Augmented Lagrangian preconditioners for fictitious domain solvers

The fictitious domain approach is a powerful technique for solving partial differential equations posed on complicated and possibly moving domains by embedding the physical domain of interest into a simpler computational domain. We present a family of augmented Lagrangian-based preconditioners for efficiently solving multiple saddle-point linear systems arising from the finite element discretization of fictitious domain formulations. We provide a detailed spectral analysis of the proposed preconditioners, deriving lower and upper bounds for the eigenvalues of the preconditioned system, and showing their independence with respect to discretization parameters when ideal versions of the preconditioners are employed. The robustness and efficiency of the preconditioners, when used in combination with flexible GMRES, are validated through extensive numerical experiments in both two and three dimensions and with different geometries. Cheaper and modified variants of these preconditioners are also considered, in order to reduce the application cost. In addition, we extend our approach to the more challenging case of fluid-structure interaction (FSI) problems, demonstrating that the present methodology remains robust and can effectively handle such complex scenarios. The relevant computational aspects related to the memory distributed implementation (based on the C++ finite element library DEAL.II) of these methodologies will be discussed.