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Unfitted finite element methods, e.g., extended finite element techniques orthe so-called finite cell method, have a great potential for large scalesimulations, since they avoid the generation of body-fitted meshes and the useof graph partitioning techniques, two main bottlenecks for problems withnon-trivial geometries. However, the linear systems that arise from thesediscretizations can be much more ill-conditioned, due to the so-called smallcut cell problem. The state-of-the-art approach is to rely on sparse directmethods, which have quadratic complexity and are thus not well suited for largescale simulations. In order to solve this situation, in this work weinvestigate the use of domain decomposition preconditioners (balancing domaindecomposition by constraints) for unfitted methods. We observe that astraightforward application of these preconditioners to the unfitted case has avery poor behavior. As a result, we propose a {customization} of the classicalBDDC methods based on the stiffness weighting operator and an improveddefinition of the coarse degrees of freedom in the definition of thepreconditioner. These changes lead to a robust and algorithmically scalablesolver able to deal with unfitted grids. A complete set of complex 3D numericalexperiments show the good performance of the proposed preconditioners.

Robust and scalable domain decomposition solvers for unfitted finite element methods