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

 

Piecewise linear finite element methods are not localized


Author: Alan Demlow
Journal: Math. Comp. 73 (2004), 1195-1201
MSC (2000): Primary 65N30, 65N15
Published electronically: July 14, 2003
MathSciNet review: 2047084
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Abstract: Recent results of Schatz show that standard Galerkin finite element methods employing piecewise polynomial elements of degree two and higher to approximate solutions to elliptic boundary value problems are localized in the sense that the global dependence of pointwise errors is of higher order than the overall order of the error. These results do not indicate that such localization occurs when piecewise linear elements are used. We show via simple one-dimensional examples that Schatz's estimates are sharp in that localization indeed does not occur when piecewise linear elements are used.


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Additional Information

Alan Demlow
Affiliation: Department of Mathematics, Malott Hall, Cornell University, Ithaca, New York 14853
Email: ard11@cornell.edu

DOI: http://dx.doi.org/10.1090/S0025-5718-03-01584-9
PII: S 0025-5718(03)01584-9
Received by editor(s): July 22, 2002
Received by editor(s) in revised form: December 15, 2002
Published electronically: July 14, 2003
Additional Notes: This material is based upon work supported under a National Science Foundation graduate fellowship and under NSF grant DMS-0071412.
Article copyright: © Copyright 2003 American Mathematical Society