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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Neumann-Neumann methods ： ウィキペディア英語版 Neumann–Neumann methods In mathematics, Neumann–Neumann methods are domain decomposition preconditioners named so because they solve a Neumann problem on each subdomain on both sides of the interface between the subdomains.〔A. Klawonn and O. B. Widlund, ''FETI and Neumann–Neumann iterative substructuring methods: connections and new results'', Comm. Pure Appl. Math., 54 (2001), pp. 57–90.〕 Just like all domain decomposition methods, so that the number of iterations does not grow with the number of subdomains, Neumann–Neumann methods require the solution of a coarse problem to provide global communication. The balancing domain decomposition is a Neumann–Neumann method with a special kind of coarse problem. More specifically, consider a domain Ω, on which we wish to solve the Poisson equation :$-\backslash Delta\; u\; =\; f,\; \backslash qquad\; u|\_\; =\; 0$ for some function ''f''. Split the domain into two non-overlapping subdomains Ω1 and Ω2 with common boundary Γ and let ''u''1 and ''u''2 be the values of ''u'' in each subdomain. At the interface between the two subdomains, the two solutions must satisfy the matching conditions :$u\_1\; =\; u\_2,\; \backslash qquad\; \backslash partial\_nu\_1\; =\; \backslash partial\_nu\_2$ where ''n'' is the unit normal vector to Γ. An iterative method for approximating each ui satisfying the matching conditions is to first solve the decoupled problems (i=1,2) :$-\backslash Delta\; u\_i^\; =\; f\_i,\; \backslash qquad\; u\_i^|\_\; =\; 0,\; \backslash quad\; u^\_i|\_\backslash Gamma\; =\; \backslash lambda^$ for some function λ(k) on Γ. We then solve the two Neumann problems :$-\backslash Delta\backslash psi\_i^\; =\; 0,\; \backslash qquad\; \backslash psi\_i^|\_\; =\; 0,\; \backslash quad\; \backslash partial\_n\backslash psi\_i^\; =\; \backslash partial\_nu\_1^\; -\; \backslash partial\_nu\_2^.$ We then obtain the next iterate by setting :$\backslash lambda^\; =\; \backslash lambda^\; -\; \backslash omega(\backslash theta\_1\backslash psi\_1^|\_\backslash Gamma\; -\; \backslash theta\_2\backslash psi\_2^|\_\backslash Gamma)$ for some parameters ω, θ1 and θ2. This procedure can be viewed as a Richardson iteration for the iterative solution of the equations arising from the Schur complement method.〔A. Quarteroni and A. Valli, ''Domain Decomposition Methods for Partial Differential Equations'', Oxford Science Publications 1999.〕 This continuous iteration can be discretized by the finite element method and then solved—in parallel—on a computer. The extension to more subdomains is straightforward, but using this method as stated as a preconditioner for the Schur complement system is not scalable with the number of subdomains; hence the need for a global coarse solve. ==See also== * Neumann–Dirichlet method 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Neumann–Neumann methods」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.