  In this paper we study a monolithic Newton-Krylov solver with exact Jacobian
for the solution of incompressible FSI problems. A main focus of this work is
on the use of geometric multigrid preconditioners with modified Richardson
smoothers preconditioned by an additive Schwarz algorithm. The definition of
the subdomains in the Schwarz smoother is driven by the natural splitting
between fluid and solid. The monolithic approach guarantees the automatic
satisfaction of the stress balance and the kinematic conditions across the
fluid-solid interface. The enforcement of the incompressibility conditions both
for the fluid and for the solid parts is taken care of by using inf-sup stable
finite element pairs without stabilization terms. A suitable Arbitrary
Lagrangian Eulerian (ALE) operator is chosen in order to avoid mesh
entanglement while solving for large displacements of the moving fluid domain.
Numerical results of two and three-dimensional benchmark tests with Newtonian
fluids and nonlinear hyperelastic solids show a robust performance of our fully
incompressible solver especially for the more challenging
direct-to-steady-state problems.
