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 $c(H)(1-C\frac{\delta^{2m-1}}{H^{2m-1}})^{2}$, where $H$ is the diameter of subdomains and $\delta$ 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.

翻译：在本文中，我们提出了一种基于两层块预条件的Jacobi-Davidson (BPJD)方法，用于高效地解决由有限元逼近的 $2m$ 阶($m=1,2$) 对称椭圆特征值问题所得到的离散特征值问题。我们的方法可以有效地计算前几个特征值对和特征函数，特别是包括多个和聚类的特征值。该方法通过使用重叠区域分解 (DD) 构建新的高效预处理器，可以高度并行化。每次迭代只需要计算几个小型并行子问题和一个相当小的特征值问题。我们的理论分析表明，该方法的收敛速度由 $c(H)(1-C\frac{\delta^{2m-1}}{H^{2m-1}})^{2}$ 限制，其中 $H$ 是子域的直径，$\delta$ 是子域之间的重叠大小。常数 $C$ 不依赖于网格大小 $h$ 和目标特征值之间的间隔，表明我们的方法是最优且聚类鲁棒的。同时，$H$ 依赖的常数 $c(H)$ 会随着 $H\to 0$ 单调递减到 $1$，这意味着更多的子域会导致更好的收敛速度。我们给出了支持我们理论的数值结果。