Stability and optimal convergence of unfitted extended finite element methods with Lagrange multipliers for the Stokes equations

TL;DR

This paper analyzes and proves the stability and optimal convergence of unfitted extended finite element methods with Lagrange multipliers for solving the Stokes equations on non-fitting meshes, including various stabilization techniques.

Contribution

It generalizes and adapts existing stabilized XFEM approaches for Stokes problems, providing theoretical stability and convergence results for multiple variants.

Findings

Proved stability and optimal convergence of the methods.

Numerical tests demonstrate the effectiveness of the approaches.

Multiple stabilization strategies improve approximation quality.

Abstract

We study a fictitious domain approach with Lagrange multipliers to discretize Stokes equations on a mesh that does not fit the boundaries. A mixed finite element method is used for fluid flow. Several stabilization terms are added to improve the approximation of the normal trace of the stress tensor and to avoid the inf-sup conditions between the spaces of the velocity and the Lagrange multipliers. We generalize first an approach based on eXtended Finite Element Method due to Haslinger-Renard involving a Barbosa-Hughes stabilization and a robust reconstruction on the badly cut elements. Secondly, we adapt the approach due to Burman-Hansbo involving a stabilization only on the Lagrange multiplier. Multiple choices for the finite elements for velocity, pressure and multiplier are considered. Additional stabilization on pressure (Brezzi-Pitk\"aranta, Interior Penalty) is added, if needed. We prove the stability and the optimal convergence of several variants of these methods under appropriate assumptions. Finally, we perform numerical tests to illustrate the capabilities of the methods.