= Quadratic-barrier error estimate for a discrete Laplacian
{title2=$\|e\|_\infty\leq D^2\|\tau\|_\infty/4$}
For a normalized two-dimensional <discrete Laplacian> satisfying the <discrete maximum principle>, suppose quadratic functions are differentiated exactly and $L_h1=0$. In a domain of diameter $D$, choose $x_0$ in the domain and $\phi(x)=|x-x_0|^2$. Then $0\leq\phi\leq D^2$ and $L_h\phi=4$. Applying the <discrete elliptic barrier> argument to zero-boundary error $L_he=\tau$ gives the displayed bound. A mere sign pattern of the stencil does not determine this constant: rescaling the whole operator changes the error-to-residual ratio without changing the domain diameter.
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