Analysis of hybrid discontinuous Galerkin methods for linearized Navier–Stokes equations

Abstract

In this article, we introduce and analyze arbitrary-order, locally conservative hybrid discontinuous Galkerin methods for linearized Navier–Stokes equations. The unknowns of the global system are reduced to trace variables on the skeleton of a triangulation and the average of pressure on each cell via embedded static condensation. We prove that the lifting operator associated with trace variables is injective for any polynomial degree. This generalizes the result in (Y. Jeon and E.-J. Park, Numerische Mathematik 123 [2013], no. 1, pp. 97–119), where quadratic and cubic rectangular elements are analyzed. Moreover, optimal error estimates in the energy norm are obtained by introducing nonstandard projection operators for the hybrid DG method. Several numerical results are presented to show the performance of the algorithm and to validate the theory developed in the article.