{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,7]],"date-time":"2026-10-07T21:34:08Z","timestamp":1791408848164,"version":"4.3.3"},"reference-count":28,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Numer. Anal."],"published-print":{"date-parts":[[2008,1]]},"abstract":"<jats:p>In the theory of domain decomposition methods, it is often assumed that each subdomain is the union of a small set of coarse triangles or tetrahedra. In this study, extensions to the existing theory which accommodate subdomains with much less regular shapes are presented; the subdomains are required only to be John domains. Attention is focused on overlapping Schwarz preconditioners for problems in two dimensions with a coarse space component of the preconditioner, which allows for good results even for coefficients which vary considerably. It is shown that the condition number of the domain decomposition method is bounded by $C(1+H\/\\delta)(1+\\log(H\/h))^2$, where the constant C is independent of the number of subdomains and possible jumps in coefficients between subdomains. Numerical examples are provided which confirm the theory and demonstrate very good performance of the method for a variety of subregions including those obtained when a mesh partitioner is used for the domain decomposition.<\/jats:p>","DOI":"10.1137\/070685841","type":"journal-article","created":{"date-parts":[[2008,5,4]],"date-time":"2008-05-04T04:04:04Z","timestamp":1209873844000},"page":"2153-2168","source":"Crossref","is-referenced-by-count":80,"title":["Domain Decomposition for Less Regular Subdomains: Overlapping Schwarz in Two Dimensions"],"prefix":"10.1137","volume":"46","author":[{"given":"Clark R.","family":"Dohrmann","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Axel","family":"Klawonn","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Olof B.","family":"Widlund","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2008,5,2]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1016\/j.aim.2005.09.004"},{"key":"R2","doi-asserted-by":"crossref","unstructured":"B. Bojarski,\n                      Remarks on Sobolev imbedding inequalities\n                      , in Complex Analysis, Joensuu 1987, Lecture Notes in Math. 1351, Springer-Verlag, Berlin, 1988, pp. 52\u201368.","DOI":"10.1007\/BFb0081242"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1007\/BF02162087"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1986-0842125-3"},{"key":"R5","first-page":"83","volume":"8","author":"Brenner S.","year":"2000","journal-title":"East-West J. Numer. Math."},{"key":"R6","doi-asserted-by":"crossref","unstructured":"S. C. Brenner and L. R. Scott,\n                      The Mathematical Theory of Finite Element Methods\n                      , Texts Appl. Math. 15, 2nd ed., Springer-Verlag, New York, 2002.","DOI":"10.1007\/978-1-4757-3658-8"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.4310\/MRL.1995.v2.n5.a5"},{"key":"R8","unstructured":"C. R. Dohrmann, A. Klawonn, and O. B. Widlund,\n                      A family of energy minimizing coarse spaces for overlapping Schwarz preconditioners\n                      , in Proceedings of the 17th International Conference on Domain Decomposition Methods in Science and Engineering, Strobl, Austria, 2006, U. Langer, M. Discacciati, D. Keyes, O. Widlund, and W. Zulehner, eds., Lect. Notes Comput. Sci. Eng. 60, Springer-Verlag, Berlin, 2007."},{"key":"R9","unstructured":"C. R. Dohrmann and O. B. Widlund,\n                      An Overlapping Schwarz Algorithm for Almost Incompressible Elasticity\n                      , manuscript, 2007."},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1007\/s002110050172"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1137\/0731086"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.2307\/1970227"},{"key":"R13","doi-asserted-by":"crossref","unstructured":"I. G. Graham, P. Lechner, and R. Scheichl,\n                      Domain decomposition for multiscale PDEs\n                      , Numer. Math. (2007), pp. 589\u2013626.","DOI":"10.1007\/s00211-007-0074-1"},{"key":"R14","unstructured":"P. Haj\u0142asz,\n                      Sobolev inequalities, truncation method, and John domains\n                      , in Papers on Analysis, Rep. Univ. Jyv\u00e4skyl\u00e4 Dep. Math. Stat. 83, University of Jyv\u00e4skyl\u00e4, Jyv\u00e4skyl\u00e4, 2001, pp. 109\u2013126."},{"key":"R15","unstructured":"G. Karypis and V. Kumar,\n                      METIS Version\n                      4.0, University of Minnesota, Department of Computer Science, Minneapolis, MN, 1998."},{"key":"R16","unstructured":"A. Klawonn, O. Rheinbach, and O. B. Widlund,\n                      An analysis of a FETI\u2013DP algorithm on irregular subdomains in the plane\n                      , SIAM J. 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