Cut finite element method for divergence-free approximation of incompressible flow: a lagrange multiplier approach

In this note, we design a cut finite element method for a low order divergence-free element applied to a boundary value problem subject to Stokes' equations. For the imposition of Dirichlet boundary conditions, we consider either Nitsche's method or a stabilized Lagrange multiplier method. In both cases, the normal component of the velocity is constrained using a multiplier, different from the standard pressure approximation. The divergence of the approximate velocities is pointwise zero over the whole mesh domain, and we derive optimal error estimates for the velocity and pressures, where the error constant is independent of how the physical domain intersects the computational mesh, and of the regularity of the pressure multiplier imposing the divergence-free condition.