Geodesic
A numerical method for determining the inhomogeneity boundary in the Dirichlet problem for the Laplace equation in a piecewise-homogeneous medium
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 50 (2010) no. 8, pp. 1462-1470 
S. V. Gavrilov; A. M. Denisov. A numerical method for determining the inhomogeneity boundary in the Dirichlet problem for the Laplace equation in a piecewise-homogeneous medium. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 50 (2010) no. 8, pp. 1462-1470. http://geodesic.mathdoc.fr/item/ZVMMF_2010_50_8_a8/
@article{ZVMMF_2010_50_8_a8,
     author = {S. V. Gavrilov and A. M. Denisov},
     title = {A numerical method for determining the inhomogeneity boundary in the {Dirichlet} problem for the {Laplace} equation in a piecewise-homogeneous medium},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1462--1470},
     year = {2010},
     volume = {50},
     number = {8},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2010_50_8_a8/}
}
TY  - JOUR
AU  - S. V. Gavrilov
AU  - A. M. Denisov
TI  - A numerical method for determining the inhomogeneity boundary in the Dirichlet problem for the Laplace equation in a piecewise-homogeneous medium
JO  - Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
PY  - 2010
SP  - 1462
EP  - 1470
VL  - 50
IS  - 8
UR  - http://geodesic.mathdoc.fr/item/ZVMMF_2010_50_8_a8/
LA  - ru
ID  - ZVMMF_2010_50_8_a8
ER  - 
%0 Journal Article
%A S. V. Gavrilov
%A A. M. Denisov
%T A numerical method for determining the inhomogeneity boundary in the Dirichlet problem for the Laplace equation in a piecewise-homogeneous medium
%J Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki
%D 2010
%P 1462-1470
%V 50
%N 8
%U http://geodesic.mathdoc.fr/item/ZVMMF_2010_50_8_a8/
%G ru
%F ZVMMF_2010_50_8_a8

Voir la notice de l'article provenant de la source Math-Net.Ru

The Dirichlet problem for Laplace’s equation in a two-dimensional domain filled with a piecewise homogeneous medium is considered. The boundary of the inhomogeneity is assumed to be unknown. The inverse problem of determining the inhomogeneity boundary from additional information on the solution of the Dirichlet problem is considered. A numerical method based on the linearization of the nonlinear operator equation for the unknown boundary is proposed for solving the inverse problem. The results of numerical experiments are presented.

[1] Zakharov E. E., Kalinin A. V., “Chislennoe reshenie trekhmernoi zadachi Dirikhle v kusochno-odnorodnoi srede metodom granichnykh integralnykh uravnenii”, Zh. vychisl. matem. i matem. fiz., 49:7 (2009), 1197–1206 | MR

[2] Denisov A. M., Zakharov E. V., Kalinin A. V., Kalinin V. V., “Chislennye metody resheniya nekotorykh obratnykh zadach elektrofiziologii serdtsa”, Differents. ur-niya, 45:7 (2009), 1014–1022 | MR | Zbl

[3] McIver M., “Characterization of surface-breaking cracks in metal sheets by using AC electric fields”, Proc. Roy Soc. A, 421 (1989), 179–194 | DOI | Zbl

[4] Friedman A., Vogelius M., “Determining cracks by boundary measurements”, Indiana Math. J., 38 (1989), 527–556 | DOI | MR | Zbl

[5] McIver M., “An inverse problem in electromagnetic crack detection”, IMA J. Appl. Math., 47 (1991), 127–145 | DOI | MR | Zbl

[6] Nishimura N., Kobayashi S., “A boundary integral-equation method for an inverse problem related to crack detection”, Int. J. Numer. Meth. Engng., 32 (1991), 1371–1387 | DOI | Zbl

[7] Bryan K., Vogelius M., “A uniqueness result concerning the identification of a collection of cracks from finitely many electrostatic boundary measurements”, SIAM. J. Math. Analys., 23 (1992), 950–958 | DOI | MR | Zbl

[8] Bryan K., Vogelius M., “A computational algorithm to determine crack locations from electrostatic boundary measurements the case of multiple cracks”, Int. J. Engng. Sci., 32 (1994), 579–603 | DOI | MR | Zbl

[9] Atkinson C., Aparicio N. D., “An inverse problem method for crack detection in elastic materials under anti-plane strain”, Proc. Roy. Soc. A, 445 (1994), 637–652 | DOI | Zbl

[10] Alessandrini G., Beretta E., Santosa F., Vessella S., “Stability in crack determination from electrostatic measurements at the boundary numerical investigation”, Inverse Problems, 11 (1995), L17–24 | DOI | MR

[11] Kress R., “Inverse boundary value problems in potential theory”, Cubo Math. Educational, 3 (2001), 393–414 | MR | Zbl

[12] Kress R., “Inverse Dirichlet problem and conformal mapping”, Math. Comp. Simulat., 66 (2004), 255–265 | DOI | MR | Zbl