{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T23:42:32Z","timestamp":1776728552836,"version":"3.51.2"},"reference-count":24,"publisher":"American Mathematical Society (AMS)","issue":"241","license":[{"start":{"date-parts":[[2002,12,5]],"date-time":"2002-12-05T00:00:00Z","timestamp":1039046400000},"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>Applying high order finite elements to unilateral contact variational inequalities may provide more accurate computed solutions, compared with linear finite elements. Up to now, there was no significant progress in the mathematical study of their performances. The main question is involved with the modeling of the nonpenetration Signorini condition on the discrete solution along the contact region. In this work we describe two nonconforming quadratic finite element approximations of the Poisson-Signorini problem, responding to the crucial practical concern of easy implementation, and we present the numerical analysis of their efficiency. By means of Falk\u2019s Lemma we prove optimal and quasi-optimal convergence rates according to the regularity of the exact solution.<\/p>","DOI":"10.1090\/s0025-5718-01-01413-2","type":"journal-article","created":{"date-parts":[[2002,11,1]],"date-time":"2002-11-01T09:59:50Z","timestamp":1036144790000},"page":"83-104","source":"Crossref","is-referenced-by-count":35,"title":["Quadratic finite element approximation of the Signorini problem"],"prefix":"10.1090","volume":"72","author":[{"given":"Z.","family":"Belhachmi","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"F.","family":"Belgacem","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2001,12,5]]},"reference":[{"key":"1","series-title":"Pure and Applied Mathematics, Vol. 65","volume-title":"Sobolev spaces","author":"Adams, Robert A.","year":"1975"},{"issue":"3","key":"2","doi-asserted-by":"publisher","first-page":"813","DOI":"10.1137\/0731044","article-title":"Parallel numerical solution of variational inequalities","volume":"31","author":"Benassi, M.","year":"1994","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"3","doi-asserted-by":"crossref","unstructured":"F. Ben Belgacem. Numerical Simulation of some Variational Inequalities Arisen from Unilateral Contact Problems by the Finite Element Method, SIAM J. Numer. Anal., 37: 1198-1216, 2000.","DOI":"10.1137\/S0036142998347966"},{"key":"4","unstructured":"F. Ben Belgacem. Mixed Finite Element Methods for Signorini\u2019s Problem, submitted."},{"key":"5","unstructured":"F. Ben Belgacem ans S. C. Brenner. Some Nonstandard Finite Element Estimates with Applications to 3D Poisson and Signorini Problems, Electronic Transactions in Numerical Analysis, 12:134-148, 2001."},{"issue":"2","key":"6","doi-asserted-by":"publisher","first-page":"287","DOI":"10.1142\/S0218202599000154","article-title":"Extension of the mortar finite element method to a variational inequality modeling unilateral contact","volume":"9","author":"Ben Belgacem, Faker","year":"1999","journal-title":"Math. Models Methods Appl. Sci.","ISSN":"https:\/\/id.crossref.org\/issn\/0218-2025","issn-type":"print"},{"key":"7","isbn-type":"print","first-page":"13","article-title":"A new nonconforming approach to domain decomposition: the mortar element method","author":"Bernardi, C.","year":"1994","ISBN":"https:\/\/id.crossref.org\/isbn\/0582238005"},{"issue":"4","key":"8","doi-asserted-by":"publisher","first-page":"431","DOI":"10.1007\/BF01404345","article-title":"Error estimates for the finite element solution of variational inequalities","volume":"28","author":"Brezzi, Franco","year":"1977","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"key":"9","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"},{"key":"10","unstructured":"P. Coorevits, P. Hild, K. Lhalouani and T. Sassi. Mixed Finite Element Method for Unilateral Problems: Convergence Analysis and Numerical Studies, Math. of Comp., posted on May 21, 2001, PII: S0025-5718(01)01318-7 (to appear in print)."},{"key":"11","series-title":"Travaux et Recherches Math\\'{e}matiques, No. 21","volume-title":"Les in\\'{e}quations en m\\'{e}canique et en physique","author":"Duvaut, G.","year":"1972"},{"key":"12","doi-asserted-by":"crossref","first-page":"963","DOI":"10.1090\/S0025-5718-1974-0391502-8","article-title":"Error estimates for the approximation of a class of variational inequalities","volume":"28","author":"Falk, Richard S.","year":"1974","journal-title":"Math. Comput."},{"key":"13","doi-asserted-by":"publisher","first-page":"153","DOI":"10.1002\/sapm1939181153","article-title":"A function related to the series for \ud835\udc52^{\ud835\udc52^{\ud835\udc65}}","volume":"18","author":"Epstein, Leo F.","year":"1939","journal-title":"J. Math. Phys. Mass. Inst. Tech.","ISSN":"https:\/\/id.crossref.org\/issn\/0097-1421","issn-type":"print"},{"key":"14","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":"15","doi-asserted-by":"publisher","first-page":"99","DOI":"10.1093\/qmath\/os-10.1.99","article-title":"An algebraic property of Laplace\u2019s differential equation","volume":"10","author":"Taussky, Olga","year":"1939","journal-title":"Quart. J. Math. Oxford Ser.","ISSN":"https:\/\/id.crossref.org\/issn\/0033-5606","issn-type":"print"},{"key":"16","series-title":"Handbook of Numerical Analysis, IV","isbn-type":"print","volume-title":"Handbook of numerical analysis. Vol. IV","year":"1996","ISBN":"https:\/\/id.crossref.org\/isbn\/0444817948"},{"key":"17","unstructured":"P. Hild. Probl\u00e8mes de contact unilat\u00e9ral et maillages incompatibles, Th\u00e8se de l\u2019Universit\u00e9 Paul Sabatier, Toulouse 3, 1998."},{"key":"18","series-title":"SIAM Studies in Applied Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1137\/1.9781611970845","volume-title":"Contact problems in elasticity: a study of variational inequalities and finite element methods","volume":"8","author":"Kikuchi, N.","year":"1988","ISBN":"https:\/\/id.crossref.org\/isbn\/0898712025"},{"key":"19","series-title":"Pure and Applied Mathematics","isbn-type":"print","volume-title":"An introduction to variational inequalities and their applications","volume":"88","author":"Kinderlehrer, David","year":"1980","ISBN":"https:\/\/id.crossref.org\/isbn\/0124073506"},{"issue":"1","key":"20","first-page":"23","article-title":"Nonconforming mixed variational formulation and domain decomposition for unilateral problems","volume":"7","author":"Lhalouani, K.","year":"1999","journal-title":"East-West J. Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0928-0200","issn-type":"print"},{"key":"21","series-title":"Travaux et Recherches Math\\'{e}matiques, No. 17","volume-title":"Probl\\`emes aux limites non homog\\`enes et applications. Vol. 1","author":"Lions, J.-L.","year":"1968"},{"issue":"5-6","key":"22","doi-asserted-by":"publisher","first-page":"805","DOI":"10.1080\/03605309208820864","article-title":"R\u00e9gularit\u00e9 des solutions d\u2019un probl\u00e8me m\u00eal\u00e9 Dirichlet-Signorini dans un domaine polygonal plan","volume":"17","author":"Moussaoui, Mohand","year":"1992","journal-title":"Comm. Partial Differential Equations","ISSN":"https:\/\/id.crossref.org\/issn\/0360-5302","issn-type":"print"},{"key":"23","series-title":"Prentice-Hall Series in Automatic Computation","volume-title":"An analysis of the finite element method","author":"Strang, Gilbert","year":"1973"},{"key":"24","doi-asserted-by":"crossref","unstructured":"Z.-H. Zhong. Finite Element Procedures for Contact-Impact Problems, Oxford University Press, 1993.","DOI":"10.1093\/oso\/9780198563839.001.0001"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2003-72-241\/S0025-5718-01-01413-2\/S0025-5718-01-01413-2.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2003-72-241\/S0025-5718-01-01413-2\/S0025-5718-01-01413-2.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T23:08:14Z","timestamp":1776726494000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2003-72-241\/S0025-5718-01-01413-2\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2001,12,5]]},"references-count":24,"journal-issue":{"issue":"241","published-print":{"date-parts":[[2003,1]]}},"alternative-id":["S0025-5718-01-01413-2"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-01-01413-2","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":[[2001,12,5]]}}}