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>A Mixed Variational Formulation for the Wellposedness and Numerical
Approximation of a PDE Model Arising in a 3-D Fluid-Structure Interation
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A Mixed Variational Formulation for the Wellposedness and Numerical
Approximation of a PDE Model Arising in a 3-D Fluid-Structure Interation
We will present qualitative and numerical results on a partial differentialequation (PDE) system which models a certain fluid-structure dynamics. Thewellposedness of this PDE model is established by means of constructing for ita nonstandard semigroup generator representation; this representation isessentially accomplished by an appropriate elimination of the pressure. Thiscoupled PDE model involves the Stokes system which evolves on a threedimensional domain $\mathcal{O}$ being coupled to a fourth order plateequation, possibly with rotational inertia parameter $\rho >0$, which evolveson a flat portion $\Omega$ of the boundary of $\mathcal{O}$. The coupling on$\Omega$ is implemented via the Dirichlet trace of the Stokes system fluidvariable - and so the no-slip condition is necessarily not in play - and viathe Dirichlet boundary trace of the pressure, which essentially acts as aforcing term on this elastic portion of the boundary. We note here thatinasmuch as the Stokes fluid velocity does not vanish on $\Omega$, the pressurevariable cannot be eliminated by the classic Leray projector; instead, thepressure is identified as the solution of a certain elliptic boundary valueproblem. Eventually, wellposedness of this fluid-structure dynamics is attainedthrough a certain nonstandard variational (``inf-sup") formulation.Subsequently we show how our constructive proof of wellposedness naturallygives rise to a certain mixed finite element method for numericallyapproximating solutions of this fluid-structure dynamics.
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