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

  • Authors:

  • John M. Neuberger;Nándor Sieben;James W. Swift

  • Affiliations:

  • Department of Mathematics and Statistics, Northern Arizona University, Flagstaff, AZ;Department of Mathematics and Statistics, Northern Arizona University, Flagstaff, AZ;Department of Mathematics and Statistics, Northern Arizona University, Flagstaff, AZ

  • Venue:
  • Journal of Computational and Applied Mathematics
  • Year:
  • 2006

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Abstract

In this paper, we numerically solve the eigenvalue problem Au 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.