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 with continuous normal component of the flux across the mesh interfaces. Continuous finite element spaces are used to approximate the solution in traditional LSFEMs. This will introduce unnecessary error and serious overshooting. In Liu and Zhang (Comput Methods Appl Mech Eng 366:113041, 2020), 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 Liu and Zhang (2020). Similar to the LSFEM developed in Liu and Zhang (2020), 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.

本文针对线性双曲输运问题，开发了两种仅通量最小二乘有限元方法（LSFEM）。输运方程通常具有不连续的解和不连续的流入边界条件，但具有穿过网格界面的通量连续的法线分量。连续有限元空间用于逼近传统LSFEM中的解决方案。这将引入不必要的错误和严重的超调。在Liu和Zhang（Comput Methods Appl Mech Eng 366：113041，2020）中，我们通过引入新的通量变量来分离通量和解的连续性要求，从而重新公式化方程。意识到Raviart-Thomas混合元素空间具有足够的自由度来近似通量及其发散，我们从系统中消除了该解决方案，并获得了两种仅助焊剂的配方，并开发了相应的LSFEM。然后，使用溶液与通量的关系，通过简单的后处理方法回收溶液。这两种仅通量的LSFEMs使用的自由度比我们在Liu和Zhang（2020）中开发的方法要少。与在Liu和Zhang（2020）中开发的LSFEM相似，这两种仅通量的LSFEM都比传统的连续多项式逼近方法更好地处理不连续解。我们显示了所提出的方法的存在性，唯一性，先验和后验误差估计。通过由最小二乘后验误差估计器驱动的自适应网格细化，即使当网格未以不连续性对齐时，也可以精确地近似求解。如果使用溶液的分段恒定重建，则超调现象非常轻微。进行了广泛的数值测试，以证明本文开发的方法的有效性。