A Stabilized Cut Finite Element Method for the Three Field Stokes Problem

TL;DR

This paper introduces a stabilized Nitsche-based fictitious domain finite element method for the three field Stokes problem, ensuring stability and optimal convergence regardless of boundary-mesh intersection.

Contribution

It develops a novel stabilized unfitted finite element scheme for the three field Stokes problem with proven stability and convergence properties.

Findings

The scheme is inf-sup stable.

Optimal convergence is achieved independent of boundary intersection.

The condition number of the system matrix is bounded regardless of boundary location.

Abstract

We propose a Nitsche-based fictitious domain method for the three field Stokes problem in which the boundary of the domain is allowed to cross through the elements of a fixed background mesh. The dependent variables of velocity, pressure and extra-stress tensor are discretised on the background mesh using linear finite elements. This equal order approximation is stabilized using a continuous interior penalty (CIP) method. On the unfitted domain boundary, Dirichlet boundary conditions are weakly enforced using Nitsche's method. We add CIP-like ghost penalties in the boundary region and prove that our scheme is inf-sup stable and that it has optimal convergence properties independent of how the domain boundary intersects the mesh. Additionally, we demonstrate that the condition number of the system matrix is bounded independently of the boundary location. We corroborate our theoretical findings with numerical examples.