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

Laurent Bourgeois; Jérémi Dardé

doi:10.3934/ipi.2014.8.23

Article Contents

2014, Volume 8, Issue 1: 23-51. Doi: 10.3934/ipi.2014.8.23

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

Laurent Bourgeois^1, and
Jérémi Dardé^2,

1.
Laboratoire POEMS, ENSTA ParisTech, 828, Boulevard des Maréchaux, 91762, Palaiseau Cedex
2.
Institut de Mathématiques, Université de Toulouse, 118, Route de Narbonne, F-31062 Toulouse Cedex 9

Received: December 31, 2012

Revised: May 31, 2013

Published: January 2014

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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.

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Mathematics Subject Classification: 35A15, 35M30, 35R25, 35R30, 35R35, 65M60.

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