Ferienakademie 2010, Course 4
Beyond Classical Finite Elements

To infinity and beyond!
(Buzz Lightyear, 1995)

Course language: English (suitable approximations will do, as well)

Course Content:

If you ask engineers to explain Finite Elements, they will probably start talking about rods and shells, as well as stability and stiffness. In contrast, when a mathematician is asked, you might learn about integrals, weak (and sometimes weird) forms, and of function spaces invented by Russians. Computer scientists will probably not know too much about Finite Elements in the first place, but they are likely to tell you about adaptive triangular meshes, complicated data structures, and how to assemble (and solve) large systems of equations. If there's so much to learn about classical Finite Elements, should we really go beyond?

Definitely yes, because "beyond" does not always mean "more difficult", but leads to new perspectives, different points of view, but also "old friends" and regular "guests". Hence, anything that resembles a Finite Element method will be on topic, and we'll not stop at putting these methods to use, nor at discussing their applications and implementations. We will discuss

• Multilevel Finite Element methods to improve time-to-solution and improve multiscale modelling,
• Discontinuous Galerkin methods to describe phenoma, for example, in geoscience (from earthquakes to tsunamis),
• Isogeometric elements to closely couple geometric models to numerical simulation,
• Finite Cell methods to model the stability of bone structures (and medical implants),

Can anyone become an expert in all of these topics? That's actually not the point, as our main goal is to bring together students from mathematics, informatics, physics, and engineering - and to learn how to present disciplinary and interdisciplinary topics to fellow students from other fields. Hence, you should bring with you a wish to meet and work with students from the other fields, to learn not only about new methods but also about different expertise and "culture" in the different disciplines - and how to successfully combine these.

In turn, the lecturers will promise not to fall into the cliches listed in the beginning (well ... at least not too strongly), and that there is also a decent amount of fun involved - not only beyond Finite Elements, but also beneath and between the mountains, and definitely not below your expectations!