# Eigenvectors of discrete Laplace matrix for 2D unit square under Neumann boundary condition

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.

• It's still the Laplacian, but corrected for the nodes on the boundary. Or rather, a different numerical approach for the Laplacian. – Dylan Jun 15 '18 at 8:01