In this paper we study the variational method and integral equation methods for a conical diffraction problem for imperfectly conducting gratings modeled by the impedance boundary value problem of the Helmholtz equation in periodic structures. We justify the strong ellipticity of the sesquilinear form corresponding to the variational formulation and prove the uniqueness of solutions at any frequency. Convergence of the finite element method using the transparent boundary condition (Dirichlet-to-Neumann mapping) is verified. The boundary integral equation method is also discussed.

翻译：本文研究了不完全导电光栅锥形衍射问题的变分方法与积分方程方法，该问题由周期结构中亥姆霍兹方程的阻抗边值问题建模。我们证明了变分公式对应的双线性形式的强椭圆性，并证明了任意频率下解的唯一性。验证了采用透明边界条件（Dirichlet-to-Neumann映射）的有限元方法的收敛性。此外还讨论了边界积分方程方法。