{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,12]],"date-time":"2026-06-12T17:59:56Z","timestamp":1781287196459,"version":"3.54.1"},"reference-count":14,"publisher":"American Mathematical Society (AMS)","issue":"361","license":[{"start":{"date-parts":[[2026,6,10]],"date-time":"2026-06-10T00:00:00Z","timestamp":1781049600000},"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 This work introduces a locally refined version of the Adini finite element for the planar biharmonic equation on rectangular partitions with at most one hanging node per edge. If global continuity of the discrete functions is enforced, for such method there is some freedom in assigning the normal derivative degree of freedom at the hanging nodes. It is proven that the convergence order\n \n \n \n \n h<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n h^2<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n known for regular solutions and regular partitions is lost for any such choice, and that assigning an average of the normal derivatives at the neighbouring regular vertices is the only choice that achieves a superlinear order, namely\n \n \n \n \n h<\/mml:mi>\n \n 3<\/mml:mn>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n h^{3\/2}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n on uniformly refined meshes. On adaptive meshes, the method behaves like a first-order scheme. Furthermore, the reliability and efficiency of an explicit residual-based error estimator are shown up to the best approximation of the Hessian by certain piecewise polynomial functions.\n <\/p>","DOI":"10.1090\/mcom\/4117","type":"journal-article","created":{"date-parts":[[2025,6,10]],"date-time":"2025-06-10T15:59:16Z","timestamp":1749571156000},"page":"2129-2153","source":"Crossref","is-referenced-by-count":0,"title":["The Adini finite element on locally refined meshes"],"prefix":"10.1090","volume":"95","author":[{"given":"D.","family":"Gallistl","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2025,6,10]]},"reference":[{"key":"1","unstructured":"A. Adini and R. W. Clough. Analysis of plate bending by the finite element method. NSF report, G-7337, 1961."},{"issue":"4","key":"2","doi-asserted-by":"publisher","first-page":"556","DOI":"10.1002\/mma.1670020416","article-title":"On the boundary value problem of the biharmonic operator on domains with angular corners","volume":"2","author":"Blum, H.","year":"1980","journal-title":"Math. Methods Appl. Sci.","ISSN":"https:\/\/id.crossref.org\/issn\/0170-4214","issn-type":"print"},{"key":"3","series-title":"Texts in Applied Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-0-387-75934-0","volume-title":"The mathematical theory of finite element methods","volume":"15","author":"Brenner, Susanne C.","year":"2008","ISBN":"https:\/\/id.crossref.org\/isbn\/9780387759333","edition":"3"},{"issue":"2","key":"4","doi-asserted-by":"publisher","first-page":"309","DOI":"10.1007\/s00211-012-0513-5","article-title":"A posteriori error estimates for nonconforming finite element methods for fourth-order problems on rectangles","volume":"124","author":"Carstensen, Carsten","year":"2013","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"2-3","key":"5","first-page":"215","article-title":"Hanging nodes in the unifying theory of a posteriori finite element error control","volume":"27","author":"Carstensen, C.","year":"2009","journal-title":"J. Comput. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0254-9409","issn-type":"print"},{"key":"6","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":"4","key":"7","doi-asserted-by":"publisher","first-page":"1779","DOI":"10.1093\/imanum\/dru054","article-title":"Morley finite element method for the eigenvalues of the biharmonic operator","volume":"35","author":"Gallistl, Dietmar","year":"2015","journal-title":"IMA J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0272-4979","issn-type":"print"},{"key":"8","doi-asserted-by":"publisher","first-page":"355","DOI":"10.5802\/smai-jcm.115","article-title":"A posteriori error estimates for nonconforming discretizations of singularly perturbed biharmonic operators","volume":"10","author":"Gallistl, Dietmar","year":"2024","journal-title":"SMAI J. Comput. Math."},{"key":"9","series-title":"Recherches en Math\\'{e}matiques Appliqu\\'{e}es [Research in Applied Mathematics]","isbn-type":"print","volume-title":"Singularities in boundary value problems","volume":"22","author":"Grisvard, P.","year":"1992","ISBN":"https:\/\/id.crossref.org\/isbn\/2225827702"},{"issue":"272","key":"10","doi-asserted-by":"publisher","first-page":"2169","DOI":"10.1090\/S0025-5718-10-02360-4","article-title":"A new error analysis for discontinuous finite element methods for linear elliptic problems","volume":"79","author":"Gudi, Thirupathi","year":"2010","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"3","key":"11","doi-asserted-by":"publisher","first-page":"1366","DOI":"10.1007\/s10915-016-0237-2","article-title":"Capacity of the Adini element for biharmonic equations","volume":"69","author":"Hu, Jun","year":"2016","journal-title":"J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0885-7474","issn-type":"print"},{"key":"12","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":"13","series-title":"Lecture Notes on Numerical Methods in Engineering and Sciences","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4020-8743-1","volume-title":"Structural analysis with the finite element method---linear statics. Vol. 2","author":"O\u00f1ate, Eugenio","year":"2013","ISBN":"https:\/\/id.crossref.org\/isbn\/9781402087424"},{"key":"14","doi-asserted-by":"crossref","unstructured":"R. Verf\u00fcrth. A Posteriori Error Estimation Techniques for Finite Element Methods. Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford, 2013.","DOI":"10.1093\/acprof:oso\/9780199679423.001.0001"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.ams.org\/mcom\/2026-95-361\/S0025-5718-2025-04117-6\/S0025-5718-2025-04117-6.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,6,12]],"date-time":"2026-06-12T17:53:02Z","timestamp":1781286782000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2026-95-361\/S0025-5718-2025-04117-6\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,6,10]]},"references-count":14,"journal-issue":{"issue":"361","published-print":{"date-parts":[[2026,9]]}},"alternative-id":["S0025-5718-2025-04117-6"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/4117","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":[[2025,6,10]]}}}