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

Abstract. 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. In the setting of [S. Bertoluzza and D. Prada, A polygonal discontinuous Galerkin method with minus one stabilization, ESAIM Math. Mod. Numer. Anal. (DOI: 10.1051/m2an/2020059)], the stabilization is obtained by penalizing, in each mesh element K, a residual in the norm of the dual of HpKq. 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.

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