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An observation on Korn's inequality for nonconforming finite element methods

Author(s): Kent-Andre Mardal; Ragnar Winther.
Journal: Math. Comp. 75 (2006), 1-6.
MSC (2000): Primary 65N30, 74S05
Posted: October 21, 2005
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Abstract: By utilizing a simple observation on traces of rigid motions we are able to strengthen a result of Brenner (2004) on Korn's inequality for nonconforming finite element methods. The approach here is tightly connected to the theory developed in Brenner's work. Our motivation for the analysis was the desire to show that a robust Darcy-Stokes element satisfies Korn's inequality, and to achieve this the stronger result seems necessary.

References:

1.: S. C. Brenner, Korn's inequalities for piecewise vector fields, Math. Comp. 73 (2004), 1067-1087. MR 2047078 (2005c:65096)
2.: M. Crouzeix and P. A. Raviart, Conforming and non-conforming finite element methods for solving the stationary Stokes equations, RAIRO Anal. Numér. 7 (1973), 33-76. MR 0343661 (49:8401)
3.: R. S. Falk, Nonconforming Finite Element Methods for the Equations of Linear Elasticity, Math. Comp. 57 (1991), 529-550. MR 1094947 (92a:65290)
4.: K.A. Mardal, X.-C. Tai, and R. Winther, A robust finite element method for Darcy-Stokes flow, SIAM J. Numer. Anal. 40 (2002), 1605-1631. MR 1950614 (2003m:76110)

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Additional Information:

Kent-Andre Mardal
Affiliation: Department of Scientific Computing, Simula Research Laboratory and Department of Informatics, University of Oslo, P.O. Box 134, 1325 Lysaker, Norway
Email: kent-and@simula.no

Ragnar Winther
Affiliation: Centre of Mathematics for Applications and Department of Informatics, University of Oslo, P.O. Box 1053, Blindern, 0316 Oslo, Norway
Email: ragnar.winther@cma.uio.no

DOI: 10.1090/S0025-5718-05-01783-7
PII: S 0025-5718(05)01783-7
Keywords: Korn's inequality, piecewise $H^1$ vector fields, nonconforming finite elements
Received by editor(s): October 11, 2004
Received by editor(s) in revised form: November 27, 2004
Posted: October 21, 2005
Copyright of article: Copyright 2005, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.


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