Analysis of Galerkin-mixed FEMs for incompressible miscible flow in porous media has been investigated extensively in the last several decades. Of particular interest in practical applications is the lowest-order Galerkin-mixed method, { in which a linear Lagrange FE approximation is used for the concentration and the lowest-order Raviart-Thomas FE approximation is used for the velocity/pressure. The previous works only showed the first-order accuracy of the method in $L^2$-norm in spatial direction,} which however is not optimal and valid only under certain extra restrictions on both time step and spatial mesh. In this paper, we provide new and optimal $L^2$-norm error estimates of Galerkin-mixed FEMs for all three components in a general case. In particular, for the lowest-order Galerkin-mixed FEM, we show unconditionally the second-order { accuracy in $L^2$-norm} for the concentration. Numerical results for both two and three-dimensional models are presented to confirm our theoretical analysis. More important is that our approach can be extended to the analysis of mixed FEMs for many strongly coupled systems to obtain optimal error estimates for all components.

中文翻译：

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多孔介质中不可压缩混相流动的 Galerkin 混合有限元新分析

在过去的几十年中，对多孔介质中不可压缩混相流动的 Galerkin 混合有限元分析进行了广泛的研究。在实际应用中特别感兴趣的是最低阶 Galerkin 混合方法，{ 其中线性拉格朗日有限元近似用于浓度，最低阶 Raviart-Thomas 有限元近似用于速度/压力。以前的工作仅显示了该方法在空间方向 $L^2$-norm 中的一阶精度，} 然而，这不是最优的，仅在时间步长和空间网格的某些额外限制下才有效。在本文中，我们在一般情况下为所有三个组件提供了 Galerkin 混合 FEM 的新的和最佳的 $L^2$-范数误差估计。特别是，对于最低阶 Galerkin 混合 FEM，我们无条件地显示了浓度的二阶 {accuracy in $L^2$-norm}。给出了二维和三维模型的数值结果以证实我们的理论分析。更重要的是，我们的方法可以扩展到许多强耦合系统的混合 FEM 分析，以获得所有组件的最佳误差估计。