Convergence of a two-stage Richardson process for nonlinear equations
type: Item
instance of: scholarly article
title: Convergence of a two-stage Richardson process for nonlinear equations
zbMATH Open document ID: 0597.65048
DOI: 10.1007/BF01933747
author: B. George
published in: BIT
publication date: 1986-01-01T00:00:00Z
review text: One considers a nonlinear equation \(A(x)=b\), where \(A: R^ n\to R^ n\) is differentiable and can be decomposed as \(A=M-N\), and the equations \(M(y)=c\) can be easily solved. The iterative process is of the form \(x_{k+1}=(1-\omega)x_{k-1}+(\alpha z_ k+x_ k)\) with \(\alpha,\omega >0\) and \(Mz_ k=r_ k+q_ k\), \(r_ k=b-A(x_ k)\), \(\| q_ k\| \leq \delta \| r_ k\|\), \(0\leq \delta <1\). As a numerical example, one solves the equation obtained by approximating the differential equation \(-\Delta u+u^ 3=f\) with nonhomogeneous Dirichlet conditions on the boundary of the unit square, using finite differences.
Mathematics Subject Classification ID: 65H10
Mathematics Subject Classification ID: 65N22
Mathematics Subject Classification ID: 35J65
zbMATH DE Number: 3963104
zbMATH Keywords: two-stage Richardson iterative processes
zbMATH Keywords: convergence
zbMATH Keywords: preconditioning
zbMATH Keywords: numerical example
reviewed by: G. Marinescu
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01933747
OpenAlex ID: W2043040770
description: scientific article; zbMATH DE number 3963104