Numerical methods for incompressible miscible flow in porous media have been studied extensively in the last several decades. In practical applications, the lowest‐order Galerkin‐mixed method is the most popular one, where the linear Lagrange element is used for the concentration and the lowest order Raviart–Thomas mixed element pair is used for the Darcy velocity and pressure. The existing error estimate of the method in L²‐norm is in the order in spatial direction, which however is not optimal and valid only under certain extra restrictions on both time step and spatial meshes, excluding the most commonly used mesh h = h_p = h_c. This paper focuses on new and optimal error estimates of a linearized Crank–Nicolson lowest‐order Galerkin‐mixed finite element method (FEM), where the second‐order accuracy for the concentration in both time and spatial directions is established unconditionally. The key to our optimal error analysis is an elliptic quasi‐projection. Moreover, we propose a simple one‐step recovery technique to obtain a new numerical Darcy velocity and pressure of second‐order accuracy. Numerical results for both two and three‐dimensional models are provided to confirm our theoretical analysis.

中文翻译：

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多孔介质中不可压缩混溶流动的Crank–Nicolson最低阶Galerkin混合有限元最优误差分析

在过去的几十年中，对多孔介质中不可压缩混溶流动的数值方法进行了广泛的研究。在实际应用中，最低阶的Galerkin混合方法是最受欢迎的方法，其中线性拉格朗日元素用于浓度，最低阶的Raviart–Thomas混合元素对用于达西速度和压力。L ²范数中该方法的现有误差估计在空间方向上是按顺序排列的，但是，这不是最佳的，并且仅在时间步和空间网格都受到某些额外限制的情况下才有效，除了最常用的网格h  =  h _p  =  h _c。本文重点研究线性化的Crank-Nicolson最低阶Galerkin混合有限元方法（FEM）的新的和最佳误差估计，其中无条件建立浓度和时间方向的二阶精度。最优误差分析的关键是椭圆准投影。此外，我们提出了一种简单的一步恢复技术来获得新的达西速度数值和二阶精度压力。提供了二维和三维模型的数值结果，以证实我们的理论分析。