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Title: Review of the methods of reflections
Authors: Ciaramella, GabrieleGander, Martin J.Halpern, LaurenceSalomon, Julien
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Issue Date: 2017
Published in: Oberwolfach Preprints (OWP), Volume 2017-27, ISSN 1864-7596
Publisher: Oberwolfach : Mathematisches Forschungsinstitut Oberwolfach
Abstract: The methods of reflections were invented to obtain approximate solutions of the motion of more than one particle in a given environment, provided that one can represent the solution for one particle rather easily. This motivation is quite similar to the motivation of the Schwarz domain decomposition method, which was invented to prove existence and uniqueness of solutions of the Laplace equation on complicated domains, which are composed of simpler ones, for which existence and uniqueness of solutions was known. Like for Schwarz methods, there is also an alternating and a parallel method of reflections, but interestingly, the parallel method is not always convergent. We carefully trace in this paper the historical development of these methods of reflections, give several precise mathematical formulations, an equivalence result with the alternating Schwarz method for two particles, and also an analysis for a one dimensional model problem with three particles of the alternating, parallel, and a recent averaged parallel method of reflections.
Keywords: Alternating method of reections; Parallel method of reections; Averaged parallel method of reections; alternating Schwarz method; stationary iterative methods; Laplace's equation
DDC: 510
License: CC BY-SA 4.0 Unported
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Appears in Collections:Mathematik

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