High-frequency eigenvalue error of the discrete Laplacian ($N^{-2}({\lambda }_{N-1,N-1}-{\Lambda }_{N-1,N-1})\to 2{\pi }^2-8$)

For $-\Delta +I$ on the unit square with zero Dirichlet boundary conditions, the continuous eigenvalues are ${\lambda }_{m,n}=1+{\pi }^2(m^2+n^2)$, whereas the mesh-$1/N$ five-point Laplacian gives ${\Lambda }_{m,n}=1+4N^2[{\sin }^2(m\pi /(2N))+{\sin }^2(n\pi /(2N))]$. The inequality $\sin z\le z$ yields ${\Lambda }_{m,n}\le {\lambda }_{m,n}$. For fixed indices, Taylor's theorem gives an error of order $N^{-2}$; for indices $m=n=N-1$, division by $N^2$ gives the positive limit $2{\pi }^2-8$. Thus an error estimate for fixed smooth eigenfunctions need not be uniform over all grid-resolvable eigenfunctions.