  We present a family of discretizations for the Variable Eddington Factor
(VEF) equations that have high-order accuracy on curved meshes and efficient
preconditioned iterative solvers. The VEF discretizations are combined with a
high-order Discontinuous Galerkin transport discretization to form an effective
high-order, linear transport method. The VEF discretizations are derived by
extending the unified analysis of Discontinuous Galerkin methods for elliptic
problems to the VEF equations. This framework is used to define analogs of the
interior penalty, second method of Bassi and Rebay, minimal dissipation local
Discontinuous Galerkin, and continuous finite element methods. The analysis of
subspace correction preconditioners, which use a continuous operator to
iteratively precondition the discontinuous discretization, is extended to the
case of the non-symmetric VEF system. Numerical results demonstrate that the
VEF discretizations have arbitrary-order accuracy on curved meshes, preserve
the thick diffusion limit, and are effective on a proxy problem from thermal
radiative transfer in both outer transport iterations and inner preconditioned
linear solver iterations. In addition, a parallel weak scaling study of the
interior penalty VEF discretization demonstrates the scalability of the method
out to 1152 processors.
