In theory, the spectrum is discrete because the Laplacian is elliptic and the domain is compact (bounded and closed), so the resolvant is compact hence the discretness of the spectrum. Now for computing these eigenvalue , we can use the separation of variables method to reduce the PDE equation to a pair of ordinary equations. The eigenvalues are computed using the boundary conditions.

In theory, the spectrum is discrete because the Laplacian is elliptic and the domain is compact (bounded and closed), so the resolvant is compact hence the discreteness of the spectrum. 

Now for computing these eigenvalues, we can use the separation of variables method to reduce the PDE equation to a pair of ordinary equations. The eigenvalues are computed using the boundary conditions.‌