  In this paper, we propose a linear and monolithic finite element method for
the approximation of an incompressible viscous fluid interacting with an
elastic and deforming plate. We use the arbitrary Lagrangian-Eulerian (ALE)
approach that works in the reference domain, meaning that no re-meshing is
needed during the numerical simulation. For time discretization, we employ the
backward Euler method. For space discretization, we respectively use P1-bubble,
P1, and P1 finite elements for the approximation of the fluid velocity,
pressure, and structure displacement. We show that our method fulfills the
geometrical conservation law and dissipates the total energy on the discrete
level. Moreover, we prove the (optimal) linear convergence with respect to the
sizes of the time step $\tau$ and the mesh $h$. We present numerical
experiments involving a substantially deforming fluid domain that do validate
our theoretical results. A comparison with a fully implicit (thus nonlinear)
scheme indicates that our semi-implicit linear scheme is faster and as accurate
as the fully implicit one, at least in stable configurations.
