Divergence-free cut finite element methods for Stokes flow

Abstract: We develop two unfitted finite element methods for the Stokes equations using $H^{\text{div}}$-conforming finite elements. Both methods achieve optimal convergence for velocity, ensure pointwise divergence-free velocity fields, and produce well-posed linear systems, regardless of the boundary's position relative to the computational mesh. The first method is a cut finite element discretization of the Stokes equations based on Brezzi-Douglas-Marini (BDM) elements, incorporating interior penalty terms to enforce tangential continuity of velocity at interior mesh edges. The second method involves a cut finite element discretization of a three-field formulation of the Stokes problem, utilizing Raviart-Thomas (RT) space for velocity. We introduce mixed ghost penalty stabilization terms for both methods to ensure stability and to preserve the divergence-free property of the $H^{\text{div}}$-conforming elements, even on unfitted meshes. Boundary conditions in both methods are imposed weakly, which presents challenges: 1) The divergence-free property of the RT and BDM finite elements may be compromised depending on how the normal component of the velocity field at the boundary is imposed. 2) Pressure robustness is influenced by the accuracy of boundary condition enforcement and may fail even if the incompressibility condition holds pointwise. We explore two approaches for weakly imposing the normal component of the boundary velocity: using a penalty parameter with Nitsche's method or a Lagrange multiplier method. We demonstrate that specific conditions on the velocity space are necessary when employing Nitsche's method or penalty. While pressure robustness can be maintained with both approaches by minimizing boundary errors, this comes at the cost of increased condition numbers in the resulting linear systems, whether the mesh is fitted or unfitted to the boundary.