We study the electromagnetic scattering by the partly covered cavity with high wave numbers, which is governed by a Helmholtz equation with the transparent boundary condition. First, we develop a fast high-order algorithm for the scattering by the partly covered cavity filled with the homogeneous medium. The Helmholtz equation is discretized using a fourth-order finite-difference scheme and then reduced into a small aperture system by the fast Fourier transformation and Gaussian elimination. The existence and uniqueness of the numerical solution are given. By means of the immersed interface method, we also propose a fast high-order method for solving the scattering by the partly covered cavity with inhomogeneous media. A preconditioned iterative method based on Krylov subspace is applied to accelerate the convergence. Numerical experiments with different types of media are presented to demonstrate the efficiency and validity of the proposed methods.

我们研究了具有高波数的部分覆盖腔的电磁散射，这是由具有透明边界条件的亥姆霍兹方程控制的。首先，我们为填充有均匀介质的部分覆盖腔的散射开发了一种快速的高阶算法。使用四阶有限差分方案将Helmholtz方程离散化，然后通过快速傅里叶变换和高斯消除将其简化为小孔径系统。给出了数值解的存在性和唯一性。通过沉浸式界面方法，我们还提出了一种快速的高阶方法，用于解决部分覆盖的腔体在非均匀介质中的散射问题。提出了一种基于Krylov子空间的预处理迭代方法，以加快收敛速度​​。