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

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

**Nov. 15, 2011, 2:30 p.m. S2 Z75**

In this talk, we present the multiharmonic analysis of a distributed parabolic control problem in a time-periodic setting. In particular, the control is assumed to be distributed and time-periodic. After eliminating the control from the optimality system, we arrive at the reduced optimality system for the state and the co-state that is nothing but a coupled system of a forward and a backward parabolic partial differential equation. The state and the co-state are approximated by a truncated Fourier series expansion in time and the Fourier coefficients are discretized by the finite element method. This leads to a large system of algebraic equations. Fortunately, this system decouples into smaller systems each of them defining the cosine and sine Fourier coefficients for the state and co-state with respect to a single frequency. For these smaller systems we construct preconditioners resulting in a fast converging MINRES solver with a parameter independent convergence rate. All these systems can be solved totally in parallel. Finally, we also present a complete discretization error analysis in space and time.