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

Authors: Huo-Yuan Duan and Roger C. E. Tan
Journal: Math. Comp. 80 (2011), 119-140
MSC (2000): Primary 26D10, 46E35, 65M60, 65N30
Published electronically: July 8, 2010
MathSciNet review: 2728974
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Abstract: The purpose of this paper is to propose a proof for the Poincaré-Friedrichs inequality for piecewise $ H^1$ functions on anisotropic meshes. By verifying suitable assumptions involved in the newly proposed proof, we show that the Poincaré-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 Poincaré-Friedrichs inequality for the Crouzeix-Raviart nonconforming linear finite element. Counterexamples show that the maximum angle condition is only sufficient.

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Additional Information

Huo-Yuan Duan
Affiliation: School of Mathematical Sciences, Nankai University, 94 Weijin Street, Nankai District, Tianjin 300071, People’s Republic of China

Roger C. E. Tan
Affiliation: Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543

Keywords: Poincaré-Friedrichs inequality of piecewise $H^{1}$ function, discontinuous Galerkin finite element method, shape-regular condition, anisotropic mesh, Crouzeix-Raviart nonconforming linear element, the maximum angle condition
Received by editor(s): June 8, 2007
Received by editor(s) in revised form: August 12, 2008
Published electronically: July 8, 2010
Additional Notes: The authors were supported by the NUS academic research grant R-146-000-064-112.
Article copyright: © Copyright 2010 American Mathematical Society

American Mathematical Society