On the Poincaré-Friedrichs inequality for piecewise 𝐻¹ functions in anisotropic discontinuous Galerkin finite element methods

Duan, Huo-Yuan; Tan, Roger

doi:10.1090/S0025-5718-2010-02296-3

On the Poincaré-Friedrichs inequality for piecewise $H^1$ functions in anisotropic discontinuous Galerkin finite element methods
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by Huo-Yuan Duan and Roger C. E. Tan;

Math. Comp. 80 (2011), 119-140

DOI: https://doi.org/10.1090/S0025-5718-2010-02296-3

Published electronically: July 8, 2010

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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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