# PhD Project: Multiharmonic Approach to Parabolic Initial-Boundary Value and Control Problems

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

**Feb. 1, 2011, 3:30 p.m. P 004**

In this talk, we will discuss linear parabolic problems by the multiharmonic

approach, which means that the solution is expanded in a Fourier series. This

is a useful technique for solving many practical problems, where the excitation

is time-periodic, or even time-harmonic. By switching from the time to the

frequency domain, a linear problem can be reduced to the solution of a linear

elliptic system of the Fourier coecients, which is simplier than solving the

time-dependent problem by time-integration methods.

The model problem, which we consider here, is the one-dimensional heat

equation with homogeneous Dirichlet boundary conditions. We will discuss the

more common line variational formulation as well as the space-time variational

formulation, into which we insert the multiharmonic ansatz and hence derive

an elliptic system for the Fourier coecients. Moreover, we will also consider

the solution of an optimal control problem by the multiharmonic approach,

where the state has to solve the heat equation with the control as right hand

side. We will formulate the optimality system of the model control problem

and then insert the multiharmonic ansatz into this KKT-system.