Discrete Concepts in PDE Constrained Optimization

Abstract

In the present chapter we give an introduction to discrete concepts for optimization problems with PDE constraints. As models for the state we consider elliptic and parabolic PDEs which are well understood from the analytical point of view. This allows to focus on structural aspects in discretization. We discuss and compare the approaches First discretize, then optimize and First optimize, then discretize, and introduce a variational discrete concept which avoids explicit discretization of the controls. We investigate problems with general constraints on the control, and also consider pointwise bounds on the state, and on the gradient of the state. We present error analysis for the variational discrete concept and accomplish our analytical findings with numerical examples which confirm our analytical results.