Multigrid Computation of Axisymmetric Electromagnetic Fields

The focus of this paper is on boundary value problems for Maxwell's equations that feature cylindrical symmetry both of the domain R³ and the data. Thus, by resorting to cylindrical coordinates, a reduction to two dimensions is possible. However, cylindrical coordinates introduce a potentially malicious singularity at the axis rendering the variational problems degenerate. As a consequence, the analysis of multigrid solvers along the lines of variational multigrid theory confronts severe difficulties. Line relaxation in radial direction and semicoarsening can successfully reign in the degeneracy. In addition, the lack of H¹-ellipticity of the double-curl operator entails using special hybrid smoothing procedures. All these techniques combined yield a fast multigrid solver. The theoretical investigation of the method relies on blending generalized Fourier techniques and modern variational multigrid theory. We first determine invariant subspaces of the multigrid iteration operator and analyze the smoothers therein. Under certain assumptions on the material parameters we manage to show uniform convergence of a symmetric V-cycle.