Analysis of the Coupling of Lagrange and Arnold--Falk--Winther Finite Elements for a Fluid-Solid Interaction Problem in Three Dimensions

Abstract

We introduce and analyze a new finite element method for a three-dimensional fluid-solid interaction problem. The media are governed by the acoustic and elastodynamic equations in a time-harmonic regime, and the transmission conditions are given by the equilibrium of forces and the equality of the corresponding normal displacements. We employ a dual-mixed variational formulation in the solid, in which the Cauchy stress tensor and the rotation are the only unknowns, and maintain the usual primal formulation in the fluid. The main novelty of our method, with respect to previous approaches for a two-dimensional version of this problem, consists of the introduction of the first transmission condition as part of the definition of the space to which the stress of the solid and the pressure of the fluid belong. As a consequence, and since the second transmission condition becomes natural, no Lagrange multipliers on the coupling boundary are needed, which certainly leads to a much simpler variational formulation....