# Numerics IV: Numerical methods for nonlinear partial differential equations

## News

- The
**results of the second exam**can be found here. If you want to have a look at your marked exam, please make an appointment. - The second exam will be on Thursday, March 30th from 14:00 to 16:00 in the room 025/026 of Arnimallee 6.
- The
**results of the first exam**can be found here. The post-exam review (Klausureinsicht) is on Tuesday, February 28th from 12:00 to 13:00 in room 121 in Arnimallee 6. Please bring an ID card with your picture on it.

## Dates

Lecture | Thu, 14-16h | Arnimallee 6, SR 007/008 |

Tutorial | Tue, 14-16h | Takustr. 9, 049 |

Exam | Thu, Feb 16th, 14-16h | Arnimallee 6, SR 007/008 |

2nd Exam | Thu, Mar 30th, 14-16h | Arnimallee 6, SR 025/026 |

## General Information

### Description

The mathematical description of many physical phenomena naturally leads to nonlinear partial differential equations. While, in some cases, these are commonly approximated by linear equations (e.g., heat flow), this is often not possible, because the structure of the problem if far from being linear.

In this lecture we will consider various aspects of nonlinear partial differential equations. On the one hand we will discuss mathematical modelling and analytical questions. On the other hand we will introduce basic numerical methods for such problems, ranging from discretization to efficient iterative methods.

The problems considered will, e.g., cover applications form material science (phase field models), and bio-mechanics.

## Target Audience

Advanced students in the Master Program Mathematics and of BMS. Various possible topics for a Master thesis will come up during this course.

## Prerequisites

Basic knowledge on partial differential equations and their numerical solution (e.g. Numerics III).

## 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.
- 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 two.
- The exercises consist of theoretical and numerical problems that should be solved by the teams.
- Programming exercises should be solved using Matlab (available at the students computer pool at the institute). Both types of exercises are rated separately.
- The solutions have to be handed in before the tutorial.
- The solutions of the numerical problems should be delivered by email to tobias.kies@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.

### Exam

- The exam will take place on Thursday, the 16th of February, from 14h-16h in the room SR 007/008 in Arnimallee 6.
- The test will be in written form and take 90 minutes. We ask you to already be there at 14:00.
- Please bring the following: an id card with your picture on it (passport, Personalausweis or drivers license are okay but your student id is not enough), a pen to write with (no pencil, no red pens) and some paper to write on.
- Optionally you may also bring any notes you have (it does not matter whether they are handwritten or not) and any books you may find helpful. You can also bring a non-programmable calculator (although we will make sure that you can solve the questions also without one). Please note that you may
**not**share any of the documents/books/devices that you have brought with others.

### Criteria for the Certificate

Necessary and sufficient for a certificate are:

- exam: passing the exam
- tutorial score: at least 60% points in theoretical as well as programming exercises
- participation: 85% attendence in the tutorial (not checked)

Certificates are graded according to the result of the exam.

### Exercises

- Exercise 1
- Exercise 2
- Exercise 3
- Exercise 4
- Exercise 5
- Exercise 6
- Exercise 7
- Exercise 8 and additional resources
- Exercise 9
- Exercise 10 and additional resources
- Exercise 11 and additional resources
- Exercise 12
- Exercise 13

## 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. Carsten Gräser | graeser@mi.fu-berlin.de | Arnimallee 6, Room 121 consultation-hour: Mon 11:00-12:00 |

Tobias Kies | tobias.kies@fu-berlin.de | Arnimallee 9, Room 009 consultation-hour: on request |