Eigenmodes of isospectral drums

Recently it was proved that there exist nonisometric planar regions that have identical Laplace spectra. That is, one cannot ''hear the shape of a drum.'' The simplest isospectral regions known are bounded by polygons with reentrant corners. While the isospectrality can be proven mathematically analytical techniques are unable to produce the eigenvalues themselves. Furthermore, standard numerical methods for computing the eigenvalues, such as adaptive finite elements, are highly inefficient. Physical experiments have been performed to measure the spectra, but the accuracy and flexibility of this method are limited. We describe an algorithm due to Descloux and Tolley [Comput. Methods Appl. Mech. Engrg., 39 (1983), pp. 37-53] that blends singular finite elements with domain decomposition and show that, with a modification that doubles its accuracy, this algorithm can be used to compute efficiently the eigenvalues for polygonal regions. We present results accurate to 12 digits for the most famous pair of isospectral drums, as well as results for another pair.