Babuška-Osborn techniques in discontinuous Galerkin methods: $L^2$-norm error estimates for unstructured meshes
Provides theoretical guarantees for discontinuous Galerkin methods on general meshes, addressing a gap in error analysis for non-uniform grids.
The authors prove inf-sup stability of interior penalty discontinuous Galerkin schemes on unstructured meshes, yielding quasi-best approximations and $L^2$-norm error estimates independent of global maximum meshsize.
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.