A MIXED VARIATIONAL FORMULATION FOR THE WELLPOSEDNESS AND NUMERICAL APPROXIMATION OF A PDE MODEL ARISING IN A 3-D FLUID-STRUCTURE INTERACTION

We present qualitative and numerical results on a partial differential equation (PDE) system which models a certain fluid-structure dynamics. Wellposedness is established by constructing for it a nonstandard semigroup generator representation; this representation is accomplished by an appropriate elimination of the pressure. This coupled PDE model involves the Stokes system which evolves on a three dimensional domain 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 O. The coupling on Omega is implemented via the Dirichlet trace of the Stokes system fluid variable - and so the no-slip condition is necessarily not in play - and via the Dirichlet boundary trace of the pressure, which essentially acts as a forcing term on Omega. We note that as the Stokes fluid velocity does not vanish on Omega, the pressure variable cannot be eliminated by the classic Leray projector; instead, it is identified as the solution of an elliptic boundary value problem. Eventually, wellposedness of the system is attained through a nonstandard variational ("inf-sup") formulation. Subsequently we show how our constructive proof of wellposedness naturally gives rise to a mixed finite element method for numerically approximating solutions of this fluid-structure dynamics.