A mixed variational framework for the radiative transfer equation

We present a rigorous variational framework for the analysis and discretization of the radiative transfer equation. Existence and uniqueness of weak solutions are established under rather general assumptions on the coefficients. Moreover, weak solutions are shown to be regular and hence also strong solutions of the radiative transfer equation. The relation of the proposed variational method to other approaches, including least-squares and even-parity formulations, is discussed. Moreover, the approximation by Galerkin methods is investigated, and simple conditions are given, under which stable quasi-optimal discretizations can be obtained. For illustration, the approximation by a finite element P _N approximation is discussed in some detail.