2022/09/19 by Ulrich Langer, Richard Löscher, Langer, Ulrich +5 · 1 citation
Computer Science · Engineering · Mathematics · #35J05 #49J20 #49M05 #65M15 #65M60 #65N22 #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods in inverse problems #Optimization and Control (math.OC)
paper · pdf · doi:10.48550/arxiv.2209.08811
openalex publication_date 2022/09/19 · openalex created_date 2022/09/21 · openalex updated_date 2026/07/28
We analyze the finite element discretization of distributed elliptic optimal control problems with variable energy regularization, where the usual L2(Ω) norm regularization term with a constant regularization parameter \varrho is replaced by a suitable representation of the energy norm in H-1(Ω) involving a variable, mesh-dependent regularization parameter \varrho(x). It turns out that the error between the computed finite element state \widetildeu\varrho h and the desired state u (target) is optimal in the L2(Ω) norm provided that \varrho(x) behaves like the local mesh size squared. This is especially important when adaptive meshes are used in order to approximate discontinuous target functions. The adaptive scheme can be driven by the computable and localizable error norm ‖ \widetildeu\varrho h - u‖L2(Ω) between the finite element state \widetildeu\varrho h and the target u. The numerical results not only illustrate our theoretical findings, but also show that the iterative solvers for the discretized reduced optimality system are very efficient and robust.