A Polygonal Discontinuous Galerkin Method with Minus One Stabilization

TL;DR

This paper introduces a novel polygonal Discontinuous Galerkin method for the Poisson equation, utilizing a residual-based stabilization and virtual element inspired auxiliary space, with proven stability and optimal error bounds.

Contribution

It presents a new stabilized DG method on polygonal meshes using a residual flux penalty and a virtual element inspired auxiliary space, with theoretical stability and error analysis.

Findings

Numerical tests confirm theoretical error estimates.

Method achieves stability on meshes with small edges.

Optimal convergence rates are demonstrated.

Abstract

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