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Applicable Analysis
An International Journal
Volume 103, 2024 - Issue 5
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Research Article

Uniform convergence of the LDG method for singularly perturbed problems

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Pages 927-935 | Received 26 Oct 2022, Accepted 21 Apr 2023, Published online: 16 May 2023
 

Abstract

In this paper, we consider a one-dimensional singularly perturbed problem of the convection–diffusion type. The problem is solved numerically by the local discontinuous Galerkin (LDG) method on a Bakhvalov-type mesh. Here we propose new numerical fluxes and new penalty parameters in the LDG method and prove the supercloseness of the LDG method in an energy norm. Besides, a variant of the energy norm is proposed. It is proved that the method is convergent uniformly in the perturbation parameter with an improved order of k + 1 in the new norm (k is the degree of the piecewise polynomial used in the LDG method).

MATHEMATICS SUBJECT CLASSIFICATION:

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The current research was partly supported by the NSFC (11771257) and Shandong Provincial NSF (ZR2021MA004).

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