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 $2m$th ($m = 1, 2$) 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 $c(H)(1-C\frac{δ^{2m-1}}{H^{2m-1}})^{2}$, where $H$ is the diameter of subdomains and $δ$ is the overlapping size among subdomains. The constant $C$ is independent of the mesh size $h$ and the internal gaps among the target eigenvalues, demonstrating that our method is optimal and cluster robust. Meanwhile, the $H$-dependent constant $c(H)$ decreases monotonically to $1$, as $H \to 0$, which means that more subdomains lead to the better convergence rate. Numerical results supporting our theory are given.
Local Multilevel Preconditioned Jacobi-Davidson Method for Elliptic Eigenvalue Problems on Adaptive Meshes
School of Mathematical Science, Tongji University（同济大学数学科学学院）；Key Laboratory of Intelligent Computing and Applications, Tongji University, Ministry of Education（同济大学教育部智能计算与应用重点实验室）
Adaptive Multilevel Methods for the Maxwell Eigenvalue Problem
School of Mathematical Sciences, Tongji University（同济大学数学科学学院）；Key Laboratory of Intelligent Computing and Applications (Tongji University), Ministry of Education（教育部智能计算与应用重点实验室（同济大学））
Randomized Block Davidson Eigensolvers for Plane-Wave Density-Functional Theory
Paul Scherrer Institute（保罗谢勒研究所）；École Polytechnique Fédérale de Lausanne (EPFL)（洛桑联邦理工学院）；Swiss National Supercomputing Centre (CSCS)（瑞士国家超级计算中心）；ETH Zurich（苏黎世联邦理工学院）
An all-at-once preconditioner for evolutionary partial differential equations
在[McDonald，Pestana and Wathen，\textit{SIAM J. Sci. Comput.}，40（2018），pp. A1012-A1033]中提出了一种适用于发展偏微分方程的块循环预条件子，证明了预条件子矩阵是可对角化的，并且在热方程的情况下具有单位元加低秩分解.本文通过在块循环预条件子的右上方引入一个小参数$ε>0$，推广了块循环预条件子。在理论上，我们证明了：（i）广义预条件子保持了对角化和单位元加低秩分解，（ii）对于足够小的$ε$，新预条件子的所有特征值都聚集在1处，（iii）新预条件子的所有特征值都聚集在1处，（iv）新预条件子的所有特征值都聚集在1处，（iv）新预条件子的所有特征值都聚集在1处。（iii）当$ε$取小于或与时间步长的平方根相当时，预处理系统的GMRES方法具有与线性系的大小无关的线性收敛速率。数值结果证实了所提出的预处理器的效率，并表明推广提高了块循环预处理器的性能。
