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.

翻译：在过去十年中，有限元方法和运动学耦合格式已被广泛研究用于不可压缩流固耦合问题，这类格式将流体速度与结构位移解耦。尽管这些方法已知具有稳定性且易于实现，但其最优误差分析始终面临挑战。以往工作主要依赖于经典椭圆投影技术，该技术仅适用于抛物型问题，无法在标准$L^2$范数下得到流固耦合问题数值解的最优收敛性。本文针对不可压缩流固耦合薄结构模型，提出了一种新的稳定的全离散运动学耦合格式，并基于新引入的耦合非平稳Ritz投影，建立了流固耦合问题数值分析的新方法，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众多其他流固耦合模型，并将为推动该领域研究提供基础性工具。