{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,11]],"date-time":"2026-04-11T04:39:34Z","timestamp":1775882374738,"version":"3.50.1"},"reference-count":55,"publisher":"Walter de Gruyter GmbH","issue":"2","funder":[{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["CA 151\/22-2"],"award-info":[{"award-number":["CA 151\/22-2"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2021,4,1]]},"abstract":"Abstract<\/jats:title>\n This article on nonconforming schemes for m<\/jats:italic> harmonic problems\nsimultaneously treats the Crouzeix\u2013Raviart (\n \n \n \n m<\/m:mi>\n =<\/m:mo>\n 1<\/m:mn>\n <\/m:mrow>\n <\/m:math>\n \n {m=1}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>) and the\nMorley finite elements (\n \n \n \n m<\/m:mi>\n =<\/m:mo>\n 2<\/m:mn>\n <\/m:mrow>\n <\/m:math>\n \n {m=2}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>) for the original and for modified right-hand side F<\/jats:italic>\nin the dual space \n \n \n \n \n V<\/m:mi>\n *<\/m:mo>\n <\/m:msup>\n :=<\/m:mo>\n \n \n H<\/m:mi>\n \n -<\/m:mo>\n m<\/m:mi>\n <\/m:mrow>\n <\/m:msup>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \u03a9<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {V^{*}:=H^{-m}(\\Omega)}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> to the energy space \n \n \n \n V<\/m:mi>\n :=<\/m:mo>\n \n \n H<\/m:mi>\n 0<\/m:mn>\n m<\/m:mi>\n <\/m:msubsup>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \u03a9<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {V:=H^{m}_{0}(\\Omega)}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>.\nThe smoother \n \n \n \n J<\/m:mi>\n :<\/m:mo>\n \n \n V<\/m:mi>\n nc<\/m:mi>\n <\/m:msub>\n \u2192<\/m:mo>\n V<\/m:mi>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {J:V_{\\mathrm{nc}}\\to V}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> in this paper is a companion operator, that is a\nlinear and bounded right-inverse to the nonconforming interpolation operator \n \n \n \n \n I<\/m:mi>\n nc<\/m:mi>\n <\/m:msub>\n :<\/m:mo>\n \n V<\/m:mi>\n \u2192<\/m:mo>\n \n V<\/m:mi>\n nc<\/m:mi>\n <\/m:msub>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {I_{\\mathrm{nc}}:V\\to V_{\\mathrm{nc}}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>, and modifies the discrete right-hand side \n \n \n \n \n F<\/m:mi>\n h<\/m:mi>\n <\/m:msub>\n :=<\/m:mo>\n \n F<\/m:mi>\n \u2218<\/m:mo>\n J<\/m:mi>\n <\/m:mrow>\n \u2208<\/m:mo>\n \n V<\/m:mi>\n nc<\/m:mi>\n *<\/m:mo>\n <\/m:msubsup>\n <\/m:mrow>\n <\/m:math>\n \n {F_{h}:=F\\circ J\\in V_{\\mathrm{nc}}^{*}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>. The\nbest-approximation property of the modified scheme from Veeser et al. (2018)\nis recovered and complemented with an analysis of the convergence rates\nin weaker Sobolev norms. Examples with oscillating data\nshow that the original method may fail to\nenjoy the best-approximation property but can also be better than the\nmodified scheme. The a posteriori analysis of this paper concerns data oscillations\nof various types in a class of right-hand sides \n \n \n \n F<\/m:mi>\n \u2208<\/m:mo>\n \n V<\/m:mi>\n *<\/m:mo>\n <\/m:msup>\n <\/m:mrow>\n <\/m:math>\n \n {F\\in V^{*}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula>. The reliable error estimates\ninvolve explicit constants and can be recommended for explicit error control of the piecewise energy norm. The efficiency follows solely up to data oscillations and examples\nillustrate this can be problematic.<\/jats:p>","DOI":"10.1515\/cmam-2021-0029","type":"journal-article","created":{"date-parts":[[2021,3,11]],"date-time":"2021-03-11T00:38:17Z","timestamp":1615423097000},"page":"289-315","source":"Crossref","is-referenced-by-count":19,"title":["A Priori and a Posteriori Error Analysis of the Crouzeix\u2013Raviart and Morley FEM with Original and Modified Right-Hand Sides"],"prefix":"10.1515","volume":"21","author":[{"given":"Carsten","family":"Carstensen","sequence":"first","affiliation":[{"name":"Department of Mathematics , Humboldt-Universit\u00e4t zu Berlin , 10099 Berlin , Germany ; and Distinguished Visiting Professor, Department of Mathematics, Indian institute of Technology Bombay, Powai, Mumbai-400076, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Neela","family":"Nataraj","sequence":"additional","affiliation":[{"name":"Department of Mathematics , Indian Institute of Technology Bombay , Powai , Mumbai 400076 , India"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2021,3,11]]},"reference":[{"key":"2023033111200672659_j_cmam-2021-0029_ref_001","doi-asserted-by":"crossref","unstructured":"S. Agmon,\nLectures on Elliptic Boundary Value Problems,\nAMS Chelsea, Providence, 2010.","DOI":"10.1090\/chel\/369"},{"key":"2023033111200672659_j_cmam-2021-0029_ref_002","doi-asserted-by":"crossref","unstructured":"D. N. Arnold and F. Brezzi,\nMixed and nonconforming finite element methods: Implementation, postprocessing and error estimates,\nRAIRO Mod\u00e9l. Math. Anal. Num\u00e9r. 19 (1985), no. 1, 7\u201332.","DOI":"10.1051\/m2an\/1985190100071"},{"key":"2023033111200672659_j_cmam-2021-0029_ref_003","doi-asserted-by":"crossref","unstructured":"D. N. Arnold and R. S. Falk,\nA uniformly accurate finite element method for the Reissner\u2013Mindlin plate,\nSIAM J. Numer. Anal. 26 (1989), no. 6, 1276\u20131290.","DOI":"10.1137\/0726074"},{"key":"2023033111200672659_j_cmam-2021-0029_ref_004","doi-asserted-by":"crossref","unstructured":"R. Becker, S. Mao and Z. Shi,\nA convergent nonconforming adaptive finite element method with quasi-optimal complexity,\nSIAM J. Numer. 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Methods Partial Differential Equations 31 (2015), no. 2, 367\u2013396.","DOI":"10.1002\/num.21892"},{"key":"2023033111200672659_j_cmam-2021-0029_ref_008","doi-asserted-by":"crossref","unstructured":"S. C. Brenner and L.-Y. Sung,\n\n \n \n \n C\n 0\n \n \n \n {C^{0}}\n \n interior penalty methods for fourth order elliptic boundary value problems on polygonal domains,\nJ. Sci. Comput. 22\/23 (2005), 83\u2013118.","DOI":"10.1007\/s10915-004-4135-7"},{"key":"2023033111200672659_j_cmam-2021-0029_ref_009","doi-asserted-by":"crossref","unstructured":"C. Carstensen,\nLectures on adaptive mixed finite element methods,\nMixed Finite Element Technologies,\nCISM Courses and Lect. 509,\nSpringer, Vienna (2009), 1\u201356.","DOI":"10.1007\/978-3-211-99094-0_1"},{"key":"2023033111200672659_j_cmam-2021-0029_ref_010","doi-asserted-by":"crossref","unstructured":"C. Carstensen, S. Bartels and S. Jansche,\nA posteriori error estimates for nonconforming finite element methods,\nNumer. 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Math.,\nSpringer, Berlin, 2001.","DOI":"10.1007\/978-3-642-61798-0"},{"key":"2023033111200672659_j_cmam-2021-0029_ref_037","unstructured":"P. Grisvard,\nSingularities in Boundary Value Problems,\nRech. Math. Appl. 22,\nMasson, Paris, 1992."},{"key":"2023033111200672659_j_cmam-2021-0029_ref_038","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":"2023033111200672659_j_cmam-2021-0029_ref_039","doi-asserted-by":"crossref","unstructured":"J. Hu and Z. Shi,\nA new a posteriori error estimate for the Morley element,\nNumer. Math. 112 (2009), no. 1, 25\u201340.","DOI":"10.1007\/s00211-008-0205-3"},{"key":"2023033111200672659_j_cmam-2021-0029_ref_040","doi-asserted-by":"crossref","unstructured":"J. Hu, Z. Shi and J. Xu,\nConvergence and optimality of the adaptive Morley element method,\nNumer. 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Algorithms 42 (2006), no. 3\u20134, 309\u2013323.","DOI":"10.1007\/s11075-006-9046-2"},{"key":"2023033111200672659_j_cmam-2021-0029_ref_048","unstructured":"L. Tartar,\nAn Introduction to Sobolev Spaces and Interpolation Spaces,\nLect. Notes Unione Mat. Ital. 3,\nSpringer, Berlin, 2007."},{"key":"2023033111200672659_j_cmam-2021-0029_ref_049","doi-asserted-by":"crossref","unstructured":"R. Vanselow,\nNew results concerning the DWR method for some nonconforming FEM,\nAppl. Math. 57 (2012), no. 6, 551\u2013568.","DOI":"10.1007\/s10492-012-0033-8"},{"key":"2023033111200672659_j_cmam-2021-0029_ref_050","doi-asserted-by":"crossref","unstructured":"A. Veeser and P. Zanotti,\nQuasi-optimal nonconforming methods for symmetric elliptic problems. I\u2014Abstract theory,\nSIAM J. Numer. Anal. 56 (2018), no. 3, 1621\u20131642.","DOI":"10.1137\/17M1116362"},{"key":"2023033111200672659_j_cmam-2021-0029_ref_051","doi-asserted-by":"crossref","unstructured":"A. Veeser and P. 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