  The paper proposes and analyzes an efficient second-order in time numerical
approximation for the Allen-Cahn equation, which is a second order nonlinear
equation arising from the phase separation model. We firstly present a fully
discrete interior penalty discontinuous Galerkin (IPDG) finite element method,
which is based on the modified Crank-Nicolson scheme and a mid-point
approximation of the nonliner term $f(u)$. We then derive the stability
analysis and error estimates for the proposed IPDG method under some regularity
assumptions on the initial function $u_0$. There are two key works in our
analysis, one is to establish unconditionally energy-stable scheme for the
discrete solutions. The other is to use a discrete spectrum estimate to handle
the midpoint of the discrete solutions $u^m$ and $u^{m+1}$ in the nonlinear
term, instead of using the standard Gronwall inequality technique. This
discrete spectrum estimate is not trivial to obtain since the IPDG space and
the conforming $H^1$ space are not contained in each other. We obtain that all
our error bounds depend on reciprocal of the perturbation parameter $\epsilon$
only in some lower polynomial order, instead of exponential order. These
sharper error bounds are the key elements in proving the convergence of our
numerical solution to the mean curvature flow. Finally, numerical experiments
are also provided to show the performance of the presented approach and method.
