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In this paper, we propose a two-level block preconditioned Jacobi-Davidson(BPJD) method for efficiently solving discrete eigenvalue problems resultingfrom finite element approximations of $2m$th ($m = 1, 2$) order symmetricelliptic eigenvalue problems. Our method works effectively to compute the firstseveral eigenpairs, including both multiple and clustered eigenvalues withcorresponding eigenfunctions, particularly. The method is highly parallelizableby constructing a new and efficient preconditioner using an overlapping domaindecomposition (DD). It only requires computing a couple of small scale parallelsubproblems and a quite small scale eigenvalue problem per iteration. Ourtheoretical analysis reveals that the convergence rate of the method is boundedby $c(H)(1-C\frac{\delta^{2m-1}}{H^{2m-1}})^{2}$, where $H$ is the diameter ofsubdomains and $\delta$ is the overlapping size among subdomains. The constant$C$ is independent of the mesh size $h$ and the internal gaps among the targeteigenvalues, 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 convergencerate. Numerical results supporting our theory are given.

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