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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 Discrete Poisson equation ： ウィキペディア英語版 Discrete Poisson equation In mathematics, the discrete Poisson equation is the finite difference analog of the Poisson equation. In it, the discrete Laplace operator takes the place of the Laplace operator. The discrete Poisson equation is frequently used in numerical analysis as a stand-in for the continuous Poisson equation, although it is also studied in its own right as a topic in discrete mathematics. ==On a two-dimensional rectangular grid== Using the finite difference numerical method to discretize the 2-dimensional Poisson equation (assuming a uniform spatial discretization, $\backslash Delta\; x=\backslash Delta\; y$) on an ''m'' × ''n'' grid gives the following formula:〔.〕 :$$ ( ^2 u )_ = \frac (u_ + u_ + u_ + u_ - 4 u_) = g_ where $2\; \backslash le\; i\; \backslash le\; m-1$ and $2\; \backslash le\; j\; \backslash le\; n-1$. The preferred arrangement of the solution vector is to use natural ordering which, prior to removing boundary elements, would look like: :$$ \vec = \begin u_ , u_ , \ldots , u_ , u_ , u_ , \ldots , u_ , \ldots , u_ \end^T This will result in an ''mn'' × ''mn'' linear system: :$A\backslash vec\; =\; \backslash vec$ where :$$ A = \begin ~D & -I & ~0 & ~0 & ~0 & \ldots & ~0 \\ -I & ~D & -I & ~0 & ~0 & \ldots & ~0 \\ ~0 & -I & ~D & -I & ~0 & \ldots & ~0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\ ~0 & \ldots & ~0 & -I & ~D & -I & ~0 \\ ~0 & \ldots & \ldots & ~0 & -I & ~D & -I \\ ~0 & \ldots & \ldots & \ldots & ~0 & -I & ~D \end, $I$ is the ''m'' × ''m'' identity matrix, and $D$, also ''m'' × ''m'', is given by: :$$ D = \begin ~4 & -1 & ~0 & ~0 & ~0 & \ldots & ~0 \\ -1 & ~4 & -1 & ~0 & ~0 & \ldots & ~0 \\ ~0 & -1 & ~4 & -1 & ~0 & \ldots & ~0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \ddots & \vdots \\ ~0 & \ldots & ~0 & -1 & ~4 & -1 & ~0 \\ ~0 & \ldots & \ldots & ~0 & -1 & ~4 & -1 \\ ~0 & \ldots & \ldots & \ldots & ~0 & -1 & ~4 \end, 〔Golub, Gene H. and C. F. Van Loan, ''Matrix Computations, 3rd Ed.'', The Johns Hopkins University Press, Baltimore, 1996, pages 177–180.〕 and $\backslash vec$ is defined by :$$ \vec = -\Delta x^2\begin g_ , g_ , \ldots , g_ , g_ , g_ , \ldots , g_ , \ldots , g_ \end^T. For each $u\_$ equation, the columns of $D$ correspond to a block of $m$ components in $u$: :$$ \begin u_ , & u_ , & \ldots, & u_ , & u_ , & u_ , & \ldots , & u_ \end^ while the columns of $I$ to the left and right of $D$ each correspond to other blocks of $m$ components within $u$: :$$ \begin u_ , & u_ , & \ldots, & u_ , & u_ , & u_ , & \ldots , & u_ \end^ and :$$ \begin u_ , & u_ , & \ldots, & u_ , & u_ , & u_ , & \ldots , & u_ \end^ respectively. From the above, it can be inferred that there are $n$ block columns of $m$ in $A$. It is important to note that prescribed values of $u$ (usually lying on the boundary) would have their corresponding elements removed from $I$ and $D$. For the common case that all the nodes on the boundary are set, we have $2\; \backslash le\; i\; \backslash le\; m\; -\; 1$ and $2\; \backslash le\; j\; \backslash le\; n\; -\; 1$, and the system would have the dimensions (''m'' − 2)(''n'' − 2) × (''m'' − 2)(''n'' − 2), where $D$ and $I$ would have dimensions (''m'' − 2) × (''m'' − 2). 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「Discrete Poisson equation」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.