A posteriori error estimates of hp spectral element methods for optimal control problems with L2-norm state constraint

In this paper, we investigate a distributed optimal control problem governed by elliptic partial differential equations with L-2-norm constraint on the state variable. Firstly, the control problem is approximated by hp spectral element methods, which combines the advantages of the finite element methods with spectral methods; then, the optimality conditions of continuous system and discrete system are presented, respectively. Next, hp a posteriori error estimates are derived for the coupled state and control approximation. In the end, a projection gradient iterative algorithm is given, which solves the optimal control problems efficiently. Numerical experiments are carried out to confirm that the numerical results are in good agreement with the theoretical results.