Restricted Additive Schwarz Method for Nonlinear Complementarity Problem with an M-Function

Xu, Hongru; Huang, Ke-Kun; Xie, Shuilian

doi:10.1007/978-3-642-22694-6_7

Cited by 3 publications

(4 citation statements)

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“…We test three problems as follows. Except comparing with the PSOR and classic additive Schwarz method as in [18], we will discuss the effect of overlap and different number of subdomains on the Algorithm 2.1. Problem 1.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…Problem 1. We consider the following problem which similar to the problem presented in [18]. Let Ω = (0, 1) × (0, 1) and consider the following test problem:…”

Section: Numerical Resultsmentioning

confidence: 99%

“…Up to now, much effort have been made to study the convergence theory for RAS, and extended RAS for some other kinds of linear systems, see, e.g., [4,5,7,9] and the references therein. [18] presented the RAS method for a kind of NCP, but…”

Section: Introductionmentioning

confidence: 99%

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Restricted Additive Schwarz Method for a Kind of Nonlinear Complementarity Problem

Xu¹

2014

JCM

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In this paper, a new Schwarz method called restricted additive Schwarz method (RAS) is presented and analyzed for a kind of nonlinear complementarity problem (NCP). The method is proved to be convergent by using weighted maximum norm. Besides, the effect of overlap on RAS is also considered. Some preliminary numerical results are reported to compare the performance of RAS and other known methods for NCP. Mathematics subject classification: 65N30, 65M60the authors did not prove the convergence of the proposed method. The purpose of this paper is to extend the RAS method to NCP (1.1) and establish its convergence results. Moreover, we discuss the effect of overlap on proposed method.The paper is organized as follows: in Section 2, we give some preliminaries and present the RAS method. In Section 3, we estimate the weighted max-norm bounds for iteration errors and establish global convergence theorem for RAS. In Section 4, we discuss the effect of overlap on the RAS method. In Section 5, we present some numerical results and give some conclusions in Section 6.

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Section: Numerical Resultsmentioning

confidence: 99%

“…Problem 1. We consider the following problem which similar to the problem presented in [18]. Let Ω = (0, 1) × (0, 1) and consider the following test problem:…”

Section: Numerical Resultsmentioning

confidence: 99%

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Restricted Additive Schwarz Method for a Kind of Nonlinear Complementarity Problem

Xu¹

2014

JCM

Self Cite

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“…To our knowledge, very few papers deal with the restricted methods for variational inequalities. RAS methods for complementarity problems have been introduced in [35][36][37]39]. Also, algebraic variants of the multiplicative Schwarz iteration and of the additive Schwarz method have been introduced in [38,40], respectively, for complementarity problems.…”

Section: Introductionmentioning

confidence: 99%

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

Badea

2019

Comput Optim Appl

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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. KeywordsDomain decomposition methods • Additive Schwarz methods • Restricted additive Schwarz methods • Schwarz-Richardson methods • Nonlinear monotone inequalities B Lori Badea

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Restricted Additive Schwarz Method for Some Inequalities Perturbed by a Lipschitz Operator

Badea

2018

Lecture Notes in Computational Science and Engineering

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Restricted Additive Schwarz Method for Nonlinear Complementarity Problem with an M-Function

Cited by 3 publications

References 5 publications

Restricted Additive Schwarz Method for a Kind of Nonlinear Complementarity Problem

Restricted Additive Schwarz Method for a Kind of Nonlinear Complementarity Problem

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

Restricted Additive Schwarz Method for Some Inequalities Perturbed by a Lipschitz Operator

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