  Maxwell's equations describe the evolution of electromagnetic fields,
together with constraints on the divergence of the magnetic and electric flux
densities. These constraints correspond to fundamental physical laws: the
nonexistence of magnetic monopoles and the conservation of charge,
respectively. However, one or both of these constraints may be violated when
one applies a finite element method to discretize in space. This is a
well-known and longstanding problem in computational electromagnetics.
  We use domain decomposition to construct a family of primal hybrid finite
element methods for Maxwell's equations, where the Lagrange multipliers are
shown to correspond to a numerical trace of the magnetic field and a numerical
flux of the electric flux density. Expressing the charge-conservation
constraint in terms of this numerical flux, we show that both constraints are
strongly preserved. As a special case, these methods include a hybridized
version of N\'ed\'elec's method, implying that it preserves the constraints
more strongly than previously recognized. These constraint-preserving
properties are illustrated using numerical experiments in both the time domain
and frequency domain. Additionally, we observe a superconvergence phenomenon,
where hybrid post-processing yields an improved estimate of the magnetic field.
