We present numerical techniques for solving the problem of fluid structure interaction
with a compressible elastic material in a laminar incompressible viscous flow via
fully coupled monolithic Arbitrary Lagrangian-Eulerian (ALE) formulation. The mathematical
description and the numerical schemes are designed in such a way that more complicated
constitutive relations can be easily incorporated. The whole domain of interest is treated as
one continuum and we utilize the well known Q2P1 finite element pair for discretization in
space to gain high accuracy. We perform numerical comparisons for different time stepping
schemes, including variants of the Fractional-Step-q -scheme, Backward Euler and Crank-
Nicholson scheme for both solid and fluid parts. The resulting nonlinear discretized algebraic
system is solved by a quasi-Newton method which approximates the Jacobian matrices by the
divided differences approach and the resulting linear systems are solved by a geometric multigrid
approach. In the numerical examples, a cylinder with attached flexible beam is allowed to
freely rotate around its axis which requires a special numerical treatment. By identifying the
center of the cylinder with one grid point of the computational mesh we prescribe a Dirichlet
type boundary condition for the velocity and the displacement of the structure at this point,
which allows free rotation around this point. We present numerical studies for different problem
parameters on various mesh types and compare the results with experimental values from
a corresponding benchmarking experime