{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:19:17Z","timestamp":1787329157124,"version":"build-2736575974"},"reference-count":17,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Numer. Anal."],"published-print":{"date-parts":[[2008,1]]},"abstract":"<jats:p>A new adaptive finite element method for solving the Stokes equations is developed, which is shown to converge with the best possible rate. The method consists of 3 nested loops. The outermost loop consists of an adaptive finite element method for solving the pressure from the (elliptic) Schur complement system that arises by eliminating the velocity. Each of the arising finite element problems is a Stokes-type problem, with the pressure being sought in the current trial space and the divergence-free constraint being reduced to orthogonality of the divergence to this trial space. Such a problem is still continuous in the velocity field. In the middle loop, its solution is approximated using the Uzawa scheme. In the innermost loop, the solution of the elliptic system for the velocity field that has to be solved in each Uzawa iteration is approximated by an adaptive finite element method.<\/jats:p>","DOI":"10.1137\/06066566x","type":"journal-article","created":{"date-parts":[[2008,2,8]],"date-time":"2008-02-08T18:10:00Z","timestamp":1202494200000},"page":"747-775","source":"Crossref","is-referenced-by-count":34,"title":["An Optimal Adaptive Finite Element Method for the Stokes Problem"],"prefix":"10.1137","volume":"46","author":[{"given":"Yaroslav","family":"Kondratyuk","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Rob","family":"Stevenson","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2008,2,8]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142901392134"},{"key":"R2","unstructured":"P. Binev,\n                      Private communication\n                      , 2007."},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-003-0492-7"},{"key":"R4","first-page":"391","volume":"28","author":"Binev P.","year":"2002","journal-title":"Serdica Math. J.","ISSN":"https:\/\/id.crossref.org\/issn\/1310-6600","issn-type":"print"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-003-0493-6"},{"key":"R6","unstructured":"S. Dahlke,\n                      Besov regularity for the Stokes problem\n                      , in Advances in Multivariate Approximation, W. Haussmann, K. Jetter, and M. Reimer, eds., Math. Res. 107, Wiley-VCH, Berlin, 1999, pp. 129\u2013138."},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1137\/S003614290139233X"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1137\/0733054"},{"key":"R9","unstructured":"Y. Kondratyuk,\n                      Adaptive Finite Element Algorithms for the Stokes Problem: Convergence Rates and Optimal Computational Complexity\n                      , Preprint 1346, Department of Mathematics, Utrecht University, The Netherlands, 2006."},{"key":"R10","unstructured":"Y. Kondratyuk and R. P. Stevenson,\n                      An Optimal Adaptive Finite Element Method for the Stokes Problem\u2014Extended Version\n                      , Technical report, Department of Mathematics, Utrecht University, The Netherlands, 2007; available at http:\/\/staff.science.uva.nl\/~rstevens\/publ.html."},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1145\/76909.76912"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142999360044"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-004-0522-0"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-07-01959-X"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1007\/s10208-005-0183-0"},{"key":"R16","unstructured":"R. Verf\u00fcrth,\n                      A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques\n                      , Wiley-Teubner, Chichester, UK, 1996."},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1007\/s11425-006-2005-5"}],"container-title":["SIAM Journal on Numerical Analysis"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/06066566X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:11:52Z","timestamp":1787325112000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/06066566X"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,1]]},"references-count":17,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2008,1]]}},"alternative-id":["10.1137\/06066566X"],"URL":"https:\/\/doi.org\/10.1137\/06066566x","relation":{},"ISSN":["0036-1429","1095-7170"],"issn-type":[{"value":"0036-1429","type":"print"},{"value":"1095-7170","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008,1]]}}}