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 unfitted finite element discretization with the new stabilization terms. 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 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Darcy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。我们证明，标准鬼罚稳定化（为稳定性和控制所得线性系统矩阵条件数，常在切割有限元方法的弱形式中添加）会破坏所选单元对的无散度性质。因此，我们提出了两种对标准稳定化策略的修正：使用宏单元和针对压力的新稳定化项。通过将计算网格分解为宏单元，并仅在宏单元的内部边上施加鬼罚项，稳定化仅在必要时生效。通过修改压力的标准稳定化项，我们在不失去对线性系统矩阵条件数控制的情况下恢复了散度的最优逼近。我们推导了带新稳定化项的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实现：1）近似速度和压力的最优收敛速率；2）适定的线性系统，其系统矩阵条件数与拟合有限元离散化相当；3）近似散度的最优收敛速率，且散度场近似为无散速度场的逐点无散度逼近。这三个性质均独立于界面相对于计算网格的位置。