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

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11 Citations (Scopus)

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

Fingerprint

Eigenvalues and eigenfunctions
Eigenfunctions
Boundary conditions
Eigenvalue Estimates
Grid
Symmetry
Extrapolate
Computing
Eigenspace
Eigenvalues and Eigenvectors
Zero
Neumann Boundary Conditions
Symmetric matrix
Dirichlet
Eigenvalue Problem
Spacing
Fractal
Multiplicity
Eigenvalue
Fractals

Keywords

  • Eigenvalue problem
  • Snowflake
  • Symmetry

ASJC Scopus subject areas

  • Applied Mathematics
  • Computational Mathematics
  • Numerical Analysis

Cite this

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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.",
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T2 - A new grid and symmetry

AU - Neuberger, John M

AU - Sieben, Nandor

AU - Swift, James W

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N2 - 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.

AB - 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.

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