Inf-sup stability of geometrically unfitted Stokes finite elements

TL;DR

This paper proves inf-sup stability for various unfitted finite element methods applied to Stokes problems on non-fitted meshes, ensuring stability and optimal error estimates in 2D and 3D.

Contribution

It establishes the inf-sup stability property for well-known unfitted Stokes finite elements, enabling reliable error analysis for these methods.

Findings

Stability holds for several 2D and 3D unfitted Stokes elements.

Provides optimal error estimates for unfitted finite element methods.

Results apply when the background mesh is shape-regular and sufficiently fine.

Abstract

The paper shows an inf-sup stability property for several well-known 2D and 3D Stokes elements on triangulations which are not fitted to a given smooth or polygonal domain. The property implies stability and optimal error estimates for a class of unfitted finite element methods for the Stokes and Stokes interface problems, such as Nitsche-XFEM or cutFEM. The error analysis is presented for the Stokes problem. All assumptions made in the paper are satisfied once the background mesh is shape-regular and fine enough.