A Two-Level Preconditioned Helmholtz Subspace Iterative Method for Maxwell Eigenvalue Problems

Abstract

In this paper, based on a domain decomposition method, we shall propose a two-level preconditioned Helmholtz subspace iterative (PHSI) method for solving algebraic eigenvalue problems resulting from edge element approximation of Maxwell eigenvalue problems. The two-level PHSI method may compute simple eigenpairs, multiple eigenpairs, and clustered eigenpairs. It is very efficient and robust for solving Maxwell eigenvalue problems since the method reduces solutions of fine eigenvalue problems to a series of solutions of parallel preconditioned systems and Helmholtz projected systems. After solving a small dimensional eigenvalue problem in a trial subspace, we finally get the total error reduction of the first several discrete eigenvalues as , where is a positive parameter, is the overlapping size among subdomains, and is the diameter of subdomains. The positive number is independent of the mesh size and internal gaps among the first several Maxwell eigenvalues, i.e., the method is optimal corresponding to the degrees of freedom and cluster robust. The number increases with increasing, and decreases with increasing. The -dependent constant decreases monotonically to 1 as , which means the more the number of subdomains, the better the convergence rate. Numerical results supporting our theory are given.