{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,8,6]],"date-time":"2025-08-06T13:41:34Z","timestamp":1754487694987},"reference-count":35,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2022,1,12]],"date-time":"2022-01-12T00:00:00Z","timestamp":1641945600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2022,1,12]],"date-time":"2022-01-12T00:00:00Z","timestamp":1641945600000},"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":["Numer. Math."],"published-print":{"date-parts":[[2022,2]]},"DOI":"10.1007\/s00211-021-01258-9","type":"journal-article","created":{"date-parts":[[2022,1,12]],"date-time":"2022-01-12T00:02:41Z","timestamp":1641945761000},"page":"591-627","update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Mixed and hybrid Petrov\u2013Galerkin finite element discretization for optimal control of the wave equation"],"prefix":"10.1007","volume":"150","author":[{"given":"Gilbert","family":"Peralta","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Karl","family":"Kunisch","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2022,1,12]]},"reference":[{"key":"1258_CR1","doi-asserted-by":"crossref","unstructured":"Anselmann, M., Bause, M., Becher, S., Matthies, G.: Galerkin-collocation approximation in time for the wave equation and its post-processing. ESAIM Math. Model. Numer. Anal. 54, 2099\u20132123 (2020)","DOI":"10.1051\/m2an\/2020033"},{"key":"1258_CR2","doi-asserted-by":"publisher","first-page":"1484","DOI":"10.1137\/100819333","volume":"50","author":"T Apel","year":"2012","unstructured":"Apel, T., Flaig, T.G.: 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":"1258_CR3","doi-asserted-by":"publisher","first-page":"7","DOI":"10.1051\/m2an\/1985190100071","volume":"19","author":"D Arnold","year":"1985","unstructured":"Arnold, D., Brezzi, F.: Mixed and nonconforming finite element methods: implementation, post-processing and error estimates. RAIRO Model. Math. Anal. Numer. 19, 7\u201332 (1985)","journal-title":"RAIRO Model. Math. Anal. Numer."},{"key":"1258_CR4","doi-asserted-by":"publisher","first-page":"333","DOI":"10.2478\/cmam-2005-0016","volume":"5","author":"C Bahriawati","year":"2005","unstructured":"Bahriawati, C., Carstensen, C.: Three Matlab implementations of the lowest-order Raviart\u2013Thomas MFEM with a posteriori error control. Comput. Methods Appl. Math. 5, 333\u2013361 (2005)","journal-title":"Comput. Methods Appl. Math."},{"key":"1258_CR5","first-page":"1","volume":"1","author":"W Bangerth","year":"1999","unstructured":"Bangerth, W., Rannacher, R.: Adaptive finite element techniques for the acoustic wave equation. J. Comput. Acoustics 1, 1\u201317 (1999)","journal-title":"J. Comput. Acoustics"},{"key":"1258_CR6","doi-asserted-by":"publisher","first-page":"564","DOI":"10.1137\/0713048","volume":"13","author":"GA Baker","year":"1976","unstructured":"Baker, G.A.: Error estimates for finite element methods for second order hyperbolic equations. SIAM J. Numer. Anal. 13, 564\u2013576 (1976)","journal-title":"SIAM J. Numer. Anal."},{"key":"1258_CR7","doi-asserted-by":"publisher","first-page":"141","DOI":"10.1093\/imanum\/8.1.141","volume":"8","author":"J Barzilai","year":"1988","unstructured":"Barzilai, J., Borwein, J.: Two-point step size gradient method. IMA J. Numer. Anal. 8, 141\u2013148 (1988)","journal-title":"IMA J. Numer. Anal."},{"key":"1258_CR8","doi-asserted-by":"publisher","first-page":"595","DOI":"10.1090\/mcom\/3464","volume":"89","author":"M Bause","year":"2020","unstructured":"Bause, M., K\u00f6cher, U., Radu, F.A., Schieweck, F.: Post-processed Galerkin approximation of improved order for wave equations. Math. Comput. 89, 595\u2013627 (2020)","journal-title":"Math. Comput."},{"key":"1258_CR9","doi-asserted-by":"publisher","first-page":"1053","DOI":"10.1137\/S0036142998345499","volume":"37","author":"E B\u00e8cache","year":"2000","unstructured":"B\u00e8cache, E., Joly, P., Tsogka, C.: An analysis of new mixed finite elements for the approximation of wave propagation problems. SIAM J. Numer. Anal. 37, 1053\u20131084 (2000)","journal-title":"SIAM J. Numer. Anal."},{"key":"1258_CR10","doi-asserted-by":"publisher","first-page":"311","DOI":"10.1007\/s002110050064","volume":"68","author":"JH Brandts","year":"1994","unstructured":"Brandts, J.H.: Superconvergence and a posteriori error estimation for triangular mixed finite elements. Numer. Math. 68, 311\u2013324 (1994)","journal-title":"Numer. Math."},{"key":"1258_CR11","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-3172-1","volume-title":"Mixed and Hybrid Finite Element Methods","author":"F Brezzi","year":"1991","unstructured":"Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, New York (1991)"},{"key":"1258_CR12","doi-asserted-by":"publisher","first-page":"957","DOI":"10.1016\/j.finel.2010.06.011","volume":"46","author":"Y Chen","year":"2010","unstructured":"Chen, Y., Lu, Z.: Error estimates for parabolic optimal control problem by fully discrete mixed finite element methods. Finite Elem. Anal. Des. 46, 957\u2013965 (2010)","journal-title":"Finite Elem. Anal. Des."},{"key":"1258_CR13","doi-asserted-by":"publisher","first-page":"205","DOI":"10.1016\/0045-7825(90)90165-I","volume":"8","author":"LC Cowsar","year":"1990","unstructured":"Cowsar, L.C., Dupont, T.F., Wheeler, M.F.: A priori estimates for mixed finite element methods for the wave equations. Comput. Methods Appl. Mech. Eng. 8, 205\u2013222 (1990)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"1258_CR14","doi-asserted-by":"publisher","first-page":"492","DOI":"10.1137\/0733026","volume":"33","author":"LC Cowsar","year":"1996","unstructured":"Cowsar, L.C., Dupont, T.F., Wheeler, M.F.: A priori estimates for mixed finite element approximations of second-order hyperbolic equations with absorbing boundary conditions. SIAM J. Numer. Anal. 33, 492\u2013504 (1996)","journal-title":"SIAM J. Numer. Anal."},{"key":"1258_CR15","doi-asserted-by":"publisher","first-page":"21","DOI":"10.1007\/s00211-004-0569-y","volume":"100","author":"YH Dai","year":"2005","unstructured":"Dai, Y.H., Fletcher, R.: Projected Barzilai\u2013Borwein methods for large-scale box constrained quadratic programming. Numer. Math. 100, 21\u201347 (2005)","journal-title":"Numer. Math."},{"key":"1258_CR16","volume-title":"Space-Time Methods: Applications to Partial Differential Equations","author":"W D\u00f6rfler","year":"2019","unstructured":"D\u00f6rfler, W., Findeisen, S., Wieners, C., Ziegler, D.: Parallel adaptive discontinuous Galerkin discretizations in space and time for linear elastic and acoustic waves. In: Langer, U., Steinbach, O. (eds.) Space-Time Methods: Applications to Partial Differential Equations. De Gruyter, Berlin (2019)"},{"key":"1258_CR17","doi-asserted-by":"publisher","first-page":"427","DOI":"10.1007\/s00211-018-0966-2","volume":"140","author":"H Egger","year":"2018","unstructured":"Egger, H., Radu, B.: Super-convergence and post-processing for mixed finite element approximations of the wave equation. B. Numer. Math. 140, 427\u2013447 (2018)","journal-title":"B. Numer. Math."},{"key":"1258_CR18","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-4355-5","volume-title":"Theory and Practice of Finite Elements","author":"A Ern","year":"2004","unstructured":"Ern, A., Guermond, J.-L.: Theory and Practice of Finite Elements. Springer, New York (2004)"},{"key":"1258_CR19","doi-asserted-by":"publisher","first-page":"465","DOI":"10.1515\/cmam-2018-0190","volume":"19","author":"J Ernesti","year":"2019","unstructured":"Ernesti, J., Wieners, C.: Space-time discontinuous Petrov\u2013Galerkin methods for linear wave equations in heterogeneous media. Comput. Methods Appl. Math. 19, 465\u2013481 (2019)","journal-title":"Comput. Methods Appl. Math."},{"key":"1258_CR20","doi-asserted-by":"publisher","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":"1258_CR21","doi-asserted-by":"publisher","first-page":"348","DOI":"10.1134\/S1995423912040076","volume":"5","author":"T Hou","year":"2012","unstructured":"Hou, T.: Error estimates and superconvergence of semidiscrete mixed methods for optimal control problems governed by hyperbolic equations. Numer. Anal. Appl. 5, 348\u2013362 (2012)","journal-title":"Numer. Anal. Appl."},{"key":"1258_CR22","doi-asserted-by":"publisher","first-page":"1698","DOI":"10.1137\/S0036142901388068","volume":"40","author":"EW Jenkins","year":"2002","unstructured":"Jenkins, E.W., Rivi\u00e8re, B., Wheeler, M.F.: A priori error estimates for mixed finite element approximations fo the acoustic wave equation. SIAM J. Numer. Anal. 40, 1698\u20131715 (2002)","journal-title":"SIAM J. Numer. Anal."},{"key":"1258_CR23","doi-asserted-by":"publisher","first-page":"830","DOI":"10.1137\/090766541","volume":"49","author":"A Kr\u00f6ner","year":"2011","unstructured":"Kr\u00f6ner, A., Kunisch, K., Vexler, B.: Semismooth Newton methods for optimal control of the wave equation with control constraints. SIAM J. Control Optim. 49, 830\u2013858 (2011)","journal-title":"SIAM J. Control Optim."},{"key":"1258_CR24","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898719628","volume-title":"ARPACK USERS GUIDE: Solution of Large Scale Eigenvalue Problems by Implicitly Restarted Arnoldi Methods","author":"RB Lehoucq","year":"1998","unstructured":"Lehoucq, R.B., Sorensen, D.C., Yang, C.: ARPACK USERS GUIDE: Solution of Large Scale Eigenvalue Problems by Implicitly Restarted Arnoldi Methods. SIAM, Philadelphia, PA (1998)"},{"key":"1258_CR25","doi-asserted-by":"crossref","unstructured":"Meidner, D., Vexler, B.: A priori error estimates for the space-time finite element discretization of parabolic optimal control problems. Part I: Problems without control constraints. SIAM J. Control Optim. 47, 1150\u20131177 (2008)","DOI":"10.1137\/070694016"},{"key":"1258_CR26","doi-asserted-by":"publisher","first-page":"2183","DOI":"10.1137\/100809611","volume":"49","author":"D Meidner","year":"2011","unstructured":"Meidner, D., Vexler, B.: A priori error analysis of the Petrov\u2013Galerkin Crank\u2013Nicholson Scheme for parabolic optimal control problems. SIAM J. Control. Optim. 49, 2183\u20132211 (2011)","journal-title":"SIAM J. Control. Optim."},{"key":"1258_CR27","doi-asserted-by":"publisher","first-page":"235","DOI":"10.1007\/BF01385506","volume":"61","author":"ChG Makridakis","year":"1992","unstructured":"Makridakis, Ch.G.: On mixed finite element methods for linear elastodynamics. Numer. Math. 61, 235\u2013260 (1992)","journal-title":"Numer. Math."},{"key":"1258_CR28","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-5561-1","volume-title":"Semigroups of Linear Operators and Applications to Partial Differential Equations","author":"A Pazy","year":"1983","unstructured":"Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983)"},{"key":"1258_CR29","unstructured":"Peralta, G.: Error estimates for mixed and hybrid FEM for elliptic optimal control problems with penalizations, submitted"},{"key":"1258_CR30","volume-title":"Numerical Approximations of Partial Differential Equations","author":"A Quarteroni","year":"2008","unstructured":"Quarteroni, A., Valli, A.: Numerical Approximations of Partial Differential Equations. Springer, Heidelberg (2008)"},{"key":"1258_CR31","doi-asserted-by":"publisher","first-page":"292","DOI":"10.1007\/BFb0064470","volume":"606","author":"PA Raviart","year":"1977","unstructured":"Raviart, P.A., Thomas, J.M.: A mixed finite element method for second order elliptic problems. Lect. Notes Math 606, 292\u2013315 (1977)","journal-title":"Lect. Notes Math"},{"key":"1258_CR32","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-27206-1","volume-title":"Mesoscopic Theories of Heat Transport in Nanosystems","author":"A Sellitto","year":"2016","unstructured":"Sellitto, A., Cimmelli, V.A., Jou, D.: Mesoscopic Theories of Heat Transport in Nanosystems. Spinger, New York (2016)"},{"key":"1258_CR33","volume-title":"Optimal Control of Partial Differential Equations: Theory","author":"F Tr\u00f6ltzsch","year":"2010","unstructured":"Tr\u00f6ltzsch, F.: Optimal Control of Partial Differential Equations: Theory. American Mathematical Society, Providence, Rhode Island, Methods and Applications (2010)"},{"key":"1258_CR34","doi-asserted-by":"publisher","first-page":"735","DOI":"10.1002\/nme.2289","volume":"75","author":"X Xing","year":"2008","unstructured":"Xing, X., Chen, Y.: Error estimates of mixed methods for optimal control problems governed by parabolic equations. Int. J. Numer. Method Eng. 75, 735\u2013754 (2008)","journal-title":"Int. J. Numer. Method Eng."},{"key":"1258_CR35","doi-asserted-by":"publisher","first-page":"1812","DOI":"10.1016\/j.cam.2009.09.018","volume":"233","author":"X Xing","year":"2010","unstructured":"Xing, X., Chen, Y., Yi, N.: Error estimates of mixed finite element methods for quadratic control problems. J Comput. Appl. Math. 233, 1812\u20131820 (2010)","journal-title":"J Comput. Appl. Math."}],"container-title":["Numerische Mathematik"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-021-01258-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00211-021-01258-9\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-021-01258-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,2,9]],"date-time":"2022-02-09T17:03:08Z","timestamp":1644426188000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00211-021-01258-9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,1,12]]},"references-count":35,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2022,2]]}},"alternative-id":["1258"],"URL":"https:\/\/doi.org\/10.1007\/s00211-021-01258-9","relation":{},"ISSN":["0029-599X","0945-3245"],"issn-type":[{"value":"0029-599X","type":"print"},{"value":"0945-3245","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,1,12]]},"assertion":[{"value":"20 May 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"1 April 2021","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"9 November 2021","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 January 2022","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}