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

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 $L^{\mathrm{\infty }}(0,T;L^2)$ 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 $L^{\mathrm{\infty }}(0,T;H^1)$ and $L^{\mathrm{\infty }}(0,T;L^2)$ 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 $L^{\mathrm{\infty }}(0,T;L^2)$ norm.