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Long-term stability of variable stepsize approximations of semigroups

Authors: Nikolai Bakaev and Alexander Ostermann
Journal: Math. Comp. 71 (2002), 1545-1567
MSC (2000): Primary 65M12, 65L20
Published electronically: August 3, 2001
MathSciNet review: 1933044
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Abstract: This paper is concerned with the stability of rational one-step approximations of $C_0$ semigroups. Particular emphasis is laid on long-term stability bounds. The analysis is based on a general Banach space framework and allows variable stepsize sequences. Under reasonable assumptions on the stepsize sequence, asymptotic stability bounds for general $C_0$ semigroups are derived. The bounds are typical in the sense that they contain, in general, a factor that grows with the number of steps. Under additional hypotheses on the approximation, more favorable stability bounds are obtained for the subclass of holomorphic semigroups.

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Additional Information

Nikolai Bakaev
Affiliation: Department of Mathematics, Air Force Technical University, Planetnaya 3, Moscow 125190, Russia

Alexander Ostermann
Affiliation: Section de mathématiques, Université de Genève, C.P. 240, CH-1211 Genève 24, Switzerland

Received by editor(s): July 10, 2000
Received by editor(s) in revised form: December 26, 2000
Published electronically: August 3, 2001
Additional Notes: The work of the first author was supported by the Swiss National Science Foundation under Grant 20-56577.99.
The second author was on leave from Universität Innsbruck, Institut für Technische Mathematik, Geometrie und Bauinformatik, Technikerstraße 13, A-6020 Innsbruck, Austria
Article copyright: © Copyright 2001 American Mathematical Society