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.

中文翻译：

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椭圆算子多重聚类特征值的两级块预条件雅可比-戴维森方法

SIAM 数值分析杂志，第 62 卷，第 2 期，第 998-1019 页，2024 年 4 月
。摘要。在本文中，我们提出了一种两级块预条件雅可比-戴维森（BPJD）方法，用于有效解决由第（[math]）阶对称椭圆特征值问题的有限元近似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，特别是包括具有相应特征函数的多个特征值和聚类特征值。通过使用重叠域分解 (DD) 构建新的高效预处理器，该方法具有高度可并行性。它只需要每次迭代计算几个小规模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] 限制，其中 [math] 是子域的直径，[math] 是子域之间的重叠大小。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且具有聚类鲁棒性。同时，依赖于[math]的常数[math]单调减小到1，如[math]，这意味着更多的子域导致更好的收敛速度。给出了支持我们理论的数值结果。