On the Poincaré-Friedrichs inequality for piecewise $H^\{1\}$ functions in anisotropic discontinuous Galerkin finite element methods

Abstract

The purpose of this paper is to propose a proof for the Poincare-Friedrichs inequality for piecewise H 1 functions on anisotropic meshes. By verifying suitable assumptions involved in the newly proposed proof, we show that the Poincare-Friedrichs inequality for piecewise H 1 functions holds independently of the aspect ratio which characterizes the shape-regular condition in finite element analysis. In addition, under the maximum angle condition, we establish the Poincare-Friedrichs inequality for the Crouzeix-Raviart non-conforming linear finite element. Counterexamples show that the maximum angle condition is only sufficient.