A stabilized Nitsche cut finite element method for the Oseen problem
This work provides a robust numerical method for fluid flow problems on complex geometries without requiring body-fitted meshes, which is valuable for computational fluid dynamics practitioners.
The paper proposes a stabilized Nitsche cut finite element method for the Oseen problem that handles boundaries cutting through background mesh elements, achieving inf-sup stability and optimal error estimates regardless of boundary cuts. Numerical examples in 2D and 3D confirm the theory, and the method is applied to transient Navier-Stokes equations on complex geometries.
We propose a stabilized Nitsche-based cut finite element formulation for the Oseen problem in which the boundary of the domain is allowed to cut through the elements of an easy-to-generate background mesh. Our formulation is based on the continuous interior penalty (CIP) method of Burman et al. [1] which penalizes jumps of velocity and pressure gradients over inter-element faces to counteract instabilities arising for high local Reynolds numbers and the use of equal order interpolation spaces for the velocity and pressure. Since the mesh does not fit the boundary, Dirichlet boundary conditions are imposed weakly by a stabilized Nitsche-type approach. The addition of CIP-like ghost-penalties in the boundary zone allows to prove that our method is inf-sup stable and to derive optimal order a priori error estimates in an energy-type norm, irrespective of how the boundary cuts the underlying mesh. All applied stabilization techniques are developed with particular emphasis on low and high Reynolds numbers. Two- and three-dimensional numerical examples corroborate the theoretical findings. Finally, the proposed method is applied to solve the transient incompressible Navier-Stokes equations on a complex geometry.