Adaptive guaranteed lower eigenvalue bounds with optimal convergence rates

IF 2.1 2区 数学 Q1 MATHEMATICS, APPLIED
Carsten Carstensen, Sophie Puttkammer
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引用次数: 0

Abstract

Guaranteed lower Dirichlet eigenvalue bounds (GLB) can be computed for the m-th Laplace operator with a recently introduced extra-stabilized nonconforming Crouzeix–Raviart (\(m=1\)) or Morley (\(m=2\)) finite element eigensolver. Striking numerical evidence for the superiority of a new adaptive eigensolver motivates the convergence analysis in this paper with a proof of optimal convergence rates of the GLB towards a simple eigenvalue. The proof is based on (a generalization of) known abstract arguments entitled as the axioms of adaptivity. Beyond the known a priori convergence rates, a medius analysis is enfolded in this paper for the proof of best-approximation results. This and subordinated \(L^2\) error estimates for locally refined triangulations appear of independent interest. The analysis of optimal convergence rates of an adaptive mesh-refining algorithm is performed in 3D and highlights a new version of discrete reliability.

Abstract Image

具有最佳收敛率的自适应保证特征下限值
使用最近引入的超稳定非顺应 Crouzeix-Raviart (\(m=1\)) 或 Morley (\(m=2\)) 有限元特征值求解器,可以计算 m-th 拉普拉斯算子的有保证的下 Dirichlet 特征值边界(GLB)。新的自适应特征值求解器优越性的惊人数值证据促使本文进行收敛分析,并证明了 GLB 对简单特征值的最佳收敛率。该证明基于题为适应性公理的已知抽象论证(的概括)。除了已知的先验收敛率之外,本文还包含了中值分析,用于证明最佳逼近结果。这和局部细化三角剖分的从属(L^2\)误差估计具有独立的意义。自适应网格细化算法的最佳收敛率分析在三维中进行,并突出了离散可靠性的新版本。
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来源期刊
Numerische Mathematik
Numerische Mathematik 数学-应用数学
CiteScore
4.10
自引率
4.80%
发文量
72
审稿时长
6-12 weeks
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
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