This thesis presents a discussion of both a direct and inverse problem related to the scattering of time-harmonic electromagnetic waves by impenetrable obstacles contained in an anisotropic or inhomogeneous medium of bounded support.

The direct scattering problem is first discussed in the setting of a homogeneous background. A variational formulation is derived, and it is shown via the Fredholm alternative and a uniqueness result that the problem is well-posed. This variational formulation is then used to implement a finite element method for approximating the problem. We prove an optimal error estimate for the scheme, discuss implementation and provide numerical examples illustrating our comments.

We then turn to the problem of computing the field scattered by a bounded object in a layered background. The main difficulty in extending our previous analysis is verifying uniqueness of a weak solution of this problem. This is done by choosing suitable radiations conditions and using asymptotic methods to analyze the far-field behavior of the fundamental solution. Once uniqueness is verified, the analysis of the previous problem can be applied to prove optimal convergence rates for an appropriate finite element method.

The inverse problem focuses on adapting the regularized sampling method which locates the support of the scatterer to the setting of the two-layered background. This technique is well suited for the inverse problem because, as we show, uniquely determining the matrix anisotropy from the near or far-field pattern of the scattered field is not possible. The inverse problem is further complicated by using limited aperture data measured on a line segment in the upper layer. We provide a theoretical analysis for the regularized sampling method in this context and give several computational examples.