In the present work, we apply semi-discretization proposed by the first author in [13] to Lavrentiev-regularized state constrained elliptic control problems. We extend the results of [17] and prove weak convergence of the adjoint states and multipliers of the regularized problems to their counterparts of the original problem. Further, we prove error estimates for finite element discretizations of the regularized problem and investigate the overall error imposed by the finite element discretization of the regularized problem compared to the continuous solution of the original problem. Finally we present numerical results which confirm our analytical findings
