The Aubin-Nitsche trick for the finite element method of Dirichlet boundary value problem is a well-known technique to obtain a higher order a priori L2 error estimation than that of [image omitted] estimates by considering the regularly dual problem. However, as far as the authors determine, when the dual problem is singular, it was not known at all up to now whether the a priori order of L2 error is still higher than that of [image omitted] error. In this paper, we propose a technique for getting a priori L2 error estimation by some verified numerical computations for the finite element projection. This enables us to obtain a higher order L2 a priori error than that of [image omitted] error, even though the associated dual problem is singular. Note that our results are not a posteriori estimates but the determination of a priori constants.
