Analysis and Numerical Approximation to Interactive Dynamics of Navier Stokes-Plate Interaction PDE System

TL;DR

This paper analyzes the well-posedness of a fluid-structure interaction PDE system involving Navier-Stokes equations and a deformable membrane, and develops a finite element method for its numerical approximation with validated error bounds.

Contribution

It provides the first combined theoretical analysis and numerical scheme for Navier-Stokes-plate interaction systems with proven error estimates.

Findings

Existence and uniqueness of weak solutions under small data.

A finite element scheme with Picard iterations for nonlinearities.

Error bounds between FEM solutions and theoretical solutions.

Abstract

We consider a Navier-Stokes fluid-plate interaction (FSI) system which describes the evolutions of the fluid contained within a 3D cavity, as it interacts with a deformable elastic membrane on the ``free" upper boundary of the cavity. These models arise in various aeroelastic and biomedical applications as well as in the control of ocular pressure, and sloshing phenomena. We analyze the well-posedness of weak solutions to the stationary ($\lambda$-parametrized) coupled PDE system by way of invoking the nonlinear generalization of the abstract variational formulations which was introduced in \cite{girault2012finite}, wherein an inf-sup approach is followed to show existence-uniqueness of solutions under a small data assumption. In addition, we provide a numerical approximation scheme of the infinite dimensional coupled system via a finite element method approximation (FEM). The numerical results use a standard conforming scheme and handle the introduced nonlinearities via Picard iterations. Numerical results are obtained for an appropriate test problem satisfying the necessary boundary conditions and coupling. Moreover, error bounds between the FEM and theoretical solution in terms of the characteristic mesh size are supplied in appropriate Sobolev norms which agree with the established literature. These FEM approximations of the coupled system with their associated error bounds validate the theoretical findings.