A Two-Level Block Preconditioned Jacobi-Davidson Method for Computing Multiple and Clustered Eigenvalues
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In this paper, we propose a two-level block preconditioned Jacobi-Davidson (BPJD) method for solving discrete eigenvalue problems resulting from finite element approximations of 2mth (m=1,2) order elliptic eigenvalue problems. A new and efficient preconditioner is constructed by an overlapping domain decomposition (DD). This method may compute the first several eigenpairs, including simple, multiple and clustered cases. In each iteration, we only need to solve a couple of parallel subproblems and one small scale eigenvalue problem. The rigorous theoretical analysis reveals that the convergence rate of the two-level BPJD method for the first several eigenvalues is bounded by c(H)(1-Cδ^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 internal gaps among target eigenvalues, which means that our method is optimal and cluster robust. Meanwhile, the H-dependent constant c(H) decreases monotonically to 1, as H→ 0, which means that more subdomains lead to the better convergence rate. Numerical results supporting our theory are given.
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