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.

翻译：我们引入了一种新的稳定方法,用于对多边形模贝的Poisson问题采用不连续的Galerkin方法,这种方法可以产生多元近似度美元的最佳趋同率。在设定[S.Bertoluzza和D.Prada,一种多角不连续的Galerkin方法,一个稳定,ESAIM Math.Mod.Numer.Anal.(DOI:10.1051/m2an/2020059)]时,通过对每个网状元素($10.51/m2an/2020059)的残值($K)进行处罚,从而实现稳定。这一负面规范是通过引入新的辅助空间实现的代数法性规范。我们进行了一个耗资1美元的清晰的稳定性和误差分析,证明了整个方法的易乱性。理论结论在一系列数字实验中得到了证明。