A two-level preconditioned Helmholtz-Jacobi-Davidson method for the Maxwell eigenvalue problem

Abstract

In this paper, based on a domain decomposition method, we propose an efficient two-level preconditioned Helmholtz-Jacobi-Davidson (PHJD) method for solving the algebraic eigenvalue problem resulting from the edge element approximation of the Maxwell eigenvalue problem. In order to eliminate the components in orthogonal complement space of the eigenvalue, we shall solve a parallel preconditioned system and a Helmholtz projection system together in fine space. After one coarse space correction in each iteration and minimizing the Rayleigh quotient in a small dimensional Davidson space, we finally get the error reduction of this two-level PHJD method as γ = c ( H ) ( 1 − C δ 2 H 2 ) \gamma =c(H)(1-C\frac {\delta ^{2}}{H^{2}}) , where C C is a constant independent of the mesh size h h and the diameter of subdomains H H , δ \delta is the overlapping size among the subdomains, and c ( H ) c(H) decreasing monotonically to 1 1 as H → 0 H\to 0 , which means the greater the number of subdomains, the better the convergence rate. Numerical results supporting our theory are given.

Convergence analysis of the Newton-Schur method for the symmetric elliptic eigenvalue problem

“…The first method proposed by Xu and Zhou in [42,43] needs O(H 2 ) = h, another two level methods based on inverse iteration can be optimal under the condition O(H 4 ) = h, see [18,45]. Recently, the method proposed by Wang and Xu in [36,37] is optimal with no assumptions between h and H. For Maxwell eigenvalue problem, similar results can be found in [26].…”

A Two-Level Block Preconditioned Jacobi-Davidson Method for Computing Multiple and Clustered Eigenvalues

“…A rigorous theoretical analysis of the two-level PJD method for the principal eigenvalue of 2mth (m = 1, 2) order elliptic operators is presented in [28]. Recently, we also present a two-level preconditioned Helmholtz-Jacobi-Davidson (PHJD) method for the Maxwell eigenvalue problem (see [14]). The two-level PHJD method works well in practical computations and it is proved to be optimal and scalable.…”

“…The two-level PHJD method works well in practical computations and it is proved to be optimal and scalable. However, for the two-level PJD method in [27,28] or the two-level PHJD method in [14], the theoretical analysis is only valid for the simple principal eigenvalue. In this paper, we try to give a rigorous analysis for the non-principal eigenvalues, including multiple and clustered eigenvalues.…”

“…We must emphasize that the theoretical analysis from simple eigenvalue to multiple and clustered eigenvalues is nontrivial for the two-level BPJD method. Firstly, it is necessary to ensure that all of constants appearing in the convergence rate do not depend on internal gaps of target eigenvalues; Secondly, since the dimension of eigenspace corresponding to the multiple eigenvalue is greater than 1, many techniques developed for simple eigenvalues in [14,27,28] are not applicable; Thirdly, to measure the distance, the Hausdorff distance or the gap needs to be used in the theoretical analysis rather than the vector norm, which brings more difficulties. In this paper, we overcome these difficulties, and try to prove that the convergence result is optimal, scalable and cluster robust, i.e., (1.1) is true.…”