  A general problem of current interest is the analysis of diffusion problems
in singularly perturbed domains, within which small subdomains are removed from
the domain interior and boundary conditions imposed on the resulting holes. One
major application is to intracellular diffusion, where the holes could
represent organelles or biochemical substrates. In this paper we use a
combination of matched asymptotic analysis and Green's function methods to
calculate the so-called accumulation time for relaxation to steady state in 2D
domains. The standard measure of the relaxation rate is in terms of the
principal nonzero eigenvalue of the negative Laplacian. However, this global
measure does not account for possible differences in the relaxation rate at
different spatial locations, is independent of the initial conditions, and
relies on the assumption that the eigenvalues have sufficiently large spectral
gaps. As previously established for diffusion-based morphogen gradient
formation, the accumulation time provides a better measure of the relaxation
process.
