{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T23:38:31Z","timestamp":1776728311894,"version":"3.51.2"},"reference-count":21,"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>\n                    Averaging techniques are popular tools in adaptive finite element methods since they provide efficient a posteriori error estimates by a simple postprocessing. In the second paper of our analysis of their reliability, we consider conforming\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"h\">\n                        <mml:semantics>\n                          <mml:mi>h<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">h<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -FEM of higher (i.e., not of lowest) order in two or three space dimensions. In this paper, reliablility is shown for conforming higher 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 local averaging techniques. Theoretical and numerical evidence supports that the reliability is up to the smoothness of given right-hand sides.\n                  <\/p>","DOI":"10.1090\/s0025-5718-02-01412-6","type":"journal-article","created":{"date-parts":[[2002,9,20]],"date-time":"2002-09-20T15:46:54Z","timestamp":1032536814000},"page":"971-994","source":"Crossref","is-referenced-by-count":37,"title":["Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids. Part II: Higher order FEM"],"prefix":"10.1090","volume":"71","author":[{"given":"S\u00f6ren","family":"Bartels","sequence":"first","affiliation":[]},{"given":"Carsten","family":"Carstensen","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\u2013137.","DOI":"10.1023\/A:1019155918070"},{"issue":"7","key":"2","doi-asserted-by":"publisher","first-page":"1073","DOI":"10.1002\/nme.1620370702","article-title":"Validation of a posteriori error estimators by numerical approach","volume":"37","author":"Babu\u0161ka, I.","year":"1994","journal-title":"Internat. J. Numer. Methods Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-5981","issn-type":"print"},{"issue":"4","key":"3","first-page":"237","article-title":"A feed-back approach to error control in finite element methods: basic analysis and examples","volume":"4","author":"Becker, R.","year":"1996","journal-title":"East-West J. Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0928-0200","issn-type":"print"},{"key":"4","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-07233-2","volume-title":"Finite elements","author":"Braess, Dietrich","year":"1997","ISBN":"https:\/\/id.crossref.org\/isbn\/0521581877"},{"issue":"6","key":"5","doi-asserted-by":"publisher","first-page":"2431","DOI":"10.1137\/S0036142994264079","article-title":"A posteriori error estimators for the Raviart-Thomas element","volume":"33","author":"Braess, D.","year":"1996","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"6","series-title":"Texts in Applied Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-4338-8","volume-title":"The mathematical theory of finite element methods","volume":"15","author":"Brenner, Susanne C.","year":"1994","ISBN":"https:\/\/id.crossref.org\/isbn\/0387941932"},{"key":"7","doi-asserted-by":"crossref","unstructured":"[Ca] C. Carstensen: Quasi interpolation and a posteriori error analysis in finite element method, M2AN 33 (1999) 1187\u20131202.","DOI":"10.1051\/m2an:1999140"},{"key":"8","unstructured":"[CB] C. Carstensen, S. Bartels: Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids. Part I: Low order conforming, nonconforming, and mixed FEM, Math. Comp., posted on February 4, 2002, PII S 0025-5718(02)01402-3 (to appear in print)."},{"key":"9","unstructured":"[CF] C. Carstensen, S.A. Funken: Constants in Cl\u00e9ment-interpolation error and residual based a posteriori error estimates in Finite Element Methods. East-West J. Numer. Anal. 8 (2000), 153\u2013175."},{"key":"10","doi-asserted-by":"crossref","unstructured":"[CMS] C. Carstensen, S. Maischak, E.P. Stephan: A posteriori error estimate and \u210e-adaptive algorithm on surfaces for Symm\u2019s integral equation. Numer. Math. (2001) Published online May 30, 2001, DOI 10.1007\/s002110100287.","DOI":"10.1007\/s002110100287"},{"issue":"3-4","key":"11","doi-asserted-by":"publisher","first-page":"345","DOI":"10.1016\/S0045-7825(99)00046-8","article-title":"The discrete artificial boundary condition on a polygonal artificial boundary for the exterior problem of Poisson equation by using the direct method of lines","volume":"179","author":"Han, Houde","year":"1999","journal-title":"Comput. Methods Appl. Mech. Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0045-7825","issn-type":"print"},{"key":"12","doi-asserted-by":"crossref","first-page":"77","DOI":"10.1051\/m2an\/197509R200771","article-title":"Approximation by finite element functions using local regularization","volume":"9","author":"Cl\u00e9ment, Ph.","year":"1975","journal-title":"Rev. Fran\\c{c}aise Automat. Informat. Recherche Op\\'{e}rationnelle S\\'{e}r. Rouge Anal. Num\\'{e}r.","ISSN":"https:\/\/id.crossref.org\/issn\/0397-9342","issn-type":"print"},{"key":"13","series-title":"Studies in Mathematics and its Applications, Vol. 4","isbn-type":"print","volume-title":"The finite element method for elliptic problems","author":"Ciarlet, Philippe G.","year":"1978","ISBN":"https:\/\/id.crossref.org\/isbn\/0444850287"},{"key":"14","isbn-type":"print","doi-asserted-by":"publisher","first-page":"105","DOI":"10.1017\/S0962492900002531","article-title":"Introduction to adaptive methods for differential equations","author":"Eriksson, Kenneth","year":"1995","ISBN":"https:\/\/id.crossref.org\/isbn\/0521482550"},{"key":"15","series-title":"Springer Series in Computational Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-61623-5","volume-title":"Finite element methods for Navier-Stokes equations","volume":"5","author":"Girault, Vivette","year":"1986","ISBN":"https:\/\/id.crossref.org\/isbn\/3540157964"},{"issue":"1-4","key":"16","doi-asserted-by":"publisher","first-page":"289","DOI":"10.1016\/S0045-7825(96)01087-0","article-title":"The partition of unity finite element method: basic theory and applications","volume":"139","author":"Melenk, J. M.","year":"1996","journal-title":"Comput. Methods Appl. Mech. Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0045-7825","issn-type":"print"},{"issue":"5","key":"17","doi-asserted-by":"publisher","first-page":"625","DOI":"10.1002\/num.1690100509","article-title":"Some remarks on Zienkiewicz-Zhu estimator","volume":"10","author":"Rodr\u00edguez, Rodolfo","year":"1994","journal-title":"Numer. Methods Partial Differential Equations","ISSN":"https:\/\/id.crossref.org\/issn\/0749-159X","issn-type":"print"},{"key":"18","unstructured":"[V] R. Verf\u00fcrth: A review of a posteriori error estimation and adaptive mesh-refinement techniques. Wiley-Teubner (1996)."},{"issue":"4","key":"19","first-page":"82","article-title":"Asymptotically exact a posteriori error estimators for elements of bi-odd degree","volume":"13","author":"Yu, De Hao","year":"1991","journal-title":"Chinese J. Numer. Math. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0899-4358","issn-type":"print"},{"issue":"2","key":"20","first-page":"64","article-title":"Asymptotically exact a posteriori error estimator for elements of bi-even degree","volume":"13","author":"Yu, De Hao","year":"1991","journal-title":"Chinese J. Numer. Math. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0899-4358","issn-type":"print"},{"issue":"2","key":"21","doi-asserted-by":"publisher","first-page":"337","DOI":"10.1002\/nme.1620240206","article-title":"A simple error estimator and adaptive procedure for practical engineering analysis","volume":"24","author":"Zienkiewicz, O. C.","year":"1987","journal-title":"Internat. J. Numer. Methods Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-5981","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2002-71-239\/S0025-5718-02-01412-6\/S0025-5718-02-01412-6.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2002-71-239\/S0025-5718-02-01412-6\/S0025-5718-02-01412-6.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T23:00:16Z","timestamp":1776726016000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2002-71-239\/S0025-5718-02-01412-6\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2002,2,4]]},"references-count":21,"journal-issue":{"issue":"239","published-print":{"date-parts":[[2002,7]]}},"alternative-id":["S0025-5718-02-01412-6"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-02-01412-6","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":[[2002,2,4]]}}}