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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 Schur complement method ： ウィキペディア英語版 Schur complement method In numerical analysis, the Schur complement method, named after Issai Schur, is the basic and the earliest version of non-overlapping domain decomposition method, also called iterative substructuring. A finite element problem is split into non-overlapping subdomains, and the unknowns in the interiors of the subdomains are eliminated. The remaining Schur complement system on the unknowns associated with subdomain interfaces is solved by the conjugate gradient method. == The method and implementation == Suppose we want to solve the Poisson equation :$-\backslash Delta\; u\; =\; f,\; \backslash qquad\; u|\_\; =\; 0$ on some domain Ω. When we discretize this problem we get an ''N''-dimensional linear system ''AU = F''. The Schur complement method splits up the linear system into sub-problems. To do so, divide Ω into two subdomains Ω1, Ω2 which share an interface Γ. Let ''U''1, ''U''2 and ''U''Γ be the degrees of freedom associated with each subdomain and with the interface. We can then write the linear system as : where ''F''1, ''F''2 and ''F''Γ are the components of the load vector in each region. The Schur complement method proceeds by noting that we can find the values on the interface by solving the smaller system :$\backslash Sigma\; U\_\backslash Gamma\; =\; F\_\backslash Gamma\; -\; A\_A\_^F\_1\; -\; A\_A\_^F\_2,$ for the interface values ''U''Γ, where we define the ''Schur complement'' matrix :$\backslash Sigma\; =\; A\_\; -\; A\_A\_^A\_\; -\; A\_A\_^A\_.$ The important thing to note is that the computation of any quantities involving $A\_^$ or $A\_^$ involves solving decoupled Dirichlet problems on each domain, and these can be done in parallel. Consequently, we need not store the Schur complement matrix explicitly; it is sufficient to know how to multiply a vector by it. Once we know the values on the interface, we can find the interior values using the two relations :$A\_U\_1\; =\; F\_1\; -\; A\_U\_\backslash Gamma,\; \backslash qquad\; A\_U\_2\; =\; F\_2\; -\; A\_U\_\backslash Gamma,$ which can both be done in parallel. The multiplication of a vector by the Schur complement is a discrete version of the Poincaré–Steklov operator, also called the Dirichlet to Neumann mapping. 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「Schur complement method」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.