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On the discrete approximation of eigenvalue problems with holomorphic parameter dependence†

Published online by Cambridge University Press:  14 February 2012

Hansgeorg Jeggle  and
Wolfgang Wendland
Hansgeorg Jeggle
Affiliation:
Fachbereich Mathematik der Technischen Universität Berlin
Wolfgang Wendland
Affiliation:
Fachbereich Mathematik der Technischen Hochschule Darmstadt
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Here eigenvalue problems A(λ)u = 0 and their approximations Ai(λ)νi = 0 are studied where the densely denned closed semi-Fredholm operators A and Ai depend holomorphically on the parameter λ. Two different kinds of approximations are established. One is based on a generalisation of the spectral projection and the other on a suitable linearisation of the problem. To this end generalised eigenvectors and suitable product spaces are introduced which provide a representation formula for the principal part of A -1 and A -1, respectively, in the neighbourhood of poles. The convergence of the methods is shown in the framework of discrete convergence theory. The results generalise the corresponding results for linearly dependent A, Ai in two directions: both methods are available for arbitrary holomorphic dependence on the parameter λ, and the first method provides convergent approximations to the whole generalised eigenspace which works also in cases of linear parameter dependence when the usual method fails.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 78 , Issue 1-2 , 1977 , pp. 1 - 29
Copyright
Copyright © Royal Society of Edinburgh 1977

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