{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,21]],"date-time":"2026-04-21T16:42:34Z","timestamp":1776789754531,"version":"3.51.2"},"reference-count":35,"publisher":"American Mathematical Society (AMS)","issue":"266","license":[{"start":{"date-parts":[[2009,11,6]],"date-time":"2009-11-06T00:00:00Z","timestamp":1257465600000},"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":"<p>\n                    We consider a large class of residuum based\n                    <italic>a posteriori<\/italic>\n                    eigenvalue\/eigenvector estimates and present an abstract framework for proving their asymptotic exactness. Equivalence of the estimator and the error is also established. To demonstrate the strength of our abstract approach we present a detailed study of hierarchical error estimators for Laplace eigenvalue problems in planar polygonal regions. To this end we develop new error analysis for the Galerkin approximation which avoids the use of the strengthened Cauchy-Schwarz inequality and the saturation assumption, and gives reasonable and explicitly computable upper bounds on the discretization error. A brief discussion is also given concerning the design of estimators which are in the same spirit, but are based on different\n                    <italic>a posteriori<\/italic>\n                    techniques\u2014notably, those of gradient recovery type.\n                  <\/p>","DOI":"10.1090\/s0025-5718-08-02181-9","type":"journal-article","created":{"date-parts":[[2009,12,1]],"date-time":"2009-12-01T13:09:23Z","timestamp":1259672963000},"page":"739-770","source":"Crossref","is-referenced-by-count":30,"title":["On estimators for eigenvalue\/eigenvector approximations"],"prefix":"10.1090","volume":"78","author":[{"given":"Luka","family":"Grubi\u0161i\u0107","sequence":"first","affiliation":[]},{"given":"Jeffrey","family":"Ovall","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2008,11,6]]},"reference":[{"issue":"1","key":"1","doi-asserted-by":"publisher","first-page":"195","DOI":"10.1090\/S0002-9939-03-07004-7","article-title":"An optimal Poincar\u00e9 inequality in \ud835\udc3f\u00b9 for convex domains","volume":"132","author":"Acosta, Gabriel","year":"2004","journal-title":"Proc. Amer. Math. Soc.","ISSN":"https:\/\/id.crossref.org\/issn\/0002-9939","issn-type":"print"},{"key":"2","series-title":"Pure and Applied Mathematics (New York)","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1002\/9781118032824","volume-title":"A posteriori error estimation in finite element analysis","author":"Ainsworth, Mark","year":"2000","ISBN":"https:\/\/id.crossref.org\/isbn\/047129411X"},{"key":"3","isbn-type":"print","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1017\/S0962492900002610","article-title":"Hierarchical bases and the finite element method","author":"Bank, Randolph E.","year":"1996","ISBN":"https:\/\/id.crossref.org\/isbn\/0521572347"},{"key":"4","unstructured":"R. E. Bank. PLTMG: A software package for solving elliptic partial differential equations, users\u2019 guide 9.0. Technical report, University of California, San Diego, 2004."},{"issue":"6","key":"5","doi-asserted-by":"publisher","first-page":"2294","DOI":"10.1137\/S003614290139874X","article-title":"Asymptotically exact a posteriori error estimators. I. Grids with superconvergence","volume":"41","author":"Bank, Randolph E.","year":"2003","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"6","key":"6","doi-asserted-by":"publisher","first-page":"2313","DOI":"10.1137\/S0036142901398751","article-title":"Asymptotically exact a posteriori error estimators. II. General unstructured grids","volume":"41","author":"Bank, Randolph E.","year":"2003","journal-title":"SIAM J. Numer. 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Preprint LAPACK Working Note 170."},{"issue":"8","key":"14","doi-asserted-by":"publisher","first-page":"1219","DOI":"10.1142\/S0218202503002878","article-title":"A posteriori error estimates for the finite element approximation of eigenvalue problems","volume":"13","author":"Dur\u00e1n, Ricardo G.","year":"2003","journal-title":"Math. Models Methods Appl. Sci.","ISSN":"https:\/\/id.crossref.org\/issn\/0218-2025","issn-type":"print"},{"key":"15","series-title":"Applied Mathematical Sciences","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-4355-5","volume-title":"Theory and practice of finite elements","volume":"159","author":"Ern, Alexandre","year":"2004","ISBN":"https:\/\/id.crossref.org\/isbn\/0387205748"},{"key":"16","unstructured":"L. Grubi\u0161i\u0107. Ritz value estimates and applications in Mathematical Physics. Ph.D. Thesis, Fernuniversit\u00e4t in Hagen, 2005. Available through dissertation.de Verlag im Internet."},{"issue":"4","key":"17","doi-asserted-by":"publisher","first-page":"1097","DOI":"10.1137\/050626533","article-title":"On eigenvalue and eigenvector estimates for nonnegative definite operators","volume":"28","author":"Grubi\u0161i\u0107, Luka","year":"2006","journal-title":"SIAM J. Matrix Anal. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0895-4798","issn-type":"print"},{"key":"18","doi-asserted-by":"crossref","unstructured":"L. Grubi\u0161i\u0107. A posteriori estimates for eigenvalue\/vector approximations. PAMM Proc. Appl. Math. Mech., 6(1):59\u201362, 2006.","DOI":"10.1002\/pamm.200610016"},{"key":"19","unstructured":"L. Grubi\u0161i\u0107. 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Wiley-Teubner Series Advances in Numerical Mathematics. John Wiley & Sons Ltd., Chichester, 1996."},{"issue":"4","key":"35","doi-asserted-by":"publisher","first-page":"695","DOI":"10.1051\/m2an:1999158","article-title":"Error estimates for some quasi-interpolation operators","volume":"33","author":"Verf\u00fcrth, R\u00fcdiger","year":"1999","journal-title":"M2AN Math. Model. Numer. 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