# American Institute of Mathematical Sciences

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

 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.
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.  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.  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.  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.  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.  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.   Google Scholar [7] C. Alvarez, C. Conca, L. Fritz and O. Kavian, Identification of immersed obstacles via boundary measurements,, Inverse Problems, 21 (2005), 1531.  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).  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.  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.  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.  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.  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.  doi: 10.1137/070704332.  Google Scholar [14] F. Caubet and M. Dambrine, Localization of small obstacles in Stokes flow,, Inverse Problems, 28 (2012).  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.  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).  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).  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).  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).  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.  doi: 10.1080/00036811.2010.549481.  Google Scholar [21] H. Brezis, Analyse Fonctionnelle, Théorie et Applications,, Dunod, (1983).   Google Scholar [22] R. Lattès and J.-L. Lions, Méthode de Quasi-Réversibilité et Applications,, Dunod, (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.  doi: 10.1137/0151085.  Google Scholar [24] P.-G. Ciarlet, The Finite Element Method for Elliptic Problems,, North Holland, (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.  doi: 10.1007/s00211-005-0662-x.  Google Scholar [26] G. Duvaut and J.-L. Lions, Les Inéquations en Mécanique et en Physique,, Dunod, (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.  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.   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).  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.  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, (2005).   Google Scholar [33] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods,, Springer, (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,, , (2012).   Google Scholar [35] V. Girault and P.-A. Raviart, Finite Element Approximation of the Navier-Stokes Equations,, Springer-Verlag, (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.  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.  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.  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.  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.  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.   Google Scholar [7] C. Alvarez, C. Conca, L. Fritz and O. Kavian, Identification of immersed obstacles via boundary measurements,, Inverse Problems, 21 (2005), 1531.  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).  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.  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.  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.  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.  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.  doi: 10.1137/070704332.  Google Scholar [14] F. Caubet and M. Dambrine, Localization of small obstacles in Stokes flow,, Inverse Problems, 28 (2012).  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.  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).  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).  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).  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).  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.  doi: 10.1080/00036811.2010.549481.  Google Scholar [21] H. Brezis, Analyse Fonctionnelle, Théorie et Applications,, Dunod, (1983).   Google Scholar [22] R. Lattès and J.-L. Lions, Méthode de Quasi-Réversibilité et Applications,, Dunod, (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.  doi: 10.1137/0151085.  Google Scholar [24] P.-G. Ciarlet, The Finite Element Method for Elliptic Problems,, North Holland, (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.  doi: 10.1007/s00211-005-0662-x.  Google Scholar [26] G. Duvaut and J.-L. Lions, Les Inéquations en Mécanique et en Physique,, Dunod, (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.  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.   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).  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.  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, (2005).   Google Scholar [33] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods,, Springer, (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,, , (2012).   Google Scholar [35] V. Girault and P.-A. Raviart, Finite Element Approximation of the Navier-Stokes Equations,, Springer-Verlag, (1979).   Google Scholar
 [1] Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056 [2] Gang Bao, Mingming Zhang, Bin Hu, Peijun Li. An adaptive finite element DtN method for the three-dimensional acoustic scattering problem. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020351 [3] Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $L^2-$norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077 [4] Wenbin Li, Jianliang Qian. Simultaneously recovering both domain and varying density in inverse gravimetry by efficient level-set methods. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020073 [5] Lingfeng Li, Shousheng Luo, Xue-Cheng Tai, Jiang Yang. A new variational approach based on level-set function for convex hull problem with outliers. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020070 [6] Li-Bin Liu, Ying Liang, Jian Zhang, Xiaobing Bao. A robust adaptive grid method for singularly perturbed Burger-Huxley equations. Electronic Research Archive, 2020, 28 (4) : 1439-1457. doi: 10.3934/era.2020076 [7] Zexuan Liu, Zhiyuan Sun, Jerry Zhijian Yang. A numerical study of superconvergence of the discontinuous Galerkin method by patch reconstruction. Electronic Research Archive, 2020, 28 (4) : 1487-1501. doi: 10.3934/era.2020078 [8] Yuxia Guo, Shaolong Peng. A direct method of moving planes for fully nonlinear nonlocal operators and applications. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020462 [9] Noah Stevenson, Ian Tice. A truncated real interpolation method and characterizations of screened Sobolev spaces. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5509-5566. doi: 10.3934/cpaa.2020250 [10] Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319 [11] Helmut Abels, Andreas Marquardt. On a linearized Mullins-Sekerka/Stokes system for two-phase flows. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020467 [12] Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348 [13] Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380 [14] Huu-Quang Nguyen, Ya-Chi Chu, Ruey-Lin Sheu. On the convexity for the range set of two quadratic functions. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020169 [15] Xuhui Peng, Rangrang Zhang. Approximations of stochastic 3D tamed Navier-Stokes equations. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5337-5365. doi: 10.3934/cpaa.2020241 [16] Guido Cavallaro, Roberto Garra, Carlo Marchioro. Long time localization of modified surface quasi-geostrophic equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020336 [17] Yi-Hsuan Lin, Gen Nakamura, Roland Potthast, Haibing Wang. Duality between range and no-response tests and its application for inverse problems. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020072 [18] Kha Van Huynh, Barbara Kaltenbacher. Some application examples of minimization based formulations of inverse problems and their regularization. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020074 [19] Xuefei He, Kun Wang, Liwei Xu. Efficient finite difference methods for the nonlinear Helmholtz equation in Kerr medium. Electronic Research Archive, 2020, 28 (4) : 1503-1528. doi: 10.3934/era.2020079 [20] Anton A. Kutsenko. Isomorphism between one-Dimensional and multidimensional finite difference operators. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020270

2019 Impact Factor: 1.373