A note on the discrete Aleksandrov-Bakelman maximum principle
The authors derive a discrete maximum-principle-type a priori estimate for difference operators of the form \[ Lu(x)= \sum_y a(x,y)u(y) \] acting on mesh functions \(u: E\to R\), where \(E\) is an arbitrary discrete set in \(\mathbb{R}^n\). The coefficient \(a(.,.)\) is defined on \(E\times E\) and vanishes except for finitely many \(y\), for each \(x\) value. If \(Lu+ f\geq 0\) in the interior \(D^0= \{x\in D\mid a(x,y)= 0 \forall y\neq D\}\) of a bounded set \(D\) in \(E\), then the a priori estimate \[ \max_D u\leq C\cdot \text{diam}(D) \Biggl\{\sum_{x\in D^0} {|f(x)|^n V(x)\over \text{det }A(x)}\Biggr\}^{1/n} \] is valid provided that \(L\) is positive and balanced in \(D^0\) and \(u\leq 0\) on \(D- D^0\), where \(A(x)= 0.5 \sum_y a(x,y)(y- x)\otimes (y- x)\) and \(V(x)\) denotes a local characteristic of the mesh.
discrete Aleksandrov-Bakelman maximum principle
second-order elliptic partial differential operators
scientific article; zbMATH DE number 1472436