= Solution
For a sufficiently smooth function, <Taylor expansion> of the four neighboring values gives
$$
\Delta_hu=\Delta u+\frac{h^2}{12}(u_{xxxx}+u_{yyyy})+O(h^4).
$$
The reaction term $\lambda u$ is sampled exactly at the grid point. On an exact solution, therefore,
$$
\boxed{(\Delta_h+\lambda)u
=\frac{h^2}{12}(u_{xxxx}+u_{yyyy})+O(h^4).}
$$
The method has \b[second-order local accuracy for the <partial differential equation>]. The original printed equations multiply this expression by $h^2$, so their unscaled residual is $O(h^4)$; that scaling does not make the <Laplacian> approximation fourth order. Resonance and inverse-matrix bounds are separate issues from this local truncation calculation.
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