Residual-based a posteriori error estimates for hp finite element solutions of semilinear Neumann boundary optimal control problems

In this paper, we investigate residual-based a posteriori error estimates for the hp finite element approximation of semilinear Neumann boundary elliptic optimal control problems. By using the hp finite element approximation for both the state and the co-state and the hp discontinuous Galerkin finite element approximation for the control, we derive a posteriori error bounds in L2-H1 norms for the Neumann boundary optimal control problems governed by semilinear elliptic equations. We also give L2-L2 a posteriori error estimates for the optimal control problems. Such estimates, which are apparently not available in the literature, can be used to construct reliable adaptive finite element approximations for the semilinear Neumann boundary optimal control problems.