  In this work, we consider the modeling of inclusions in the material using an
unfitted finite element method. In the unfitted methods, structured background
meshes are used and only the underlying finite element space is modified to
incorporate the discontinuities, such as inclusions. Hence, the unfitted
methods provide a more flexible framework for modeling the materials with
multiple inclusions. We employ the method of Lagrange multipliers for enforcing
the interface conditions between the inclusions and matrix, this gives rise to
the linear system of equations of saddle point type. We utilize the Uzawa
method for solving the saddle point system and propose preconditioning
strategies for primal and dual systems.
  For the dual systems, we review and compare the preconditioning strategies
that are developed for FETI and SIMPLE methods. While for the primal system, we
employ a tailored multigrid method specifically developed for the unfitted
meshes. Lastly, the comparison between the proposed preconditioners is made
through several numerical experiments.
