Residual-based a posteriori error estimates for the hp version of the finite element discretization of the elliptic Robin boundary control problem

Abstract

Optimal control problems governed by partial differential equations have become a very active and successful research area. So, in this paper, we analyzed a priori and a posteriori error estimates for the hp finite element discretization of elliptic Robin boundary control problems. With the discrete and continuous optimality conditions of the problem, we constructed the error estimators. Based on the residual of the model equations for the coupled state and control approximations, the upper error bound is proved using Scott–Zhang-type quasi interpolation estimates. In order to provide the optimality, lower error bound is shown using some polynomial inverse estimates in weighted Sobolev spaces. Such estimators can be used to construct reliable adaptive methods for optimal control problems.