A priori and a posteriori error analysis of hp spectral element discretization for optimal control problems with elliptic equations

Abstract

. A priori and a posteriori error estimates of hp spectral element method for an elliptic optimal control problem with L2-norm control constraint are investigated in this paper. We then present the optimality conditions and set up hp spectral element discretization scheme. Based on some important interpolation operators and suitable immediate variable, a priori error estimates for this problem in hp spectral element discretization are established carefully. Furthermore, using some interpolation operators, a rigorous posteriori error estimates of hp spectral element approximation are also proved for control and the coupled state approximation in L2−H1-norm and L2−L2-norm, respectively. Such estimators can be used to construct reliable adaptive spectral element methods for optimal control problems. Finally, the error analysis results are confirmed by numerical results.