Advanced Finite Elements

Advanced finite element techniques such as, e.g.,
- Mixed and Hybrid Finite Elements
- Discontinuous Galerkin Methods
- Nonconforming Methods
- Adaptive Finite Element Method
- Isogeometric analysis
- Reduced Basis
- Treatment of nonlinear PDEs
- Modern Iterative Solvers and Preconditioning
- Applications in Solid Mechanics and Incompressible Fluid Mechanics

The main goal of this module is to deepen the understanding of the derivation and analysis of advanced finite element techniques and suitable efficient solvers. The discussion is accompanied by relevant examples from solid and fluid mechanics, which enables students to develop some initial competence for choosing appropriate discretization techniques for different physical problems. At the end of this module, students are able to engage in current research topics and to study advanced finite element literature independently.

Einführung in die Programmierung (MA8003) , Numerik gewöhnlicher Differentialgleichungen (MA2304), Numerical Methods for Partial Differential Equations (MA 3303), The theory and implementation of conforming finite elements for elliptic second order PDEs is supposed to be known.
Helpful but not necessary: Functional Analysis (MA3001)

lectures, tutorials, project teams,
The module is offered as lectures with accompanying practice sessions. In the lectures, the contents will be presented in a talk with demonstrative examples, as well as through discussion with the students. The lectures should motivate the students to carry out their own analysis of the themes presented and to independently study the relevant literature. Corresponding to each lecture, practice sessions will be offered, in which exercise sheets and solutions will be available. In this way, students can deepen their understanding of the methods and concepts taught in the lectures and independently check their progress.

blackboard, slides, assignment sheets, lab exercises

Daniele Antonio Di Pietro and Alexandre Ern:
Mathematical Aspects of Discontinuous Galerkin Methods.
Mathematics and Applications 69, Springer, Heidelberg, 2012.
Alexandre Ern and Jean-Luc Guermond:
Theory and practice of finite elements.
Applied Mathematical Sciences 159, Springer, New York, 2004.
Alfio Quarteroni and Alberto Valli:
Numerical approximation of partial differential equations.
Springer Series in Computational Mathematics 23, Springer, Berlin, 1994.
J. Austin Cottrell; Thomas J.R. Hughes; Yuri Bazilevs
Isogeometric Analysis: Toward Integration of CAD and FEA
Wiley, 2009
D. Boffi, F. Brezzi, M. Fortin
Mixed Finite Element Methods and Applications
Springer, 2013
Hesthaven, Rozza, Stamm
Certified Reduced Basis Methods for Parametrized Partial Differential Equations
Springer 2015
Alfio Quarteroni, Andrea Manzoni, Federico Negri
Reduced Basis Methods for Partial Differential Equations
Springer 2016

The exam will be oral (30 minutes), since this gives more flexibility in asking questions concerning all of the individual topics of the lecture. Students should demonstrate that they have gained deep knowledge of definitions, main tools, and results of the advanced finite element methods covered in the lecture. The students are expected to be able to derive the methods, to explain their properties, and to apply them to specific examples. The students' understanding of these topics will be evaluated by problems asking them to discuss the above material on small concrete examples. Moreover, they will be asked to state certain characteristic properties of the learned constructions and methods. In addition, they will be asked to provide proofs or proof sketches for fundamental theorems discussed in the course. Also, aspects of implementation will be part of the exam, e.g. writing down pseudo-code samples of a few lines length or orally explaining some more involved approaches. No learning aids or electrical devices are permitted during the exam.