# Numerik III

## News and Important Remarks

After consultation with Maren-Wanda Wolf (mawolf{at}math.fu-berlin.de), you can have a look on your Klausur in Arnimallee 6, room 122, until Monday, 2.11..

Here you can find the results of the Nachklausur.

You can have a look on your Klausur on Friday, 17.7. between 12.00 and 14.00 o'clock in Arnimallee 6, room 131 (Secretary Nordt).

Here you can find the results of the Klausur.

Attention: in the inverse estimate in problem 2 on exercise sheet 10 there is a constant missing, here you can find a corrected version.

## Dates

Lecture | Tue, 8:30 - 10:00, Tue, 14:15 - 15:45 |
Arnimallee 6, SR 025/026, Takustr. 9, SR 049 |

Tutorial | Tue, 16:15 - 17:45 | Arnimallee 6, SR 025/026 |

Klausur | Tue, 07.07.15, 14:15 - 15:45 | Takustr. 9, SR 049 |

Nachklausur | Fr, 09.10.15, 10.15 - 11.45 | Arnimallee 6, SR 031 |

## General Information

### Description

Mathematical modelling of spatial or spatial/temporal phenomena such as porous medium flow, solidification of melts, weather prediction, etc. typically leads to partial differential equations (pdes). After some remarks on the modelling with and classification of pdes, the course will concentrate on elliptic problems. Starting with a brief introduction to the classical theory (existence and uniqueness of solutions, Green's functions) and assiciated difference methods we will mainly focus on weak solutions and their approximation by finite element methods. Adaptivity and multigrid methods will be also discussed.

### Target Audience

Students of Bachelor and Master courses in Mathematics and of BMS

### Prerequisites:

Prerequisites for this course are basic knowledge in calculus (Analysis I-III) and Numerical Analysis (Numerik I). Some knowledge in Functional Analysis will help a lot.

## Registration

All participants should register at the KVV, so that we know who is participating. The overall number of participants is also necessary to justify the equipment of the course. In addition, depending on your program of study, you have to register in the Campus Management.

## Exercises and Criteria for a Certificate

### Tutorial & Exercises

- The tutorial offers the possibility to discuss and better understand the presented material and exercises. Furthermore solutions of exercises are presented by the students.
- A sheet with exercises will be handed out each Tuesday during the lecture. These exercises are also available electronically on this web page (see below).
- The exercises are intended to be solved by teams of three members.
- The exercises consist of theoretical and numerical problems. The latter should be solved using Matlab (available at the students computer pool at the institute). Both types of exercises are rated seperately.
- The solutions have to be finished before the tutorial two weeks after they were handed out.
- The solutions of the numerical problems should be delivered by email to mawolf{at}math.fu-berlin.de. Note that a complete solution for a numerical experiment consists of a running Matlab code, a program executing the required test runs, and protocols of the execution of these test runs. Delivering a correct and running code without knowing what is going on in the code will be regarded and rated as attempt of deception.
- The solutions of the theoretical problems have to be presented in the tutorial on request (without regard to preference and presence) by a member of a given group.

### Klausur

- There will be a Klausur on 07.07.2015.
- Students who fail in the Klausur have a second chance in the Nachklausur, which will take place on 09.10.2015.

### Criteria for the Certificate

Neccessary and sufficient for a certificate are:

- written exam: passing the Klausur or the Nachklausur (the better grade counts)
- active participation: at least as much success as failure in the presentation of the solution of theoretical problems and 50% of the maximal number of points for numerical problems
- constant participation: 85% attendence in the tutorial (not checked)

Certificates are graded according to the result of the Klausur.

### Exercises

- exercise 1
- exercise 2
- exercise 3
- exercise 4
- exercise 5
- exercise 6
- exercise 7
- exercise 8, basis.m, quadrature.m
- exercise 9, uniform_grid.m, generate_grid.m, circle.mat, assemble_mass.m, assemble_stiff.m
- exercise 10, hierarchic_basis.m, refine_grid.m, assemble_P1.m

## Accompanying Material

Here you can find a manuscript on hierarchical error estimates for the laplacian.

### Lecture Notes:

The lecture notes are available here.

The lecture notes Numerics II.

The lecture notes Numerics I, in german only.

## Literature

F. John: Partial Differential Equations. Springer (1982)

M. Renardy, R. C. Rogers: An introduction to partial differential equations, Springer, 2. Auflage (2004)

A. Quarteroni, R. Sacco, F. Saleri: Numerische Mathematik 2. Springer (2002)

D. Braess: Finite Elemente. Springer, 3. Auflage (2002)

P. A. Raviart, J. M. Thomas: Introduction à l'analyse numérique des équations aux dérivées partielles, Dunod (1998)

## Contact

Prof. Dr. Ralf Kornhuber | Arnimalle 6, Room 130 Secretary Frau Nordt: Arnimallee 6, Room 131 Consultation-Hour: Tue, 11-12 email: kornhube{at}math.fu-berlin.de |

Maren-Wanda Wolf | Arnimalle 6, Room 122 Consultation-Hour: Thu, 14-15 email: mawolf{at}math.fu-berlin.de |