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An \(\alpha \)-robust analysis of finite element method for space-time fractional diffusion equation

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Abstract

This paper primarily lies in presenting an \(\alpha \)-robust analysis of finite element method for space-time fractional diffusion equation. To this end, we firstly develop finite element approximation to fractional Laplacian and provide a d-dimensional fast Fourier transform (FFT)-based fast algorithm to derive spatial discretization of space-time fractional diffusion problem. Then we study the \(\alpha \)-robust stability and error analyses of full discrete scheme of space-time fractional diffusion problem using L1 scheme on graded temporal meshes. Different from current many existing works, the established stability and error bounds will not blow-up under energy norm as \(\alpha \rightarrow 1^{-}\), and the present error analysis illustrates that a choice of time mesh graded factor \(\gamma =(2-\alpha )/\alpha \) shall yield an optimal rate of convergence \(\mathcal {O}(N^{-(2-\alpha )})\) in temporal direction, where N is the number of temporal meshes. Eventually, some numerical tests are given to show the efficiency and \(\alpha \)-robust behavior of the proposed scheme.

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Acknowledgements

The authors are very grateful to the reviewers for carefully reading this paper and for their suggestions and comments which have improved the paper.

Funding

Professor Li was supported by the National Natural Science Foundation of China (No. 12101089) and the Natural Science Foundation of Sichuan Province (No. 2022NSFSC1844).

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Authors and Affiliations

  1. School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 611731, China

    Yi Yang & Jin Huang

  2. School of Mathematics, Chengdu Normal University, Chengdu, 611130, China

    Hu Li

Authors
  1. Yi Yang
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  2. Jin Huang
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  3. Hu Li
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Contributions

The author Y. Yang contributed to the conception of the study, performed the experiment and wrote the manuscript. The authors J. Huang and H. Li contributed significantly to analysis and manuscript preparation, and they helped perform the analysis with constructive discussions.

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Correspondence to Jin Huang.

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Yang, Y., Huang, J. & Li, H. An \(\alpha \)-robust analysis of finite element method for space-time fractional diffusion equation. Numer Algor (2024). https://doi.org/10.1007/s11075-024-01789-w

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