The Problem Of The Electromagnetic Scattering From A Two-dimensional Large Open Cavity

In recent decades, considerable research has been devoted to numerical solutions of electromagnetic cavity problems because of significant industrial applications. The paper is concerned with a time harmonic scattering problem of electromagnetic waves from a two-dimensional open cavity embedded in the infinite ground plane, which is governed by a Helmholtz type equation with nonlocal transparent boundary condition on the aperture. For electromagnetic cavity problems, without loss of generalities, we focus primarily on high wave number problems in our discussion. A fundamental issue for Helmholtz type problems is the stability analysis with the explicit dependency on wavenumber k. Although the classic Fredholm alternative theory yields the unique solvability for Helmholtz type problems, it gives no indication of how the solution depends on the wave number k. The wavenumber-dependent stability analysis turns out to be a great challenge, for both the geometry and the type of boundary conditions strongly affect the bounds. Firstly, we present some stability estimates with the explicit dependency of wave number k for the Helmholtz type cavity problem, which improves the previous results. For mostly k, the bound is proved to be o(k4/3)||g||2/1,Γ+o(k2/3)|g||Γ· Especially when the plane wave is incident on the electromagnetic cavity, though||g||= O(k), the bound is still proved to be O(k2) in the same conditions. A variety of numerical methods, including finite difference methods, finite element methods, boundary element methods, and hybrid methods, have been developed for different applications in the engineering community. The computation for large cavity problems is especially challenging due to the highly oscillatory nature of the fields. Then we propose a continuous interior penalty finite element method to discretize the Helmholtz equation. Some stability estimates and error estimates are derived, by using the trick of so-called’stability-error iterative improvement’. Finally, numerical results are presented to verify our theoretical results and to illustrate the effectiveness of the CIP-FEM for simulating such large-cavity problems.