{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T07:47:32Z","timestamp":1776844052103,"version":"3.51.2"},"reference-count":30,"publisher":"American Mathematical Society (AMS)","issue":"274","license":[{"start":{"date-parts":[[2011,11,18]],"date-time":"2011-11-18T00:00:00Z","timestamp":1321574400000},"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":"<p>\n                    Quasi-optimal a\u00a0posteriori error estimates in\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper L Superscript normal infinity Baseline left-parenthesis 0 comma upper T semicolon upper L squared left-parenthesis normal upper Omega right-parenthesis right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msup>\n                              <mml:mi>L<\/mml:mi>\n                              <mml:mi mathvariant=\"normal\">\n                                \u221e\n                                \n                              <\/mml:mi>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>T<\/mml:mi>\n                            <mml:mo>;<\/mml:mo>\n                            <mml:msup>\n                              <mml:mi>L<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi mathvariant=\"normal\">\n                              \u03a9\n                              \n                            <\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">L^\\infty (0,T;L^2(\\Omega ))<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    are derived for the finite element approximation of Allen-Cahn equations. The estimates depend on the inverse of a small parameter only in a low order polynomial and are valid past topological changes of the evolving interface. The error analysis employs an elliptic reconstruction of the approximate solution and applies to a large class of conforming, nonconforming, mixed, and discontinuous Galerkin methods. Numerical experiments illustrate the theoretical results.\n                  <\/p>","DOI":"10.1090\/s0025-5718-2010-02444-5","type":"journal-article","created":{"date-parts":[[2010,12,29]],"date-time":"2010-12-29T14:39:48Z","timestamp":1293633588000},"page":"761-780","source":"Crossref","is-referenced-by-count":25,"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"],"prefix":"10.1090","volume":"80","author":[{"given":"S\u00f6ren","family":"Bartels","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"R\u00fcdiger","family":"M\u00fcller","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2010,11,18]]},"reference":[{"key":"1","doi-asserted-by":"crossref","unstructured":"[AC79] Sam Allen and John Cahn. A microscopic theory for antiphase boundary motion and its application to antiphase domain coarsening. Acta Metall., 27:1084\u20131095, 1979.","DOI":"10.1016\/0001-6160(79)90196-2"},{"issue":"2","key":"2","doi-asserted-by":"publisher","first-page":"637","DOI":"10.1512\/iumj.1993.42.42028","article-title":"The spectrum of the Cahn-Hilliard operator for generic interface in higher space dimensions","volume":"42","author":"Alikakos, Nicholas D.","year":"1993","journal-title":"Indiana Univ. Math. 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Preprint available at www.bartels.ins.uni-bonn.de, 2008."},{"key":"6","unstructured":"[BMO09a] S\u00f6ren Bartels, R\u00fcdiger M\u00fcller, and Christoph Ortner. Robust a posteriori error estimates for the approximation of Cahn-Hilliard equations past topological changes. In preparation, 2009."},{"key":"7","unstructured":"[BMO09b] S\u00f6ren Bartels, R\u00fcdiger M\u00fcller, and Christoph Ortner. Robust a priori and a posteriori error analysis for the approximation of Allen-Cahn and Ginzburg-Landau equations past topological changes. Preprint available at www.bartels.ins.uni-bonn.de, 2009."},{"key":"8","series-title":"Mathematical Notes","isbn-type":"print","volume-title":"The motion of a surface by its mean curvature","volume":"20","author":"Brakke, Kenneth A.","year":"1978","ISBN":"https:\/\/id.crossref.org\/isbn\/0691082049"},{"issue":"218","key":"9","doi-asserted-by":"publisher","first-page":"465","DOI":"10.1090\/S0025-5718-97-00837-5","article-title":"A posteriori error estimate for the mixed finite element method","volume":"66","author":"Carstensen, Carsten","year":"1997","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"2","key":"10","doi-asserted-by":"publisher","first-page":"233","DOI":"10.1007\/s002110100378","article-title":"A posteriori error estimates for nonconforming finite element methods","volume":"92","author":"Carstensen, Carsten","year":"2002","journal-title":"Numer. 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