A new maximality argument for a coupled fluid-structure interaction, with implications for a divergence-free finite element method

George Avalos¹ ; Matthew Dvorak²

¹ Department of Mathematics University of Nebraska-Lincoln Lincoln, NE 68588, U.S.A.
² Department of Mathematics University of Nebraska-Lincoln Lincoln NE 68588, U.S.A.

Applicationes Mathematicae, Tome 35 (2008) no. 3, pp. 259-280.

Résumé

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.

Détail

DOI : 10.4064/am35-3-2

Keywords: consider coupled pde model various fluid structure interactions seen nature has recently shown authors contemp math model admits explicit semigroup generator representation mathcal mathcal subset mathbf rightarrow mathbf where mathbf associated space fluid structure initial however argument maximality criterion indirect did provide explicit solution phi mathcal equation lambda i mathcal phi given mathbf lambda present work reconsiders proof maximality fluid structure generator mathcal gives explicit method solving said fluid structure equation involves nonstandard usage babu brezzi theorem subsequently finite element method approximating solutions fluid structure dynamics developed based explicit proof maximality