Journal of Computational Physics
Computer Music Journal
Shape recognition using eigenvalues of the Dirichlet Laplacian
Pattern Recognition
Eigenfrequencies of fractal drums
Journal of Computational and Applied Mathematics
Drums, curve descriptors and affine invariant region matching
Image and Vision Computing
Laplace-Beltrami spectra as 'Shape-DNA' of surfaces and solids
Computer-Aided Design
Computing eigenmodes ofelliptic operators using radial basis functions
Computers & Mathematics with Applications
Shape of a drum, a constructive approach
ICS'10 Proceedings of the 14th WSEAS international conference on Systems: part of the 14th WSEAS CSCC multiconference - Volume II
Shape of a drum, a constructive approach
WSEAS Transactions on Mathematics
Boundary Quasi-Orthogonality and Sharp Inclusion Bounds for Large Dirichlet Eigenvalues
SIAM Journal on Numerical Analysis
Singularities and treatments of elliptic boundary value problems
Mathematical and Computer Modelling: An International Journal
Eigenvalue estimates for saddle point matrices of Hermitian and indefinite leading blocks
Journal of Computational and Applied Mathematics
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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.