Accurate modeling of moving boundaries and interfaces is a difficulty present in many situations of computational mechanics. We use a new approach, X-Mesh, to simulate with the finite element method the interaction between two immiscible flows while keeping an accurate and sharp description of the interface without remeshing. The surface tension between the two fluids is added thanks to the ghost fluid method which when coupled with X-Mesh allows us to be exactly sharp in the pressure jump. This reduces the parasitic currents due to the surface tension down to the numerical precision.