Volume 41, pp. 21-41, 2014.

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

Elias Jarlebring and Stefan Güttel

Abstract

We present a new algorithm for the iterative solution of nonlinear operator eigenvalue problems arising from partial differential equations (PDEs). This algorithm combines automatic spatial resolution of linear operators with the infinite Arnoldi method for nonlinear matrix eigenproblems proposed by Jarlebring et al. [Numer. Math., 122 (2012), pp. 169–195]. 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 with its chebop functionality although our function representation is entirely based on coefficients instead of function values. Our approach also allows nonlinearity 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.

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Key words

Arnoldi's method, nonlinear eigenvalue problems, partial differential equations, Krylov subspaces, delay-differential equations, Chebyshev polynomials

AMS subject classifications

65F15, 65N35, 65N25

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