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Supercloseness of weak Galerkin method on Bakhvalov-type mesh for a singularly perturbed problem in 1D

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Abstract

In this paper, we analyze supercloseness in an energy norm of a weak Galerkin (WG) method on a Bakhvalov-type mesh for a singularly perturbed two-point boundary value problem. For this aim, a special approximation is designed according to the specific structures of the mesh, the WG finite element space and the WG scheme. More specifically, in the interior of each element, the approximation consists of a Gauß–Lobatto interpolant inside the layer and a Gauß–Radau projection outside the layer. On the boundary of each element, the approximation equals the true solution. Besides, with the help of over-penalization technique inside the layer, we prove uniform supercloseness of order k + 1 for the WG method. Numerical experiments verify the supercloseness result and test the influence of different penalization parameters inside the layer.

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Acknowledgements

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

Funding

This research is supported by National Natural Science Foundation of China (11771257), Shandong Provincial Natural Science Foundation, China (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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Correspondence to Jin Zhang.

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Liu, X., Zhang, J. Supercloseness of weak Galerkin method on Bakhvalov-type mesh for a singularly perturbed problem in 1D. Numer Algor 93, 367–395 (2023). https://doi.org/10.1007/s11075-022-01420-w

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