  We consider the sharp interface limit of the Allen-Cahn equation with
homogeneous Neumann boundary condition in a two-dimensional domain $\Omega$, in
the situation where an interface has developed and intersects $\partial\Omega$.
Here a parameter $\varepsilon>0$ in the equation, which is related to the
thickness of the diffuse interface, is sent to zero. The limit problem is given
by mean curvature flow with a $90$\textdegree-contact angle condition and
convergence using strong norms is shown for small times. Here we assume that a
smooth solution to this limit problem exists on $[0,T]$ for some $T>0$ and that
it can be parametrized suitably. With the aid of asymptotic expansions we
construct an approximate solution for the Allen-Cahn equation and estimate the
difference of the exact and approximate solution with the aid of a spectral
estimate for the linearized Allen-Cahn operator.
