THE USE OF PLANE WAVES TO APPROXIMATE WAVE PROPAGATION IN ANISOTROPIC MEDIA

In this paper we extend the standard Ultra Weak Variational Formulation (UWVF) of Maxwell’s equations in an isotropic medium to the case of an anisotropic medium. We verify that the underlying theoretical framework carries over to anisotropic media (however error estimates are not yet available) and completely describe the new scheme. We then consider TM mode scattering, show how this results in a Helmholtz equation in two dimensions with an anisotropic coefficient and demonstrate how to formulate the UWVF for it. In one special case, convergence can be proved. We then show some numerical results that suggest that the UWVF can successfully simulate wave propagation in anisotropic media. Mathematics subject classification: 65N30, 65N12 Electromagnetic wave propagation in anisotropic media arises in several applications including ground penetrating radar [3], microwave interaction with wood [13] and biological materials [20]. This paper is devoted to developing a method for approximating the electromagnetic field propagating in anisotropic media, with particular attention to microwave interactions with anisotropic (e.g. wooden) scatterers. This implies that the wavelength of the radiation is neither very large nor very small compared to features of scatterers located in the medium. We shall develop a Discontinuous Galerkin (DG) method for the anisotropic Maxwell system with the novelty that local solutions of the anisotropic Maxwell system on each element are used as basis functions. This requires us to impose the restriction that the matrix electromagnetic parameters ǫ (permittivity) and � (permeability) must be piecewise constant on each element in the mesh. More precisely, the method we shall develop is an extension of the Ultra Weak