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Domain optimization for the Navier-Stokes equation by an embedding domain method

Slawig, Thomas

Preprint-Reihe des Instituts für Mathematik, Technische Universität Berlin

Domain optimization problems for the two-dimensional stationary flow of incompressible linear-viscous fluids, i.e. the Navier-Stokes equations, are studied. An embedding domain technique which provides an equivalent formulation of the problem on a fixed domain is introduced. Existence of a solution to the domain optimization problem and Fréchet differentiability with respect to the variation of the domain of tracking type cost functionals for the velocity field are proved. A simply computable formula for the derivative of the cost functional is presented. Numerical examples show the advantages of the embedding domain method and the reliability of the derivative formula.