TY - JOUR

T1 - The computation of the spectra of highly oscillatory fredholm integral operators

AU - BRUNNER, Hermann

AU - Iserles, Arieh

AU - Nørsett, Syvert P.

N1 - Copyright:
Copyright 2012 Elsevier B.V., All rights reserved.

PY - 2011/12

Y1 - 2011/12

N2 - We are concerned with the computation of spectra of highly oscillatory Fredholm problems, in particular with the Fox-Li operator whereω>> 1. Our main tool is the finite section method: an eigenfunction is expanded in an orthonormal basis of the underlying space, resulting in an algebraic eigenvalue problem. We consider two competing bases: a basis of Legendre polynomials and a basis consisting of modified Fourier functions (cosines and shifted sines), and derive detailed asymptotic estimates of the rate of decay of the coefficients. Although the Legendre basis enjoys in principle much faster convergence, this does not lead to much smaller matrices. Since the computation of Legendre coefficients is expensive, while modified Fourier coefficients can be computed efficiently with FFT, we deduce that modified Fourier expansions, implemented in a manner that takes advantage of their structure, present a considerably more effective tool for the computation of highly oscillatory Fredholm spectra.

AB - We are concerned with the computation of spectra of highly oscillatory Fredholm problems, in particular with the Fox-Li operator whereω>> 1. Our main tool is the finite section method: an eigenfunction is expanded in an orthonormal basis of the underlying space, resulting in an algebraic eigenvalue problem. We consider two competing bases: a basis of Legendre polynomials and a basis consisting of modified Fourier functions (cosines and shifted sines), and derive detailed asymptotic estimates of the rate of decay of the coefficients. Although the Legendre basis enjoys in principle much faster convergence, this does not lead to much smaller matrices. Since the computation of Legendre coefficients is expensive, while modified Fourier coefficients can be computed efficiently with FFT, we deduce that modified Fourier expansions, implemented in a manner that takes advantage of their structure, present a considerably more effective tool for the computation of highly oscillatory Fredholm spectra.

UR - http://www.scopus.com/inward/record.url?scp=84865305869&partnerID=8YFLogxK

U2 - 10.1216/JIE-2011-23-4-467

DO - 10.1216/JIE-2011-23-4-467

M3 - Article

AN - SCOPUS:84865305869

SN - 0897-3962

VL - 23

SP - 467

EP - 519

JO - Journal of Integral Equations and Applications

JF - Journal of Integral Equations and Applications

IS - 4

ER -