A p-robust polygonal discontinuous Galerkin method with minus one stabilization

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 p. The stabilization is obtained by penalizing, in each mesh element K, a residual in the norm of the dual of H-1(K). This negative norm is algebraically realized via the introduction of new auxiliary spaces. We carry out a p-explicit stability and error analysis, proving p-robustness of the overall method. The theoretical findings are demonstrated in a series of numerical experiments.