A robust nonconforming -element

Authors:
Trygve K. Nilssen, Xue-Cheng Tai and Ragnar Winther

Journal:
Math. Comp. **70** (2001), 489-505

MSC (2000):
Primary 65N12, 65N30

DOI:
https://doi.org/10.1090/S0025-5718-00-01230-8

Published electronically:
February 23, 2000

MathSciNet review:
1709156

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 -element which is -conforming. We show that the new finite element method converges in the energy norm uniformly in the perturbation parameter.

**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**

Retrieve articles in *Mathematics of Computation*
with MSC (2000):
65N12,
65N30

Retrieve articles in all journals with MSC (2000): 65N12, 65N30

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