{"title":"On Error Estimates for Discretization Operators for the Solution of the Poisson Equation","authors":"A. B. Utesov","doi":"10.1134/s0012266124010117","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> A discretization operator for the solution of the Poisson equation with the right-hand side\nfrom the Korobov class is constructed and its error is estimated in the <span>\\(L^{p} \\)</span>-metric, <span>\\(2\\leq p\\leq \\infty \\)</span>. It is proved that for <span>\\(p=2 \\)</span> the resulting error estimate for the discretization\noperator is order sharp on the power scale. An error in calculating the trigonometric Fourier\ncoefficients used when constructing the discretization operator is also found. It should be noted\nthat the obtained estimate in one case is better than previously known estimates of the errors of\ndiscretization operators constructed from the values of the right-hand side of the equation at the\nnodes of the modified Korobov grid and the Smolyak grid, and in the other case it coincides with\nthem up to constants.\n</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1134/s0012266124010117","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
A discretization operator for the solution of the Poisson equation with the right-hand side
from the Korobov class is constructed and its error is estimated in the \(L^{p} \)-metric, \(2\leq p\leq \infty \). It is proved that for \(p=2 \) the resulting error estimate for the discretization
operator is order sharp on the power scale. An error in calculating the trigonometric Fourier
coefficients used when constructing the discretization operator is also found. It should be noted
that the obtained estimate in one case is better than previously known estimates of the errors of
discretization operators constructed from the values of the right-hand side of the equation at the
nodes of the modified Korobov grid and the Smolyak grid, and in the other case it coincides with
them up to constants.
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