{"title":"关于三维非结构化有限元离散的离散格林函数的实在性","authors":"Andrew P. Miller","doi":"10.1007/s10092-023-00556-y","DOIUrl":null,"url":null,"abstract":"<p>The aim of this paper is twofold. First, we prove <span>\\(L^p\\)</span> estimates for a regularized Green’s function in three dimensions. We then establish new estimates for the discrete Green’s function and obtain some positivity results. In particular, we prove that the discrete Green’s functions with singularity in the interior of the domain cannot be bounded uniformly with respect of the mesh parameter <i>h</i>. Actually, we show that at the singularity the discrete Green’s function is of order <span>\\(h^{-1}\\)</span>, which is consistent with the behavior of the continuous Green’s function. In addition, we also show that the discrete Green’s function is positive and decays exponentially away from the singularity. We also provide numerically persistent negative values of the discrete Green’s function on Delaunay meshes which then implies a discrete Harnack inequality cannot be established for unstructured finite element discretizations.</p>","PeriodicalId":9522,"journal":{"name":"Calcolo","volume":null,"pages":null},"PeriodicalIF":1.4000,"publicationDate":"2023-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"On the positivity of the discrete Green’s function for unstructured finite element discretizations in three dimensions\",\"authors\":\"Andrew P. Miller\",\"doi\":\"10.1007/s10092-023-00556-y\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>The aim of this paper is twofold. First, we prove <span>\\\\(L^p\\\\)</span> estimates for a regularized Green’s function in three dimensions. We then establish new estimates for the discrete Green’s function and obtain some positivity results. In particular, we prove that the discrete Green’s functions with singularity in the interior of the domain cannot be bounded uniformly with respect of the mesh parameter <i>h</i>. Actually, we show that at the singularity the discrete Green’s function is of order <span>\\\\(h^{-1}\\\\)</span>, which is consistent with the behavior of the continuous Green’s function. In addition, we also show that the discrete Green’s function is positive and decays exponentially away from the singularity. We also provide numerically persistent negative values of the discrete Green’s function on Delaunay meshes which then implies a discrete Harnack inequality cannot be established for unstructured finite element discretizations.</p>\",\"PeriodicalId\":9522,\"journal\":{\"name\":\"Calcolo\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.4000,\"publicationDate\":\"2023-12-22\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Calcolo\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s10092-023-00556-y\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Calcolo","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s10092-023-00556-y","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
On the positivity of the discrete Green’s function for unstructured finite element discretizations in three dimensions
The aim of this paper is twofold. First, we prove \(L^p\) estimates for a regularized Green’s function in three dimensions. We then establish new estimates for the discrete Green’s function and obtain some positivity results. In particular, we prove that the discrete Green’s functions with singularity in the interior of the domain cannot be bounded uniformly with respect of the mesh parameter h. Actually, we show that at the singularity the discrete Green’s function is of order \(h^{-1}\), which is consistent with the behavior of the continuous Green’s function. In addition, we also show that the discrete Green’s function is positive and decays exponentially away from the singularity. We also provide numerically persistent negative values of the discrete Green’s function on Delaunay meshes which then implies a discrete Harnack inequality cannot be established for unstructured finite element discretizations.
期刊介绍:
Calcolo is a quarterly of the Italian National Research Council, under the direction of the Institute for Informatics and Telematics in Pisa. Calcolo publishes original contributions in English on Numerical Analysis and its Applications, and on the Theory of Computation.
The main focus of the journal is on Numerical Linear Algebra, Approximation Theory and its Applications, Numerical Solution of Differential and Integral Equations, Computational Complexity, Algorithmics, Mathematical Aspects of Computer Science, Optimization Theory.
Expository papers will also appear from time to time as an introduction to emerging topics in one of the above mentioned fields. There will be a "Report" section, with abstracts of PhD Theses, news and reports from conferences and book reviews. All submissions will be carefully refereed.