The contribution addresses the specific physical sensitivity inherent in the interaction of slender structures like membranes or shells and instationary incompressible flows and its consequences for numerical solution schemes. Different partitioned strong coupling approaches are presented and compared with respect to their efficiency. In addition to the classical block Gauss–Seidel iteration and a Newton–Krylov approach, a two-level coupling scheme is introduced. It is based on a coarse grid predictor followed by the usual iterative fine-scale solution. The performance of the three coupling schemes is investigated applying a two-dimensional model problem of a membrane roof. Three-dimensional numerical examples demonstrate the mechanical and numerical sensitivity for these especially delicate fluid–structure interaction problems. Copyright © 2010 John Wiley & Sons, Ltd.