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

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