Grid approximation of a singularly perturbed boundary-value problem for a quasi-linear elliptic equation in the completely degenerate case
type: Item
instance of: scholarly article
title: Grid approximation of a singularly perturbed boundary-value problem for a quasi-linear elliptic equation in the completely degenerate case
zbMATH Open document ID: 0780.65063
author: G. I. Shishkin
published in: Computational Mathematics and Mathematical Physics
publication date: 1992-09-27T00:00:00Z
review text: The author is concerned with the equation \[ \varepsilon^ 2L_ 2u(x) + \varepsilon L_ 1u(x)-g(x,u(x)) =0 \] for \(x = (x_ 1,\dots,x_ n)\), \(0<x_ 1 < d\), \(x_ i \in \mathbb{R}\), \(i = 2,\dots,n\), subjected to a Dirichlet boundary condition. Here \(0 < \varepsilon \leq 1\) denotes a parameter, \(L_ 2\) is a second order elliptic operator, \(L_ 1\) is a first order differential operator, and for the function \(g\) the condition \(\partial/\partial u(g(x)) \geq \text{const} > 0\) is assumed. The coefficients of the operators \(L_ 2\), \(L_ 1\) can depend on \(u\) and on the first order derivatives of \(u\), the parameter \(\varepsilon\) may be involved in the boundary condition. By means of a differential inequality the author obtains a priori estimations for the solution \(u\) and its derivatives. Next he takes a positive integer \(N\) sufficiently large, \(\delta_ 0 = \min\{\varepsilon\ln N,d/2\}\) and constructs a non-uniform mesh on \([0,d]\) with \(h_ 1 = \delta_ 0/N\) on \([0,\delta_ 0]\cup[d-\delta_ 0,d]\) and \(h_ 2 = d/N\) on \((\delta_ 0,d-\delta_ 0)\). For \(0 \leq x_ i \leq 1\) \((i = 2,\dots,n)\) the mesh is uniformly independent of \(\varepsilon\). On this special network the author defines difference schemes in a standard manner. Under some regularity assumptions he proves that the convergence of the solutions of difference schemes is uniform with respect to the parameter \(\varepsilon\). The same is discussed for an iterative difference scheme with an additional parameter \(t\) playing the role of time. The number of iterations is characterized as well.
Mathematics Subject Classification ID: 65N06
Mathematics Subject Classification ID: 65N12
Mathematics Subject Classification ID: 35J70
Mathematics Subject Classification ID: 35B25
zbMATH DE Number: 56759
zbMATH Keywords: singularly perturbed boundary value problem
zbMATH Keywords: quasilinear
zbMATH Keywords: complete degeneration
zbMATH Keywords: degenerate elliptic equations
zbMATH Keywords: finite difference method
zbMATH Keywords: difference schemes
zbMATH Keywords: convergence
reviewed by: S. Burys
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 56759