{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,3]],"date-time":"2026-05-03T07:14:11Z","timestamp":1777792451613,"version":"3.51.4"},"reference-count":9,"publisher":"American Mathematical Society (AMS)","issue":"234","license":[{"start":{"date-parts":[[2001,2,23]],"date-time":"2001-02-23T00:00:00Z","timestamp":982886400000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    Finite element methods for some elliptic fourth order singular perturbation problems are discussed. We show that if such problems are discretized by the nonconforming Morley method, in a regime close to second order elliptic equations, then the error deteriorates. In fact, a counterexample is given to show that the Morley method diverges for the reduced second order equation. As an alternative to the Morley element we propose to use a nonconforming\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper H squared\">\n                        <mml:semantics>\n                          <mml:msup>\n                            <mml:mi>H<\/mml:mi>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:msup>\n                          <mml:annotation encoding=\"application\/x-tex\">H^2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -element which is\n                    <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>\n                    -conforming. We show that the new finite element method converges in the energy norm uniformly in the perturbation parameter.\n                  <\/p>","DOI":"10.1090\/s0025-5718-00-01230-8","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:17:46Z","timestamp":1027707466000},"page":"489-505","source":"Crossref","is-referenced-by-count":92,"title":["A robust nonconforming \ud835\udc3b\u00b2-element"],"prefix":"10.1090","volume":"70","author":[{"given":"Trygve","family":"Nilssen","sequence":"first","affiliation":[]},{"given":"Xue-Cheng","family":"Tai","sequence":"additional","affiliation":[]},{"given":"Ragnar","family":"Winther","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2000,2,23]]},"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"},{"key":"2","series-title":"Texts in Applied Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-4338-8","volume-title":"The mathematical theory of finite element methods","volume":"15","author":"Brenner, Susanne C.","year":"1994","ISBN":"https:\/\/id.crossref.org\/isbn\/0387941932"},{"key":"3","series-title":"Studies in Mathematics and its Applications, Vol. 4","isbn-type":"print","volume-title":"The finite element method for elliptic problems","author":"Ciarlet, Philippe G.","year":"1978","ISBN":"https:\/\/id.crossref.org\/isbn\/0444850287"},{"issue":"6","key":"4","doi-asserted-by":"publisher","first-page":"1486","DOI":"10.1137\/0727086","article-title":"Equivalence of finite element methods for problems in elasticity","volume":"27","author":"Falk, Richard S.","year":"1990","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"5","series-title":"Monographs and Studies in Mathematics","isbn-type":"print","volume-title":"Elliptic problems in nonsmooth domains","volume":"24","author":"Grisvard, P.","year":"1985","ISBN":"https:\/\/id.crossref.org\/isbn\/0273086472"},{"key":"6","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1051\/m2an\/197509R100091","article-title":"Some nonconforming finite elements for the plate bending problem","volume":"9","author":"Lascaux, P.","year":"1975","journal-title":"Rev. Fran\\c{c}aise Automat. Informat. Recherche Op\\'{e}rationnelle S\\'{e}r. Rouge Anal. Num\\'{e}r.","ISSN":"https:\/\/id.crossref.org\/issn\/0397-9342","issn-type":"print"},{"key":"7","unstructured":"M. Wang, The necessity and sufficiency of the patch test for the convergence of nonconforming finite elements, Preprint, Dept. of Math., Penn. State University (1999)."},{"issue":"161","key":"8","doi-asserted-by":"publisher","first-page":"47","DOI":"10.2307\/2007363","article-title":"On the finite element method for singularly perturbed reaction-diffusion problems in two and one dimensions","volume":"40","author":"Schatz, A. H.","year":"1983","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"2","key":"9","first-page":"113","article-title":"Error estimates for the Morley element","volume":"12","author":"Shi, Zhong Ci","year":"1990","journal-title":"Math. Numer. Sinica","ISSN":"https:\/\/id.crossref.org\/issn\/0254-7791","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2001-70-234\/S0025-5718-00-01230-8\/S0025-5718-00-01230-8.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2001-70-234\/S0025-5718-00-01230-8\/S0025-5718-00-01230-8.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T22:34:47Z","timestamp":1776724487000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2001-70-234\/S0025-5718-00-01230-8\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2000,2,23]]},"references-count":9,"journal-issue":{"issue":"234","published-print":{"date-parts":[[2001,4]]}},"alternative-id":["S0025-5718-00-01230-8"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-00-01230-8","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":[[2000,2,23]]}}}