EIGENMODES OF ISOSPECTRAL DRUMS

Recently it was proved that there exist nonisometric planar regions th at have identical Laplace spectra. That is, one cannot ''hear the shap e of a drum.'' The simplest isospectral regions known are bounded by p olygons with reentrant corners. While the isospectrality can be proven mathematically analytical techniques are unable to produce the eigenv alues themselves. Furthermore, standard numerical methods for computin g the eigenvalues, such as adaptive finite elements, are highly ineffi cient. Physical experiments have been performed to measure the spectra , but the accuracy and flexibility of this method are limited. We desc ribe an algorithm due to Descloux and Tolley [Comput. Methods Appl. Me ch. Engrg., 39 (1983), pp. 37-53] that blends singular finite elements with domain decomposition and show that, with a modification that dou bles its accuracy, this algorithm can be used to compute efficiently t he eigenvalues for polygonal regions. We present results accurate to 1 2 digits for the most famous pair of isospectral drums, as well as res ults for another pair.