Stability and error estimates of a linear numerical scheme approximating nonlinear fluid-structure interactions

TL;DR

This paper introduces a linear, monolithic finite element method for simulating incompressible viscous fluid interactions with elastic plates, ensuring energy conservation and optimal convergence without re-meshing.

Contribution

The paper presents a novel linear, semi-implicit finite element scheme for fluid-structure interaction that maintains energy stability and achieves optimal convergence rates.

Findings

Method satisfies geometrical conservation law

Achieves optimal linear convergence in space and time

Comparable accuracy to fully implicit schemes in stable cases

Abstract

In this paper, we propose a linear and monolithic finite element method for the approximation of an incompressible viscous fluid interacting with an elastic and deforming plate. We use the arbitrary Lagrangian-Eulerian (ALE) approach that works in the reference domain, meaning that no re-meshing is needed during the numerical simulation. For time discretization, we employ the backward Euler method. For space discretization, we respectively use P1-bubble, P1, and P1 finite elements for the approximation of the fluid velocity, pressure, and structure displacement. We show that our method fulfills the geometrical conservation law and dissipates the total energy on the discrete level. Moreover, we prove the (optimal) linear convergence with respect to the sizes of the time step $\tau$ and the mesh $h$. We present numerical experiments involving a substantially deforming fluid domain that do validate our theoretical results. A comparison with a fully implicit (thus nonlinear) scheme indicates that our semi-implicit linear scheme is faster and as accurate as the fully implicit one, at least in stable configurations.