Analysis of hybridized discontinuous Galerkin methods without elliptic regularity assumptions

TL;DR

This paper develops new stability and error analysis techniques for hybridized discontinuous Galerkin methods that do not rely on elliptic regularity assumptions, broadening their applicability to various PDEs.

Contribution

It introduces novel inf-sup conditions based on stabilized saddle point structures, enabling optimal error estimates without elliptic regularity assumptions.

Findings

Applicable to Poisson, convection-reaction-diffusion, Stokes, and Oseen equations

Provides stability and optimal error bounds without elliptic regularity

Extends the theoretical foundation of HDG methods

Abstract

In this paper we present new stability and optimal error analyses of hybridized discontinuous Galerkin (HDG) methods which do not require elliptic regularity assumptions. To obtain error estimates without elliptic regularity assumptions, we use new inf-sup conditions based on stabilized saddle point structures of HDG methods. We show that this approach can be applied to obtain optimal error estimates of HDG methods for the Poisson equations, the convection-reaction-diffusion equations, the Stokes equations, and the Oseen equations.