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The local discontinuous Galerkin method for a singularly perturbed convection–diffusion problem with characteristic and exponential layers

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Abstract

A singularly perturbed convection–diffusion problem, posed on the unit square in \({\mathbb {R}}^2\), is studied; its solution has both exponential and characteristic boundary layers. The problem is solved numerically using the local discontinuous Galerkin (LDG) method on Shishkin meshes. Using tensor-product piecewise polynomials of degree at most \(k>0\) in each variable, the error between the LDG solution and the true solution is proved to converge, uniformly in the singular perturbation parameter, at a rate of \(O\left( \left( N^{-1}\ln N\right) ^{k+1/2}\right) \) in an associated energy norm, where N is the number of mesh intervals in each coordinate direction. (This is the first uniform convergence result proved for the LDG method applied to a problem with characteristic boundary layers.) Furthermore, we prove that this order of convergence increases to \(O\left( \left( N^{-1}\ln N\right) ^{k+1}\right) \) when one measures the energy-norm difference between the LDG solution and a local Gauss-Radau projection of the true solution into the finite element space. This uniform supercloseness property implies an optimal \(L^2\) error estimate of order \(\left( N^{-1}\ln N\right) ^{k+1}\) for our LDG method. Numerical experiments show the sharpness of our theoretical results.

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Authors and Affiliations

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

    Yao Cheng

  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. Martin Stynes
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Correspondence to Martin Stynes.

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This paper was partly written during a visit by Yao Cheng to Beijing Computational Science Research Center in July 2022. The research of Y. Cheng was supported by NSFC grant 11801396 and Natural Science Foundation of Jiangsu Province grant BK20170374. The research of M. Stynes was partly supported by NSFC grants 12171025 and NSAF-U2230402.

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Cheng, Y., Stynes, M. The local discontinuous Galerkin method for a singularly perturbed convection–diffusion problem with characteristic and exponential layers. Numer. Math. 154, 283–318 (2023). https://doi.org/10.1007/s00211-023-01361-z

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  • DOI: https://doi.org/10.1007/s00211-023-01361-z

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