  Finite element methods and kinematically coupled schemes that decouple the
fluid velocity and structure displacement have been extensively studied for
incompressible fluid-structure interaction (FSI) over the past decade. While
these methods are known to be stable and easy to implement, optimal error
analysis has remained challenging. Previous work has primarily relied on the
classical elliptic projection technique, which is only suitable for parabolic
problems and does not lead to optimal convergence of numerical solutions to the
FSI problems in the standard $L^2$ norm. In this article, we propose a new
stable fully discrete kinematically coupled scheme for incompressible FSI
thin-structure model and establish a new approach for the numerical analysis of
FSI problems in terms of a newly introduced coupled non-stationary Ritz
projection, which allows us to prove the optimal-order convergence of the
proposed method in the $L^2$ norm. The methodology presented in this article is
also applicable to numerous other FSI models and serves as a fundamental tool
for advancing research in this field.
