RESUMEN

This paper proposes a thorough investigation of the convergence of the volume averaging method described by Whitaker [The Method of Volume Averaging, Kluwer Academic, Norwell, MA, 1999] as applied to convection-diffusion problems inside a cylinder. A spectral description of volume averaging brings to the fore new perspectives about the mathematical analysis of those approximations. 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.