Accuracy and convergence properties of the finite difference multigrid solution of an optimal control optimality system

Abstract

The finite difference multigrid solution of an optimal control problem associated with an elliptic equation is considered. Stability of the finite difference optimality system and optimal-order error estimates in the discrete L(2) norm and in the discrete H(1) norm under minimum smoothness requirements on the exact solution are proved. Sharp convergence factor estimates of the two grid method for the optimality system are obtained by means of local Fourier analysis. A multigrid convergence theory is provided which guarantees convergence of the multigrid process towards weak solutions of the optimality system.