Analysis of a family of HDG methods for second order elliptic problems

TL;DR

This paper analyzes a family of HDG methods for second order elliptic problems, providing error estimates and a flux postprocessing technique, with numerical validation in 2D.

Contribution

It introduces a new error analysis for HDG methods with minimal regularity assumptions and a flux postprocessing to improve conservation.

Findings

Error estimates for flux and potential are established.

A local flux postprocessing improves conservation properties.

Numerical experiments confirm theoretical results.

Abstract

In this paper, we analyze a family of hybridizable discontinuous Galerkin (HDG) methods for second order elliptic problems in two and three dimensions. The methods use piecewise polynomials of degree $k\geqslant 0$ for both the flux and numerical trace, and piecewise polynomials of degree $k+1 $ for the potential. We establish error estimates for the numerical flux and potential under the minimal regularity condition. Moreover, we construct a local postprocessing for the flux, which produces a numerical flux with better conservation. Numerical experiments in two-space dimensions confirm our theoretical results.