{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T04:29:33Z","timestamp":1776832173708,"version":"3.51.2"},"reference-count":20,"publisher":"American Mathematical Society (AMS)","issue":"236","license":[{"start":{"date-parts":[[2001,11,17]],"date-time":"2001-11-17T00:00:00Z","timestamp":1005955200000},"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>We present an a posteriori error estimator for a mixed finite element method for the Reissner-Mindlin plate model. The finite element method we deal with, was analyzed by Dur\u00e1n and Liberman in 1992 and can also be seen as a particular example of the general family analyzed by Brezzi, Fortin, and Stenberg in 1991. The estimator is based on the evaluation of the residual of the finite element solution. We show that the estimator yields local lower and global upper bounds of the error in the numerical solution in a natural norm for the problem, which includes the <inline-formula content-type=\"math\/mathml\">\n<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper H Superscript 1\">\n  <mml:semantics>\n    <mml:msup>\n      <mml:mi>H<\/mml:mi>\n      <mml:mn>1<\/mml:mn>\n    <\/mml:msup>\n    <mml:annotation encoding=\"application\/x-tex\">H^1<\/mml:annotation>\n  <\/mml:semantics>\n<\/mml:math>\n<\/inline-formula> norms of the terms corresponding to the deflection and the rotation and a dual norm for the shearing force. The estimates are valid uniformly with respect to the plate thickness.<\/p>","DOI":"10.1090\/s0025-5718-00-01289-8","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T22:13:53Z","timestamp":1027721633000},"page":"1383-1396","source":"Crossref","is-referenced-by-count":14,"title":["A posteriori error estimator for a mixed finite element method for Reissner-Mindlin plate"],"prefix":"10.1090","volume":"70","author":[{"given":"Elsa","family":"Liberman","sequence":"first","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2000,11,17]]},"reference":[{"issue":"1","key":"1","doi-asserted-by":"publisher","first-page":"7","DOI":"10.1051\/m2an\/1985190100071","article-title":"Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates","volume":"19","author":"Arnold, D. N.","year":"1985","journal-title":"RAIRO Mod\\'{e}l. Math. Anal. Num\\'{e}r.","ISSN":"https:\/\/id.crossref.org\/issn\/0764-583X","issn-type":"print"},{"issue":"6","key":"2","doi-asserted-by":"publisher","first-page":"1276","DOI":"10.1137\/0726074","article-title":"A uniformly accurate finite element method for the Reissner-Mindlin plate","volume":"26","author":"Arnold, Douglas N.","year":"1989","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"1","key":"3","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/0045-7825(87)90114-9","article-title":"A feedback finite element method with a posteriori error estimation. I. The finite element method and some basic properties of the a posteriori error estimator","volume":"61","author":"Babu\u0161ka, I.","year":"1987","journal-title":"Comput. Methods Appl. Mech. Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0045-7825","issn-type":"print"},{"key":"4","doi-asserted-by":"crossref","unstructured":"I. Babus\u0306ka and W. C. Rheinboldt, A posteriori error estimators in the finite element method, Int. J. Numer. Methods Eng.  12 (1978), 1587-1615.","DOI":"10.1002\/nme.1620121010"},{"issue":"170","key":"5","doi-asserted-by":"publisher","first-page":"283","DOI":"10.2307\/2007953","article-title":"Some a posteriori error estimators for elliptic partial differential equations","volume":"44","author":"Bank, R. E.","year":"1985","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"6","first-page":"491","article-title":"On the convergence of a four-node plate bending element based on Mindlin-Reissner plate theory and a mixed interpolation","author":"Bathe, K.-J.","year":"1985"},{"key":"7","doi-asserted-by":"crossref","unstructured":"K. J. Bathe and F. Brezzi, A simplified analysis of two plate bending elements\u2014the MITC4 and MITC9 elements, NUMETA 87 (G. N. Pande and J. Middleton, eds.), Numerical Techniques for Engineering Analysis and Design, vol. 1, Martinus Nijhoff, Dordrecht, 1987.","DOI":"10.1007\/978-94-009-3653-9_46"},{"issue":"8","key":"8","doi-asserted-by":"publisher","first-page":"1787","DOI":"10.1002\/nme.1620280806","article-title":"Mixed-interpolated elements for Reissner-Mindlin plates","volume":"28","author":"Brezzi, Franco","year":"1989","journal-title":"Internat. J. Numer. Methods Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-5981","issn-type":"print"},{"key":"9","doi-asserted-by":"crossref","unstructured":"K. J. Bathe and E. N. Dvorkin, A four-node plate bending element based on Mindlin Reissner plate theory and a mixed interpolation, J. Numer. Methods Engrg.  21 (1985),367-383.","DOI":"10.1002\/nme.1620210213"},{"key":"10","first-page":"571","article-title":"On a problem of Goursat","volume":"44","author":"B\u01cedescu, Radu","year":"1939","journal-title":"Gaz. 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