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 H1 (in space) in general. Moreover, the space of H1-regular fields is not dense in the space of solutions. Thus an H1-conforming Finite Element Method can fail, even with mesh refinement. The situation is different than in the case of the Laplace problem or of the Lamé system, for which mesh refinement or the addition of conforming singular functions work. To cope with this difficulty, the Singular Complement Method is introduced. This method consists of adding some well-chosen test functions. These functions are derived from the singular solutions of the Laplace problem. Also, the SCM preserves the interesting features of the original method: easiness of implementation, low memory requirements, small cost in terms of the CPU time. To ascertain its validity, some concrete problems are solved numerically.