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# Wissenschaftliches Rechnen (Scientific Computing) I

### Priv.-Doz. Dr. Christian Rieger

Assistant: NN
Time: Di. 10:15-12:00 + Do. 8:15-10:00
Place: Wegeler Str. 10 / Zeichensaal
Link to Basis: click here

Partial differential equations (pdes) are essential in many practical applications including engineering, physics and economics. The solution to such pdes are often not available in closed form and hence have to be computed numerically.
(The modeling perspective can be discussed in more details in an accompanying seminar.)

The specific topics of this lecture follow the module handbook for the Bachelor programme in mathematics. The goal of this lecture is to derive numerical discretizations of boundary values problems of second order elliptic equations. The main focus of this lecture will be on the "finite element" method. We will introduce this method and discuss both theoretical and practical aspects.

The main focus will be on the theoretical aspects which include error estimates both a priori and a posteriori.
We will also discuss properties of the resulting system of linear algebraic equations and their efficient numerical treatment.

### Prerequisites:

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 but not required, as we will recall the basic concepts from functional analysis as they are needed in the sequel of the lecture. Furthermore some basic programming skills will be helpful since there will be a few practical exercises.

### Exercise classes:

The exercise classes take place on
• Thursday, 14:15 - 16:00 in the in the building beside the Mathematikzentrum, room N0.003
• Friday, 12:15 - 14:00 in the HRZ, room 6.020

### Exercise sheets:

Attendance Sheet 0
Attendance Sheet 0.1
Sheet 1
Sheet 2
Sheet 3
Sheet 4
Sheet 5
Sheet 6
Sheet 7
Sheet 8 (updated) and mesh.txt
(You can check your solution for programming exercise d) with exercise_d_solution.txt )
Sheet 9
Sheet 10
Sheet 11 and uniformMesh3.txt
Sheet 12
Sheet 13

### 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)

### Literature recommendation for the lecture (more literature will be provided during the lecture)

• 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