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

 

Best approximation property in the $ W^1_{\infty}$ norm for finite element methods on graded meshes


Authors: A. Demlow, D. Leykekhman, A. H. Schatz and L. B. Wahlbin
Journal: Math. Comp. 81 (2012), 743-764
MSC (2010): Primary 65N30, 65N15
Published electronically: September 29, 2011
MathSciNet review: 2869035
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Abstract: We consider finite element methods for a model second-order elliptic equation on a general bounded convex polygonal or polyhedral domain. Our first main goal is to extend the best approximation property of the error in the $ W^1_{\infty }$ norm, which is known to hold on quasi-uniform meshes, to more general graded meshes. We accomplish it by a novel proof technique. This result holds under a condition on the grid which is mildly more restrictive than the shape regularity condition typically enforced in adaptive codes. The second main contribution of this work is a discussion of the properties of and relationships between similar mesh restrictions that have appeared in the literature.


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Additional Information

A. Demlow
Affiliation: Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506
Email: alan.demlow@uky.edu.

D. Leykekhman
Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269
Email: leykekhman@math.uconn.edu

A. H. Schatz
Affiliation: Department of Mathematics, Cornell University, Ithaca, New York 14853.
Email: schatz@math.cornell.edu

L. B. Wahlbin
Affiliation: Department of Mathematics, Cornell University, Ithaca, New York 14853.
Email: wahlbin@math.cornell.edu

DOI: http://dx.doi.org/10.1090/S0025-5718-2011-02546-9
PII: S 0025-5718(2011)02546-9
Keywords: Maximum norm, finite element, optimal error estimates
Received by editor(s): March 2, 2010
Received by editor(s) in revised form: October 27, 2010
Published electronically: September 29, 2011
Additional Notes: The first author was partially supported by NSF grant DMS-0713770.
The second author was partially supported by NSF grant DMS-0811167.
The third author was partially supported by NSF grant DMS-0612599.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.