{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T06:44:02Z","timestamp":1776840242589,"version":"3.51.2"},"reference-count":31,"publisher":"American Mathematical Society (AMS)","issue":"239","license":[{"start":{"date-parts":[[2003,2,4]],"date-time":"2003-02-04T00:00:00Z","timestamp":1044316800000},"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>Averaging techniques are popular tools in adaptive finite element methods for the numerical treatment of second order partial differential equations since they provide efficient a\u00a0posteriori error estimates by a simple postprocessing. In this paper, their reliablility is shown for conforming, nonconforming, and mixed low order finite element methods in a model situation: the Laplace equation with mixed boundary conditions. Emphasis is on possibly unstructured grids, nonsmoothness of exact solutions, and a wide class of averaging techniques. Theoretical and numerical evidence supports that the reliability is up to the smoothness of given right-hand sides.<\/p>","DOI":"10.1090\/s0025-5718-02-01402-3","type":"journal-article","created":{"date-parts":[[2002,9,20]],"date-time":"2002-09-20T15:46:54Z","timestamp":1032536814000},"page":"945-969","source":"Crossref","is-referenced-by-count":120,"title":["Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids. Part I: Low order conforming, nonconforming, and mixed FEM"],"prefix":"10.1090","volume":"71","author":[{"given":"Carsten","family":"Carstensen","sequence":"first","affiliation":[]},{"given":"S\u00f6ren","family":"Bartels","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2002,2,4]]},"reference":[{"key":"1","doi-asserted-by":"crossref","unstructured":"[ACF]  J. Alberty, C. Carstensen, S.A. Funken: Remarks around 50 lines of Matlab: short finite element implementation. Numer. Algorithms 20 (1999) 117-137.","DOI":"10.1023\/A:1019155918070"},{"issue":"4","key":"2","doi-asserted-by":"publisher","first-page":"385","DOI":"10.1007\/s002110050222","article-title":"Error estimators for a mixed method","volume":"74","author":"Alonso, A.","year":"1996","journal-title":"Numer. 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