{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T23:42:43Z","timestamp":1776728563788,"version":"3.51.2"},"reference-count":8,"publisher":"American Mathematical Society (AMS)","issue":"241","license":[{"start":{"date-parts":[[2002,12,4]],"date-time":"2002-12-04T00:00:00Z","timestamp":1038960000000},"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>It is well known that discrete solutions to the convection-diffusion equation contain nonphysical oscillations when boundary layers are present but not resolved by the discretisation. However, except for one-dimensional problems, there is little analysis of this phenomenon. In this paper, we present an analysis of the two-dimensional problem with constant flow aligned with the grid, based on a Fourier decomposition of the discrete solution. For Galerkin bilinear finite element discretisations, we derive closed form expressions for the Fourier coefficients, showing them to be weighted sums of certain functions which are oscillatory when the mesh P\u00e9clet number is large. The oscillatory functions are determined as solutions to a set of three-term recurrences, and the weights are determined by the boundary conditions. These expressions are then used to characterise the oscillations of the discrete solution in terms of the mesh P\u00e9clet number and boundary conditions of the problem.<\/p>","DOI":"10.1090\/s0025-5718-01-01392-8","type":"journal-article","created":{"date-parts":[[2002,11,1]],"date-time":"2002-11-01T09:59:50Z","timestamp":1036144790000},"page":"263-288","source":"Crossref","is-referenced-by-count":13,"title":["A characterisation of oscillations in the discrete two-dimensional convection-diffusion equation"],"prefix":"10.1090","volume":"72","author":[{"given":"Howard","family":"Elman","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Alison","family":"Ramage","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2001,12,4]]},"reference":[{"key":"1","unstructured":"H.C. Elman and A. Ramage, An analysis of smoothing effects of upwinding strategies for the convection-diffusion equation, Tech. Report UMCP-CSD:CS-TR-4160, University of Maryland, College Park MD 20742, 2000."},{"key":"2","unstructured":"P.M. Gresho and R.L. Sani, Incompressible flow and the finite element method, John Wiley and Sons, Chichester, 1999."},{"key":"3","isbn-type":"print","volume-title":"Numerical solution of partial differential equations by the finite element method","author":"Johnson, Claes","year":"1987","ISBN":"https:\/\/id.crossref.org\/isbn\/0521345146"},{"key":"4","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"},{"key":"5","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":"1-2","key":"6","doi-asserted-by":"publisher","first-page":"99","DOI":"10.1016\/0045-7825(94)90213-5","article-title":"Numerical crosswind smear in the streamline diffusion method","volume":"113","author":"Semper, Bill","year":"1994","journal-title":"Comput. Methods Appl. Mech. Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0045-7825","issn-type":"print"},{"key":"7","unstructured":"M.R. Spiegel, Mathematical handbook of formulas and tables, Schaum\u2019s outline series, McGraw-Hill, New York, 1990."},{"issue":"3","key":"8","doi-asserted-by":"publisher","first-page":"490","DOI":"10.1137\/1019071","article-title":"The methods of cyclic reduction, Fourier analysis and the FACR algorithm for the discrete solution of Poisson\u2019s equation on a rectangle","volume":"19","author":"Swarztrauber, Paul N.","year":"1977","journal-title":"SIAM Rev.","ISSN":"https:\/\/id.crossref.org\/issn\/1095-7200","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2003-72-241\/S0025-5718-01-01392-8\/S0025-5718-01-01392-8.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2003-72-241\/S0025-5718-01-01392-8\/S0025-5718-01-01392-8.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T23:08:30Z","timestamp":1776726510000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2003-72-241\/S0025-5718-01-01392-8\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2001,12,4]]},"references-count":8,"journal-issue":{"issue":"241","published-print":{"date-parts":[[2003,1]]}},"alternative-id":["S0025-5718-01-01392-8"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-01-01392-8","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":[[2001,12,4]]}}}