Convergence of the Galerkin Method for Two-Dimensional Electromagnetic Problems

The application of the Galerkin method to a class of two-dimensional time-harmonic electromagnetic scattering and radiation problems in exterior domains is studied. The scatterers and sources consist of a periodic grating on the one hand and of bounded objects on the other. In both cases Maxwell’s equations reduce to an elliptic system of two coupled partial differential equations for two components of the electromagnetic field. A region is chosen which encloses all scatterers and sources and outside of which there is homogeneous material. Then on this region a boundary value problem and its variational formulation are derived which are equivalent to the original problem on the unbounded domain. The boundary operator, which represents the influence of outer space, leads to a perturbation of the sesquilinear form of the variational problem which is not weakly continuous on the canonical Hilbert space. Using a theorem of Michlin the convergence of the Galerkin method is proved. This provides at the same time a new proof of existence of solutions of these types of scattering problems that is different from the limiting absorption method.