Robust and scalable domain decomposition solvers for unfitted finite element methods

Abstract

Unfitted finite element methods, e.g., extended finite element techniques or the so-called finite cell method, have a great potential for large scale simulations, since they avoid the generation of body-fitted meshes and the use of graph partitioning techniques, two main bottlenecks for problems with non-trivial geometries. However, the linear systems that arise from these discretizations can be much more ill-conditioned, due to the so-called small cut cell problem. The state-of-the-art approach is to rely on sparse direct methods, which have quadratic complexity and are thus not well suited for large scale simulations. In order to solve this situation, in this work we investigate the use of domain decomposition preconditioners (balancing domain decomposition by constraints) for unfitted methods. We observe that a straightforward application of these preconditioners to the unfitted case has a very poor behavior. As a result, we propose a customization of the classical BDDC methods based on the stiffness weighting operator and an improved definition of the coarse degrees of freedom in the definition of the preconditioner. These changes lead to a robust and algorithmically scalable solver able to deal with unfitted grids. A complete set of complex 3D numerical experiments shows the good performance of the proposed preconditioners.