{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:34:57Z","timestamp":1787330097451,"version":"build-2736575974"},"reference-count":25,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Matrix Anal. Appl."],"published-print":{"date-parts":[[2019,1]]},"abstract":"<jats:p>We consider the iterative solution of linear systems with a symmetric saddle point system matrix. We address a family of solution techniques that exploit the knowledge of a preconditioner (or approximate solution procedure) both for the top left block of the matrix on the one hand and for the Schur complement resulting from its elimination on the other hand. This includes many \u201csegregated\u201d or \u201cSchur complement\u201d iterations such as the inexact Uzawa method and pressure correction techniques, and also many \u201cblock\u201d preconditioners, based on the approximate block factorization of the system matrix. An analysis is developed which proves convergence in norm of stationary iterations. It is more rigorous than eigenvalue analyses which ignore nonnormality effects, while being more general than previous norm analyses. The analysis also clarifies the relations that exist between the many members of this family of methods and offers practical guidelines to select the scheme most appropriate to a situation at hand.<\/jats:p>","DOI":"10.1137\/18m1208836","type":"journal-article","created":{"date-parts":[[2019,1,22]],"date-time":"2019-01-22T13:59:16Z","timestamp":1548165556000},"page":"122-146","source":"Crossref","is-referenced-by-count":18,"title":["Convergence of Some Iterative Methods for Symmetric Saddle Point Linear Systems"],"prefix":"10.1137","volume":"40","author":[{"given":"Yvan","family":"Notay","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2019,1,22]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1007\/BF03024946"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1007\/BF01405194"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1017\/S0962492904000212"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-006-0679-9"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1988-0917816-8"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142994273343"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1002\/fld.1650080802"},{"key":"atypb8","unstructured":"L. Chen and Y. Wu,\n                      Convergence Analysis for A Class of Iterative Methods for Solving Saddle Point Systems\n                      ,arXiv:1710.03409v3[math.NA], 2018."},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1137\/16M1106304"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2007.09.026"},{"key":"atypb11","unstructured":"H. Elman, D. Silvester, and A. Wathen,\n                      Finite Elements and Fast Iterative Solvers\n                      , Oxford University Press, Oxford, UK, 2005."},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1137\/0731085"},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479894275030"},{"key":"atypb14","unstructured":"C. Hirsch,\n                      Numerical Computation of Internal and External Flows, $2$nd Ed.\n                      , Elsevier, Amsterdam, 2003."},{"key":"atypb15","doi-asserted-by":"publisher","DOI":"10.1137\/100818509"},{"key":"atypb17","first-page":"123","volume":"37","author":"Notay Y.","year":"2010","journal-title":"Electron. Trans. Numer. Anal."},{"key":"atypb18","doi-asserted-by":"publisher","DOI":"10.1137\/130911962"},{"key":"atypb19","unstructured":"Y. Notay,\n                      Analysis of two-grid methods: The nonnormal case\n                      , Tech. report GANMN 18-01, Universit\u00e9 Libre de Bruxelles, Brussels, Belgium, 2018.http:\/\/homepages.ulb.ac.be\/~ynotay\/."},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.1016\/0017-9310(72)90054-3"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1137\/130934921"},{"key":"atypb22","unstructured":"D. Silvester, H. Elman, and A. Ramage,\n                      Incompressible Flow and Iterative Solver Software (IFISS) Version 3.2\n                      ,http:\/\/www.manchester.ac.uk\/ifiss\/(2012)."},{"key":"atypb23","doi-asserted-by":"publisher","DOI":"10.1016\/j.apnum.2003.11.012"},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1007\/s002110050356"},{"key":"atypb25","unstructured":"H. Uzawa,\n                      Iterative methods for concave programming\n                      , in Studies in Linear and Nonlinear Programming, K. J. Arrow, L. Hurwicz, and H. Uzawa, eds., Stanford University Press, Stansford, CA, 1958, pp. 154-165."},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-01-01324-2"}],"container-title":["SIAM Journal on Matrix Analysis and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/18M1208836","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:47:23Z","timestamp":1787327243000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/18M1208836"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,1]]},"references-count":25,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2019,1]]}},"alternative-id":["10.1137\/18M1208836"],"URL":"https:\/\/doi.org\/10.1137\/18m1208836","relation":{},"ISSN":["0895-4798","1095-7162"],"issn-type":[{"value":"0895-4798","type":"print"},{"value":"1095-7162","type":"electronic"}],"subject":[],"published":{"date-parts":[[2019,1]]}}}