翻訳と辞書 Words near each other ・ Cædwalla of Wessex ・ Cælin ・ Cæremoniale Episcoporum ・ Cæsar Bang ・ Cæsar Clement ・ Cæsar Peter Møller Boeck ・ Cèdre Gouraud Forest ・ Cèdric Gogoua ・ Cèilidh ・ Cèlia Sànchez-Mústich ・ Cèmuhî language ・ Cère ・ Cère, Landes ・ Cèsar Martinell i Brunet ・ Cèze ・ Céa's lemma ・ Céaux ・ Céaux-d'Allègre ・ Cébaco ・ Cébaco (corregimiento) ・ Cébazan ・ Cébazat ・ Céchi ・ Cécil von Renthe-Fink ・ Cécile ・ Cécile Argiolas ・ Cécile Aubry ・ Cécile Bayiha ・ Cécile Brunschvicg ・ Cécile Bruyère Dictionary Lists mini英和辞書 mini和英辞書 Webster 1913 Latin-English FOLDOC Wikipedia English ウィキペディア

翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Céa's lemma ： ウィキペディア英語版 Céa's lemma Céa's lemma is a lemma in mathematics. It is an important tool for proving error estimates for the finite element method applied to elliptic partial differential equations. ==Lemma statement== Let $V$ be a real Hilbert space with the norm $\backslash |\backslash cdot\backslash |.$ Let $a:V\backslash times\; V\backslash to\; \backslash mathbb\; R$ be a bilinear form with the properties * $|a(v,\; w)|\; \backslash le\; \backslash gamma\; \backslash |v\backslash |\backslash ,\backslash |w\backslash |$ for some constant $\backslash gamma>0$ and all $v,\; w$ in $V$ (continuity) * $a(v,\; v)\; \backslash ge\; \backslash alpha\; \backslash |v\backslash |^2$ for some constant $\backslash alpha>0$ and all $v$ in $V$ (coercivity or $V$-ellipticity). Let $L:V\backslash to\; \backslash mathbb\; R$ be a bounded linear operator. Consider the problem of finding an element $u$ in $V$ such that : $a(u,\; v)=L(v)\backslash ,$ for all $v$ in $V.\backslash ,$ Consider the same problem on a finite-dimensional subspace $V\_h$ of $V,$ so, $u\_h$ in $V\_h$ satisfies : $a(u\_h,\; v)=L(v)\backslash ,$ for all $v$ in $V\_h.\backslash ,$ By the Lax–Milgram theorem, each of these problems has exactly one solution. Céa's lemma states that : $\backslash |u-u\_h\backslash |\backslash le\; \backslash frac\backslash |u-v\backslash |$ for all $v$ in $V\_h.$ That is to say, the subspace solution $u\_h$ is "the best" approximation of $u$ in $V\_h,$ up to the constant $\backslash gamma/\backslash alpha.$ The proof is straightforward : $\backslash alpha\backslash |u-u\_h\backslash |^2\; \backslash le\; a(u-u\_h,u-u\_h)\; =\; a(u-u\_h,u-v)\; +\; a(u-u\_h,v\; -\; u\_h)\; =\; a(u-u\_h,u-v)$ \le \gamma\|u-u_h\|\|u-v\| for all $v$ in $V\_h.$ We used the $a$-orthogonality of $u-u\_h$ and $V\_h$ : $a(u-u\_h,v)\; =\; 0,\; \backslash \; \backslash forall\; \backslash \; v$ in $V\_h$ which follows directly from $V\_h\; \backslash subset\; V$ : $a(u,\; v)\; =\; L(v)\; =\; a(u\_h,\; v)$ for all $v$ in $V\_h$. Note: Céa's lemma holds on complex Hilbert spaces also, one then uses a sesquilinear form $a(\backslash cdot,\; \backslash cdot)$ instead of a bilinear one. The coercivity assumption then becomes $|a(v,\; v)|\; \backslash ge\; \backslash alpha\; \backslash |v\backslash |^2$ for all $v$ in $V$ (notice the absolute value sign around $a(v,\; v)$). 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「Céa's 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.