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We consider elliptic distributed optimal control problems with energy regularization. Here the standardL2{L_{2}}-norm regularization is replaced by theH-1{H^{-1}}-norm leading to more focused controls. In this case, the optimality system can be reduced to a single singularly perturbed diffusion-reaction equation known as differential filter in turbulence theory. We investigate the error between the finite element approximationuϱ⁢h{u_{\varrho h}}to the state  u  and the desired stateu¯{\overline{u}}in terms of the mesh-size  h  and the regularization parameter ϱ. The choiceϱ=h2{\varrho=h^{2}}ensures optimal convergence the rate of which only depends on the regularity of the target functionu¯{\overline{u}}. The resulting symmetric and positive definite system of finite element equations is solved by the conjugate gradient (CG) method preconditioned by algebraic multigrid (AMG) or balancing domain decomposition by constraints (BDDC). We numerically study robustness and efficiency of the AMG preconditioner with respect to  h , ϱ, and the number of subdomains (cores)  p . Furthermore, we investigate the parallel performance of the BDDC preconditioned CG solver.

Robust Discretization and Solvers for Elliptic Optimal Control Problems with Energy Regularization