On the L2 a Priori error estimates to the finite element solution of elliptic problems with singular adjoint operator

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.