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Numerical solution to the time-dependent Maxwell equations in axisymmetric singular domains: The singular complement method. (English) Zbl 1033.65086

Summary: We present a method to solve numerically the axisymmetric time-dependent Maxwell equations in a singular domain. F. Assous, P. Ciarlet jun., and S. Labrunie [Math. Methods Appl. Sci. 25, 49–78 (2002; Zbl 0995.35070); Math. Methods Appl. Sci. 26, 861–896 (2003)] exposed the mathematical tools and an in-depth study of the problems posed in the meridian half-plane. The numerical method and experiments based on this theory are now described here. It is also the generalization to axisymmetric problems of the singular complement method that we developed to solve Maxwell equations in 2D singular domains [cf. F. Assous, P. Ciarlet jun. and E. Garcia, C. R. Acad. Sci., Paris, Sér. I, Math. 330, 391–396 (2000; Zbl 1007.78015)]. It is based on a splitting of the space of solutions in a regular subspace, and a singular one, derived from the singular solutions of the Laplace problem. Numerical examples are finally given, to illustrate our purpose. In particular, they show how the singular complement method captures the singular part of the solution.

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
35Q60 PDEs in connection with optics and electromagnetic theory
78A25 Electromagnetic theory (general)
78M10 Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory
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