Abstract
Using techniques from numerical analysis, we have approximated — jointly with J. W. Neuberger and R. J. Renka — the smallest 50 eigenvalues, along with their associated eigenfunctions, of the Dirichlet Laplacian on the Koch snowflake domain. Physically, these correspond to the frequencies, and the associated normal modes, of the Koch snowflake drum. Several graphical representations of these eigenfunctions, as well as the magnitudes of their gradients, have been produced using Cray supercomputers and Silicon Graphics machines. We briefly describe here the numerical methods used in the computations and present pictures of a selected set of the eigenfunctions. We also compare the graphical images of the boundary behavior (of the magnitude) of the gradient with the known mathematical results (of Lapidus and Pang). This work is partly motivated by the physical experiments of Sapoval et al. related to the formation of fractal structures.
Research partially supported by the National Science Foundation under grant DMS-9207098 as well as by a grant from the University of California at San Diego’s Supercomputer Center.
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Griffith, C.A., Lapidus, M.L. (1997). Computer Graphics and the Eigenfunctions for the Koch Snowflake Drum. In: Andersson, S.I., Lapidus, M.L. (eds) Progress in Inverse Spectral Geometry. Trends in Mathematics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8938-4_7
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DOI: https://doi.org/10.1007/978-3-0348-8938-4_7
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