{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:46:21Z","timestamp":1787319981716,"version":"build-2736575974"},"reference-count":18,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Control Optim."],"published-print":{"date-parts":[[2009,1]]},"abstract":"<jats:p>There are two main possibilities for the numerical computation of optimal control problems with constraints given by partial differential equations: One may consider first the discretized problem and then build the optimality condition. The other possibility is to formulate first the optimality condition on the continuous level and then discretize. Both approaches may lead to different discrete adjoint equations because discretization and building the adjoint do not commute in general. This type of inconsistency takes place when conventional stabilized finite elements for flow problems, as for instance, streamline diffusion (SUPG), are used, due to its nonsymmetry. Consequently, the computed control is significantly affected by the way of defining the discrete optimality condition. Hence, there is a need for symmetric stabilization so that discretization and building the adjoint commute. We formulate the use of this kind of stabilization and give a quasi-optimal a priori estimate in the context of optimal control problems for the Oseen system. In particular, we show that local projection stabilization and edge-oriented stabilization result to be quasi-optimal for optimal control problems.<\/jats:p>","DOI":"10.1137\/060653494","type":"journal-article","created":{"date-parts":[[2009,2,13]],"date-time":"2009-02-13T18:17:12Z","timestamp":1234549032000},"page":"672-687","source":"Crossref","is-referenced-by-count":41,"title":["Optimal Control in Fluid Mechanics by Finite Elements with Symmetric Stabilization"],"prefix":"10.1137","volume":"48","author":[{"given":"M.","family":"Braack","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2009,2,13]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1016\/j.finel.2004.06.001"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1007\/s10092-001-8180-4"},{"key":"R3","doi-asserted-by":"crossref","unstructured":"R. Becker and M. Braack,\n                      A two-level stabilization scheme for the Navier-Stokes equations,\n                      in Numer. Math. Adv. Appl., E. A. M. Feistauer, ed., ENUMATH 2003, Springer, 2004, pp. 123\u2013130.","DOI":"10.1007\/978-3-642-18775-9_9"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-007-0067-0"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1137\/050631227"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2006.07.011"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1016\/j.compfluid.2005.02.001"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1002\/fld.1160"},{"key":"R9","doi-asserted-by":"publisher","DOI":"10.1016\/0045-7825(82)90071-8"},{"key":"R10","doi-asserted-by":"publisher","DOI":"10.1137\/040617686"},{"key":"R11","unstructured":"S. Collis and M. Heinkenschloss,\n                      Analysis of the streamline upwind\/Petrov Galerkin method applied to the solution of optimal control problems,\n                      Technical report 02-01, Rice University, Houston, TX, 2002."},{"key":"R12","first-page":"1019","volume":"39","author":"Ded\u00e9 L.","year":"2005","journal-title":"'er."},{"key":"R13","doi-asserted-by":"crossref","unstructured":"J. Douglas and T. Dupont,\n                      Interior penalty procedures for elliptic and parabolic Galerkin methods,\n                      in Computing Methods in Applied Sciences (2nd Int. Symp., Versailles, 1975), Lecture Notes in Physics 58, Springer, New York, 1976, pp. 207\u2013216.","DOI":"10.1007\/BFb0120591"},{"key":"R14","doi-asserted-by":"crossref","unstructured":"T. Hughes, L. Franca, and M. Balestra,\n                      A new finite element formulation for computational fluid dynamics: V. circumvent the Babuska-Brezzi condition: A stable Petrov-Galerkin formulation for the Stokes problem accommodating equal order interpolation,\n                      Comput. Methods Appl. Mech. Engrg., 59 (1986) pp. 89\u201399.","DOI":"10.1016\/0045-7825(86)90025-3"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1986-0842120-4"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2003.01.001"},{"key":"R17","doi-asserted-by":"crossref","unstructured":"J. L. Lions,\n                      Optimal Control of Systems Governed by Partial Differential Equations,\n                      Number 170 in Die Grundlehren der mathematischen Wissenschaften, Springer-Verlag, Berlin, 1971.","DOI":"10.1007\/978-3-642-65024-6"},{"key":"R18","doi-asserted-by":"crossref","unstructured":"J. Nitsche,\n                      \u00dcber ein Variationsprinzip zur L\u00f6sung von Dirichlet-Problemen bei Verwendung von Teilr\u00e4umen, die keinen Randbedingungen unterworfen sind,\n                      Abh. Math. Sem. Univ. Hamburg, 36 (1970\/71), pp. 9\u201315.","DOI":"10.1007\/BF02995904"}],"container-title":["SIAM Journal on Control and Optimization"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/060653494","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T12:54:24Z","timestamp":1787316864000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/060653494"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,1]]},"references-count":18,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2009,1]]}},"alternative-id":["10.1137\/060653494"],"URL":"https:\/\/doi.org\/10.1137\/060653494","relation":{},"ISSN":["0363-0129","1095-7138"],"issn-type":[{"value":"0363-0129","type":"print"},{"value":"1095-7138","type":"electronic"}],"subject":[],"published":{"date-parts":[[2009,1]]}}}