An analysis of HDG methods for convection dominated diffusion problems

TL;DR

This paper provides an a priori error analysis of HDG methods for convection-dominated diffusion problems, demonstrating convergence rates and stability properties, with numerical verification of theoretical findings.

Contribution

It establishes convergence rates for HDG methods with different stabilization parameters and analyzes their stability and implementation aspects.

Findings

L2 error converges at order k + 1/2 for non-aligned meshes

Optimal convergence rate achieved when meshes are aligned with flux

Spectral condition number is independent of diffusion coefficient

Abstract

In this paper, we establish an a priori error analysis of HDG methods with two types of stabilization parameter applied to convection dominated diffusion problem. We show that, using polynomials of degree no greater than k, L2 error of the scalar variable of HDG methods converges in order k + 1/2 if the meshes are not aligned with the flux. This result implies that the HDG approximation converges as fast as conventional DG methods. In the case that the meshes are aligned with the flux, we show that HDG methods with the second type of stabilization parameter achieves optimal rate of convergence. We also discuss preferred form for implementation of HDG methods. In appendix, we show that the spectral condition number of stiffness matrix of HDG methods is independent of the diffusion coefficient. Numerical experiments are presented to verify our theoretical results.