A new maximality argument for a coupled fluid-structure interaction, with implications for a divergence-free finite element method
Volume 35 / 2008
Abstract
We consider a coupled PDE model of various fluid-structure interactions seen in nature. It has recently been shown by the authors [Contemp. Math. 440, 2007] that this model admits of an explicit semigroup generator representation $\mathcal{A}:D(\mathcal{A})\subset \mathbf{H} \rightarrow \mathbf{H}$, where $\mathbf{H}$ is the associated space of fluid-structure initial data. However, the argument for the maximality criterion was indirect, and did not provide for an explicit solution $\Phi \in D(\mathcal{A})$ of the equation $(\lambda I-\mathcal{A} )\Phi =F$ for given $F\in \mathbf{H}$ and $\lambda >0$. The present work reconsiders the proof of maximality for the fluid-structure generator $\mathcal{A}$, and gives an explicit method for solving the said fluid-structure equation. This involves a nonstandard usage of the Babuška–Brezzi Theorem. Subsequently, a finite element method for approximating solutions of the fluid-structure dynamics is developed, based upon our explicit proof of maximality.
