Discrete Poincar\'e and Trace Inequalities for the Hybridizable Discontinuous Galerkin Method

Abstract

In this paper, we derive discrete Poincar\'e and trace inequalities for the hybridizable discontinuous Galerkin (HDG) method. We employ the Crouzeix-Raviart space as a bridge, connecting classical discrete functional tools from Brenner's foundational work \cite{brenner2003poincare} with hybridizable finite element spaces comprised of piecewise polynomial functions defined both within the element interiors and on the mesh skeleton. This approach yields custom-tailored inequalities that underpin the stability analysis of HDG discretizations. The resulting framework is then used to demonstrate the well-posedness and robustness of HDG-based numerical schemes for second-order elliptic problems, even under minimal regularity assumptions on the source term and boundary data.