Boundary value problems for diffusion in singularly perturbed domains is a topic of
considerable current interest. Applications include intracellular diffusive transport and the spread
of pollutants or heat from localized sources. In a previous paper, we introduced a new method for
characterizing the approach to steady state in the case of two-dimensional (2D) diffusion. This was
based on a local measure of the relaxation rate known as the accumulation time T(x). The latter
was calculated by solving the diffusion equation in Laplace space using a combination of matched
asymptotics and Green's function methods. We thus obtained an asymptotic expansion of T(x) in
powers of \nu = - 1/ ln \epsilon , where \epsilon specifies the relative size of the holes. In this paper, we develop the
corresponding theory for three-dimensional (3D) diffusion. The analysis is a nontrivial extension of
the 2D case due to differences in the singular nature of the Laplace transformed Green's function.
In particular, the asymptotic expansion of the solution of the 3D diffusion equation in Laplace space
involves terms of order O((\epsilon /s)
n), where s is the Laplace variable. These s-singularities have to
be removed by partial series resummations in order to obtain an asymptotic expansion of T(x) in
powers of \