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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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Maximum-norm estimates for resolvents of elliptic finite element operators
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by Nikolai Yu. Bakaev, Vidar Thomée and Lars B. Wahlbin PDF
Math. Comp. 72 (2003), 1597-1610 Request permission

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
  • 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