Inverse Problems and Imaging (IPI)

The "exterior approach" to solve the inverse obstacle problem for the Stokes system

Pages: 23 - 51, Volume 8, Issue 1, February 2014      doi:10.3934/ipi.2014.8.23

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Laurent Bourgeois - Laboratoire POEMS, ENSTA ParisTech, 828, Boulevard des Maréchaux, 91762, Palaiseau Cedex, France (email)
Jérémi Dardé - Institut de Mathématiques, Université de Toulouse, 118, Route de Narbonne, F-31062 Toulouse Cedex 9, France (email)

Abstract: We apply an ``exterior approach" based on the coupling of a method of quasi-reversibility and of a level set method in order to recover a fixed obstacle immersed in a Stokes flow from boundary measurements. Concerning the method of quasi-reversibility, two new mixed formulations are introduced in order to solve the ill-posed Cauchy problems for the Stokes system by using some classical conforming finite elements. We provide some proofs for the convergence of the quasi-reversibility methods on the one hand and of the level set method on the other hand. Some numerical experiments in $2D$ show the efficiency of the two mixed formulations and of the exterior approach based on one of them.

Keywords:  Stokes system, level set method, quasi-reversibility method, finite element method, inverse obstacle problem.
Mathematics Subject Classification:  35A15, 35M30, 35R25, 35R30, 35R35, 65M60.

Received: January 2013;      Revised: June 2013;      Available Online: March 2014.