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
Day 2
Babuška-Nečas and Banach Closed Range Theorems.
Day 3
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.
Center for Simulation and Data Science (JARA-CSD)
RWTH Aachen University