A method for solving an exterior three-dimensional boundary value problem for the Laplace equation

We develop and experimentally study the algorithms for solving three-dimensionalmixed boundary value problems for the Laplace equation in unbounded domains. These algorithms are based on the combined use of the finite elementmethod and an integral representation of the solution in a homogeneous spac...

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Bibliographic Details
Published in:Journal of applied and industrial mathematics 2016-04, Vol.10 (2), p.277-287
Main Authors: Savchenko, A. O., Il’in, V. P., Butyugin, D. S.
Format: Article
Language:English
Subjects:
Algorithms
Boundaries
Boundary conditions
Boundary value problems
Computational mathematics
Convergence
Integrals
Iterative methods
Laplace transforms
Mathematical analysis
Mathematical models
Mathematics
Mathematics and Statistics
Representations
Studies
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Summary:We develop and experimentally study the algorithms for solving three-dimensionalmixed boundary value problems for the Laplace equation in unbounded domains. These algorithms are based on the combined use of the finite elementmethod and an integral representation of the solution in a homogeneous space. The proposed approach consists in the use of the Schwarz alternating method with consecutive solution of the interior and exterior boundary value problems in the intersecting subdomains on whose adjoining boundaries the iterated interface conditions are imposed. The convergence of the iterative method is proved. The convergence rate of the iterative process is studied analytically in the case when the subdomains are spherical layers with the known exact representations of all consecutive approximations. In this model case, the influence of the algorithm parameters on the method efficiency is analyzed. The approach under study is implemented for solving a problem with a sophisticated configuration of boundaries while using a high precision finite elementmethod to solve the interior boundary value problems. The convergence rate of the iterations and the achieved accuracy of the computations are illustrated with some numerical experiments.
ISSN:1990-4789
1990-4797
DOI:10.1134/S1990478916020125