We can also find eigenvalues and eigenvectors of the laplacian
using the finite element method. Essentially, we solve the generalized
eigenvalue equation Eu=(lambda)Gu, where G is the Grammian matrix of inner
products. This yields a spectrum of eigenvalues, as well as the corresponding
eigenvectors. As we use larger and larger matrices (corresponding to the better
and better spline approximations), we approach the complete spectrum of the
laplacian. Also, the approximate eigenvalues converge to the actual eigenvalue.
There is also an issue of multiplicity of eigenvalues, due to symmetry of the
gasket. There are many patterns in the multiplicities; these have been well
understood by others.
Here is a graph of the eigenfunction with eigenvalue 240.1707: