{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,30]],"date-time":"2026-04-30T00:29:02Z","timestamp":1777508942369,"version":"3.51.4"},"reference-count":29,"publisher":"Walter de Gruyter GmbH","issue":"3","funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS 1620058"],"award-info":[{"award-number":["DMS 1620058"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS 1619892"],"award-info":[{"award-number":["DMS 1619892"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2018,7,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>We devise a novel framework for the error analysis of finite element\napproximations to low-regularity solutions in nonconforming settings\nwhere the discrete trial and test spaces are not subspaces of their\nexact counterparts. The key is to use face-to-cell extension\noperators so as to give a weak meaning to the normal or tangential\ntrace on each mesh face individually for vector fields with minimal\nregularity and then to prove the consistency of this new formulation\nby means of some recently-derived mollification operators that\ncommute with the usual derivative operators. We illustrate the\ntechnique on Nitsche\u2019s boundary penalty method applied to a scalar\ndiffusion equation and to the time-harmonic Maxwell\u2019s equations. In\nboth cases, the error estimates are robust in the case of\nheterogeneous material properties. We also revisit the error\nanalysis framework proposed by Gudi where a trimming operator is\nintroduced to map discrete test functions into conforming test\nfunctions. This technique also gives error estimates for minimal\nregularity solutions, but the constants depend on the material\nproperties through contrast factors.<\/jats:p>","DOI":"10.1515\/cmam-2017-0058","type":"journal-article","created":{"date-parts":[[2017,12,15]],"date-time":"2017-12-15T22:15:44Z","timestamp":1513376144000},"page":"451-475","source":"Crossref","is-referenced-by-count":10,"title":["Abstract Nonconforming Error Estimates and Application to Boundary Penalty Methods for Diffusion Equations and Time-Harmonic Maxwell\u2019s Equations"],"prefix":"10.1515","volume":"18","author":[{"given":"Alexandre","family":"Ern","sequence":"first","affiliation":[{"name":"Universit\u00e9 Paris-Est and INRIA , CERMICS (ENPC) , 77455 Marne-la-Vall\u00e9e Cedex 2 ; and INRIA Paris, 2 rue Simone Iff, 75589 Paris , France"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jean-Luc","family":"Guermond","sequence":"additional","affiliation":[{"name":"Department of Mathematics , Texas A&M University 3368 TAMU , College Station , TX 77843 , USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2017,12,15]]},"reference":[{"key":"2025051309544635639_j_cmam-2017-0058_ref_001_w2aab3b7e2812b1b6b1ab2ab1Aa","doi-asserted-by":"crossref","unstructured":"C.  Amrouche, C.  Bernardi, M.  Dauge and V.  Girault,\nVector potentials in three-dimensional non-smooth domains,\nMath. Methods Appl. Sci. 21 (1998), no. 9, 823\u2013864.","DOI":"10.1002\/(SICI)1099-1476(199806)21:9<823::AID-MMA976>3.0.CO;2-B"},{"key":"2025051309544635639_j_cmam-2017-0058_ref_002_w2aab3b7e2812b1b6b1ab2ab2Aa","unstructured":"I.  Babu\u0161ka and A. K.  Aziz,\nSurvey lectures on the mathematical foundations of the finite element method,\nThe Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations\n(Baltimore 1972),\nAcademic Press, New York (1972), 1\u2013359."},{"key":"2025051309544635639_j_cmam-2017-0058_ref_003_w2aab3b7e2812b1b6b1ab2ab3Aa","doi-asserted-by":"crossref","unstructured":"S.  Badia, R.  Codina, T.  Gudi and J.  Guzm\u00e1n,\nError analysis of discontinuous Galerkin methods for the Stokes problem under minimal regularity,\nIMA J. Numer. Anal. 34 (2014), no. 2, 800\u2013819.","DOI":"10.1093\/imanum\/drt022"},{"key":"2025051309544635639_j_cmam-2017-0058_ref_004_w2aab3b7e2812b1b6b1ab2ab4Aa","doi-asserted-by":"crossref","unstructured":"L.  Beir\u00e3o da Veiga, F.  Brezzi, A.  Cangiani, G.  Manzini, L. D.  Marini and A.  Russo,\nBasic principles of virtual element methods,\nMath. Models Methods Appl. Sci. 23 (2013), no. 1, 199\u2013214.","DOI":"10.1142\/S0218202512500492"},{"key":"2025051309544635639_j_cmam-2017-0058_ref_005_w2aab3b7e2812b1b6b1ab2ab5Aa","doi-asserted-by":"crossref","unstructured":"C.  Bernardi and F.  Hecht,\nError indicators for the mortar finite element discretization of the Laplace equation,\nMath. Comp. 71 (2002), no. 240, 1371\u20131403.","DOI":"10.1090\/S0025-5718-01-01401-6"},{"key":"2025051309544635639_j_cmam-2017-0058_ref_006_w2aab3b7e2812b1b6b1ab2ab6Aa","doi-asserted-by":"crossref","unstructured":"A.  Bonito, J.-L.  Guermond and F.  Luddens,\nRegularity of the maxwell equations in heterogeneous media and Lipschitz domains,\nJ. Math. Anal. Appl. 408 (2013), 498\u2013512.","DOI":"10.1016\/j.jmaa.2013.06.018"},{"key":"2025051309544635639_j_cmam-2017-0058_ref_007_w2aab3b7e2812b1b6b1ab2ab7Aa","doi-asserted-by":"crossref","unstructured":"D.  Braess,\nFinite Elements. Theory, Fast Solvers, and Applications in Elasticity Theory, 3rd ed.,\nCambridge University Press, Cambridge, 2007.","DOI":"10.1017\/CBO9780511618635"},{"key":"2025051309544635639_j_cmam-2017-0058_ref_008_w2aab3b7e2812b1b6b1ab2ab8Aa","doi-asserted-by":"crossref","unstructured":"S. C.  Brenner and L. R.  Scott,\nThe Mathematical Theory of Finite Element Methods, 2nd ed.,\nTexts Appl. Math. 15,\nSpringer, New York, 2002.","DOI":"10.1007\/978-1-4757-3658-8"},{"key":"2025051309544635639_j_cmam-2017-0058_ref_009_w2aab3b7e2812b1b6b1ab2ab9Aa","doi-asserted-by":"crossref","unstructured":"H.  Brezis,\nFunctional Analysis, Sobolev Spaces and Partial Differential Equations,\nUniversitext,\nSpringer, New York, 2011.","DOI":"10.1007\/978-0-387-70914-7"},{"key":"2025051309544635639_j_cmam-2017-0058_ref_010_w2aab3b7e2812b1b6b1ab2ac10Aa","doi-asserted-by":"crossref","unstructured":"C.  Carstensen and M.  Schedensack,\nMedius analysis and comparison results for first-order finite element methods in linear elasticity,\nIMA J. Numer. Anal. 35 (2015), no. 4, 1591\u20131621.","DOI":"10.1093\/imanum\/dru048"},{"key":"2025051309544635639_j_cmam-2017-0058_ref_011_w2aab3b7e2812b1b6b1ab2ac11Aa","doi-asserted-by":"crossref","unstructured":"J.  C\u00e9a,\nApproximation variationnelle des probl\u00e8mes aux limites,\nAnn. Inst. Fourier (Grenoble) 14 (1964), 345\u2013444.","DOI":"10.5802\/aif.181"},{"key":"2025051309544635639_j_cmam-2017-0058_ref_012_w2aab3b7e2812b1b6b1ab2ac12Aa","doi-asserted-by":"crossref","unstructured":"D. A.  Di Pietro and A.  Ern,\nMathematical Aspects of Discontinuous Galerkin Methods,\nMath. Appl. 69,\nSpringer, Berlin, 2012.","DOI":"10.1007\/978-3-642-22980-0"},{"key":"2025051309544635639_j_cmam-2017-0058_ref_013_w2aab3b7e2812b1b6b1ab2ac13Aa","doi-asserted-by":"crossref","unstructured":"A.  Ern and J.-L.  Guermond,\nTheory and Practice of Finite Elements,\nAppl. Math. Sci. 159,\nSpringer, New York, 2004.","DOI":"10.1007\/978-1-4757-4355-5"},{"key":"2025051309544635639_j_cmam-2017-0058_ref_014_w2aab3b7e2812b1b6b1ab2ac14Aa","doi-asserted-by":"crossref","unstructured":"A.  Ern and J.-L.  Guermond,\nDiscontinuous Galerkin methods for Friedrichs\u2019 systems. I. General theory,\nSIAM J. Numer. Anal. 44 (2006), no. 2, 753\u2013778.","DOI":"10.1137\/050624133"},{"key":"2025051309544635639_j_cmam-2017-0058_ref_015_w2aab3b7e2812b1b6b1ab2ac15Aa","doi-asserted-by":"crossref","unstructured":"A.  Ern and J.-L.  Guermond,\nMollification in strongly Lipschitz domains with application to continuous and discrete de Rham complexes,\nComput. Methods Appl. Math. 16 (2016), no. 1, 51\u201375.","DOI":"10.1515\/cmam-2015-0034"},{"key":"2025051309544635639_j_cmam-2017-0058_ref_016_w2aab3b7e2812b1b6b1ab2ac16Aa","doi-asserted-by":"crossref","unstructured":"A.  Ern and J.-L.  Guermond,\nFinite element quasi-interpolation and best approximation,\nM2AN Math. Model. Numer. Anal. 51 (2017), no. 4, 1367\u20131385.","DOI":"10.1051\/m2an\/2016066"},{"key":"2025051309544635639_j_cmam-2017-0058_ref_017_w2aab3b7e2812b1b6b1ab2ac17Aa","doi-asserted-by":"crossref","unstructured":"K. O.  Friedrichs,\nThe identity of weak and strong extensions of differential operators,\nTrans. Amer. Math. Soc. 55 (1944), 132\u2013151.","DOI":"10.1090\/S0002-9947-1944-0009701-0"},{"key":"2025051309544635639_j_cmam-2017-0058_ref_018_w2aab3b7e2812b1b6b1ab2ac18Aa","doi-asserted-by":"crossref","unstructured":"T.  Gudi,\nA new error analysis for discontinuous finite element methods for linear elliptic problems,\nMath. Comp. 79 (2010), no. 272, 2169\u20132189.","DOI":"10.1090\/S0025-5718-10-02360-4"},{"key":"2025051309544635639_j_cmam-2017-0058_ref_019_w2aab3b7e2812b1b6b1ab2ac19Aa","doi-asserted-by":"crossref","unstructured":"S.  Hofmann, M.  Mitrea and M.  Taylor,\nGeometric and transformational properties of Lipschitz domains, Semmes\u2013Kenig\u2013Toro domains, and other classes of finite perimeter domains,\nJ. Geom. Anal. 17 (2007), no. 4, 593\u2013647.","DOI":"10.1007\/BF02937431"},{"key":"2025051309544635639_j_cmam-2017-0058_ref_020_w2aab3b7e2812b1b6b1ab2ac20Aa","doi-asserted-by":"crossref","unstructured":"F.  Jochmann,\nRegularity of weak solutions of Maxwell\u2019s equations with mixed boundary-conditions,\nMath. Methods Appl. Sci. 22 (1999), no. 14, 1255\u20131274.","DOI":"10.1002\/(SICI)1099-1476(19990925)22:14<1255::AID-MMA83>3.0.CO;2-N"},{"key":"2025051309544635639_j_cmam-2017-0058_ref_021_w2aab3b7e2812b1b6b1ab2ac21Aa","doi-asserted-by":"crossref","unstructured":"J.  Leray,\nSur le mouvement d\u2019un liquide visqueux emplissant l\u2019espace,\nActa Math. 63 (1934), no. 1, 193\u2013248.","DOI":"10.1007\/BF02547354"},{"key":"2025051309544635639_j_cmam-2017-0058_ref_022_w2aab3b7e2812b1b6b1ab2ac22Aa","unstructured":"J.  Ne\u010das,\nSur une m\u00e9thode pour r\u00e9soudre les \u00e9quations aux d\u00e9riv\u00e9es partielles de type elliptique, voisine de la variationnelle,\nAnn. Sc. Norm. Super. Pisa 16 (1962), 305\u2013326."},{"key":"2025051309544635639_j_cmam-2017-0058_ref_023_w2aab3b7e2812b1b6b1ab2ac23Aa","doi-asserted-by":"crossref","unstructured":"J.  Nitsche,\n\u00dcber ein Variationsprinzip zur L\u00f6sung von Dirichlet-Problemen bei Verwendung von Teilr\u00e4umen, die keinen Randbedingungen unterworfen sind,\nAbh. Math. Semin. Univ. Hambg. 36 (1971), 9\u201315.","DOI":"10.1007\/BF02995904"},{"key":"2025051309544635639_j_cmam-2017-0058_ref_024_w2aab3b7e2812b1b6b1ab2ac24Aa","unstructured":"J.  Sch\u00f6berl,\nCommuting quasi-interpolation operators for mixed finite elements,\nTechnical Report ISC-01-10-MATH, Texas A&M University, 2001."},{"key":"2025051309544635639_j_cmam-2017-0058_ref_025_w2aab3b7e2812b1b6b1ab2ac25Aa","doi-asserted-by":"crossref","unstructured":"J.  Sch\u00f6berl,\nA posteriori error estimates for Maxwell equations,\nMath. Comp. 77 (2008), no. 262, 633\u2013649.","DOI":"10.1090\/S0025-5718-07-02030-3"},{"key":"2025051309544635639_j_cmam-2017-0058_ref_026_w2aab3b7e2812b1b6b1ab2ac26Aa","unstructured":"S.  Sobolev,\nSur un th\u00e9or\u00e8me d\u2019analyse fonctionnelle,\nRec. Math. (N. S.) 4(46) (1938), 471\u2013497."},{"key":"2025051309544635639_j_cmam-2017-0058_ref_027_w2aab3b7e2812b1b6b1ab2ac27Aa","doi-asserted-by":"crossref","unstructured":"G.  Strang,\nVariational crimes in the finite element method,\nThe Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations,\nAcademic Press, New York (1972), 689\u2013710.","DOI":"10.1016\/B978-0-12-068650-6.50030-7"},{"key":"2025051309544635639_j_cmam-2017-0058_ref_028_w2aab3b7e2812b1b6b1ab2ac28Aa","unstructured":"A.  Veeser and P.  Zanotti,\nPrivate communication, 2016."},{"key":"2025051309544635639_j_cmam-2017-0058_ref_029_w2aab3b7e2812b1b6b1ab2ac29Aa","doi-asserted-by":"crossref","unstructured":"R.  Verf\u00fcrth,\nA Posteriori Error Estimation Techniques for Finite Element Methods,\nNumer. Math. Sci. Comput.,\nOxford University Press, Oxford, 2013.","DOI":"10.1093\/acprof:oso\/9780199679423.001.0001"}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/view\/journals\/cmam\/18\/3\/article-p451.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/cmam-2017-0058\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/cmam-2017-0058\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,5,13]],"date-time":"2025-05-13T09:55:16Z","timestamp":1747130116000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/cmam-2017-0058\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,12,15]]},"references-count":29,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2017,12,15]]},"published-print":{"date-parts":[[2018,7,1]]}},"alternative-id":["10.1515\/cmam-2017-0058"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2017-0058","relation":{},"ISSN":["1609-9389","1609-4840"],"issn-type":[{"value":"1609-9389","type":"electronic"},{"value":"1609-4840","type":"print"}],"subject":[],"published":{"date-parts":[[2017,12,15]]}}}