SIAM Journal on Numerical Analysis, Volume 59, Issue 4, Page 1875-1895, January 2021.
The time discrete scheme of characteristics type is especially effective for convection-dominated diffusion problems. The scheme has been used in various engineering areas with different approximations in spatial direction. The lowest-order mixed method is the most popular one for miscible flow in porous media. The method is based on a linear Lagrange approximation to the concentration and the zero-order Raviart--Thomas approximation to the pressure/velocity. However, the analysis for the lowest-order characteristics-mixed finite element method (FEM) has not been well done, although significant effort has been made in the last several decades. In all previous works, only first-order accuracy in spatial direction was proved under certain time-step and mesh size restrictions. The main purpose of this paper is to establish optimal error estimates, i.e., the second order in the $L^2$-norm for the concentration and the first order for the pressure/velocity, while the concentration is the more important physical component. For this purpose, an elliptic quasi-projection is introduced in our analysis to clean up the pollution of the numerical velocity through the nonlinear dispersion-diffusion tensor and the concentration-dependent viscosity. Moreover, the numerical pressure/velocity of the second-order accuracy can be obtained by re-solving the (elliptic) pressure equation at a given time level with a higher-order approximation. Numerical results presented in this paper confirm our theoretical analysis.

中文翻译：

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多孔介质中不可压缩混相流动的最低阶特征混合有限元分析

SIAM 数值分析杂志，第 59 卷，第 4 期，第 1875-1895 页，2021 年 1 月。
特征类型的时间离散方案对于以对流为主的扩散问题特别有效。该方案已用于在空间方向上具有不同近似值的各种工程领域。最低阶混合方法是多孔介质中混相流动最流行的方法。该方法基于浓度的线性拉格朗日近似和压力/速度的零阶 Raviart-Thomas 近似。然而，对最低阶特征混合有限元法 (FEM) 的分析还没有很好地完成，尽管在过去的几十年里已经做出了很大的努力。在以前的所有工作中，在某些时间步长和网格大小限制下，仅证明了空间方向的一阶精度。本文的主要目的是建立最优误差估计，即。即，$L^2$-范数中的二阶浓度和一阶压力/速度，而浓度是更重要的物理分量。为此，我们在分析中引入了椭圆准投影，以通过非线性色散扩散张量和浓度相关粘度来清除数值速度的污染。此外，二阶精度的数值压力/速度可以通过用高阶近似重新求解给定时间水平的（椭圆）压力方程来获得。本文中的数值结果证实了我们的理论分析。在我们的分析中引入了椭圆准投影，以通过非线性色散扩散张量和浓度相关的粘度来清除数值速度的污染。此外，二阶精度的数值压力/速度可以通过用高阶近似重新求解给定时间水平的（椭圆）压力方程来获得。本文中的数值结果证实了我们的理论分析。在我们的分析中引入了椭圆准投影，以通过非线性色散扩散张量和浓度相关的粘度来清除数值速度的污染。此外，二阶精度的数值压力/速度可以通过用高阶近似重新求解给定时间水平的（椭圆）压力方程来获得。本文中的数值结果证实了我们的理论分析。