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In this paper, we study the least-squares finite element methods (LSFEM) forthe linear hyperbolic transport equations. The linear transport equationnaturally allows discontinuous solutions and discontinuous inflow conditions,while the normal component of the flux across the mesh faces needs to becontinuous. Traditional LSFEMs using continuous finite element approximationswill introduce unnecessary extra error for discontinuous solutions and boundaryconditions. In order to separate the continuity requirements, a new fluxvariable is introduced. With this reformulation, the continuities of the fluxand 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 boundarycondition strongly or weakly.  With the reformulation, the new LSFEMs can handle discontinuous solutions andboundary conditions much better than the traditional LSFEMs with continuouspolynomial approximations. With least-squares functionals as a posteriori errorestimators, the adaptive methods can naturally identify error sources includingsingularity and non-matching discontinuity. For discontinuity aligned mesh, noextra error is introduced. If an $RT_0 \times P_0$ pair is used to approximatethe flux and the solution, the new adaptive LSFEMs can approximatediscontinuous solutions with almost no overshooting even when the mesh is notaligned with discontinuity.  Existence and uniqueness of the solutions and a priori and a posteriori errorestimates are established for the proposed methods. Extensive numerical testsare performed to show the effectiveness of the methods developed in the paper.

Adaptive Least-Squares Finite Element Methods for Linear Transport Equations Based on an H(div) Flux Reformulation