Computing Eigenfunctions on the Koch Snowflake: A New Grid and Symmetry
This work provides more accurate numerical eigenvalues for the Laplacian on a fractal domain, which is of interest to researchers studying spectral geometry on fractals.
The authors numerically solve the eigenvalue problem for the Laplacian on the Koch Snowflake with Dirichlet and Neumann boundary conditions, achieving improved accuracy over prior work by Lapidus et al. through a new boundary condition imposition and grid refinement.
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 & 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.