{"title":"多项式轨迹在四面体上的稳定提升","authors":"Charles Parker, Endre Süli","doi":"10.1007/s10208-024-09670-x","DOIUrl":null,"url":null,"abstract":"<p>On the reference tetrahedron <span>\\(K\\)</span>, we construct, for each <span>\\(k \\in {\\mathbb {N}}_0\\)</span>, a right inverse for the trace operator <span>\\(u \\mapsto (u, \\partial _{\\textbf{n}} u, \\ldots , \\partial _{\\textbf{n}}^k u)|_{\\partial K}\\)</span>. The operator is stable as a mapping from the trace space of <span>\\(W^{s, p}(K)\\)</span> to <span>\\(W^{s, p}(K)\\)</span> for all <span>\\(p \\in (1, \\infty )\\)</span> and <span>\\(s \\in (k+1/p, \\infty )\\)</span>. Moreover, if the data is the trace of a polynomial of degree <span>\\(N \\in {\\mathbb {N}}_0\\)</span>, then the resulting lifting is a polynomial of degree <i>N</i>. One consequence of the analysis is a novel characterization for the range of the trace operator.\n</p>","PeriodicalId":55151,"journal":{"name":"Foundations of Computational Mathematics","volume":"19 1","pages":""},"PeriodicalIF":2.5000,"publicationDate":"2024-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Stable Liftings of Polynomial Traces on Tetrahedra\",\"authors\":\"Charles Parker, Endre Süli\",\"doi\":\"10.1007/s10208-024-09670-x\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>On the reference tetrahedron <span>\\\\(K\\\\)</span>, we construct, for each <span>\\\\(k \\\\in {\\\\mathbb {N}}_0\\\\)</span>, a right inverse for the trace operator <span>\\\\(u \\\\mapsto (u, \\\\partial _{\\\\textbf{n}} u, \\\\ldots , \\\\partial _{\\\\textbf{n}}^k u)|_{\\\\partial K}\\\\)</span>. The operator is stable as a mapping from the trace space of <span>\\\\(W^{s, p}(K)\\\\)</span> to <span>\\\\(W^{s, p}(K)\\\\)</span> for all <span>\\\\(p \\\\in (1, \\\\infty )\\\\)</span> and <span>\\\\(s \\\\in (k+1/p, \\\\infty )\\\\)</span>. Moreover, if the data is the trace of a polynomial of degree <span>\\\\(N \\\\in {\\\\mathbb {N}}_0\\\\)</span>, then the resulting lifting is a polynomial of degree <i>N</i>. One consequence of the analysis is a novel characterization for the range of the trace operator.\\n</p>\",\"PeriodicalId\":55151,\"journal\":{\"name\":\"Foundations of Computational Mathematics\",\"volume\":\"19 1\",\"pages\":\"\"},\"PeriodicalIF\":2.5000,\"publicationDate\":\"2024-07-29\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Foundations of Computational Mathematics\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s10208-024-09670-x\",\"RegionNum\":1,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"COMPUTER SCIENCE, THEORY & METHODS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Foundations of Computational Mathematics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10208-024-09670-x","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
Stable Liftings of Polynomial Traces on Tetrahedra
On the reference tetrahedron \(K\), we construct, for each \(k \in {\mathbb {N}}_0\), a right inverse for the trace operator \(u \mapsto (u, \partial _{\textbf{n}} u, \ldots , \partial _{\textbf{n}}^k u)|_{\partial K}\). The operator is stable as a mapping from the trace space of \(W^{s, p}(K)\) to \(W^{s, p}(K)\) for all \(p \in (1, \infty )\) and \(s \in (k+1/p, \infty )\). Moreover, if the data is the trace of a polynomial of degree \(N \in {\mathbb {N}}_0\), then the resulting lifting is a polynomial of degree N. One consequence of the analysis is a novel characterization for the range of the trace operator.
期刊介绍:
Foundations of Computational Mathematics (FoCM) will publish research and survey papers of the highest quality which further the understanding of the connections between mathematics and computation. The journal aims to promote the exploration of all fundamental issues underlying the creative tension among mathematics, computer science and application areas unencumbered by any external criteria such as the pressure for applications. The journal will thus serve an increasingly important and applicable area of mathematics. The journal hopes to further the understanding of the deep relationships between mathematical theory: analysis, topology, geometry and algebra, and the computational processes as they are evolving in tandem with the modern computer.
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