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November  2013, 7(4): 1271-1293. doi: 10.3934/ipi.2013.7.1271

Factorization method for the inverse Stokes problem

Armin Lechleiter 1, and  Tobias Rienmüller 1,

1. 

Zentrum für Technomathematik, Universität Bremen, 28359 Bremen, Germany, Germany

Received  January 2013 Revised  August 2013 Published  November 2013

We propose an imaging technique for the detection of porous inclusions in a stationary flow based on the Factorization method. The stationary flow is described by the Stokes-Brinkmann equations, a standard model for a flow through a (partially) porous medium, involving the deformation tensor of the flow and the permeability tensor of the porous inclusion. On the boundary of the domain we prescribe Robin boundary conditions that provide the freedom to model, e.g., in- or outlets for the flow. The direct Stokes-Brinkmann problem to find a velocity field and a pressure for given boundary data is a mixed variational problem lacking coercivity due to the indefinite pressure part. It is well-known that indefinite problems are difficult to tackle theoretically using Factorization methods. Interestingly, the Factorization method can nevertheless be applied to this non-coercive problem, as long as one uses data consisting only of velocity measurements. We provide numerical experiments showing the feasibility of the proposed technique.
Keywords: Inverse boundary value problem, factorization method., Stokes equation.
Mathematics Subject Classification: Primary: 35R30, 65N21; Secondary: 76D0.
Citation: 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
References:
[1]

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M. Krotkiewski, I. Ligaarden, K.-A. Lie and D. W. Schmid, On the importance of the Stokes-Brinkman equations for computing effective permeability in carbonate karst reservoirs, Commun. Comput. Phys., 10 (2011), 1315-1332. Google Scholar

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W. C. H. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000.  Google Scholar

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C. L. M. H. Navier, Sur les lois du mouvement des fluides, Mem. Acad. R. Sci. Inst. Fr., 6 (1827), 389-440. Google Scholar

[23]

P. Popov, Y. Efendiev and G. Qin, Multiscale modeling and simulations of flows in naturally fractured karst reservoirs, Commun. Comput. Phys., 6 (2009), 162-184. doi: 10.4208/cicp.2009.v6.p162.  Google Scholar

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C. Pozrikidis, Boundary Integral and Singularity Methods for Linearized Viscous Flow, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511624124.  Google Scholar

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V. Tsiporin, Charakterisierung Eines Gebiets Durch Spektraldaten eines Dirichletproblems zur Stokesgleichung, (German) [Characterization of a Domain via the Spectral data of a Dirichlet Problem for the Stokes Equation], PhD thesis, Georg-August-Universität Göttingen, 2003. Google Scholar

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Q. M. Z. Zia and R. Potthast, Flow and shape reconstructions from remote measurements, Math. Meth. Appl. Sci., 36 (2013), 1171-1186. doi: 10.1002/mma.2670.  Google Scholar

show all references

References:
[1]

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

[2]

C. J. Alves, R. Kress and A. L. Silvestre, Integral equations for an inverse boundary value problem for the two-dimensional stokes equations, Journal of Inverse and Ill-Posed Problems, 15 (2007), 461-481. doi: 10.1515/jiip.2007.026.  Google Scholar

[3]

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

[4]

A. Ballerini, Stable determination of a body immersed in a fluid: The nonlinear stationary case, Applicable Analysis, 92 (2013), 460-481. doi: 10.1080/00036811.2011.628173.  Google Scholar

[5]

M. Badra, F. Caubet and M. Dambrine, Detecting an obstacle immersed in a fluid by shape optimization methods, M3AS, 21 (2011), 2069-2101. doi: 10.1142/S0218202511005660.  Google Scholar

[6]

S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Springer Verlag, New York, 2008. doi: 10.1007/978-0-387-75934-0.  Google Scholar

[7]

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

[8]

A. Ern and J. L. Guermond, Theory and Practice of Finite Elements, Springer Verlag, New York, 2004.  Google Scholar

[9]

L. C. Evans, Partial Differential Equations, American Mathematical Society, Providence, 1998.  Google Scholar

[10]

G. P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Steady-State Problems, Springer Verlag, New York, 2011. doi: 10.1007/978-0-387-09620-9.  Google Scholar

[11]

M. Hanke and M. Brühl, Recent progress in electrical impedance tomography, Inverse Problems, 90 (2003), 65-90. doi: 10.1088/0266-5611/19/6/055.  Google Scholar

[12]

N. Hyvönen, H. Hakula and S. Pursiainen, Numerical implementation of the factorization method within the complete electrode model of impedance tomography, Inverse Problems and Imaging, 1 (2007), 299-317. doi: 10.3934/ipi.2007.1.299.  Google Scholar

[13]

H. Haddar and G. Migliorati, Numerical analysis of the Factorization Method for EIT with piecewise constant uncertain background, Inverse Problems, 29 (2013), 065009. doi: 10.1088/0266-5611/29/6/065009.  Google Scholar

[14]

H. Heck, G. Uhlmann and J.-N. Wang, Reconstruction of obstacles immersed in an incompressible fluid, Inverse Problems and Imaging, 1 (2007), 63-76. doi: 10.3934/ipi.2007.1.63.  Google Scholar

[15]

G. C. Hsiao and W. L. Wendland, Boundary Integral Equations, Springer Verlag, New York, 2008. doi: 10.1007/978-3-540-68545-6.  Google Scholar

[16]

A. Kirsch and N. Grinberg, The Factorization Method for Inverse Problems, Oxford University Press, Oxford, 2008.  Google Scholar

[17]

A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, Springer Verlag, New York, 2011. doi: 10.1007/978-1-4419-8474-6.  Google Scholar

[18]

M. Krotkiewski, I. Ligaarden, K.-A. Lie and D. W. Schmid, On the importance of the Stokes-Brinkman equations for computing effective permeability in carbonate karst reservoirs, Commun. Comput. Phys., 10 (2011), 1315-1332. Google Scholar

[19]

O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, London, 1969.  Google Scholar

[20]

M. Lewicka and S. Müller, The uniform Korn-Poincaré inequality in thin domains, Annales de l'Institut Henri Poincare - Non Linear Analysis, 28 (2011), 443-469. doi: 10.1016/j.anihpc.2011.03.003.  Google Scholar

[21]

W. C. H. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000.  Google Scholar

[22]

C. L. M. H. Navier, Sur les lois du mouvement des fluides, Mem. Acad. R. Sci. Inst. Fr., 6 (1827), 389-440. Google Scholar

[23]

P. Popov, Y. Efendiev and G. Qin, Multiscale modeling and simulations of flows in naturally fractured karst reservoirs, Commun. Comput. Phys., 6 (2009), 162-184. doi: 10.4208/cicp.2009.v6.p162.  Google Scholar

[24]

C. Pozrikidis, Boundary Integral and Singularity Methods for Linearized Viscous Flow, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511624124.  Google Scholar

[25]

V. Tsiporin, Charakterisierung Eines Gebiets Durch Spektraldaten eines Dirichletproblems zur Stokesgleichung, (German) [Characterization of a Domain via the Spectral data of a Dirichlet Problem for the Stokes Equation], PhD thesis, Georg-August-Universität Göttingen, 2003. Google Scholar

[26]

Q. M. Z. Zia and R. Potthast, Flow and shape reconstructions from remote measurements, Math. Meth. Appl. Sci., 36 (2013), 1171-1186. doi: 10.1002/mma.2670.  Google Scholar

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