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Sobolev space preconditioning for Newton's method using domain decomposition
Author(s) -
Axelsson O.,
Faragó I.,
Karátson J.
Publication year - 2002
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.293
Subject(s) - mathematics , sobolev space , conjugate gradient method , domain decomposition methods , piecewise , mathematical analysis , convergence (economics) , newton's method , boundary (topology) , piecewise linear function , mathematical optimization , finite element method , nonlinear system , physics , quantum mechanics , economics , thermodynamics , economic growth
An inner–outer iteration is constructed for ill‐conditioned non‐linear elliptic boundary value problems, using a damped inexact Newton Method for the outer and a conjugate gradient method for the inner iteration. The focus is on efficient preconditioning for the inner iteration. Sobolev space background is used to construct preconditioners as discretizations of appropriately chosen piecewise constant coefficient elliptic operators. The combination of this theoretical approach with a suitable domain decomposition idea results in well‐structured preconditioners that are able to compensate for the sharp gradients of the coefficients. Furthermore, convergence estimates and mesh independence of the condition numbers are direct consequences of the method. Copyright © 2002 John Wiley & Sons, Ltd.