Divergence-free cut finite element methods for Stokes flow

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.