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February  2014, 8(1): 23-51. doi: 10.3934/ipi.2014.8.23

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, France
2. 

Institut de Mathématiques, Université de Toulouse, 118, Route de Narbonne, F-31062 Toulouse Cedex 9, France

Received  January 2013 Revised  June 2013 Published  March 2014

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: quasi-reversibility method, level set method, inverse obstacle problem., Stokes system, finite element method.
Mathematics Subject Classification: 35A15, 35M30, 35R25, 35R30, 35R35, 65M6.
Citation: Laurent Bourgeois, Jérémi Dardé. The "exterior approach" to solve the inverse obstacle problem for the Stokes system. Inverse Problems & Imaging, 2014, 8 (1) : 23-51. doi: 10.3934/ipi.2014.8.23
References:
[1]

C. Fabre and G. Lebeau, Prolongement unique des solutions de Stokes, Commun. Partial Differ. Eq., 21 (1996), 573-596. doi: 10.1080/03605309608821198.  Google Scholar

[2]

C.-L. Lin, G. Uhlmann and J.-N. Wang, Optimal three-ball inequalities and quantitative uniqueness for the Stokes system, Discrete Contin. Dyn. Syst., 28 (2010), 1273-1290. doi: 10.3934/dcds.2010.28.1273.  Google Scholar

[3]

M. Boulakia, A.-C. Egloffe and C. Grandmont, Stability estimates for a Robin coefficient in the two-dimensional Stokes system, Mathematical Control and Related Fields, 3 (2013), 21-49. doi: 10.3934/mcrf.2013.3.21.  Google Scholar

[4]

A. L. Bukhgeim, Extension of solutions of elliptic equations from discrete sets, J. Inverse Ill-Posed Probl., 1 (1993), 17-32. doi: 10.1515/jiip.1993.1.1.17.  Google Scholar

[5]

J. Cheng, M. Choulli and J. Lin, Stable determination of a boundary coefficient in an elliptic equation, Math. Models Methods Appl. Sci., 18 (2008), 107-123. doi: 10.1142/S0218202508002620.  Google Scholar

[6]

A. Ben Abda, I. Ben Saad and M. Hassine, Data completion for the Stokes system, CRAS Mécanique, 337 (2009), 703-708. Google Scholar

[7]

C. Alvarez, C. Conca, L. Fritz and O. Kavian, Identification of immersed obstacles via boundary measurements, Inverse Problems, 21 (2005), 1531-1552. doi: 10.1088/0266-5611/21/5/003.  Google Scholar

[8]

A. Ballerini, Stable determination of an immersed body in a stationary Stokes fluid, Inverse Problems, 26 (2010), 125015. doi: 10.1088/0266-5611/26/12/125015.  Google Scholar

[9]

N. F. M. Martins and A. L. Silvestre, An iterative MFS approach for the detection of immersed obstacles, Engineering Analysis with Boundary Elements, 32 (2008), 517-524. doi: 10.1016/j.enganabound.2007.10.011.  Google Scholar

[10]

C. Alvarez, C. Conca, R. Lecaros and J. H. Ortega, On the identification of a rigid body immersed in a fluid: A numerical approach, Engineering Analysis with Boundary Elements, 32 (2008), 919-925. doi: 10.1016/j.enganabound.2007.02.007.  Google Scholar

[11]

M. Badra, F. Caubet and M. Dambrine, Detecting an obstacle immersed in a fluid by shape optimization methods, Math. Models Methods Appl. Sci., 21 (2011), 2069-2101. doi: 10.1142/S0218202511005660.  Google Scholar

[12]

F. Caubet, M. Dambrine, D. Kateb and C. D. Timimoun, A Kohn-Vogelius formulation to detect an obstacle immersed in a fluid, Inverse Problems and Imaging, 7 (2013), 123-157. doi: 10.3934/ipi.2013.7.123.  Google Scholar

[13]

A. Ben Abda, M. Hassine, M. Jaoua and M. Masmoudi, Topological sensitivity analysis for the location of small cavities in Stokes flow, SIAM J. Control and Optimization, 48 (2009), 2871-2900. doi: 10.1137/070704332.  Google Scholar

[14]

F. Caubet and M. Dambrine, Localization of small obstacles in Stokes flow, Inverse Problems, 28 (2012), 105007. doi: 10.1088/0266-5611/28/10/105007.  Google Scholar

[15]

L. Bourgeois and J. Dardé, A quasi-reversibility approach to solve the inverse obstacle problem, Inverse Problems and Imaging, 4 (2010), 351-377. doi: 10.3934/ipi.2010.4.351.  Google Scholar

[16]

J. Dardé, The exterior approach: A new framework to solve inverse obstacle problems, Inverse Problems, 28 (2012), 015008. doi: 10.1088/0266-5611/28/1/015008.  Google Scholar

[17]

C. Conca, P. Cumsille, J. Ortega and L. Rosier, On the detection of a moving obstacle in an ideal fluid by a boundary measurement, Inverse Problems, 24 (2008), 045001. doi: 10.1088/0266-5611/24/4/045001.  Google Scholar

[18]

C. Conca, M. Malik and A. Munnier, Detection of a moving rigid solid in a perfect fluid, Inverse Problems, 26 (2010), 095010. doi: 10.1088/0266-5611/26/9/095010.  Google Scholar

[19]

C. Conca, E. Schwindt and T. Takahashi, On the identifiability of a rigid body moving in a stationary viscous fluid, Inverse Problems, 28 (2012), 015005. doi: 10.1088/0266-5611/28/1/015005.  Google Scholar

[20]

L. Bourgeois and J. Dardé, About identification of defects in an elastic-plastic medium from boundary measurements in the antiplane case, Applicable Analysis, 90 (2011), 1481-1497. doi: 10.1080/00036811.2010.549481.  Google Scholar

[21]

H. Brezis, Analyse Fonctionnelle, Théorie et Applications, Dunod, Paris, 1983.  Google Scholar

[22]

R. Lattès and J.-L. Lions, Méthode de Quasi-Réversibilité et Applications, Dunod, Paris, 1967.  Google Scholar

[23]

M. V. Klibanov and F. Santosa, A computational quasi-reversibility method for cauchy problems for Laplace's equation, SIAM J. Appl. Math., 51 (1991), 1653-1675. doi: 10.1137/0151085.  Google Scholar

[24]

P.-G. Ciarlet, The Finite Element Method for Elliptic Problems, North Holland, Amsterdam, 1978.  Google Scholar

[25]

W. Ming and J. Xu, The Morley element for fourth order elliptic equations in any dimensions, Numerische Mathematik, 103 (2006), 155-169. doi: 10.1007/s00211-005-0662-x.  Google Scholar

[26]

G. Duvaut and J.-L. Lions, Les Inéquations en Mécanique et en Physique, Dunod, Paris, 1972.  Google Scholar

[27]

L. Bourgeois, A mixed formulation of quasi-reversibility to solve the Cauchy problem for Laplace's equation, Inverse Problems, 21 (2005), 1087-1104. doi: 10.1088/0266-5611/21/3/018.  Google Scholar

[28]

J. Dardé, A. Hannukaiinen and N. Hyvönen, An $H_{ d i v}$-based mixed quasi-reversibility method for solving elliptic Cauchy problems, SIAM J. Num. Anal., 51 (2013), 2123-2148. Google Scholar

[29]

L. Bourgeois and J. Dardé, A duality-based method of quasi-reversibility to solve the Cauchy problem in the presence of noisy data, Inverse Problems, 26 (2010), 095016. doi: 10.1088/0266-5611/26/9/095016.  Google Scholar

[30]

I. Ekeland and R. Temam, Analyse Convexe et Problèmes Variationnels, Dunod, 1974.  Google Scholar

[31]

S. Osher and J. A. Sethian, Front propagating with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comp. Phys., 79 (1988), 12-49. doi: 10.1016/0021-9991(88)90002-2.  Google Scholar

[32]

A. Henrot and M. Pierre, Variation et Optimisation de Formes, Une Analyse Géométrique, Springer, Paris, 2005.  Google Scholar

[33]

F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer, New-York, 1991. doi: 10.1007/978-1-4612-3172-1.  Google Scholar

[34]

F. Hecht, A. Le Hyaric, J. Morice, K. Ohtsuka and O. Pironneau, Freefem++ Manual, http://www.freefem.org/ff++/ftp/freefem++doc.pdf, 2012. Google Scholar

[35]

V. Girault and P.-A. Raviart, Finite Element Approximation of the Navier-Stokes Equations, Springer-Verlag, Berlin, 1979.  Google Scholar

show all references

References:
[1]

C. Fabre and G. Lebeau, Prolongement unique des solutions de Stokes, Commun. Partial Differ. Eq., 21 (1996), 573-596. doi: 10.1080/03605309608821198.  Google Scholar

[2]

C.-L. Lin, G. Uhlmann and J.-N. Wang, Optimal three-ball inequalities and quantitative uniqueness for the Stokes system, Discrete Contin. Dyn. Syst., 28 (2010), 1273-1290. doi: 10.3934/dcds.2010.28.1273.  Google Scholar

[3]

M. Boulakia, A.-C. Egloffe and C. Grandmont, Stability estimates for a Robin coefficient in the two-dimensional Stokes system, Mathematical Control and Related Fields, 3 (2013), 21-49. doi: 10.3934/mcrf.2013.3.21.  Google Scholar

[4]

A. L. Bukhgeim, Extension of solutions of elliptic equations from discrete sets, J. Inverse Ill-Posed Probl., 1 (1993), 17-32. doi: 10.1515/jiip.1993.1.1.17.  Google Scholar

[5]

J. Cheng, M. Choulli and J. Lin, Stable determination of a boundary coefficient in an elliptic equation, Math. Models Methods Appl. Sci., 18 (2008), 107-123. doi: 10.1142/S0218202508002620.  Google Scholar

[6]

A. Ben Abda, I. Ben Saad and M. Hassine, Data completion for the Stokes system, CRAS Mécanique, 337 (2009), 703-708. Google Scholar

[7]

C. Alvarez, C. Conca, L. Fritz and O. Kavian, Identification of immersed obstacles via boundary measurements, Inverse Problems, 21 (2005), 1531-1552. doi: 10.1088/0266-5611/21/5/003.  Google Scholar

[8]

A. Ballerini, Stable determination of an immersed body in a stationary Stokes fluid, Inverse Problems, 26 (2010), 125015. doi: 10.1088/0266-5611/26/12/125015.  Google Scholar

[9]

N. F. M. Martins and A. L. Silvestre, An iterative MFS approach for the detection of immersed obstacles, Engineering Analysis with Boundary Elements, 32 (2008), 517-524. doi: 10.1016/j.enganabound.2007.10.011.  Google Scholar

[10]

C. Alvarez, C. Conca, R. Lecaros and J. H. Ortega, On the identification of a rigid body immersed in a fluid: A numerical approach, Engineering Analysis with Boundary Elements, 32 (2008), 919-925. doi: 10.1016/j.enganabound.2007.02.007.  Google Scholar

[11]

M. Badra, F. Caubet and M. Dambrine, Detecting an obstacle immersed in a fluid by shape optimization methods, Math. Models Methods Appl. Sci., 21 (2011), 2069-2101. doi: 10.1142/S0218202511005660.  Google Scholar

[12]

F. Caubet, M. Dambrine, D. Kateb and C. D. Timimoun, A Kohn-Vogelius formulation to detect an obstacle immersed in a fluid, Inverse Problems and Imaging, 7 (2013), 123-157. doi: 10.3934/ipi.2013.7.123.  Google Scholar

[13]

A. Ben Abda, M. Hassine, M. Jaoua and M. Masmoudi, Topological sensitivity analysis for the location of small cavities in Stokes flow, SIAM J. Control and Optimization, 48 (2009), 2871-2900. doi: 10.1137/070704332.  Google Scholar

[14]

F. Caubet and M. Dambrine, Localization of small obstacles in Stokes flow, Inverse Problems, 28 (2012), 105007. doi: 10.1088/0266-5611/28/10/105007.  Google Scholar

[15]

L. Bourgeois and J. Dardé, A quasi-reversibility approach to solve the inverse obstacle problem, Inverse Problems and Imaging, 4 (2010), 351-377. doi: 10.3934/ipi.2010.4.351.  Google Scholar

[16]

J. Dardé, The exterior approach: A new framework to solve inverse obstacle problems, Inverse Problems, 28 (2012), 015008. doi: 10.1088/0266-5611/28/1/015008.  Google Scholar

[17]

C. Conca, P. Cumsille, J. Ortega and L. Rosier, On the detection of a moving obstacle in an ideal fluid by a boundary measurement, Inverse Problems, 24 (2008), 045001. doi: 10.1088/0266-5611/24/4/045001.  Google Scholar

[18]

C. Conca, M. Malik and A. Munnier, Detection of a moving rigid solid in a perfect fluid, Inverse Problems, 26 (2010), 095010. doi: 10.1088/0266-5611/26/9/095010.  Google Scholar

[19]

C. Conca, E. Schwindt and T. Takahashi, On the identifiability of a rigid body moving in a stationary viscous fluid, Inverse Problems, 28 (2012), 015005. doi: 10.1088/0266-5611/28/1/015005.  Google Scholar

[20]

L. Bourgeois and J. Dardé, About identification of defects in an elastic-plastic medium from boundary measurements in the antiplane case, Applicable Analysis, 90 (2011), 1481-1497. doi: 10.1080/00036811.2010.549481.  Google Scholar

[21]

H. Brezis, Analyse Fonctionnelle, Théorie et Applications, Dunod, Paris, 1983.  Google Scholar

[22]

R. Lattès and J.-L. Lions, Méthode de Quasi-Réversibilité et Applications, Dunod, Paris, 1967.  Google Scholar

[23]

M. V. Klibanov and F. Santosa, A computational quasi-reversibility method for cauchy problems for Laplace's equation, SIAM J. Appl. Math., 51 (1991), 1653-1675. doi: 10.1137/0151085.  Google Scholar

[24]

P.-G. Ciarlet, The Finite Element Method for Elliptic Problems, North Holland, Amsterdam, 1978.  Google Scholar

[25]

W. Ming and J. Xu, The Morley element for fourth order elliptic equations in any dimensions, Numerische Mathematik, 103 (2006), 155-169. doi: 10.1007/s00211-005-0662-x.  Google Scholar

[26]

G. Duvaut and J.-L. Lions, Les Inéquations en Mécanique et en Physique, Dunod, Paris, 1972.  Google Scholar

[27]

L. Bourgeois, A mixed formulation of quasi-reversibility to solve the Cauchy problem for Laplace's equation, Inverse Problems, 21 (2005), 1087-1104. doi: 10.1088/0266-5611/21/3/018.  Google Scholar

[28]

J. Dardé, A. Hannukaiinen and N. Hyvönen, An $H_{ d i v}$-based mixed quasi-reversibility method for solving elliptic Cauchy problems, SIAM J. Num. Anal., 51 (2013), 2123-2148. Google Scholar

[29]

L. Bourgeois and J. Dardé, A duality-based method of quasi-reversibility to solve the Cauchy problem in the presence of noisy data, Inverse Problems, 26 (2010), 095016. doi: 10.1088/0266-5611/26/9/095016.  Google Scholar

[30]

I. Ekeland and R. Temam, Analyse Convexe et Problèmes Variationnels, Dunod, 1974.  Google Scholar

[31]

S. Osher and J. A. Sethian, Front propagating with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations, J. Comp. Phys., 79 (1988), 12-49. doi: 10.1016/0021-9991(88)90002-2.  Google Scholar

[32]

A. Henrot and M. Pierre, Variation et Optimisation de Formes, Une Analyse Géométrique, Springer, Paris, 2005.  Google Scholar

[33]

F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer, New-York, 1991. doi: 10.1007/978-1-4612-3172-1.  Google Scholar

[34]

F. Hecht, A. Le Hyaric, J. Morice, K. Ohtsuka and O. Pironneau, Freefem++ Manual, http://www.freefem.org/ff++/ftp/freefem++doc.pdf, 2012. Google Scholar

[35]

V. Girault and P.-A. Raviart, Finite Element Approximation of the Navier-Stokes Equations, Springer-Verlag, Berlin, 1979.  Google Scholar

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