{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,27]],"date-time":"2026-05-27T12:59:09Z","timestamp":1779886749201,"version":"3.53.1"},"reference-count":18,"publisher":"American Mathematical Society (AMS)","issue":"321","license":[{"start":{"date-parts":[[2020,9,4]],"date-time":"2020-09-04T00:00:00Z","timestamp":1599177600000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["CA 151\/22-2"],"award-info":[{"award-number":["CA 151\/22-2"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n All previously known optimal adaptive least-squares finite element methods (LSFEMs) combine two marking strategies with a separate\n \n \n \n \n L<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n L^2<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n data approximation as a consequence of the natural norms equivalent to the least-squares functional. The algorithm and its analysis in this paper circumvent the natural norms in a div-LSFEM model problem with lowest-order conforming and mixed finite element functions and allow for a simple collective D\u00f6rfler marking for the first time. A refined analysis provides discrete reliability and quasi-orthogonality in the weaker norms\n \n \n \n \n \n L<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n \n \u00d7\n \n <\/mml:mo>\n \n H<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msup>\n <\/mml:mrow>\n L^2\\times H^1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n rather than\n \n \n \n \n H<\/mml:mi>\n (<\/mml:mo>\n div<\/mml:mi>\n )<\/mml:mo>\n \n \u00d7\n \n <\/mml:mo>\n \n H<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msup>\n <\/mml:mrow>\n H(\\operatorname {div})\\times H^1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and replaces data approximation terms by data oscillations. The optimal convergence rates then follow for the lowest-order version from the axioms of adaptivity for the newest-vertex bisection without restrictions on the initial mesh-size in any space dimension.\n <\/p>","DOI":"10.1090\/mcom\/3474","type":"journal-article","created":{"date-parts":[[2019,6,26]],"date-time":"2019-06-26T09:27:58Z","timestamp":1561541278000},"page":"89-103","source":"Crossref","is-referenced-by-count":10,"title":["Collective marking for adaptive least-squares finite element methods with optimal rates"],"prefix":"10.1090","volume":"89","author":[{"given":"Carsten","family":"Carstensen","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2019,9,4]]},"reference":[{"key":"1","doi-asserted-by":"crossref","unstructured":"P. Binev, W. Dahmen, and R. 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Math. 136 (2017), 1097\u20131115, DOI 10.1007\/s00211-017-0866-x.","DOI":"10.1007\/s00211-017-0866-x"},{"issue":"6","key":"13","doi-asserted-by":"publisher","first-page":"2644","DOI":"10.1137\/16M1068050","article-title":"Axioms of adaptivity with separate marking for data resolution","volume":"55","author":"Carstensen, C.","year":"2017","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"14","doi-asserted-by":"crossref","unstructured":"C. Carstensen and J. Storn, Asymptotic Exactness of the least-squares finite element residual, SIAM J. Numer. Anal. 56 (2018), 2008\u20132028, DOI 10.1137\/17M1125972, \\url{https:\/\/doi.org\/10.1137\/17M1125972}.","DOI":"10.1137\/17M1125972"},{"key":"15","doi-asserted-by":"crossref","unstructured":"J. M. Cascon, C. Kreuzer, R. H. Nochetto, and K. G. Siebert, Quasi-optimal convergence rate for an adaptive finite element method, SIAM J. Numer. 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