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<p>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 <i>k</i>, the method achieves supercloseness of almost <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10915_2025_3056_Article_IEq1.gif" Format="GIF" Height="22" Rendition="HTML" Resolution="72" Type="Linedraw" Width="38" /> </InlineMediaObject> <EquationSource Format="TEX">\(k+\frac{1}{2}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>+</mo> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </math></EquationSource> </InlineEquation> order in an energy norm. Numerical experiments confirm the theoretical accuracy and efficiency of the proposed method.</p>

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Supercloseness of the HDG Method on Shishkin Mesh for a Singularly Perturbed Convection Diffusion Problem in 2D

  • 期刊论文

  • 发表日期：

    2025年9月22日

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  • Xiaoqi Ma,
  • Jin Zhang

摘要

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 \(k+\frac{1}{2}\) k + 1 2 order in an energy norm. Numerical experiments confirm the theoretical accuracy and efficiency of the proposed method.