Solving Vlasov-Maxwell equations in singular geometries (Q1010027)

\noindent The present paper is devoted to the solution of the time-dependent Vlasov-Maxwell equations in singular geometries, i.e., when the boundary includes reentrant corners or edges that generate strong fields. This singularities require a careful computation of the electromagnetic field in their neighborhood. The authors develop a method, the so-called singular complement method, which consists in the splitting of the space of solutions into a two-term, possibly orthogonal sum. The first subspace is made of regular fields, and coincides with the whole space of solutions, provided that the domain is either convex or regular, i.e., with a smooth boundary. So one can compute the regular part of the solution using a classical method. The second part of the solution is the subspace of singular fields. It is computed with the help of specifically designed methods which originate from relations between the electromagnetic singularities and the singularities of the Laplace operator. The present work extends previous works of the authors, aimed at solving divergenc-free electromagnetic problems [\textit{A. Assous, P. Ciarlet jun.} and \textit{J. Segré}, J. Comput. Phys. 161, No.~1, 218--249 (2000; Zbl 1007.78014)] to systems with electric fields with divergences. The numerical algorithms used to solve the time-dependent Vlasov-Maxwell equations in singular geometries are described in detail. Finally, numerical experiments are shortly presented.