Discrete maximum principle

A discrete elliptic operator satisfies a discrete maximum principle when a grid function with nonnegative discrete Laplacian cannot have a positive interior maximum unless it is constant. It yields uniqueness and max-norm stability estimates for Dirichlet finite-difference problems.

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.

A finite elliptic stencil has boundary-reaching connectivity when every unknown node has a directed path of strictly positive neighbor coefficients to a prescribed boundary node. For $L{\psi }_p={\sum }_{r\ne p}{c}_{p,r}{\psi }_r-c_{p,p}{\psi }_p$, with $c_{p,r}\ge 0$ and $c_{p,p}\ge {\sum }_{r\ne p}{c}_{p,r}$, a nonnegative interior maximum of a subsolution $L\psi \ge 0$ propagates along every positive-weight edge. The path condition carries it to the boundary, proving the discrete maximum principle. Connectedness of the underlying geometric domain alone does not imply this graph property.

For the five-point discretization $L_h$ of $-\Delta$ on the unit square, the displayed quadratic polynomial has $L_{h}B=1$ exactly, nonnegative boundary values and $0\le B\le 1/8$. If $L_{h}e=\tau$ and $e$ has zero boundary values, the discrete maximum principle applied to $\pm e-\Vert \tau {\Vert }_{\infty }B$ gives $|e|\le \Vert \tau {\Vert }_{\infty }B\le \Vert \tau {\Vert }_{\infty }/8$. The same estimate holds for $L_h+I$, since $(L_h+I)B=1+B\ge 1$. This turns a uniform local truncation error bound into a pointwise global error bound without inverting the full matrix.