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

Xiaoqi Ma, Jin Zhang

TL;DR

This work addresses parameter-uniform convergence for a HDG discretization of a 2D singularly perturbed convection–diffusion problem on a Shishkin mesh. By combining a solution decomposition with a novel error-control technique for layer convection terms and a carefully chosen stabilization, it proves almost $k+\tfrac{1}{2}$-order supercloseness in the energy norm for polynomial degree $k$, along with projection-error bounds. Theoretical results are corroborated by numerical experiments across a range of $\varepsilon$ and mesh sizes, demonstrating the method’s accuracy and efficiency in convection-dominated regimes. Overall, the paper provides a rigorous parametric-robust foundation for HDG on layer-adapted meshes in 2D 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.

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

TL;DR

This work addresses parameter-uniform convergence for a HDG discretization of a 2D singularly perturbed convection–diffusion problem on a Shishkin mesh. By combining a solution decomposition with a novel error-control technique for layer convection terms and a carefully chosen stabilization, it proves almost -order supercloseness in the energy norm for polynomial degree , along with projection-error bounds. Theoretical results are corroborated by numerical experiments across a range of and mesh sizes, demonstrating the method’s accuracy and efficiency in convection-dominated regimes. Overall, the paper provides a rigorous parametric-robust foundation for HDG on layer-adapted meshes in 2D 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 , the method achieves supercloseness of almost order in an energy norm. Numerical experiments confirm the theoretical accuracy and efficiency of the proposed method.

Paper Structure

This paper contains 10 sections, 6 theorems, 59 equations, 4 tables.

Table of Contents

  1. Introduction
  2. A priori information, Shishkin mesh and HDG method
  3. A priori information
  4. Shishkin mesh
  5. The HDG method
  6. Projection and projection errors
  7. Projection
  8. Projection errors
  9. Supercloseness
  10. Numerical experiment

Key Result

Theorem 1

We decompose the exact solution $u$ of problem eq:S-1 into $u = S+E_{1}+E_{2}+E_{3}$, where for all $(x, y)\in \Omega$ the component functions have the regularity for $0\le i+j\le k+2$. Here constant C depends on $\bm{\beta}$, $c$ and $f$.

Theorems & Definitions (11)

  • Theorem 1
  • proof
  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • Lemma 2
  • proof
  • Theorem 3
  • Theorem 4
  • ...and 1 more