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:我们根据混合的离差元素对离差元素对达西界面问题进行切分的限定元素。 根据混合的离差元素对数 $\ textbf{RT ⁇ 0\time $0$, $\ textbf{BDM}1\time $0$, $ textbf{RT}1\time $1美元, 和 $ textbf{RT} k$, 我们根据混合的离差元素对达西界面问题进行切分解。 我们显示标准幽灵罚款稳定化, 通常在削减的有限元素方法中增加, 以稳定并控制由此产生的线性系统矩阵的条件数量。 因此, 我们建议对标准的稳定化战略进行两次更正; 使用宏观的不偏差和新的稳定化条件。 通过将计算模型分解成宏观的离差, 并且只对内部的离差值的离差数值, 以内部的离差变差值 3 的离差值 。 我们的离差值 稳定度 的数值 值 值 值 值 值 值 值 值 值 值 的 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值 值