A mixed variational formulation for the wellposedness and numerical approximation of a PDE model arising in a 3-D fluid-structure interaction
George Avalos, Thomas J. Clark · Evolution equations and control theory · 2014
We present qualitative and numerical results on a partialdifferential equation (PDE) system which models a certainfluid-structure dynamics. Wellposedness is established byconstructing for it a nonstandard semigroup generatorrepresentation; this representation is accomplished by anappropriate elimination of the pressure. This coupled PDE modelinvolves the Stokes system which evolves on a three dimensionaldomain $\mathcal{O}$ coupled to a fourth order plate equation,possibly with rotational inertia parameter $\rho>0$. This plate PDE evolves on a flat portion $\Omega$ of the boundary of$\mathcal{O}$. The coupling on $\Omega$ is implemented via theDirichlet trace of the Stokes system fluid variable - and so theno-slip condition is necessarily not in play - and via the Dirichletboundary trace of the pressure, which essentially acts as a forcingterm on $\Omega$. We note that as the Stokes fluid velocity does notvanish on $\Omega$, the pressure variable cannot be eliminated bythe classic Leray projector; instead, it is identified as thesolution of an elliptic boundary value problem. Eventually,wellposedness of the system is attained through a nonstandardvariational (``inf-sup') formulation. Subsequently we show how ourconstructive proof of wellposedness naturally gives rise to a mixedfinite element method for numerically approximating solutions ofthis fluid-structure dynamics.