{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T22:43:49Z","timestamp":1776725029157,"version":"3.51.2"},"reference-count":11,"publisher":"American Mathematical Society (AMS)","issue":"230","license":[{"start":{"date-parts":[[2000,3,3]],"date-time":"2000-03-03T00:00:00Z","timestamp":952041600000},"content-version":"am","delay-in-days":366,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>The aim of this paper is to display numerical results that show the interest of some multilevel methods for problems of parabolic type. These schemes are based on multilevel spatial splittings and the use of different time steps for the various spatial components. The spatial discretization we investigate is of spectral Fourier type, so the approximate solution naturally splits into the sum of a low frequency component and a high frequency one. The time discretization is of implicit\/explicit Euler type for each spatial component. Based on a posteriori estimates, we introduce adaptive one-level and multilevel algorithms. Two problems are considered: the heat equation and a nonlinear problem. Numerical experiments are conducted for both problems using the one-level and the multilevel algorithms. The multilevel method is up to 70% faster than the one-level method.<\/p>","DOI":"10.1090\/s0025-5718-99-01087-x","type":"journal-article","created":{"date-parts":[[2002,7,26]],"date-time":"2002-07-26T18:13:45Z","timestamp":1027707225000},"page":"547-581","source":"Crossref","is-referenced-by-count":3,"title":["Adaptive multilevel methods in space and time for parabolic problems-the periodic case"],"prefix":"10.1090","volume":"69","author":[{"given":"J.","family":"Burie","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"M.","family":"Marion","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[1999,3,3]]},"reference":[{"key":"1","unstructured":"J.-B. Burie, Th\u00e8se, Ecole Centrale de Lyon, 1996."},{"issue":"4","key":"2","doi-asserted-by":"publisher","first-page":"1574","DOI":"10.1137\/S0036142994267989","article-title":"Multilevel methods in space and time for the Navier-Stokes equations","volume":"34","author":"Burie, J. B.","year":"1997","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"3","series-title":"Springer Series in Computational Physics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-84108-8","volume-title":"Spectral methods in fluid dynamics","author":"Canuto, Claudio","year":"1988","ISBN":"https:\/\/id.crossref.org\/isbn\/0387173714"},{"key":"4","unstructured":"A. Debussche, T. Dubois and R. Temam, The nonlinear Galerkin method: A multi-scale method applied to the simulation of homogenous turbulent flows, ICASE Report No 93-93 (1993)."},{"issue":"1","key":"5","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"},{"issue":"3","key":"6","doi-asserted-by":"publisher","first-page":"706","DOI":"10.1137\/0732033","article-title":"Adaptive finite element methods for parabolic problems. II. Optimal error estimates in \ud835\udc3f_{\u221e}\ud835\udc3f\u2082 and \ud835\udc3f_{\u221e}\ud835\udc3f_{\u221e}","volume":"32","author":"Eriksson, Kenneth","year":"1995","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"6","key":"7","doi-asserted-by":"publisher","first-page":"1729","DOI":"10.1137\/0732078","article-title":"Adaptive finite element methods for parabolic problems. IV. Nonlinear problems","volume":"32","author":"Eriksson, Kenneth","year":"1995","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"4","key":"8","doi-asserted-by":"publisher","first-page":"611","DOI":"10.1051\/m2an\/1985190406111","article-title":"Time discretization of parabolic problems by the discontinuous Galerkin method","volume":"19","author":"Eriksson, Kenneth","year":"1985","journal-title":"RAIRO Mod\\'{e}l. Math. Anal. Num\\'{e}r.","ISSN":"https:\/\/id.crossref.org\/issn\/0764-583X","issn-type":"print"},{"issue":"1","key":"9","doi-asserted-by":"publisher","first-page":"93","DOI":"10.1051\/m2an\/1988220100931","article-title":"Modelling of the interaction of small and large eddies in two-dimensional turbulent flows","volume":"22","author":"Foias, C.","year":"1988","journal-title":"RAIRO Mod\\'{e}l. Math. Anal. Num\\'{e}r.","ISSN":"https:\/\/id.crossref.org\/issn\/0764-583X","issn-type":"print"},{"issue":"4","key":"10","doi-asserted-by":"publisher","first-page":"1058","DOI":"10.1137\/0732048","article-title":"Numerics and hydrodynamic stability: toward error control in computational fluid dynamics","volume":"32","author":"Johnson, Claes","year":"1995","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"4","key":"11","doi-asserted-by":"publisher","first-page":"1170","DOI":"10.1137\/0732054","article-title":"Error estimates on a new nonlinear Galerkin method based on two-grid finite elements","volume":"32","author":"Marion, Martine","year":"1995","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2000-69-230\/S0025-5718-99-01087-X\/S0025-5718-99-01087-X.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2000-69-230\/S0025-5718-99-01087-X\/S0025-5718-99-01087-X.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,20]],"date-time":"2026-04-20T22:16:24Z","timestamp":1776723384000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2000-69-230\/S0025-5718-99-01087-X\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1999,3,3]]},"references-count":11,"journal-issue":{"issue":"230","published-print":{"date-parts":[[2000,4]]}},"alternative-id":["S0025-5718-99-01087-X"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-99-01087-x","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":[[1999,3,3]]}}}