Lecture SS 18 Wissenschaftliches Rechnen II
Partial differential equations (PDEs) are essential in many practical applications including engineering, physics and economics. In the previous lecture “Wissenschaftliches Rechnen I” we discussed the finite element method for the numerical solution of such PDEs. The specific topics of this lecture follow the module handbook for the Bachelor programme in mathematics. The goal of this lecture is to discuss PDEs with (small) parameters. Those parameters introduce new challenges for the numerical solution and its error analysis. We will briefly discuss those aspects. Furthermore, we will treat parabolic PDEs that are time dependent for which the heat equation is a prototype. For the efficient numerical solution of PDEs we will also discuss generalized finite elements and meshfree methods. Finally, we will discuss methods for the solution to PDEs on scattered point sets. Here, we will focus on reproducing kernel Hilbert space methods. Those methods are also applied in some machine learning applications. We will provide some links to those methods.
Prerequisites:
This lecture builds on the lecture “Wissenschaftliches Rechnen I” held by Priv. Doz. Dr. Christian Rieger in the last semester. Knowledge of the covered topics is sufficient, but attendance of the previous course is not mandatory. The lecture assumes some basic knowledge of numerical methods. In particular some basic knowledge of numerical linear algebra (sparse matrices and iterative solvers) is helpful. Some basic knowledge from functional analysis (including also elementary measure theory) is also very helpful. Furthermore some basic programming skills will be helpful since there will be a few practical exercises.
Exercise class:
The exercise class takes place on
- Thursday, 16:15 - 18:00 in the in the Mathematikzentrum, room 0.008
Exercise sheets:
- Attendance Sheet 0
- Sheet 1
- Sheet 2
- Sheet 3
- Sheet 4
- Sheet 5
- Sheet 6
- Sheet 7
- Sheet 8
- Sheet 9 (corrected version)
Literature recommendation for prerequisites:
- Hohmann, A., Deuflhard, P., (2003) Numerical Analysis in Modern Scientific Computing: An Introduction. Springer
- Alt, H.W. (translated by Nuernberg, R.)(2016) Linear Functional Analysis: An application oriented introduction. Springer London Universitext, ISBN-13: 9781447172796 (for basics in functional analysis)
- Braess, D. (2007). Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics. Cambridge: Cambridge University Press. doi:10.1017/CBO9780511618635
- Groosmann C., Roos H.-G., Stynes, M., (2007) Numerical Treatment of Partial Differential Equations., Springer