Dynamic Ritz Projection for Finite Element Methods in Fluid-Structure Interaction

TL;DR

This paper introduces a dynamic Ritz projection tailored for fluid-structure interaction problems, enabling proof of optimal-order convergence of finite element methods in the $L^ abla(0,T;L^2)$ norm despite interface condition challenges.

Contribution

It develops a new dynamic Ritz projection satisfying interface conditions, proving its approximation properties and enabling optimal convergence analysis for FSI finite element methods.

Findings

Established existence and uniqueness of the dynamic Ritz projection.

Derived error estimates for the projection.

Proved optimal-order convergence of finite element methods in $L^ abla(0,T;L^2)$ norm.

Abstract

Regardless of the development of various finite element methods for fluid-structure interaction (FSI) problems, optimal-order convergence of finite element discretizations of the FSI problems in the $L^\infty(0,T;L^2)$ norm has not been proved due to the incompatibility between standard Ritz projections and the interface conditions in the FSI problems. To address this issue, we define a dynamic Ritz projection (which satisfies a dynamic interface condition) associated to the FSI problem and study its approximation properties in the $L^\infty(0,T;H^1)$ and $L^\infty(0,T;L^2)$ norms. Existence and uniqueness of the dynamic Ritz projection of the solution, as well as estimates of the error between the solution and its dynamic Ritz projection, are established. By utilizing the established results, we prove optimal-order convergence of finite element methods for the FSI problem in the $L^\infty(0,T;L^2)$ norm.