A fast preconditioned iterative method for the electromagnetic scattering by multiple cavities with high wave numbers

Abstract

We study the problem of electromagnetic scattering by multiple open cavities embedded in an infinite ground plane with high wave numbers. The problem can be described by a series of Helmholtz equations with coupled boundary conditions. We develop a sixth-order finite difference scheme to discretize the coupled Helmholtz equations. By Gaussian elimination in the vertical direction and Fourier transform in the horizontal direction, we can reduce the multiple cavity scattering problem to an aperture linear system. However, in the situation of high wave numbers, the condition number of the coefficient matrix of the reduced linear system is especially large and the system tends to be ill-conditioned. The convergence histories of most iterative methods become oscillating which consume considerable computations and memory spaces. In order to overcome the difficulty caused by high wave numbers, we develop an efficient preconditioned iterative method based on the Krylov subspace, which greatly improves the eigenvalue distributions and reduces the number of iterations. Numerical experiments show the validity and efficiency of the proposed sixth-order fast preconditioned algorithm for solving the scattering by multiple cavities with high wave numbers.