2023/12/19 by Ulrich Langer, Richard Löscher, Langer, Ulrich +5
Computer Science · Engineering · #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Computational Fluid Dynamics and Aerodynamics #FOS: Mathematics #Numerical Analysis (math.NA)
paper · pdf · doi:10.48550/arxiv.2312.12282
openalex publication_date 2023/12/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We propose, analyze, and test new iterative solvers for large-scale systems of linear algebraic equations arising from the finite element discretization of reduced optimality systems defining the finite element approximations to the solution of elliptic tracking-type distributed optimal control problems with both the standard L2 and the more general energy regularizations. If we aim at an approximation of the given desired state yd by the computed finite element state yh that asymptotically differs from yd in the order of the best L2 approximation under acceptable costs for the control, then the optimal choice of the regularization parameter \varrho is linked to the mesh-size h by the relations \varrho=h4 and \varrho=h2 for the L2 and the energy regularization, respectively. For this setting, we can construct efficient parallel iterative solvers for the reduced finite element optimality systems. These results can be generalized to variable regularization parameters adapted to the local behavior of the mesh-size that can heavily change in case of adaptive mesh refinement. Similar results can be obtained for the space-time finite element discretization of the corresponding parabolic and hyperbolic optimal control problems.