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Accuracy and convergence properties of the finite difference multigrid solution of an optimal control optimality system. (English) Zbl 1031.49029

Summary: 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.

MSC:

49M25 Discrete approximations in optimal control
65N06 Finite difference methods for boundary value problems involving PDEs
49K20 Optimality conditions for problems involving partial differential equations
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
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