An inf-sup approach to semigroup wellposedness for a compressible flow - incompressible fluid interactive PDE system

TL;DR

This paper establishes the wellposedness of a coupled compressible-incompressible fluid PDE system using an inf-sup approach, and develops a finite element method with error estimates for numerical approximation.

Contribution

It introduces a novel semigroup approach to prove wellposedness and connects this theory to a practical FEM with convergence guarantees.

Findings

Wellposedness proven via semigroup generator construction.

Finite element method developed with error estimates.

Numerical simulations demonstrate method effectiveness.

Abstract

This work presents qualitative and numerical results on a system of partial differential equations (PDEs) which models certain fluid-fluid interaction dynamics. This system models a compressible fluid in a domain $\Omega^+ \subset \mathbb{R}^2$, coupled to an incompressible fluid modeled by Stokes flow in domain $\Omega^- \subset \mathbb{R}^2$, with the strong coupling implemented through certain boundary conditions on the shared interface, $\Gamma$. The wellposedness of this system is established by means of constructing for it a semigroup generator representation. This representation is accomplished by eliminating one of the pressure variables via identifying it as the solution of a certain boundary value problem, while the wellposedness is established via a nonstandard usage of the Babuska-Brezzi Theorem. In later sections, we demonstrate how the earlier constructive proof of wellposedness naturally lends itself to a certain finite element method (FEM), by which to numerically approximate solutions of the given coupled PDE system. This FEM is provided with error estimates and associated rates of convergence.