Numerical solution to the time-dependent Maxwell equations in two-dimensional singular domains: The Singular Complement Method

In this paper, we present a method to solve numerically the time-dependent Maxwell equations in nonsmooth and nonconvex domains. Indeed, the solution is not of regularity H-1 (in space) in general. Moreover, the space of H-1- regular fields is not dense in the space of solutions. Thus an H-1-conformi ng Finite Element Method can fail, even with mesh refinement. The situation is different than in the case of the Laplace problem or of the Lame system , for which mesh refinement or the addition of conforming singular function s work. To cope with this difficulty, the Singular Complement Method is int roduced. This method consists of adding some well-chosen test functions. Th ese functions are derived from the singular solutions of the Laplace proble m. Also, the SCM preserves the interesting features of the original method: easiness of implementation, low memory requirements, small cost in terms o f the CPU time. To ascertain its validity, some concrete problems are solve d numerically. (C) 2000 Academic Press.