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

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.

💡 Research Summary

This paper addresses the numerical simulation of a fluid‑structure interaction (FSI) problem in which a three‑dimensional incompressible viscous fluid, governed by the unsteady Stokes equations, is coupled to a two‑dimensional elastic plate modeled by the Kirchhoff‑Love theory. The main difficulty in such problems lies in the fourth‑order plate equation, which traditionally forces the use of H²‑conforming finite elements (e.g., Argyris, Hermite) or non‑conforming P₂‑Morley elements. Both approaches increase implementation complexity and hinder flexible coupling or domain‑decomposition strategies.

The authors overcome this obstacle by introducing an auxiliary variable (z = -\Delta w), where (w) denotes the plate displacement. This reformulation replaces the original biharmonic equation (\Delta^{2} w) with a coupled system of two second‑order equations: \