Supercloseness of the HDG method on Shishkin mesh for a singularly perturbed convection diffusion problem in 2D

Xiaoqi Ma; Jin Zhang

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arXiv · 2406.18948

Supercloseness of the HDG method on Shishkin mesh for a singularly perturbed convection diffusion problem in 2D

Xiaoqi Ma·Jin Zhang

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Abstract

This paper presents the first analysis of parameter-uniform convergence for a hybridizable discontinuous Galerkin (HDG) method applied to a singularly perturbed convection-diffusion problem in 2D using a Shishkin mesh. The primary difficulty lies in accurately estimating the convection term in the layer, where existing methods often fall short. To address this, a novel error control technique is employed, along with reasonable assumptions regarding the stabilization function. The results show that, with polynomial degrees not exceeding $k$, the method achieves supercloseness of almost $k+\frac{1}{2}$ order in an energy norm. Numerical experiments confirm the theoretical accuracy and efficiency of the proposed method.

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Xiaoqi Ma, Jin Zhang. 2024-06-27. Supercloseness of the HDG method on Shishkin mesh for a singularly perturbed convection diffusion problem in 2D. https://arxiv.org/abs/2406.18948

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