A divergence preserving cut finite element method for Darcy flow

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.