A CutFEM divergence-free discretization for the Stokes problem

TL;DR

This paper presents a novel CutFEM discretization for the Stokes problem that ensures divergence-free velocity approximation on unfitted meshes, incorporating stabilization techniques for improved accuracy and stability.

Contribution

It introduces a divergence-free CutFEM scheme with boundary penalization and ghost penalty stabilization, providing stability and optimal error estimates for unfitted mesh discretizations.

Findings

Scheme is stable under small and anisotropic cuts.

Velocity field is divergence-free outside an O(h) neighborhood of the boundary.

Grad-div stabilization allows heavy penalty scaling while maintaining optimal error rates.

Abstract

We construct and analyze a CutFEM discretization for the Stokes problem based on the Scott-Vogelius pair. The discrete piecewise polynomial spaces are defined on macro-element triangulations which are not fitted to the smooth physical domain. Boundary conditions are imposed via penalization through the help of a Nitsche-type discretization, whereas stability with respect to small and anisotropic cuts of the bulk elements is ensured by adding local ghost penalty stabilization terms. We show stability of the scheme as well as a divergence--free property of the discrete velocity outside an $O(h)$ neighborhood of the boundary. To mitigate the error caused by the violation of the divergence-free condition, we introduce local grad-div stabilization. The error analysis shows that the grad-div parameter can scale like $O(h^{-1})$, allowing a rather heavy penalty for the violation of mass conservation, while still ensuring optimal order error estimates.