Commun. Comput. Phys., 9 (2011), pp. 497-519.


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

Sigal Gottlieb 1*, Jae-Hun Jung 2, Saeja Kim 1

1 Mathematics Department, University of Massachusetts Dartmouth, North Dartmouth, MA 02747, USA.
2 Mathematics Department, Suny Buffalo, Buffalo, New York 14260-2900, USA.

Received 30 November 2009; Accepted (in revised version) 17 May 2010
Available online 17 September 2010
doi:10.4208/cicp.301109.170510s

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 post-processing. In this paper we review the history of the Gibbs phenomenon and the story of its resolution.

AMS subject classifications: 42A10, 41A10, 41A25

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Key words: Gibbs phenomenon, post-processing, Galerkin approximation, collocation approximation, spectral methods, exponential accuracy.

*Corresponding author.
Email: sgottlieb@umassd.edu (S. Gottlieb), jaehun@buffalo.edu (J.-H. Jung), skim@umassd.edu (S. Kim)
 

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