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Computing Eigenfunctions on the Koch Snowflake: A New Grid and Symmetry | | John M. Neuberger
; Nandor Sieben
; James W. Swift
; | | Date: |
5 Oct 2010 | | Abstract: | In this paper we numerically solve the eigenvalue problem $Delta u + lambda
u = 0$ 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 $h
ightarrow 0$ 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. | | Source: | arXiv, 1010.0775 | | Services: | Forum | Review | PDF | Favorites | |
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