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This spectral point of view is complementary with the Lyapunov--Schmidt reduction technique and provides a precise framework for investigating convergence. It is shown for convection-diffusion inside a cylinder that the spectral convergence of the volume averageddescription depends on the chosen averaging operator, as well as on the boundary conditions. A remarkable result states that only part of the eigenmodes among the infinite discrete spectrum of the full solution can be captured by averaging methods. This leads to a general convergence theorem (which was already examined with the use of the center manifold theorem [G. N. Mercer and A. J. Roberts, SIAM J. Appl. Math., 50 (1990), pp. 1547--1565] and investigated with Lyapunov--Schmidt reduction techniques [S. Chakraborty and V. Balakotaiah, Chem. Engrg. Sci., 57 (2002), pp. 2545--2564] in similar contexts). Moreover, a necessary and sufficient condition for an eigenvalue to be captured is given. We then investigate specific averaging operators, the convergence of which is found to be exponential.<\/jats:p>","DOI":"10.1137\/040610015","type":"journal-article","created":{"date-parts":[[2005,10,17]],"date-time":"2005-10-17T21:00:17Z","timestamp":1129582817000},"page":"122-152","source":"Crossref","is-referenced-by-count":2,"title":["Convergence of the Generalized Volume Averaging Method on a Convection-Diffusion Problem: A Spectral Perspective"],"prefix":"10.1137","volume":"66","author":[{"given":"C.","family":"Pierre","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"F.","family":"Plouraboue","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"M.","family":"Quintard","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2006,7,31]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1098\/rspa.1956.0065"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1109\/10.284920"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1115\/1.3138623"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1098\/rsta.1995.0025"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1137\/S0036139901368863"},{"key":"R6","volume-title":"Asymptotic analysis for periodic structures","author":"Bensoussan Alain","year":"1978"},{"key":"R7","doi-asserted-by":"publisher","DOI":"10.1063\/1.1721476"},{"key":"R8","first-page":"463","volume":"306","author":"Bourgeat Alain","year":"1988","journal-title":"C. 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