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

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

**Jan. 22, 2013, 3:30 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.