{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T18:50:37Z","timestamp":1776797437087,"version":"3.51.2"},"reference-count":32,"publisher":"American Mathematical Society (AMS)","issue":"294","license":[{"start":{"date-parts":[[2015,12,17]],"date-time":"2015-12-17T00:00:00Z","timestamp":1450310400000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n We use the elliptic reconstruction technique in combination with a duality approach to prove a posteriori error estimates for fully discrete backward Euler scheme for linear parabolic equations. As an application, we combine our result with the residual based estimators from the a posteriori estimation for elliptic problems to derive space-error indicators and thus a fully practical version of the estimators bounding the error in the\n \n \n \n \n \n \n L<\/mml:mi>\n <\/mml:mrow>\n \n \n \u221e\n \n <\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n (<\/mml:mo>\n 0<\/mml:mn>\n ,<\/mml:mo>\n T<\/mml:mi>\n ;<\/mml:mo>\n \n \n L<\/mml:mi>\n <\/mml:mrow>\n 2<\/mml:mn>\n <\/mml:msub>\n (<\/mml:mo>\n \n \u03a9\n \n <\/mml:mi>\n )<\/mml:mo>\n )<\/mml:mo>\n <\/mml:mrow>\n \\mathrm {L}_{\\infty }(0,T;\\mathrm {L}_2(\\varOmega ))<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n norm. These estimators, which are of optimal order, extend those introduced by Eriksson and Johnson in 1991 by taking into account the error induced by the mesh changes and allowing for a more flexible use of the elliptic estimators. For comparison with previous results we derive also an energy-based a posteriori estimate for the\n \n \n \n \n \n \n L<\/mml:mi>\n <\/mml:mrow>\n \n \n \u221e\n \n <\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n (<\/mml:mo>\n 0<\/mml:mn>\n ,<\/mml:mo>\n T<\/mml:mi>\n ;<\/mml:mo>\n \n \n L<\/mml:mi>\n <\/mml:mrow>\n 2<\/mml:mn>\n <\/mml:msub>\n (<\/mml:mo>\n \n \u03a9\n \n <\/mml:mi>\n )<\/mml:mo>\n )<\/mml:mo>\n <\/mml:mrow>\n \\mathrm {L}_{\\infty }(0,T;\\mathrm {L}_2(\\varOmega ))<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -error which simplifies a previous one given by Lakkis and Makridakis in 2006. We then compare both estimators (duality vs. energy) in practical situations and draw conclusions.\n <\/p>","DOI":"10.1090\/s0025-5718-2014-02912-8","type":"journal-article","created":{"date-parts":[[2014,12,17]],"date-time":"2014-12-17T09:19:15Z","timestamp":1418807955000},"page":"1537-1569","source":"Crossref","is-referenced-by-count":14,"title":["A comparison of duality and energy a posteriori estimates for L_{\u221e}(0,T;L\u2082(\\varOmega)) in parabolic problems"],"prefix":"10.1090","volume":"84","author":[{"given":"Omar","family":"Lakkis","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Charalambos","family":"Makridakis","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Tristan","family":"Pryer","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2014,12,17]]},"reference":[{"issue":"254","key":"1","doi-asserted-by":"publisher","first-page":"511","DOI":"10.1090\/S0025-5718-05-01800-4","article-title":"A posteriori error estimates for the Crank-Nicolson method for parabolic equations","volume":"75","author":"Akrivis, Georgios","year":"2006","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"2","series-title":"Pure and Applied Mathematics (New York)","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1002\/9781118032824","volume-title":"A posteriori error estimation in finite element analysis","author":"Ainsworth, Mark","year":"2000","ISBN":"https:\/\/id.crossref.org\/isbn\/047129411X"},{"issue":"251","key":"3","doi-asserted-by":"publisher","first-page":"1117","DOI":"10.1090\/S0025-5718-04-01697-7","article-title":"A posteriori analysis of the finite element discretization of some parabolic equations","volume":"74","author":"Bergam, A.","year":"2005","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"274","key":"4","doi-asserted-by":"publisher","first-page":"761","DOI":"10.1090\/S0025-5718-2010-02444-5","article-title":"Quasi-optimal and robust a posteriori error estimates in \ud835\udc3f^{\u221e}(\ud835\udc3f\u00b2) for the approximation of Allen-Cahn equations past singularities","volume":"80","author":"Bartels, S\u00f6ren","year":"2011","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"5","doi-asserted-by":"publisher","first-page":"153","DOI":"10.1002\/sapm1939181153","article-title":"A function related to the series for \ud835\udc52^{\ud835\udc52^{\ud835\udc65}}","volume":"18","author":"Epstein, Leo F.","year":"1939","journal-title":"J. Math. Phys. Mass. Inst. Tech.","ISSN":"https:\/\/id.crossref.org\/issn\/0097-1421","issn-type":"print"},{"key":"6","isbn-type":"print","volume-title":"Finite elements","author":"Braess, Dietrich","year":"2001","ISBN":"https:\/\/id.crossref.org\/isbn\/0521011957","edition":"2"},{"key":"7","unstructured":"[BS07] Susanne C. Brenner and L. Ridgway Scott, The Mathematical Theory of Finite Element Methods, third ed., Texts in Applied Mathematics, vol. 15, Springer-Verlag, New York, 2007."},{"issue":"3","key":"8","doi-asserted-by":"publisher","first-page":"437","DOI":"10.1051\/m2an:2004021","article-title":"A posteriori error analysis of the fully discretized time-dependent Stokes equations","volume":"38","author":"Bernardi, Christine","year":"2004","journal-title":"M2AN Math. Model. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0764-583X","issn-type":"print"},{"issue":"247","key":"9","doi-asserted-by":"publisher","first-page":"1167","DOI":"10.1090\/S0025-5718-04-01634-5","article-title":"An adaptive finite element algorithm with reliable and efficient error control for linear parabolic problems","volume":"73","author":"Chen, Zhiming","year":"2004","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"10","series-title":"Studies in Mathematics and its Applications, Vol. 4","isbn-type":"print","volume-title":"The finite element method for elliptic problems","author":"Ciarlet, Philippe G.","year":"1978","ISBN":"https:\/\/id.crossref.org\/isbn\/0444850287"},{"issue":"5","key":"11","doi-asserted-by":"publisher","first-page":"2524","DOI":"10.1137\/07069047X","article-title":"Quasi-optimal convergence rate for an adaptive finite element method","volume":"46","author":"Cascon, J. Manuel","year":"2008","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"3","key":"12","doi-asserted-by":"publisher","first-page":"805","DOI":"10.1137\/S0036142900375438","article-title":"Postprocessing the linear finite element method","volume":"40","author":"de Frutos, Javier","year":"2002","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"3","key":"13","doi-asserted-by":"publisher","first-page":"2157","DOI":"10.1137\/070708792","article-title":"A posteriori error estimates in the maximum norm for parabolic problems","volume":"47","author":"Demlow, Alan","year":"2009","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"271","key":"14","doi-asserted-by":"publisher","first-page":"1233","DOI":"10.1090\/S0025-5718-10-02346-X","article-title":"Sharply local pointwise a posteriori error estimates for parabolic problems","volume":"79","author":"Demlow, Alan","year":"2010","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"2","key":"15","doi-asserted-by":"publisher","first-page":"185","DOI":"10.1007\/s00211-010-0349-9","article-title":"Convergence and quasi-optimality of an adaptive finite element method for controlling \ud835\udc3f\u2082 errors","volume":"117","author":"Demlow, Alan","year":"2011","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"1","key":"16","doi-asserted-by":"publisher","first-page":"43","DOI":"10.1137\/0728003","article-title":"Adaptive finite element methods for parabolic problems. I. A linear model problem","volume":"28","author":"Eriksson, Kenneth","year":"1991","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"2","key":"17","doi-asserted-by":"publisher","first-page":"353","DOI":"10.1051\/m2an:2008048","article-title":"A posteriori error analysis of Euler-Galerkin approximations to coupled elliptic-parabolic problems","volume":"43","author":"Ern, Alexandre","year":"2009","journal-title":"M2AN Math. Model. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0764-583X","issn-type":"print"},{"key":"18","series-title":"Graduate Studies in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1090\/gsm\/019","volume-title":"Partial differential equations","volume":"19","author":"Evans, Lawrence C.","year":"1998","ISBN":"https:\/\/id.crossref.org\/isbn\/0821807722"},{"key":"19","doi-asserted-by":"crossref","unstructured":"[GL10] Emmanuil H. Georgoulis and Omar Lakkis, A posteriori error bounds for discontinuous galerkin methods for quasilinear parabolic problems, Numerical Mathematics and Advanced Applications 2009 (Gunilla Kreiss, Per L\u00f6tstedt, Axel M\u00e5lqvist, and Maya Neytcheva, eds.), Springer Berlin Heidelberg, 2010, pp. 351\u2013358 (English).","DOI":"10.1007\/978-3-642-11795-4_37"},{"issue":"2","key":"20","doi-asserted-by":"publisher","first-page":"427","DOI":"10.1137\/080722461","article-title":"A posteriori error control for discontinuous Galerkin methods for parabolic problems","volume":"49","author":"Georgoulis, Emmanuil H.","year":"2011","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"4","key":"21","doi-asserted-by":"publisher","first-page":"1375","DOI":"10.1093\/imanum\/drr026","article-title":"Design and convergence analysis for an adaptive discretization of the heat equation","volume":"32","author":"Kreuzer, Christian","year":"2012","journal-title":"IMA J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0272-4979","issn-type":"print"},{"issue":"3","key":"22","doi-asserted-by":"publisher","first-page":"275","DOI":"10.1016\/0377-0427(94)90034-5","article-title":"A recursive approach to local mesh refinement in two and three dimensions","volume":"55","author":"Kossaczk\u00fd, Igor","year":"1994","journal-title":"J. Comput. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0377-0427","issn-type":"print"},{"issue":"256","key":"23","doi-asserted-by":"publisher","first-page":"1627","DOI":"10.1090\/S0025-5718-06-01858-8","article-title":"Elliptic reconstruction and a posteriori error estimates for fully discrete linear parabolic problems","volume":"75","author":"Lakkis, Omar","year":"2006","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"4","key":"24","doi-asserted-by":"publisher","first-page":"421","DOI":"10.1002\/num.10053","article-title":"Local a posteriori error estimates and adaptive control of pollution effects","volume":"19","author":"Liao, Xiaohai","year":"2003","journal-title":"Numer. Methods Partial Differential Equations","ISSN":"https:\/\/id.crossref.org\/issn\/0749-159X","issn-type":"print"},{"issue":"1","key":"25","doi-asserted-by":"publisher","first-page":"246","DOI":"10.1093\/imanum\/drq019","article-title":"Gradient recovery in adaptive finite-element methods for parabolic problems","volume":"32","author":"Lakkis, Omar","year":"2012","journal-title":"IMA J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0272-4979","issn-type":"print"},{"issue":"4","key":"26","doi-asserted-by":"publisher","first-page":"1585","DOI":"10.1137\/S0036142902406314","article-title":"Elliptic reconstruction and a posteriori error estimates for parabolic problems","volume":"41","author":"Makridakis, Charalambos","year":"2003","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"3-4","key":"27","doi-asserted-by":"publisher","first-page":"223","DOI":"10.1016\/S0045-7825(98)00121-2","article-title":"Adaptive finite elements for a linear parabolic problem","volume":"167","author":"Picasso, Marco","year":"1998","journal-title":"Comput. Methods Appl. Mech. Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0045-7825","issn-type":"print"},{"key":"28","series-title":"Lecture Notes in Computational Science and Engineering","isbn-type":"print","volume-title":"Design of adaptive finite element software","volume":"42","author":"Schmidt, Alfred","year":"2005","ISBN":"https:\/\/id.crossref.org\/isbn\/354022842X"},{"key":"29","series-title":"Springer Series in Computational Mathematics","isbn-type":"print","volume-title":"Galerkin finite element methods for parabolic problems","volume":"25","author":"Thom\u00e9e, Vidar","year":"2006","ISBN":"https:\/\/id.crossref.org\/isbn\/9783540331216","edition":"2"},{"key":"30","unstructured":"[Ver96] R\u00fcdiger Verf\u00fcrth, A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, Wiley-Teubner, Chichester-Stuttgart, 1996."},{"issue":"3","key":"31","doi-asserted-by":"publisher","first-page":"195","DOI":"10.1007\/s10092-003-0073-2","article-title":"A posteriori error estimates for finite element discretizations of the heat equation","volume":"40","author":"Verf\u00fcrth, R.","year":"2003","journal-title":"Calcolo","ISSN":"https:\/\/id.crossref.org\/issn\/0008-0624","issn-type":"print"},{"key":"32","doi-asserted-by":"publisher","first-page":"723","DOI":"10.1137\/0710062","article-title":"A priori \ud835\udc3f\u2082 error estimates for Galerkin approximations to parabolic partial differential equations","volume":"10","author":"Wheeler, Mary Fanett","year":"1973","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2015-84-294\/S0025-5718-2014-02912-8\/S0025-5718-2014-02912-8.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2015-84-294\/S0025-5718-2014-02912-8\/S0025-5718-2014-02912-8.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T18:23:35Z","timestamp":1776795815000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2015-84-294\/S0025-5718-2014-02912-8\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,12,17]]},"references-count":32,"journal-issue":{"issue":"294","published-print":{"date-parts":[[2015,7]]}},"alternative-id":["S0025-5718-2014-02912-8"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-2014-02912-8","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2014,12,17]]}}}