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arXiv · 1010.0775

Computing Eigenfunctions on the Koch Snowflake: A New Grid and Symmetry

John M. Neuberger·Nandor Sieben·James W. Swift
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Abstract

In this paper we numerically solve the eigenvalue problem $Δu + λu = 0$ on the fractal region defined by the Koch Snowflake, with zero-Dirichlet or zero-Neumann boundary conditions. The Laplacian with boundary conditions is approximated by a large symmetric matrix. The eigenvalues and eigenvectors of this matrix are computed by ARPACK. We impose the boundary conditions in a way that gives improved accuracy over the previous computations of Lapidus, Neuberger, Renka & Griffith. We extrapolate the results for grid spacing $h$ to the limit $h \rightarrow 0$ in order to estimate eigenvalues of the Laplacian and compare our results to those of Lapdus et al. We analyze the symmetry of the region to explain the multiplicity-two eigenvalues, and present a canonical choice of the two eigenfunctions that span each two-dimensional eigenspace.

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John M. Neuberger, Nandor Sieben, James W. Swift. 2010-10-05. Computing Eigenfunctions on the Koch Snowflake: A New Grid and Symmetry. https://arxiv.org/abs/1010.0775

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