The Wind ShiftsPub Date : 2022-02-22DOI: 10.2307/j.ctv2b29sxn.28
F. Nietzsche
{"title":"Thus","authors":"F. Nietzsche","doi":"10.2307/j.ctv2b29sxn.28","DOIUrl":"https://doi.org/10.2307/j.ctv2b29sxn.28","url":null,"abstract":"Interpolation estimates in one and more dimensions are covered exhaustively in Chapter 1 of [EG04]. The simplest estimates control the error in W = W 1,p of one-dimensional continuous degree-d piecewise-polynomial Lagrange interpolation of functions u ∈ W 1+d,p. The Hölder inequality | b a u(x)v(x)dx| ≤ u L p v L q 1 p + 1 q = 1 (a key step in the discovery that L q is the dual of L p) implies |u(x) − u(y)| = | y x u ′ (t)dt| ≤ |x − y| 1/q (b a |u ′ | p) 1/p = |x − y| 1/q u ′ L p. Last week we used the case p = 1, q = ∞. Most finite element analyses require the case p = q = 2. This estimate implies that any u ∈ W 1,p is bounded if p ≥ 1 and continuous if p > 1. If y is a point where |u(y)| achieves its global minimum, then b a |u(t)| p dt ≥ |b − a||u(y)| p or |u(y)| ≤ |b − a| −1/p u L p , so at any point x, |u(x)| ≤ |u(x)−u(y)|+|u(y)| ≤ |b−a| −1/p u L p +|b−a| 1/q u ′ L p = |b−a| −1/p u W 1,q. First consider the case of linear interpolation d = 1. The continuous piecewise-linear interpolant P to equidistant grid values u j of u ∈ W 1,p is given by P (x) = θu j+1 + (1 − θ)u j = x − x j h u j+1 + x j+1 − x h u j","PeriodicalId":447163,"journal":{"name":"The Wind Shifts","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114551236","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Wind ShiftsPub Date : 2022-02-22DOI: 10.2307/j.ctv2b29sxn.137
{"title":"Notes from the Art Institute","authors":"","doi":"10.2307/j.ctv2b29sxn.137","DOIUrl":"https://doi.org/10.2307/j.ctv2b29sxn.137","url":null,"abstract":"","PeriodicalId":447163,"journal":{"name":"The Wind Shifts","volume":"41 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130102132","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Wind ShiftsPub Date : 2022-02-22DOI: 10.2307/j.ctv2b29sxn.52
{"title":"Lament for the chilero from Las Cruces","authors":"","doi":"10.2307/j.ctv2b29sxn.52","DOIUrl":"https://doi.org/10.2307/j.ctv2b29sxn.52","url":null,"abstract":"","PeriodicalId":447163,"journal":{"name":"The Wind Shifts","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132845378","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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