A Posteriori Error Estimates For Hp Finite Element Solutions Of Convex Optimal Control Problems

In this paper,hp-version of the finite element approximation of convex optimal control problems is investigated and hp-a posteriori error estimates are presented. The control variable is approximated by piecewise polynomials, and both the state y and the co-state p by piecewise continuous polynomials.J.M.Melenk obtains two important operators in the analysis of hp-version of the finite element method: one is quasi-interpolation operator of Clement type, another is quasi-interpolation operator of Scott-Zhang type. These two operators are suitable in obtaining the error estimates of control variable in L2 norm and of both the state and the co-state in H1 norm in optimal control problems. But in many applications,we need to compute the error estimates of both control and state in L2 norm,therefore we need two new operators.In order to derive the error estimates of the state in L2 norm, we use the technique employed by J.M.Melenk in deriving the operators of hp-version to obtain two new operators.Then we obtain hp-a posteriori upper error estimators for the coupled control and state approximations in L2-H1 norms and in L2-L2 norms respectively. Finally, we derive hp-a posteriori lower error estimators for the error of the control problems under certain conditions,and show that hp-a posteriori error estimators obtained is relatively reliable and efficient in adaptive finite element approximation.