By Prof. Dr. Christian Kreuzer. Universidad Técnica de Dortmund
Seminar Date: 2022-03-10
We consider finite element solutions to quadratic optimization problems, where the state depends on the control via a well-posed linear partial differential equation. Exploiting the structure of a suitably reduced optimality system, we prove that the combined state and adjoint state errors of the variational discretization is bounded by the best approximation error in the underlying discrete spaces. The constant in this bound depends on the inverse square root of the Tikhonov regularization parameter. Furthermore, if the operators of control action and observation are compact, this quasi-best approximation constant converges to the quasi-best approximation constant of the state equation and thus becomes independent of the Tikhonov parameter as the mesh size tends to 0. We give quantitative relationships between mesh size and Tikhonov parameter ensuring this independence. We also derive generalizations of these results for discretized control variables and bounded controls.