We develop two unfitted finite element methods for the Stokes equations based on $\mathbf{H}^{\text{div}}$-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 formulation 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 resulting discrete problems are stable and the divergence-free property of the $\mathbf{H}^{\text{div}}$-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：我们针对基于$\mathbf{H}^{\text{div}}$相容有限元的斯托克斯方程，发展了两种非贴体有限元方法。第一种方法采用Brezzi-Douglas-Marini单元对斯托克斯方程进行切割有限元离散，并通过内部惩罚项强制网格内部边界的速度切向连续性。第二种方法基于包含涡量、速度和压力的三场形式斯托克斯问题，采用Raviart-Thomas空间离散速度场，并构建了切割有限元离散格式。针对两种方法，我们引入混合虚拟惩罚稳定项，使得离散问题保持稳定，同时保证非贴体网格中$\mathbf{H}^{\text{div}}$相容单元的无散度特性得以保留。通过数值比较两种方法，结果表明：无论计算边界与网格的相对位置如何，两种方法均能获得鲁棒的离散问题、速度场的最优收敛阶以及逐点无散度的速度场。