On the Discrete Poincaré–Friedrichs Inequalities for Nonconforming Approximations of the Sobolev Space H ¹

ABSTRACT

We present a direct proof of the discrete Poincaré–Friedrichs inequalities for a class of nonconforming approximations of the Sobolev space H ¹(Ω), indicate optimal values of the constants in these inequalities, and extend the discrete Friedrichs inequality onto domains only bounded in one direction. We consider a polygonal domain Ω in two or three space dimensions and its shape-regular simplicial triangulation. The nonconforming approximations of H ¹(Ω) consist of functions from H ¹ on each element such that the mean values of their traces on interelement boundaries coincide. The key idea is to extend the proof of the discrete Poincaré–Friedrichs inequalities for piecewise constant functions used in the finite volume method. The results have applications in the analysis of nonconforming numerical methods, such as nonconforming finite element or discontinuous Galerkin methods.

Keywords:

• Nonconforming approximation
• Poincaré–Friedrichs inequalities
• Sobolev space H ¹

Mathematics Subject Classification:

• 65N30
• 46E35

ACKNOWLEDGMENT

The author would like to thank his Ph.D. advisor Danielle Hilhorst from the University of Paris-Sud and Professor Robert Eymard from the University of Marne-la-Vallée for their valuable advice and hints.