Mixed aggregated finite element methods for the unfitted discretization of the Stokes problem

TL;DR

This paper develops and analyzes aggregated finite element methods for the Stokes problem on unfitted meshes, addressing ill-conditioning and stability issues with new stabilization techniques and providing comprehensive numerical validation.

Contribution

It introduces aggregated mixed finite element spaces with stabilization for unfitted Stokes discretization, ensuring stability and optimal error estimates.

Findings

Condition number bounds are independent of small cut cells

Stability and optimal error estimates are proven theoretically

Numerical experiments confirm the effectiveness of the proposed methods

Abstract

In this work, we consider unfitted finite element methods for the numerical approximation of the Stokes problem. It is well-known that this kind of methods lead to arbitrarily ill-conditioned systems. In order to solve this issue, we consider the recently proposed aggregated finite element method, originally motivated for coercive problems. However, the well-posedness of the Stokes problem is far more subtle and relies on a discrete inf-sup condition. We consider mixed finite element methods that satisfy the discrete version of the inf-sup condition for body-fitted meshes, and analyze how the discrete inf-sup is affected when considering the unfitted case. We propose different aggregated mixed finite element spaces combined with simple stabilization terms, which can include pressure jumps and/or cell residuals, to fix the potential deficiencies of the aggregated inf-sup. We carry out a complete numerical analysis, which includes stability, optimal a priori error estimates, and condition number bounds that are not affected by the small cut cell problem. For the sake of conciseness, we have restricted the analysis to hexahedral meshes and discontinuous pressure spaces. A thorough numerical experimentation bears out the numerical analysis. The aggregated mixed finite element method is ultimately applied to two problems with non-trivial geometries.