Computing eigenfunctions on the Koch Snowflake: A new grid and symmetry

In this paper, we numerically solve the eigenvalue problem Δ u + λ 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...

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Bibliographic Details
Published in:Journal of computational and applied mathematics 2006-06, Vol.191 (1), p.126-142
Main Authors: Neuberger, John M., Sieben, Nándor, Swift, James W.
Format: Article
Language:English
Subjects:
Eigenvalue problem
Exact sciences and technology
Group theory
Group theory and generalizations
Mathematical analysis
Mathematics
Numerical analysis
Numerical analysis. Scientific computation
Partial differential equations
Partial differential equations, boundary value problems
Sciences and techniques of general use
Snowflake
Symmetry
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Summary:In this paper, we numerically solve the eigenvalue problem Δ u + λ 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 and Griffith. We extrapolate the results for grid spacing h to the limit h → 0 in order to estimate eigenvalues of the Laplacian and compare our results to those of Lapidus 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.
ISSN:0377-0427
1879-1778
DOI:10.1016/j.cam.2005.03.075