  This paper develops and analyzes an interior penalty discontinuous Galerkin
(IPDG) method using piecewise linear polynomials for the indefinite time
harmonic Maxwell equations with the impedance boundary condition in the three
dimensional space. The main novelties of the proposed IPDG method include the
following: first, the method penalizes not only the jumps of the tangential
component of the electric field across the element faces but also the jumps of
the tangential component of its vorticity field; second, the penalty parameters
are taken as complex numbers of negative imaginary parts. For the differential
problem, we prove that the sesquilinear form associated with the Maxwell
problem satisfies a generalized weak stability (i.e., inf-sup condition) for
star-shaped domains.Such a generalized weak stability readily infers
wave-number explicit a priori estimates for the solution of the Maxwell
problem, which plays an important role in the error analysis for the IPDG
method. For the proposed IPDG method, we show that the discrete sesquilinear
form satisfies a coercivity for all positive mesh size $h$ and wave number $k$
and for general domains including non-star-shaped ones. In turn, the coercivity
easily yields the well-posedness and stability estimates (i.e., a priori
estimates) for the discrete problem without imposing any mesh constraint. Based
on these discrete stability estimates, by adapting a nonstandard error estimate
technique of Fung and Wu (2009), we derive both the energy-norm and the
$L^2$-norm error estimates for the IPDG method in all mesh parameter regimes
including pre-asymptotic regime (i.e., $k^2 h\gtrsim 1$). Numerical experiments
are also presented to gauge the theoretical results and to numerically examine
the pollution effect (with respect to $k$) in the error bounds.
