Eigenfrequencies of fractal drums

Lehel Banjai

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Submission date: 16. Feb. 2005 (revised version: March 2005)
Pages: 20
published in: Journal of computational and applied mathematics, 198 (2007) 1, p. 1-18 
DOI number (of the published article): 10.1016/j.cam.2005.11.015
Bibtex
MSC-Numbers: 65N25, 65N35, 35P99, 30C20
Keywords and phrases: fractals, eigenvalues, spectral methods, conformal transplantation
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Abstract:
A method for the computation of eigenfrequencies and eigenmodes of fractal drums is presented. The approach involves first mapping the unit disk to a polygon approximating the fractal and then solving a weighted eigenvalue problem on the unit disk by a spectral collocation method. The numerical computation of the complicated conformal mapping was made feasible by the use of the fast multipole method as described in [1]. The linear system arising from the spectral discretization is large and dense. To circumvent this problem we devise a fast method for the inversion of such a system. Consequently the eigenvalue problem is solved iteratively. We obtain 8 digits for the first eigenvalue of the Koch snowflake and at least 5 digits for eigenvalues up to the 20th. Numerical results for two more fractals are shown.

[1] L. Banjai and L. N. Trefethen. A multipole method for Schwarz-Christoffel mapping of polygons with thousands of sides. SIAM J. Sci. Comput., 25(3):1042-1065, 2003.