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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 interaction

Thomas J. Clark, George Avalos

2014
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Evolution Equations and Control Theory
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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
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... otational inertia parameter ρ>0. This plate PDE evolves on a flat portion Ω of the boundary of O. The coupling on Ω 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 Ω. We note that as the Stokes fluid velocity does not vanish on Ω, 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. Keywords Fluid-structure interaction, 3D linearized Navier-Stokes, Kirchhoff plate, finite element method, Babuška-Brezzi theorem Disciplines Mathematics Comments Abstract We will present qualitative and numerical results on a partial differential equation (PDE) system which models a certain fluid-structure dynamics. The wellposedness of this PDE model is established by means of constructing for it a nonstandard semigroup generator representation; this representation is essentially accomplished by an appropriate elimination of the pressure. This coupled PDE model involves the Stokes system which evolves on a three dimensional domain O being coupled to a fourth order plate equation, possibly with rotational inertia parameter ρ > 0, which evolves on a flat portion Ω of the boundary of O. The coupling on Ω 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 this elastic portion of the boundary. We note here that inasmuch as the Stokes fluid velocity does not vanish on Ω, the pressure variable cannot be eliminated by the classic Leray projector; instead, the pressure is identified as the solution of a certain elliptic boundary value problem. Eventually, wellposedness of this fluid-structure dynamics is attained through a certain nonstandard variational ("inf-sup") formulation. Subsequently we show how our constructive proof of wellposedness naturally gives rise to a certain mixed finite element method for numerically approximating solutions of this fluid-structure dynamics.

doi:10.3934/eect.2014.3.557
fatcat:fq5um6fsrnetbm7orvztcxw4ti