Error estimates for the discretization of elliptic control problems with pointwise control and state constraints

Abstract

A family of elliptic optimal control problems with pointwise constraints on control and state is considered. We are interested in approximation of the optimal solution by a finite element discretization of the involved partial differential equations. The discretization error for a problem with mixed state constraints is estimated in the semidiscrete case and in the fully discrete scheme with the convergence of order h|ln?h| and h 1/2, respectively. However, considering the unregularized continuous problem and the discrete regularized version, and choosing suitable relation between the regularization parameter and the mesh size, i.e., ?~h 2, a convergence order arbitrary close to 1, i.e., h 1?β is obtained. Therefore, we benefit from tuning the involved parameters.