Numerical solution to the 3D Static Maxwell equations in axisymmetric singular domains with arbitrary data

TL;DR

This paper introduces a numerical method for solving 3D static Maxwell equations in axisymmetric domains with singularities, effectively capturing singular behaviors and extending the Singular Complement Method to 3D problems.

Contribution

The paper develops a variational formulation and approximation method for 3D axisymmetric Maxwell equations with singularities, generalizing the Singular Complement Method.

Findings

Method accurately captures singular parts of the solution

Numerical experiments validate the approach

Extension of Singular Complement Method to 3D axisymmetric problems

Abstract

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 \cite{AsCLS03}, \cite{CiLa11}. 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.