In this paper, we introduce a new stabilization for discontinuous Galerkin methods for the Poisson problem on polygonal meshes, which induces optimal convergence rates in the polynomial approximation degree [Formula: see text]. The stabilization is obtained by penalizing, in each mesh element [Formula: see text], a residual in the norm of the dual of [Formula: see text]. This negative norm is algebraically realized via the introduction of new auxiliary spaces. We carry out a [Formula: see text]-explicit stability and error analysis, proving [Formula: see text]-robustness of the overall method. The theoretical findings are demonstrated in a series of numerical experiments.
Discontinuous Galerkin methods through the lens of variational multiscale analysis
