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 with the new stabilization term the proposed cut finite element discretization results in 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 computational mesh into macro-elements and applying ghost penalty terms only on interior edges of macro-elements. 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 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：我们研究基于混合有限元对$\textbf{RT}_k\times Q_k$（$k\geq 0$）的Darcy界面问题切割有限元离散格式，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，为提升稳定性和控制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化项会破坏所考虑单元对的散度自由性质。为此，我们针对压力提出新型稳定化项，证明该方案在保持系统矩阵条件数可控的前提下，可恢复散度的最优逼近精度。我们证明采用新型稳定化项的切割有限元离散格式能够获得螺线管速度场的逐点散度自由逼近。基于$\textbf{RT}_k\times Q_k$（$k\geq 0$）推导了所提非拟合有限元离散格式的先验误差估计。此外，通过将计算网格分解为宏单元并仅在宏单元内部边界施加鬼罚项，采用$\textbf{RT}_0\times Q_0$、$\textbf{RT}_1\times Q_1$及$\textbf{BDM}_1\times Q_0$（其中$\textbf{BDM}$为Brezzi-Douglas-Marini空间）单元对的数值实验表明：新方法具备以下特性：1）近似速度和压力具有最优收敛阶；2）系统矩阵条件数尺度与拟合有限元离散格式相当，确保线性系统适定性；3）螺线管速度场的逐点散度自由逼近可使近似散度达到最优收敛阶。上述三种性质均与界面相对于计算网格的位置无关。