A Domain Decomposition Analysis for a Two-scale Linear Transport Problem

We present a domain decomposition theory on an interface problem for the linear transport equation between a diffusive and a non-diffusive region. To leading order, i.e. up to an error of the order of the mean free path in the diffusive region, the solution in the non-diffusive region is independent of the density in the diffusion region. However, the diffusive and the non-diffusive regions are coupled at the interface at the next order of approximation. In particular, our algorithm avoids interating the diffusion and transport solutions as is done in most other methods — see for example Bal-Maday [Math. Modelling and Numer. Anal. to appear]. Our analysis is based instead on an accurate description of the boundary layer at the interface matching the phase-space density of particles leaving the non-diffusion region to the bulk density that solves the diffusive equation. 1. The interface problem Consider the steady, linear transport equation with isotropic scattering and slab geometry: (1.1) μ∂xΨ(x, μ) + σ(x)Ψ(x, μ) = σ(x)c(x)Ψ(x) , where Ψ(x) = 1 2 ∫ 1