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\n If\n \n \n \n \n u<\/mml:mi>\n \n \u00af\n \n <\/mml:mo>\n <\/mml:mover>\n \\overline {u}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n denotes a local, spatial average of\n \n \n \n u<\/mml:mi>\n u<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , then\n \n \n \n \n \n u<\/mml:mi>\n \n \n \u2032\n \n <\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n =<\/mml:mo>\n u<\/mml:mi>\n \n \u2212\n \n <\/mml:mo>\n \n u<\/mml:mi>\n \n \u00af\n \n <\/mml:mo>\n <\/mml:mover>\n <\/mml:mrow>\n u^{\\prime }=u-\\overline {u}<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is the associated fluctuation. Consider a time relaxation term added to the usual finite element method. The simplest case for the model advection equation\n \n \n \n \n \n u<\/mml:mi>\n \n t<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n +<\/mml:mo>\n \n a<\/mml:mi>\n \n \u2192\n \n <\/mml:mo>\n <\/mml:mover>\n \n \u22c5\n \n <\/mml:mo>\n \n \u2207\n \n <\/mml:mi>\n u<\/mml:mi>\n =<\/mml:mo>\n f<\/mml:mi>\n (<\/mml:mo>\n x<\/mml:mi>\n ,<\/mml:mo>\n t<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n u_{t}+\\overrightarrow {a}\\cdot \\nabla u=f(x,t)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is\n \n \\[\n \n \n \n (<\/mml:mo>\n \n u<\/mml:mi>\n \n h<\/mml:mi>\n ,<\/mml:mo>\n t<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n +<\/mml:mo>\n \n a<\/mml:mi>\n \n \u2192\n \n <\/mml:mo>\n <\/mml:mover>\n \n \u22c5\n \n <\/mml:mo>\n \n \u2207\n \n <\/mml:mi>\n \n u<\/mml:mi>\n \n h<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n ,<\/mml:mo>\n \n v<\/mml:mi>\n \n h<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n )<\/mml:mo>\n +<\/mml:mo>\n \n \u03c7\n \n <\/mml:mi>\n (<\/mml:mo>\n \n u<\/mml:mi>\n \n h<\/mml:mi>\n <\/mml:mrow>\n \n \n \u2032\n \n <\/mml:mi>\n <\/mml:mrow>\n <\/mml:msubsup>\n ,<\/mml:mo>\n \n v<\/mml:mi>\n \n h<\/mml:mi>\n <\/mml:mrow>\n \n \n \u2032\n \n <\/mml:mi>\n <\/mml:mrow>\n <\/mml:msubsup>\n )<\/mml:mo>\n =<\/mml:mo>\n (<\/mml:mo>\n f<\/mml:mi>\n (<\/mml:mo>\n x<\/mml:mi>\n ,<\/mml:mo>\n t<\/mml:mi>\n )<\/mml:mo>\n ,<\/mml:mo>\n \n v<\/mml:mi>\n \n h<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msub>\n )<\/mml:mo>\n .<\/mml:mo>\n <\/mml:mrow>\n (u_{h,t}+\\overrightarrow {a}\\cdot \\nabla u_{h},v_{h})+\\chi (u_{h}^{\\prime },v_{h}^{\\prime })=(f(x,t),v_{h}).<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n \\]\n <\/disp-formula>\n We analyze the error in this and (more importantly) higher order extensions and show that the added time relaxation term not only suppresses excess energy in marginally resolved scales but also increases the accuracy of the resulting finite element approximation.\n <\/p>","DOI":"10.1090\/s0025-5718-09-02316-3","type":"journal-article","created":{"date-parts":[[2010,2,4]],"date-time":"2010-02-04T11:19:48Z","timestamp":1265282388000},"page":"619-648","source":"Crossref","is-referenced-by-count":15,"title":["On the accuracy of the finite element method plus time relaxation"],"prefix":"10.1090","volume":"79","author":[{"given":"J.","family":"Connors","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"W.","family":"Layton","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2009,12,16]]},"reference":[{"key":"1","doi-asserted-by":"crossref","unstructured":"[AL99] N. 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Gustafsson, Quasioptimal finite element approximation of first order hyperbolic and convection-diffusion equations, pp. 273-281 in: Analytical and Numerical Approaches to Asymptotic Problems in Analysis, (eds: A. O. H. Axelsson, L. S. Frank, and A. van der Sluis) North Holland, Amsterdam, 1980.","DOI":"10.1016\/S0304-0208(08)71115-0"},{"key":"5","series-title":"Scientific Computation","isbn-type":"print","volume-title":"Mathematics of large eddy simulation of turbulent flows","author":"Berselli, L. 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