# 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] Laurent Bourgeois, Jérémi Dardé. A quasi-reversibility approach to solve the inverse obstacle problem. Inverse Problems & Imaging, 2010, 4 (3) : 351-377. doi: 10.3934/ipi.2010.4.351 [2] Jérémi Dardé. Iterated quasi-reversibility method applied to elliptic and parabolic data completion problems. Inverse Problems & Imaging, 2016, 10 (2) : 379-407. doi: 10.3934/ipi.2016005 [3] Zhenlin Guo, Ping Lin, Guangrong Ji, Yangfan Wang. Retinal vessel segmentation using a finite element based binary level set method. Inverse Problems & Imaging, 2014, 8 (2) : 459-473. doi: 10.3934/ipi.2014.8.459 [4] Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271 [5] Yinnian He, Yanping Lin, Weiwei Sun. Stabilized finite element method for the non-stationary Navier-Stokes problem. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 41-68. doi: 10.3934/dcdsb.2006.6.41 [6] Mohsen Tadi. A computational method for an inverse problem in a parabolic system. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 205-218. doi: 10.3934/dcdsb.2009.12.205 [7] Wangtao Lu, Shingyu Leung, Jianliang Qian. An improved fast local level set method for three-dimensional inverse gravimetry. Inverse Problems & Imaging, 2015, 9 (2) : 479-509. doi: 10.3934/ipi.2015.9.479 [8] Cornel M. Murea, H. G. E. Hentschel. A finite element method for growth in biological development. Mathematical Biosciences & Engineering, 2007, 4 (2) : 339-353. doi: 10.3934/mbe.2007.4.339 [9] Martin Burger, José A. Carrillo, Marie-Therese Wolfram. A mixed finite element method for nonlinear diffusion equations. Kinetic & Related Models, 2010, 3 (1) : 59-83. doi: 10.3934/krm.2010.3.59 [10] Jiaping Yu, Haibiao Zheng, Feng Shi, Ren Zhao. Two-grid finite element method for the stabilization of mixed Stokes-Darcy model. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 387-402. doi: 10.3934/dcdsb.2018109 [11] Christos V. Nikolopoulos, Georgios E. Zouraris. Numerical solution of a non-local elliptic problem modeling a thermistor with a finite element and a finite volume method. Conference Publications, 2007, 2007 (Special) : 768-778. doi: 10.3934/proc.2007.2007.768 [12] Jun Lai, Ming Li, Peijun Li, Wei Li. A fast direct imaging method for the inverse obstacle scattering problem with nonlinear point scatterers. Inverse Problems & Imaging, 2018, 12 (3) : 635-665. doi: 10.3934/ipi.2018027 [13] Zhousheng Ruan, Sen Zhang, Sican Xiong. Solving an inverse source problem for a time fractional diffusion equation by a modified quasi-boundary value method. Evolution Equations & Control Theory, 2018, 7 (4) : 669-682. doi: 10.3934/eect.2018032 [14] Binjie Li, Xiaoping Xie, Shiquan Zhang. New convergence analysis for assumed stress hybrid quadrilateral finite element method. Discrete & Continuous Dynamical Systems - B, 2017, 22 (7) : 2831-2856. doi: 10.3934/dcdsb.2017153 [15] Kun Wang, Yinnian He, Yueqiang Shang. Fully discrete finite element method for the viscoelastic fluid motion equations. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 665-684. doi: 10.3934/dcdsb.2010.13.665 [16] Junjiang Lai, Jianguo Huang. A finite element method for vibration analysis of elastic plate-plate structures. Discrete & Continuous Dynamical Systems - B, 2009, 11 (2) : 387-419. doi: 10.3934/dcdsb.2009.11.387 [17] So-Hsiang Chou. An immersed linear finite element method with interface flux capturing recovery. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2343-2357. doi: 10.3934/dcdsb.2012.17.2343 [18] Donald L. Brown, Vasilena Taralova. A multiscale finite element method for Neumann problems in porous microstructures. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1299-1326. doi: 10.3934/dcdss.2016052 [19] Qingping Deng. A nonoverlapping domain decomposition method for nonconforming finite element problems. Communications on Pure & Applied Analysis, 2003, 2 (3) : 297-310. doi: 10.3934/cpaa.2003.2.297 [20] Runchang Lin. A robust finite element method for singularly perturbed convection-diffusion problems. Conference Publications, 2009, 2009 (Special) : 496-505. doi: 10.3934/proc.2009.2009.496

2018 Impact Factor: 1.469