Optimal [math] Error Analysis of a Loosely Coupled Finite Element Scheme for Thin-Structure Interactions

Abstract

SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1782-1813, August 2024.
Abstract. Finite element methods and kinematically coupled schemes that decouple the fluid velocity and structure displacement have been extensively studied for incompressible fluid-structure interactions (FSIs) 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 for the FSI problems in the standard [math] norm. In this article, we propose a new stable fully discrete kinematically coupled scheme for the incompressible FSI thin-structure model and establish a new approach for the numerical analysis of FSI problems in terms of a newly introduced coupled nonstationary Ritz projection, which allows us to prove the optimal-order convergence of the proposed method in the [math] 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.