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We propose a Discontinuous Galerkin method for the Poisson equation onpolygonal tessellations in two dimensions, stabilized by penalizing, locally ineach element $K$, a residual term involving the fluxes, measured in the norm ofthe dual of $H^1(K)$. The scalar product corresponding to such a norm isnumerically realized via the introduction of a (minimal) auxiliary spaceinspired by the Virtual Element Method. Stability and optimal error estimatesin the broken $H^1$ norm are proven under a weak shape regularity assumptionallowing the presence of very small edges.  The results of numerical tests confirm the theoretical estimates.

A Polygonal Discontinuous Galerkin Method with Minus One Stabilization