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Balanced-Norm and Energy-Norm Error Analyses for a Backward Euler/FEM Solving a Singularly Perturbed Parabolic Reaction-Diffusion Problem

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Abstract

In the derivation of error bounds, uniformly in the singular perturbation parameter \(\varepsilon \), for finite element methods (FEMs) applied to elliptic singularly perturbed linear reaction-diffusion problems, the usual energy norm is unsatisfactory since it is essentially no stronger than the \(L^2\) norm. Consequently various researchers have analysed errors in FEM solutions, uniformly in \(\varepsilon \), using balanced norms whose \(H^1\) component is weighted correctly to maintain its influence. But the derivation of energy and balanced-norm error bounds for FEM solutions of singularly perturbed reaction-diffusion problems is confined almost entirely to steady-state elliptic problems — little has been proved for time-dependent parabolic singularly perturbed problems. The present paper addresses this gap in the literature: the backward Euler method in time, combined with a bilinear FEM on a spatial Shishkin mesh, is applied to solve a parabolic singularly perturbed reaction-diffusion problem, and energy-norm and balanced-norm error estimates, which are uniform in the singular perturbation parameter \(\varepsilon \), are derived — these results are stronger than any previous results of the same type. Furthermore, numerical experiments demonstrate the sharpness of our error bounds.

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Acknowledgements

We are grateful to two unknown reviewers who provided several perceptive and helpful comments that guided us in improving the clarity of the paper.

Funding

The research of Xiangyun Meng is supported in part by the National Natural Science Foundation of China under grant 12101039 and by the Fundamental Research Funds for the Central Universities under grant 2020RC101. The research of Martin Stynes is supported in part by the National Natural Science Foundation of China under grants 12171025 and NSAF-U1930402.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Beijing Jiaotong University, Beijing, 100044, China

    Xiangyun Meng

  2. Applied and Computational Mathematics Division, Beijing Computational Science Research Center, Beijing, 100193, China

    Martin Stynes

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  1. Xiangyun Meng
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Correspondence to Martin Stynes.

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The research of Xiangyun Meng is supported in part by the Fundamental Research Funds for the Central Universities under grant 2020RC101 and the National Natural Science Foundation of China under grants 12101039. The research of Martin Stynes is supported in part by the National Natural Science Foundation of China under grants 12171025 and NSAF-U1930402.

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Meng, X., Stynes, M. Balanced-Norm and Energy-Norm Error Analyses for a Backward Euler/FEM Solving a Singularly Perturbed Parabolic Reaction-Diffusion Problem. J Sci Comput 92, 67 (2022). https://doi.org/10.1007/s10915-022-01931-7

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  • DOI: https://doi.org/10.1007/s10915-022-01931-7

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