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

TL;DR

This paper presents a new p-robust stabilization technique for polygonal discontinuous Galerkin methods applied to the Poisson problem, achieving optimal convergence rates through residual penalization in dual norms.

Contribution

It introduces a novel stabilization method using residual penalization in dual norms, ensuring p-robustness and optimal convergence in polygonal DG methods.

Findings

Achieves p-robust stability and convergence

Demonstrates optimal convergence rates numerically

Introduces auxiliary spaces for negative norm realization

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 $H^1(K)$. 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.