  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.~Neum\"{u}ller 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.
