A spatially adaptive iterative method for a class of nonlinear operator eigenproblems

Jarlebring, Elias and Güttel, Stefan (2012) A spatially adaptive iterative method for a class of nonlinear operator eigenproblems. [MIMS Preprint]

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Abstract

We present a new algorithm for the iterative solution of nonlinear operator eigenvalue problems arising from partial differential equations. This algorithm combines automatic spatial resolution of linear operators with the infinite Arnoldi method for nonlinear matrix eigenproblems proposed in [E. JARLEBRING, W. MICHIELS, AND K. MEERBERGEN, A linear eigenvalue algorithm for the nonlinear eigenvalue problem, Numer. Math., 122 (2012)]. The iterates in this infinite Arnoldi method are functions, and each iteration requires the solution of an inhomogeneous differential equation. This formulation is independent of the spatial representation of the functions, which allows us to employ a dynamic representation with an accuracy of about the level of machine precision at each iteration, similar to what is done in the Chebfun system [Z. BATTLES AND L. N. TREFETHEN, An extension of MATLAB to continuous functions and operators, SIAM J. Sci. Comput., 25 (2004)] with its chebop functionality [T. A. DRISCOLL, F. BORNEMANN, AND L. N. TREFETHEN, The chebop system for automatic solution of differential equations, BIT, 48 (2008)], although our function representation is entirely based on coefficients instead of function values. Our approach also allows for nonlinearities in the boundary conditions of the PDE. The algorithm is illustrated with several examples, e.g., the study of eigenvalues of a vibrating string with delayed boundary feedback control.

Item Type: MIMS Preprint
Additional Information: Accepted for publication in Electronic Transactions on Numerical Analysis, 2013
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis
Depositing User: Stefan Güttel
Date Deposited: 05 Nov 2013
Last Modified: 08 Nov 2017 18:18
URI: http://eprints.maths.manchester.ac.uk/id/eprint/2040

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