369
Views
37
CrossRef citations to date
0
Altmetric
Original Articles

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

Pages 925-952 | Published online: 16 Aug 2006
 

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.

Mathematics Subject Classification:

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.

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.