Discrete elliptic barrier

For a monotone discrete elliptic operator, a nonnegative mesh function $\varphi$ with $L_h\varphi \ge d>0$ can bound errors through the discrete maximum principle. If $L_{h}e=\tau$ and the boundary error is zero, the subsolutions $\pm e+(\Vert \tau {\Vert }_{\infty }/d)\varphi$ yield $\Vert e{\Vert }_{\infty }\le (\Vert \tau {\Vert }_{\infty }/d){\max }_{\partial \Omega }\varphi$. On different subregions one may use different lower bounds for $L_h\varphi$ and different truncation-error bounds, taking the maximum of their ratios. Nonnegativity, or an explicit subtraction of the minimum when $L_{h}1=0$, is essential for the unshifted estimate.

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.