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Balanced-norm error estimate of the local discontinuous Galerkin method on layer-adapted meshes for reaction-diffusion problems

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Abstract

We study the local discontinuous Galerkin (LDG) method on layer-adapted meshes for singularly perturbed problems. For these problems of reaction-diffusion type, the balanced norm is suitable to capture the contribution of boundary layer component of the solution. However, it is rather difficult to derive optimal balanced-norm error estimate of the LDG method in the general case. That is, one can only obtain suboptimal error estimate of the LDG method in the mentioned “artificial” norm by using the traditional technique. In this paper, we propose a novel projection which combines the local Gauss-Radau projection with the local L2 projection in a subtle way. We obtain an improved balanced-error estimate. In particular, this estimate is optimal in the case that the regular component of the solution belongs to the finite element space. Numerical experiments are also given to test the theoretical results.

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Acknowledgements

The authors gratefully acknowledge the editor and the two anonymous referees for their time and invaluable suggestions, which considerably improved the quality of the paper.

Funding

This research was supported by National Natural Science Foundation of China (No. 11801396), Natural Science Foundation of Jiangsu Province (No. BK20170374), and Natural Science Foundation of the Jiangsu Higher Education Institutions of China (No. 17KJB110016).

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Authors and Affiliations

  1. School of Mathematical Sciences, Suzhou University of Science and Technology, Suzhou, 215009, Jiangsu Province, China

    Yao Cheng & Li Yan

  2. International Education School, Suzhou University of Science and Technology, Jiangsu, 215009, China

    Yanjie Mei

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  1. Yao Cheng
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  2. Li Yan
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  3. Yanjie Mei
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Correspondence to Yao Cheng.

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Cheng, Y., Yan, L. & Mei, Y. Balanced-norm error estimate of the local discontinuous Galerkin method on layer-adapted meshes for reaction-diffusion problems. Numer Algor 91, 1597–1626 (2022). https://doi.org/10.1007/s11075-022-01316-9

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