A Two-Level Block Preconditioned Jacobi–Davidson Method for Multiple and Clustered Eigenvalues of Elliptic Operators

IF 2.8 2区数学 Q1 MATHEMATICS, APPLIED

SIAM Journal on Numerical Analysis Pub Date : 2024-04-22 DOI:10.1137/23m1580711

Qigang Liang, Wei Wang, Xuejun Xu

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引用次数: 0

Abstract

SIAM Journal on Numerical Analysis, Volume 62, Issue 2, Page 998-1019, April 2024.
Abstract. In this paper, we propose a two-level block preconditioned Jacobi–Davidson (BPJD) method for efficiently solving discrete eigenvalue problems resulting from finite element approximations of [math]th ([math]) order symmetric elliptic eigenvalue problems. Our method works effectively to compute the first several eigenpairs, including both multiple and clustered eigenvalues with corresponding eigenfunctions, particularly. The method is highly parallelizable by constructing a new and efficient preconditioner using an overlapping domain decomposition (DD). It only requires computing a couple of small scale parallel subproblems and a quite small scale eigenvalue problem per iteration. Our theoretical analysis reveals that the convergence rate of the method is bounded by [math], where [math] is the diameter of subdomains and [math] is the overlapping size among subdomains. The constant [math] is independent of the mesh size [math] and the internal gaps among the target eigenvalues, demonstrating that our method is optimal and cluster robust. Meanwhile, the [math]-dependent constant [math] decreases monotonically to 1, as [math], which means that more subdomains lead to the better convergence rate. Numerical results supporting our theory are given.

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椭圆算子多特征值和聚类特征值的两级块预处理雅各比-戴维森方法

SIAM 数值分析期刊》第 62 卷第 2 期第 998-1019 页，2024 年 4 月。摘要本文提出了一种两级块预条件雅各比-戴维森（BPJD）方法，用于高效求解有限元逼近[math]th（[math]）阶对称椭圆特征值问题所产生的离散特征值问题。我们的方法可以有效地计算前几个特征对，包括多特征值和簇特征值以及相应的特征函数。通过使用重叠域分解（DD）构建一个新的高效预处理器，该方法具有很高的并行性。它每次迭代只需要计算几个小规模的并行子问题和一个相当小规模的特征值问题。我们的理论分析表明，该方法的收敛速度受 [math] 约束，其中 [math] 是子域直径，[math] 是子域间的重叠大小。常数[math]与网格大小[math]和目标特征值之间的内部间隙无关，这表明我们的方法是最优的，并且具有集群鲁棒性。同时，与[math]相关的常数[math]随着[math]的增大单调递减到1，这意味着更多的子域会带来更好的收敛速度。本文给出了支持我们理论的数值结果。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

SIAM Journal on Numerical Analysis 数学-应用数学

CiteScore

4.80

自引率

6.90%

发文量

110

审稿时长

4-8 weeks

期刊介绍： SIAM Journal on Numerical Analysis (SINUM) contains research articles on the development and analysis of numerical methods. Topics include the rigorous study of convergence of algorithms, their accuracy, their stability, and their computational complexity. Also included are results in mathematical analysis that contribute to algorithm analysis, and computational results that demonstrate algorithm behavior and applicability.