EU Regional School - Mathematical Theory of Finite Elements (A Crash Course for Engineers)

by Prof. Leszek Demkowicz (Oden Institute for Computational Engineering and Sciences The University of Texas at Austin)

Wednesday 21 May 2025, 09:00 → Friday 23 May 2025, 16:00 Europe/Berlin

Raum GRS001 (Schinkelstraße 2a, 52062 Aachen )

 

We review fundamentals of Galerkin and conforming Finite Element (FE) methods using model diffusion-convection-reaction and linear elasticity problems. We discuss the possibility of different variational formulations leading to different energy spaces and corresponding conforming elements. The course is focusing on the famous inf-sup stability condition and the concept of discrete stability. We review the classical results of Babuška, Mikhlin and Brezzi, and finish with a short exposition of the Discontinuous Petrov Galerkin (DPG) method. The three day-long course consists of three 1.5 hour lectures per day accompanied with a Q/Q session afterwards and it is based on [2]. You may also consult [4, 1, 3].

Day 1

1. Classical calculus of variations. Concept of a variational formulation.
2. Abstract variational problem.
3. Diffusion-convection-reaction model problem. Different variational formulations.
4. Distributional derivatives and different energy spaces.
5. Bubnov- and Petrov-Galerkin methods.

Day 2

1. Babuška-Nečas and Banach Closed Range Theorems.

2. Riesz Representation and Lax Milgram Theorems.
3. Babuška Theorem and concept of discrete stability.
4. Ritz method.
5. Exact sequence elements.

Day 3

1. Examples of coercive problems.
2. Compact perturbations of coercive problems. Mikchlin’s theory of asymptotic stability.
3. Mixed problems, Brezzi’s theory.
4. The idea of optimal testing, the DPG method.

References

[1] L. Demkowicz. Lecture notes on Energy Spaces. Technical Report 13, ICES, 2018.

[2] L. Demkowicz. Mathematical Theory of Finite Elements. SIAM, 2024.

[3] L. Demkowicz and J. Gopalakrishnan. Encyclopedia of Computational Mechanics, Second Edition, chapter Discontinuous Petrov-Galerkin (DPG) Method. Wiley, 2018. Eds. Erwin Stein, René de Borst, Thomas J. R. Hughes, see also ICES Report 2015/20.

[4] J.T. Oden and L.F. Demkowicz. Applied Functional Analysis for Science and Engineering. Chapman & Hall/CRC Press, Boca Raton, 2018. Third edition.