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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 Bramble–Hilbert lemma ： ウィキペディア英語版 Bramble–Hilbert lemma In mathematics, particularly numerical analysis, the Bramble–Hilbert lemma, named after James H. Bramble and Stephen Hilbert, bounds the error of an approximation of a function $\backslash textstyle\; u$ by a polynomial of order at most $\backslash textstyle\; m-1$ in terms of derivatives of $\backslash textstyle\; u$ of order $\backslash textstyle\; m$. Both the error of the approximation and the derivatives of $\backslash textstyle\; u$ are measured by $\backslash textstyle\; L^$ norms on a bounded domain in $\backslash textstyle\; \backslash mathbb^$. This is similar to classical numerical analysis, where, for example, the error of linear interpolation $\backslash textstyle\; u$ can be bounded using the second derivative of $\backslash textstyle\; u$. However, the Bramble–Hilbert lemma applies in any number of dimensions, not just one dimension, and the approximation error and the derivatives of $\backslash textstyle\; u$ are measured by more general norms involving averages, not just the maximum norm. Additional assumptions on the domain are needed for the Bramble–Hilbert lemma to hold. Essentially, the boundary of the domain must be "reasonable". For example, domains that have a spike or a slit with zero angle at the tip are excluded. Lipschitz domains are reasonable enough, which includes convex domains and domains with continuously differentiable boundary. The main use of the Bramble–Hilbert lemma is to prove bounds on the error of interpolation of function $\backslash textstyle\; u$ by an operator that preserves polynomials of order up to $\backslash textstyle\; m-1$, in terms of the derivatives of $\backslash textstyle\; u$ of order $\backslash textstyle\; m$. This is an essential step in error estimates for the finite element method. The Bramble–Hilbert lemma is applied there on the domain consisting of one element (or, in some superconvergence results, a small number of elements). ==The one-dimensional case== Before stating the lemma in full generality, it is useful to look at some simple special cases. In one dimension and for a function $\backslash textstyle\; u$ that has $\backslash textstyle\; m$ derivatives on interval $\backslash textstyle\; \backslash left(\; a,b\backslash right)$, the lemma reduces to :$\backslash inf\_-v^\backslash bigr\backslash Vert\_\backslash leq\; C\backslash left(\; m\backslash right)\; \backslash left(\; b-a\backslash right)\; ^\backslash bigl\backslash Vert\; u^\backslash bigr\backslash Vert\_,$ where $\backslash textstyle\; P\_$ is the space of all polynomials of order at most $\backslash textstyle\; m-1$. In the case when $\backslash textstyle\; p=\backslash infty$, $\backslash textstyle\; m=2$, $\backslash textstyle\; k=0$, and $\backslash textstyle\; u$ is twice differentiable, this means that there exists a polynomial $\backslash textstyle\; v$ of degree one such that for all $\backslash textstyle\; x\backslash in\backslash left(\; a,b\backslash right)$, :$\backslash left\backslash vert\; u\backslash left(\; x\backslash right)\; -v\backslash left(\; x\backslash right)\; \backslash right\backslash vert\; \backslash leq\; C\backslash left(\; b-a\backslash right)\; ^\backslash sup\_\backslash left\backslash vert\; u^\backslash right\backslash vert.$ This inequality also follows from the well-known error estimate for linear interpolation by choosing $\backslash textstyle\; v$ as the linear interpolant of $\backslash textstyle\; u$. 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「Bramble–Hilbert lemma」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.