{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T10:26:06Z","timestamp":1776767166790,"version":"3.51.2"},"reference-count":20,"publisher":"American Mathematical Society (AMS)","issue":"232","license":[{"start":{"date-parts":[[2000,5,20]],"date-time":"2000-05-20T00:00:00Z","timestamp":958780800000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>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.<\/p>","DOI":"10.1090\/s0025-5718-99-01164-3","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:17:31Z","timestamp":1027707451000},"page":"1341-1354","source":"Crossref","is-referenced-by-count":33,"title":["An additive Schwarz method for variational inequalities"],"prefix":"10.1090","volume":"69","author":[{"given":"Lori","family":"Badea","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Junping","family":"Wang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[1999,5,20]]},"reference":[{"issue":"1","key":"1","doi-asserted-by":"publisher","first-page":"61","DOI":"10.1007\/BF01395872","article-title":"A generalization of the Schwarz alternating method to an arbitrary number of subdomains","volume":"55","author":"Badea, Lori","year":"1989","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"1","key":"2","doi-asserted-by":"publisher","first-page":"179","DOI":"10.1137\/0728010","article-title":"On the Schwarz alternating method with more than two subdomains for nonlinear monotone problems","volume":"28","author":"Badea, Lori","year":"1991","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"195","key":"3","doi-asserted-by":"publisher","first-page":"1","DOI":"10.2307\/2938660","article-title":"Convergence estimates for product iterative methods with applications to domain decomposition","volume":"57","author":"Bramble, James H.","year":"1991","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"2","key":"4","doi-asserted-by":"publisher","first-page":"378","DOI":"10.1137\/0728021","article-title":"Geometry related convergence results for domain decomposition algorithms","volume":"28","author":"Chan, Tony F.","year":"1991","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"5","isbn-type":"print","volume-title":"Domain decomposition methods","year":"1989","ISBN":"https:\/\/id.crossref.org\/isbn\/0898712335"},{"key":"6","isbn-type":"print","volume-title":"Iterative methods for large linear systems","year":"1990","ISBN":"https:\/\/id.crossref.org\/isbn\/0124074758"},{"key":"7","isbn-type":"print","first-page":"3","article-title":"Towards a unified theory of domain decomposition algorithms for elliptic problems","author":"Dryja, Maksymilian","year":"1990","ISBN":"https:\/\/id.crossref.org\/isbn\/089871253X"},{"issue":"5-6","key":"8","doi-asserted-by":"publisher","first-page":"449","DOI":"10.1080\/01630569208816491","article-title":"Parallel algorithms of Schwarz variant for variational inequalities","volume":"13","author":"Hoffmann, K.-H.","year":"1992","journal-title":"Numer. Funct. Anal. Optim.","ISSN":"https:\/\/id.crossref.org\/issn\/0163-0563","issn-type":"print"},{"issue":"2","key":"9","doi-asserted-by":"publisher","first-page":"167","DOI":"10.1007\/BF03325426","article-title":"Monotone multigrid methods for elliptic variational inequalities. I","volume":"69","author":"Kornhuber, Ralf","year":"1994","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"key":"10","isbn-type":"print","first-page":"297","article-title":"Overlapping domain decomposition methods for the simplified Dirichlet-Signorini problem","author":"Kuznetsov, Yu. A.","year":"1992","ISBN":"https:\/\/id.crossref.org\/isbn\/044489702X"},{"issue":"1","key":"11","first-page":"75","article-title":"Block relaxation methods for algebraic obstacle problems with \ud835\udc40-matrices","volume":"2","author":"Kuznetsov, Yu. A.","year":"1994","journal-title":"East-West J. Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0928-0200","issn-type":"print"},{"key":"12","series-title":"Contemporary Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1090\/conm\/157","volume-title":"Domain decomposition methods in science and engineering","volume":"157","year":"1994","ISBN":"https:\/\/id.crossref.org\/isbn\/0821851586"},{"issue":"1","key":"13","doi-asserted-by":"publisher","first-page":"77","DOI":"10.1007\/BF01442171","article-title":"A multilevel iterative method for symmetric, positive definite linear complementarity problems","volume":"11","author":"Mandel, Jan","year":"1984","journal-title":"Appl. Math. Optim.","ISSN":"https:\/\/id.crossref.org\/issn\/0095-4616","issn-type":"print"},{"key":"14","isbn-type":"print","first-page":"1","article-title":"On the Schwarz alternating method. I","author":"Lions, P.-L.","year":"1988","ISBN":"https:\/\/id.crossref.org\/isbn\/0898712203"},{"key":"15","isbn-type":"print","first-page":"47","article-title":"On the Schwarz alternating method. II. Stochastic interpretation and order properties","author":"Lions, P.-L.","year":"1989","ISBN":"https:\/\/id.crossref.org\/isbn\/0898712335"},{"key":"16","isbn-type":"print","first-page":"202","article-title":"On the Schwarz alternating method. III. A variant for nonoverlapping subdomains","author":"Lions, P.-L.","year":"1990","ISBN":"https:\/\/id.crossref.org\/isbn\/089871253X"},{"key":"17","isbn-type":"print","volume-title":"Fourth International Symposium on Domain Decomposition Methods for Partial Differential Equations","year":"1991","ISBN":"https:\/\/id.crossref.org\/isbn\/0898712785"},{"issue":"4","key":"18","first-page":"341","article-title":"Parallel algorithms for variational inequalities based on domain decomposition","volume":"4","author":"L\u00fc, Tao","year":"1991","journal-title":"Systems Sci. Math. Sci.","ISSN":"https:\/\/id.crossref.org\/issn\/1000-9590","issn-type":"print"},{"key":"19","series-title":"Bericht [Report]","isbn-type":"print","volume-title":"Block relaxation methods for algebraic obstacle problems with $M$-matrices: theory and applications","volume":"63","author":"Tarvainen, Pasi","year":"1994","ISBN":"https:\/\/id.crossref.org\/isbn\/9513403831"},{"issue":"2","key":"20","doi-asserted-by":"publisher","first-page":"600","DOI":"10.1137\/S0036142995288920","article-title":"On monotone and geometric convergence of Schwarz methods for two-sided obstacle problems","volume":"35","author":"Zeng, Jinping","year":"1998","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2000-69-232\/S0025-5718-99-01164-3\/S0025-5718-99-01164-3.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2000-69-232\/S0025-5718-99-01164-3\/S0025-5718-99-01164-3.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T22:26:43Z","timestamp":1776724003000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2000-69-232\/S0025-5718-99-01164-3\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1999,5,20]]},"references-count":20,"journal-issue":{"issue":"232","published-print":{"date-parts":[[2000,10]]}},"alternative-id":["S0025-5718-99-01164-3"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-99-01164-3","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[1999,5,20]]}}}