow up when the order of fractional derivative \(\alpha \rightarrow 1^{-}\). The 3D numerical examples are proposed to verify the efficiency and accuracy of the ADI method. The CPU time is also provided, which shows the proposed method is very efficient for 3D subdiffusion problems.

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Data Availability

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

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Acknowledgements

The authors are grateful to the referees for many helpful suggestions.

Funding

The work was supported by National Natural Science Foundation of China (12226337, 12226340, 12126321), Scientific Research Fund of Hunan Provincial Education Department (21B0550), Hunan Provincial Natural Science Foundation (2022JJ50083, 2023JJ50164), and Hunan Provincial Innovation Foundation For Postgraduate (CX20231110).

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Authors and Affiliations

  1. College of Science, Hunan University of Technology, Zhuzhou, 412008, China

    Ziyi Zhou, Haixiang Zhang & Xuehua Yang

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  1. Ziyi Zhou
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  2. Haixiang Zhang
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Contributions

Haixiang Zhang provided the methodology along with the model problem. Ziyi Zhou implemented the scheme, obtained the error analysis and the numerical experiments and writing the manuscript. Xuehua Yang revised the manuscript for important content.

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Correspondence to Haixiang Zhang.

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Zhou, Z., Zhang, H. & Yang, X. \(H^{1}\)-norm error analysis of a robust ADI method on graded mesh for three-dimensional subdiffusion problems. Numer Algor 96, 1533–1551 (2024). https://doi.org/10.1007/s11075-023-01676-w

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