Additive and restricted additive Schwarz–Richardson methods for inequalities with nonlinear monotone operators

Lori Badea ()
Additional contact information
Lori Badea: Institute of Mathematics of the Romanian Academy, Francophone Center for Mathematics in Bucharest

Computational Optimization and Applications, 2019, vol. 74, issue 2, No 2, 345-385

Abstract: Abstract The main aim of this paper is to analyze in a comparative way the convergence of some additive and additive Schwarz–Richardson methods for inequalities with nonlinear monotone operators. We first consider inequalities perturbed by a Lipschitz operator in the framework of a finite dimensional Hilbert space and prove that they have a unique solution if a certain condition is satisfied. For these inequalities, we introduce additive and restricted additive Schwarz methods as subspace correction algorithms and prove their convergence, under a certain convergence condition, and estimate the error. The convergence of the restricted additive methods does not depend on the number of the used subspaces and we prove that the convergence rate of the additive methods depends only on a reduced number of subspaces which corresponds to the minimum number of colors required to color the subdomains such that the subdomains having the same color do not intersect with each other, but not on the actual number of subdomains. The convergence condition of the algorithms is more restrictive than the existence and uniqueness condition of the solution. We then introduce new additive and restricted additive Schwarz algorithms that have a better convergence and whose convergence condition is identical to the condition of existence and uniqueness of the solution. The additive and restricted additive Schwarz–Richardson algorithms for inequalities with nonlinear monotone operators are obtained by taking the Lipschitz operator of a particular form and the convergence results are deducted from the previous ones. In the finite element space, the introduced algorithms are additive and restricted additive Schwarz–Richardson methods in the usual sense. Numerical experiments carried out for three problems confirm the theoretical predictions.

Keywords: Domain decomposition methods; Additive Schwarz methods; Restricted additive Schwarz methods; Schwarz–Richardson methods; Nonlinear monotone inequalities; 65N55; 65K15; 65N30; 65J15 (search for similar items in EconPapers)
Date: 2019
References: View references in EconPapers View complete reference list from CitEc
Citations:

Downloads: (external link)
http://link.springer.com/10.1007/s10589-019-00116-7 Abstract (text/html)
Access to the full text of the articles in this series is restricted.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:coopap:v:74:y:2019:i:2:d:10.1007_s10589-019-00116-7

Ordering information: This journal article can be ordered from
http://www.springer.com/math/journal/10589

DOI: 10.1007/s10589-019-00116-7

Access Statistics for this article

Computational Optimization and Applications is currently edited by William W. Hager

More articles in Computational Optimization and Applications from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().