= High-frequency eigenvalue error of the discrete Laplacian
{title2=$N^{-2}(\lambda_{N-1,N-1}-\Lambda_{N-1,N-1})\to2\pi^2-8$}
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 $\sin z\leq z$ yields $\Lambda_{m,n}\leq\lambda_{m,n}$. 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>.
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