{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T13:50:56Z","timestamp":1776865856959,"version":"3.51.2"},"reference-count":18,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2007,2,14]],"date-time":"2007-02-14T00:00:00Z","timestamp":1171411200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2007,2,14]],"date-time":"2007-02-14T00:00:00Z","timestamp":1171411200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Adv Comput Math"],"published-print":{"date-parts":[[2007,7]]},"DOI":"10.1007\/s10444-007-9031-x","type":"journal-article","created":{"date-parts":[[2007,2,13]],"date-time":"2007-02-13T15:04:52Z","timestamp":1171379092000},"page":"95-106","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":8,"title":["Asymptotic expansion and extrapolation for the Eigenvalue approximation of the biharmonic eigenvalue problem by Ciarlet\u2013Raviart scheme"],"prefix":"10.1007","volume":"27","author":[{"given":"Wei","family":"Chen","sequence":"first","affiliation":[]},{"given":"Qun","family":"Lin","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2007,2,14]]},"reference":[{"key":"9031_CR1","doi-asserted-by":"crossref","unstructured":"Babu\u0161ka, I., Osborn, J.: Eigenvalue problems. In: Ciarlet, P.G., Lions, J.L. (eds.) Handbook of Numerical Analysis, vol. II, Finite element method (Part I) (1991)","DOI":"10.1016\/S1570-8659(05)80042-0"},{"key":"9031_CR2","doi-asserted-by":"publisher","first-page":"97","DOI":"10.1007\/s006070050053","volume":"63","author":"P.E. Bj\u00f8rstad","year":"1999","unstructured":"Bj\u00f8rstad, P.E., Tj\u00f8stheim, B.P.: High precision solutions of two fourth order eigenvalue problems. Computing 63, 97\u2013107 (1999)","journal-title":"Computing"},{"key":"9031_CR3","doi-asserted-by":"publisher","first-page":"121","DOI":"10.1090\/S0025-5718-99-01072-8","volume":"69","author":"D. Boffi","year":"2000","unstructured":"Boffi, D., Brezzi, F., Gastaldi, L.: On the problem of spurious eigenvalues in the approximation of linear elliptic problems in mixed form. Math. Comput. 69 121\u2013140 (2000)","journal-title":"Math. Comput."},{"key":"9031_CR4","volume-title":"Springer Series in Computational Mathematics, vol.\u00a015","author":"F. Brezzi","year":"1991","unstructured":"Brezzi, F., Fortin, M.: Mixed and hybrid finite element method. In: Springer Series in Computational Mathematics, vol.\u00a015. Springer, Berlin Heidelberg New York (1991)"},{"key":"9031_CR5","doi-asserted-by":"publisher","first-page":"1505","DOI":"10.1098\/rspa.2000.0573","volume":"456","author":"B.M. Brown","year":"2000","unstructured":"Brown, B.M., Davies, E.B., Jimack, P.K., Mihajlovi\u0107, M.D.: A numerical investigation of the solution of a class of fourth order eigenvalue problem. Proc. Roy. Soc. London Ser. A 456, 1505\u20131521 (2000)","journal-title":"Proc. Roy. Soc. London Ser. A"},{"key":"9031_CR6","volume-title":"Finite Element Method for Elliptic Problems","author":"P.G. Ciarlet","year":"1978","unstructured":"Ciarlet, P.G.: Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)"},{"key":"9031_CR7","doi-asserted-by":"crossref","first-page":"125","DOI":"10.1016\/B978-0-12-208350-1.50009-1","volume-title":"Symposium on Mathematical Aspects of Finite Elements in Partial DIfferential Equations","author":"P.G. Ciarlet","year":"1974","unstructured":"Ciarlet, P.G., Raviart, P.A.: A mixed finite element method for the biharmonic equation. In: de. Boor, C. (ed.) Symposium on Mathematical Aspects of Finite Elements in Partial DIfferential Equations, pp. 125\u2013143. Academic, New York (1974)"},{"key":"9031_CR8","doi-asserted-by":"publisher","first-page":"107","DOI":"10.1023\/A:1014291224961","volume":"15","author":"V. Heuveline","year":"2001","unstructured":"Heuveline, V., Rannacher, R.: A posteriori error control for finite element approximations of elliptic eigenvalue problems. Adv. Comput. Math. 15, 107\u2013138 (2001)","journal-title":"Adv. Comput. Math."},{"key":"9031_CR9","doi-asserted-by":"publisher","first-page":"333","DOI":"10.1007\/s002110100386","volume":"93","author":"Q. Hu","year":"2002","unstructured":"Hu, Q., Zou, J.: Two new variants of nonlinear inexact Uzawa algorithms for saddle-point problems. Numer. Math. 93, 333\u2013359 (2002)","journal-title":"Numer. Math."},{"key":"9031_CR10","doi-asserted-by":"crossref","first-page":"399","DOI":"10.2977\/prims\/1195189071","volume":"14","author":"K. Ishihara","year":"1978","unstructured":"Ishihara, K.: A mixed finite element method for the biharmonic eigenvalue problem of plate bending. Publ. Res. Inst. Math. Sci. 14, 399\u2013414 (1978)","journal-title":"Publ. Res. Inst. Math. Sci."},{"key":"9031_CR11","unstructured":"Lin, Q., Lin, J.: High Performance FEMs. China Sci. Technol. Press (2006)"},{"key":"9031_CR12","first-page":"230","volume-title":"Proc. Sys. Sci. & Sys. Engrg.","author":"Q. Lin","year":"1991","unstructured":"Lin, Q., Li, J., Zhou, A.: A rectangle test for Ciarlet\u2013Raviart scheme. In: Proc. Sys. Sci. & Sys. Engrg., pp. 230\u2013231. Great Wall Culture Publishing Co., China (1991)"},{"key":"9031_CR13","first-page":"1","volume":"158","author":"Q. Lin","year":"1984","unstructured":"Lin, Q., Lu, T.: Asymptotic expansions for finite element eigenvalues and finite element. Bonner Math. Schriften 158, 1\u201310 (1984)","journal-title":"Bonner Math. Schriften"},{"key":"9031_CR14","volume-title":"High Efficiency FEM Construction and Analysis","author":"Q. Lin","year":"1996","unstructured":"Lin, Q., Yan, N.: High Efficiency FEM Construction and Analysis. Hebei University Press, China (1996)"},{"key":"9031_CR15","doi-asserted-by":"publisher","first-page":"427","DOI":"10.2307\/2007651","volume":"36","author":"B. Mercier","year":"1981","unstructured":"Mercier, B., Osborn, J., Rappaz, J., Raviart, P.A.: Eigenvalue approximation by mixed and hybird methods. Math. Comput. 36, 427\u2013453 (1981)","journal-title":"Math. Comput."},{"key":"9031_CR16","doi-asserted-by":"crossref","unstructured":"Osborn, J.: Spectral approximation for compact operators. Math. Comput. 29, 712\u2013725","DOI":"10.1090\/S0025-5718-1975-0383117-3"},{"key":"9031_CR17","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1007\/BF02684402","volume":"59","author":"C. Wieners","year":"1997","unstructured":"Wieners, C.: Bounds for the N lowest eigenvalues of fourth-order boundary value problems. Computing 59, 29\u201341 (1997)","journal-title":"Computing"},{"key":"9031_CR18","doi-asserted-by":"publisher","first-page":"17","DOI":"10.1090\/S0025-5718-99-01180-1","volume":"70","author":"J. Xu","year":"2001","unstructured":"Xu, J., Zhou, A.: A two-grid discretization scheme for eigenvalue problems. Math. Comput. 70, 17\u201325 (2001)","journal-title":"Math. Comput."}],"container-title":["Advances in Computational Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-007-9031-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10444-007-9031-x\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-007-9031-x","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10444-007-9031-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,5,18]],"date-time":"2022-05-18T01:11:33Z","timestamp":1652836293000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10444-007-9031-x"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,2,14]]},"references-count":18,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2007,7]]}},"alternative-id":["9031"],"URL":"https:\/\/doi.org\/10.1007\/s10444-007-9031-x","relation":{},"ISSN":["1019-7168","1572-9044"],"issn-type":[{"value":"1019-7168","type":"print"},{"value":"1572-9044","type":"electronic"}],"subject":[],"published":{"date-parts":[[2007,2,14]]},"assertion":[{"value":"14 February 2007","order":1,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}