A note on the discrete Aleksandrov-Bakelman maximum principle
type: Item
instance of: scholarly article
title: A note on the discrete Aleksandrov-Bakelman maximum principle
zbMATH Open document ID: 0963.65107
author: Hung-Ju Kuo
author: Neil S. Trudinger
published in: Taiwanese Journal of Mathematics
publication date: 2001-07-03T00:00:00Z
review text: 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.
reviewed by: Ulrich Langer
Mathematics Subject Classification ID: 65N12
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 65N06
zbMATH DE Number: 1472436
zbMATH Keywords: discrete Aleksandrov-Bakelman maximum principle
zbMATH Keywords: second-order elliptic partial differential operators
zbMATH Keywords: a priori estimates
zbMATH Keywords: stability
zbMATH Keywords: difference operators
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.11650/twjm/1500407198
OpenAlex ID: W1534932719
DOI: 10.11650/TWJM/1500407198
description: scientific article; zbMATH DE number 1472436