Energy norm error estimates for averaged discontinuous Galerkin methods: multidimensional case

TL;DR

This paper develops an analysis for averaged discontinuous Galerkin methods applied to elliptic problems, deriving error estimates in the energy norm without requiring extra smoothness assumptions.

Contribution

It introduces a natural framework for overpenalized interior penalty bilinear forms using local averages and distribution theory, avoiding heuristic flux and penalty choices.

Findings

Error estimates between local averages and exact solutions in $H^1$-seminorm

A conforming analysis framework without extra smoothness assumptions

Derivation of overpenalized IP bilinear forms from lower order perturbations

Abstract

A mathematical analysis is presented for a class of interior penalty (IP) discontinuous Galerkin approximations of elliptic boundary value problems. In the framework of the present theory one can derive some overpenalized IP bilinear forms in a natural way avoiding any heuristic choice of fluxes and penalty terms. The main idea is to start from bilinear forms for the local average of discontinuous approximations which are rewritten using the theory of distributions. It is pointed out that a class of overpenalized IP bilinear forms can be obtained using a lower order perturbation of these. Also, error estimations can be derived between the local averages of the discontinuous approximations and the analytic solution in the $H^1$-seminorm. Using the local averages, the analysis is performed in a conforming framework without any assumption on extra smoothness for the solution of the original boundary value problem.