For a stopped diffusion process in a time dependent domain, we obtain the asymptotics of the triplet exit time/exit position/overshoot for the discretely stopped Euler scheme. Here, the overshoot means the distance to the boundary of the process when it exits the domain. As a first consequence of this result, we obtain an expansion for the weak error. From the expansion and the sensitivity of the underlying Dirichlet problem with respect to the domain, we finally derive a procedure to improve the convergence by suitably restraining the domain.