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

We consider the finite element discretization and the iterative solution of singularly perturbed elliptic reaction-diffusion equations in three-dimensional computational domains. These equations arise from the optimality conditions for elliptic distributed optimal control problems with energy regularization that were recently studied by M.~Neumuller and O.~Steinbach (2020). We provide quasi-optimal a priori finite element error estimates which depend both on the mesh size $h$ and on the regularization parameter $\varrho$. The choice $\varrho = h^2$ ensures optimal convergence which only depends on the regularity of the target function. For the iterative solution, we employ an algebraic multigrid preconditioner and a balancing domain decomposition by constraints (BDDC) preconditioner. We numerically study robustness and efficiency of the proposed algebraic preconditioners with respect to the mesh size $h$, the regularization parameter $\varrho$, and the number of subdomains (cores) $p$. Furthermore, we investigate the parallel performance of the BDDC preconditioned conjugate gradient solver.