Quadratic-barrier error estimate for a discrete Laplacian ($\Vert e{\Vert }_{\infty }\le D^2\Vert \tau {\Vert }_{\infty }/4$)

For a normalized two-dimensional discrete Laplacian satisfying the discrete maximum principle, suppose quadratic functions are differentiated exactly and $L_{h}1=0$. In a domain of diameter $D$, choose $x_0$ in the domain and $\varphi (x)=|x-x_0{|}^2$. Then $0\le \varphi \le D^2$ and $L_h\varphi =4$. Applying the discrete elliptic barrier argument to zero-boundary error $L_{h}e=\tau$ 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.