= Maximum-norm convergence of the corrected nine-point Poisson scheme
{title2=$\|U-u\|_\infty\le\tfrac18\|\tau\|_\infty$}
On the unit square with exact <Dirichlet boundary data>, let $D_9U=B_hf$ and suppose the normalized residual on the exact smooth solution has <supremum norm> at most $\varepsilon$. The error satisfies $D_9e=-\tau$ and vanishes on the boundary. The barrier $q=[x(1-x)+y(1-y)]/4$ has $D_9q=-1$, $q\ge0$ on the boundary, and maximum $1/8$. Apply the <discrete maximum principle> to $e-\varepsilon q$ and $-e-\varepsilon q$, obtaining $|e|\le\varepsilon q$. Thus $\|e\|_\infty\le\varepsilon/8$, and fourth-order residual gives fourth-order <supremum norm> nodal convergence. The proof needs a uniform smoothness bound and a compatible boundary discretization.
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