Eigenvectors of discrete Laplace matrix for 2D unit square with free boundary is simply $$ \phi(x,y)= \cos(\frac{\pi}{n} kx) \cos(\frac{\pi}{m} ly) $$ It is easy to see that its 2nd order derivative equals itself (scaled).
For example, the Laplace matrix of a 4 by 4 grid is
The numerically computed eigenvectors are consistent with the expression $\cos(\frac{\pi}{n} kx) \cos(\frac{\pi}{m} ly)$.
My question is what is the following matrix:
Precisely put, what is the differential equation on a continuous 2D unit square domain corresponds to this discrete operator?
The value 4 corresponds to the inner nodes, value 3 for the boundary nodes, value 2 for the corner nodes. Is this the Laplacian under Neumann boundary condiction?
Second question: what are the eigenvectors of such matrix? Of course one can compute them numerically, but is there an analytic expression for these eigenvectors? I find this related entry and this though they did not resolve my question.