On the solution of Maxwell's equations in axisymmetric domains with edges

Abstract

In this paper we present the basic mathematical tools for treating boundary value problems for the Maxwell equations in three-dimensional axisymmetric domains with reentrant edges using the Fourier-finite-element method. We consider both the classical and the regularized time-harmonic Maxwell equations subject to perfect conductor boundary conditions. The partial Fourier decomposition reduces the three-dimensional boundary value problem into an infinite sequence of two-dimensional boundary value problems in the plane meridian domain of the axsiymmetric domain. Here, suitable weighted Sobolev spaces that characterize the solutions of the reduced problems are given, and their trace properties on the rotational axis are proved. In these spaces, it is proved that the reduced problems are well posed, and the asymptotic behavior of the solutions near reentrant comers of the meridian domain is explicitly described by suitable singularity functions. Finally, a finite number of the two-dimensional problems is considered and treated using H 1 -conforming finite elements. An approximation of the solution of the three-dimensional problem is obtained by Fourier synthesis. For domains with reentrant edges, the singular field method is employed to compensate the singular behavior of the solutions of the reduced problems. Emphases are given to convergence analysis of the combined approximations in H 1 under different regularity assumptions on the solution.