Sharply localized pointwise and estimates for finite element methods for quasilinear problems
Author:
Alan Demlow
Journal:
Math. Comp. 76 (2007), 17251741
MSC (2000):
Primary 65N30, 65N15
Published electronically:
April 23, 2007
MathSciNet review:
2336265
Fulltext PDF Free Access
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Abstract: We establish pointwise and estimates for finite element methods for a class of secondorder quasilinear elliptic problems defined on domains in . These estimates are localized in that they indicate that the pointwise dependence of the error on global norms of the solution is of higher order. Our pointwise estimates are similar to and rely on results and analysis techniques of Schatz for linear problems. We also extend estimates of Schatz and Wahlbin for pointwise differences in pointwise errors to quasilinear problems. Finally, we establish estimates for the error in , where is a subdomain. These negative norm estimates are novel for linear as well as for nonlinear problems. Our analysis heavily exploits the fact that Galerkin error relationships for quasilinear problems may be viewed as perturbed linear error relationships, thus allowing easy application of properly formulated results for linear problems.
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 J. P. KRASOVSKII, Properties of green's functions and generalized solutions of elliptic boundary value problems, Soviet Math. Dokl., 10 (1969), pp. 5458.
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 D. LEYKEKHMAN, Pointwise localized error estimates for parabolic finite element equations, Numer. Math., 96 (2004), pp. 583600. MR 2028727 (2004k:65175)
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 A. H. SCHATZ, Pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids. I. Global estimates, Math. Comp., 67 (1998), pp. 877899. MR 1464148 (98j:65082)
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 , Asymptotically exact a posteriori estimators for the pointwise gradient error on each element in irregular meshes. II. The piecewise linear case, Math. Comp., 73 (2004), pp. 517523 (electronic). MR 2028417 (2004i:65127)
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Additional Information
Alan Demlow
Affiliation:
Department of Mathematics, University of Kentucky, 715 Patterson Office Tower, Lexington, Kentucky 40506–0027
Email:
demlow@ms.uky.edu
DOI:
http://dx.doi.org/10.1090/S0025571807019837
PII:
S 00255718(07)019837
Keywords:
Finite element methods,
quasilinear elliptic problems,
local error analysis,
pointwise error analysis
Received by editor(s):
November 7, 2005
Received by editor(s) in revised form:
July 4, 2006
Published electronically:
April 23, 2007
Additional Notes:
This material is based upon work supported under a National Science Foundation postdoctoral research fellowship and by the Deutsche Forschungsgemeinschaft.
Article copyright:
© Copyright 2007
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
