Transport approximations in partially diffusive media

This paper concerns the analysis of approximations of transport equations in diffusive media. Firstly, we consider a variational formulation for the first-order transport equation that has the correct diffusive behavior in the limit of small mean free paths. The associated bilinear form is shown to be coercive on a classical Hilbert space in transport theory with a constant of coercivity independent of the mean free path. This allows us to obtain the diffusion approximation of transport as an orthogonal projection onto a subspace of functions that are independent of the angular variable. Similarly, projections onto functions that are independent of the angular variable only in subsets of the full domain can be interpreted as a transport-diffusion coupling method. Convergence results based on averaging lemmas and error estimates are presented. Secondly, we address the problem of extended non-scattering layers or filaments surrounded by highly scattering media and derive generalized diffusion equations to model transport in such geometries.