# Hierarchical Solution Concepts for Flow Control Problems

### From DFG-SPP 1253

## Project leaders

FB Mathematik

Uni Hamburg

Tel: 040 42838-4079

Tel: 040 42838-5115 (Sekr.)

Fax: 040 42838-5117

Email: michael.hinze@uni-hamburg.de

Homepage: http://www.math.uni-hamburg.de/home/hinze

FB Mathematik

Uni Dortmund

Tel: 0231 755-3075

Tel: 0231 755-6840 (Frau Lamprecht) (Sekr.)

Fax: 0231 755-5933

Email: stefan.turek@mathematik.uni-dortmund.de

Homepage:

## DFG funded assistants

Michael Koester (Uni Dortmund)

michael.koester@mathematik.tu-dortmund.de

Andreas Günther (Uni Hamburg)

## Description

The goal of the project is the development of a generally applicable hierarchical (multigrid) solution framework for flow control problems which allows their numerical solution requiring a computational effort of only a small multiple of that of the flow simulation itself. To achieve our goal we combine

High performance scientific computing techniques in flow simulation, Discrete concepts in space and time which are tailored to the structure of the first-order necessary optimality conditions of the underlying optimization problem (Karush-Kuhn-Tucker system, short KKT system), and Sophisticated optimization algorithms combined with multigrid concepts which allow to exploit the structure of the underlying optimization problem. We investigate two different multigrid approaches. APPROACH I tackles the KKT system all-at-once, and APPROACH II exploits the structure of the KKT system and reduces it to a nonlinear integral equation (variational inequality) for the control u, which then is tackled by a multigrid approach tailored to integral equations.

Within this first application period we concentrate on methodic and algorithmic aspects which are validated at optimization problems for 2D time dependent flows with prototypical character.