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

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 infite 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 effciency of the two mixed formulations and of the exterior approach based on one of them.

[1]  H. Brezis.  Analyse fonctionnelle : théorie et applications , 1983 .

[2]  Jaime H. Ortega,et al.  Identification of immersed obstacles via boundary measurements , 2005 .

[3]  Jérémi Dardé,et al.  A quasi-reversibility approach to solve the inverse obstacle problem , 2010 .

[4]  Michael V. Klibanov,et al.  A computational quasi-reversiblility method for Cauchy problems for Laplace's equation , 1991 .

[5]  C. Fabre,et al.  Prolongement Unique Des Solutions , 1996 .

[6]  Michel Fortin,et al.  Mixed and Hybrid Finite Element Methods , 2011, Springer Series in Computational Mathematics.

[7]  Antoine Henrot,et al.  Variation et optimisation de formes : une analyse géométrique , 2005 .

[8]  On the identification of a rigid body immersed in a fluid: A numerical approach , 2008 .

[9]  J. Lions,et al.  Les inéquations en mécanique et en physique , 1973 .

[10]  STABLE DETERMINATION OF A BOUNDARY COEFFICIENT IN AN ELLIPTIC EQUATION , 2008 .

[11]  J. Dardé.  The ‘exterior approach’: a new framework to solve inverse obstacle problems , 2011 .

[12]  Robert Lattès,et al.  Méthode de quasi-réversibilbilité et applications , 1967 .

[13]  Laurent Bourgeois,et al.  A mixed formulation of quasi-reversibility to solve the Cauchy problem for Laplace's equation , 2005 .

[14]  Antti Hannukainen,et al.  An $H_\mathsf{div}$-Based Mixed Quasi-reversibility Method for Solving Elliptic Cauchy Problems , 2013, SIAM J. Numer. Anal..

[15]  Jenn-Nan Wang,et al.  Optimal three-ball inequalities and quantitative uniqueness for the Stokes system , 2008, 0812.3730.

[16]  Fabien Caubet,et al.  Localization of small obstacles in Stokes flow , 2012 .

[17]  Jérémi Dardé,et al.  A duality-based method of quasi-reversibility to solve the Cauchy problem in the presence of noisy data , 2010 .

[18]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[19]  S. Osher,et al.  Algorithms Based on Hamilton-Jacobi Formulations , 1988 .

[20]  A. L. BUKHGEIM.  Extension of Solutions of elliptic equations from discrete sets , 1993 .

[21]  Jaime H. Ortega,et al.  On the detection of a moving obstacle in an ideal fluid by a boundary measurement , 2008 .

[22]  Amel Ben Abda,et al.  Data completion for the Stokes system , 2009 .

[23]  P. Raviart,et al.  Finite Element Approximation of the Navier-Stokes Equations , 1979 .

[24]  Nuno F. M. Martins,et al.  An iterative MFS approach for the detection of immersed obstacles , 2008 .

[25]  C. Grandmont,et al.  Stability estimates for a Robin coefficient in the two-dimensional Stokes system , 2012, 1202.1263.

[26]  Fabien Caubet,et al.  A Kohn-Vogelius formulation to detect an obstacle immersed in a fluid , 2013 .

[27]  L. Bourgeois,et al.  About identification of defects in an elastic-plastic medium from boundary measurements in the antiplane case , 2011 .

[28]  Fabien Caubet,et al.  DETECTING AN OBSTACLE IMMERSED IN A FLUID BY SHAPE OPTIMIZATION METHODS , 2011 .

[29]  Wang Ming,et al.  The Morley element for fourth order elliptic equations in any dimensions , 2006, Numerische Mathematik.

[30]  J. Sethian,et al.  Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations , 1988 .

[31]  R. Temam,et al.  Analyse convexe et problèmes variationnels , 1974 .

[32]  C. Conca,et al.  On the identifiability of a rigid body moving in a stationary viscous fluid , 2011 .

[33]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[34]  A. Ballerini.  Stable determination of an immersed body in a stationary Stokes fluid , 2010, 1003.0301.

[35]  A. Ben Abda,et al.  Topological Sensitivity Analysis for the Location of Small Cavities in Stokes Flow , 2009, SIAM J. Control. Optim..

[36]  Detection of a moving rigid solid in a perfect fluid , 2010, 1007.0234.