A Stabilized Cut Finite Element Method for the Three Field Stokes Problem

Abstract: We propose a Nitsche-based fictitious domain method for the three field Stokes problem in which the boundary of the domain is allowed to cross through the elements of a fixed background mesh. The dependent variables of velocity, pressure and extra-stress tensor are discretised on the background mesh using linear finite elements. This equal order approximation is stabilized using a continuous interior penalty (CIP) method. On the unfitted domain boundary, Dirichlet boundary conditions are weakly enforced using Nitsche's method. We add CIP-like ghost penalties in the boundary region and prove that our scheme is inf-sup stable and that it has optimal convergence properties independent of how the domain boundary intersects the mesh. Additionally, we demonstrate that the condition number of the system matrix is bounded independently of the boundary location. We corroborate our theoretical findings with numerical examples.