The aggregated unfitted finite element method for elliptic problems

TL;DR

This paper introduces an aggregated unfitted finite element method that improves the conditioning of linear systems in elliptic problems, enabling efficient large-scale computations with optimal convergence.

Contribution

It presents a novel cell aggregation technique that enhances unfitted finite element spaces, ensuring condition numbers scale like standard methods and facilitating practical applications.

Findings

Condition number scales as in standard finite element methods.

Method achieves optimal finite element convergence order.

Numerical experiments confirm theoretical results in 2D and 3D.

Abstract

Unfitted finite element techniques are valuable tools in different applications where the generation of body-fitted meshes is difficult. However, these techniques are prone to severe ill conditioning problems that obstruct the efficient use of iterative Krylov methods and, in consequence, hinders the practical usage of unfitted methods for realistic large scale applications. In this work, we present a technique that addresses such conditioning problems by constructing enhanced finite element spaces based on a cell aggregation technique. The presented method, called aggregated unfitted finite element method, is easy to implement, and can be used, in contrast to previous works, in Galerkin approximations of coercive problems with conforming Lagrangian finite element spaces. The mathematical analysis of the new method states that the condition number of the resulting linear system matrix scales as in standard finite elements for body-fitted meshes, without being affected by small cut cells, and that the method leads to the optimal finite element convergence order. These theoretical results are confirmed with 2D and 3D numerical experiments.