{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,22]],"date-time":"2026-08-22T04:09:46Z","timestamp":1787371786181,"version":"build-2736575974"},"reference-count":47,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Numer. Anal."],"published-print":{"date-parts":[[2008,1]]},"abstract":"<jats:p>A postprocessing technique for mixed finite-element methods for the incompressible Navier\u2013Stokes equations is analyzed. The postprocess, which amounts to solving a (linear) Stokes problem, is shown to increase the order of convergence of the method to which it is applied by one unit (times the logarithm of the mesh diameter). In proving the error bounds, some superconvergence results are also obtained. Contrary to previous analysis of the postprocessing technique, in the present paper we take into account the loss of regularity suffered by the solutions of the Navier\u2013Stokes equations at the initial time in the absence of nonlocal compatibility conditions of the data. As in [H. G. Heywood and R. Rannacher, SIAM J. Numer. Anal., 25 (1988), pp. 489\u2013512], where the same hypothesis is assumed, no better than fifth-order convergence is achieved.<\/jats:p>","DOI":"10.1137\/06064458","type":"journal-article","created":{"date-parts":[[2007,12,20]],"date-time":"2007-12-20T18:32:01Z","timestamp":1198175521000},"page":"201-230","source":"Crossref","is-referenced-by-count":17,"title":["The Postprocessed Mixed Finite-Element Method for the Navier\u2013Stokes Equations: Refined Error Bounds"],"prefix":"10.1137","volume":"46","author":[{"given":"Javier","family":"de Frutos","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Bosco","family":"Garc\u00eda-Archilla","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Julia","family":"Novo","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2007,12,20]]},"reference":[{"key":"R1","unstructured":"R. Adams,\n                      Sobolev Spaces\n                      , Academic Press, New York, 1975."},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1007\/s002110050056"},{"key":"R3","first-page":"374","volume":"23","author":"Arnold D. N.","year":"1984","journal-title":"Calcolo","ISSN":"https:\/\/id.crossref.org\/issn\/0008-0624","issn-type":"print"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1093\/imanum\/drl010"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1142\/S021820250500042X"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1137\/040602821"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1093\/imanum\/dri019"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1137\/0728032"},{"key":"R9","doi-asserted-by":"crossref","unstructured":"P. G. Ciarlet,\n                      The Finite Element Method for Elliptic Problems\n                      , North-Holland, Amsterdam, 1978.","DOI":"10.1115\/1.3424474"},{"key":"R10","doi-asserted-by":"crossref","unstructured":"P. Constantin and C. Foias,\n                      Navier\u2013Stokes Equations\n                      , Chicago Lectures in Math., The University of Chicago, Chicago, 1988.","DOI":"10.7208\/chicago\/9780226764320.001.0001"},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1051\/m2an\/197307R300331"},{"key":"R12","unstructured":"P. J. Davis,\n                      Gamma function and related functions\n                      , in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, M. Abramowitz and I. A. Stegun, eds., Dover, New York, 1992 (reprint of the 1972 edition)."},{"key":"R13","doi-asserted-by":"publisher","DOI":"10.1051\/m2an:2004037"},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1051\/m2an\/1988220100931"},{"key":"R15","doi-asserted-by":"publisher","DOI":"10.1007\/BF01047831"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1007\/s002110000188"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142999351984"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.1016\/S0168-9274(99)00086-0"},{"key":"R19","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142900375438"},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1016\/S0045-7825(02)00419-X"},{"key":"R21","doi-asserted-by":"publisher","DOI":"10.1002\/nme.1310"},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2007.03.015"},{"key":"R23","unstructured":"J. de Frutos, B. Garc\u00eda-Archilla, and J. Novo,\n                      Postprocessing Finite-Element Methods for the Navier\u2013Stokes Equations: The Fully Discrete Case\n                      , manuscript."},{"key":"R24","doi-asserted-by":"publisher","DOI":"10.1007\/BF00276188"},{"key":"R25","doi-asserted-by":"crossref","unstructured":"G. P. Galdi,\n                      An Introduction to the Mathematical Theory of the Navier\u2013Stokes Equations. Vol\n                      . I.\n                      Linearized Steady Problems\n                      , Springer-Verlag, New York, 1994.","DOI":"10.1007\/978-1-4757-3866-7"},{"key":"R26","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142995296096"},{"key":"R27","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-99-01057-1"},{"key":"R28","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142998335893"},{"key":"R29","doi-asserted-by":"crossref","unstructured":"V. Girault and P. A. Raviart,\n                      Finite Element Methods for Navier\u2013Stokes Equations\n                      , Springer-Verlag, Berlin, 1986.","DOI":"10.1007\/978-3-642-61623-5"},{"key":"R30","doi-asserted-by":"publisher","DOI":"10.1051\/m2an:2001145"},{"key":"R31","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142901385659"},{"key":"R32","doi-asserted-by":"publisher","DOI":"10.1002\/num.20077"},{"key":"R33","doi-asserted-by":"publisher","DOI":"10.1007\/s10444-004-7640-1"},{"key":"R34","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-005-0632-3"},{"key":"R35","unstructured":"D. Henry,\n                      Geometric Theory of Semilinear Parabolic Equations\n                      , Springer-Verlag, Berlin, 1991."},{"key":"R36","doi-asserted-by":"publisher","DOI":"10.1137\/0719018"},{"key":"R37","doi-asserted-by":"publisher","DOI":"10.1137\/0725032"},{"key":"R38","doi-asserted-by":"publisher","DOI":"10.1137\/0727022"},{"key":"R39","doi-asserted-by":"publisher","DOI":"10.1016\/0045-7930(73)90027-3"},{"key":"R40","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142901396375"},{"key":"R41","doi-asserted-by":"publisher","DOI":"10.1016\/S0168-9274(96)00038-4"},{"key":"R42","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142901390500"},{"key":"R43","doi-asserted-by":"publisher","DOI":"10.1080\/026811199282047"},{"key":"R44","doi-asserted-by":"publisher","DOI":"10.1016\/S0377-0427(99)00056-4"},{"key":"R45","doi-asserted-by":"publisher","DOI":"10.1007\/s00466-003-0465-9"},{"key":"R46","doi-asserted-by":"publisher","DOI":"10.1007\/s11071-005-2822-z"},{"key":"R47","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142903430931"}],"container-title":["SIAM Journal on Numerical Analysis"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/06064458","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:09:35Z","timestamp":1787324975000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/06064458"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2007,12,20]]},"references-count":47,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2008,1]]}},"alternative-id":["10.1137\/06064458"],"URL":"https:\/\/doi.org\/10.1137\/06064458","relation":{},"ISSN":["0036-1429","1095-7170"],"issn-type":[{"value":"0036-1429","type":"print"},{"value":"1095-7170","type":"electronic"}],"subject":[],"published":{"date-parts":[[2007,12,20]]}}}