Divergence-free cut finite element methods for Stokes flow

We develop two unfitted cut finite element methods for the Stokes equations based on $\mathbf{H}^{\text{div}}$-conforming finite elements which exhibit optimal convergence order for the velocity, pointwise divergence-free velocity fields, and well-posed linear systems, independently of the position of the boundary relative to the computational mesh. The first method is based on the Brezzi-Douglas-Marini (BDM) elements and involves interior penalty terms to enforce tangential continuity of the velocity at interior edges in the mesh. The second method is a 3-field formulation involving the vorticity, velocity, and pressure and uses the Raviart-Thomas (RT) 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 on unfitted meshes. In both methods boundary conditions are imposed weakly. We show that imposing Dirichlet boundary conditions weakly introduces additional challenges; 1) The divergence-free property of the RT and the BDM finite elemens may be lost depending on how the normal component of the velocity field at the boundary is imposed. 2) Pressure robustness is affected by how well the boundary conditions are satisfied and may not hold even if the incompressibility condition holds pointwise. We study two approaches of weakly imposing the normal component of the velocity at the boundary; we either use a penalty parameter and Nitsche's method or a Lagrange multiplier method. We show that appropriate conditions on the velocity space has to be imposed when Nitsche's method is used. Pressure robustness can hold with both approaches by reducing the error at the boundary but this impacts the condition numbers of linear systems, independent of if the mesh is fitted or unfitted to the boundary.