{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:16:50Z","timestamp":1787329010354,"version":"build-2736575974"},"reference-count":25,"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":[[2009,1]]},"abstract":"<jats:p>The least-squares approximations of an optimal control problem governed by the Stokes equations are considered, which leads to an unconstrained coupled optimization problem by the Lagrange multiplier method. The least-squares functionals for the two- and three-dimensional first-order coupled optimality systems are employed by modifying those functionals in [Z. Cai, T. A. Manteuffel, and S. F. McCormick, SIAM J. Numer. Anal., 34 (1997), pp. 1727\u20131741]. The established ellipticity and continuity in a product $H^1$ norm yield the optimal discretization error estimates in the finite element spaces. For numerical tests, we apply V-cycle multigrid methods to the whole discrete algebraic system.<\/jats:p>","DOI":"10.1137\/070701157","type":"journal-article","created":{"date-parts":[[2009,4,1]],"date-time":"2009-04-01T18:17:15Z","timestamp":1238609835000},"page":"1524-1545","source":"Crossref","is-referenced-by-count":5,"title":["First-Order System Least-Squares Methods for an Optimal Control Problem by the Stokes Flow"],"prefix":"10.1137","volume":"47","author":[{"given":"Soorok","family":"Ryu","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Hyung-Chun","family":"Lee","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Sang Dong","family":"Kim","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2009,4,1]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.3160170104"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1016\/0898-1221(95)00074-9"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1016\/S0362-546X(97)00152-1"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1080\/10618569808940839"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1994-1257573-4"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1016\/j.camwa.2004.10.004"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1137\/040607848"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142996313592"},{"key":"R9","doi-asserted-by":"crossref","unstructured":"P. Bochev, T. A. Manteuffel, and S. F. McCormick,\n                      Analysis of velocity-flux least-squares principles for the Navier\u2013Stokes equations: Part\n                      II, SIAM J. Numer. Anal., 36 (1999) pp. 1125\u20131144.","DOI":"10.1137\/S0036142997324976"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-97-00848-X"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1137\/S003614299527299X"},{"key":"R12","doi-asserted-by":"crossref","unstructured":"P. G. Ciarlet,\n                      The Finite Element Method for Elliptic Problems\n                      , North\u2013Holland, Amsterdam, 1978.","DOI":"10.1115\/1.3424474"},{"key":"R13","doi-asserted-by":"crossref","unstructured":"V. Girault and P. A. 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