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A robust nonconforming $H^2$-element


Authors: Trygve K. Nilssen, Xue-Cheng Tai and Ragnar Winther
Journal: Math. Comp. 70 (2001), 489-505
MSC (2000): Primary 65N12, 65N30
Published electronically: February 23, 2000
MathSciNet review: 1709156
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Abstract:

Finite element methods for some elliptic fourth order singular perturbation problems are discussed. We show that if such problems are discretized by the nonconforming Morley method, in a regime close to second order elliptic equations, then the error deteriorates. In fact, a counterexample is given to show that the Morley method diverges for the reduced second order equation. As an alternative to the Morley element we propose to use a nonconforming $H^2$-element which is $H^1$-conforming. We show that the new finite element method converges in the energy norm uniformly in the perturbation parameter.


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Additional Information

Trygve K. Nilssen
Affiliation: Department of Mathematics, University of Bergen, Johannes Brunsgt. 12, 5007 Bergen, Norway
Email: Trygve.Nilssen@mi.uib.no

Xue-Cheng Tai
Affiliation: Department of Mathematics, University of Bergen, Johannes Brunsgt. 12, 5007 Bergen, Norway
Email: Xue-Cheng.Tai@mi.uib.no

Ragnar Winther
Affiliation: Department of Informatics and Department of Mathematics, University of Oslo, P.O. Box 1080 Blindern, 0316 Oslo, Norway
Email: ragnar@ifi.uio.no

DOI: https://doi.org/10.1090/S0025-5718-00-01230-8
Keywords: singular perturbation problems, nonconforming finite elements, uniform error estimates.
Received by editor(s): March 9, 1999
Received by editor(s) in revised form: June 8, 1999
Published electronically: February 23, 2000
Additional Notes: This work was partially supported by the Research Council of Norway (NFR), under grant 128224/431, and by ELF Petroleum Norway AS
Article copyright: © Copyright 2000 American Mathematical Society