The solution to the Poisson equation with Dirichlet boundary condition on a non-convex domain generally does not have $H^2$ regularity. Therefore, the $H^1$ error of the FEM solution generally cannot achieve $O(h)$. However, what happens if the solution belongs to $H^2$? In this case, the $H^1$ error of the FEM solution becomes $O(h)$. But what about the $L^2$ error? We present several results concerning this problem.