Fourth-order correction of the nine-point Poisson stencil

For a smooth solution of $\Delta u=f$, the corrected formula ${\Gamma }_{9}u=h^{2}f+(h^2/12){\Gamma }_{5}f$ has unscaled residual $O(h^6)$ and normalized local truncation error $O(h^4)$. The inverse interior matrix has Euclidean operator norm $O(h^{-2})$; on $O(h^{-2})$ grid points this gives unweighted Euclidean nodal error $O(h^3)$. Uniform derivatives up to the boundary are required for a grid-independent remainder bound.