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

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

**Feb. 1, 2011, 2:30 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 coeffcients, 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 coeffcients. 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.