An additive Schwarz method

for variational inequalities

Authors:
Lori Badea and Junping Wang

Journal:
Math. Comp. **69** (2000), 1341-1354

MSC (1991):
Primary 65K10, 65J99, 35R35, 35J60, 49D27, 49D37

Published electronically:
May 20, 1999

MathSciNet review:
1665946

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This paper proposes an additive Schwarz method for variational inequalities and their approximations by finite element methods. The Schwarz domain decomposition method is proved to converge with a geometric rate depending on the decomposition of the domain. The result is based on an abstract framework of convergence analysis established for general variational inequalities in Hilbert spaces.

**1.**Lori Badea,*A generalization of the Schwarz alternating method to an arbitrary number of subdomains*, Numer. Math.**55**(1989), no. 1, 61–81. MR**987156**, 10.1007/BF01395872**2.**Lori Badea,*On the Schwarz alternating method with more than two subdomains for nonlinear monotone problems*, SIAM J. Numer. Anal.**28**(1991), no. 1, 179–204. MR**1083331**, 10.1137/0728010**3.**James H. Bramble, Joseph E. Pasciak, Jun Ping Wang, and Jinchao Xu,*Convergence estimates for product iterative methods with applications to domain decomposition*, Math. Comp.**57**(1991), no. 195, 1–21. MR**1090464**, 10.1090/S0025-5718-1991-1090464-8**4.**Tony F. Chan, Thomas Y. Hou, and P.-L. Lions,*Geometry related convergence results for domain decomposition algorithms*, SIAM J. Numer. Anal.**28**(1991), no. 2, 378–391. MR**1087510**, 10.1137/0728021**5.**Tony F. Chan, Roland Glowinski, Jacques Périaux, and Olof B. Widlund (eds.),*Domain decomposition methods*, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1989. MR**991999****6.**David R. Kincaid and Linda J. Hayes (eds.),*Iterative methods for large linear systems*, Academic Press, Inc., Boston, MA, 1990. Papers from the conference held at the University of Texas at Austin, Austin, Texas, October 19–21, 1988. MR**1038083****7.**Maksymilian Dryja and Olof B. Widlund,*Towards a unified theory of domain decomposition algorithms for elliptic problems*, Third International Symposium on Domain Decomposition Methods for Partial Differential Equations (Houston, TX, 1989) SIAM, Philadelphia, PA, 1990, pp. 3–21. MR**1064335****8.**K.-H. Hoffmann and J. Zou,*Parallel algorithms of Schwarz variant for variational inequalities*, Numer. Funct. Anal. Optim.**13**(1992), no. 5-6, 449–462. MR**1187905**, 10.1080/01630569208816491**9.**Ralf Kornhuber,*Monotone multigrid methods for elliptic variational inequalities. I*, Numer. Math.**69**(1994), no. 2, 167–184. MR**1310316**, 10.1007/BF03325426**10.**Yu. A. Kuznetsov and P. Neittaanmäki,*Overlapping domain decomposition methods for the simplified Dirichlet-Signorini problem*, Computational and applied mathematics, II (Dublin, 1991) North-Holland, Amsterdam, 1992, pp. 297–306. MR**1204694****11.**Yu. A. Kuznetsov, P. Neittaanmäki, and P. Tarvainen,*Block relaxation methods for algebraic obstacle problems with 𝑀-matrices*, East-West J. Numer. Math.**2**(1994), no. 1, 75–89. MR**1274553****12.**Alfio Quarteroni, Jacques Périaux, Yuri A. Kuznetsov, and Olof B. Widlund (eds.),*Domain decomposition methods in science and engineering*, Contemporary Mathematics, vol. 157, American Mathematical Society, Providence, RI, 1994. MR**1262599****13.**Jan Mandel,*A multilevel iterative method for symmetric, positive definite linear complementarity problems*, Appl. Math. Optim.**11**(1984), no. 1, 77–95. MR**726977**, 10.1007/BF01442171**14.**P.-L. Lions,*On the Schwarz alternating method. I*, First International Symposium on Domain Decomposition Methods for Partial Differential Equations (Paris, 1987) SIAM, Philadelphia, PA, 1988, pp. 1–42. MR**972510****15.**P.-L. Lions,*On the Schwarz alternating method. II. Stochastic interpretation and order properties*, Domain decomposition methods (Los Angeles, CA, 1988) SIAM, Philadelphia, PA, 1989, pp. 47–70. MR**992003****16.**P.-L. Lions,*On the Schwarz alternating method. III. A variant for nonoverlapping subdomains*, Third International Symposium on Domain Decomposition Methods for Partial Differential Equations (Houston, TX, 1989) SIAM, Philadelphia, PA, 1990, pp. 202–223. MR**1064345****17.**Roland Glowinski, Yuri A. Kuznetsov, Gérard Meurant, Jacques Périaux, and Olof B. Widlund (eds.),*Fourth International Symposium on Domain Decomposition Methods for Partial Differential Equations*, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1991. MR**1106444****18.**Tao Lü, Chin Bo Liem, and Tsi Min Shih,*Parallel algorithms for variational inequalities based on domain decomposition*, Systems Sci. Math. Sci.**4**(1991), no. 4, 341–348. MR**1142773****19.**Pasi Tarvainen,*Block relaxation methods for algebraic obstacle problems with 𝑀-matrices: theory and applications*, Bericht [Report], vol. 63, Universität Jyväskylä, Mathematisches Institut, Jyväskylä, 1994. Dissertation, University of Jyväskylä, Jyväskylä, 1994. MR**1407743****20.**Jinping Zeng and Shuzi Zhou,*On monotone and geometric convergence of Schwarz methods for two-sided obstacle problems*, SIAM J. Numer. Anal.**35**(1998), no. 2, 600–616 (electronic). MR**1618862**, 10.1137/S0036142995288920

Retrieve articles in *Mathematics of Computation of the American Mathematical Society*
with MSC (1991):
65K10,
65J99,
35R35,
35J60,
49D27,
49D37

Retrieve articles in all journals with MSC (1991): 65K10, 65J99, 35R35, 35J60, 49D27, 49D37

Additional Information

**Lori Badea**

Affiliation:
Institute of Mathematics, Romanian Academy of Sciences, Bucharest, Romania

Email:
lbadea@stoilow.imar.ro

**Junping Wang**

Affiliation:
Department of Mathematics, University of Wyoming, Laramie, Wyoming 82071

Email:
junping@uwyo.edu

DOI:
http://dx.doi.org/10.1090/S0025-5718-99-01164-3

Keywords:
variational inequalities,
obstacle problems,
finite element methods,
domain decomposition methods

Received by editor(s):
December 16, 1997

Received by editor(s) in revised form:
September 22, 1998

Published electronically:
May 20, 1999

Additional Notes:
The research of Wang is supported in part by National Science Foundation Grant # DMS-9706985

Article copyright:
© Copyright 2000
American Mathematical Society