Computing Eigenfunctions on the Koch Snowflake: A New Grid and Symmetry

TL;DR

This paper introduces a new numerical method for computing eigenfunctions of the Laplacian on the Koch Snowflake fractal, improving accuracy and analyzing symmetry to understand eigenvalue multiplicities.

Contribution

It develops a novel grid and symmetry-based approach for solving the eigenvalue problem on fractals, with enhanced accuracy over previous methods.

Findings

Improved eigenvalue estimates for the Koch Snowflake Laplacian.

Identification of symmetry-related eigenvalue multiplicities.

Comparison showing better accuracy than prior computations.

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 $h \rightarrow 0$ 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.