A Revisiting of the Pressure Elimination for a Fluid-Structure PDE Interaction and Its Implications

TL;DR

This paper introduces a new pressure elimination technique for fluid-structure PDE interactions, enabling explicit semigroup representation and well-posedness proofs in general geometries, with implications for finite element methods.

Contribution

The paper develops a novel pressure elimination method for fluid-structure PDEs, providing explicit semigroup representation and extending well-posedness results to Lipschitz domains.

Findings

Explicit $C_{0}$-semigroup generator for the coupled PDE system.

Proof of well-posedness in general Lipschitz geometries.

Finite element method with convergence rates for static FSI models.

Abstract

In this paper we construct a novel technique for eliminating and recovering the pressure for a fluid-structure interaction model. This pressure elimination methodology is valid for general bounded Lipschitz domains. The specific fluid-structure interaction (FSI) that we consider is a well-known model of Stokes flow coupled to a system of linear elasticity, which constitutes a coupled parabolic-hyperbolic system. The coupling between the two distinct PDE dynamics occurs across a boundary interface, with each of the components evolving on its own distinct geometry, with the domains of each being Lipschitz. Our new pressure elimination technique admits of an explicit $C_{0}$-semigroup generator representation $\mathcal{A}: D(\mathcal{A}) \subset \mathbf{H} \to \mathbf{H}$, where $\mathbf{H}$ is the associated finite energy space of fluid-structure initial data. This leads to a novel proof of well-posedness in the explicit semigroup sense of the continuous PDE, now valid in general geometries. Subsequently, we illustrate an immediate consequence of our semigroup well-posedness result; namely a finite element method (FEM) with associated rates of convergence for a static version of the FSI, posed on polygonal domains.