{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T13:50:58Z","timestamp":1776865858753,"version":"3.51.2"},"reference-count":31,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2021,6,29]],"date-time":"2021-06-29T00:00:00Z","timestamp":1624924800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2021,6,29]],"date-time":"2021-06-29T00:00:00Z","timestamp":1624924800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"crossref","award":["11625101"],"award-info":[{"award-number":["11625101"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Adv Comput Math"],"published-print":{"date-parts":[[2021,8]]},"DOI":"10.1007\/s10444-021-09874-7","type":"journal-article","created":{"date-parts":[[2021,6,29]],"date-time":"2021-06-29T15:06:32Z","timestamp":1624979192000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Optimal superconvergence analysis for the Crouzeix-Raviart and the Morley elements"],"prefix":"10.1007","volume":"47","author":[{"given":"Jun","family":"Hu","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-0321-3098","authenticated-orcid":false,"given":"Limin","family":"Ma","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Rui","family":"Ma","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2021,6,29]]},"reference":[{"issue":"1","key":"9874_CR1","doi-asserted-by":"publisher","first-page":"7","DOI":"10.1051\/m2an\/1985190100071","volume":"19","author":"DN Arnold","year":"1985","unstructured":"Arnold, DN, Brezzi, F: Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates. ESAIM Math. Model. Numer. Anal. 19(1), 7\u201332 (1985)","journal-title":"ESAIM Math. Model. Numer. Anal."},{"key":"9874_CR2","unstructured":"Douglas, N.A., Falk, R.S., Winther, R.: Differential complexes and stability of finite element methods I. The de Rham Complex. In: Compatible Spatial Discretizations, pp 23\u201346. Springer, New York (2006)"},{"issue":"6","key":"9874_CR3","doi-asserted-by":"publisher","first-page":"2294","DOI":"10.1137\/S003614290139874X","volume":"41","author":"RE Bank","year":"2003","unstructured":"Bank, R.E., Xu, J.: Asymptotically exact a posteriori error estimators, part I: Grids with superconvergence. SIAM J. Numer. Anal. 41(6), 2294\u20132312 (2003)","journal-title":"SIAM J. Numer. Anal."},{"issue":"175","key":"9874_CR4","doi-asserted-by":"publisher","first-page":"103","DOI":"10.1090\/S0025-5718-1986-0842125-3","volume":"47","author":"JH Bramble","year":"1986","unstructured":"Bramble, J.H., Pasciak, J.E., Schatz, A.H.: The construction of preconditioners for elliptic problems by substructuring I. Math. Comput. 47(175), 103\u2013134 (1986)","journal-title":"Math. Comput."},{"issue":"3","key":"9874_CR5","doi-asserted-by":"publisher","first-page":"311","DOI":"10.1007\/s002110050064","volume":"68","author":"JH Brandts","year":"1994","unstructured":"Brandts, J.H.: Superconvergence and a posteriori error estimation for triangular mixed finite elements. Numer. Math. 68(3), 311\u2013324 (1994)","journal-title":"Numer. Math."},{"issue":"1","key":"9874_CR6","doi-asserted-by":"publisher","first-page":"39","DOI":"10.1016\/S0168-9274(99)00034-3","volume":"34","author":"JH Brandts","year":"2000","unstructured":"Brandts, J.H.: Superconvergence for triangular order k = 1 Raviart-Thomas mixed finite elements and for triangular standard quadratic finite element methods. Appl. Numer. Mathe. 34(1), 39\u201358 (2000)","journal-title":"Appl. Numer. Mathe."},{"key":"9874_CR7","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-75934-0","volume-title":"The Mathematical Theory of Finite Element Methods","author":"S Brenner","year":"2008","unstructured":"Brenner, S., Scott, R: The Mathematical Theory of Finite Element Methods, 3rd ed. Springer, New York (2008)","edition":"3rd ed"},{"key":"9874_CR8","unstructured":"Brezzi, F.: On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. Revue fran\u00e7aise d\u2019automatique, informatique, recherche op\u00e9rationnelle. Analyse num\u00e9rique"},{"issue":"2","key":"9874_CR9","doi-asserted-by":"publisher","first-page":"217","DOI":"10.1007\/BF01389710","volume":"47","author":"F Brezzi","year":"1985","unstructured":"Brezzi, F., Douglas, J., Marini, L.D.: Two families of mixed finite elements for second order elliptic problems. Numer. Math. 47(2), 217\u2013235 (1985)","journal-title":"Numer. Math."},{"key":"9874_CR10","doi-asserted-by":"crossref","unstructured":"Brezzi, F., Raviart, P.-A.: Mixed finite element methods for 4th order elliptic equations. Topics in numerical analysis, III (Proc. Roy Irish Acad Conf) (1976)","DOI":"10.1016\/B978-0-12-496952-0.50019-2"},{"key":"9874_CR11","unstructured":"Chen, C., Huang, Y.: High accuracy theory of finite element methods (1995)"},{"issue":"2","key":"9874_CR12","doi-asserted-by":"publisher","first-page":"125","DOI":"10.1007\/s002110050084","volume":"69","author":"H Chen","year":"2013","unstructured":"Chen, H., Bo, L.: Superconvergence analysis and error expansion for the Wilson nonconforming finite element. Numer. Math. 69(2), 125\u2013140 (2013)","journal-title":"Numer. Math."},{"issue":"R3","key":"9874_CR13","doi-asserted-by":"publisher","first-page":"33","DOI":"10.1051\/m2an\/197307R300331","volume":"7","author":"M Crouzeix","year":"1973","unstructured":"Crouzeix, M., Raviart, P.-A.: Conforming and nonconforming finite element methods for solving the stationary Stokes equations. Revue fran\u00e7aise d\u2019automatique informatique recherche op\u00e9rationnelle, Math\u00e9,matique 7(R3), 33\u201375 (1973)","journal-title":"Revue fran\u00e7aise d\u2019automatique informatique recherche op\u00e9rationnelle, Math\u00e9,matique"},{"issue":"169","key":"9874_CR14","doi-asserted-by":"publisher","first-page":"39","DOI":"10.1090\/S0025-5718-1985-0771029-9","volume":"44","author":"J Douglas","year":"1985","unstructured":"Douglas, J., Roberts, J.E.: Global estimates for mixed methods for second order elliptic problems. Math. Comput. 44(169), 39\u201352 (1985)","journal-title":"Math. Comput."},{"issue":"2-4","key":"9874_CR15","doi-asserted-by":"publisher","first-page":"121","DOI":"10.1007\/BF02575724","volume":"26","author":"J Douglas","year":"1989","unstructured":"Douglas, J., Wang, J.: Superconvergence of mixed finite element methods on rectangular domains. Calcolo 26(2-4), 121\u2013133 (1989)","journal-title":"Calcolo"},{"issue":"1","key":"9874_CR16","doi-asserted-by":"publisher","first-page":"287","DOI":"10.1007\/BF01385626","volume":"58","author":"R Dur\u00e1n","year":"1990","unstructured":"Dur\u00e1n, R.: Superconvergence for rectangular mixed finite elements. Numer. Math. 58(1), 287\u2013298 (1990)","journal-title":"Numer. Math."},{"key":"9874_CR17","doi-asserted-by":"publisher","first-page":"196","DOI":"10.1007\/s10915-014-9821-5","volume":"61","author":"J Hu","year":"2014","unstructured":"Hu, J., Huang, Y., Lin, Q.: Lower bounds for eigenvalues of elliptic operators: By nonconforming finite element methods. J. Sci. Comput. 61, 196\u2013221 (2014)","journal-title":"J. Sci. Comput."},{"issue":"2","key":"9874_CR18","doi-asserted-by":"publisher","first-page":"410","DOI":"10.1007\/s10915-014-9899-9","volume":"63","author":"J Hu","year":"2015","unstructured":"Hu, J., Ma, R.: The Enriched Crouzeix-Raviart elements are equivalent to the Raviart-Thomas elements. J. Sci. Comput. 63(2), 410\u2013425 (2015)","journal-title":"J. Sci. Comput."},{"issue":"3","key":"9874_CR19","doi-asserted-by":"publisher","first-page":"491","DOI":"10.1007\/s00211-015-0729-2","volume":"132","author":"J Hu","year":"2016","unstructured":"Hu, J., Ma, R.: Superconvergence of both the Crouzeix-Raviart and Morley elements. Numer. Math. 132(3), 491\u2013509 (2016)","journal-title":"Numer. Math."},{"issue":"6","key":"9874_CR20","first-page":"561","volume":"23","author":"H Jun","year":"2005","unstructured":"Jun, H., Shi, Z.-C.: Constrained quadrilateral nonconforming rotated Q1 element. J. Comput. Math. 23(6), 561\u2013586 (2005)","journal-title":"J. Comput. Math."},{"issue":"263","key":"9874_CR21","first-page":"1253","volume":"77","author":"H Yunqing","year":"1986","unstructured":"Yunqing, H., Xu, J.: Superconvergence of quadratic finite elements on mildly structured grids. Math. Comput. 77(263), 1253\u20131268 (1986)","journal-title":"Math. Comput."},{"issue":"1","key":"9874_CR22","doi-asserted-by":"publisher","first-page":"43","DOI":"10.1007\/BF01436186","volume":"21","author":"J Claes","year":"1973","unstructured":"Claes, J.: On the convergence of a mixed finite-element method for plate bending problems. Numer. Math. 21(1), 43\u201362 (1973)","journal-title":"Numer. Math."},{"key":"9874_CR23","first-page":"257","volume":"45","author":"W Krendl","year":"2016","unstructured":"Krendl, W., Rafetseder, K., Zulehner, W.: A decomposition result for biharmonic problems and the Hellan-Herrmann-Johnson method. Electron. Trans. Numer. Anal. 45, 257\u2013282 (2016)","journal-title":"Electron. Trans. Numer. Anal."},{"issue":"2","key":"9874_CR24","doi-asserted-by":"publisher","first-page":"792","DOI":"10.1137\/17M112587X","volume":"56","author":"Y Li","year":"2018","unstructured":"Li, Y.: Global superconvergence of the lowest order mixed finite element on mildly structured meshes. SIAM J. Numer. Anal. 56(2), 792\u2013815 (2018)","journal-title":"SIAM J. Numer. Anal."},{"key":"9874_CR25","doi-asserted-by":"crossref","unstructured":"Lin, Q., Tobiska, L., Zhou, A.: Superconvergence and extrapolation of non-conforming low order finite elements applied to the Poisson equation. IMA Journal of Numerical Analysis, 25(1) (2005)","DOI":"10.1093\/imanum\/drh008"},{"issue":"3\u20134","key":"9874_CR26","first-page":"349","volume":"118","author":"JL Lions","year":"1972","unstructured":"Lions, J.L., Magenes, E.: Non-homogeneous boundary value problems and applications. Lithos 118(3\u20134), 349\u2013364 (1972)","journal-title":"Lithos"},{"issue":"3","key":"9874_CR27","doi-asserted-by":"publisher","first-page":"407","DOI":"10.1007\/s00211-008-0190-6","volume":"111","author":"S Mao","year":"2009","unstructured":"Mao, S., Shi, Z.: High accuracy analysis of two nonconforming plate elements. Numer. Math. 111(3), 407\u2013443 (2009)","journal-title":"Numer. Math."},{"issue":"3","key":"9874_CR28","doi-asserted-by":"publisher","first-page":"493","DOI":"10.1137\/0722029","volume":"22","author":"LD Marini","year":"1985","unstructured":"Marini, L.D.: An inexpensive method for the evaluation of the solution of the lowest order Raviart-Thomas mixed method. SIAM J. Numer. Anal. 22 (3), 493\u2013496 (1985)","journal-title":"SIAM J. Numer. Anal."},{"issue":"3","key":"9874_CR29","first-page":"322","volume":"3","author":"P Ming","year":"2006","unstructured":"Ming, P., Shi, Z., Yun, X.: Superconvergence studies of quadrilateral nonconforming rotated Q1 elements. Int. J. Numer. Anal. Model. 3(3), 322\u2013332 (2006)","journal-title":"Int. J. Numer. Anal. Model."},{"issue":"2","key":"9874_CR30","doi-asserted-by":"publisher","first-page":"149","DOI":"10.1017\/S0001925900004546","volume":"19","author":"LSD Morley","year":"1968","unstructured":"Morley, L.S.D.: The triangular equilibrium element in the solution of plate bending problems. Aeronaut. Quarterly 19(2), 149\u2013169 (1968)","journal-title":"Aeronaut. Quarterly"},{"key":"9874_CR31","first-page":"292","volume-title":"A mixed finite element method for second order elliptic problems","author":"P-A Raviart","year":"1977","unstructured":"Raviart, P.-A., Thomas, J.-M.: A mixed finite element method for second order elliptic problems, pp 292\u2013315. Springer, Berlin (1977)"}],"container-title":["Advances in Computational Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-021-09874-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10444-021-09874-7\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-021-09874-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,1,2]],"date-time":"2023-01-02T00:03:25Z","timestamp":1672617805000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10444-021-09874-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,6,29]]},"references-count":31,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2021,8]]}},"alternative-id":["9874"],"URL":"https:\/\/doi.org\/10.1007\/s10444-021-09874-7","relation":{},"ISSN":["1019-7168","1572-9044"],"issn-type":[{"value":"1019-7168","type":"print"},{"value":"1572-9044","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,6,29]]},"assertion":[{"value":"21 October 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"6 May 2021","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"29 June 2021","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"52"}}