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.

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.

On the unit square with exact Dirichlet boundary data, let $D_{9}U=B_{h}f$ and suppose the normalized residual on the exact smooth solution has supremum norm at most $\epsilon$. The error satisfies $D_{9}e=-\tau$ and vanishes on the boundary. The barrier $q=[x(1-x)+y(1-y)]/4$ has $D_{9}q=-1$, $q\ge 0$ on the boundary, and maximum $1/8$. Apply the discrete maximum principle to $e-\epsilon q$ and $-e-\epsilon q$, obtaining $|e|\le \epsilon q$. Thus $\Vert e{\Vert }_{\infty }\le \epsilon /8$, and fourth-order residual gives fourth-order supremum norm nodal convergence. The proof needs a uniform smoothness bound and a compatible boundary discretization.