We propose a new numerical method to compute the singular solution of the Maxwell equations in axisymmetric domains, as for example in non convex polygonal domains. As geometrical singularities are mainly related to the space dependent part of the model, we focus on the static field computation. We then introduce a new approach, that consists in decomposing the domain into two or more subdomains, and to derive an ad hoc variational formulation in each subdomain. The interface conditions are then imposed with a method deduced from a Nitsche method coupled with a specific “exchange” approach. An advantage of this domain decomposition method is that it does not require neither overlapping nor iteration process. Another advantage is that no particular mesh refinement is needed near the geometrical singularities. Numerical examples will be shown.

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AbstractWe 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.