{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T07:49:32Z","timestamp":1776844172295,"version":"3.51.2"},"reference-count":32,"publisher":"American Mathematical Society (AMS)","issue":"256","license":[{"start":{"date-parts":[[2007,5,26]],"date-time":"2007-05-26T00:00:00Z","timestamp":1180137600000},"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                    We derive a\u00a0posteriori error estimates for fully discrete approximations to solutions of linear parabolic equations. The space discretization uses finite element spaces that are allowed to change in time. Our main tool is an appropriate adaptation of the elliptic reconstruction technique, introduced by Makridakis and Nochetto. We derive novel a posteriori estimates for the norms of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"normal upper L Subscript normal infinity Baseline left-parenthesis 0 comma upper T semicolon normal upper L 2 left-parenthesis normal upper Omega right-parenthesis right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi mathvariant=\"normal\">L<\/mml:mi>\n                              <mml:mi mathvariant=\"normal\">\n                                \u221e\n                                \n                              <\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo>\n                              \u2061\n                              \n                            <\/mml:mo>\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:msub>\n                              <mml:mi mathvariant=\"normal\">L<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo>\n                              \u2061\n                              \n                            <\/mml:mo>\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\">\\operatorname {L}_\\infty (0,T;\\operatorname {L}_2(\\Omega ))<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and the higher order spaces,\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"normal upper L Subscript normal infinity Baseline left-parenthesis 0 comma upper T semicolon normal upper H Superscript 1 Baseline left-parenthesis normal upper Omega right-parenthesis right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi mathvariant=\"normal\">L<\/mml:mi>\n                              <mml:mi mathvariant=\"normal\">\n                                \u221e\n                                \n                              <\/mml:mi>\n                            <\/mml:msub>\n                            <mml:mo>\n                              \u2061\n                              \n                            <\/mml:mo>\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 mathvariant=\"normal\">H<\/mml:mi>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo>\n                              \u2061\n                              \n                            <\/mml:mo>\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\">\\operatorname {L}_\\infty (0,T;\\operatorname {H}^1(\\Omega ))<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    and\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"normal upper H Superscript 1 Baseline left-parenthesis 0 comma upper T semicolon normal upper L 2 left-parenthesis normal upper Omega right-parenthesis right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msup>\n                              <mml:mi mathvariant=\"normal\">H<\/mml:mi>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:msup>\n                            <mml:mo>\n                              \u2061\n                              \n                            <\/mml:mo>\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:msub>\n                              <mml:mi mathvariant=\"normal\">L<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo>\n                              \u2061\n                              \n                            <\/mml:mo>\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\">\\operatorname {H}^1(0,T;\\operatorname {L}_2(\\Omega ))<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , with optimal orders of convergence.\n                  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Wiley-Teubner, Chichester-Stuttgart, 1996."},{"issue":"224","key":"31","doi-asserted-by":"publisher","first-page":"1335","DOI":"10.1090\/S0025-5718-98-01011-4","article-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","volume":"67","author":"Verf\u00fcrth, R.","year":"1998","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"4","key":"32","doi-asserted-by":"publisher","first-page":"487","DOI":"10.1002\/(SICI)1098-2426(199807)14:4<487::AID-NUM4>3.0.CO;2-G","article-title":"A posteriori error estimates for nonlinear problems: \ud835\udc3f^{\ud835\udc5f}(0,\ud835\udc47;\ud835\udc4a^{1,\ud835\udf0c}(\u03a9))-error estimates for finite element discretizations of parabolic equations","volume":"14","author":"Verf\u00fcrth, R.","year":"1998","journal-title":"Numer. 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