Finite difference approximations to the Dirichlet problem for elliptic systems
type: Item
instance of: scholarly article
title: Finite difference approximations to the Dirichlet problem for elliptic systems
zbMATH Open document ID: 0602.65068
DOI: 10.1007/BF01389626
author: C. U. Huy
author: W. Walter
author: P. Joseph Mckenna
published in: Numerische Mathematik
publication date: 1986-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/133109
review text: The authors consider systems of difference equations \[ (1)\quad \Delta^ hu+f(x,u)=0,\quad in\quad \Omega_ h\subset {\mathbb{R}}^ m,\quad u\in {\mathbb{R}}^ n,\quad u=0\quad on\quad \Gamma_ h \] where \(\Delta^ h\) is usual difference approximation of the Laplacian and \(\partial f_ i/\partial u_ j\geq 0\), \(i\neq j\), \(\partial f_ i/\partial u_ i\geq - L\). Based on inverse isotonity of \(I-\lambda \Delta^ h\), \(\lambda\geq 0\), monotone iterative schemes for solving (1) are constructed. These schemes are motivated by Crank-Nicholson and alternating direction implicit discretization of the corresponding parabolic problem. Unlike the stability problem the monotonity condition obtained in the article for the Crank-Nicholson scheme only distinguishes the purely implicit case, when no restriction on the time step (iteration parameter) arises. The necessity of this condition is supported by a numerical experiment. The authors use their schemes to obtain upper and lower solutions converging to minimal and maximal solutions, respectively, and to check uniqueness of the numerical solutions. Numerical results are given for a system from mathematical biology \((m=1\), \(n=2)\).
reviewed by: Vladimir D. Korneev
Mathematics Subject Classification ID: 65N06
Mathematics Subject Classification ID: 65N12
Mathematics Subject Classification ID: 35J05
zbMATH DE Number: 3972119
zbMATH Keywords: Poisson equation
zbMATH Keywords: systems of difference equations
zbMATH Keywords: inverse isotonity
zbMATH Keywords: monotone iterative schemes
zbMATH Keywords: Crank-Nicholson
zbMATH Keywords: alternating direction implicit discretization
zbMATH Keywords: stability
zbMATH Keywords: upper and lower solutions
zbMATH Keywords: minimal and maximal solutions
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 3972119