  In this work we develop a fictitious domain method for the Stokes problem
which allows computations in domains whose boundaries do not depend on the
mesh. The method is based on the ideas of Xfem and has been first introduced
for the Poisson problem. The fluid part is treated by a mixed finite element
method, and a Dirichlet condition is imposed by a Lagrange multiplier on an
immersed structure localized by a level-set function. A stabilization technique
is carried out in order to get the convergence for this multiplier. The latter
represents the forces that the fluid applies on the structure. The aim is to
perform fluid-structure simulations for which these forces have a central role.
We illustrate the capacities of the method by extending it to the
incompressible Navier-Stokes equations coupled with a moving rigid solid.
