Numerical Solution to the 3D Static Maxwell Equations in Axisymmetric Singular Domains with Arbitrary Data

We propose a numerical method to solve the three-dimensional static Maxwell equations in a singular axisymmetric domain, generated by the rotation of a singular polygon around one of its sides. The mathematical tools and an in-depth study of the problem set in the meridian half-plane are exposed in [F. Assous, P. Ciarlet, Jr., S. Labrunie and J. Segre, Numerical solution to the time-dependent Maxwell equations in axisymmetric singular domains: the singular complement method, J. Comput. Phys. 191 (2003), no. 1, 147-176] and [P. Ciarlet, Jr. and S. Labrunie, Numerical solution of Maxwell's equations in axisymmetric domains with the Fourier singular complement method, Differ. Equ. Appl. 3 (2011), no. 1, 113 155]. Here, we derive a variational formulation and the corresponding approximation method. Numerical experiments are proposed, and show that the approach is able to capture the singular part of the solution. This article can also be viewed as a generalization of the Singular Complement Method to three-dimensional axisymmetric problems.