  In this paper, two flux-only least-squares finite element methods (LSFEM) for
the linear hyperbolic transport problem are developed. The transport equation
often has discontinuous solutions and discontinuous inflow boundary conditions,
but the normal component of the flux across the mesh interfaces is continuous.
In traditional LSFEMs, the continuous finite element space is used to
approximate the solution. This will cause unnecessary error around the
discontinuity and serious overshooting. In arXiv:1807.01524 [math.NA], we
reformulate the equation by introducing a new flux variable to separate the
continuity requirements of the flux and the solution. Realizing that the
Raviart-Thomas mixed element space has enough degrees of freedom to approximate
both the flux and its divergence, we eliminate the solution from the system and
get two flux-only formulations, and develop corresponding LSFEMs. The solution
then is recovered by simple post-processing methods using its relation with the
flux. These two versions of flux-only LSFEMs use less DOFs than the method we
developed in arXiv:1807.01524 [math.NA].
  Similar to the LSFEM developed in arXiv:1807.01524 [math.NA], both flux-only
LSFEMs can handle discontinuous solutions better than the traditional
continuous polynomial approximations. We show the existence, uniqueness, a
priori and a posteriori error estimates of the proposed methods. With adaptive
mesh refinements driven by the least-squares a posteriori error estimators, the
solution can be accurately approximated even when the mesh is not aligned with
discontinuity. The overshooting phenomenon is very mild if a piecewise constant
reconstruction of the solution is used. Extensive numerical tests are done to
show the effectiveness of the methods developed in the paper.
