Finite element methods and kinematically coupled schemes that decouple the fluid velocity and structure's 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 kinematically coupled scheme for incompressible FSI thin-structure model and establish a new framework 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.

翻译：过去十年间，有限元方法及解耦流体速度与结构位移的运动学耦合方案在不可压缩流固耦合问题中得到了广泛研究。尽管这些方法具有稳定且易于实现的优点，但其最优误差分析始终面临挑战。现有工作主要依赖经典椭圆投影技术，该技术仅适用于抛物型问题，无法在标准$L^2$范数下实现流固耦合问题数值解的最优收敛。本文针对不可压缩流固耦合薄结构模型提出一种新型运动学耦合格式，通过引入全新的耦合非稳态Ritz投影建立了流固耦合问题数值分析的新框架，从而证明了所提方法在$L^2$范数下的最优阶收敛性。本文提出的方法论同样适用于其他多种流固耦合模型，可作为推动该领域研究的基础工具。