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：本文提出了一种两层块预条件Jacobi-Davidson（BPJD）方法，用于高效求解$2m$阶（$m=1,2$）对称椭圆特征值问题的有限元近似离散特征值问题。该方法能有效计算前几个特征对，尤其适用于多重及密集特征值及其对应特征函数的求解。通过构建一种基于重叠区域分解（DD）的新型高效预条件子，该方法具有高度并行性，每次迭代仅需计算少量小规模并行子问题和一个极小规模特征值问题。理论分析表明，该方法的收敛速率以$c(H)(1-C\frac{\delta^{2m-1}}{H^{2m-1}})^{2}$为上界，其中$H$为子区域直径，$\delta$为子区域间重叠尺寸。常数$C$独立于网格尺寸$h$及目标特征值内部间隙，证明该方法具有最优性和簇鲁棒性。同时，依赖$H$的常数$c(H)$随$H \to 0$单调递减至1，这意味着子区域越多，收敛速率越优。文中给出了支持理论分析的数值结果。