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Balanced and energy norm error bounds for a spatial FEM with Crank-Nicolson and BDF2 time discretisation applied to a singularly perturbed reaction-diffusion problem

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Abstract

A singularly perturbed parabolic problem posed on the unit square \(\Omega \) in \(\mathbb {R}^2\) is considered. Its solution typically has a boundary layer on each side of \(\Omega \) for time \(t>0\). To solve it numerically, a Galerkin FEM on a spatial Shishkin mesh is used, while for the time discretisation, a uniform temporal mesh is employed, on which we consider the Crank-Nicolson and BDF2 schemes. Energy-norm and balanced-norm error bounds on the Crank-Nicolson solution are derived, and for the BDF2 solution, a balanced-norm error bound is proved. These are the first balanced-norm error bounds for these discretisations in the singularly perturbed reaction-diffusion research literature. Numerical results demonstrate that all our error bounds are sharp.

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Funding

The research of Xiangyun Meng is supported by 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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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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  2. Martin Stynes
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Martin Stynes designed the study, performed the research, and wrote the manuscript; Xiangyun Meng designed the study, performed the research, and wrote the manuscript.

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Correspondence to Martin Stynes.

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Meng, X., Stynes, M. Balanced and energy norm error bounds for a spatial FEM with Crank-Nicolson and BDF2 time discretisation applied to a singularly perturbed reaction-diffusion problem. Numer Algor 95, 1155–1176 (2024). https://doi.org/10.1007/s11075-023-01603-z

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  • DOI: https://doi.org/10.1007/s11075-023-01603-z

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