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

¹ Institut Universitaire de France, Département de Mathématiques et Applications, École Normale Supérieure Paris, 45 rue d'Ulm, 75230 Paris Cedex 05, France. golse@dma.ens.fr.
² Department of Mathematics, University of Wisconsin-Madison, Madison, Wisconsin 53706, USA. jin@math.wisc.edu.
³ Department of Mathematics, Institute of Physical Sciences and Technology, University of Maryland, College Park, Maryland 20742, USA. lvrmr@math.umd.edu.

Received: 17 February 2003

Abstract

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 diffusive 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 iterating the diffusion and transport solutions as is done in most other methods — see for example Bal–Maday (2002). 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-diffusive region to the bulk density that solves the diffusion equation.