# Multiharmonic Finite Element Analysis of a Time-Periodic Parabolic Optimal Control Problem

## Dipl.-Ing.^{in} Monika Wolfmayr

**Jan. 22, 2013, 2:30 p.m. S2 059**

In this talk, we present the multiharmonic analysis of a distributed parabolic optimal control problem in a time-periodic setting including existence and uniqueness results of the solution of some weak space-time variational formulation for the parabolic time-periodic boundary value problem appearing in the constraints for the optimal control problem. Since the cost functional is quadratic, the optimal control problem is uniquely solvable as well. In order to solve the optimal control problem, we state its optimality system and discretize it by the multiharmonic finite element method leading to a system of linear algebraic equations which decouples into smaller systems. We construct preconditioners for these systems which yield robust and fast convergence rates for the preconditioned minimal residual method. All systems can be solved totally in parallel. Moreover, we present an AMLI preconditioner which leads to a robust and fast solver of optimal complexity. Finally, a complete analysis for the error introduced by the multiharmonic finite element discretization is presented as well as some numerical results confirming our theoretical findings including examples with different desired states. In addition, we consider jumping material coefficients in one of the examples.