Carsten Carstensen, Benedikt Gräßle, Ngoc Tien Tran
{"title":"保证特征值下限的自适应混合高阶方法","authors":"Carsten Carstensen, Benedikt Gräßle, Ngoc Tien Tran","doi":"10.1007/s00211-024-01407-w","DOIUrl":null,"url":null,"abstract":"<p>The higher-order guaranteed lower eigenvalue bounds of the Laplacian in the recent work by Carstensen et al. (Numer Math 149(2):273–304, 2021) require a parameter <span>\\(C_{\\text {st},1}\\)</span> that is found <i>not</i> robust as the polynomial degree <i>p</i> increases. This is related to the <span>\\(H^1\\)</span> stability bound of the <span>\\(L^{2}\\)</span> projection onto polynomials of degree at most <i>p</i> and its growth <span>\\(C_{\\textrm{st, 1}}\\propto (p+1)^{1/2}\\)</span> as <span>\\(p \\rightarrow \\infty \\)</span>. A similar estimate for the Galerkin projection holds with a <i>p</i>-robust constant <span>\\(C_{\\text {st},2}\\)</span> and <span>\\(C_{\\text {st},2} \\le 2\\)</span> for right-isosceles triangles. This paper utilizes the new inequality with the constant <span>\\(C_{\\text {st},2}\\)</span> to design a modified hybrid high-order eigensolver that directly computes guaranteed lower eigenvalue bounds under the idealized hypothesis of exact solve of the generalized algebraic eigenvalue problem and a mild explicit condition on the maximal mesh-size in the simplicial mesh. A key advance is a <i>p</i>-robust parameter selection. The analysis of the new method with a different fine-tuned volume stabilization allows for a priori quasi-best approximation and improved <span>\\(L^{2}\\)</span> error estimates as well as a stabilization-free reliable and efficient a posteriori error control. The associated adaptive mesh-refining algorithm performs superior in computer benchmarks with striking numerical evidence for optimal higher empirical convergence rates.</p>","PeriodicalId":49733,"journal":{"name":"Numerische Mathematik","volume":null,"pages":null},"PeriodicalIF":2.1000,"publicationDate":"2024-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Adaptive hybrid high-order method for guaranteed lower eigenvalue bounds\",\"authors\":\"Carsten Carstensen, Benedikt Gräßle, Ngoc Tien Tran\",\"doi\":\"10.1007/s00211-024-01407-w\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>The higher-order guaranteed lower eigenvalue bounds of the Laplacian in the recent work by Carstensen et al. (Numer Math 149(2):273–304, 2021) require a parameter <span>\\\\(C_{\\\\text {st},1}\\\\)</span> that is found <i>not</i> robust as the polynomial degree <i>p</i> increases. This is related to the <span>\\\\(H^1\\\\)</span> stability bound of the <span>\\\\(L^{2}\\\\)</span> projection onto polynomials of degree at most <i>p</i> and its growth <span>\\\\(C_{\\\\textrm{st, 1}}\\\\propto (p+1)^{1/2}\\\\)</span> as <span>\\\\(p \\\\rightarrow \\\\infty \\\\)</span>. A similar estimate for the Galerkin projection holds with a <i>p</i>-robust constant <span>\\\\(C_{\\\\text {st},2}\\\\)</span> and <span>\\\\(C_{\\\\text {st},2} \\\\le 2\\\\)</span> for right-isosceles triangles. This paper utilizes the new inequality with the constant <span>\\\\(C_{\\\\text {st},2}\\\\)</span> to design a modified hybrid high-order eigensolver that directly computes guaranteed lower eigenvalue bounds under the idealized hypothesis of exact solve of the generalized algebraic eigenvalue problem and a mild explicit condition on the maximal mesh-size in the simplicial mesh. A key advance is a <i>p</i>-robust parameter selection. The analysis of the new method with a different fine-tuned volume stabilization allows for a priori quasi-best approximation and improved <span>\\\\(L^{2}\\\\)</span> error estimates as well as a stabilization-free reliable and efficient a posteriori error control. The associated adaptive mesh-refining algorithm performs superior in computer benchmarks with striking numerical evidence for optimal higher empirical convergence rates.</p>\",\"PeriodicalId\":49733,\"journal\":{\"name\":\"Numerische Mathematik\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":2.1000,\"publicationDate\":\"2024-05-06\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Numerische Mathematik\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s00211-024-01407-w\",\"RegionNum\":2,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Numerische Mathematik","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00211-024-01407-w","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
摘要
卡斯滕森等人最近的研究(《数值数学》149(2):273-304, 2021)中的拉普拉卡矩的高阶保证下特征值边界需要一个参数 \(C_{\text{st},1}\),随着多项式度数 p 的增加,这个参数并不稳定。这与(L^{2}\)投影到最多 p 阶多项式的(H^1)稳定性约束及其增长(C_{\textrm{st, 1}}\propto (p+1)^{1/2}\) as \(p \rightarrow \infty \)有关。对于直角等腰三角形,Galerkin 投影的类似估计值也是成立的,并且有一个 p-robust 常量 \(C_{\text {st},2}\) 和 \(C_{\text {st},2} \le 2\) 。本文利用带有常数 \(C_{\text {st},2}\) 的新不等式设计了一种改进的混合高阶特征值求解器,它可以在精确求解广义代数特征值问题的理想化假设下直接计算有保证的特征值下限值,并对简单网格中的最大网格尺寸设定了温和的显式条件。一个关键的进步是p-稳健参数选择。通过对新方法与不同的微调体积稳定的分析,可以实现先验的准最佳近似和改进的误差估计,以及无稳定的可靠高效的后验误差控制。相关的自适应网格细化算法在计算机基准测试中表现出色,并有显著的数值证据证明其具有更高的最佳经验收敛率。
Adaptive hybrid high-order method for guaranteed lower eigenvalue bounds
The higher-order guaranteed lower eigenvalue bounds of the Laplacian in the recent work by Carstensen et al. (Numer Math 149(2):273–304, 2021) require a parameter \(C_{\text {st},1}\) that is found not robust as the polynomial degree p increases. This is related to the \(H^1\) stability bound of the \(L^{2}\) projection onto polynomials of degree at most p and its growth \(C_{\textrm{st, 1}}\propto (p+1)^{1/2}\) as \(p \rightarrow \infty \). A similar estimate for the Galerkin projection holds with a p-robust constant \(C_{\text {st},2}\) and \(C_{\text {st},2} \le 2\) for right-isosceles triangles. This paper utilizes the new inequality with the constant \(C_{\text {st},2}\) to design a modified hybrid high-order eigensolver that directly computes guaranteed lower eigenvalue bounds under the idealized hypothesis of exact solve of the generalized algebraic eigenvalue problem and a mild explicit condition on the maximal mesh-size in the simplicial mesh. A key advance is a p-robust parameter selection. The analysis of the new method with a different fine-tuned volume stabilization allows for a priori quasi-best approximation and improved \(L^{2}\) error estimates as well as a stabilization-free reliable and efficient a posteriori error control. The associated adaptive mesh-refining algorithm performs superior in computer benchmarks with striking numerical evidence for optimal higher empirical convergence rates.
期刊介绍:
Numerische Mathematik publishes papers of the very highest quality presenting significantly new and important developments in all areas of Numerical Analysis. "Numerical Analysis" is here understood in its most general sense, as that part of Mathematics that covers:
1. The conception and mathematical analysis of efficient numerical schemes actually used on computers (the "core" of Numerical Analysis)
2. Optimization and Control Theory
3. Mathematical Modeling
4. The mathematical aspects of Scientific Computing