Goal-oriented error estimation and adaptivity for fluid–structure interaction using exact linearized adjoints

We develop duality-based a posteriori error estimates for functional outputs of solutions of steady fluid–structure-interaction problems. The crucial complication in obtaining these estimates pertains to the derivation of the coupled dual (exact linearized adjoint) problem owing to the free-boundary character of fluid–structure interaction. We present two approaches to derive the dual problem. In the domain-map linearization approach, the fluid subproblem is first transformed to a fixed reference domain, after which one essentially linearizes with respect to the domain transformation map. In the shape-linearization approach, fluid unknowns are fixed in the current configuration and a very weak formulation of the fluid subproblem is then linearized using shape-derivative techniques. We show that the dual problems correspond to coupled fluid–structure problems with nonstandard coupling conditions. Furthermore, we present numerical experiments that demonstrate the consistency of the dual-based error estimates and their usefulness in goal-oriented adaptive mesh-refinement.