COPYRIGHT 2008, PRINCETON UNIVERSITY PRESS

442 chapter 17100V(x, y)0020x40020yFigure 17.2 The analytic (Fourier series) solution of Laplace’s equation showingGibbs-overshoot oscillations near x = 0. The solution shown here uses 21 terms, yet theoscillations remain even if a large number of terms is summed.17.3.1 Polynomial Expansion As an AlgorithmIt is worth pointing out that even though a product of separate functions of x and yis an acceptable form for a solution to Laplace’s equation, this does not mean thatthe solution to realistic problems will have this form. Indeed, a realistic solutioncan be expressed as an infinite sum of such products, but the sum is no longerseparable. Worse than that, as an algorithm, we must stop the sum at some point,yet the series converges so painfully slowly that many terms are needed, and soround-off error may become a problem. In addition, the sinh functions in (17.18)overflow for large n, which can be avoided somewhat by expressing the quotientof the two sinh functions in terms of exponentials and then taking a large n limit:sinh(nπy/L)sinh(nπ)= enπ(y/L−1) − e −nπ(y/L+1)1 − e −2nπ → e nπ(y/L−1) . (17.19)A third problem with the “analytic” solution is that a Fourier series convergesonly in the mean square (Figure 17.2). This means that it converges to the average ofthe left- and right-hand limits in the regions where the solution is discontinuous[Krey 98], such as in the corners of the box. Explicitly, what you see in Figure 17.2is a phenomenon known as the Gibbs overshoot that occurs when a Fourier serieswith a finite number of terms is used to represent a discontinuous function. Rather−101COPYRIGHT 2008, PRINCET O N UNIVE R S I T Y P R E S SEVALUATION COPY ONLY. NOT FOR USE IN COURSES.ALLpup_06.04 — 2008/2/15 — Page 442