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Local energy estimates for the finite element method on sharply varying grids

Authors: Alan Demlow, Johnny Guzmán and Alfred H. Schatz
Journal: Math. Comp. 80 (2011), 1-9
MSC (2010): Primary 65N30, 65N15
Published electronically: June 28, 2010
MathSciNet review: 2728969
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Abstract: Local energy error estimates for the finite element method for elliptic problems were originally proved in 1974 by Nitsche and Schatz. These estimates show that the local energy error may be bounded by a local approximation term, plus a global ``pollution'' term that measures the influence of solution quality from outside the domain of interest and is heuristically of higher order. However, the original analysis of Nitsche and Schatz is restricted to quasi-uniform grids. We present local a priori energy estimates that are valid on shape regular grids, an assumption which allows for highly graded meshes and which matches much more closely the typical practical situation. Our chief technical innovation is an improved superapproximation result.

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Additional Information

Alan Demlow
Affiliation: Department of Mathematics, University of Kentucky, 715 Patterson Office Tower, Lexington, Kentucky 40506–0027

Johnny Guzmán
Affiliation: Division of Applied Mathematics, Brown University, 182 George Street, Providence, Rhode Island 02906

Alfred H. Schatz
Affiliation: Department of Mathematics, Malott Hall, Cornell University, Ithaca, New York 14853

Received by editor(s): August 14, 2008
Received by editor(s) in revised form: July 16, 2009
Published electronically: June 28, 2010
Additional Notes: The first author was partially supported by NSF grant DMS-0713770.
The second author was partially supported by NSF grant DMS-0503050.
The third author was partially supported by NSF grant DMS-0612599.
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.