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.

翻译：本文首次分析了在Shishkin网格上应用可杂交间断伽辽金（HDG）方法求解二维奇异摄动对流扩散问题的参数一致收敛性。主要困难在于准确估计边界层内的对流项，现有方法在此常存在不足。为解决该问题，本文采用了一种新颖的误差控制技术，并对稳定化函数提出了合理假设。结果表明，在多项式次数不超过$k$的情况下，该方法能在能量范数下实现近$k+\frac{1}{2}$阶的超逼近性。数值实验验证了所提方法的理论精度与计算效率。