Generalizing Raviart-Thomas elements to PN transport

The Raviart-Thomas finite elements have so far been successfully employed for the spatial discretization of the mixed dual formulation of the diffusion equation, providing the basis for a fast diffusion and simplified spherical harmonic (or SPN) solver. We here consider generalizing this approach to the transport equation with spherical harmonic (or PN) angular discretization. This is challenging because in the transport case the well-known even/odd parity decomposition of the angular flux yields mixed unknowns that are both scalar, while Raviart-Thomas elements are designed to simultaneously approximate one scalar and one vector unknown. In this paper we first investigate the possibility of defining a vector unknown within the mixed formulation of the transport equation, so as to make Raviart-Thomas elements suitable for the spatial discretization. In view of the failure of such an approach, we then turn our attention to a second-order primal alternative originally developed by Hennart for the diffusion case. Introducing non-conforming finite elements in the second-order primal formulation of the transport equation, we obtain satisfying results on a severe fixed-source test problem. (authors)