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For fixed mode numbers $k,l$, expansion of the sines giveswhereas the mixed Kronecker product correction givesThus both eigenvalue approximations are second order for fixed modes. The leading nine-point error includes an additional positive term ${\pi }^4{h}^2{k}^2{l}^2/6$.

There is also an exact comparison, not just an asymptotic one. Since $\sin x<x$ for $0<x<\pi /2$, and both sine factors in the correction are positive,Both estimates lie on the less-negative side of the continuum Laplacian eigenvalue, so the five-point formula is the better eigenvalue estimate for every corresponding interior mode. The five-point versus nine-point Laplacian eigenvalue accuracy comparison does not contradict the useful isotropy or special Poisson accuracy properties of a nine-point stencil. Those properties concern a different error criterion. The fixed-mode expansion is not uniform for modes with $k$ or $l$ comparable to $h^{-1}$; wavelengths close to the mesh scale need not have small relative spectral error.