{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T20:09:36Z","timestamp":1776802176628,"version":"3.51.2"},"reference-count":26,"publisher":"American Mathematical Society (AMS)","issue":"304","license":[{"start":{"date-parts":[[2017,5,3]],"date-time":"2017-05-03T00:00:00Z","timestamp":1493769600000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["1217081 and 1522707"],"award-info":[{"award-number":["1217081 and 1522707"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100007289","name":"Purdue Research Foundation","doi-asserted-by":"publisher","award":["1217081 and 1522707"],"award-info":[{"award-number":["1217081 and 1522707"]}],"id":[{"id":"10.13039\/100007289","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>In this paper, we study a modified residual-based a posteriori error estimator for the nonconforming linear finite element approximation to the interface problem. The reliability of the estimator is analyzed by a new and direct approach without using the Helmholtz decomposition. It is proved that the estimator is reliable with constant independent of the jump of diffusion coefficients across the interfaces, without the assumption that the diffusion coefficient is quasi-monotone. Numerical results for one test problem with intersecting interfaces are also presented.<\/p>","DOI":"10.1090\/mcom\/3151","type":"journal-article","created":{"date-parts":[[2016,2,24]],"date-time":"2016-02-24T17:15:01Z","timestamp":1456334101000},"page":"617-636","source":"Crossref","is-referenced-by-count":13,"title":["Residual-based a posteriori error estimate for interface problems: Nonconforming linear elements"],"prefix":"10.1090","volume":"86","author":[{"given":"Zhiqiang","family":"Cai","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Cuiyu","family":"He","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Shun","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2016,5,3]]},"reference":[{"issue":"6","key":"1","doi-asserted-by":"publisher","first-page":"2320","DOI":"10.1137\/S0036142903425112","article-title":"Robust a posteriori error estimation for nonconforming finite element approximation","volume":"42","author":"Ainsworth, Mark","year":"2005","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"4","key":"2","doi-asserted-by":"publisher","first-page":"1777","DOI":"10.1137\/060665993","article-title":"A posteriori error estimation for discontinuous Galerkin finite element approximation","volume":"45","author":"Ainsworth, Mark","year":"2007","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"5-6","key":"3","doi-asserted-by":"publisher","first-page":"723","DOI":"10.1016\/S0045-7825(02)00593-5","article-title":"Energy norm a posteriori error estimation for discontinuous Galerkin methods","volume":"192","author":"Becker, Roland","year":"2003","journal-title":"Comput. Methods Appl. Mech. Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0045-7825","issn-type":"print"},{"issue":"4","key":"4","doi-asserted-by":"publisher","first-page":"579","DOI":"10.1007\/PL00005393","article-title":"Adaptive finite element methods for elliptic equations with non-smooth coefficients","volume":"85","author":"Bernardi, C.","year":"2000","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"2","key":"5","doi-asserted-by":"publisher","first-page":"233","DOI":"10.1007\/s002110100378","article-title":"A posteriori error estimates for nonconforming finite element methods","volume":"92","author":"Carstensen, Carsten","year":"2002","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"1","key":"6","doi-asserted-by":"publisher","first-page":"68","DOI":"10.1137\/050628854","article-title":"Framework for the a posteriori error analysis of nonconforming finite elements","volume":"45","author":"Carstensen, Carsten","year":"2007","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"5","key":"7","doi-asserted-by":"publisher","first-page":"1761","DOI":"10.1137\/100805133","article-title":"Discontinuous Galerkin finite element methods for interface problems: a priori and a posteriori error estimations","volume":"49","author":"Cai, Zhiqiang","year":"2011","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"3","key":"8","doi-asserted-by":"publisher","first-page":"2132","DOI":"10.1137\/080717407","article-title":"Recovery-based error estimator for interface problems: conforming linear elements","volume":"47","author":"Cai, Zhiqiang","year":"2009","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"2","key":"9","doi-asserted-by":"publisher","first-page":"578","DOI":"10.1137\/080742993","article-title":"Flux recovery and a posteriori error estimators: conforming elements for scalar elliptic equations","volume":"48","author":"Cai, Zhiqiang","year":"2010","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"1","key":"10","doi-asserted-by":"publisher","first-page":"30","DOI":"10.1137\/080722631","article-title":"Recovery-based error estimators for interface problems: mixed and nonconforming finite elements","volume":"48","author":"Cai, Zhiqiang","year":"2010","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"2","key":"11","doi-asserted-by":"publisher","first-page":"476","DOI":"10.1002\/num.20629","article-title":"Robust residual- and recovery-based a posteriori error estimators for interface problems with flux jumps","volume":"28","author":"Cai, Zhiqiang","year":"2012","journal-title":"Numer. Methods Partial Differential Equations","ISSN":"https:\/\/id.crossref.org\/issn\/0749-159X","issn-type":"print"},{"issue":"1","key":"12","doi-asserted-by":"publisher","first-page":"151","DOI":"10.1137\/100803857","article-title":"Robust equilibrated residual error estimator for diffusion problems: conforming elements","volume":"50","author":"Cai, Zhiqiang","year":"2012","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"211","key":"13","doi-asserted-by":"publisher","first-page":"1017","DOI":"10.2307\/2153481","article-title":"Error estimators for nonconforming finite element approximations of the Stokes problem","volume":"64","author":"Dari, Enzo","year":"1995","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"4","key":"14","doi-asserted-by":"publisher","first-page":"385","DOI":"10.1051\/m2an\/1996300403851","article-title":"A posteriori error estimators for nonconforming finite element methods","volume":"30","author":"Dari, E.","year":"1996","journal-title":"RAIRO Mod\\'{e}l. Math. Anal. Num\\'{e}r.","ISSN":"https:\/\/id.crossref.org\/issn\/0764-583X","issn-type":"print"},{"issue":"3","key":"15","doi-asserted-by":"publisher","first-page":"313","DOI":"10.1007\/s002110050172","article-title":"Multilevel Schwarz methods for elliptic problems with discontinuous coefficients in three dimensions","volume":"72","author":"Dryja, Maksymilian","year":"1996","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"key":"16","series-title":"Numerical Mathematics and Scientific Computation","isbn-type":"print","volume-title":"Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics","author":"Elman, Howard C.","year":"2005","ISBN":"https:\/\/id.crossref.org\/isbn\/9780198528685"},{"key":"17","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":"2","key":"18","doi-asserted-by":"publisher","first-page":"237","DOI":"10.1051\/m2an\/1996300202371","article-title":"Element-oriented and edge-oriented local error estimators for nonconforming finite element methods","volume":"30","author":"Hoppe, Ronald H. W.","year":"1996","journal-title":"RAIRO Mod\\'{e}l. Math. Anal. Num\\'{e}r.","ISSN":"https:\/\/id.crossref.org\/issn\/0764-583X","issn-type":"print"},{"key":"19","doi-asserted-by":"publisher","first-page":"101","DOI":"10.1080\/00036817408839086","article-title":"On the Poisson equation with intersecting interfaces","volume":"4","author":"Kellogg, R. Bruce","year":"1974","journal-title":"Applicable Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0003-6811","issn-type":"print"},{"issue":"257","key":"20","doi-asserted-by":"publisher","first-page":"43","DOI":"10.1090\/S0025-5718-06-01903-X","article-title":"A posteriori error analysis for locally conservative mixed methods","volume":"76","author":"Kim, Kwang Y.","year":"2007","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"256","key":"21","doi-asserted-by":"publisher","first-page":"1659","DOI":"10.1090\/S0025-5718-06-01872-2","article-title":"Energy norm a posteriori error estimates for mixed finite element methods","volume":"75","author":"Lovadina, Carlo","year":"2006","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"4","key":"22","doi-asserted-by":"publisher","first-page":"1394","DOI":"10.1137\/S0036142903433790","article-title":"A local a posteriori error estimator based on equilibrated fluxes","volume":"42","author":"Luce, R.","year":"2004","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"1","key":"23","doi-asserted-by":"publisher","first-page":"47","DOI":"10.1023\/A:1014221125034","article-title":"A posteriori error estimators for elliptic equations with discontinuous coefficients","volume":"16","author":"Petzoldt, Martin","year":"2002","journal-title":"Adv. Comput. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1019-7168","issn-type":"print"},{"issue":"3","key":"24","doi-asserted-by":"publisher","first-page":"489","DOI":"10.1051\/m2an:2002022","article-title":"A posteriori error estimates with post-processing for nonconforming finite elements","volume":"36","author":"Schieweck, Friedhelm","year":"2002","journal-title":"M2AN Math. Model. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0764-583X","issn-type":"print"},{"key":"25","series-title":"Numerical Mathematics and Scientific Computation","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1093\/acprof:oso\/9780199679423.001.0001","volume-title":"A posteriori error estimation techniques for finite element methods","author":"Verf\u00fcrth, R\u00fcdiger","year":"2013","ISBN":"https:\/\/id.crossref.org\/isbn\/9780199679423"},{"issue":"3","key":"26","doi-asserted-by":"publisher","first-page":"397","DOI":"10.1007\/s10915-010-9410-1","article-title":"Guaranteed and fully robust a posteriori error estimates for conforming discretizations of diffusion problems with discontinuous coefficients","volume":"46","author":"Vohral\u00edk, Martin","year":"2011","journal-title":"J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0885-7474","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2017-86-304\/S0025-5718-2016-03151-8\/S0025-5718-2016-03151-8.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2017-86-304\/S0025-5718-2016-03151-8\/S0025-5718-2016-03151-8.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T19:04:17Z","timestamp":1776798257000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2017-86-304\/S0025-5718-2016-03151-8\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,5,3]]},"references-count":26,"journal-issue":{"issue":"304","published-print":{"date-parts":[[2017,3]]}},"alternative-id":["S0025-5718-2016-03151-8"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/3151","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":[[2016,5,3]]}}}