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Sharply localized pointwise and $ W_\infty^{-1}$ estimates for finite element methods for quasilinear problems


Author: Alan Demlow
Journal: Math. Comp. 76 (2007), 1725-1741
MSC (2000): Primary 65N30, 65N15
DOI: https://doi.org/10.1090/S0025-5718-07-01983-7
Published electronically: April 23, 2007
MathSciNet review: 2336265
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Abstract: We establish pointwise and $ W_\infty^{-1}$ estimates for finite element methods for a class of second-order quasilinear elliptic problems defined on domains $ \Omega$ in $ \mathbb{R}^n$. 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 $ e(x_1)-e(x_2)$ in pointwise errors to quasilinear problems. Finally, we establish estimates for the error in $ W_\infty^{-1}(D)$, where $ D \subset \Omega$ 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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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: https://doi.org/10.1090/S0025-5718-07-01983-7
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.

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