Error Estimates of hp Spectral Element Methods in Nonlinear Optimal Control Problem

The main purpose of this paper is to discuss hp spectral element method for optimal control problem governed by a nonlinear elliptic equation with \(L^2\)-norm constraint for control variable. We then set up its weak formulation and hp spectral element approximation scheme. A priori error estimates of hp spectral element approximation based on some suitable projection operators are proved carefully. Using some properties of projection operators, a posteriori error estimates for both the state and the control approximation under some reasonable assumptions are established rigorously. Such estimates are useful tools, which can be used to construct reliable adaptive spectral element methods for optimal control problems.