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Journal of Inverse and Ill-posed Problems

Editor-in-Chief: Kabanikhin, Sergey I.


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1569-3945
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Volume 24, Issue 6

Issues

Numerical solution of an elliptic 3-dimensional Cauchy problem by the alternating method and boundary integral equations

Ihor Borachok / Roman Chapko / B. Tomas JohanssonORCID iD: http://orcid.org/0000-0001-9066-7922
Published Online: 2016-01-28 | DOI: https://doi.org/10.1515/jiip-2015-0053

Abstract

We consider the Cauchy problem for the Laplace equation in 3-dimensional doubly-connected domains, that is the reconstruction of a harmonic function from knowledge of the function values and normal derivative on the outer of two closed boundary surfaces. We employ the alternating iterative method, which is a regularizing procedure for the stable determination of the solution. In each iteration step, mixed boundary value problems are solved. The solution to each mixed problem is represented as a sum of two single-layer potentials giving two unknown densities (one for each of the two boundary surfaces) to determine; matching the given boundary data gives a system of boundary integral equations to be solved for the densities. For the discretisation, Weinert’s method [24] is employed, which generates a Galerkin-type procedure for the numerical solution via rewriting the boundary integrals over the unit sphere and expanding the densities in terms of spherical harmonics. Numerical results are included as well.

Keywords: Alternating method; Cauchy problem; integral equation; Laplace equation

MSC 2010: 35R25; 35J05; 65R20

References

  • [1]

    Abramowitz M. and Stegun I. A., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards Applied Mathematics Series, Washington, 1972. Google Scholar

  • [2]

    Babenko C., Chapko R. and Johansson B. T., Numerical solution of the Cauchy problem for the Laplace equation in simply connected domains via the alternating method and boundary integrals, Advances in Boundary Integral Methods, University of Brighton, Brighton (2015), 200–209. Google Scholar

  • [3]

    Baranger T. N., Johansson B. T. and Rischette R., On the alternating method for Cauchy problems and its finite element discretisation, Applied Inverse Problems (Gothenburg 2011), Springer Proc. Math. Stat. 48, Springer, New York (2013), 183–197. Google Scholar

  • [4]

    Berntsson F., Kozlov V. A., Mpinganzima L. and Turesson B. O., An alternating iterative procedure for the Cauchy problem for the Helmholtz equation, Inverse Probl. Sci. Eng. 22 (2014), 45–62. Google Scholar

  • [5]

    Cakoni F. and Kress R., Integral equations for inverse problems in corrosion detection from partial Cauchy data, Inverse Probl. Imaging 1 (2007), 229–245. Google Scholar

  • [6]

    Cao H., Klibanov M. V. and Pereverzev S. V., A Carleman estimate and the balancing principle in the quasi-reversibility method for solving the Cauchy problem for the Laplace equation, Inverse Problems 25 (2009), 1–21. Google Scholar

  • [7]

    Chapko R. and Johansson B. T., A direct integral equation method for a Cauchy problem for the Laplace equation in 3-dimensional semi-infinite domains, CMES Comput. Model. Eng. Sci. 85 (2012), 105–128. Google Scholar

  • [8]

    Chapko R. and Johansson B. T., On the numerical solution of a Cauchy problem for the Laplace equation via a direct integral equation approach, Inverse Probl. Imaging 6 (2012), 25–36. Google Scholar

  • [9]

    Chapko R., Johansson B. T. and Protsyuk O., A direct boundary integral equation method for the numerical construction of harmonic functions in three-dimensional layered domains containing a cavity, Int. J. Comput. Math. 89 (2012), 1448–1462. Google Scholar

  • [10]

    Ganesh M. and Graham I. G., A high-order algorithm for obstacle scattering in three dimensions, J. Comput. Phys. 198 (2004), 211–242. Google Scholar

  • [11]

    Ganesh M., Graham I. G. and Sivaloganathan J. A., New spectral boundary integral collocation method for three-dimensional potential problems, SIAM J. Numer. Anal. 35 (1998), 778–805. Google Scholar

  • [12]

    Graham I. G. and Sloan I. H., Fully discrete spectral boundary integral methods for Helmholtz problems on smooth closed surfaces in 3, Numer. Math. 92 (2002), 289–323. Google Scholar

  • [13]

    Hào D. N. and Quyen T. N. T., Finite element methods for coefficient identification in an elliptic equation, Appl. Anal. 93 (2014), 1533–1566. Google Scholar

  • [14]

    Hasanov A. and Slodička M., An analysis of inverse source problems with final time measured output data for the heat conduction equation: A semigroup approach, Appl. Math. Lett. 26 (2013), 207–214. Google Scholar

  • [15]

    Hsiao G. and Wendland W., Boundary Integral Equations, Springer, Berlin, 2008. Google Scholar

  • [16]

    Ivanyshyn O. and Kress R., Identification of sound-soft 3D obstacles from phaseless data, Inverse Probl. Imaging 4 (2010), 131–149. Google Scholar

  • [17]

    Kabanikhin S. I. and Karchevsky A. L., Optimization method for solving the Cauchy problem for an elliptic equation, J. Inverse Ill-Posed Probl. 3 (1995), 21–46. Google Scholar

  • [18]

    Kabanikhin S. I. and Shishlenin M. A., Direct and iteration methods for solving inverse and ill-posed problems, Sib. Èlektron. Mat. Izv. 5 (2008), 595–608. Google Scholar

  • [19]

    Kozlov V. A. and Maz’ya V. G., On iterative procedures for solving ill-posed boundary value problems that preserve differential equations (in Russian), Algebra i Analiz 1 (1989), 144–170; translation in Leningrad Math. J. 1 (1990), 1207–1228. Google Scholar

  • [20]

    Lesnic D., Elliott L. and Ingham D. B., An iterative boundary element method for solving numerically the Cauchy problem for the Laplace equation, Eng. Anal. Bound. Elem. 20 (1997), 123–133. Google Scholar

  • [21]

    Marin L., Elliott L., Heggs P. J., Ingham D. B., Lesnic D. and Wen X., An alternating iterative algorithm for the Cauchy problem associated to the Helmholtz equation, Comput. Methods Appl. Mech. Engrg. 192 (2003), 709–722. Google Scholar

  • [22]

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

  • [23]

    Morozov V. A., On the solution of functional equations by the method of regularization (in Russian), Dokl. Akad. Nauk SSSR 167 (1966), 510–512; translation in Sov. Math. Dokl. 7 (1966), 414–417. Google Scholar

  • [24]

    Wienert L., Die numerische Approximation von Randintegraloperatoren für die Helmholtzgleichung im ,3 Ph.D. thesis, University of Göttingen, 1990. Google Scholar

About the article

Received: 2015-05-21

Revised: 2015-09-12

Accepted: 2015-12-23

Published Online: 2016-01-28

Published in Print: 2016-12-01


Citation Information: Journal of Inverse and Ill-posed Problems, Volume 24, Issue 6, Pages 711–725, ISSN (Online) 1569-3945, ISSN (Print) 0928-0219, DOI: https://doi.org/10.1515/jiip-2015-0053.

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