One of the main difficulties that has to be faced with fictitious domain approximation of FSI with immersed thin-walled solids is related to the potential lack of mass conservation across the interface. This is a consequence of the fact that the discrete pressure does not allow for strong discontinuities across the interface. Different approaches have been proposed in the literature to circumvent this issue. Penalized grad-div interfacial stabi- lization is known to enhance mass conservation across the interface, but at the price of degradation of the system matrix conditioning, which drastically limits the applicability of the method. Enhancing mass conservation in fictitious domain methods by using globally discontinuous pressure is an alternative to XFEM, but inf-sup stability prevents the use of low order elements for the velocity, unless stabilization terms are introduced that com- promise local mass conservation. Similar observations can be made on the combination of unfitted mesh methods and divergence free approximations, with the exception of the cut-FEM method reported in [1]. In this talk, we will present a low order fictitious domain stabilized finite element method [2], which mitigates the above mentioned issue with the addition of a single velocity constraint (in the spirit of [3]). We provide a complete a priori numerical analysis of the method under minimal regularity assumptions. A com- prehensive numerical study illustrates the capabilities of the proposed method, including comparisons with alternative fitted and unfitted mesh methods. REFERENCES [1] E. Burman, P. Hansbo, and M. G. Larson. Cut finite element method for divergence free approximation of incompressible flow: optimal error estimates and pressure in- dependence, 2022. arXiv:2207.04734. [2] D. Corti, G. Delay, M.A. Fern ́andez, F. Vergnet, M. Vidrascu. Low-order fictitious domain method with enhanced mass conservation for an interface Stokes problem. ESAIM: M2AN, 2024, In press. [3] K. Ohmori and N. Saito. Flux-free finite element method with Lagrange multipliers for two-fluid flows. J. Sci. Comput., 32(2):147–173, 2007.