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Computing eigenfunctions on the Koch snowflake: a new grid and symmetry. (English) Zbl 1094.35085

Summary: 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 and 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 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.

MSC:

35P10 Completeness of eigenfunctions and eigenfunction expansions in context of PDEs
31C20 Discrete potential theory
20C35 Applications of group representations to physics and other areas of science
35J25 Boundary value problems for second-order elliptic equations
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs

Software:

ARPACK
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Online Encyclopedia of Integer Sequences:

a(n) = 1 + (4*9^n - 9*4^n) / 5.

References:

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