{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T07:42:46Z","timestamp":1776843766525,"version":"3.51.2"},"reference-count":10,"publisher":"American Mathematical Society (AMS)","issue":"224","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    Using the abstract framework of [R. Verf\u00fcrth: A posteriori error estimates for nonlinear problems. Finite element discretizations of elliptic equations. Math. Comput. 62 (206), 445 \u2013 475 (1994)] we analyze a residual a posteriori error estimator for space-time finite element discretizations of quasilinear parabolic pdes. The estimator gives global upper and local lower bounds on the error of the numerical solution. The finite element discretizations in particular cover the so-called\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"theta\">\n                        <mml:semantics>\n                          <mml:mi>\n                            \u03b8\n                            \n                          <\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">\\theta<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -scheme, which includes the implicit and explicit Euler methods and the Crank-Nicholson scheme.\n                  <\/p>","DOI":"10.1090\/s0025-5718-98-01011-4","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:14:44Z","timestamp":1027707284000},"page":"1335-1360","source":"Crossref","is-referenced-by-count":45,"title":["A posteriori error estimates for nonlinear problems. \ud835\udc3f^{\ud835\udc5f}(0,\ud835\udc47;\ud835\udc3f^{\ud835\udf0c}(\u03a9))-error estimates for finite element discretizations of parabolic equations"],"prefix":"10.1090","volume":"67","author":[{"given":"R.","family":"Verf\u00fcrth","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[1998]]},"reference":[{"key":"1","series-title":"Pure and Applied Mathematics, Vol. 65","volume-title":"Sobolev spaces","author":"Adams, Robert A.","year":"1975"},{"key":"2","isbn-type":"print","doi-asserted-by":"publisher","first-page":"9","DOI":"10.1007\/978-3-663-11336-2_1","article-title":"Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems","author":"Amann, Herbert","year":"1993","ISBN":"https:\/\/id.crossref.org\/isbn\/3815420458"},{"issue":"1","key":"3","doi-asserted-by":"publisher","first-page":"31","DOI":"10.1051\/m2an\/1991250100311","article-title":"Estimateurs a posteriori d\u2019erreur pour le calcul adaptatif d\u2019\u00e9coulements quasi-newtoniens","volume":"25","author":"Baranger, Jacques","year":"1991","journal-title":"RAIRO Mod\\'{e}l. Math. Anal. Num\\'{e}r.","ISSN":"https:\/\/id.crossref.org\/issn\/0764-583X","issn-type":"print"},{"key":"4","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"},{"key":"5","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1051\/m2an\/197509R200771","article-title":"Approximation by finite element functions using local regularization","volume":"9","author":"Cl\u00e9ment, Ph.","year":"1975","journal-title":"Rev. Fran\\c{c}aise Automat. Informat. Recherche Op\\'{e}rationnelle S\\'{e}r. Rouge Anal. Num\\'{e}r.","ISSN":"https:\/\/id.crossref.org\/issn\/0397-9342","issn-type":"print"},{"key":"6","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-58090-1","volume-title":"Mathematical analysis and numerical methods for science and technology. Vol. 5","author":"Dautray, Robert","year":"1992","ISBN":"https:\/\/id.crossref.org\/isbn\/354050205X"},{"issue":"6","key":"7","doi-asserted-by":"publisher","first-page":"1729","DOI":"10.1137\/0732078","article-title":"Adaptive finite element methods for parabolic problems. IV. Nonlinear problems","volume":"32","author":"Eriksson, Kenneth","year":"1995","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"6","key":"8","doi-asserted-by":"publisher","first-page":"1750","DOI":"10.1137\/0732079","article-title":"Adaptive finite element methods for parabolic problems. V. Long-time integration","volume":"32","author":"Eriksson, Kenneth","year":"1995","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"206","key":"9","doi-asserted-by":"publisher","first-page":"445","DOI":"10.2307\/2153518","article-title":"A posteriori error estimates for nonlinear problems. Finite element discretizations of elliptic equations","volume":"62","author":"Verf\u00fcrth, R.","year":"1994","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"10","unstructured":"[10] \\bysame, A posteriori error estimates for nonlinear problems. Finite element discretizations of parabolic equations. Report 180, Ruhr-Universit\u00e4t Bochum (1995) http:\/\/www.num1.ruhr-uni-bochum.de\/rv\/preprints.html"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/1998-67-224\/S0025-5718-98-01011-4\/S0025-5718-98-01011-4.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/1998-67-224\/S0025-5718-98-01011-4\/S0025-5718-98-01011-4.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T21:51:12Z","timestamp":1776721872000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/1998-67-224\/S0025-5718-98-01011-4\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1998]]},"references-count":10,"journal-issue":{"issue":"224","published-print":{"date-parts":[[1998,10]]}},"alternative-id":["S0025-5718-98-01011-4"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-98-01011-4","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":[[1998]]}}}