{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T03:24:36Z","timestamp":1776828276452,"version":"3.51.2"},"reference-count":24,"publisher":"American Mathematical Society (AMS)","issue":"233","license":[{"start":{"date-parts":[[2001,3,3]],"date-time":"2001-03-03T00:00:00Z","timestamp":983577600000},"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>\n                    In this paper we derive an\n                    <italic>a posteriori<\/italic>\n                    error bound for the Lagrange\u2013Galerkin discretisation of an unsteady (linear) convection-diffusion problem, assuming only that the underlying space-time mesh is nondegenerate. The proof of this error bound is based on strong stability estimates of an associated dual problem, together with the Galerkin orthogonality of the finite element method. Based on this\n                    <italic>a posteriori<\/italic>\n                    bound, we design and implement the corresponding adaptive algorithm to ensure global control of the error with respect to a user-defined tolerance.\n                  <\/p>","DOI":"10.1090\/s0025-5718-00-01187-x","type":"journal-article","created":{"date-parts":[[2005,7,11]],"date-time":"2005-07-11T17:02:26Z","timestamp":1121101346000},"page":"77-106","source":"Crossref","is-referenced-by-count":33,"title":["Adaptive Lagrange\u2013Galerkin methods for unsteady convection-diffusion problems"],"prefix":"10.1090","volume":"70","author":[{"given":"Paul","family":"Houston","sequence":"first","affiliation":[]},{"given":"Endre","family":"S\u00fcli","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2000,3,3]]},"reference":[{"key":"1","doi-asserted-by":"crossref","unstructured":"A.M. Baptista, E.E. Adams and P. Gresho. Benchmarks for the transport equation: the convection-diffusion forum and beyond. In, Lynch and Davies, editors, Quantitative Skill Assessment for Coastal Ocean Models, AGU Coastal and Estuarine Studies, 47:241\u2013268, 1995.","DOI":"10.1029\/CE047p0241"},{"key":"2","unstructured":"R. Becker and R. Rannacher. Weighted a posteriori error control in FE methods. Technical Report 96-1, Institut f\u00fcr Angewandte Mathematik, Universit\u00e4t Heidelberg, Heidelberg, Germany, 1996."},{"key":"3","unstructured":"M. Bercovier and O. Pironneau. Characteristics and the finite element method. In T. Kawai, editor, Proceedings of the Fourth International Symposium on Finite Element Methods in Flow Problems, pp 67\u201373. North\u2013Holland, 1982."},{"key":"4","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":"5","unstructured":"E. Burman. Adaptive Finite Element Methods for Compressible Two-Phase Flow. PhD thesis, Chalmers University of Technology, G\u00f6teborg, 1998."},{"issue":"5","key":"6","doi-asserted-by":"publisher","first-page":"871","DOI":"10.1137\/0719063","article-title":"Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures","volume":"19","author":"Douglas, Jim, Jr.","year":"1982","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"1","key":"7","doi-asserted-by":"publisher","first-page":"43","DOI":"10.1137\/0728003","article-title":"Adaptive finite element methods for parabolic problems. I. A linear model problem","volume":"28","author":"Eriksson, Kenneth","year":"1991","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"8","doi-asserted-by":"crossref","unstructured":"K. Eriksson and C. Johnson. Adaptive streamline diffusion finite element methods for time dependent convection diffusion problems. Technical Report 1993-23, Department of Mathematics, Chalmers University of Technology, G\u00f6teborg, Sweden, 1993.","DOI":"10.2307\/2153160"},{"key":"9","isbn-type":"print","doi-asserted-by":"publisher","first-page":"105","DOI":"10.1017\/S0962492900002531","article-title":"Introduction to adaptive methods for differential equations","author":"Eriksson, Kenneth","year":"1995","ISBN":"https:\/\/id.crossref.org\/isbn\/0521482550"},{"key":"10","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"},{"key":"11","unstructured":"P. Hansbo and C. Johnson. Streamline diffusion finite element methods for fluid flow. von Karman Institute Lectures, 1995."},{"key":"12","unstructured":"P. Houston. Lagrange\u2013Galerkin Methods for Unsteady Convection-Diffusion Problems: A Posteriori Error Analysis and Adaptivity. PhD thesis, University of Oxford, 1996."},{"key":"13","doi-asserted-by":"crossref","unstructured":"P. Houston, J. Mackenzie, E. S\u00fcli and G. Warnecke. A posteriori error analysis for numerical approximations of Friedrichs systems. Numer. Math. 82:433\u2013470, 1999.","DOI":"10.1007\/s002110050426"},{"key":"14","unstructured":"P. Houston, R. Rannacher and E. S\u00fcli. A posteriori error analysis for stabilised finite element approximations of transport problems. Comput. Methods Appl. Mech. Engrg. (to appear)."},{"key":"15","unstructured":"P. Houston and E. S\u00fcli. Adaptive Lagrange\u2013Galerkin methods for unsteady convection-dominated diffusion problems. Oxford University Computing Laboratory Technical Report NA95\/24, 1995 (http:\/\/www.comlab.ox.ac.uk\/oucl\/publications\/natr\/NA-95-24.html)."},{"key":"16","unstructured":"P. Houston and E. S\u00fcli. On the design of an artificial diffusion model for the Lagrange\u2013Galerkin method on unstructured triangular grids. Oxford University Computing Laboratory Technical Report NA96\/07, 1996 (http:\/\/www.comlab.ox.ac.uk\/oucl\/publications\/natr\/NA-96-07.html)."},{"key":"17","unstructured":"P. Houston and E. S\u00fcli. A posteriori error analysis for linear convection-diffusion problems under weak mesh regularity assumptions. Oxford University Computing Laboratory Technical Report NA97\/03, 1997 (http:\/\/www.comlab.ox.ac.uk\/oucl\/publications\/natr\/NA-97-03.html)."},{"key":"18","unstructured":"P. Houston and E. S\u00fcli. Local mesh design for the numerical solution of hyperbolic problems. In M. Baines, editor, Numerical Methods for Fluid Dynamics VI, pp 17\u201330. ICFD, 1998."},{"issue":"3","key":"19","doi-asserted-by":"publisher","first-page":"309","DOI":"10.1007\/BF01396435","article-title":"On the transport-diffusion algorithm and its applications to the Navier-Stokes equations","volume":"38","author":"Pironneau, O.","year":"1981","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"key":"20","unstructured":"R. Sandboge. Adaptive Finite Element Methods for Reactive Flow Problems. PhD thesis, Chalmers University of Technology, G\u00f6teborg, 1996."},{"issue":"190","key":"21","doi-asserted-by":"publisher","first-page":"483","DOI":"10.2307\/2008497","article-title":"Finite element interpolation of nonsmooth functions satisfying boundary conditions","volume":"54","author":"Scott, L. Ridgway","year":"1990","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"22","doi-asserted-by":"crossref","unstructured":"E. S\u00fcli. A posteriori error analysis and adaptivity for finite element approximations of hyperbolic problems. In D. Kr\u00f6ner, M. Ohlberger and C. Rohde, editors, An Introduction to Recent Developments in Theory and Numerics for Conservation Laws, Volume 5 of Lecture notes in Computational Science and Engineering, pp. 123\u2013144. Springer\u2013Verlag, 1998.","DOI":"10.1007\/978-3-642-58535-7_4"},{"key":"23","isbn-type":"print","first-page":"441","article-title":"Finite element methods for hyperbolic problems: a posteriori error analysis and adaptivity","author":"S\u00fcli, Endre","year":"1997","ISBN":"https:\/\/id.crossref.org\/isbn\/0198500149"},{"key":"24","doi-asserted-by":"crossref","unstructured":"R. Verf\u00fcrth. Error estimates for some quasi-interpolation operators. 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