We present a numerical method to simulate the movements of a thin valve immersed in an incompressible viscous fluid. The fluid and structure meshes are not matching: the kinematic continuity is imposed using Lagrange multipliers. The method therefore belongs to the ``Fictitious Domain'' (FD) family. This approach allows very large displacements. We propose a partitioned fluid-structure algorithm which keeps the fluid and structure solvers independent and which is able to manage contact without assuming that the structure solvers include contact capabilities. Various numerical tests are proposed. In particular, when the displacement is moderate, we show that the results are in good agreement with Arbitrary Lagrangian Eulerian (ALE) simulations. We then present results with contact on a rigid wall. Finally we show that the proposed approach can be mixed with ALE when the wall is elastic.