Uniform Stability and Error Analysis for Some Discontinuous Galerkin Methods

TL;DR

This paper establishes uniform stability and error estimates for hybrid and weak Galerkin methods, demonstrating their convergence to conforming methods as stabilization parameters vary.

Contribution

It introduces new stability and convergence estimates for HDG and WG methods, linking their limits to conforming methods using Brezzi theory.

Findings

Stability and error estimates are uniform across stabilization parameters.

HDG converges to a primal conforming method as stabilization vanishes.

WG converges to a mixed conforming method in the limit.

Abstract

In this paper, we provide a number of new estimates on the stability and convergence of both hybrid discontinuous Galerkin (HDG) and weak Galerkin (WG) methods. By using the standard Brezzi theory on mixed methods, we carefully define appropriate norms for the various discretization variables and then establish that the stability and error estimates hold uniformly with respect to stabilization and discretization parameters. As a result, by taking appropriate limit of the stabilization parameters, we show that the HDG method converges to a primal conforming method and the WG method converge to a mixed conforming method.