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

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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 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.

Original languageEnglish (US)
Pages (from-to)126-142
Number of pages17
JournalJournal of Computational and Applied Mathematics
Volume191
Issue number1
DOIs
StatePublished - Jun 15 2006

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Keywords

  • Eigenvalue problem
  • Snowflake
  • Symmetry

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics

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