FETI-DP methods for Optimal Control Problems

where μ , and λ are the Lamé parameters. The state (displacement field) is sought in V = H1 0 (Ω ,∂ΩD) 2 = {y ∈ H1(Ω)2 : y = 0 on ∂ΩD}, where Ω ⊂ R2 and ∂ΩD is part of its boundary. For simplicity, we consider the case of volume control, i.e., U = L2(Ω). Dual-primal FETI methods were first introduced by Farhat, Lesoinne, Le Tallec, Pierson, and Rixen [3] and have successfully scaled to 105 processor cores [6]. In [8] a first convergence bound for scalar problems in 2D was provided. Numerical scalability for FETI-DP methods applied to linear elasticity problems was first proven in [7]. Balancing Neumann-Neumann domain decomposition methods for the optimal control of scalar problems have been considered in Heinkenschloss and Nguyen [5, 4]. There, local optimal control problems on non-overlapping subdomains are considered and a Balancing Neumann-Neumann preconditioner is constructed for the indefinite Schur complement. Of course also multigrid methods have been considered for optimal control problems, see, e.g., [10]. A review of block approaches to optimal control problems can be found in [9]. A recent block approach can be found in [11].