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

Ulrich Langer, Olaf Steinbach*, Huidong Yang

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


We consider elliptic distributed optimal control problems with energy regularization. Here the standard L 2 -norm regularization is replaced by the 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 approximation u Qh to the state u and the desired state u ¯ in terms of the mesh-size h and the regularization parameter Q. The choice Q = h 2 ensures optimal convergence the rate of which only depends on the regularity of the target function 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, Q, and the number of subdomains (cores) p. Furthermore, we investigate the parallel performance of the BDDC preconditioned CG solver.

Original languageEnglish
Pages (from-to)97-111
Number of pages15
JournalComputational Methods in Applied Mathematics
Issue number1
Publication statusPublished - 1 Jan 2022


  • A Priori Error Estimates
  • Elliptic Optimal Control Problems
  • Energy Regularization
  • Fast Solvers
  • Finite Element Discretization
  • Parallelization

ASJC Scopus subject areas

  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics


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