Image: Eigenmodes of GWW Isospectral Domains

Description: A plot of the eigenmode amplitudes of the Laplace operator (with Dirichlet boundary conditions) on two domains proposed by Gordon, Webb and Wolpert. Cyan areas have positive amplitude, Yellow areas have negative amplitude, and black areas have zero amplitude. Note the eigenvalues are all equal, while the domains are not rigid-transformation equivalent. These were all calculated using a home-made Finite-Element solver in Python, which first makes a triangulation of the domains using a wrapper of the Triangle library, and then solves a discretized diagonalization problem based on the variational formulation of the Laplacian diagonalization problem, in the subspace of continuous piecewise linear functions (where the "pieces"\domains where the function is linear are simply the aforementioned triangles). The diagonalization itself uses Scipy's sparse implicitly restarted Lanczos method, and the array manipulations were done using numpy.
Title: Eigenmodes of GWW Isospectral Domains
Credit: Own work
Author: Rotem Shalev
Usage Terms: Creative Commons Attribution 4.0
License: CC BY 4.0
License Link: https://creativecommons.org/licenses/by/4.0
Attribution Required?: Yes