| Consider a three-dimensional (3D) problem defined on a domain symmetric by rotation around an axis with data independent of the angular component. By using cylindrical coordinates, we can then reduce this axisymmetric 3D problem into a two-dimensional (2 D) one. The advantage of such dimension reduction is that the discretization of the 3D problem results in a linear system of the same size as the 2D one saving computational time significantly. Due to the Jacobian arising from change of variables, however, we must work in weighted Sobolev spaces, where the weight function is the radial component r, once this dimension reduction is done. In this dissertation, we analyze the time harmonic Maxwell equations under axial symmetry. In particular, we provide an edge finite element analysis and a multigrid analysis of the so-called "meridian" problem, a problem arising from the axisymmetric Maxwell equations. New commuting projectors in weighted spaces are introduced, and a dual mixed problem in weighted spaces, which is interesting in its own right, is analyzed. These will provide the main ingredients for the analysis of the meridian problem. |
