# Numerics III

## News

- Students enrolled at FU that registered in Campus Management (CM) can fulfill the criteria for successful participation, such as participation, active participation and individual exam, separately in different semesters.
- The written exam is open for examination on Friday, July 15, 1:30 - 2:30 p.m. in A6, Room 141 (Sandra Lindstädt).
- The results can be found here.
- Two typos have been removed from Problem 2 on 1. Exercise sheet.
- A typo has been removed from Problem 2c) on 2. Exercise sheet.
- The tutorial on May 26 is cancelled due to a holiday. As announced on the 4. Exercise sheet, the solutions have to be handed in by May 24, as usual. Reference solutions will be distributed on May 25 by e-mail. The 5. Exercise sheet will be out tomorrow and due on June 1, as usual, but, as an exception, treatment is voluntary, because no previous discussion can occur.
- On the 4. Exercise sheet in Problem 2b) the solution u_1 and the right-hand side f_1 have been corrected.
- In light of the written exam in the afternoon, the lecture on July 11, 10-12 a.m., is cancelled.

## Dates

Lecture | Mon, 10-12h | A3/SR 120 (Arnimallee 3-5) | Prof. Dr. Ana Djurdjevac |

Mon, 14-16h | A3/SR 120 (Arnimallee 3-5) | Prof. Dr. Ana Djurdjevac | |

Tutorial | Do, 16-18h | A6/SR 007/008 Seminarraum (Arnimallee 6) | Prof. Dr. Ralf Kornhuber |

Exam | July 11, 14-16h | Rostlaube, Habelschwerdter Allee 45, Hörsaal 1 a | |

Second exam | October 10, 14-16h | Arnimallee 3, Hörsaal 001 |

## 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

Basic knowledge in calculus (Analysis I-III) and Numerical Analysis (Numerik I). Some knowledge in Functional Analysis will help but is not necessary.

## Registration

All participants should register at the KVV, so that we know the number of participants and can, e.g., contact them by e-mail. In addition, depending on your program of study, you might have to register in the Campus Management (we have no access to these data).

## 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.
- Each week a sheet with exercises will be made available electronically on this web page (see below).
- The exercises are intended to be solved by teams of 2-3 members.
- The exercises consist of theoretical and numerical problems. Programming exercises should be solved using Matlab. Octave can also be used if Matlab is not available. Both types of exercises are rated separately.
- The solutions of the numerical problems and of theoretical problems should be delivered by email to ralf.kornhuber@fu-berlin.de. Please use LateX/Word/Scans(Smartphone apps) for theoretical problems. A complete solution for a numerical experiment (programming exercises) consists of a running Matlab/Octave 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.

### Individual Exam

There will be a written exam on July 11, 14-16h and a second exam on October 10, 14-16h.

### Criteria for the Certificate

Necessary and sufficient conditions for a certificate are:

- individual exam: passing the written exam
- active participation: a) submission of presentable solutions of at least 60 % of both, theoretical and programming problems and b) actual presentation of at least 2 solutions in the tutorial,
- physical participation: presence in lectures and tutorials (not officially monitored)

The grade of final certificate is the grade of the individual exam.

### Exercises

- Exercise 1
- Exercise 2
- Exercise 3
- Exercise 4
- Exercise 5
- Exercise 6
- Exercise 7
- Exercise 8
- Exercise 9
- Exercise 10

### Lecture Notes

The lecture notes are available here.

### Accompanying Material

Lars Gårding, The Dirichlet Problem The Mathematical Intelligencer volume 2, pages 43–53 (1979)

### Matlab

Here you can find an introduction to Matlab (in German, sorry).

## Literature

- D. Braess: Finite Elemente. Springer, 3rd edition (2002)
- P. Knabner, L. Angermann: Numerik partieller Differentialgleichungen. Springer (2000)
- P. Deuflhard, M. Weiser: Numerische Mathematik 3. De Gruyter (2011)
- J. Wloka: Partielle Differentialgleichungen. Teubner (1982)
- D. Werner: Funktionalanalysis. Springer, Berlin (2000)
- H. Alt: Lineare Funktionalanalysis. Sprinter, 6th edition (2012)
- W. Rudin: Functional Analysis. McGraw-Hill, 2nd edition (1991)
- L. Evans: Partial Differential Equations. AMS, 19th volume (1998)
- F. John: Partial Differential Equations. Springer (1982)
- M. Renardy, R. C. Rogers: An introduction to partial differential equations. Springer, 2nd volume (2004)
- A. Quarteroni, R. Sacco, F. Saleri: Numerische Mathematik 2. Springer (2002)
- P. A. Raviart, J. M. Thomas: Introduction à l'analyse numérique des équations aux dérivées partielles. Dunod (1998)

## Contact

Prof. Dr. Ana Djurdjevac | adjurdjevac@zedat.fu-berlin.de | Arnimallee 9 |

Prof. Dr. Ralf Kornhuber | ralf.kornhuber@fu-berlin.de | Arnimallee 6, Room 130 |