  In this paper, we study the least-squares finite element methods (LSFEM) for
the linear hyperbolic transport equations. The linear transport equation
naturally allows discontinuous solutions and discontinuous inflow conditions,
while the normal component of the flux across the mesh faces needs to be
continuous. Traditional LSFEMs using continuous finite element approximations
will introduce unnecessary extra error for discontinuous solutions and boundary
conditions. In order to separate the continuity requirements, a new flux
variable is introduced. With this reformulation, the continuities of the flux
and the solution can be handled separately in natural
$H(\mbox{div};\Omega)\times L^2(\Omega)$ conforming finite element spaces.
Several variants of the methods are developed to handle the inflow boundary
condition strongly or weakly.
  With the reformulation, the new LSFEMs can handle discontinuous solutions and
boundary conditions much better than the traditional LSFEMs with continuous
polynomial approximations. With least-squares functionals as a posteriori error
estimators, the adaptive methods can naturally identify error sources including
singularity and non-matching discontinuity. For discontinuity aligned mesh, no
extra error is introduced. If an $RT_0 \times P_0$ pair is used to approximate
the flux and the solution, the new adaptive LSFEMs can approximate
discontinuous solutions with almost no overshooting even when the mesh is not
aligned with discontinuity.
  Existence and uniqueness of the solutions and a priori and a posteriori error
estimates are established for the proposed methods. Extensive numerical tests
are performed to show the effectiveness of the methods developed in the paper.
