Dimensionally reduced models: Derivation and analyses

In this chapter, we consider some special situations in which the three-dimensional (3D) Maxwell equations can be reformulated as two-dimensional (2D) models. More precisely, the computational domain boils down to a subset of ℝ², with respect to a suitable system of coordinates (cylindrical, spherical, cartesian). Nevertheless, the electric and magnetic fields, and other vector quantities, still belong to ℝ³. Under suitable symmetry assumptions, one gets a single set of 2D equations or, equivalently, a single 2D variational formulation. In the general case, the electromagnetic field would be the solution to an infinite set of 2D equations, or variational formulations, obtained by Fourier analysis.