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We prove the inf-sup stability of a discontinuous Galerkin scheme for secondorder elliptic operators in (unbalanced) mesh-dependent norms for quasi-uniformmeshes for all spatial dimensions. This results in a priori error bounds inthese norms. As an application we examine a problem with rough source termwhere the solution can not be characterised as a weak solution and showquasi-optimal error control.

Analysis of Discontinuous Galerkin Methods using Mesh-Dependent Norms and Applications to Problems with Rough Data