= Discrete elliptic barrier
For a monotone discrete elliptic operator, a nonnegative mesh function $\phi$ with $L_h\phi\geq d>0$ can bound errors through the <discrete maximum principle>. If $L_he=\tau$ and the boundary error is zero, the subsolutions $\pm e+(\|\tau\|_\infty/d)\phi$ yield $\|e\|_\infty\leq(\|\tau\|_\infty/d)\max_{\partial\Omega}\phi$. On different subregions one may use different lower bounds for $L_h\phi$ and different truncation-error bounds, taking the maximum of their ratios. Nonnegativity, or an explicit subtraction of the minimum when $L_h1=0$, is essential for the unshifted estimate.
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