Computing Eigenvalues of Isospectral Drums

Toby Driscoll (Cornell)

Recently, Gordon, Webb, and Wolpert proved that one cannot ``hear the shape of a drum''---they constructed a pair of nonisometric planar regions that have identical Laplace spectra. Their discovery has been documented by artices in Science, Science News, and the Math Monthly. The simplest form of their example is as a pair of eight-sided nonconvex polygons. Since Gordon et al.'s work, many other isospectral pairs have been found, all in the form of nonconvex polygons.

While several elegant mathematical techniques can be used to prove the isospectrality of the drums, none can produce the spectrum itself. Standard numerical methods for determining the eigenvalues are in general inefficient because of the reentrant corners. However, there is an algorithm, originally due to Descloux and Tolley, that blends domain decomposition with finite elements. With a modification that doubles its accuracy, this method can be used to compute the eigenvalues of polygons efficiently and accurately. The first twenty-five eigenvalues of the Gordon, Webb, and Wolpert drums have been found to twelve digits, and eigenfunctions can be computed to similar accuracy. The algorithm clearly performs better than finite elements and other methods applied to this problem.

For more information, see the URL http://cam.cornell.edu/~driscoll/.