{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T07:55:50Z","timestamp":1776844550489,"version":"3.51.2"},"reference-count":16,"publisher":"American Mathematical Society (AMS)","issue":"288","license":[{"start":{"date-parts":[[2014,10,23]],"date-time":"2014-10-23T00:00:00Z","timestamp":1414022400000},"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 consider a space-time variational formulation for linear parabolic partial differential equations. We introduce an associated Petrov-Galerkin truth finite element discretization with favorable discrete inf-sup constant\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"beta Subscript delta\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>\n                              \u03b2\n                              \n                            <\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi>\n                                \u03b4\n                                \n                              <\/mml:mi>\n                            <\/mml:mrow>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">\\beta _{\\delta }<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , the inverse of which enters into error estimates:\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"beta Subscript delta\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>\n                              \u03b2\n                              \n                            <\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi>\n                                \u03b4\n                                \n                              <\/mml:mi>\n                            <\/mml:mrow>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">\\beta _{\\delta }<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is unity for the heat equation;\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"beta Subscript delta\">\n                        <mml:semantics>\n                          <mml:msub>\n                            <mml:mi>\n                              \u03b2\n                              \n                            <\/mml:mi>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mi>\n                                \u03b4\n                                \n                              <\/mml:mi>\n                            <\/mml:mrow>\n                          <\/mml:msub>\n                          <mml:annotation encoding=\"application\/x-tex\">\\beta _{\\delta }<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    decreases only linearly in time for non-coercive (but asymptotically stable) convection operators. The latter in turn permits effective long-time\n                    <italic>a posteriori<\/italic>\n                    error bounds for reduced basis approximations, in sharp contrast to classical (pessimistic) exponentially growing energy estimates. The paper contains a full analysis and various extensions for the formulation introduced briefly by Urban and Patera (2012) as well as numerical results for a model reaction-convection-diffusion equation.\n                  <\/p>","DOI":"10.1090\/s0025-5718-2013-02782-2","type":"journal-article","created":{"date-parts":[[2013,10,23]],"date-time":"2013-10-23T08:52:30Z","timestamp":1382518350000},"page":"1599-1615","source":"Crossref","is-referenced-by-count":90,"title":["An improved error bound for reduced basis approximation of linear parabolic problems"],"prefix":"10.1090","volume":"83","author":[{"given":"Karsten","family":"Urban","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Anthony","family":"Patera","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2013,10,23]]},"reference":[{"issue":"9","key":"1","doi-asserted-by":"publisher","first-page":"667","DOI":"10.1016\/j.crma.2004.08.006","article-title":"An \u2018empirical interpolation\u2019 method: application to efficient reduced-basis discretization of partial differential equations","volume":"339","author":"Barrault, Maxime","year":"2004","journal-title":"C. 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Sci.","ISSN":"https:\/\/id.crossref.org\/issn\/0218-2025","issn-type":"print"},{"key":"6","first-page":"305","article-title":"Sur une m\u00e9thode pour r\u00e9soudre les \u00e9quations aux d\u00e9riv\u00e9es partielles du type elliptique, voisine de la variationnelle","volume":"16","author":"Ne\u010das, Jind\u0159ich","year":"1962","journal-title":"Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3)","ISSN":"https:\/\/id.crossref.org\/issn\/0391-173X","issn-type":"print"},{"issue":"3","key":"7","doi-asserted-by":"publisher","first-page":"157","DOI":"10.1007\/s10092-009-0005-x","article-title":"Reduced basis approximation and a posteriori error estimation for the time-dependent viscous Burgers\u2019 equation","volume":"46","author":"Nguyen, Ngoc-Cuong","year":"2009","journal-title":"Calcolo","ISSN":"https:\/\/id.crossref.org\/issn\/0008-0624","issn-type":"print"},{"issue":"3","key":"8","doi-asserted-by":"publisher","first-page":"423","DOI":"10.1093\/imanum\/dri044","article-title":"Reduced-basis output bound methods for parabolic problems","volume":"26","author":"Rovas, D. V.","year":"2006","journal-title":"IMA J. Numer. 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Urban, Space-time reduced basis methods for time-periodic partial differential problems, Proceedings of MATHMOD 2012 \u2013 7th Vienna International Conference on Mathematical Modelling, Vienna, February 15-17, 2012, 481, 1\u20136."},{"key":"12","unstructured":"T. Tonn, Reduced-Basis Method (RBM) for Non-Affine Elliptic Parametrized PDEs (Motivated by Optimization in Hydromechnanics), Ph.D. thesis, Ulm University, Germany, 2012."},{"issue":"3-4","key":"13","doi-asserted-by":"publisher","first-page":"203","DOI":"10.1016\/j.crma.2012.01.026","article-title":"A new error bound for reduced basis approximation of parabolic partial differential equations","volume":"350","author":"Urban, Karsten","year":"2012","journal-title":"C. R. Math. Acad. Sci. Paris","ISSN":"https:\/\/id.crossref.org\/issn\/1631-073X","issn-type":"print"},{"key":"14","unstructured":"S. Vallagh\u00e9, A. Le-Hyaric, M. Fouquemberg, and C. 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(2012)."}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2014-83-288\/S0025-5718-2013-02782-2\/S0025-5718-2013-02782-2.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2014-83-288\/S0025-5718-2013-02782-2\/S0025-5718-2013-02782-2.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T17:57:26Z","timestamp":1776794246000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2014-83-288\/S0025-5718-2013-02782-2\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,10,23]]},"references-count":16,"journal-issue":{"issue":"288","published-print":{"date-parts":[[2014,7]]}},"alternative-id":["S0025-5718-2013-02782-2"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-2013-02782-2","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":[[2013,10,23]]}}}