We discuss the numerical approximation of fluid-structure interaction (FSI) problems with a particular concern on two hemodynamics applications. For the first application, blood is modeled as an incompressible fluid and it interacts with an elastic wall containing a geometric orifice which mimics a leaky mitral valve. The goal is to simulate hemodynamics conditions encountered in patients with mitral regurgitation in order to show strengths and limitations of echocardiographic methods to assess the severity of the pathology. To solve the coupled problem, we propose a monolithic approach which consists in preconditioning the linear system obtained after the space-time discretization and linearization of the FSI problem with a suitable diagonal scaling combined with an ILU preconditioner. For the second application, this algorithm is extend to allow the presence of structures with different mechanical properties with the aim of testing different configurations for the Left Ventricle Assist Device aortic anastomosis. In the proposed method, fluid-structure and structure-structure interactions are treated in the same way.