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Optimal Balanced-Norm Error Estimate of the LDG Method for Reaction–Diffusion Problems I: The One-Dimensional Case

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Abstract

A singularly perturbed reaction–diffusion problem in 1D is solved numerically by a local discontinuous Galerkin (LDG) finite element method. For this type of problem the standard energy norm is too weak to capture the contribution of the boundary layer component of the true solution, so balanced norms have been used by many authors to give more satisfactory error bounds for solutions computed using various types of finite element method. But for the LDG method, up to now no optimal-order balanced-norm error estimate has been derived. In this paper, we consider an LDG method with central numerical flux on a Shishkin mesh. Using the superconvergence property of the local \(L^2\) projector and some local coupled projections around the two transition points of the mesh, we prove an optimal-order balanced-norm error estimate for the computed solution; that is, when piecewise polynomials of degree k are used on a Shishkin mesh with N mesh intervals, in the balanced norm we establish \(O((N^{-1}\ln N)^{k+1})\) convergence when k is even and \(O((N^{-1}\ln N)^{k})\) when k is odd. Numerical experiments confirm the sharpness of these error bounds.

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Funding

The research of Yao Cheng was supported by NSFC Grant 11801396, Natural Science Foundation of Jiangsu Province Grant BK20170374 and QingLan Project of Jiangsu Province. The research of Xuesong Wang was supported by NSFC Grant 11801396 and Graduate Student Scientific Research Innovation Projects of Jiangsu Province (KYCX22_3254). The research of Martin Stynes is supported in part by the National Natural Science Foundation of China under Grants 12171025 and NSAF-U2230402.

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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 & Xuesong Wang

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

    Martin Stynes

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  1. Yao Cheng
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  2. Xuesong Wang
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  3. Martin Stynes
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Correspondence to Martin Stynes.

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Cheng, Y., Wang, X. & Stynes, M. Optimal Balanced-Norm Error Estimate of the LDG Method for Reaction–Diffusion Problems I: The One-Dimensional Case. J Sci Comput 100, 50 (2024). https://doi.org/10.1007/s10915-024-02602-5

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  • DOI: https://doi.org/10.1007/s10915-024-02602-5

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