Finite element approximation for a reformulation of a 3D fluid-2D plate interaction system

TL;DR

This paper introduces a finite element method for a coupled 3D fluid and 2D elastic plate interaction, reformulating the plate equation to avoid complex elements, and provides stability, error estimates, and numerical validation.

Contribution

The reformulation of the fourth-order plate equation into coupled second-order equations simplifies finite element approximation and avoids complex element requirements.

Findings

Established well-posedness and stability of the discretized system.

Derived a priori error estimates for the numerical scheme.

Numerical experiments confirmed theoretical convergence rates.

Abstract

We study a finite element approximation of a coupled fluid-structure interaction consisting of a three-dimensional incompressible viscous fluid governed by the unsteady Stokes equations and a two-dimensional elastic plate. To avoid the use of $H^2-$conforming or nonconforming $\mathbb{P}_2$-Morley plate elements, the fourth-order plate equation is reformulated into a system of coupled second-order equations using an auxiliary variable. The coupling condition is enforced using a Lagrange multiplier representing the trace of the mean-zero fluid pressure on the interface. We establish well-posedness and stability results for the time-discrete and fully-discrete problems, and derive a priori error estimates. A partitioned domain decomposition algorithm based on a fixed-point iteration is employed for the numerical solution. Numerical experiments verify the theoretical rates of convergence in space and time using manufactured solutions, and demonstrate the applicability of the method to a physical problem.