Uniform convergence of the LDG method for singularly perturbed problems

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).

KEYWORDS:

• Singularly perturbed
• locally discontinuous Galerkin
• Bakhvalov-type
• supercloseness
• convergent uniformly

MATHEMATICS SUBJECT CLASSIFICATION:

• 03-08

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).