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 𝓐:D(𝓐)⊂ H → H, where 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 Φ ∈ D(𝓐) of the equation (λI-𝓐)Φ =F for given F ∈ H and λ > 0. The present work reconsiders the proof of maximality for the fluid-structure generator 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-am35-3-2,
author = {George Avalos and Matthew Dvorak},
title = {A new maximality argument for a coupled fluid-structure interaction, with implications for a divergence-free finite element method},
journal = {Applicationes Mathematicae},
volume = {35},
year = {2008},
pages = {259-280},
zbl = {1194.35024},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-am35-3-2}
}
George Avalos; Matthew Dvorak. A new maximality argument for a coupled fluid-structure interaction, with implications for a divergence-free finite element method. Applicationes Mathematicae, Tome 35 (2008) pp. 259-280. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-am35-3-2/