Spectral element discretization of optimal control problems, in Spectral and High Order Methods for Partial Differential Equations

In this work we consider the numerical solution of a distributed optimal control problem associated with an elliptic partial differential equation. We approximate the optimality system by the spectral element method and derive a posteriori error estimates with respect to the cost functional. Then we use an hN adaptive refinement technique to reduce this error: the error indicator is used to mark what elements must be refined. The choice between an h or N refinement is based on the use of a predicted error reduction algorithm. Numerical results show the way this algorithm works.