Smoothing methods and semismooth methods for nondifferentiable operator equations

Citation
Xj. Chen et al., Smoothing methods and semismooth methods for nondifferentiable operator equations, SIAM J NUM, 38(4), 2000, pp. 1200-1216
Citations number
42
Categorie Soggetti
Mathematics
Journal title
SIAM JOURNAL ON NUMERICAL ANALYSIS
ISSN journal
00361429 → ACNP
Volume
38
Issue
4
Year of publication
2000
Pages
1200 - 1216
Database
ISI
SICI code
0036-1429(20001110)38:4<1200:SMASMF>2.0.ZU;2-R
Abstract
We consider superlinearly convergent analogues of Newton methods for nondif ferentiable operator equations in function spaces. The superlinear converge nce analysis of semismooth methods for nondifferentiable equations describe d by a locally Lipschitzian operator in R-n is based on Rademacher's theore m which does not hold in function spaces. We introduce a concept of slant d ifferentiability and use it to study superlinear convergence of smoothing m ethods and semismooth methods in a uni ed framework. We show that a functio n is slantly differentiable at a point if and only if it is Lipschitz conti nuous at that point. An application to the Dirichlet problems for a simple class of nonsmooth elliptic partial differential equations is discussed.