A posteriori error estimates of hp spectral element method for parabolic optimal control problems

In this paper, we investigate the spectral element approximation for the optimal control problem of parabolic equation, and present a hp spectral element approximation scheme for the parabolic optimal control problem. For improve the accuracy of the algorithm and construct an adaptive finite element approximation. Under the Scott-Zhang type quasi-interpolation operator, a $L^2(H^1)-L^2(L^2)$ posteriori error estimates of the hp spectral element approximated solutions for both the state variables and the control variable are obtained. Adopting two auxiliary equations and stability results, a $L^2(L^2)-L^2(L^2)$ posteriori error estimates are derived for the hp spectral element approximation of optimal parabolic control problem.