  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 $H^1(K)$. 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.
