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

DOI:
https://doi.org/10.1090/S0025-5718-01-01389-8

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 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 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

Email:
bakaev@math.unige.ch, bakaev@postman.ru

**Alexander Ostermann**

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

Email:
Alexander.Ostermann@math.unige.ch

DOI:
https://doi.org/10.1090/S0025-5718-01-01389-8

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