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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Can a finite element method
perform arbitrarily badly?


Authors: Ivo Babuska and John E. Osborn
Journal: Math. Comp. 69 (2000), 443-462
MSC (1991): Primary :, 65N15, 65N30
Published electronically: February 24, 1999
MathSciNet review: 1648351
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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we construct elliptic boundary value problems whose standard finite element approximations converge arbitrarily slowly in the energy norm, and show that adaptive procedures cannot improve this slow convergence. We also show that the $L_{2}$-norm and the nodal point errors converge arbitrarily slowly. With the $L_{2}$-norm two cases need to be distinguished, and the usual duality principle does not characterize the error completely. The constructed elliptic problems are one dimensional.


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Additional Information

Ivo Babuska
Affiliation: Texas Institute for Computational and Applied Mathematics, University of Texas at Austin, Austin, TX 78712

John E. Osborn
Affiliation: Department of Mathematics, University of Maryland, College Park, MD 20742
Email: jeo@math.umd.edu

DOI: http://dx.doi.org/10.1090/S0025-5718-99-01085-6
PII: S 0025-5718(99)01085-6
Keywords: Finite element methods, convergence, adaptivity, rough coefficients
Received by editor(s): May 5, 1998
Published electronically: February 24, 1999
Additional Notes: The first author was supported in part by NSF Grant #DMS-95-01841.
Article copyright: © Copyright 2000 American Mathematical Society