A Posteriori Error Estimates for $hp$ Spectral Element Approximation of Elliptic Control Problems with Integral Control and State Constraints

Abstract

This paper investigates an optimal control problem governed by an elliptic equation with integral control and state constraints. The control problem is approximated by the $hp$ spectral element method with high accuracy and geometric flexibility. Optimality conditions of the continuous and discrete optimal control problems are presented, respectively. The a posteriori error estimates both for the control and state variables are established in detail. In addition, illustrative numerical examples are carried out to demonstrate the accuracy of theoretical results and the validity of the proposed method.