Spectral Element Discretization of Optimal Control Problems

Gaudio, Loredana

Spectral Element Discretization of Optimal Control Problems

Sunday 14th March 2010

Quarteroni, A.

In this thesis we investigate hN-adaptive algorithms for the solution of optimal control problems, discretized by means of spectral element methods. Here h indicated the maximum diameter of the spectral element, while N is the maximum polynomial degree used within every element. The goal is to achieve a suitable accuracy on the error of a given cost functional (which represents the output of physical interest), according to a criterion of minimization of the total number of degrees of freedom. We apply this strategy to environmental applications, specifically on atmospheric pollution problems, however the same strategy here proposed can be extended to many engineering problems of practical interest. For the analysis of the optimal control problem, we use the Lagrangian functional approach. Both the state and the adjoint equations are discretized by Legendre-Galerkin spectral element method. The proposed hN-adaptive algorithms are based on the splitting of the error on the cost functional in two components: an iteration error and a discretization error. For linear optimal control problems, the iteration error is minimized in the iterative process for the optimal control solution. Instead, the discretization error is reduced via the adaptive strategy, through the use of a posteriori error estimate obtained by means of the dual weighted residual method. The choice between h or N refinement is based on a predicted error reduction algorithm. Numerical examples highlight the efficiency of the proposed error estimate and the associated hN-adaptive strategy. Key words: optimal control problem, spectral element method, a posteriori error estimates, mesh renement, hN-adaptivity