# 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 coecients, 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 coecients. 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.