{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,14]],"date-time":"2026-05-14T22:46:14Z","timestamp":1778798774497,"version":"3.51.4"},"reference-count":35,"publisher":"Springer Science and Business Media LLC","issue":"5","license":[{"start":{"date-parts":[[2017,8,10]],"date-time":"2017-08-10T00:00:00Z","timestamp":1502323200000},"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":["Found Comput Math"],"published-print":{"date-parts":[[2018,10]]},"DOI":"10.1007\/s10208-017-9364-x","type":"journal-article","created":{"date-parts":[[2017,8,10]],"date-time":"2017-08-10T18:23:13Z","timestamp":1502389393000},"page":"1109-1130","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":25,"title":["Runge\u2013Kutta Time Discretization of Nonlinear Parabolic Equations Studied via Discrete Maximal Parabolic Regularity"],"prefix":"10.1007","volume":"18","author":[{"given":"Peer C.","family":"Kunstmann","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Buyang","family":"Li","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Christian","family":"Lubich","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2017,8,10]]},"reference":[{"key":"9364_CR1","volume-title":"Sobolev Spaces","author":"RA Adams","year":"2003","unstructured":"R. A. Adams and J. J. F. Fournier, Sobolev Spaces, 2nd ed., Academic Press, Amsterdam, 2003.","edition":"2"},{"key":"9364_CR2","doi-asserted-by":"crossref","first-page":"1527","DOI":"10.1090\/mcom\/3228","volume":"86","author":"G Akrivis","year":"2017","unstructured":"G. Akrivis, B. Li, and C. Lubich, Combining maximal regularity and energy estimates for time discretizations of quasilinear parabolic equations. Math. Comp., 86 (2017), pp. 1527\u20131552.","journal-title":"Math. Comp."},{"key":"9364_CR3","doi-asserted-by":"crossref","first-page":"429","DOI":"10.1007\/s00211-011-0363-6","volume":"118","author":"G Akrivis","year":"2011","unstructured":"G. Akrivis, C. Makridakis, and R. H. Nochetto, Galerkin and Runge\u2013Kutta methods: unified formulation, a posteriori error estimates and nodal superconvergence. Numer. Math., 118 (2011), pp. 429\u2013456.","journal-title":"Numer. Math."},{"key":"9364_CR4","doi-asserted-by":"crossref","first-page":"669","DOI":"10.1081\/NFA-120016264","volume":"23","author":"A Ashyralyev","year":"2002","unstructured":"A. Ashyralyev, S. Piskarev, and L. Weis, On well-posedness of difference schemes for abstract parabolic equations in $$L_p([0, T];E)$$ L p ( [ 0 , T ] \u037e E ) spaces, Numer. Funct. Anal. Optim., 23 (2002), pp. 669\u2013693.","journal-title":"Numer. Funct. Anal. Optim."},{"key":"9364_CR5","volume-title":"Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, Applied Mathematical Sciences 183","author":"F Boyer","year":"2013","unstructured":"F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, Applied Mathematical Sciences 183, Springer, New York, 2013."},{"key":"9364_CR6","doi-asserted-by":"crossref","unstructured":"Y. Z. Chen and L. C. Wu, Second Order Elliptic Equations and Elliptic Systems, Translations of Mathematical Monographs, Volume 174, AMS, 1998.","DOI":"10.1090\/mmono\/174"},{"key":"9364_CR7","doi-asserted-by":"crossref","unstructured":"F. Cobos, Real interpolation and compactness, Revista Matematica de la Universidad Complutense de Madrid, 2 (1989), n\u00famero suplementario.","DOI":"10.5209\/rev_REMA.1989.v2.18076"},{"key":"9364_CR8","doi-asserted-by":"crossref","first-page":"394","DOI":"10.1093\/imanum\/drr017","volume":"32","author":"G Dziuk","year":"2012","unstructured":"G. Dziuk, C. Lubich, and D. Mansour, Runge-Kutta time discretization of parabolic differential equations on evolving surfaces. IMA J. Numer. Anal., 32 (2012), pp. 394\u2013416.","journal-title":"IMA J. Numer. Anal."},{"key":"9364_CR9","volume-title":"Partial Differential Equations, second edition, Graduate Studies in Mathematics 19","author":"LC Evans","year":"2010","unstructured":"L. C. Evans, Partial Differential Equations, second edition, Graduate Studies in Mathematics 19, American Mathematical Society, Providence, RI, 2010."},{"key":"9364_CR10","doi-asserted-by":"crossref","first-page":"441","DOI":"10.1007\/s00211-005-0585-6","volume":"100","author":"X Feng","year":"2005","unstructured":"X. Feng, M. von Oehsen, and A. Prohl, Rate of convergence of regularization procedures and finite element approximations for the total variation flow. Numer. Math., 100 (2005), pp. 441\u2013456.","journal-title":"Numer. Math."},{"key":"9364_CR11","doi-asserted-by":"crossref","first-page":"533","DOI":"10.1051\/m2an:2003041","volume":"37","author":"X Feng","year":"2003","unstructured":"X. Feng and A. Prohl, Analysis of total variation flow and its finite element approximations. ESAIM Math. Model. Numer. Anal., 37 (2003), pp. 533\u2013556.","journal-title":"ESAIM Math. Model. Numer. Anal."},{"key":"9364_CR12","doi-asserted-by":"crossref","first-page":"677","DOI":"10.1137\/040616553","volume":"44","author":"M Geissert","year":"2006","unstructured":"M. Geissert, Discrete maximal $$L^p$$ L p regularity for finite element operators, SIAM J. Numer. Anal., 44 (2006), pp. 677\u2013698.","journal-title":"SIAM J. Numer. Anal."},{"key":"9364_CR13","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1007\/s00211-007-0110-1","volume":"108","author":"M Geissert","year":"2007","unstructured":"M. Geissert, Applications of discrete maximal $$L^p$$ L p regularity for finite element operators, Numer. Math., 108 (2007), pp. 121\u2013149.","journal-title":"Numer. Math."},{"key":"9364_CR14","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-61798-0","volume-title":"Elliptic Partial Differential Equations of Second Order, reprint of the","author":"D Gilbarg","year":"2001","unstructured":"D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, reprint of the third edition, Springer\u2013Verlag, Berlin Heidelberg New York, 2001.","edition":"3"},{"key":"9364_CR15","unstructured":"E. Hairer and G. Wanner, Solving Ordinary Differential Equations II: Stiff and Differential\u2013Algebraic Problems, 2 $$^\\text{nd}$$ nd revised ed., Springer\u2013Verlag, Berlin Heidelberg, Springer Series in Computational Mathematics v. 14, 2002."},{"key":"9364_CR16","first-page":"241","volume":"234","author":"T Kemmochi","year":"2016","unstructured":"T. Kemmochi, Discrete maximal regularity for abstract Cauchy problems. Studia Math., 234 (2016), pp. 241\u2013263.","journal-title":"Studia Math."},{"key":"9364_CR17","unstructured":"T. Kemmochi and N. Saito, Discrete maximal regularity and the finite element method for parabolic equations. Preprint. arXiv:1602.06864"},{"key":"9364_CR18","doi-asserted-by":"crossref","first-page":"3600","DOI":"10.1137\/15M1040918","volume":"54","author":"B Kov\u00e1cs","year":"2016","unstructured":"B. Kov\u00e1cs, B. Li, and C. Lubich, A-stable time discretizations preserve maximal parabolic regularity, SIAM J. Numer. Anal., 54 (2016), pp. 3600\u20133624.","journal-title":"SIAM J. Numer. Anal."},{"key":"9364_CR19","doi-asserted-by":"crossref","unstructured":"P. C. Kunstmann, Maximal $$L_p$$ L p -regularity for second order elliptic operators with uniformly continuous coefficients on domains, Progress in Nonlinear Differential Equations and Applications, Vol. 55, 293-305, Birkh\u00e4user 2003.","DOI":"10.1007\/978-3-0348-8085-5_20"},{"key":"9364_CR20","first-page":"65","volume":"2004","author":"PC Kunstmann","year":"1855","unstructured":"P. C. Kunstmann and L. Weis, Maximal $$L^p$$ L p regularity for Parabolic Equations, Fourier Multiplier Theorems and $$H^1$$ H 1 -Functional Calculus. Functional Analytic Methods for Evolution Equations, Lecture Notes in Mathematics, vol. 1855 (2004), pp. 65\u2013311.","journal-title":"Functional Analytic Methods for Evolution Equations, Lecture Notes in Mathematics"},{"key":"9364_CR21","doi-asserted-by":"crossref","first-page":"923","DOI":"10.1007\/s00211-016-0821-2","volume":"135","author":"D Leykekhman","year":"2017","unstructured":"D. Leykekhman and B. Vexler, Discrete maximal parabolic regularity for Galerkin finite element methods. Numer. Math., 135 (2017), pp 923\u2013952.","journal-title":"Numer. Math."},{"key":"9364_CR22","unstructured":"B. Li, Convergence of a decoupled mixed FEM for the dynamic Ginzburg\u2013Landau equations in nonsmooth domains with incompatible initial data. arXiv:1605.01208"},{"key":"9364_CR23","doi-asserted-by":"crossref","first-page":"489","DOI":"10.1007\/s00211-015-0698-5","volume":"131","author":"B Li","year":"2015","unstructured":"B. Li, Maximum-norm stability and maximal $$L^p$$ L p regularity of FEMs for parabolic equations with Lipschitz continuous coefficients, Numer. Math., 131 (2015), pp. 489-516.","journal-title":"Numer. Math."},{"key":"9364_CR24","first-page":"622","volume":"10","author":"B Li","year":"2013","unstructured":"B. Li and W. Sun, Error analysis of linearized semi-implicit Galerkin finite element methods for nonlinear parabolic equations. Int. J. Numer. Anal. & Modeling, 10 (2013), pp. 622\u2013633.","journal-title":"Int. J. Numer. Anal. & Modeling"},{"key":"9364_CR25","doi-asserted-by":"crossref","first-page":"2623","DOI":"10.1137\/13093769X","volume":"52","author":"B Li","year":"2014","unstructured":"B. Li and W. Sun, Linearized FE approximations to a nonlinear gradient flow. SIAM J. Numer. Anal., 52 (2014), pp. 2623\u20132646.","journal-title":"SIAM J. Numer. Anal."},{"key":"9364_CR26","doi-asserted-by":"crossref","first-page":"1418","DOI":"10.1137\/140958803","volume":"53","author":"B Li","year":"2015","unstructured":"B. Li and W. Sun, Regularity of the diffusion-dispersion tensor and error analysis of FEMs for a porous media flow, SIAM J. Numer. Anal., 53 (2015), pp. 1418\u20131437.","journal-title":"SIAM J. Numer. Anal."},{"key":"9364_CR27","doi-asserted-by":"crossref","first-page":"1071","DOI":"10.1090\/mcom\/3133","volume":"86","author":"B Li","year":"2017","unstructured":"B. Li and W. Sun, Maximal $$L^p$$ L p analysis of finite element solutions for parabolic equations with nonsmooth coefficients in convex polyhedra, Math. Comp., 86 (2017), pp. 1071\u20131102.","journal-title":"Math. Comp."},{"key":"9364_CR28","doi-asserted-by":"crossref","first-page":"340","DOI":"10.1016\/0022-0396(78)90005-0","volume":"30","author":"A Lichnewsky","year":"1978","unstructured":"A. Lichnewsky and R. Temam, Pseudosolutions of the time-dependent minimal surface problem. J. Differential Equations, 30 (1978), pp. 340\u2013364.","journal-title":"J. Differential Equations"},{"key":"9364_CR29","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1007\/BF02684796","volume":"19","author":"J-L Lions","year":"1964","unstructured":"J.-L. Lions and J. Peetre, Sur une classe d\u2019espaces d\u2019interpolation. Publications math\u00e9matiques de l\u2019IHES, 19 (1964), pp. 5\u201368.","journal-title":"Publications math\u00e9matiques de l\u2019IHES"},{"key":"9364_CR30","doi-asserted-by":"crossref","first-page":"601","DOI":"10.1090\/S0025-5718-1995-1284670-0","volume":"64","author":"C Lubich","year":"1995","unstructured":"C. Lubich and A. Ostermann, Runge\u2013Kutta approximations of quasi-linear parabolic equations, Math. Comp., 64 (1995), pp. 601\u2013627.","journal-title":"Math. Comp."},{"key":"9364_CR31","volume-title":"Analytic Semigroups and Optimal Regularity in Parabolic Problems","author":"A Lunardi","year":"1995","unstructured":"A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems. Birkh\u00e4user Verlag, Basel (1995)."},{"key":"9364_CR32","doi-asserted-by":"crossref","first-page":"367","DOI":"10.1016\/S0168-9274(01)00161-1","volume":"42","author":"A Ostermann","year":"2002","unstructured":"A. Ostermann and M. Thalhammer, Convergence of Runge\u2013Kutta methods for nonlinear parabolic equations, Appl. Numer. Math., 42 (2002), pp. 367\u2013380.","journal-title":"Appl. Numer. Math."},{"key":"9364_CR33","doi-asserted-by":"crossref","first-page":"161","DOI":"10.32917\/hmj\/1206128381","volume":"23","author":"J Pr\u00fcss","year":"1993","unstructured":"J. Pr\u00fcss and H. Sohr, Imaginary powers of elliptic second order differential operators in $$L^p$$ L p -spaces, Hiroshima Math. J., 23 (1993), pp. 161-192.","journal-title":"Hiroshima Math. J."},{"key":"9364_CR34","doi-asserted-by":"crossref","first-page":"793","DOI":"10.1007\/s00028-011-0111-5","volume":"11","author":"E Spadaro","year":"2011","unstructured":"E. Spadaro and U. Stefanelli, A variational view at the time-dependent minimal surface equation. J. Evolution Equations, 11 (2011), pp. 793\u2013809.","journal-title":"J. Evolution Equations"},{"key":"9364_CR35","doi-asserted-by":"crossref","first-page":"735","DOI":"10.1007\/PL00004457","volume":"319","author":"L Weis","year":"2001","unstructured":"L. Weis, Operator-valued Fourier multiplier theorems and maximal $$L_p$$ L p -regularity, Math. Ann., 319 (2001), pp. 735\u2013758.","journal-title":"Math. Ann."}],"container-title":["Foundations of Computational Mathematics"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10208-017-9364-x\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10208-017-9364-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10208-017-9364-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,10,14]],"date-time":"2020-10-14T21:03:23Z","timestamp":1602709403000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10208-017-9364-x"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,8,10]]},"references-count":35,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2018,10]]}},"alternative-id":["9364"],"URL":"https:\/\/doi.org\/10.1007\/s10208-017-9364-x","relation":{},"ISSN":["1615-3375","1615-3383"],"issn-type":[{"value":"1615-3375","type":"print"},{"value":"1615-3383","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,8,10]]}}}