  In this work, we develop a new algorithm to solve large-scale incompressible
time-dependent fluid--structure interaction (FSI) problems using a matrix-free
finite element method in arbitrary Lagrangian--Eulerian (ALE) frame of
reference. We derive a semi-implicit time integration scheme which improves the
geometry-convective explicit (GCE) scheme for problems involving the
interaction between incompressible hyperelastic solids and incompressible
fluids. The proposed algorithm relies on the reformulation of the time-discrete
problem as a generalized Stokes problem with strongly variable coefficients,
for which optimal preconditioners have recently been developed. The resulting
algorithm is scalable, optimal, and robust: we test our implementation on model
problems that mimic classical Turek benchmarks in two and three dimensions, and
investigate timing and scalability results.
