ON THE POINCARE-FRIEDRICHS INEQUALITY FOR PIECEWISE H1 FUNCTIONS IN ANISOTROPIC DISCONTINUOUS GALERKIN FINITE ELEMENT METHODS

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.