Past lectures

Abstract

The lectures discuss "adaptivity" in solving differential equations by discretization with emphasis on Galerkin finite element methods. The key issues are "a posteriori error estimation" and "automatic mesh adaptation". Beside the traditional approach of "energy-error control" a new duality-based technique for "goal-oriented error estimation" will be discussed in detail. This method aims at economical computation of certain quantities of physical interest, which is typically required in the design cycles of technical applications. For example, the drag coefficient of a body immersed in a viscous flow is computed, then it is minimized by varying certain control parameters and finally the stability of the resulting flow is investigated by solving an eigenvalue problem.

Tentative contents

1. The goals of numerical simulation

2. Approaches to a posteriori error estimation

3. How to organize mesh adaptation?

4. How far does theoretical analysis carry?

5. Application to non-linear problems or "the power of elementary calculus"

6. Application in optimal control

7. Application in stability analysis

8. Applications in structural mechanics

9. Applications in fluid mechanics

10. Miscellaneous and open problems

It is planned to complement the course by practical training based on software packages like MATHLAB or DEAL-II.