Numerical Analysis and Applications

, Volume 11, Issue 4, pp 346–358

# A New Non-Overlapping Domain Decomposition Method for a 3D Laplace Exterior Problem

• V. M. Sveshnikov
• A. O. Savchenko
• A. V. Petukhov
Article

## Abstract

We propose a method for solving three-dimensional boundary value problems for Laplace’s equation in an unbounded domain. It is based on non-overlapping decomposition of the exterior domain into two subdomains so that the initial problem is reduced to two subproblems, namely, exterior and interior boundary value problems on a sphere. To solve the exterior boundary value problem, we propose a singularity isolation method. To match the solutions on the interface between the subdomains (the sphere), we introduce a special operator equation approximated by a system of linear algebraic equations. This system is solved by iterative methods in Krylov subspaces. The performance of the method is illustrated by solving model problems.

## Keywords

exterior boundary value problems non-overlapping decomposition computation of integrals with singularities iterative methods in Krylov subspaces

## References

1. 1.
Quarteroni, A. and Valli, A., Domain DecompositionMethods for Partial Differential Equations, Oxford: Clarendon Press, 1999.
2. 2.
Dolean, V., Jolivet, P., and Nataf, F., An Introduction to Domain Decomposition Methods: Algorithms, Theory and Parallel Implementation, Philadelphia, USA: SIAM, 2015.
3. 3.
Savchenko, A.O., Il’in, V.P., and Butyugin, D.S., A Method of Solving an Exterior Three-Dimensional Boundary Value Problem for the Laplace Equation, J. Appl. Ind. Math., 2016, vol. 10, no. 2, pp. 41–53.
4. 4.
De-hao, Yu. and Ji-ming, Wu, A Nonoverlapping Domain DecompositionMethod for Exterior 3-D Problem, J. Comput. Math., 2001, vol. 19, no. 1, pp. 77–86.Google Scholar
5. 5.
Langer, U. and Steinbach, O., Coupled Finite and Boundary Element Domain Decomposition Methods, in Boundary Element Analysis. Mathematical Aspects and Applications, Schanz, M. and Steinbach, O., Eds., Heidelberg: Springer, 2006, pp. 61–95.
6. 6.
Il’in, V.P., Metody i tekhnologii konechnykh elementov (Methods and Technologies of Finite Elements), Novosibirsk: Publ. House of ICM&MG SB RAS, 2007.Google Scholar
7. 7.
Sveshnikov, V.M., Construction of Direct and Iterative DecompositionMethods, J. Appl. Ind. Math., 2010, vol. 4, no. 3, pp. 431–440.
8. 8.
Korneev, V.D. and Sveshnikov, V.M., Parallel Algorithms and Domain Decomposition Techniques for Solving Three-Dimensional Boundary Value Problems on Quasi-Structured Grids, Num. An. Appl., 2016, vol. 9, no. 2, pp. 141–149.
9. 9.
Il’in, V.P., Metody konechnykh raznostei i konechnykh ob’yomov dlya ellipticheskikh uravnenii (Finite Difference and Finite Volume Methods for Elliptic Equations), Novosibirsk: Publ. House of ICM&MG SB RAS, 2001.Google Scholar
10. 10.
Lebedev, V.I. and Agoshkov, V.I., Operatory Puankare–Steklova i ikh prilozheniya v analize (Poincare–Steklov Operators and Their Applications in Analysis),Moscow: OVM Akad. Nauk SSSR, 1983.
11. 11.
Koshlyakov, N.S., Gliner, E.B., and Smirnov, M.M., Uravneniya v chastnykh proizvodnykh matematicheskoi fiziki (Partial Differential Equations ofMathematical Physics),Moscow: Vysshaya shkola, 1970.Google Scholar
12. 12.
Gradshteyn, I.S. and Ryzhik, I.M., Tablitsy integralov, summ, ryadov i proizvedenii (Tables of Integrals, Series, and Products),Moscow: Fizmatgiz, 1963.Google Scholar
13. 13.
Lebedev, V.I., Funktsional’nyi analiz i vychislitel’naya matematika (Functional Analysis in Computational Mathematics),Moscow: Fizmatlit, 2005.Google Scholar
14. 14.
15. 15.
Faddeev, D.K. and Faddeeva, V.N., Vychislitel’nye metody lineynoi algebry (Computational Methods of Linear Algebra),Moscow: Fizmatlit, 1963.Google Scholar

## Authors and Affiliations

• V. M. Sveshnikov
• 1
• 2
Email author
• A. O. Savchenko
• 1
• A. V. Petukhov
• 1
1. 1.Institute of Computational Mathematics and Mathematical Geophysics, Siberian BranchRussian Academy of SciencesNovosibirskRussia
2. 2.Novosibirsk State UniversityNovosibirskRussia