We study cut finite element discretizations of a Darcy interface problem based on the mixed finite element pairs $\textbf{RT}_0\times Q_0$, $\textbf{BDM}_1\times Q_0$, and $\textbf{RT}_1\times Q_1$. Here $Q_k$ is the space of discontinuous polynomial functions of degree k, $\textbf{RT}_{k}$ is the Raviart-Thomas space, and $\textbf{BDM}_k$ is the Brezzi-Douglas-Marini space. 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, destroys the divergence-free property of the considered element pairs. 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. We derive a priori error estimates for the proposed unfitted finite element discretization based on $\textbf{RT}_k\times Q_k$, $k\geq 0$. Numerical experiments indicate that with the new method we have 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 it does 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 to the computational mesh.

翻译：我们研究了基于混合有限元对$\textbf{RT}_0\times Q_0$、$\textbf{BDM}_1\times Q_0$和$\textbf{RT}_1\times Q_1$的达西界面问题的切割有限元离散化。其中$Q_k$是次数为k的不连续多项式函数空间，$\textbf{RT}_{k}$是Raviart-Thomas空间，$\textbf{BDM}_k$是Brezzi-Douglas-Marini空间。我们表明，在弱形式中常添加的标准鬼罚稳定化（用于切割有限元方法的稳定性并控制所得线性系统矩阵的条件数）会破坏所考虑元素对的无散度性质。因此，我们提出了对标准稳定化策略的两个修正：使用宏单元和针对压力的新稳定项。通过将计算网格分解为宏单元并仅在宏单元的内边上应用鬼罚项，稳定化仅在需要时激活。通过修改压力的标准稳定项，我们恢复了散度的最优逼近，而不会失去对线性系统矩阵条件数的控制。我们推导了基于$\textbf{RT}_k\times Q_k$（$k\geq 0$）的所提议非拟合有限元离散化的先验误差估计。数值实验表明，使用新方法我们能够获得：1）近似速度和压力的最优收敛阶；2）适定线性系统，其中系统矩阵的条件数尺度与拟合有限元离散化相同；3）近似散度的最优收敛阶，以及对无散速度场的逐点散度自由逼近。这三个性质均独立于界面相对于计算网格的位置。