{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,9,14]],"date-time":"2022-09-14T14:27:58Z","timestamp":1663165678932},"reference-count":37,"publisher":"American Mathematical Society (AMS)","issue":"333","license":[{"start":{"date-parts":[[2022,9,21]],"date-time":"2022-09-21T00:00:00Z","timestamp":1663718400000},"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":"
Asymptotic expansions are derived for eigenvalues produced by both the Crouzeix-Raviart element and the enriched Crouzeix\u2013Raviart element. The expansions are optimal in the sense that extrapolation eigenvalues based on them admit a fourth order convergence provided that exact eigenfunctions are smooth enough. The major challenge in establishing the expansions comes from the fact that the canonical interpolation of both nonconforming elements lacks a crucial superclose property, and the nonconformity of both elements. The main idea is to employ the relation between the lowest-order mixed Raviart\u2013Thomas element and the two nonconforming elements, and consequently make use of the superclose property of the canonical interpolation of the lowest-order mixed Raviart\u2013Thomas element. To overcome the difficulty caused by the nonconformity, the commuting property of the canonical interpolation operators of both nonconforming elements is further used, which turns the consistency error problem into an interpolation error problem. Then, a series of new results are obtained to show the final expansions.<\/p>","DOI":"10.1090\/mcom\/3635","type":"journal-article","created":{"date-parts":[[2021,2,3]],"date-time":"2021-02-03T16:24:44Z","timestamp":1612369484000},"page":"75-109","source":"Crossref","is-referenced-by-count":1,"title":["Asymptotic expansions of eigenvalues by both the Crouzeix\u2013Raviart and enriched Crouzeix\u2013Raviart elements"],"prefix":"10.1090","volume":"91","author":[{"given":"Jun","family":"Hu","sequence":"first","affiliation":[]},{"given":"Limin","family":"Ma","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2021,9,21]]},"reference":[{"key":"1","first-page":"93","article-title":"Asymptotic lower bounds for eigenvalues by nonconforming finite element methods","volume":"17","author":"Armentano, Mar\u00eda G.","year":"2004","journal-title":"Electron. Trans. Numer. Anal."},{"issue":"4","key":"2","doi-asserted-by":"publisher","first-page":"447","DOI":"10.1007\/BF01399326","article-title":"Direct and inverse error estimates for finite elements with mesh refinements","volume":"33","author":"Babu\u0161ka, I.","year":"1979","journal-title":"Numer. Math.","ISSN":"http:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"6","key":"3","doi-asserted-by":"publisher","first-page":"1249","DOI":"10.1137\/0724082","article-title":"Estimates for the errors in eigenvalue and eigenvector approximation by Galerkin methods, with particular attention to the case of multiple eigenvalues","volume":"24","author":"Babu\u0161ka, I.","year":"1987","journal-title":"SIAM J. Numer. Anal.","ISSN":"http:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"186","key":"4","doi-asserted-by":"publisher","first-page":"275","DOI":"10.2307\/2008468","article-title":"Finite element-Galerkin approximation of the eigenvalues and eigenvectors of selfadjoint problems","volume":"52","author":"Babu\u0161ka, I.","year":"1989","journal-title":"Math. Comp.","ISSN":"http:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"4","key":"5","first-page":"321","article-title":"Finite element eigenvalue computation on domains with reentrant corners using Richardson extrapolation","volume":"8","author":"Blum, H.","year":"1990","journal-title":"J. Comput. Math.","ISSN":"http:\/\/id.crossref.org\/issn\/0254-9409","issn-type":"print"},{"issue":"2","key":"6","doi-asserted-by":"publisher","first-page":"350","DOI":"10.1002\/mma.3452","article-title":"A posteriori error analysis for nonconforming approximation of multiple eigenvalues","volume":"40","author":"Boffi, Daniele","year":"2017","journal-title":"Math. Methods Appl. Sci.","ISSN":"http:\/\/id.crossref.org\/issn\/0170-4214","issn-type":"print"},{"issue":"3","key":"7","doi-asserted-by":"publisher","first-page":"311","DOI":"10.1007\/s002110050064","article-title":"Superconvergence and a posteriori error estimation for triangular mixed finite elements","volume":"68","author":"Brandts, Jan H.","year":"1994","journal-title":"Numer. Math.","ISSN":"http:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"1","key":"8","doi-asserted-by":"publisher","first-page":"95","DOI":"10.1007\/s10444-007-9031-x","article-title":"Asymptotic expansion and extrapolation for the eigenvalue approximation of the biharmonic eigenvalue problem by Ciarlet-Raviart scheme","volume":"27","author":"Chen, Wei","year":"2007","journal-title":"Adv. Comput. Math.","ISSN":"http:\/\/id.crossref.org\/issn\/1019-7168","issn-type":"print"},{"key":"9","first-page":"33","article-title":"Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I","volume":"7","author":"Crouzeix, M.","year":"1973","journal-title":"Rev. Fran\\c{c}aise Automat. Informat. Recherche Op\\'{e}rationnelle S\\'{e}r. Rouge"},{"key":"10","doi-asserted-by":"crossref","unstructured":"A. Damle and G. C. Peterson, Understanding the eigenstructure of various triangles, SIAM Undergrad. Res. Online 3 (2010), 187\u2013208.","DOI":"10.1137\/10S010612"},{"issue":"4","key":"11","first-page":"327","article-title":"Quadrature and extrapolation for the variable coefficient elliptic eigenvalue problem","volume":"3","author":"Ding, Yan Heng","year":"1990","journal-title":"Systems Sci. Math. Sci.","ISSN":"http:\/\/id.crossref.org\/issn\/1000-9590","issn-type":"print"},{"issue":"169","key":"12","doi-asserted-by":"publisher","first-page":"39","DOI":"10.2307\/2007791","article-title":"Global estimates for mixed methods for second order elliptic equations","volume":"44","author":"Douglas, Jim, Jr.","year":"1985","journal-title":"Math. Comp.","ISSN":"http:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"3","key":"13","doi-asserted-by":"publisher","first-page":"287","DOI":"10.1007\/BF01385626","article-title":"Superconvergence for rectangular mixed finite elements","volume":"58","author":"Dur\u00e1n, Ricardo","year":"1990","journal-title":"Numer. Math.","ISSN":"http:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"1","key":"14","doi-asserted-by":"publisher","first-page":"196","DOI":"10.1007\/s10915-014-9821-5","article-title":"Lower bounds for eigenvalues of elliptic operators: by nonconforming finite element methods","volume":"61","author":"Hu, Jun","year":"2014","journal-title":"J. Sci. Comput.","ISSN":"http:\/\/id.crossref.org\/issn\/0885-7474","issn-type":"print"},{"issue":"2","key":"15","doi-asserted-by":"publisher","first-page":"A797--A821","DOI":"10.1137\/19M1261997","article-title":"Asymptotically exact a posteriori error estimates of eigenvalues by the Crouzeix-Raviart element and enriched Crouzeix-Raviart element","volume":"42","author":"Hu, Jun","year":"2020","journal-title":"SIAM J. Sci. Comput.","ISSN":"http:\/\/id.crossref.org\/issn\/1064-8275","issn-type":"print"},{"issue":"3","key":"16","doi-asserted-by":"publisher","first-page":"491","DOI":"10.1007\/s00211-015-0729-2","article-title":"Superconvergence of both the Crouzeix-Raviart and Morley elements","volume":"132","author":"Hu, Jun","year":"2016","journal-title":"Numer. Math.","ISSN":"http:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"2","key":"17","doi-asserted-by":"publisher","first-page":"410","DOI":"10.1007\/s10915-014-9899-9","article-title":"The enriched Crouzeix-Raviart elements are equivalent to the Raviart-Thomas elements","volume":"63","author":"Hu, Jun","year":"2015","journal-title":"J. Sci. Comput.","ISSN":"http:\/\/id.crossref.org\/issn\/0885-7474","issn-type":"print"},{"issue":"4","key":"18","doi-asserted-by":"publisher","first-page":"Paper No. 52, 25","DOI":"10.1007\/s10444-021-09874-7","article-title":"Optimal superconvergence analysis for the Crouzeix-Raviart and the Morley elements","volume":"47","author":"Hu, Jun","year":"2021","journal-title":"Adv. Comput. Math.","ISSN":"http:\/\/id.crossref.org\/issn\/1019-7168","issn-type":"print"},{"issue":"263","key":"19","doi-asserted-by":"publisher","first-page":"1253","DOI":"10.1090\/S0025-5718-08-02051-6","article-title":"Superconvergence of quadratic finite elements on mildly structured grids","volume":"77","author":"Huang, Yunqing","year":"2008","journal-title":"Math. Comp.","ISSN":"http:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"2","key":"20","doi-asserted-by":"publisher","first-page":"435","DOI":"10.1002\/num.20268","article-title":"Approximation and eigenvalue extrapolation of biharmonic eigenvalue problem by nonconforming finite element methods","volume":"24","author":"Jia, Shanghui","year":"2008","journal-title":"Numer. Methods Partial Differential Equations","ISSN":"http:\/\/id.crossref.org\/issn\/0749-159X","issn-type":"print"},{"issue":"2","key":"21","doi-asserted-by":"publisher","first-page":"163","DOI":"10.1137\/1026033","article-title":"Eigenvalues of the Laplacian in two dimensions","volume":"26","author":"Kuttler, J. R.","year":"1984","journal-title":"SIAM Rev.","ISSN":"http:\/\/id.crossref.org\/issn\/0036-1445","issn-type":"print"},{"issue":"2","key":"22","doi-asserted-by":"publisher","first-page":"792","DOI":"10.1137\/17M112587X","article-title":"Global superconvergence of the lowest-order mixed finite element on mildly structured meshes","volume":"56","author":"Li, Yu-Wen","year":"2018","journal-title":"SIAM J. Numer. Anal.","ISSN":"http:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"2","key":"23","first-page":"172","article-title":"Can we compute Laplace eigenvalues well, like computing \ud835\udf0b?","volume":"1","author":"Lin, Qun","year":"2005","journal-title":"Int. J. Inf. Syst. Sci.","ISSN":"http:\/\/id.crossref.org\/issn\/1708-296X","issn-type":"print"},{"issue":"264","key":"24","doi-asserted-by":"publisher","first-page":"2061","DOI":"10.1090\/S0025-5718-08-02098-X","article-title":"New expansions of numerical eigenvalues for -\u0394\ud835\udc62=\ud835\udf06\ud835\udf0c\ud835\udc62 by nonconforming elements","volume":"77","author":"Lin, Qun","year":"2008","journal-title":"Math. Comp.","ISSN":"http:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"1","key":"25","doi-asserted-by":"publisher","first-page":"213","DOI":"10.1016\/j.cam.2008.07.019","article-title":"New expansions of numerical eigenvalues by Wilson\u2019s element","volume":"225","author":"Lin, Qun","year":"2009","journal-title":"J. Comput. Appl. Math.","ISSN":"http:\/\/id.crossref.org\/issn\/0377-0427","issn-type":"print"},{"key":"26","unstructured":"Q. Lin and J. Lin, Finite element methods: Accuracy and Improvement, China Sci. Press, Beijing, (2006)."},{"key":"27","first-page":"1","article-title":"Asymptotic expansions for finite element eigenvalues and finite element solution","author":"Lin, Qun","year":"1984"},{"issue":"1","key":"28","doi-asserted-by":"publisher","first-page":"19","DOI":"10.1002\/(SICI)1099-0887(199901)15:1<19::AID-CNM220>3.0.CO;2-W","article-title":"High-accuracy approximations for eigenvalue problems by the Carey non-conforming finite element","volume":"15","author":"Lin, Qun","year":"1999","journal-title":"Comm. Numer. Methods Engrg.","ISSN":"http:\/\/id.crossref.org\/issn\/1069-8299","issn-type":"print"},{"issue":"8","key":"29","doi-asserted-by":"publisher","first-page":"1884","DOI":"10.1016\/j.apnum.2009.01.011","article-title":"Asymptotic error expansion and Richardson extrapolation of eigenvalue approximations for second order elliptic problems by the mixed finite element method","volume":"59","author":"Lin, Qun","year":"2009","journal-title":"Appl. Numer. Math.","ISSN":"http:\/\/id.crossref.org\/issn\/0168-9274","issn-type":"print"},{"issue":"1","key":"30","first-page":"10","article-title":"New expansions of numerical eigenvalue for -\u0394\ud835\udc62=\ud835\udf06\ud835\udf0c\ud835\udc62 by linear elements on different triangular meshes","volume":"6","author":"Lin, Qun","year":"2010","journal-title":"Int. J. Inf. Syst. Sci.","ISSN":"http:\/\/id.crossref.org\/issn\/1708-296X","issn-type":"print"},{"issue":"11","key":"31","first-page":"132","article-title":"Extrapolation of three-dimensional eigenvalue finite element approximation","volume":"41","author":"Lin, Qun","year":"2011","journal-title":"Math. Pract. Theory","ISSN":"http:\/\/id.crossref.org\/issn\/1000-0984","issn-type":"print"},{"issue":"1","key":"32","doi-asserted-by":"publisher","first-page":"65","DOI":"10.1007\/s006070200003","article-title":"High accuracy analysis of the Adini\u2019s nonconforming element","volume":"68","author":"Luo, P.","year":"2002","journal-title":"Computing","ISSN":"http:\/\/id.crossref.org\/issn\/0010-485X","issn-type":"print"},{"issue":"3","key":"33","doi-asserted-by":"publisher","first-page":"493","DOI":"10.1137\/0722029","article-title":"An inexpensive method for the evaluation of the solution of the lowest order Raviart-Thomas mixed method","volume":"22","author":"Marini, Luisa Donatella","year":"1985","journal-title":"SIAM J. Numer. Anal.","ISSN":"http:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"1","key":"34","doi-asserted-by":"publisher","first-page":"155","DOI":"10.1007\/s00211-005-0662-x","article-title":"The Morley element for fourth order elliptic equations in any dimensions","volume":"103","author":"Wang, Ming","year":"2006","journal-title":"Numer. Math.","ISSN":"http:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"key":"35","doi-asserted-by":"crossref","unstructured":"L. S. D. Morley, The triangular equilibrium element in the solution of plate bending problems, Aeronau. Quart. 19 (1968), 149\u2013169.","DOI":"10.1017\/S0001925900004546"},{"issue":"1","key":"36","doi-asserted-by":"publisher","first-page":"23","DOI":"10.1007\/BF01396493","article-title":"Nonconforming finite element methods for eigenvalue problems in linear plate theory","volume":"33","author":"Rannacher, Rolf","year":"1979","journal-title":"Numer. Math.","ISSN":"http:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"key":"37","first-page":"292","article-title":"A mixed finite element method for 2nd order elliptic problems","author":"Raviart, P.-A.","year":"1977"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.ams.org\/mcom\/earlyview\/#mcom3635\/.pdf","content-type":"unspecified","content-version":"am","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2022-91-333\/S0025-5718-2021-03635-2\/S0025-5718-2021-03635-2.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,12,10]],"date-time":"2021-12-10T16:42:36Z","timestamp":1639154556000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2022-91-333\/S0025-5718-2021-03635-2\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,9,21]]},"references-count":37,"journal-issue":{"issue":"333","published-print":{"date-parts":[[2022,1]]}},"alternative-id":["S0025-5718-2021-03635-2"],"URL":"http:\/\/dx.doi.org\/10.1090\/mcom\/3635","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["0025-5718","1088-6842"],"issn-type":[{"value":"0025-5718","type":"print"},{"value":"1088-6842","type":"electronic"}],"subject":["Applied Mathematics","Computational Mathematics","Algebra and Number Theory"],"published":{"date-parts":[[2021,9,21]]}}}