{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,1]],"date-time":"2026-07-01T19:09:10Z","timestamp":1782932950836,"version":"3.54.5"},"reference-count":32,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2012,10,23]],"date-time":"2012-10-23T00:00:00Z","timestamp":1350950400000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Numer. Math."],"published-print":{"date-parts":[[2013,5]]},"DOI":"10.1007\/s00211-012-0511-7","type":"journal-article","created":{"date-parts":[[2012,10,22]],"date-time":"2012-10-22T12:26:13Z","timestamp":1350908773000},"page":"151-182","source":"Crossref","is-referenced-by-count":23,"title":["Efficient numerical realization of discontinuous Galerkin methods for temporal discretization of parabolic problems"],"prefix":"10.1007","volume":"124","author":[{"given":"Thomas","family":"Richter","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Andreas","family":"Springer","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Boris","family":"Vexler","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2012,10,23]]},"reference":[{"issue":"5","key":"511_CR1","doi-asserted-by":"crossref","first-page":"813","DOI":"10.1080\/10556780701228532","volume":"22","author":"R Becker","year":"2007","unstructured":"Becker, R., Meidner, D., Vexler, B.: Efficient numerical solution of parabolic optimization problems by finite element methods. Optim. Methods Softw. 22(5), 813\u2013833 (2007)","journal-title":"Optim. Methods Softw."},{"key":"511_CR2","first-page":"2735","volume":"458","author":"M Besier","year":"2012","unstructured":"Besier, M., Rannacher, R.: Goal-oriented space-time adaptivity in the finite element Galerkin method for the computation of nonstationary incompressible flow. Int. J. Numer. Methods Fluids 458, 2735\u20132757 (2012)","journal-title":"Int. J. Numer. Methods Fluids"},{"key":"511_CR3","doi-asserted-by":"crossref","unstructured":"Brezinski, C., Van Iseghem, J.: A taste of Pad\u00e9 approximation. In: Acta numerica, 1995, Acta Numer., pp. 53\u2013103. Cambridge University Press, Cambridge (1995)","DOI":"10.1017\/S096249290000252X"},{"issue":"3\u20134","key":"511_CR4","first-page":"690","volume":"4","author":"K Chrysafinos","year":"2007","unstructured":"Chrysafinos, K.: Discontinuous Galerkin approximations for distributed optimal control problems constrained by parabolic PDEs. Int. J. Numer. Anal. Model. 4(3\u20134), 690\u2013712 (2007)","journal-title":"Int. J. Numer. Anal. Model."},{"key":"511_CR5","doi-asserted-by":"crossref","unstructured":"Ciarlet, P.G.: The finite element method for elliptic problems. Classics in Applied Mathematics, vol. 40. SIAM, Philadelphia, PA (2002)","DOI":"10.1137\/1.9780898719208"},{"key":"511_CR6","volume-title":"Linear Operators: Part I. Wiley Classics Library","author":"N Dunford","year":"1988","unstructured":"Dunford, N., Schwartz, J.T.: Linear Operators: Part I. Wiley Classics Library. Wiley, New York (1988)"},{"issue":"1","key":"511_CR7","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1137\/0728003","volume":"28","author":"K Eriksson","year":"1991","unstructured":"Eriksson, K., Johnson, C.: Adaptive finite element methods for parabolic problems I: a linear model problem. SIAM J. Numer. Anal. 28(1), 43\u201377 (1991)","journal-title":"SIAM J. Numer. Anal."},{"issue":"3","key":"511_CR8","doi-asserted-by":"crossref","first-page":"706","DOI":"10.1137\/0732033","volume":"32","author":"K Eriksson","year":"1995","unstructured":"Eriksson, K., Johnson, C.: Adaptive finite element methods for parabolic problems II: Optimal error estimates in $$L_\\infty L_2$$ and $$L_\\infty L_\\infty $$ . SIAM J. Numer. Anal. 32(3), 706\u2013740 (1995)","journal-title":"SIAM J. Numer. Anal."},{"key":"511_CR9","doi-asserted-by":"crossref","first-page":"611","DOI":"10.1051\/m2an\/1985190406111","volume":"19","author":"K Eriksson","year":"1985","unstructured":"Eriksson, K., Johnson, C., Thom\u00e9e, V.: Time discretization of parabolic problems by the discontinuous Galerkin method. M2AN Math. Model. Numer. Anal. 19, 611\u2013643 (1985)","journal-title":"M2AN Math. Model. Numer. Anal."},{"issue":"3","key":"511_CR10","doi-asserted-by":"crossref","first-page":"515","DOI":"10.1137\/0722031","volume":"22","author":"R Frank","year":"1985","unstructured":"Frank, R., Schneid, J., \u00dcberhuber, C.W.: Order results for implicit Runge\u2013Kutta methods applied to stiff systems. SIAM J. Numer. Anal. 22(3), 515\u2013534 (1985)","journal-title":"SIAM J. Numer. Anal."},{"issue":"1\u20132","key":"511_CR11","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1016\/S0377-0427(99)00134-X","volume":"111","author":"E Hairer","year":"1999","unstructured":"Hairer, E., Wanner, G.: Stiff differential equations solved by Radau methods. J. Comput. Appl. Math. 111(1\u20132), 93\u2013111 (1999)","journal-title":"J. Comput. Appl. Math."},{"issue":"1","key":"511_CR12","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1515\/jnum.2011.003","volume":"19","author":"S Hussain","year":"2011","unstructured":"Hussain, S., Schieweck, F., Turek, S.: Higher order Galerkin time discretizations and fast multigrid solvers for the heat equation. J. Numer. Math. 19(1), 41\u201362 (2011)","journal-title":"J. Numer. Math."},{"key":"511_CR13","doi-asserted-by":"crossref","unstructured":"Lang, J.: Adaptive Multilevel Solution of Nonlinear Parabolic PDE Systems. Lecture Notes in Computational Science and Engineering, vol. 16. Springer, Berlin (2001)","DOI":"10.1007\/978-3-662-04484-1"},{"key":"511_CR14","doi-asserted-by":"crossref","unstructured":"Lesaint, P., Raviart, P.-A.: On a finite element method for solving the neutron transport equation. In: Mathematical Aspects of Finite Elements in Partial Differential Equations, pp. 89\u2013123. Academic Press, New York (1974)","DOI":"10.1016\/B978-0-12-208350-1.50008-X"},{"issue":"1","key":"511_CR15","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\u2013time finite element methods for parabolic optimization problems. SIAM J. Control Optim. 46(1), 116\u2013142 (2007)","journal-title":"SIAM J. Control Optim."},{"issue":"3","key":"511_CR16","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\u2013time finite element approximation of parabolic optimal control problems. Part I: problems without control constraints. SIAM J. Control Optim. 47(3), 1150\u20131177 (2008)","journal-title":"SIAM J. Control Optim."},{"issue":"3","key":"511_CR17","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\u2013time finite element approximation of parabolic optimal control problems. Part II: problems with control constraints. SIAM J. Control Optim. 47(3), 1301\u20131329 (2008)","journal-title":"SIAM J. Control Optim."},{"issue":"5","key":"511_CR18","doi-asserted-by":"crossref","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\u2013Nicolson Scheme for parabolic optimal control problems. SIAM J. Control Optim. 49(5), 2183\u20132211 (2011)","journal-title":"SIAM J. Control Optim."},{"issue":"2","key":"511_CR19","doi-asserted-by":"crossref","first-page":"345","DOI":"10.1007\/s00211-011-0409-9","volume":"120","author":"I Neitzel","year":"2011","unstructured":"Neitzel, I., Vexler, B.: A priori error estimates for space-time finite element discretization of semilinear parabolic optimal control problems. Numer. Math. 120(2), 345\u2013386 (2011)","journal-title":"Numer. Math."},{"key":"511_CR20","unstructured":"Perron, O.: Die Lehre von den Kettenbr\u00fcchen. Band II. Analytisch-funktionentheoretische Kettenbr\u00fcche, 3rd edn. B. G. Teubner Verlagsgesellschaft, Stuttgart (1957)"},{"issue":"2","key":"511_CR21","doi-asserted-by":"crossref","first-page":"143","DOI":"10.1007\/s002459900102","volume":"39","author":"JP Raymond","year":"1999","unstructured":"Raymond, J.P., Zidani, H.: Hamiltonian Pontryagin\u2019s principles for control problems governed by semilinear parabolic equations. Appl. Math. Optim. 39(2), 143\u2013177 (1999)","journal-title":"Appl. Math. Optim."},{"issue":"1","key":"511_CR22","doi-asserted-by":"crossref","first-page":"369","DOI":"10.1137\/060670468","volume":"30","author":"M Schmich","year":"2008","unstructured":"Schmich, M., Vexler, B.: Adaptivity with dynamic meshes for space\u2013time finite element discretizations of parabolic equations. SIAM J. Sci. Comput. 30(1), 369\u2013393 (2008)","journal-title":"SIAM J. Sci. Comput."},{"issue":"3","key":"511_CR23","doi-asserted-by":"crossref","first-page":"837","DOI":"10.1137\/S0036142999352394","volume":"38","author":"D Sch\u00f6tzau","year":"2000","unstructured":"Sch\u00f6tzau, D., Schwab, C.: Time discretization of parabolic problems by the $$hp$$ -version of the discontinuous Galerkin finite element method. SIAM J. Numer. Anal. 38(3), 837\u2013875 (2000)","journal-title":"SIAM J. Numer. Anal."},{"key":"511_CR24","doi-asserted-by":"crossref","first-page":"475","DOI":"10.1007\/s00211-009-0285-8","volume":"115","author":"D Sch\u00f6tzau","year":"2010","unstructured":"Sch\u00f6tzau, D., Wihler, T.: A posteriori error estimation for hp-version time-stepping methods for parabolic partial differential equations. Numer. Math. 115, 475\u2013509 (2010)","journal-title":"Numer. Math."},{"key":"511_CR25","unstructured":"Gascoigne: The finite element toolkit. http:\/\/www.gascoigne.uni-hd.de"},{"key":"511_CR26","unstructured":"RoDoBo: A C++ library for optimization with stationary and nonstationary PDEs with interface to Gascoigne [25]. http:\/\/www.rodobo.uni-hd.de"},{"key":"511_CR27","unstructured":"Thom\u00e9e, V.: Galerkin Finite Element Methods for Parabolic Problems. Spinger Series in Computational Mathematics, vol. 25. Springer, Berlin (2006)"},{"issue":"4","key":"511_CR28","doi-asserted-by":"crossref","first-page":"475","DOI":"10.1007\/BF01932026","volume":"18","author":"G Wanner","year":"1978","unstructured":"Wanner, G., Hairer, E., N\u00f8rsett, S.P.: Order stars and stability theorems. BIT 18(4), 475\u2013489 (1978)","journal-title":"BIT"},{"issue":"4","key":"511_CR29","doi-asserted-by":"crossref","first-page":"449","DOI":"10.1093\/imanum\/7.4.449","volume":"7","author":"AJ Wathen","year":"1987","unstructured":"Wathen, A.J.: Realistic eigenvalue bounds for the Galerkin mass matrix. IMA J. Numer. Anal. 7(4), 449\u2013457 (1987)","journal-title":"IMA J. Numer. Anal."},{"issue":"49\u201350","key":"511_CR30","doi-asserted-by":"crossref","first-page":"6685","DOI":"10.1016\/S0045-7825(01)00258-4","volume":"190","author":"T Werder","year":"2001","unstructured":"Werder, T., Gerdes, K., Sch\u00f6tzau, D., Schwab, C.: $$hp$$ -Discontinuous Galerkin time stepping for parabolic problems. Comput. Methods Appl. Mech. Engrg. 190(49\u201350), 6685\u20136708 (2001)","journal-title":"Comput. Methods Appl. Mech. Engrg."},{"key":"511_CR31","doi-asserted-by":"crossref","unstructured":"Wloka, J.: Partial Differential Equations. Cambridge University Press, Cambridge (1987) (Translated from the German by C.B. Thomas and M.J. Thomas)","DOI":"10.1017\/CBO9781139171755"},{"issue":"3","key":"511_CR32","doi-asserted-by":"crossref","first-page":"583","DOI":"10.1137\/0721040","volume":"21","author":"TJ Ypma","year":"1984","unstructured":"Ypma, T.J.: Local convergence of inexact Newton methods. SIAM J. Numer. Anal. 21(3), 583\u2013590 (1984)","journal-title":"SIAM J. Numer. Anal."}],"container-title":["Numerische Mathematik"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-012-0511-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00211-012-0511-7\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-012-0511-7","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,7,4]],"date-time":"2019-07-04T21:45:52Z","timestamp":1562276752000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00211-012-0511-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2012,10,23]]},"references-count":32,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2013,5]]}},"alternative-id":["511"],"URL":"https:\/\/doi.org\/10.1007\/s00211-012-0511-7","relation":{},"ISSN":["0029-599X","0945-3245"],"issn-type":[{"value":"0029-599X","type":"print"},{"value":"0945-3245","type":"electronic"}],"subject":[],"published":{"date-parts":[[2012,10,23]]}}}