{"title":"A refined first-order expansion formula in Rn: Application to interpolation and finite element error estimates","authors":"Joël Chaskalovic , Franck Assous","doi":"10.1016/j.cam.2024.116274","DOIUrl":null,"url":null,"abstract":"<div><p>The aim of this paper is to derive a refined first-order expansion formula in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>, the goal being to get an optimal reduced remainder, compared to the one obtained by usual Taylor’s formula. For a given function, the formula we derived is obtained by introducing a linear combination of the first derivatives, computed at <span><math><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></math></span> equally spaced points. We show how this formula can be applied to two important applications: the interpolation error and the finite elements error estimates. In both cases, we illustrate under which conditions a significant improvement of the errors can be obtained, namely how the use of the refined expansion can reduce the upper bound of error estimates.</p></div>","PeriodicalId":2,"journal":{"name":"ACS Applied Bio Materials","volume":null,"pages":null},"PeriodicalIF":4.6000,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"ACS Applied Bio Materials","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0377042724005235","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATERIALS SCIENCE, BIOMATERIALS","Score":null,"Total":0}
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Abstract
The aim of this paper is to derive a refined first-order expansion formula in , the goal being to get an optimal reduced remainder, compared to the one obtained by usual Taylor’s formula. For a given function, the formula we derived is obtained by introducing a linear combination of the first derivatives, computed at equally spaced points. We show how this formula can be applied to two important applications: the interpolation error and the finite elements error estimates. In both cases, we illustrate under which conditions a significant improvement of the errors can be obtained, namely how the use of the refined expansion can reduce the upper bound of error estimates.
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