We study cut finite element discretizations of a Darcy interface problem based on the mixed finite element pairs $\textbf{RT}_k\times Q_k$, $k\geq 0$. Here $Q_k$ is the space of discontinuous polynomial functions of degree less or equal to $k$ and $\textbf{RT}$ is the Raviart-Thomas 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 linear system matrix, destroys the divergence-free property of the considered element pairs. Therefore, we propose new stabilization terms for the pressure and show that we recover the optimal approximation of the divergence without losing control of the condition number of the linear system matrix. We prove that the method with the new stabilization term has pointwise divergence-free approximations of solenoidal velocity fields. We derive a priori error estimates for the proposed unfitted finite element discretization based on $\textbf{RT}_k\times Q_k$, $k\geq 0$. In addition, by decomposing the mesh into macro-elements and applying ghost penalty terms only on interior edges of macro-elements, stabilization is applied very restrictively and only where needed. Numerical experiments with element pairs $\textbf{RT}_0\times Q_0$, $\textbf{RT}_1\times Q_1$, and $\textbf{BDM}_1\times Q_0$ (where $\textbf{BDM}$ is the Brezzi-Douglas-Marini space) indicate that 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：我们研究基于混合有限元对$\textbf{RT}_k\times Q_k$（$k\geq 0$）的Darcy界面问题离散化方法，其中$Q_k$为次数不超过$k$的不连续多项式函数空间，$\textbf{RT}$为Raviart-Thomas空间。研究表明，为稳定性和控制线性系统矩阵条件数而在切割有限元弱形式中常用的标准鬼罚稳定项会破坏所选单元对的无散性质。为此，我们提出压强的新稳定项，证明该方法能在不损失线性系统矩阵条件数控制的前提下恢复散度的最优逼近。我们证明，采用新稳定项的方法可对螺线管速度场实现逐点无散逼近。基于$\textbf{RT}_k\times Q_k$（$k\geq 0$）导出所提非拟合有限元离散化的先验误差估计。此外，通过将网格分解为宏单元并仅在宏单元内部边上施加鬼罚项，稳定化处理得以严格限制在必要区域。采用$\textbf{RT}_0\times Q_0$、$\textbf{RT}_1\times Q_1$和$\textbf{BDM}_1\times Q_0$（其中$\textbf{BDM}$为Brezzi-Douglas-Marini空间）单元对的数值实验表明：1）速度和压强逼近达到最优收敛阶；2）线性系统适定，系统矩阵条件数尺度与拟合有限元离散化相当；3）散度逼近达到最优收敛阶，且实现螺线管速度场的逐点无散逼近。上述三个性质均独立于界面相对于计算网格的位置。