## Maximum-norm estimates for resolvents of elliptic finite element operators

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- by Nikolai Yu. Bakaev, Vidar Thomée and Lars B. Wahlbin;
- Math. Comp.
**72**(2003), 1597-1610 - DOI: https://doi.org/10.1090/S0025-5718-02-01488-6
- Published electronically: December 3, 2002
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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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## Bibliographic Information

**Nikolai Yu. Bakaev**- Affiliation: Department of Mathematics, Institute of Economics and Business, Berzarina St. 12, Moscow 123298, Russia
- Email: bakaev@postman.ru
**Vidar Thomée**- Affiliation: Department of Mathematics, Chalmers University of Technology, S-41296 Göteborg, Sweden
- MR Author ID: 172250
- Email: thomee@math.chalmers.se
**Lars B. Wahlbin**- Affiliation: Department of mathematics, Cornell University, Ithaca New York 14853
- Email: wahlbin@math.cornell.edu
- 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 - © Copyright 2002 American Mathematical Society
- Journal: Math. Comp.
**72**(2003), 1597-1610 - MSC (2000): Primary 65M12, 65M06, 65M60
- DOI: https://doi.org/10.1090/S0025-5718-02-01488-6
- MathSciNet review: 1986795