  In this paper, we derive a method for the second-order form of the transport
equation that is both globally conservative and compatible with voids, using
Continuous Finite Element Methods (CFEM). The main idea is to use the
Least-Squares (LS) form of the transport equation in the void regions and the
Self-Adjoint Angular Flux (SAAF) form elsewhere. While the SAAF formulation is
globally conservative, the LS formulation need a correction in void. The price
to pay for this fix is the loss of symmetry of the bilinear form. We first
derive this Conservative LS (CLS) formulation in void. Second we combine the
SAAF and CLS forms and end up with an hybrid SAAF-CLS method, having the
desired properties. We show that extending the theory to near-void regions is a
minor complication and can be done without affecting the global conservation of
the scheme. Being angular discretization agnostic, this method can be applied
to both discrete ordinates (SN) and spherical harmonics (PN) methods. However,
since a globally conservative and void compatible second-order form already
exists for SN (Wang et al. 2014), but is believed to be new for PN, we focus
most of our attention on that latter angular discretization. We implement and
test our method in Rattlesnake within the Multiphysics Object Oriented
Simulation Environment (MOOSE) framework. Results comparing it to other methods
are presented.
