{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2024,5,17]],"date-time":"2024-05-17T21:46:35Z","timestamp":1715982395431},"reference-count":44,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2014,9,9]],"date-time":"2014-09-09T00:00:00Z","timestamp":1410220800000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Comput Optim Appl"],"published-print":{"date-parts":[[2015,4]]},"DOI":"10.1007\/s10589-014-9695-3","type":"journal-article","created":{"date-parts":[[2014,9,8]],"date-time":"2014-09-08T13:14:37Z","timestamp":1410182077000},"page":"719-751","update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Symmetric error estimates for discontinuous Galerkin time-stepping schemes for optimal control problems constrained to evolutionary Stokes equations"],"prefix":"10.1007","volume":"60","author":[{"given":"Konstantinos","family":"Chrysafinos","sequence":"first","affiliation":[]},{"given":"Efthimios N.","family":"Karatzas","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2014,9,9]]},"reference":[{"key":"9695_CR1","doi-asserted-by":"crossref","first-page":"303","DOI":"10.1007\/BF00271794","volume":"1","author":"F Abergel","year":"1990","unstructured":"Abergel, F., Temam, R.: On some control problems in fluid mechanics. Theor. Comput. Fluid Dyn. 1, 303\u2013326 (1990)","journal-title":"Theor. Comput. Fluid Dyn."},{"key":"9695_CR2","doi-asserted-by":"crossref","first-page":"1484","DOI":"10.1137\/100819333","volume":"50","author":"T Apel","year":"2012","unstructured":"Apel, T., Flaig, T.: Crank-Nicolson schemes for optimal control problems with evolution equations. SIAM J. Numer. Anal. 50, 1484\u20131512 (2012)","journal-title":"SIAM J. Numer. Anal."},{"key":"9695_CR3","series-title":"Frontiers in Applied Mathematics","volume-title":"Optimal Control of Viscous Flows","author":"E Casas","year":"1998","unstructured":"Casas, E.: An optimal control problem governed by the evolution Navier\u2013Stokes equations. In: Sritharan, S.S. (ed.) Optimal Control of Viscous Flows. Frontiers in Applied Mathematics. SIAM, Philadelphia (1998)"},{"key":"9695_CR4","doi-asserted-by":"crossref","first-page":"2281","DOI":"10.1137\/110829404","volume":"50","author":"E Casas","year":"2012","unstructured":"Casas, E., Chrysafinos, K.: A discontinuous Galerkin time stepping scheme for the velocity tracking problem. SIAM. J. Numer. Anal. 50, 2281\u20132306 (2012)","journal-title":"SIAM. J. Numer. Anal."},{"key":"9695_CR5","doi-asserted-by":"crossref","unstructured":"Casas, E., Chrysafinos, K.: Error estimates for the discretization of the velocity tracking problem. http:\/\/www.math.ntua.gr\/~chrysafinos . Accessed 1 Sept 2014","DOI":"10.1007\/s00211-014-0680-7"},{"key":"9695_CR6","first-page":"690","volume":"4","author":"K Chrysafinos","year":"2007","unstructured":"Chrysafinos, K.: Discontinuous Galerkin finite element approximations for distributed optimal control problems constrained to parabolic PDE\u2019s. Int. J. Numer. Anal. Model. 4, 690\u2013712 (2007)","journal-title":"Int. J. Numer. Anal. Model."},{"key":"9695_CR7","doi-asserted-by":"crossref","first-page":"327","DOI":"10.1016\/j.cam.2009.02.092","volume":"231","author":"K Chrysafinos","year":"2009","unstructured":"Chrysafinos, K.: Analysis and finite element approximations for distributed optimal control problems for implicit parabolic equations. J. Comput. Appl. Math. 231, 327\u2013348 (2009)","journal-title":"J. Comput. Appl. Math."},{"key":"9695_CR8","doi-asserted-by":"crossref","first-page":"1473","DOI":"10.3934\/dcdsb.2012.17.1473","volume":"17","author":"K Chrysafinos","year":"2012","unstructured":"Chrysafinos, K., Karatzas, E.: Symmetric error estimates for discontinuous Galerkin approximations for an optimal control problem associated to semilinear parabolic PDEs. Discret. Contin. Dynam. Syst. B 17, 1473\u20131506 (2012)","journal-title":"Discret. Contin. Dynam. Syst. B"},{"key":"9695_CR9","doi-asserted-by":"crossref","unstructured":"Chrysafinos, K., karatzas, E.: Error estimates for discontinuous Galerkin time-stepping schemes for Robin boundary control problems constrained to parabolic pdes\u2019, http:\/\/www.math.ntua.gr\/~chrysafinos . Accessed 1 Sept 2014","DOI":"10.1137\/130943108"},{"key":"9695_CR10","doi-asserted-by":"crossref","first-page":"2135","DOI":"10.1090\/S0025-5718-10-02348-3","volume":"79","author":"K Chrysafinos","year":"2010","unstructured":"Chrysafinos, K., Walkington, N.J.: Discontinuous Galerkin approximations of the Stokes and Navier\u2013Stokes equations. Math. Comput. 79, 2135\u20132167 (2010)","journal-title":"Math. Comput."},{"key":"9695_CR11","doi-asserted-by":"crossref","DOI":"10.1137\/1.9780898719208","volume-title":"The Finite Element Method for Elliptic Problems","author":"P Ciarlet","year":"2002","unstructured":"Ciarlet, P.: The Finite Element Method for Elliptic Problems. SIAM Classics, Philadelphia (2002)"},{"key":"9695_CR12","doi-asserted-by":"crossref","DOI":"10.7208\/chicago\/9780226764320.001.0001","volume-title":"Navier\u2013Stokes equations","author":"P Constantin","year":"1988","unstructured":"Constantin, P.: Navier\u2013Stokes equations. The University of Chicago press, Chicago (1988)"},{"key":"9695_CR13","volume-title":"Mathematical analysis and Numerical Methods for Science and Technology","author":"R Dautray","year":"1993","unstructured":"Dautray, R., Lions, J.-L.: Mathematical analysis and Numerical Methods for Science and Technology, vol. 6. Spinger-Verlag, Berlin (1993)"},{"key":"9695_CR14","first-page":"87","volume":"143","author":"K Deckelnick","year":"2002","unstructured":"Deckelnick, K., Hinze, M.: Error estimates in space and time for tracking type control of the instationary Stokes system. Int. Ser. Numer. Math. 143, 87\u2013103 (2002)","journal-title":"Int. Ser. Numer. Math."},{"key":"9695_CR15","doi-asserted-by":"crossref","first-page":"297","DOI":"10.1007\/s00211-003-0507-4","volume":"97","author":"K Deckelnick","year":"2004","unstructured":"Deckelnick, K., Hinze, M.: Semidiscretization and error estimates for distributed control of the instationary Navier\u2013Stokes equations. Numer. Math. 97, 297\u2013320 (2004)","journal-title":"Numer. Math."},{"key":"9695_CR16","doi-asserted-by":"crossref","first-page":"1289","DOI":"10.1016\/j.na.2005.04.035","volume":"62","author":"J-C los Reyes de","year":"2005","unstructured":"de los Reyes, J.-C., Kunisch, K.: A semi-smooth Newton method for control constrained boundary control of the Navie\u2013Stokes equtions. Nonlinear Anal. 62, 1289\u20131316 (2005)","journal-title":"Nonlinear Anal."},{"key":"9695_CR17","doi-asserted-by":"crossref","DOI":"10.1090\/gsm\/019","volume-title":"Partial Differential Equations","author":"L Evans","year":"1998","unstructured":"Evans, L.: Partial Differential Equations. AMS, Providence (1998)"},{"key":"9695_CR18","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-61623-5","volume-title":"Finite Element Methods for Navier-Stokes","author":"V Girault","year":"1986","unstructured":"Girault, V., Raviart, P.-A.: Finite Element Methods for Navier-Stokes. Springer-Verlag, New York (1986)"},{"key":"9695_CR19","volume-title":"Advances in Design and Control","author":"MD Gunzburger","year":"2003","unstructured":"Gunzburger, M.D.: Perspectives in flow control and optimization. Advances in Design and Control. SIAM, Philadelphia (2003)"},{"key":"9695_CR20","doi-asserted-by":"crossref","first-page":"1481","DOI":"10.1137\/S0036142997329414","volume":"37","author":"MD Gunzburger","year":"2000","unstructured":"Gunzburger, M.D., Manservisi, S.: Analysis and approximation of the velocity tracking problem for Navier\u2013Stokes flows with distributed control. SIAM J. Numer. Anal. 37, 1481\u20131512 (2000)","journal-title":"SIAM J. Numer. Anal."},{"key":"9695_CR21","doi-asserted-by":"crossref","first-page":"1913","DOI":"10.1137\/S0363012998337400","volume":"37","author":"MD Gunzburger","year":"2000","unstructured":"Gunzburger, M.D., Manservisi, S.: The velocity tracking problem for Navier\u2013Stokes flows with bounded distributed control. SIAM J. Control Optim. 37, 1913\u20131945 (2000)","journal-title":"SIAM J. Control Optim."},{"key":"9695_CR22","doi-asserted-by":"crossref","first-page":"594","DOI":"10.1137\/S0363012999353771","volume":"39","author":"MD Gunzburger","year":"2000","unstructured":"Gunzburger, M.D., Manservisi, S.: The velocity tracking problem for Navier\u2013Stokes flow with boundary control. SIAM J. Control Optim. 39, 594\u2013634 (2000)","journal-title":"SIAM J. Control Optim."},{"key":"9695_CR23","unstructured":"Hecht, F.: FreeFem++, 3rd edn, Version 3.13, 2011. http:\/\/www.freefem.org\/ff++ . Accessed 1 Aug 2014"},{"key":"9695_CR24","doi-asserted-by":"crossref","first-page":"275","DOI":"10.1137\/0719018","volume":"19","author":"J Heywood","year":"1982","unstructured":"Heywood, J., Rannacher, R.: Finite element approximation of the nonstationary Navier\u2013Stokes problem. I. Regularity of solutions and second order estimates for spatial discretization. SIAM J. Numer. Anal. 19, 275\u2013311 (1982)","journal-title":"SIAM J. Numer. Anal."},{"key":"9695_CR25","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1007\/s10589-005-4559-5","volume":"30","author":"M Hinze","year":"2005","unstructured":"Hinze, M.: A variational discretization concept in control constrained optimization: the linear-quadratic case. Comput. Optim. Appl. 30, 45\u201361 (2005)","journal-title":"Comput. Optim. Appl."},{"key":"9695_CR26","doi-asserted-by":"crossref","first-page":"925","DOI":"10.1137\/S0363012999361810","volume":"40","author":"M Hinze","year":"2001","unstructured":"Hinze, M., Kunisch, K.: Second order methods for optimal control of time-dependent fluid flow. SIAM J. Control Optim. 40, 925\u2013946 (2001)","journal-title":"SIAM J. Control Optim."},{"key":"9695_CR27","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1002\/zamm.200310094","volume":"84","author":"M Hinze","year":"2004","unstructured":"Hinze, M., Kunisch, K.: Second order methods for boundary control of the instationary Navier\u2013Stokes system. ZAMMZ. Angew. Math. Mech. 84, 171\u2013187 (2004)","journal-title":"ZAMMZ. Angew. Math. Mech."},{"key":"9695_CR28","doi-asserted-by":"crossref","first-page":"287","DOI":"10.1023\/A:1012869611793","volume":"16","author":"LS Hou","year":"2001","unstructured":"Hou, L.S.: Error estimates for semidiscrete finite element approximation of the evolutionary Stokes equations under minimal regularity assumptions. J. Sci. Comput. 16, 287\u2013317 (2001)","journal-title":"J. Sci. Comput."},{"issue":"5","key":"9695_CR29","doi-asserted-by":"crossref","first-page":"1726","DOI":"10.1137\/060670110","volume":"46","author":"K Kunisch","year":"2007","unstructured":"Kunisch, K., Vexler, B.: Constrained Dirichlet boundary control in $$L^2$$ L 2 for a class of evolution equations. SIAM J. Control Optim. 46(5), 1726\u20131753 (2007)","journal-title":"SIAM J. Control Optim."},{"key":"9695_CR30","volume-title":"Control Theory for Partial Differential Equations","author":"I Lasiecka","year":"2000","unstructured":"Lasiecka, I., Triggiani, R.: Control Theory for Partial Differential Equations. Cambridge University Press, Cambridge (2000)"},{"key":"9695_CR31","doi-asserted-by":"crossref","DOI":"10.1137\/1.9781611970616","volume-title":"Some Aspects of the Control of Distributed Parameter Systems","author":"J-L Lions","year":"1972","unstructured":"Lions, J.-L.: Some Aspects of the Control of Distributed Parameter Systems. SIAM Publications, Philadelphia (1972)"},{"issue":"3","key":"9695_CR32","doi-asserted-by":"crossref","first-page":"1032","DOI":"10.1137\/S0036142902397090","volume":"42","author":"W-B Liu","year":"2004","unstructured":"Liu, W.-B., Ma, H.-P., Tang, T., Yan, N.: A posteriori error estimates for DG time-stepping method for optimal control problems governed by parabolic equations. SIAM J. Numer. Anal. 42(3), 1032\u20131061 (2004)","journal-title":"SIAM J. Numer. Anal."},{"key":"9695_CR33","doi-asserted-by":"crossref","first-page":"116","DOI":"10.1137\/060648994","volume":"46","author":"D Meidner","year":"2007","unstructured":"Meidner, D., Vexler, B.: Adaptive space-time finite element methods for parabolic optimization problems. SIAM J. Control Optim. 46, 116\u2013142 (2007)","journal-title":"SIAM J. Control Optim."},{"key":"9695_CR34","doi-asserted-by":"crossref","first-page":"1150","DOI":"10.1137\/070694016","volume":"47","author":"D Meidner","year":"2008","unstructured":"Meidner, D., Vexler, B.: A priori error estimates for space-time finite element discretization of parabolic optimal control problems. Part I: problems without control constraints. SIAM J. Control Optim. 47, 1150\u20131177 (2008)","journal-title":"SIAM J. Control Optim."},{"key":"9695_CR35","doi-asserted-by":"crossref","first-page":"1301","DOI":"10.1137\/070694028","volume":"47","author":"D Meidner","year":"2008","unstructured":"Meidner, D., Vexler, B.: A priori error estimates for space-time finite element discretization of parabolic optimal control problems. Part II: problems with control constraints. SIAM J. Control Optim. 47, 1301\u20131329 (2008)","journal-title":"SIAM J. Control Optim."},{"key":"9695_CR36","volume-title":"Optimal Control of Nonlinear Parabolic Systems: Theory, Algorithms and Applications","author":"P Neittaanmaki","year":"1994","unstructured":"Neittaanmaki, P., Tiba, D.: Optimal Control of Nonlinear Parabolic Systems: Theory, Algorithms and Applications. M. Dekker, New York (1994)"},{"key":"9695_CR37","doi-asserted-by":"crossref","first-page":"345","DOI":"10.1007\/s00211-011-0409-9","volume":"120","author":"I Neitzel","year":"2012","unstructured":"Neitzel, I., Vexler, B.: A priori error estimates for space-time finite element discretization of semilinear parabolic optimal control problems. Numer. Math. 120, 345\u2013386 (2012)","journal-title":"Numer. Math."},{"key":"9695_CR38","doi-asserted-by":"crossref","first-page":"151","DOI":"10.1007\/s00211-012-0511-7","volume":"124","author":"T Richter","year":"2013","unstructured":"Richter, T., Springer, A., Vexler, B.: Efficient numerical realization of discontinuous Galerkin methods for temporal discretization of parabolic problems. Numer. Math. 124, 151\u2013182 (2013)","journal-title":"Numer. Math."},{"key":"9695_CR39","unstructured":"Springer, A., Vexler, B.: Third order convergent time discretization for parabolic optimal control problems with control constraints, Comput. Optim. Appl. 57(1), 205\u2013240 (2014)"},{"key":"9695_CR40","doi-asserted-by":"crossref","DOI":"10.1137\/1.9781611971415","volume-title":"Optimal Control of Viscous Flow","author":"SS Sritharan","year":"1998","unstructured":"Sritharan, S.S.: Optimal Control of Viscous Flow. SIAM, Philadelphia (1998)"},{"key":"9695_CR41","doi-asserted-by":"crossref","DOI":"10.1090\/chel\/343","volume-title":"Navier\u2013Stokes Equations","author":"R Temam","year":"2001","unstructured":"Temam, R.: Navier\u2013Stokes Equations. American Mathematical Society, Chelsea, Providence (2001)"},{"key":"9695_CR42","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-662-03359-3","volume-title":"Galerkin Finite Element Methods for Parabolic Problems","author":"V Thom\u00e9e","year":"1997","unstructured":"Thom\u00e9e, V.: Galerkin Finite Element Methods for Parabolic Problems. Spinger-Verlag, Berlin (1997)"},{"key":"9695_CR43","doi-asserted-by":"crossref","DOI":"10.1090\/gsm\/112","volume-title":"Optimal Control of Partial Differential Equations: Theory, Methods and Applications, Graduate Studies in Mathematics","author":"F Tr\u00f6ltzsch","year":"2010","unstructured":"Tr\u00f6ltzsch, F.: Optimal Control of Partial Differential Equations: Theory, Methods and Applications, Graduate Studies in Mathematics. AMS, Providence (2010)"},{"key":"9695_CR44","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1051\/cocv:2005029","volume":"12","author":"F Tr\u00f6ltzsch","year":"2006","unstructured":"Tr\u00f6ltzsch, F., Wachsmuth, D.: Second-order suficcient optimality conditions for the optimal control of Navier\u2013Stokes equations. ESAIM COCV 12, 93\u2013119 (2006)","journal-title":"ESAIM COCV"}],"container-title":["Computational Optimization and Applications"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10589-014-9695-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10589-014-9695-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10589-014-9695-3","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,4,16]],"date-time":"2022-04-16T14:50:21Z","timestamp":1650120621000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10589-014-9695-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,9,9]]},"references-count":44,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2015,4]]}},"alternative-id":["9695"],"URL":"https:\/\/doi.org\/10.1007\/s10589-014-9695-3","relation":{},"ISSN":["0926-6003","1573-2894"],"issn-type":[{"value":"0926-6003","type":"print"},{"value":"1573-2894","type":"electronic"}],"subject":[],"published":{"date-parts":[[2014,9,9]]}}}