= Five-point versus nine-point Laplacian eigenvalue accuracy
{title2=$\lambda_9=\lambda_5+\tfrac{h^2}{6}\lambda_{T,k}\lambda_{T,l}$}
For the <Dirichlet discrete Laplacian> on a square, let $T=h^{-2}\operatorname{tridiag}(1,-2,1)$. 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 $h^2T\otimes T/6$. 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 $\pi^4h^2(k^4+l^4)/12$ and $\pi^4h^2(k^2+l^2)^2/12$.
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