On the Poincaré-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 PoincaréFriedrichs inequality for piecewise H1 functions on anisotropic meshes. By verifying suitable assumptions involved in the newly proposed proof, we show that the Poincaré-Friedrichs inequality for piecewise H1 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 Poincaré-Friedrichs inequality for the Crouzeix-Raviart nonconforming linear finite element. Counterexamples show that the maximum angle condition is only sufficient.