  In this paper we consider the finite element approximation of Maxwell's
problem and analyse the prescription of essential boundary conditions in a weak
sense using Nitsche's method. To avoid indefiniteness of the problem, the
original equations are augmented with the gradient of a scalar field that
allows one to impose the zero divergence of the magnetic induction, even if the
exact solution for this scalar field is zero. Two finite element approximations
are considered, namely, one in which the approximation spaces are assumed to
satisfy the appropriate inf-sup condition that render the standard Galerkin
method stable, and another augmented and stabilised one that permits the use of
finite element interpolations of arbitrary order. Stability and convergence
results are provided for the two finite element formulations considered.
