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

Abstract

<abstract><p>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.</p></abstract>

Space-time spectral methods for a fourth-order parabolic optimal control problem in three control constraint cases

“…And [7] derives a posteriori error estimates. In [19], Lu et al present a hp spectral element approximation scheme for a parabolic optimal control problem, and derive a posteriori error estimates.…”