Fast computation of drum modes using the spectrum of the Neumann-to-Dirichlet map

Alex Barnett

Mathematics, Dartmouth College.

We present and analyze a new method for numerical computation of the spectrum and eigenfunctions of a planar star-shaped domain with Dirichlet boundary condition. The method is 'fast' since it is computes a cluster of eigenfunctions (numbering of order the square-root of the eigenvalue) in the time usually taken to compute a single one. In practice, with 400 wavelengths across the domain, and relative error 1e-10, this speed-up is around 1e3. It is related to the little-understood 'scaling method', but, in constrast, has a rigorous error analysis and allows higher-order accuracy. We will include some applications to quantum chaos.

Joint work with Andrew Hassell (ANU).

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