Geodesic
A new non-overlapping domain decomposition method for the 3-D Laplace exterior problem
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 21 (2018) no. 4, pp. 435-449.

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

We propose a method for solving the three-dimensional boundary value problems for the Laplace equation in an unbounded domain. It is based on the non-overlapping decomposition of the exterior domain to the two subdomains such that the initial problem is reduced to the two subproblems, namely, the exterior and the interior boundary value problems on a sphere. To solve the exterior boundary value problem, we propose a singularity isolation method. To cross-link the solutions at the interface of subdomains (a sphere), we introduce a special operator equation that is approximated by the system of linear algebraic equations. Such a system is solved by iterative methods in the Krylov subspaces. The method is illustrated by solving the model problems confirming its operability. 
@article{SJVM_2018_21_4_a6,
     author = {V. M. Sveshnikov and A. O. Savchenko and A. V. Petukhov},
     title = {A new non-overlapping domain decomposition method for the {3-D} {Laplace} exterior problem},
     journal = {Sibirskij \v{z}urnal vy\v{c}islitelʹnoj matematiki},
     pages = {435--449},
     publisher = {mathdoc},
     volume = {21},
     number = {4},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/SJVM_2018_21_4_a6/}
}
TY  - JOUR
AU  - V. M. Sveshnikov
AU  - A. O. Savchenko
AU  - A. V. Petukhov
TI  - A new non-overlapping domain decomposition method for the 3-D Laplace exterior problem
JO  - Sibirskij žurnal vyčislitelʹnoj matematiki
PY  - 2018
SP  - 435
EP  - 449
VL  - 21
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/SJVM_2018_21_4_a6/
LA  - ru
ID  - SJVM_2018_21_4_a6
ER  - 
%0 Journal Article
%A V. M. Sveshnikov
%A A. O. Savchenko
%A A. V. Petukhov
%T A new non-overlapping domain decomposition method for the 3-D Laplace exterior problem
%J Sibirskij žurnal vyčislitelʹnoj matematiki
%D 2018
%P 435-449
%V 21
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/SJVM_2018_21_4_a6/
%G ru
%F SJVM_2018_21_4_a6
V. M. Sveshnikov; A. O. Savchenko; A. V. Petukhov. A new non-overlapping domain decomposition method for the 3-D Laplace exterior problem. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 21 (2018) no. 4, pp. 435-449. http://geodesic.mathdoc.fr/item/SJVM_2018_21_4_a6/

[1] Quarteroni A., Valli A., Domain Decomposition Methods for Partial Differential Equations, Clarendon Press, Oxford, 1999 | MR | Zbl

[2] Dolean V., Jolivet P., Nataf F., An Introduction to Domain Decomposition Methods: Algorithms, Theory and Parallel Implementation, SIAM, Philadelphia, USA, 2015 | MR | Zbl

[3] Savchenko A. O., Il'in V. P., Butyugin D. S., “A method of solving an exterior three–dimensional boundary value problem for the Laplace equation”, J. of Applied and Industrial Mathematics, 10:2 (2016), 277–287 | DOI | DOI | MR | Zbl

[4] De-hao Yu., Ji-ming Wu., “A nonoverlapping domain decomposition method for exterior 3-D problem”, J. of Comput. Mathematics, 19:1 (2001), 77–86 | MR

[5] Langer U., Steinbach O., “Coupled finite and boundary element domain decomposition methods”, Boundary Element Analysis, Mathematical Aspects and Applications, Lect. Notes Appl. Comput. Mech., 29, eds. M. Schanz, O. Steinbach, Springer, Heidelberg, 2006, 61–95 | DOI | MR

[6] Ilin V. P., Metody i tekhnologii konechnykh elementov, Izd-vo IVMiMG SO RAN, Novosibirsk, 2007

[7] Sveshnikov V. M., “Construction of direct and iterative decomposition methods”, J. of Applied and Industrial Mathematics, 4:3 (2010), 431–440 | DOI | MR | Zbl

[8] Korneev V. D., Sveshnikov V. M., “Parallel algorithms and domain decomposition techniques for solving three-dimensional boundary value problems on quasistructured grids”, Numerical Analysis and Applications, 9:2 (2016), 141–149 | DOI | DOI | MR | Zbl

[9] Ilin V. P., Metody konechnykh raznostei i konechnykh ob'emov dlya ellipticheskikh uravnenii, Izd-vo IVMiMG SO RAN, Novosibirsk, 2001

[10] Lebedev V. I., Agoshkov V. I., Operatory Puankare–Steklova i ikh prilozheniya v analize, OVM AN SSSR, M., 1983 | MR

[11] Koshlyakov N. S., Gliner E. B., Smirnov M. M., Uravneniya v chastnykh proizvodnykh matematicheskoi fiziki, Vysshaya shkola, M., 1970

[12] Gradshtein I. S., Ryzhik I. M., Tablitsy integralov, summ, ryadov i proizvedenii, Fizmatgiz, M., 1963 | MR

[13] Lebedev V. I., Funktsionalnyi analiz i vychislitelnaya matematika, Fizmatlit, M., 2005

[14] http://sourceforge.net/projects/netgen-mesher/

[15] Faddeev D. K., Faddeeva V. N., Vychislitelnye metody lineinoi algebry, Fizmatlit, M., 1963 | MR