Robust discretization and solvers for elliptic optimal control problems with energy regularization
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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. Neumüller 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 ϱ. The choice ϱ = 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 ϱ, and the number of subdomains (cores) p. Furthermore, we investigate the parallel performance of the BDDC preconditioned conjugate gradient solver.
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