Analysis of a damped nonlinear multilevel method
type: Item
instance of: scholarly article
title: Analysis of a damped nonlinear multilevel method
zbMATH Open document ID: 0673.65031
DOI: 10.1007/BF01406516
author: Wolfgang Hackbusch
author: Arnold Reusken
published in: Numerische Mathematik
publication date: 1989-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/133353
review text: Let H be a real Hilbert space and n: \(H\to H'\) be a nonlinear operator which is monotone in the sense of Minty. Moreover n is assumed to be a gradient operator. Given a family of subspaces \(S_ h\), the corresponding operators \(n_ h: S_ h\to S_ h'\) define the discrete analoga of n. Specifically, the discretization of the nonlinear equation \(n(u)=f\) is considered. The uniform ellipticity of the Fréchet derivative \(Dn(u)(v,v)\geq \gamma \| v\|^ 2\) guarantees \(\| u_ 1-u_ 2\| \leq \gamma^{-1}\| f_ 1-f_ 2\|,\) if \(n(u_ i)=f_ i\). Therefore one has a generalization of Céa's lemma and convergence. The authors define a nonlinear multilevel algorithm with a fairly general class of nonlinear relaxations for the smoothing. An essential ingredient is a damping parameter which helps to control the multigrid iteration and global convergence can be proved. Numerical results show that damping may indeed be necessary in some cases to obtain convergence. Moreover they illustrate that the damped Newton nonlinear multigrid method is a very efficient method.
Mathematics Subject Classification ID: 65J15
Mathematics Subject Classification ID: 65N22
Mathematics Subject Classification ID: 35J65
Mathematics Subject Classification ID: 47J25
Mathematics Subject Classification ID: 47H05
Mathematics Subject Classification ID: 65H10
zbMATH DE Number: 4102394
zbMATH Keywords: monotone operator
zbMATH Keywords: numerical examples
zbMATH Keywords: Hilbert space
zbMATH Keywords: gradient operator
zbMATH Keywords: Céa's lemma
zbMATH Keywords: nonlinear multilevel algorithm
zbMATH Keywords: nonlinear relaxations
zbMATH Keywords: smoothing
zbMATH Keywords: damping parameter
zbMATH Keywords: multigrid iteration
zbMATH Keywords: global convergence
zbMATH Keywords: damped Newton nonlinear multigrid method
reviewed by: D. Braess
MaRDI profile type: Publication
cites work: Sharp Estimates for Multigrid Rates of Convergence with General Smoothing and Acceleration
cites work: Analysis of a Multilevel Iterative Method for Nonlinear Finite Element Equations
cites work: Q3727300
cites work: Q3874805
cites work: Q5652137
cites work: Convergence of the multigrid full approximation scheme for a class of elliptic mildly nonlinear boundary value problems
cites work: Convergence of the multilevel full approximation scheme including the V- cycle
cites work: A multigrid method for the membrane problem
OpenAlex ID: W2085160061
description: scientific article; zbMATH DE number 4102394