A monolithic ALE Newton–Krylov solver with Multigrid-Richardson–Schwarz preconditioning for incompressible Fluid-Structure Interaction

•The robustness of our solver is proved for fully incompressible 2D–3D FSI problems.•A partitioned smoother to a monolithic GMRES is more robust than a partitioned solver.•An algorithm for the direct-to-steady-state FSI problem is proposed.•Our formulation solved with the MUMPS package shows a very good convergence rate. In this paper we study a monolithic Newton–Krylov solver with exact Jacobian for the solution of fully incompressible Fluid-Structure Interaction problems of either steady-state or time-dependent type. Unlike common approaches, the enforcement of the incompressibility conditions both for the fluid and for the solid parts is taken care of by using an inf-sup stable finite element pair, without stabilization terms. The Krylov solver is preconditioned using geometric multigrid with smoothers of Richardson type, in turn preconditioned by additive Schwarz algorithms. The separate solution of fluid or solid operators occurs only at the preconditioning stage of the smoother, thus guaranteeing at each level an accurate interface momentum balance. The definition of the subdomains in the Schwarz smoother is driven by the natural splitting between fluid and solid. For each part and level, the domain is subdivided into a number of minimally overlapping subdomains. Numerical investigations of two and three-dimensional benchmark tests with Newtonian fluids and nonlinear hyperelastic solids are carried out by reporting several performance indices, including condition number estimates. A robust performance of the proposed fully incompressible solver is observed, especially for the more challenging direct-to-steady-state problems.