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We propose a stabilized Nitsche-based cut finite element formulation for theOseen problem in which the boundary of the domain is allowed to cut through theelements of an easy-to-generate background mesh. Our formulation is based onthe continuous interior penalty (CIP) method of Burman et al. [1] whichpenalizes jumps of velocity and pressure gradients over inter-element faces tocounteract instabilities arising for high local Reynolds numbers and the use ofequal order interpolation spaces for the velocity and pressure. Since the meshdoes not fit the boundary, Dirichlet boundary conditions are imposed weakly bya stabilized Nitsche-type approach. The addition of CIP-like ghost-penalties inthe boundary zone allows to prove that our method is inf-sup stable and toderive optimal order a priori error estimates in an energy-type norm,irrespective of how the boundary cuts the underlying mesh. All appliedstabilization techniques are developed with particular emphasis on low and highReynolds numbers. Two- and three-dimensional numerical examples corroborate thetheoretical findings. Finally, the proposed method is applied to solve thetransient incompressible Navier-Stokes equations on a complex geometry.

A stabilized Nitsche cut finite element method for the Oseen problem