An asymptotic analysis if performed on a family of discontinuous finite element methods (DFEMs) for spherical geometry transport. It is found that transport methods of this type transition into discrete versions of the spherical geometry diffusion equation in the thick diffusion limit with boundary conditions that may, in general, be inaccurate. A linear DFEM method has been designed such that its asymptotic diffusion boundary conditions are accurate. In a related development, the asymptotic diffusion equation is used to accelerate the transport calculation and the iterative scheme is fully described. The results of the analysis are confirmed by numerical testing of the specific case of linear elements. 5 refs., 2 figs., 1 tab.