Shape Optimization for Stokes flows: a finite element convergence analysis

Fumagalli, Ivan; Parolini, Nicola; Verani, Marco

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 4, pp. 921-951.

In this paper we analyze a two-dimensional shape optimization problem, governed by Stokes equations that are defined on a domain with a part of the boundary that is described as the graph of the control function. The state problem formulation is mapped onto a reference domain, which is independent of the control function, and the analysis is mainly led on such domain. The existence of an optimal control function is proved, and optimality conditions are derived. After the analytical inspection of the problem, finite element discretization is considered for both the control function and the state variables, and a priori convergence error estimates are derived. Numerical experiments assess the validity of the theoretical results.

Classification : 49M25, 49Q10, 65N15, 65N30
Mots clés : Shape optimization, Stokes problem, reference domain, convergence rates, finite elements

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