{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T16:19:10Z","timestamp":1776788350157,"version":"3.51.2"},"reference-count":24,"publisher":"American Mathematical Society (AMS)","issue":"264","license":[{"start":{"date-parts":[[2009,5,29]],"date-time":"2009-05-29T00:00:00Z","timestamp":1243555200000},"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":"

\n The paper explores new expansions of the eigenvalues for\n \n \n \n \n \n \u2212\n \n <\/mml:mo>\n \n \u0394\n \n <\/mml:mi>\n u<\/mml:mi>\n =<\/mml:mo>\n \n \u03bb\n \n <\/mml:mi>\n \n \u03c1\n \n <\/mml:mi>\n u<\/mml:mi>\n <\/mml:mrow>\n -\\Delta u=\\lambda \\rho u<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n in\n \n \n \n S<\/mml:mi>\n S<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n with Dirichlet boundary conditions by the bilinear element (denoted\n \n \n \n \n Q<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n Q_1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n ) and three nonconforming elements, the rotated bilinear element (denoted\n \n \n \n \n Q<\/mml:mi>\n 1<\/mml:mn>\n \n r<\/mml:mi>\n o<\/mml:mi>\n t<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msubsup>\n Q_1^{rot}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n ), the extension of\n \n \n \n \n Q<\/mml:mi>\n 1<\/mml:mn>\n \n r<\/mml:mi>\n o<\/mml:mi>\n t<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msubsup>\n Q_1^{rot}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n (denoted\n \n \n \n \n E<\/mml:mi>\n \n Q<\/mml:mi>\n 1<\/mml:mn>\n \n r<\/mml:mi>\n o<\/mml:mi>\n t<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msubsup>\n <\/mml:mrow>\n EQ_1^{rot}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n ) and Wilson\u2019s elements. The expansions indicate that\n \n \n \n \n Q<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n Q_1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and\n \n \n \n \n Q<\/mml:mi>\n 1<\/mml:mn>\n \n r<\/mml:mi>\n o<\/mml:mi>\n t<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msubsup>\n Q_1^{rot}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n provide upper bounds of the eigenvalues, and that\n \n \n \n \n E<\/mml:mi>\n \n Q<\/mml:mi>\n 1<\/mml:mn>\n \n r<\/mml:mi>\n o<\/mml:mi>\n t<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msubsup>\n <\/mml:mrow>\n EQ_1^{rot}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n and Wilson\u2019s elements provide lower bounds of the eigenvalues. By extrapolation, the\n \n \n \n \n O<\/mml:mi>\n (<\/mml:mo>\n \n h<\/mml:mi>\n 4<\/mml:mn>\n <\/mml:msup>\n )<\/mml:mo>\n <\/mml:mrow>\n O(h^4)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n convergence rate can be obtained, where\n \n \n \n h<\/mml:mi>\n h<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is the maximal boundary length of uniform rectangles. Numerical experiments are carried out to verify the theoretical analysis made.\n <\/p>","DOI":"10.1090\/s0025-5718-08-02098-x","type":"journal-article","created":{"date-parts":[[2008,7,25]],"date-time":"2008-07-25T12:55:42Z","timestamp":1216990542000},"page":"2061-2084","source":"Crossref","is-referenced-by-count":31,"title":["New expansions of numerical eigenvalues for -\u0394\ud835\udc62=\ud835\udf06\ud835\udf0c\ud835\udc62 by nonconforming elements"],"prefix":"10.1090","volume":"77","author":[{"given":"Qun","family":"Lin","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hung-Tsai","family":"Huang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zi-Cai","family":"Li","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2008,5,29]]},"reference":[{"issue":"6","key":"1","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":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"186","key":"2","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":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"3","isbn-type":"print","first-page":"641","article-title":"Eigenvalue problems","author":"Babu\u0161ka, I.","year":"1991","ISBN":"https:\/\/id.crossref.org\/isbn\/0444703659"},{"issue":"1","key":"4","doi-asserted-by":"publisher","first-page":"11","DOI":"10.1007\/BF01389427","article-title":"Asymptotic error expansion and Richardson extrapolation for linear finite elements","volume":"49","author":"Blum, H.","year":"1986","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"key":"5","doi-asserted-by":"publisher","first-page":"112","DOI":"10.1137\/0707006","article-title":"Estimation of linear functionals on Sobolev spaces with application to Fourier transforms and spline interpolation","volume":"7","author":"Bramble, J. H.","year":"1970","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"6","doi-asserted-by":"publisher","first-page":"939","DOI":"10.1137\/0710080","article-title":"Convergence of approximation methods to compute eigenelements of linear operations","volume":"10","author":"Chatelin, Fran\u00e7oise","year":"1973","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"2","key":"7","doi-asserted-by":"publisher","first-page":"125","DOI":"10.1007\/s002110050084","article-title":"Superconvergence analysis and error expansion for the Wilson nonconforming finite element","volume":"69","author":"Chen, Hong Sen","year":"1994","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"key":"8","doi-asserted-by":"crossref","first-page":"467","DOI":"10.2140\/pjm.1954.4.467","article-title":"Asymptotic lower bounds for the frequencies of certain polygonal membranes","volume":"4","author":"Forsythe, George E.","year":"1954","journal-title":"Pacific J. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0030-8730","issn-type":"print"},{"issue":"1","key":"9","first-page":"25","article-title":"Nonconforming quadrilateral rotated \ud835\udc44\u2081 element for Reissner-Mindlin plate","volume":"21","author":"Hu, Jun","year":"2003","journal-title":"J. Comput. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0254-9409","issn-type":"print"},{"issue":"2","key":"10","doi-asserted-by":"publisher","first-page":"159","DOI":"10.1007\/BF01390335","article-title":"Approximation in variationally posed eigenvalue problems","volume":"29","author":"Kolata, William G.","year":"1977","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"key":"11","series-title":"Mathematics and its Applications","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4613-3338-8","volume-title":"Combined methods for elliptic equations with singularities, interfaces and infinities","volume":"444","author":"Li, Zi Cai","year":"1998","ISBN":"https:\/\/id.crossref.org\/isbn\/0792350847"},{"issue":"6","key":"12","doi-asserted-by":"publisher","first-page":"631","DOI":"10.1007\/BF01385645","article-title":"Fourth order eigenvalue approximation by extrapolation on domains with reentrant corners","volume":"58","author":"Qun, Lin","year":"1991","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"key":"13","unstructured":"Q. Lin and J. Lin, Finite Element Methods; Accuracy and Improvement, Science Press, Beijing, 2006."},{"key":"14","unstructured":"Q. Lin and Q. Zhu, Processing and Post processing for the Finite Element Method (in Chinese), Shanghai Scientific & Technical Press., 1994."},{"key":"15","unstructured":"T. L\u00fc, C.B. Liem and T.M.Shih, The Splitting Extrapolation and Combination Techniques - New Techniques of Parallel Solutions for Multi-dimensional Problems (in Chinese), Scientific Publishers, Beijing, 1998."},{"issue":"2","key":"16","first-page":"113","article-title":"High accuracy analysis of the Wilson element","volume":"17","author":"Luo, Ping","year":"1999","journal-title":"J. Comput. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0254-9409","issn-type":"print"},{"issue":"154","key":"17","doi-asserted-by":"publisher","first-page":"427","DOI":"10.2307\/2007651","article-title":"Eigenvalue approximation by mixed and hybrid methods","volume":"36","author":"Mercier, B.","year":"1981","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"18","doi-asserted-by":"publisher","first-page":"155","DOI":"10.1007\/BF01402526","article-title":"Higher order convergence results for the Rayleigh-Ritz method applied to eigenvalue problems. II. Improved error bounds for eigenfunctions","volume":"19","author":"Pierce, J. G.","year":"1972","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"1","key":"19","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":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"key":"20","series-title":"Prentice-Hall Series in Automatic Computation","volume-title":"An analysis of the finite element method","author":"Strang, Gilbert","year":"1973"},{"key":"21","series-title":"Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 15","doi-asserted-by":"crossref","DOI":"10.1137\/1.9781611970531","volume-title":"Variational methods for eigenvalue approximation","author":"Weinberger, Hans F.","year":"1974"},{"issue":"2","key":"22","first-page":"139","article-title":"Convergence and superconvergence of Hermite bicubic element for eigenvalue problem of the biharmonic equation","volume":"19","author":"Wu, Dong-sheng","year":"2001","journal-title":"J. Comput. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0254-9409","issn-type":"print"},{"issue":"3","key":"23","first-page":"286","article-title":"Computable error bounds of finite element approximations for eigenvalue problems","volume":"16","author":"Yang, Yi Du","year":"1994","journal-title":"Math. Numer. Sinica","ISSN":"https:\/\/id.crossref.org\/issn\/0254-7791","issn-type":"print"},{"issue":"4","key":"24","first-page":"413","article-title":"A posteriori error estimates in Adini finite element for eigenvalue problems","volume":"18","author":"Yang, Yi-du","year":"2000","journal-title":"J. Comput. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0254-9409","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2008-77-264\/S0025-5718-08-02098-X\/S0025-5718-08-02098-X.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2008-77-264\/S0025-5718-08-02098-X\/S0025-5718-08-02098-X.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T15:29:19Z","timestamp":1776785359000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2008-77-264\/S0025-5718-08-02098-X\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,5,29]]},"references-count":24,"journal-issue":{"issue":"264","published-print":{"date-parts":[[2008,10]]}},"alternative-id":["S0025-5718-08-02098-X"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-08-02098-x","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2008,5,29]]}}}