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.