{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,5]],"date-time":"2026-10-05T23:01:40Z","timestamp":1791241300475,"version":"4.1.0"},"reference-count":27,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"5","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Numer. Anal."],"published-print":{"date-parts":[[2008,1]]},"abstract":"<jats:p>We analyze the simplest and most standard adaptive finite element method (AFEM), with any polynomial degree, for general second order linear, symmetric elliptic operators. As is customary in practice, the AFEM marks exclusively according to the error estimator and performs a minimal element refinement without the interior node property. We prove that the AFEM is a contraction, for the sum of the energy error and the scaled error estimator, between two consecutive adaptive loops. This geometric decay is instrumental to derive the optimal cardinality of the AFEM. We show that the AFEM yields a decay rate of the energy error plus oscillation in terms of the number of degrees of freedom as dictated by the best approximation for this combined nonlinear quantity.<\/jats:p>","DOI":"10.1137\/07069047x","type":"journal-article","created":{"date-parts":[[2008,6,18]],"date-time":"2008-06-18T18:02:38Z","timestamp":1213812158000},"page":"2524-2550","source":"Crossref","is-referenced-by-count":381,"title":["Quasi-Optimal Convergence Rate for an Adaptive Finite Element Method"],"prefix":"10.1137","volume":"46","author":[{"given":"J. Manuel","family":"Cascon","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Christian","family":"Kreuzer","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ricardo H.","family":"Nochetto","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Kunibert G.","family":"Siebert","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2008,6,18]]},"reference":[{"key":"R1","doi-asserted-by":"crossref","unstructured":"M. Ainsworth and J. T. Oden,\n                      A Posteriori Error Estimation in Finite Element Analysis\n                      , John Wiley & Sons, New York, 2000.","DOI":"10.1002\/9781118032824"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1016\/0899-8248(91)90006-G"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-003-0492-7"},{"key":"R4","first-page":"1001","volume":"28","author":"Binev P.","year":"2002","journal-title":"Serdica Math. J.","ISSN":"https:\/\/id.crossref.org\/issn\/1310-6600","issn-type":"print"},{"key":"R5","unstructured":"J. M. Casc\u00f3n, C. Kreuzer, R. H. Nochetto, and K. G. Siebert,\n                      Quasi-optimal AFEM for General Elliptic Operators\n                      , manuscript."},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-04-01634-5"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1137\/070681508"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1137\/0733054"},{"key":"R9","unstructured":"D. Gilbarg and N. S. Trudinger,\n                      Elliptic Partial Differential Equations of Second Order\n                      , Springer-Verlag, Berlin, 1983."},{"key":"R10","unstructured":"P. Grisvard,\n                      Elliptic Problems in Nonsmooth Domains\n                      , Pitman, Boston, 1985."},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1006\/jmaa.1999.6518"},{"key":"R12","doi-asserted-by":"publisher","DOI":"10.1080\/00036817408839086"},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1016\/0377-0427(94)90034-5"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1137\/0916014"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1145\/76909.76912"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1137\/04060929X"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142999360044"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1137\/S0036144502409093"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-02-01463-1"},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1142\/S0218202508002838"},{"key":"R21","unstructured":"A. Schmidt and K. G. Siebert,\n                      Design of Adaptive Finite Element Software: The Finite Element Toolboox ALBERTA\n                      , Lect. Notes Comput. Sci. Eng. 42, Springer-Verlag, Berlin, 2005."},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1990-1011446-7"},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1007\/s10208-005-0183-0"},{"key":"R24","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-07-01959-X"},{"key":"R25","doi-asserted-by":"publisher","DOI":"10.1007\/BF02684475"},{"key":"R26","doi-asserted-by":"publisher","DOI":"10.1007\/s002110100377"},{"key":"R27","unstructured":"R. Verf\u00fcrth,\n                      A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Technique\n                      , Wiley\u2013Teubner, Chichester, 1996."}],"container-title":["SIAM Journal on Numerical Analysis"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/07069047X","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:14:25Z","timestamp":1787325265000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/07069047X"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2008,1]]},"references-count":27,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2008,1]]}},"alternative-id":["10.1137\/07069047X"],"URL":"https:\/\/doi.org\/10.1137\/07069047x","relation":{},"ISSN":["0036-1429","1095-7170"],"issn-type":[{"value":"0036-1429","type":"print"},{"value":"1095-7170","type":"electronic"}],"subject":[],"published":{"date-parts":[[2008,1]]}}}