{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,6]],"date-time":"2026-10-06T21:24:02Z","timestamp":1791321842941,"version":"4.3.1"},"reference-count":37,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"5","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 for domain decomposition algorithms of the iterative substructuring family, each subdomain is typically assumed to be the union of a few coarse triangles or tetrahedra. This is an unrealistic assumption, in particular if the subdomains result from the use of a mesh partitioner, in which case they might not even have uniformly Lipschitz continuous boundaries. The purpose of this study is to derive bounds for the condition number of these preconditioned conjugate gradient methods which depend only on a parameter in an isoperimetric inequality, two geometric parameters characterizing John and uniform domains, and the maximum number of edges of any subdomain. A related purpose is to explore to what extent well-known technical tools previously developed for quite regular subdomains can be extended to much more irregular subdomains. Some of these results are valid for any John domain, while an extension theorem, which is needed in this study, requires that the subdomains have complements which are uniform. The results, so far, are complete only for problems in two dimensions. Details are worked out for a FETI\u2013DP algorithm and numerical results support the findings. Some of the numerical experiments illustrate that care must be taken when selecting the scaling of the preconditioners in the case of irregular subdomains.<\/jats:p>","DOI":"10.1137\/070688675","type":"journal-article","created":{"date-parts":[[2008,6,11]],"date-time":"2008-06-11T18:02:42Z","timestamp":1213207362000},"page":"2484-2504","source":"Crossref","is-referenced-by-count":87,"title":["An Analysis of a FETI\u2013DP Algorithm on Irregular Subdomains in the Plane"],"prefix":"10.1137","volume":"46","author":[{"given":"Axel","family":"Klawonn","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Oliver","family":"Rheinbach","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":[[2008,6,11]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1016\/j.aim.2005.09.004"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1016\/0041-5553(78)90012-5"},{"key":"R3","unstructured":"S. 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Widlund,\n                      Extending theory for domain decomposition algorithms to irregular subdomains\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. B. Widlund, and W. Zulehner, eds., Lect. Notes Comput. Sci. Engrg. 60, Springer-Verlag, 2007, pp. 255\u2013261.","DOI":"10.1007\/978-3-540-75199-1_29"},{"key":"R14","doi-asserted-by":"crossref","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. B. Widlund, and W. Zulehner, eds., Lect. Notes Comput. Sci. 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