One of the main difficulties that has to be faced with fictitious domain approximation of fluid- structure interaction 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 stabilization 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 might compromise local mass conservation. Similar observations can be made on the combination of unfitted mesh methods and divergence free approximations, with the exception of the low-roder method reported in [1, 2]. In this talk, we will discuss two low-order fictitious domain finite element methods which circumvent the above mentioned issues. The first is based in the addition of a single velocity constraint [3] (see also [4]), while the second relies on the use of the of the low- order divergence free element reported in [1]. A priori error estimates under minimal regularity assumptions are provided. A comprehensive numerical study illustrates the capabilities of the proposed methods, including comparisons with alternative fitted and unfitted mesh methods. This is joint work with E. Burman, D. Corti, G. Delay, D. Duva, F. Vergnet, M. Vidrascu.