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

Abstract

<jats:title>Abstract</jats:title><jats:p>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. Segré, Numerical solution to the time-dependent Maxwell equations in axisymmetric singular domains: the singular complement method, J. Comput. Phys. 191 2003, 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, 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.</jats:p>

A Numerical Method for 3d Time-Dependent Maxwell’s Equations in Axisymmetric Singular Domains With Arbitrary Data

“…In this section, we briefly recall for completeness, the method used to compute the singular basis x k S,j , 1 ≤ j ≤ N E and y k S,j , 1 ≤ j ≤ N B . Details can be found in [9]. For each Fourier mode indexed by k ∈ Z, and for a given singularity indexed by j, 1 ≤ j ≤ N B , the electric and magnetic singular basis x k S,j and y k S,j solve respectively the following variational formulation…”

“…Note that the right-hand side of these equations, in both electric and magnetic cases, can be obtained by computing the analytic expressions of curl k S X j and div k S X j , involved in a k S X j , v , and the same for the magnetic case, replacing S X j by S Y j , see details in [9].…”