The asymptotic diffusion limit of a linear discontinuous discretization of a two-dimensional linear transport equation

Abstract

Consider a linear transport problem, and let the mean free path and the absorption cross section be of size [epsilon]. It is well known that one obtains a diffusion problem as [epsilon] tends to zero. We discretize the transport problem on a fixed mesh, independent of [epsilon], consider again the limit [epsilon] --> 0 and ask whether one obtains an accurate discretization of the continuous diffusion problem. The answer is known to be affirmative for the linear discontinuous Galerkin finite element discretization in one space dimension. In this paper, we ask whether the same result holds in two space dimensions. We consider a linear discontinuous discretization based on rectangular meshes. Our main result is that the asymptotic limit of this discrete problem is not a discretization of the asymptotic limit of the continuous problem and thus that the discretization will be inaccurate in the asymptotic regime under consideration. We also propose a modified scheme which has the correct asymptotic behavior for spatially periodic problems, although not always for problems with boundaries. We present numerical results confirming our formal asymptotic analysis.