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Uniform convergence of optimal order for a finite element method on a Bakhvalov-type mesh for a singularly perturbed convection-diffusion equation with parabolic layers

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Abstract

This paper is to analyze a finite element method of any order on a Bakhvalov-type mesh in the case of 2D. By introducing a new interpolation according to the characteristics of layers, we show that the finite element method has uniform convergence of the optimal order with respect to the singular perturbation parameter. The result partially resolves an open problem introduced by Roos and Stynes (Comput. Methods Appl. Math. 15(4):531–550, 2015).

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Acknowledgements

We thank the anonymous referees for their valuable comments and suggestions that led us to improve this paper.

Funding

This research is supported by the National Natural Science Foundation of China (11771257) and Shandong Provincial Natural Science Foundation (ZR2021MA004).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Qilu University of Technology (Shandong Academy of Sciences), Jinan, 250353, China

    Xiaowei Liu

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

    Jin Zhang

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  1. Xiaowei Liu
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  2. Jin Zhang
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Xiaowei Liu developed the idea for the study. Jin Zhang wrote the main manuscript text. The authors revised and reviewed the manuscript.

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Correspondence to Jin Zhang.

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Liu, X., Zhang, J. Uniform convergence of optimal order for a finite element method on a Bakhvalov-type mesh for a singularly perturbed convection-diffusion equation with parabolic layers. Numer Algor 94, 459–478 (2023). https://doi.org/10.1007/s11075-023-01508-x

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