Stability and convergence analysis of the kinematically coupled scheme and its extensions for the fluid-structure interaction

In this work we analyze the stability and convergence properties of a loosely-coupled scheme, called the kinematically coupled scheme, and its extensions for the interaction between an incompressible, viscous fluid and a thin, elastic structure. We consider a benchmark problem where the structure is modeled using a general thin structure model, and the coupling between the fluid and structure is linear. We derive the energy estimates associated with the unconditional stability of an extension of the kinematically coupled scheme, called the $\beta$-scheme. Furthermore, for the first time we present \textit{a priori} estimates showing optimal, first-order in time convergence in the case when $\beta=1$. We further discuss the extensions of our results to other fluid-structure interaction problems, in particular the fluid-thick structure interaction problem. The theoretical stability and convergence results are supported with numerical examples.