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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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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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  • 1. D.N. Arnold and F. Brezzi, Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates, RAIRO Mod. Math. Anal. Num. 19 (1985), pp. 7-32. MR 87g:65126
  • 2. S.C. Brenner and L.R. Scott, The mathematical theory of finite element methods, Springer Verlag 1994. MR 95f:65001
  • 3. P.G. Ciarlet, The finite element method for elliptic problems, North-Holland Publishing Company, 1978. MR 58:25001
  • 4. R.S. Falk and M.E. Morley, Equivalence of finite element methods for problems in elasticity, SIAM J. Num. Anal. 27 (1990), pp. 1486-1505. MR 91i:65177
  • 5. P. Grisvard, Elliptic problems on nonsmooth domains, Monographs and studies in mathematics vol. 24, Pitman Publishing Inc., 1985. MR 86m:35044
  • 6. P. Lascaux and P. Lesaint, Some nonconforming finite element for the plate bending problem, RAIRO Sér. Rouge Anal. Numér. R-1 (1975), pp. 9-53. MR 54:11941
  • 7. M. Wang, The necessity and sufficiency of the patch test for the convergence of nonconforming finite elements, Preprint, Dept. of Math., Penn. State University (1999).
  • 8. A.H. Schatz and L.B. Wahlbin. On the finite element for singularly perturbed reaction-diffusion problems in two and one space dimensions, Math.Comp. 40 (1983), pp. 47-89. MR 84c:65137
  • 9. Z. Shi, Error estimates of Morley element, Math. Numer. Sinica 12 (1990), pp. 113-118. MR 91i:65182

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Additional Information

Trygve K. Nilssen
Affiliation: Department of Mathematics, University of Bergen, Johannes Brunsgt. 12, 5007 Bergen, Norway

Xue-Cheng Tai
Affiliation: Department of Mathematics, University of Bergen, Johannes Brunsgt. 12, 5007 Bergen, Norway

Ragnar Winther
Affiliation: Department of Informatics and Department of Mathematics, University of Oslo, P.O. Box 1080 Blindern, 0316 Oslo, Norway

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

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