{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T19:39:49Z","timestamp":1740166789216,"version":"3.37.3"},"reference-count":21,"publisher":"Walter de Gruyter GmbH","issue":"2","funder":[{"name":"DFG","award":["MATHEON project C33"],"award-info":[{"award-number":["MATHEON project C33"]}]},{"DOI":"10.13039\/501100003725","name":"National Research Foundation of Korea","doi-asserted-by":"crossref","award":["R31-2008-000-10049-0"],"award-info":[{"award-number":["R31-2008-000-10049-0"]}],"id":[{"id":"10.13039\/501100003725","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2016,4,1]]},"abstract":"Abstract<\/jats:title>\n A hierarchical a posteriori error estimator for the first-order finite element method (FEM) on a red-refined triangular mesh\nis presented for the 2D Poisson model problem.\nReliability and efficiency with some explicit constant is proved for triangulations with inner angles smaller than or equal to \n \n \n \n \n \u03c0<\/m:mi>\n 2<\/m:mn>\n <\/m:mfrac>\n <\/m:math>\n ${\\frac{\\pi }{2}}$<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>.\nThe error estimator does not rely on any saturation assumption and is valid even in the pre-asymptotic regime on arbitrarily coarse meshes.\nThe evaluation of the estimator is a simple post-processing of the piecewise linear FEM without any extra solve plus a higher-order approximation term.\nThe results also allow the striking observation that arbitrary local averaging of the primal variable leads to a reliable and efficient error estimation.\nSeveral numerical experiments illustrate the performance of the proposed a posteriori error estimator for computational benchmarks.<\/jats:p>","DOI":"10.1515\/cmam-2016-0010","type":"journal-article","created":{"date-parts":[[2016,3,1]],"date-time":"2016-03-01T09:42:35Z","timestamp":1456825355000},"page":"213-230","source":"Crossref","is-referenced-by-count":0,"title":["Reliable Averaging for the Primal Variable in the Courant FEM and Hierarchical Error Estimators on Red-Refined Meshes"],"prefix":"10.1515","volume":"16","author":[{"given":"Carsten","family":"Carstensen","sequence":"first","affiliation":[{"name":"Humboldt-Universit\u00e4t zu Berlin, Unter den Linden 6, 10099 Berlin, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Martin","family":"Eigel","sequence":"additional","affiliation":[{"name":"Weierstrass Institute, Mohrenstr. 39, 10117 Berlin, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2016,3,1]]},"reference":[{"key":"ref241","first-page":"116","article-title":"Convergence of simple adaptive Galerkin schemes based onh - 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