{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T16:17:54Z","timestamp":1787329074398,"version":"build-2736575974"},"reference-count":27,"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>When discussing numerical solutions of the Navier\u2013Stokes equations, especially when turbulent flows are concerned, there are at least two questions that can be raised. What is meaningful to compute? How to determine the fidelity of the computed solution with respect to the true solution? This paper takes a step towards the answer of these questions for turbulent flows. We consider long-time averages of weak solutions of the Navier\u2013Stokes equations, rather than strong solutions. We present error estimates for the time-averaged energy dissipation rate, drag and lift, most of them under the assumption of small Reynolds\/generalized Grashof number. For shear flows, we address the question of fidelity of the computed solution with respect to the true solution, in view of Kolmogorov's energy cascade theory.<\/jats:p>","DOI":"10.1137\/060649550","type":"journal-article","created":{"date-parts":[[2007,12,20]],"date-time":"2007-12-20T18:32:01Z","timestamp":1198175521000},"page":"151-179","source":"Crossref","is-referenced-by-count":4,"title":["Convergence of Time-Averaged Statistics of Finite Element Approximations of the Navier\u2013Stokes Equations"],"prefix":"10.1137","volume":"46","author":[{"given":"V.","family":"John","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"W.","family":"Layton","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"C. C.","family":"Manica","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2007,12,20]]},"reference":[{"key":"R1","unstructured":"R. A. Adams,\n                      Sobolev Spaces\n                      , Academic Press, New York, 1975."},{"key":"R2","unstructured":"L. C. Berselli, T. Iliescu, and W. Layton,\n                      Mathematics of Large Eddy Simulation of Turbulent Flows\n                      , Sci. Comput., Springer-Verlag, 2006."},{"key":"R3","doi-asserted-by":"crossref","unstructured":"F. Brezzi and M. Fortin,\n                      Mixed and Hybrid Finite Element Methods\n                      , Springer-Verlag, New York, 1991.","DOI":"10.1007\/978-1-4612-3172-1"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1103\/PhysRevLett.69.1648"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1017\/S0022112070000691"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1017\/S0022112002001386"},{"key":"R7","doi-asserted-by":"crossref","unstructured":"C. R. Doering and J. D. Gibbon,\n                      Applied Analysis of the Navier-Stokes Equations\n                      , Cambridge University Press, Cambridge, UK, 1995.","DOI":"10.1017\/CBO9780511608803"},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1016\/j.physd.2005.10.002"},{"key":"R9","doi-asserted-by":"crossref","unstructured":"G. P. Galdi,\n                      An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Volume\n                      II\n                      . 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