{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,8]],"date-time":"2026-01-08T00:35:31Z","timestamp":1767832531802,"version":"3.49.0"},"reference-count":33,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2021,10,6]],"date-time":"2021-10-06T00:00:00Z","timestamp":1633478400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2021,10,6]],"date-time":"2021-10-06T00:00:00Z","timestamp":1633478400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Sci Comput"],"published-print":{"date-parts":[[2021,11]]},"DOI":"10.1007\/s10915-021-01645-2","type":"journal-article","created":{"date-parts":[[2021,10,7]],"date-time":"2021-10-07T19:10:50Z","timestamp":1633633850000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["A Posteriori Error Estimates for Parabolic Optimal Control Problems with Controls Acting on Lower Dimensional Manifolds"],"prefix":"10.1007","volume":"89","author":[{"given":"Ram","family":"Manohar","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5910-8943","authenticated-orcid":false,"given":"Rajen Kumar","family":"Sinha","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,10,6]]},"reference":[{"key":"1645_CR1","volume-title":"Sobolev spaces","author":"RA Adams","year":"2009","unstructured":"Adams, R.A., Fournier, J.J.: Sobolev spaces, 2nd edn. Elsevier, Singapore (2009)","edition":"2"},{"key":"1645_CR2","doi-asserted-by":"crossref","unstructured":"Ciarlet, P.G.: The finite element method for elliptic problems. North-Holland, Amsterdam (1978)","DOI":"10.1115\/1.3424474"},{"key":"1645_CR3","doi-asserted-by":"publisher","first-page":"1735","DOI":"10.1137\/110843216","volume":"50","author":"E Casas","year":"2012","unstructured":"Casas, E., Clason, C., Kunisch, K.: Approximation of elliptic control problems in measure spaces with sparse solutions. SIAM J. Control Optim. 50, 1735\u20131752 (2012)","journal-title":"SIAM J. Control Optim."},{"key":"1645_CR4","doi-asserted-by":"publisher","first-page":"28","DOI":"10.1137\/120872395","volume":"51","author":"E Casas","year":"2013","unstructured":"Casas, E., Clason, C., Kunisch, K.: Parabolic control problems in measure spaces with sparse solutions. SIAM J. Control Optim. 51, 28\u201363 (2013)","journal-title":"SIAM J. Control Optim."},{"key":"1645_CR5","doi-asserted-by":"publisher","first-page":"1585","DOI":"10.1137\/050626600","volume":"45","author":"E Casas","year":"2006","unstructured":"Casas, E., Raymond, J.P.: Error estimates for the numerical approximation of Dirichlet boundary control for semilinear elliptic equations. SIAM J. Control Optim. 45, 1585\u20131611 (2006)","journal-title":"SIAM J. Control Optim."},{"key":"1645_CR6","doi-asserted-by":"publisher","first-page":"1400","DOI":"10.1137\/S0363012903430317","volume":"43","author":"C Castro","year":"2004","unstructured":"Castro, C., Zuazua, E.: Unique continuation and control for the heat equation from an oscillating lower dimensional manifold. SIAM J. Control Optim. 43, 1400\u20131434 (2004)","journal-title":"SIAM J. Control Optim."},{"key":"1645_CR7","doi-asserted-by":"publisher","first-page":"97","DOI":"10.1137\/110840133","volume":"52","author":"W Gong","year":"2014","unstructured":"Gong, W., Hinze, M., Zhou, Z.J.: A priori error analysis for finite element approximation of parabolic optimal control problems with pointwise control. SIAM J. Control Optim. 52, 97\u2013119 (2014)","journal-title":"SIAM J. Control Optim."},{"key":"1645_CR8","doi-asserted-by":"publisher","first-page":"941","DOI":"10.1007\/s10915-015-0051-2","volume":"66","author":"W Gong","year":"2016","unstructured":"Gong, W., Hinze, M., Zhou, Z.J.: Finite element method and a priori error estimates for Dirichlet boundary control problems governed by parabolic PDEs. J. Sci. Comput. 66, 941\u2013967 (2016)","journal-title":"J. Sci. Comput."},{"key":"1645_CR9","doi-asserted-by":"publisher","first-page":"2008","DOI":"10.1137\/13091213X","volume":"52","author":"W Gong","year":"2014","unstructured":"Gong, W., Wang, G.S., Yan, N.N.: Approximations of elliptic optimal control problems with controls acting on a lower dimensional manifold. SIAM J. Control Optim. 52, 2008\u20132035 (2014)","journal-title":"SIAM J. Control Optim."},{"key":"1645_CR10","doi-asserted-by":"publisher","first-page":"1229","DOI":"10.1137\/151004744","volume":"54","author":"W Gong","year":"2016","unstructured":"Gong, W., Yan, N.N.: Finite element approximations of the optimal control problems with controls acting on a lower dimensional manifold. SIAM J. Numer. Anal. 54, 1229\u20131262 (2016)","journal-title":"SIAM J. Numer. Anal."},{"key":"1645_CR11","first-page":"251","volume":"20","author":"F Hetch","year":"2012","unstructured":"Hetch, F.: New Development in FreeFem++. J. Numer. Math. 20, 251\u2013265 (2012)","journal-title":"J. Numer. Math."},{"key":"1645_CR12","unstructured":"Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE Constraints. Math. Model. Theo. Appl (2009)"},{"key":"1645_CR13","doi-asserted-by":"publisher","first-page":"231","DOI":"10.1137\/S0363012999358038","volume":"40","author":"A Khapalov","year":"2001","unstructured":"Khapalov, A.: Mobile point controls versus locally distributed ones for the controllability of the semilinear parabolic equation. SIAM J. Control Optim. 40, 231\u2013252 (2001)","journal-title":"SIAM J. Control Optim."},{"key":"1645_CR14","unstructured":"Kufner, K., John, O., Fucik, S.: Function Spaces. Nordhoff, Leiden, The Netherlands (1977)"},{"key":"1645_CR15","doi-asserted-by":"publisher","first-page":"3078","DOI":"10.1137\/140959055","volume":"52","author":"K Kunisch","year":"2014","unstructured":"Kunisch, K., Pieper, K., Vexler, B.: Measure valued directional sparsity for parabolic optimal control problems. SIAM J. Control Optim. 52, 3078\u20133108 (2014)","journal-title":"SIAM J. Control Optim."},{"key":"1645_CR16","first-page":"21","volume":"7","author":"I Lasiecka","year":"1978","unstructured":"Lasiecka, I., Malanowski, K.: On discrete-time Ritz-Galerkin approximation of control constrained optimal control problems for parabolic systems. Control Cybern. 7, 21\u201336 (1978)","journal-title":"Control Cybern."},{"key":"1645_CR17","doi-asserted-by":"publisher","first-page":"2797","DOI":"10.1137\/120885772","volume":"51","author":"D Leykekhman","year":"2013","unstructured":"Leykekhman, D., Vexler, B.: Optimal a priori error estimates of parabolic optimal control problems with pointwise control. SIAM J. Numer. Anal. 51, 2797\u20132821 (2013)","journal-title":"SIAM J. Numer. Anal."},{"key":"1645_CR18","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-65024-6","volume-title":"Optimal control of systems governed by partial differential Equations","author":"JL Lions","year":"1971","unstructured":"Lions, J.L.: Optimal control of systems governed by partial differential Equations. Springer-Verlag, Berlin (1971)"},{"key":"1645_CR19","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-65217-2","volume-title":"Nonhomogeneous Boundary Value Problems and Applications","author":"JL Lions","year":"1972","unstructured":"Lions, J.L., Magenes, E.: Nonhomogeneous Boundary Value Problems and Applications. Springer-Verlag, New York (1972)"},{"key":"1645_CR20","doi-asserted-by":"publisher","first-page":"285","DOI":"10.1023\/A:1014239012739","volume":"15","author":"WB Liu","year":"2001","unstructured":"Liu, W.B., Yan, N.N.: A posteriori error estimates for distributed convex optimal control problems. Adv. Comput. Math. 15, 285\u2013309 (2001)","journal-title":"Adv. Comput. Math."},{"key":"1645_CR21","doi-asserted-by":"publisher","first-page":"497","DOI":"10.1007\/s002110100380","volume":"93","author":"WB Liu","year":"2003","unstructured":"Liu, W.B., Yan, N.N.: A posteriori error estimates for optimal control problems governed by parabolic equations. Numer. Math. 93, 497\u2013521 (2003)","journal-title":"Numer. Math."},{"key":"1645_CR22","volume-title":"Adaptive finite element methods for optimal control governed by PDEs","author":"WB Liu","year":"2008","unstructured":"Liu, W.B., Yan, N.N.: Adaptive finite element methods for optimal control governed by PDEs. Science press, Beijing (2008)"},{"key":"1645_CR23","doi-asserted-by":"publisher","first-page":"69","DOI":"10.1007\/BF01447752","volume":"8","author":"K Malanowski","year":"1981","unstructured":"Malanowski, K.: Convergence of approximations vs. regularity of solutions for convex, control constrained optimal-control problems. Appl. Math. Optim. 8, 69\u201395 (1981)","journal-title":"Appl. Math. Optim."},{"key":"1645_CR24","doi-asserted-by":"publisher","first-page":"510","DOI":"10.1137\/0311040","volume":"11","author":"RS McKnight","year":"1973","unstructured":"McKnight, R.S., Bosarge, W.E.: The Ritz-Galerkin procedure for parabolic control problems. SIAM J. Control Optim. 11, 510\u2013524 (1973)","journal-title":"SIAM J. Control Optim."},{"key":"1645_CR25","doi-asserted-by":"publisher","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 I. Problems without control constraints. SIAM J. Control Optim. 47, 1150\u20131177 (2008)","journal-title":"SIAM J. Control Optim."},{"key":"1645_CR26","doi-asserted-by":"publisher","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 fnite element discretization of parabolic optimal control problems II. Problems with control constraints. SIAM J. Control Optim. 47, 1301\u20131329 (2008)","journal-title":"SIAM J. Control Optim."},{"key":"1645_CR27","doi-asserted-by":"publisher","first-page":"466","DOI":"10.1137\/S0036142999360044","volume":"38","author":"P Morin","year":"2000","unstructured":"Morin, P., Nochetto, R.H., Siebert, K.G.: Data oscillation and convergence of adaptive FEM. SIAM J. Numer. Anal. 38, 466\u2013488 (2000)","journal-title":"SIAM J. Numer. Anal."},{"key":"1645_CR28","doi-asserted-by":"crossref","unstructured":"Nguyen, P. A., Raymond, J. P.: Control localized on thin structures for semilinear parabolic equations, In Nonlinear Partial Differential Equations and Their Applications, Coll\u00e9ge de France Seminar, vol. XIV, Paris, 1997\/1998. Stud. Math. Appl., vol. 31, pp. 591-645. North-Holland, Amsterdam (2002)","DOI":"10.1016\/S0168-2024(02)80027-3"},{"key":"1645_CR29","first-page":"467","volume":"6","author":"PA Nguyen","year":"2001","unstructured":"Nguyen, P.A., Raymond, J.P.: Control problems for convection-diffusion equations with control localized on manifolds. ESAIM: Control Optim. Calc. Var. 6, 467\u2013488 (2001)","journal-title":"ESAIM: Control Optim. Calc. Var."},{"key":"1645_CR30","doi-asserted-by":"publisher","first-page":"23","DOI":"10.1016\/j.apnum.2018.09.015","volume":"136","author":"P Shakya","year":"2019","unstructured":"Shakya, P., Sinha, R.K.: A posteriori error analysis for finite element approximations of parabolic optimal control problems with measure data. Appl. Numer. Math. 136, 23\u201345 (2019)","journal-title":"Appl. Numer. Math."},{"key":"1645_CR31","doi-asserted-by":"crossref","unstructured":"Shakya, P., Sinha, R.K.: Finite element approximations of parabolic optimal control problem with measure data in time. Appl. Anal. 1\u201329,(2019)","DOI":"10.1016\/j.apnum.2018.09.015"},{"key":"1645_CR32","doi-asserted-by":"publisher","first-page":"353","DOI":"10.4171\/ZAA\/1203","volume":"23","author":"A R\u00f6sch","year":"2004","unstructured":"R\u00f6sch, A.: Error estimates for parabolic optimal control problems with control constraints. Z. Anal. Anwendungen 23, 353\u2013376 (2004)","journal-title":"Z. Anal. Anwendungen"},{"key":"1645_CR33","doi-asserted-by":"publisher","first-page":"173","DOI":"10.1007\/BF02417890","volume":"117","author":"R Winther","year":"1978","unstructured":"Winther, R.: Error estimates for a Galerkin approximation of a parabolic control problem. Ann. Mat. Pura. App. 117, 173\u2013206 (1978)","journal-title":"Ann. Mat. Pura. App."}],"container-title":["Journal of Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-021-01645-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10915-021-01645-2\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10915-021-01645-2.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,10,24]],"date-time":"2021-10-24T05:05:15Z","timestamp":1635051915000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10915-021-01645-2"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,10,6]]},"references-count":33,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2021,11]]}},"alternative-id":["1645"],"URL":"https:\/\/doi.org\/10.1007\/s10915-021-01645-2","relation":{},"ISSN":["0885-7474","1573-7691"],"issn-type":[{"value":"0885-7474","type":"print"},{"value":"1573-7691","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,10,6]]},"assertion":[{"value":"12 April 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 July 2021","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 September 2021","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"6 October 2021","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"39"}}