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

TL;DR

This paper demonstrates that the HDG method on Shishkin meshes achieves near-optimal convergence rates for 2D singularly perturbed convection-diffusion problems, with theoretical analysis and numerical validation.

Contribution

It provides the first parameter-uniform convergence analysis of the HDG method on Shishkin meshes for 2D singularly perturbed problems, introducing a new error control technique.

Findings

Achieves supercloseness of almost $k+\frac{1}{2}$ order in energy norm.

Validates theoretical results with numerical experiments.

Addresses convection layer estimation challenges in singular perturbation problems.

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.