{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T08:38:21Z","timestamp":1776847101425,"version":"3.51.2"},"reference-count":25,"publisher":"American Mathematical Society (AMS)","issue":"243","license":[{"start":{"date-parts":[[2004,2,3]],"date-time":"2004-02-03T00:00:00Z","timestamp":1075766400000},"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 In this work, the bilinear finite element method on a Shishkin mesh for convection-diffusion problems is analyzed in the two-dimensional setting. A superconvergence rate\n \n \n \n \n O<\/mml:mi>\n (<\/mml:mo>\n \n N<\/mml:mi>\n \n \n \u2212\n \n <\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n \n ln<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n \n \u2061\n \n <\/mml:mo>\n N<\/mml:mi>\n +<\/mml:mo>\n \n \u03f5\n \n <\/mml:mi>\n \n N<\/mml:mi>\n \n \n \u2212\n \n <\/mml:mo>\n 1.5<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n ln<\/mml:mi>\n \n \u2061\n \n <\/mml:mo>\n N<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n O(N^{-2}\\ln ^2 N + \\epsilon N^{-1.5}\\ln N)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n in a discrete\n \n \n \n \n \u03f5\n \n <\/mml:mi>\n \\epsilon<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -weighted energy norm is established under certain regularity assumptions. This convergence rate is uniformly valid with respect to the singular perturbation parameter\n \n \n \n \n \u03f5\n \n <\/mml:mi>\n \\epsilon<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . Numerical tests indicate that the rate\n \n \n \n \n O<\/mml:mi>\n (<\/mml:mo>\n \n N<\/mml:mi>\n \n \n \u2212\n \n <\/mml:mo>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n \n ln<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n \n \u2061\n \n <\/mml:mo>\n N<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n O(N^{-2}\\ln ^2 N)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is sharp for the boundary layer terms. As a by-product, an\n \n \n \n \n \u03f5\n \n <\/mml:mi>\n \\epsilon<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -uniform convergence of the same order is obtained for the\n \n \n \n \n L<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n L^2<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -norm. Furthermore, under the same regularity assumption, an\n \n \n \n \n \u03f5\n \n <\/mml:mi>\n \\epsilon<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -uniform convergence of order\n \n \n \n \n \n N<\/mml:mi>\n \n \n \u2212\n \n <\/mml:mo>\n 3<\/mml:mn>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n \n ln<\/mml:mi>\n \n 5<\/mml:mn>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n \n \u2061\n \n <\/mml:mo>\n N<\/mml:mi>\n +<\/mml:mo>\n \n \u03f5\n \n <\/mml:mi>\n \n N<\/mml:mi>\n \n \n \u2212\n \n <\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n \n ln<\/mml:mi>\n \n 1<\/mml:mn>\n \n \/<\/mml:mo>\n <\/mml:mrow>\n 2<\/mml:mn>\n <\/mml:mrow>\n <\/mml:msup>\n \n \u2061\n \n <\/mml:mo>\n N<\/mml:mi>\n <\/mml:mrow>\n N^{-3\/2}\\ln ^{5\/2} N + \\epsilon N^{-1}\\ln ^{1\/2} N<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n in the\n \n \n \n \n L<\/mml:mi>\n \n \u221e\n \n <\/mml:mi>\n <\/mml:msup>\n L^\\infty<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n norm is proved for some mesh points in the boundary layer region.\n <\/p>","DOI":"10.1090\/s0025-5718-03-01486-8","type":"journal-article","created":{"date-parts":[[2003,4,18]],"date-time":"2003-04-18T13:09:02Z","timestamp":1050671342000},"page":"1147-1177","source":"Crossref","is-referenced-by-count":94,"title":["Finite element superconvergence on Shishkin mesh for 2-D convection-diffusion problems"],"prefix":"10.1090","volume":"72","author":[{"given":"Zhimin","family":"Zhang","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2003,2,3]]},"reference":[{"key":"1","unstructured":"C. Chen and Y. Huang, High Accuracy Theory of Finite Element Methods (in Chinese), Hunan Science Press, P.R. China, 1995."},{"issue":"2","key":"2","doi-asserted-by":"publisher","first-page":"301","DOI":"10.1142\/S0218202501000878","article-title":"The \u210e\ud835\udc5d-version of the streamline diffusion finite element method in two space dimensions","volume":"11","author":"Gerdes, K.","year":"2001","journal-title":"Math. Models Methods Appl. Sci.","ISSN":"https:\/\/id.crossref.org\/issn\/0218-2025","issn-type":"print"},{"issue":"2","key":"3","doi-asserted-by":"publisher","first-page":"394","DOI":"10.1137\/0521022","article-title":"Differentiability properties of solutions of the equation -\ud835\udf00\u00b2\u0394\ud835\udc62+\ud835\udc5f\ud835\udc62=\ud835\udc53(\ud835\udc65,\ud835\udc66) in a square","volume":"21","author":"Han, H.","year":"1990","journal-title":"SIAM J. Math. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1410","issn-type":"print"},{"issue":"179","key":"4","doi-asserted-by":"publisher","first-page":"25","DOI":"10.2307\/2008248","article-title":"Crosswind smear and pointwise errors in streamline diffusion finite element methods","volume":"49","author":"Johnson, C.","year":"1987","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"5","isbn-type":"print","first-page":"121","article-title":"Boundary layers and corner singularities for a self-adjoint problem","author":"Kellogg, Bruce","year":"1995","ISBN":"https:\/\/id.crossref.org\/isbn\/082479320X"},{"key":"6","series-title":"Lecture Notes in Pure and Applied Mathematics","isbn-type":"print","volume-title":"Finite element methods","volume":"196","year":"1998","ISBN":"https:\/\/id.crossref.org\/isbn\/0824701488"},{"issue":"4","key":"7","first-page":"249","article-title":"Streamline diffusion least-squares mixed finite element methods for convection-diffusion problems","volume":"5","author":"Lazarov, R. D.","year":"1997","journal-title":"East-West J. Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0928-0200","issn-type":"print"},{"issue":"3","key":"8","doi-asserted-by":"publisher","first-page":"770","DOI":"10.1137\/S0036142999351212","article-title":"Uniform convergence and superconvergence of mixed finite element methods on anisotropically refined grids","volume":"38","author":"Li, Jichun","year":"2000","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"9","unstructured":"Q. Lin and N. Yan, Construction and Analysis of High Efficient Finite Elements (in Chinese), Hebei University Press, P.R. China, 1996."},{"issue":"2","key":"10","doi-asserted-by":"publisher","first-page":"604","DOI":"10.1006\/jmaa.2001.7550","article-title":"Asymptotic analysis and Shishkin-type decomposition for an elliptic convection-diffusion problem","volume":"261","author":"Lin\u00df, Torsten","year":"2001","journal-title":"J. Math. Anal. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0022-247X","issn-type":"print"},{"issue":"4","key":"11","doi-asserted-by":"publisher","first-page":"1520","DOI":"10.1137\/S0036142997317602","article-title":"\ud835\udc3b\ud835\udc43 FEM for reaction-diffusion equations. I. Robust exponential convergence","volume":"35","author":"Melenk, J. M.","year":"1998","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"2","key":"12","doi-asserted-by":"publisher","first-page":"379","DOI":"10.1137\/S0036141097317542","article-title":"Analytic regularity for a singularly perturbed problem","volume":"30","author":"Melenk, J. M.","year":"1999","journal-title":"SIAM J. Math. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1410","issn-type":"print"},{"key":"13","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1142\/2933","volume-title":"Fitted numerical methods for singular perturbation problems","author":"Miller, J. J. H.","year":"1996","ISBN":"https:\/\/id.crossref.org\/isbn\/9810224621"},{"key":"14","series-title":"Applied Mathematics and Mathematical Computation","isbn-type":"print","volume-title":"Numerical solution of convection-diffusion problems","volume":"12","author":"Morton, K. W.","year":"1996","ISBN":"https:\/\/id.crossref.org\/isbn\/0412564408"},{"issue":"5","key":"15","doi-asserted-by":"publisher","first-page":"291","DOI":"10.1002\/(SICI)1521-4001(199805)78:5<291::AID-ZAMM291>3.0.CO;2-R","article-title":"Layer-adapted grids for singular perturbation problems","volume":"78","author":"Roos, H.-G.","year":"1998","journal-title":"ZAMM Z. Angew. Math. Mech.","ISSN":"https:\/\/id.crossref.org\/issn\/0044-2267","issn-type":"print"},{"key":"16","series-title":"Springer Series in Computational Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-662-03206-0","volume-title":"Numerical methods for singularly perturbed differential equations","volume":"24","author":"Roos, H.-G.","year":"1996","ISBN":"https:\/\/id.crossref.org\/isbn\/3540607188"},{"issue":"2","key":"17","doi-asserted-by":"publisher","first-page":"505","DOI":"10.1137\/0733027","article-title":"Superconvergence in finite element methods and meshes that are locally symmetric with respect to a point","volume":"33","author":"Schatz, A. H.","year":"1996","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"161","key":"18","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":"216","key":"19","doi-asserted-by":"publisher","first-page":"1403","DOI":"10.1090\/S0025-5718-96-00781-8","article-title":"The \ud835\udc5d and \u210e\ud835\udc5d versions of the finite element method for problems with boundary layers","volume":"65","author":"Schwab, Christoph","year":"1996","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"20","unstructured":"G.I. Shishkin, Discrete approximation of singularly perturbed elliptic and parabolic problems (in Russian), Russian Academy of Sciences, Ural Section, Ekaterinburg, 1992."},{"issue":"1","key":"21","doi-asserted-by":"publisher","first-page":"36","DOI":"10.1006\/jmaa.1997.5581","article-title":"A uniformly convergent Galerkin method on a Shishkin mesh for a convection-diffusion problem","volume":"214","author":"Stynes, Martin","year":"1997","journal-title":"J. Math. Anal. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0022-247X","issn-type":"print"},{"key":"22","series-title":"Handbook of Numerical Analysis, II","isbn-type":"print","volume-title":"Handbook of numerical analysis. Vol. II","year":"1991","ISBN":"https:\/\/id.crossref.org\/isbn\/0444703659"},{"key":"23","series-title":"Lecture Notes in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/BFb0096835","volume-title":"Superconvergence in Galerkin finite element methods","volume":"1605","author":"Wahlbin, Lars B.","year":"1995","ISBN":"https:\/\/id.crossref.org\/isbn\/3540600116"},{"key":"24","doi-asserted-by":"crossref","unstructured":"Z. Zhang, Superconvergent approximation of singularly perturbed problems, Numer. Meth. PDEs 18 (2002), 374\u2013395.","DOI":"10.1002\/num.10001"},{"key":"25","doi-asserted-by":"publisher","first-page":"560","DOI":"10.2307\/1968940","article-title":"On the existence of a measure invariant under a transformation","volume":"40","author":"Oxtoby, J. C.","year":"1939","journal-title":"Ann. of Math. (2)","ISSN":"https:\/\/id.crossref.org\/issn\/0003-486X","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2003-72-243\/S0025-5718-03-01486-8\/S0025-5718-03-01486-8.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2003-72-243\/S0025-5718-03-01486-8\/S0025-5718-03-01486-8.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T23:20:01Z","timestamp":1776727201000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2003-72-243\/S0025-5718-03-01486-8\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2003,2,3]]},"references-count":25,"journal-issue":{"issue":"243","published-print":{"date-parts":[[2003,7]]}},"alternative-id":["S0025-5718-03-01486-8"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-03-01486-8","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":[[2003,2,3]]}}}