Analysis of the Coupling of Primal and Dual-Mixed Finite Element Methods for a Two-Dimensional Fluid-Solid Interaction Problem

Abstract

This paper deals with a time-harmonic fluid-solid interaction problem posed in the plane. More precisely, we apply the coupling of primal and dual-mixed finite element methods to compute both the pressure of the scattered wave in the linearized fluid and the elastic vibrations that take place in the solid elastic body. To this end, we solve a transmission problem holding between the cross-section of the infinitely long cylinder representing the obstacle and an annular region surrounding it. The novelty of our method lies in the use of a dual-mixed variational formulation in the obstacle, while maintaining the usual primal formulation in the fluid. In other words, we introduce a stress-pressure formulation of the problem instead of the traditional displacement-pressure encountered in the literature. As a consequence, one of the transmission conditions becomes essential, and hence we enforce it weakly by means of a Lagrange multiplier. Next, we apply the abstract framework developed in a recent work by A. Buffa, prove that our coupled variational formulation is well posed, and define the corresponding discrete scheme by using PEERS in the solid domain and standard Lagrange finite elements in the fluid domain. Then we show that the resulting Galerkin scheme is uniquely solvable and convergent and derive optimal error estimates. Finally, we illustrate our analysis with some results from computational experiments.