The work summarized in this memoir stem from one of the following two topics: the modeling and numerical simulation of coupled systems (Chapters 1-3) and the stabilized finite element methods for transient problems (Chapter 4). These works are mainly motivated by the study of the aeroelastic stability of civil engineering structures and the numerical simulation of blood flow and of cardiac electrophysiology. As regards fluid-structure interaction, we couple the Navier-Stokes equation in moving domains with the nonlinear elastodynamics equation. We investigate the stability of the equilibrium states of the system via an analysis of the harmonic solutions of a specific linear problem. In the context of the temporal simulation, we propose an exact Newton method for the solution of implicit coupling schemes. Then we address the following question: how to avoid strong coupling without compromising stability? This issue is addressed through two different perspectives: through projection based semi-implicit coupling, and with an appropriate weak treatment of the coupling conditions. We also address the numerical simulation of the ECG using a 3D mathematical model fully based on PDE/ODE. The main ingredients of this model are: phenomenological cell dynamics, bidomain equation (in the heart) and generalized Laplace equation (in the torso). Other key aspects of the modeling are highlighted, which allows us to simulate realistic 12-lead ECGs. Some time discretization schemes for the bidomain equation and the heart-torso system are analyzed. Finally, we generalize the continuous interior penalty method to the Oseen problem and the transient Navier-Stokes equation. A priori error bounds uniform on the viscosity coefficient are provided for arbitrary equal-order velocity/pressure approximations. We present an abstract error analysis for symmetric stabilization methods applied to the transient Stokes and the transient reaction-advection-diffusion equations. For Stokes we show that the small time-step stability can be removed by a suitable choice of the initial velocity approximation. For reaction-advection-diffusion we address the problem of the stencil increase (introduced by the stabilization operator) via an explicit treatment of the stabilization.
