  This paper proposes and analyzes two fully discrete mixed interior penalty
discontinuous Galerkin (DG) methods for the fourth order nonlinear
Cahn-Hilliard equation. Both methods use the backward Euler method for time
discretization and interior penalty discontinuous Galerkin methods for spatial
discretization. They differ from each other on how the nonlinear term is
treated, one of them is based on fully implicit time-stepping and the other
uses the energy-splitting time-stepping. The primary goal of the paper is to
prove the convergence of the numerical interfaces of the DG methods to the
interface of the Hele-Shaw flow. This is achieved by establishing error
estimates that depend on $\epsilon^{-1}$ only in some low polynomial orders,
instead of exponential orders. Similar to [14], the crux is to prove a discrete
spectrum estimate in the discontinuous Galerkin finite element space. However,
the validity of such a result is not obvious because the DG space is not a
subspace of the (energy) space $H^1$ and it is larger than the finite element
space. This difficult is overcome by a delicate perturbation argument which
relies on the discrete spectrum estimate in the finite element space proved in
\cite{Feng_Prohl04}. Numerical experiment results are also presented to gauge
the theoretical results and the performance of the proposed fully discrete
mixed DG methods.
