We study cut finite element discretizations of a Darcy interface problem based on the mixed finite element pairs $\textbf{RT}_0\times\text{P}_0$, $\textbf{BDM}_1\times \text{P}_0$, and $\textbf{RT}_1\times \text{P}_1$. We show that the standard ghost penalty stabilization, often added in the weak forms of cut finite element methods for stability and control of the condition number of the resulting linear system matrix, pollutes the computed velocity field so the divergence-free property of the considered elements is lost. Therefore, we propose two corrections to the standard stabilization strategy; using macro-elements and new stabilization terms for the pressure. By decomposing the computational mesh into macro-elements and applying ghost penalty terms only on interior edges of macro-elements, stabilization is active only where needed. By modifying the standard stabilization terms for the pressure we recover the optimal approximation of the divergence without losing control of the condition number of the linear system matrix. Numerical experiments indicate that with the new stabilization terms the unfitted finite element discretization, for the given element pairs, results in 1) optimal rates of convergence of the approximate velocity and pressure; 2) well-posed linear systems where the condition number of the system matrix scales as for fitted finite element discretizations; 3) optimal rates of convergence of the approximate divergence with pointwise divergence-free approximations of solenoidal velocity fields. All three properties hold independently of how the interface is positioned relative the computational mesh.

翻译：我们根据混合的有限元素配对 $\ textbf{RT ⁇ 0\time\ text{P ⁇ 0$, $\ textbf{BD ⁇ 1\time\ text{P ⁇ 0$, $ textbf{RT ⁇ 1\time\ text{P ⁇ 1美元, 以及$ textbf{RT ⁇ 1\time\time\ text{P ⁇ 1$。 我们研究根据混合的有限元素配对 $\ textbf{RT{RT ⁇ 0\time\ text{P ⁇ 0\ text{P ⁇ 0}, 削减一定数量, 标准幽灵罚款稳定化, 通常以较弱的形式添加, 用于稳定并控制由此产生的线性系统界面的条件数量, 污染计算计算速度, 这样, 我们建议对标准的稳定化策略进行两次修正, 从而丧失考虑的不偏差特性属性。 因此, 我们建议对标准的稳定战略进行两次修正; 使用宏观指数和新的稳定度的离差程度的离差值, 3 线系统比值的精确度, 保证最接近的离差的离差值的精确的系统。