Given that we have Laplace operator defined on some domain, is there a general approach to finding the eigenfunction(s) for that Laplacian?
kmcrane
@lucida: We didn't talk about algorithms for solving eigenvalue problems, but they are all basically the same, independent of what your Laplace operator looks like. In fact, eigenvector/eigenvalue algorithms typically work with any matrix $A$, independent of what it represents (i.e., it doesn't even have to be a Laplacian). The only (important!) special case is when we have a Laplacian on a regular grid, where we know the eigenfunctions ahead of time: just regularly sampled sines and cosines. This is one reason why the FFT can be made fast on regular grids, but is much more difficult on a general mesh.
Given that we have Laplace operator defined on some domain, is there a general approach to finding the eigenfunction(s) for that Laplacian?
@lucida: We didn't talk about algorithms for solving eigenvalue problems, but they are all basically the same, independent of what your Laplace operator looks like. In fact, eigenvector/eigenvalue algorithms typically work with any matrix $A$, independent of what it represents (i.e., it doesn't even have to be a Laplacian). The only (important!) special case is when we have a Laplacian on a regular grid, where we know the eigenfunctions ahead of time: just regularly sampled sines and cosines. This is one reason why the FFT can be made fast on regular grids, but is much more difficult on a general mesh.