Variational Approximation of Maxwell's Equations in Biperiodic Structures

Abstract

Consider a plane wave incident on a biperiodic diffractive structure. The diffraction problem may be modeled by Maxwell's equations with transparent boundary conditions and solved by a finite element method. In this paper, a variational approximation is studied. The well-posedness of the continuous and discretized problems is established in the following sense. In the continuous case, it is shown that the model problem attains a unique ${\cal H}^1$ solution for all but possibly a discrete set of frequencies. In the discrete case, error estimates for the variational (finite element) approximation of the model problem with or without truncation of the nonlocal boundary operators are obtained.