The aggregated unfitted finite element method for elliptic problems
For computational scientists using unfitted methods, this work provides a practical solution to severe ill-conditioning that previously hindered large-scale simulations.
The paper tackles ill-conditioning in unfitted finite element methods by introducing a cell aggregation technique that yields condition numbers scaling as in standard body-fitted meshes, with optimal convergence rates confirmed in 2D and 3D experiments.
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.