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: