A fictitious domain approach for the Stokes problem based on the extended finite element method

TL;DR

This paper introduces a fictitious domain method based on the extended finite element method for solving the Stokes problem, enabling computations in domains with non-matching boundaries and improving approximation accuracy.

Contribution

It extends the eXtended Finite Element Method to the Stokes problem, incorporating stabilization and level-set techniques for better boundary handling and interface approximation.

Findings

Convergence of the proposed method is established.

Numerical tests demonstrate the method's effectiveness.

The approach accurately captures fluid-structure interfaces.

Abstract

In the present work, we propose to extend to the Stokes problem a fictitious domain approach inspired by eXtended Finite Element Method and studied for Poisson problem in [Renard]. The method allows computations in domains whose boundaries do not match. A mixed finite element method is used for fluid flow. The interface between the fluid and the structure is localized by a level-set function. Dirichlet boundary conditions are taken into account using Lagrange multiplier. A stabilization term is introduced to improve the approximation of the normal trace of the Cauchy stress tensor at the interface and avoid the inf-sup condition between the spaces for velocity and the Lagrange multiplier. Convergence analysis is given and several numerical tests are performed to illustrate the capabilities of the method.