In this talk we consider the formulation of FSI problems based on a Fictitious Domain method with a Lagrange multiplier. The time discretization of the resulting system of partial differential equations leads to the solution of a stationary problem with a saddle point structure. We present a rigorous analysis of existence, uniqueness and stability of the solution of the continuous problem and of its finite element discretization. Moreover, we introduce splitting schemes and discuss their stability and convergence properties.