  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 show the good performance of the proposed preconditioners.
