  We apply the discontinuous Galerkin finite element method with a degree $p$
polynomial basis to the linear advection equation and derive a PDE which the
numerical solution solves exactly. We use a Fourier approach to derive
polynomial solutions to this PDE and show that the polynomials are closely
related to the $\frac{p}{p+1}$ Pad\'e approximant of the exponential function.
We show that for a uniform mesh of $N$ elements there exist $(p+1)N$
independent polynomial solutions, $N$ of which can be viewed as physical and
$pN$ as non-physical. We show that the accumulation error of the physical mode
is of order $2p+1$. In contrast, the non-physical modes are damped out
exponentially quickly. We use these results to present a simple proof of the
superconvergence of the DG method on uniform grids as well as show a connection
between spatial superconvergence and the superaccuracies in dissipation and
dispersion errors of the scheme. Finally, we show that for a class of initial
projections on a uniform mesh, the superconvergent points of the numerical
error tend exponentially quickly towards the downwind based Radau points.
