Five-point versus nine-point Laplacian eigenvalue accuracy (${\lambda }_9={\lambda }_5+\frac{h^2}{6}{\lambda }_{T,k}{\lambda }_{T,l}$)

For the Dirichlet discrete Laplacian on a square, let $T=h^{-2}\mathrm{tridiag}(1,-2,1)$. The standard axial five-point operator is $T\otimes I+I\otimes T$, and the axial-weight $2/3$, corner-weight $1/6$ nine-point operator additionally contains $h^{2}T\otimes T/6$. Its eigenvalues are therefore strictly larger than the five-point values for the same sine modes. Both lie above the corresponding negative continuum Laplacian eigenvalues, so the five-point operator has smaller spectral error. For fixed mode numbers the leading errors are respectively ${\pi }^4{h}^2(k^4+l^4)/12$ and ${\pi }^4{h}^2(k^2+l^2{)}^2/12$.