Convergence of the Generalized Volume Averaging Method on a Convection-Diffusion Problem: A Spectral Perspective

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 convectiondiffusion inside a cylinder that the spectral convergence of the volume averaged description 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.