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Maximum-norm estimates for resolvents of elliptic finite element operators

Authors: Nikolai Yu. Bakaev, Vidar Thomée and Lars B. Wahlbin
Journal: Math. Comp. 72 (2003), 1597-1610
MSC (2000): Primary 65M12, 65M06, 65M60
Published electronically: December 3, 2002
MathSciNet review: 1986795
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Abstract: Let $\Omega$ be a convex domain with smooth boundary in $R^d$. It has been shown recently that the semigroup generated by the discrete Laplacian for quasi-uniform families of piecewise linear finite element spaces on $\Omega$ is analytic with respect to the maximum-norm, uniformly in the mesh-width. This implies a resolvent estimate of standard form in the maximum-norm outside some sector in the right halfplane, and conversely. Here we show directly that such a resolvent estimate holds outside any sector around the positive real axis, with arbitrarily small angle. This is useful in the study of fully discrete approximations based on $A(\theta)$-stable rational functions, with $\theta$ small.

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Additional Information

Nikolai Yu. Bakaev
Affiliation: Department of Mathematics, Institute of Economics and Business, Berzarina St. 12, Moscow 123298, Russia

Vidar Thomée
Affiliation: Department of Mathematics, Chalmers University of Technology, S-41296 Göteborg, Sweden

Lars B. Wahlbin
Affiliation: Department of mathematics, Cornell University, Ithaca New York 14853

Keywords: Resolvent estimates, maximum-norm, elliptic, parabolic, finite elements
Received by editor(s): September 7, 2001
Received by editor(s) in revised form: March 1, 2002
Published electronically: December 3, 2002
Additional Notes: The first author was partly supported by the Swiss National Science Foundation under Grant 20-56577.99
The second and third authors were partly supported by the U.S. National Science Foundation under Grant DMS 0071412
Article copyright: © Copyright 2002 American Mathematical Society

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