A stabilized Nitsche cut finite element method for the Oseen problem

TL;DR

This paper introduces a stabilized Nitsche cut finite element method for the Oseen problem that handles complex geometries with boundaries cutting through mesh elements, ensuring stability and optimal error estimates across various Reynolds numbers.

Contribution

It develops a novel stabilized Nitsche-based cut finite element formulation with CIP-like ghost penalties for the Oseen problem, providing stability and accuracy regardless of boundary cuts.

Findings

Method is inf-sup stable and achieves optimal error estimates.

Numerical examples confirm theoretical stability and accuracy.

Successfully applied to transient Navier-Stokes equations on complex geometries.

Abstract

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.