 Sixth-order source correction of the nine-point Poisson stencil

The nine-point finite-difference stencil for the Laplacian has expansion $D_{9}u=\Delta u+h^2{\Delta }^{2}u/12+h^4(u_{xxxxxx}+u_{yyyyyy}+5u_{xxxxyy}+5u_{xxyyyy})/360+O(h^6)$. If $\Delta u=f$, differentiating this equation gives $u_{xxxxyy}+u_{xxyyyy}=f_{xxyy}$ and $u_{xxxxxx}+u_{yyyyyy}={\Delta }^{2}f-3f_{xxyy}$. Hence the corrected compact equationhas sixth-order normalized consistency of a numerical method. Source derivatives can be supplied analytically; numerical approximation of $\Delta f$ must be fourth order and of the fourth derivatives second order. With exact square-grid Dirichlet boundary data and uniformly smooth solution, the discrete maximum principle inverse bound converts this residual into sixth-order nodal convergence. The unknown stencil stays compact even if more distant known source values are used.