This paper is devoted to the solution of the instationary Maxwell equations with charges. The geometry of the domain can be singular, in the sense that its boundary can include reentrant corners or edges. The difficulties arise from the fact that those geometrical singularities generate, in their neighborhood, strong electromagnetic fields. The time-dependency of the divergence of the electric field, is addressed. To tackle this problem, some new theoretical and practical results are presented, on curl-free singular fields, and on singular fields with L2 (non-vanishing) divergence. The method, which allows to compute the instationary electromagnetic field, is based on a splitting of the spaces of solutions into a two-term direct sum. First, the subspace of regular fields: it coincides with the whole space of solutions, provided that the domain is either convex, or with a smooth boundary. Second, a singular subspace, defined and characterized via the singularities of the Laplace operator. Several numerical examples are presented, to illustrate the mathematical framework. This paper is the generalization of the singular complement method.