We develop two unfitted finite element methods for the Stokes equations based on Hdiv-conforming finite elements. The first method is a cut finite element discretization of the Stokes equations based on the Brezzi- Douglas-Marini elements and involves interior penalty terms to enforce tangential continuity of the velocity at interior edges in the mesh. The second method is a cut finite element discretization of a three-field for- mulation of the Stokes problem involving the vorticity, velocity, and pressure and uses the Raviart-Thomas space for the velocity. We present mixed ghost penalty stabilization terms for both methods so that the re- sulting discrete problems are stable and the divergence-free property of the Hdiv-conforming elements is preserved also for unfitted meshes. We compare the two methods numerically. Both methods exhibit robust discrete problems, optimal convergence order for the velocity, and pointwise divergence-free velocity fields, independently of the position of the boundary relative to the computational mesh.

翻译：我们基于Hdiv协调有限元发展了两种适用于Stokes方程的非拟合有限元方法。第一种方法采用Brezzi-Douglas-Marini单元构建Stokes方程的切割有限元离散格式，通过引入内部惩罚项确保网格内部边处速度的切向连续性。第二种方法则对包含涡量、速度和压力的Stokes问题三场方程进行切割有限元离散，速度场采用Raviart-Thomas空间。针对两种方法，我们提出混合鬼影罚稳定化项，使得离散问题在非拟合网格上仍保持稳定性，并保留Hdiv协调单元的无散度特性。通过数值实验对比两种方法，结果表明：两种方法均能获得鲁棒的离散系统、速度场的最优收敛阶以及逐点无散速度场，且这些性质与计算网格边界位置无关。