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A Review of David Gottlieb’s Work on the Resolution of the Gibbs Phenomenon

Published online by Cambridge University Press:  20 August 2015

Sigal Gottlieb ,
Jae-Hun Jung  and
Saeja Kim
Sigal Gottlieb*
Affiliation:
Mathematics Department, University of Massachusetts Dartmouth, North Dartmouth, MA 02747, USA
Jae-Hun Jung*
Affiliation:
Mathematics Department, Suny Buffalo, Buffalo, New York 14260-2900, USA
Saeja Kim*
Affiliation:
Mathematics Department, University of Massachusetts Dartmouth, North Dartmouth, MA 02747, USA
*
Corresponding author.Email:sgottlieb@umassd.edu
Email:jaehun@buffalo.edu
Email:skim@umassd.edu
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Abstract

Given a piecewise smooth function, it is possible to construct a global expansion in some complete orthogonal basis, such as the Fourier basis. However, the local discontinuities of the function will destroy the convergence of global approximations, even in regions for which the underlying function is analytic. The global expansions are contaminated by the presence of a local discontinuity, and the result is that the partial sums are oscillatory and feature non-uniform convergence. This characteristic behavior is called the Gibbs phenomenon. However, David Gottlieb and Chi-Wang Shu showed that these slowly and non-uniformly convergent global approximations retain within them high order information which can be recovered with suitable postprocessing. In this paper we review the history of the Gibbs phenomenon and the story of its resolution.

Keywords

42A1041A1041A25Gibbs phenomenonpost-processingGalerkin approximationcollocation approximationspectral methodsexponential accuracy
Type
Review Article
Information
Communications in Computational Physics , Volume 9 , Issue 3 , March 2011 , pp. 497 - 519
Copyright
Copyright © Global Science Press Limited 2011

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