An analysis of HDG methods for convection-dominated diffusion problems

¹ School of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA.
fuxxx165@math.umn.edu
² Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong, P.R. China.
weifeqiu@cityu.edu.hk
³ Department of Mathematics, University of Maryland, College Park, MD 20742-4015, USA.
wujun@umd.edu

Received: 3 October 2013
Revised: 12 March 2014

Abstract

We present the first a priori error analysis of the h-version of the hybridizable discontinuous Galkerin (HDG) methods applied to convection-dominated diffusion problems. We show that, when using polynomials of degree no greater than k, the L²-error of the scalar variable converges with order k + 1 / 2 on general conforming quasi-uniform simplicial meshes, just as for conventional DG methods. We also show that the method achieves the optimal L²-convergence order of k + 1 on special meshes. Moreover, we discuss a new way of implementing the HDG methods for which the spectral condition number of the global matrix is independent of the diffusion coefficient. Numerical experiments are presented which verify our theoretical results.