Babu\v{s}ka-Osborn techniques in discontinuous Galerkin methods: $L^2$-norm error estimates for unstructured meshes

TL;DR

This paper establishes the stability and error estimates of interior penalty discontinuous Galerkin methods on unstructured, geometrically graded meshes, extending Babuška-Osborn techniques to these schemes.

Contribution

It proves inf-sup stability and quasi-best $L^2$-norm error estimates for DG methods on general unstructured meshes, including geometrically graded ones.

Findings

Inf-sup stability of DG schemes on unstructured meshes.

Error estimates independent of global mesh size.

Numerical experiments confirming theoretical results.

Abstract

We prove the inf-sup stability of the interior penalty class of discontinuous Galerkin schemes in unbalanced mesh-dependent norms, under a mesh condition allowing for a general class of meshes, which includes many examples of geometrically graded element neighbourhoods. The inf-sup condition results in the stability of the interior penalty Ritz projection in $L^2$ as well as, for the first time, quasi-best approximations in the $L^2$-norm which in turn imply a priori error estimates that do not depend on the global maximum meshsize in that norm. Some numerical experiments are also given.